The Pennsylvania State University
The Graduate School
DESIGN OPTIMIZATION OF COMPLIANT STRUCTURES FOR RADIOFREQUENCY
ABLATION AND ADDITIVE MANUFACTURING
A Dissertation in
Mechanical Engineering
by
Bradley W. Hanks
© 2020 Bradley W. Hanks
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
August 2020
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The dissertation of Bradley W. Hanks was reviewed and approved by the following:
Mary Frecker
Professor, Mechanical Engineering & Biomedical Engineering
Dissertation Advisor
Chair of Committee
Reuben Kraft
Associate Professor, Mechanical Engineering
Jason Moore
Associate Professor, Mechanical Engineering
Edward Reutzel
Senior Research Associate, Engineering Science and Mechanics, Applied
Research Laboratory
Timothy Simpson
Paul Morrow Professor in Engineering Design and Manufacturing, Mechanical
Engineering, Industrial and Manufacturing Engineering
Karen A. Thole
Professor of Mechanical Engineering
Head of the Department of Mechanical Engineering
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Abstract
Recent advances in the capabilities of model simulation and manufacturing technology
are paving the way for systematic design tools. These tools enable designers to explore organic
and nonconventional designs that were previously not possible to analyze or manufacture.
Through design optimization, these nonconventional designs may be used to maximize part
performance in a variety of applications such as medical, aerospace, and defense industries. One
sector of nonconventional designs that have been relatively unexplored is that of compliant
mechanisms or tailored mechanical properties for metamaterial design.
Compliant mechanisms are structures which gain motion through deflection or
deformation of a material rather than traditional hinges or joints. Metamaterials gain their
properties based their structure as opposed to material composition. Using material deformation
to gain motion allows for generation of unique structures that reduce the wear commonly seen in
hinges, remove the need for lubrication in harsh environments, reduce part count, and improve
dexterity. While these advantages present unique opportunities in a variety of situations, it
remains challenging to design and optimize these nonconventional structures.
In this dissertation two systematic design optimization approaches are described for
compliant structures: a shape matching approach for a deployable surgical tool for radiofrequency
ablation (RFA) of tumors and generation of lattice structures for additive manufacturing (AM).
Radiofrequency ablation is an increasing common minimally invasive treatment option
for abdominal tumors. During the procedure, an electrode is inserted into the tumor and the
surrounding tissue is heated. To effectively destroy a tumor with RFA, it is critical that the shape
of the treatment match the shape of the tumor with a small margin around the periphery. The
shape of the treatment, or ablation zone, is largely dependent on the shape of the electrode.
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To improve treatment for endoscopic RFA of tumors, a novel compliant deployable RFA
electrode is presented along with a systematic design optimization approach for shape matching
the treatment zone and tumor. The optimization approach includes a finite element model of RFA
coupled with a genetic algorithm, an approach that may be applied to tumors throughout the
abdomen and mediastinum. A specific example of the approach is demonstrated through
pancreatic tumor ablation. The results of the approach show treatment efficiency is increased
from 25% to 87% for a 2.5cm spherical tumor. In addition, experimental validation of the finite
element RFA model is reported.
With the recent advances in AM, systematic design optimization approaches are needed
for generating structures which take advantage of the design freedom available. With the added
design freedom, lattice structures, which consist of a repeating unit cell topology patterned
throughout a structure, may be used to reduce weight or increase functionality while maintaining
part performance. For AM, nearly infinite possibilities are available for the unit cell topology that
will populate the lattice structure. However, there is a lack of fundamental understanding of unit
cell topology selection. As a result, lattice structures are used without full consideration for how
the lattice structure changes the properties of the part.
For development of tailored compliance, this dissertation presents a design for AM
(DfAM) systematic design optimization approach for generating manufacturable and customized
unit cell topologies. Using a ground structure topology optimization (GSTO) approach, novel unit
cell topologies may be generated to meet application specific needs. By incorporating a library of
unique optimization objectives, constraints, and penalties, Additive Lattice Topology
Optimization (ALTO) is an application-agnostic approach to unit cell design, demonstrated
through various case studies for unique lattice structure applications.
Two case studies are presented for multi-functional lattices that combine minimization of
strain energy and maximization of thermal conductance. The results show how DfAM
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considerations can be included in the optimization to improve manufacturability and are
demonstrated with printed examples of the optimized solution. In addition, these multi-functional
lattices highlight the importance of multi-objective optimization and its ability to generate a
Pareto-optimal set of solutions.
Three other case studies are presented for optimizing structures that match a target
constitutive matrix. The first of these three demonstrates a novel powder removability factor to
generate a unit cell topology that has the same orthogonal effective elastic moduli as a baseline
unit cell topology. The optimized solution shows improved powder removal compared to the
baseline unit cell through mass measurement, CT scan analysis, and tortuosity quantification.
Experimental compression testing of both the optimized and baseline unit cell topologies validate
the predicted results within 6%. The second of these case studies uses optimization to design
three novel unit cell topologies that utilize mechanism-like behavior to reduce stiffness without
reducing weight. This enables lattice structures with a non-uniform stiffness while maintaining a
uniform weight distribution. The third case study demonstrates generation of an orthotropic unit
cell topology, with three times the effective modulus in one direction as compared another.
As an additional part of this work, a compilation of data from analytical models,
experimental characterization, and finite-element analysis of lattice structures has resulted in the
largest public database of mechanical property information for metal lattice structures fabricated
with AM. The Lattice Unit-cell Characterization Interface for Engineers (LUCIE) offers Ashby-
style plots for unit cell topology trends in mechanical properties. It represents a compilation of 69
sources, resulting in 1400 experimental data points, 200+ finite element points, and 45+
analytical models for a total of 18 different unit cell topologies.
The culmination of this work presents research contributions for novel design
optimization approaches for RFA of tumors and DfAM of metal lattice structures through
numerous case studies and experimental validation.
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Table of Contents
List of Figures .......................................................................................................................... ix
List of Tables ........................................................................................................................... xviii
Abbreviations ........................................................................................................................... xx
Nomenclature ........................................................................................................................... xxii
Acknowledgements .................................................................................................................. xxviii
Chapter 1 Introduction ............................................................................................................. 1
1.1 Background and Motivation ....................................................................................... 1 1.1.1 Compliant Mechanisms ................................................................................... 2 1.1.2 Pancreatic Cancer ............................................................................................ 4 1.1.3 Radiofrequency Ablation ................................................................................ 7 1.1.4 Additive Manufacturing .................................................................................. 14 1.1.5 Lattice Structures ............................................................................................. 23 1.1.6 Design Optimization ....................................................................................... 36 1.1.7 Topology Optimization and Additive Manufacturing ..................................... 50
1.2 Research Objectives and Tasks .................................................................................. 53 1.3 Dissertation Contents and Structure ........................................................................... 54
Chapter 2 Optimization and Experimental Validation of an EUS-RFA Electrode .................. 56
2.1 Background and Motivation ....................................................................................... 56 2.2 Deployable Compliant EUS-RFA Electrode Design ................................................. 58 2.3 Systematic Optimization Approach ........................................................................... 61
2.3.1 Ablation Zone Prediction with a Finite Element Model ................................. 61 2.3.2 Optimization using a Genetic Algorithm ........................................................ 65 2.3.3 Results from the Systematic Optimization Approach ..................................... 70 2.3.4 Discussion and Comparison to Standard Straight Electrode ........................... 78 2.3.5 Conclusions from Systematic Optimization Approach ................................... 82
2.4 Testing and Validation of Deployable Electrode ....................................................... 83 2.4.1 Prototype Development ................................................................................... 83 2.4.2 Deployment in a Tissue Phantom .................................................................... 86 2.4.3 Ex-vivo validation of TAM ............................................................................. 93
2.5 Closing Remarks ........................................................................................................ 103
Chapter 3 Unit Cell Generation through Ground Structure Topology Optimization ............... 104
3.1 Background and Motivation ....................................................................................... 104 3.2 Additive Lattice Topology Optimization (ALTO) ..................................................... 107
3.2.1 Ground Structure Model .................................................................................. 108 3.2.2 Optimization Algorithms ................................................................................. 110
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3.2.3 ALTO Objectives, Constraints, and Penalties ................................................. 114 3.2.4 ALTO Validation ............................................................................................ 143
3.3 Closing Remarks ........................................................................................................ 152
Chapter 4 Case Studies on Multi-functional Lattices .............................................................. 153
4.1 Background and Motivation ....................................................................................... 153 4.2 Case Study 1: Generation of a Novel 3D Multi-functional Lattice ............................ 155
4.2.1 Optimization Description ................................................................................ 155 4.2.2 Optimization Results ....................................................................................... 158
4.3 Case Study 2: Generation of a Novel 2D Multi-functional Lattice ............................ 164 4.3.1 Optimization Description ................................................................................ 164 4.3.2 Optimization Results ....................................................................................... 167
4.4 Discussion of Results ................................................................................................. 170 4.5 Closing Remarks ........................................................................................................ 173
Chapter 5 Case Studies on Homogenized Lattice Structures ................................................... 174
5.1 Background and Motivation ....................................................................................... 174 5.2 Case Study 1: Generation of a 3D Unit Cell Topology with Improved Powder
Removability ............................................................................................................ 176 5.2.1 Optimization Description ................................................................................ 177 5.2.2 Optimization Results ....................................................................................... 181 5.2.3 Ground Structure Model Validation ................................................................ 184 5.2.4 Powder Removability ...................................................................................... 193
5.3 Case Study 2: Generation of a 3D Unit Cell Topologies with a Fixed Volume
Fraction .................................................................................................................... 203 5.3.1 Optimization Description ................................................................................ 204 5.3.2 Optimization Results ....................................................................................... 207 5.3.3 Fabrication and Compression Testing ............................................................. 212
5.4 Case Study 3: Generation of a 2D Orthotropic Unit Cell Topology .......................... 214 5.4.1 Optimization Description ................................................................................ 214 5.4.2 Optimization Results ....................................................................................... 218
5.5 Discussion of Results ................................................................................................. 222 5.6 Closing Remarks ........................................................................................................ 228
Chapter 6 LUCIE: Lattice Unit-cell Characterization Interface for Engineers ........................ 229
6.1 Background and Motivation ....................................................................................... 229 6.2 Methods ...................................................................................................................... 231
6.2.1 Literature Compilation .................................................................................... 232 6.2.2 Data Collection ................................................................................................ 238 6.2.3 Assumptions for Comparing Data ................................................................... 240 6.2.4 LUCIE ............................................................................................................. 242
6.3 Results and Case Studies ............................................................................................ 245 6.3.1 Results of Literature Compilation ................................................................... 245 6.3.2 LUCIE Case Study 1 ....................................................................................... 252 6.3.3 LUCIE Case Study 2 ....................................................................................... 253 6.3.4 LUCIE Case Study 3 ....................................................................................... 255
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6.4 Discussion .................................................................................................................. 258 6.5 Closing remarks ......................................................................................................... 264
Chapter 7 Conclusions and Future Work ................................................................................. 266
7.1 Summary of Objectives and Tasks ............................................................................. 266 7.1.1 Objective 1: Develop systematic optimization approach for compliant
deployable radiofrequency ablation electrodes ................................................ 266 7.1.2 Objective 2: Develop three-dimensional ground structure topology
optimization for unit cell DfAM ....................................................................... 267 7.1.3 Objective 3: Explore multi-functional lattices through case study on unit
cell generation for thermal conductance and minimum strain energy .............. 269 7.1.4 Objective 4: Explore homogenized lattices through case study on
generation of unit cells with tailored compliance ............................................. 270 7.1.5 Objective 5: Develop mechanical property database for metal lattice
structures from current literature containing analytical, finite element, and
experimental data ............................................................................................. 271 7.2 Primary Conclusions .................................................................................................. 272 7.3 Research Contributions .............................................................................................. 273
7.3.1 Electrode Optimization for Radiofrequency Ablation .................................... 274 7.3.2 Additive Lattice Topology Optimization ........................................................ 275
7.4 Recommended Future Research Directions ............................................................... 277
References ................................................................................................................................ 283
Appendix A: MATLAB Image Analysis ................................................................................. 316
Appendix B: ALTO 3D MATLAB Files ................................................................................. 319
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List of Figures
Figure 1.1 Compliant surgical forceps reduce the need for complex hinges on a small
scale, lack debris from wear, and do not require lubrication, adapted from [8] ............... 2
Figure 1.2 Examples of multi-material (left) and metal (right) compliant mechanisms
designed for AM, adapted from [13,19] ........................................................................... 4
Figure 1.3 Five year survival rates for the 10 most common cancers [22]. Pancreatic
cancer is the lowest with an average fiver year survival rate of 7%. ............................... 5
Figure 1.4 (Left) Diagram of radiofrequency ablation showing the mismatch that can
occur between a tumor and ablation zone and (Right) thermal-imaging of
radiofrequency ablation showing the ellipsoidal temperature profile around a straight
electrode, adapted from [41] ............................................................................................ 10
Figure 1.5 Basic electrode types and several commercial models of radiofrequency
ablation electrodes described by Mulier et al., images adapted from [51] ....................... 10
Figure 1.6 Metal AM system types include powder bed fusion and wire/powder feed
directed energy deposition, typically using either a laser or electron beam energy
source, adapted from [112] .............................................................................................. 17
Figure 1.7 Example of warping due to thermal contraction..................................................... 23
Figure 1.8 Various forms of cellular solids including a (top) 2D honeycomb, (middle) 3D
open-cell foam, and (bottom) 3D closed-cell foam, adapted from [136] ......................... 25
Figure 1.9 Lattice structures have been proposed for a variety of medical implants with
the intent of improving performance and functionality. Due to the variety of lattice
structures that can be used in such applications, there is a need for further research to
understand how to select or design an appropriate unit cell for the intended
application. Images adapted from A) [179], B) [177], C) [181], D) & E) [194]. ............ 27
Figure 1.10 Unit Cells are often generated based on Boolean operations with primitive
shapes such as these examples, adapted from [195] ........................................................ 28
Figure 1.11 Unit cells may be generated using implicit mathematical equations to
represent a surface. In this particular case, the surface is defined at F(x,y,z) = 0 with
parameters "a" and "b" used to tailor the shape of the surface; adapted from [195]. ....... 29
Figure 1.12 Two common methods for lattice population are (top) uniform in which the
basic lattice is not modified based on the component shape and (bottom) conformal
in which each unit cell may be deformed to follow the surface of the component,
adapted from [224] ........................................................................................................... 30
Figure 1.13 One method of generating a functionally graded lattice is by computing the
stress within the part, plotting the stress in gray scale, and then dithering the image
to generate a density map. Next, small stiff unit cells are placed in dense regions,
x
with high stress, and large unit cells are placed in less dense regions, adapted from
[229]. ................................................................................................................................ 31
Figure 1.14 Functionally graded lattice structure populated using a relative density map
based on CT imaging, adapted from [178] ....................................................................... 32
Figure 1.15 Lattice generation and sizing method using stress results from a finite
element analysis, adapted from [234] .............................................................................. 33
Figure 1.16 Diagram of genetic algorithm optimization process ............................................. 39
Figure 1.17 Cartoon graphic used to describe maximization of an objective for gradient-
based algorithms, adapted from [254] .............................................................................. 40
Figure 1.18 Minimization of compliance benchmark optimized with the SIMP method.
Varying the power-law penalization factor, p, reduces the amount of gray material in
the solution, clearly defining the boundaries of the topology. Images adapted from
[270]. ................................................................................................................................ 44
Figure 1.19 Boundary and Loading conditions for a compliant inverter as well as the
evolution of the topology during convergence, adapted from [285] ................................ 45
Figure 1.20 A variety of compliant gripper topologies generated by varying optimization
parameters, adapted from [285] ....................................................................................... 46
Figure 1.21 An example comparison of the continuum and ground structure approach for
topology optimization ...................................................................................................... 48
Figure 1.22 Potential tradeoffs exist in the ability to find design that consider the amount
of support material used while simultaneously considering the performance of the
component, adapted from [328] ....................................................................................... 51
Figure 2.1 For straight electrodes, multiple insertions are often required to overlap a
series of ablation zones to destroy tumors. The image above is for an example of a
percutaneous approach, but, for EUS-RFA, the ability to insert multiple electrodes is
limited creating the need for specialized electrodes, adapted from [51]. ......................... 57
Figure 2.2 The electrode is deployed into the tumor through an endoscopic needle, acting
as a sheath, whose tip has been positioned at the periphery of the tumor, adapted
from [339] ........................................................................................................................ 58
Figure 2.3 Two variations of the proposed electrode design are pictured in the stowed
(A,D) and deployed (B,E) configurations. The design parameters for each are shown
(C,F), adapted from [339] ................................................................................................ 60
Figure 2.4 General setup for the TAM. Due to the symmetry of the electrode, the TAM
was reduced to quarter symmetry to decrease computation time. The grounding pads
regions are the top and bottom quarter circles as well as the exterior wall of the
cylinder (not shown) , adapted from [339]. ...................................................................... 64
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Figure 2.5 Graphical description of the objective functions used for optimization, adapted
from [339] ........................................................................................................................ 68
Figure 2.6 Each design tested during the optimization is plotted in this figure according to
its corresponding objective function values, adapted from [339] .................................... 74
Figure 2.7 (Left) The side view of the temperature profile for an optimal electrode
solution for Case 1. The ablation zone is marked by the white isothermal contour
(60°C). (Right) The ablation zone surface (green) is shown compared to the target
shape (black outline) for the top and side views, adapted from [339]. ............................ 75
Figure 2.8 Each Case 2 design simulated is plotted above according to its corresponding
objective function values, adapted from [339] ................................................................. 76
Figure 2.9 (Left) The side view of the temperature profile for an optimal electrode
solution for Case 2. The ablation zone is marked by the white isothermal contour
(60°C). (Right) The ablation zone surface (green) is shown compared to the target
shape (black outline) for the top and side views, adapted from [339]. ............................ 77
Figure 2.10 Top and side views of the ablation zones for the straight electrode and
optimal solutions of Case 1 and Case 2. The green surface represents the 60°C
isothermal surface and the black circle represents the target zone, adapted from
[339]. ................................................................................................................................ 80
Figure 2.11 L-PBF 2X scale electrode prototype .................................................................... 84
Figure 2.12 Near 1X scale L-PBF electrode prototype ............................................................ 84
Figure 2.13 4-tine laser micromachined prototype .................................................................. 85
Figure 2.14 8-tine and 12-tine prototypes based on optimal solution configurations that
used for ex-vivo experimentation..................................................................................... 86
Figure 2.15 Insertion force and deployment experimental setup for early prototypes of the
EUS-RFA electrode ......................................................................................................... 87
Figure 2.16 (Left) The spread or amount of deployment is shown based on the insertion
speed. (Right) The insertion force profile is separated into three segments showing
the friction sliding through the sheath, the penetration of the central needle, and then
the deployment of the outer tines. Both graphs are for the L-PBF electrode. .................. 89
Figure 2.17 (Left) The spread or amount of deployment is shown based on the insertion
speed. (Right) The insertion force profile is separated into three segments showing
the friction sliding through the sheath, the penetration of the central needle, and then
the deployment of the outer tines. Both graphs are for the LMM electrode. ................... 90
Figure 2.18 Geometry details for 8-tine deployable electrode prototype for TAM
validation, adapted from [355] ......................................................................................... 94
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Figure 2.19 Schematic and photo of the experimental set-up for thermochromic tissue
phantom testing, adapted from [355] ............................................................................... 96
Figure 2.20: Tissue phantom and TAM ablation zone and analyzed images for the 15W
power level, adapted from [355]. (Scale 15.4 px/mm) ..................................................... 99
Figure 2.21: Tissue phantom and TAM ablation zone and analyzed images for the 20W
power level, adapted from [355]. (Scale 17.4 px/mm) ..................................................... 99
Figure 2.22: Comparison of the ablation zone size and shape by the major and minor axis
of an ellipse fitted to each ablation zone with image analysis; see examples in Figure
2.20 and Figure 2.21, adapted from [355] ........................................................................ 100
Figure 3.1 Overview of Additive Lattice Topology Optimization (ALTO) to generate
optimized unit cell topologies for lattices fabricated with AM ........................................ 108
Figure 3.2 Example of traditional thermal circuits and analogy to the ground structure
formulation for calculating the unit cell thermal conductance ......................................... 118
Figure 3.3 Schematic for three test strain cases to an RVE for calculation of the
homogenized constitutive matrix, adapted from [298] .................................................... 121
Figure 3.4 Schematic showing how powder can become trapped in internal cavities for
powder-based AM processes............................................................................................ 125
Figure 3.5 Ground structure-based pore size measurement for the PRF. On the left part of
the image, an example of the element-to-element distance measurement is made
midpoint to midpoint and then the radius of each element is subtracted to obtain the
pore size for the pore distance matrix. A few examples of some of the pore
measurements are shown on the right part of the image. In the pore matrix, each
element is measured relative to every other element. ...................................................... 126
Figure 3.6 For lattice structures, horizontal elements are allowed to span between two
vertical supports because of the generally small length-scale. However, there is an
intermediate range for which the build angle is too low to accurately build. .................. 130
Figure 3.7 Horizontal lattice members are shown in two lattice examples, 3x3x3 arrays of
octet-truss unit cells (4 mm, 7 mm) and cubic unit cells (4 mm, 6 mm). These
lattices were printed in Ti64 on a 3D Systems ProX DMP 320. Image adapted from
[361]. ................................................................................................................................ 131
Figure 3.8 Basic 2D example for determining the supported nodes for the unsupported‒N
penalty. This example comes from row three of Table 3.2 which passes the
unsupported‒N penalty but would fail the unsupported‒S penalty. The unsupported‒
S penalty follows a similar process but the node-neighbor pairs would not be
considered meaning that node 4 is not self-supported, but instead supported by a
neighboring unit cell. Horizontal elements that bridge supported nodes are
considered feasible. .......................................................................................................... 140
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Figure 3.9 Graph of penalty function that reduces complexity by encouraging elements
with a cross-sectional area larger than the relevant cross-sectional area limit. ................ 142
Figure 3.10 Schematics for the MBB beam benchmark problem and the initial ground
structure used to represent the design space..................................................................... 145
Figure 3.11 Schematics for the cantilever beam benchmark problem and the initial
ground structure used to represent the design space. ....................................................... 145
Figure 3.12 MBB beam comparison with results reported in literature. The different
solutions are shown for a fixed volume and changing the minimum feature size
limit. The line width on the bottom images is correlated to the cross-sectional area
from the GSTO solutions. Top row of images adapted from [273]. ................................ 146
Figure 3.13 Cantilever benchmark comparison with literature results. Various solutions
are shown for a fixed allowable volume and changing the minimum feature size
limit. The line width on the bottom images is correlated to the cross-sectional area
from the GSTO solutions. Top row of images adapted from [273]. ................................ 147
Figure 3.14 Schematic for thermal conductance benchmark from literature. The
continuum topology optimization approach (left) is shown compared to the GSTO
approach (right). ............................................................................................................... 149
Figure 3.15 Thermal conductance benchmark results from literature (left, middle) and
from the GSTO in ALTO show good correlation, validating the thermal conductance
formulation and optimization objective. Left and middle images adapted from [278]
and [281], respectively. .................................................................................................... 150
Figure 4.1 Case Study 1: 3D multi-functional lattice example problem, simultaneously
optimizing for (A) minimization of strain energy and (B) maximizing the thermal
conductance ...................................................................................................................... 157
Figure 4.2 Solutions to the optimization problem from Case Study 1. Design A and
Design B are the solutions without and with the DfAM penalty, respectively. ............... 159
Figure 4.3 Design A has several long horizontal overhangs reducing the
manufacturability of the unit cell. Incorporating a DfAM penalty into the
optimization, the algorithm discovered Design B which is fully self-supported. The
images depict an XZ plane through the center of the unit cell topology for improved
visibility of some of the unsupported regions. ................................................................. 159
Figure 4.4 Graphs of the solutions space for an instance of the Case Study 1
Optimization. Each point is a Pareto-optimal solution for (A) modified thermal
conductance versus strain energy objectives and (B) modified thermal conductance
versus volume fraction objectives. The red point in each graph represents Design B
shown in Figure 4.2 .......................................................................................................... 162
Figure 4.5 Printed examples of the optimized solution from the 3D multi-functional
lattice case study. The examples were printed as a single unit cell (scaled up 5x)
with FFF and in a 2x2x2 array (scaled 2x) with L-PBF. By incorporating DfAM
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considerations into the optimization, the optimized solution was print-ready, not
requiring any design interpretation. ................................................................................. 164
Figure 4.6 Case Study 2: (A) Minimize strain energy and (B) Maximize thermal
conductance. Blue dashed lines denote symmetry planes. Images adapted from
[361]. ................................................................................................................................ 166
Figure 4.7 Example convergence plot for the solution to the optimization problem with
equal weighting factors for both objectives. The result shows an increasing objective
function score until the constraints are met and then gradually decreasing until
convergence. .................................................................................................................... 168
Figure 4.8: Case Study 2 results showing the tradeoffs between minimization of strain
energy and maximization of thermal conductance. 11 different solutions are shown
beginning for incrementing the weight values by 0.1 as the solutions move
clockwise. Image adapted from [361]. ............................................................................. 168
Figure 4.9: Comparison of the objective function values for minimization of strain energy
and maximization of thermal conductivity (minimization of its’ reciprocal plotted
here). This plot shows the solution space, demonstrating the tradeoffs between
objectives. On the plot, all 101 solutions are shown, with the grey dots being non-
dominated solutions on the Pareto-front. Image adapted from [361]. .............................. 169
Figure 5.1 The octet-truss (A) is the baseline unit cell topology for matching the
constitutive matrix, colors have been added to improve visualization of 3D topology.
The initial ground structure is shown in (B), where each line represents a truss
element. With symmetry, there were a total of 120 distinct design variables. ................. 181
Figure 5.2 The two unit cells output from the optimization are shown on either side of the
octet-truss. The top images show a single unit cell and the bottom images are side
views of a 3x3 patterned array. As a result of the PRF as an objective, both the
octahedral and opti-octet open up the porous space to improve powder removal and
flowability. ....................................................................................................................... 182
Figure 5.3 Complete build plate with octet-truss and opti-octet lattice structures among a
few other lattice types. In the picture, one of the five samples of each lattice type are
labeled. The interlocked and diamond lattice are discussed as part of Case Study 2 in
Section 5.3. ....................................................................................................................... 185
Figure 5.4 Examples of the octet-truss (left) and opti-octet (right) crushed after
compression loading. Prior to compression, the samples were 40 x 40 mm cubes or a
unit cell size of 5mm as noted by the scale bars. ............................................................. 186
Figure 5.5 Average stress strain curves for the octet-truss and opti-octet truss. The solid
portion of the curves represents the region where the effective modulus was
measured, 10-50% of the peak stress. The grey band behind each curve represents
the average plus and minus the standard deviation of the experimental data. ................. 187
Figure 5.6 AM-based corrections applied to ground structure prediction. As a result of
either or both corrections, the cross-section of the strut is effectively reduced. .............. 190
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Figure 5.7 Schematic of a compression sample where the color gradient represents the
impact of edge effects from the friction with compression platens. Near the
compression platens a unit cell experiences “constrained” boundary conditions as
compared to “relaxed” boundary conditions for a unit cell near the middle of the
sample. ............................................................................................................................. 192
Figure 5.8 As-printed lineup of octet-truss powder removal specimens. The largest
specimen is a 5x5x5 array of 7 mm unit cells, approximately 35 mm in width. The
smallest specimen is a 5x5x5 array of 1.5 mm unit cells, approximately 7.5 mm in
width. ............................................................................................................................... 195
Figure 5.9 Comparison of the relative error between the as-designed and as-printed mass
for the powder removal specimens. In (B) three regions are predicted based on the
relative error (1) no powder removed, (2) partial removal, and (3) all powder
removed, shown with the red ellipses overlaid. ............................................................... 197
Figure 5.10 Qualitative CT scan comparison of a central slice for the octet-truss and opti-
octet unit cell sizes. The dashed line squares represent a single unit cell within each
image. Comparing the 4.5 mm unit cell sizes against the computer generated as-
designed cross-section, the octet-truss shows more trapped powder or partially
sintered regions as compared to the opti-octet, validating that the opti-octet
demonstrates improved powder removability. ................................................................. 200
Figure 5.11 Case Study 2 is a multi-objective optimization for creating a lattice structure
that mimics the homogenized constitutive matrix of a Diamond unit cell (A) and (B).
The unit cell graph is shown in (C) and (D). To ensure similar interconnectivity to
the Diamond unit cell, the full ground structure is reduced to the initial configuration
shown in (E) and (F). ....................................................................................................... 207
Figure 5.12 Case Study 2 resulted in three unit cells which match the scaled target
constitutive matrix of the diamond unit cell. In each solution, a fundamental
tetrahedra (grey) shape was found with decreasing diameter. For the middle and top
solutions, several non-load bearing elements (blue) form a mechanism around the
tetrahedra shape, based on the pin-joint assumption, to simultaneously match the
volume fraction and homogenization targets. .................................................................. 208
Figure 5.13 The interlocked lattice is a unique patterning of the tetrahedra base that
results in a scaled and interlocked set of two diamond lattice. In the final patterned
interlocked lattice, the unit cells only connect to diagonally neighboring unit cells. ...... 210
Figure 5.14 The Interlocked lattice is a set of two interconnected lattices. The upper
lattices were printed in Alumide® on an EOS P110. The lower left image shows the
interlocking lattices, distinguished by the blue and grey colors. ..................................... 211
Figure 5.15 Average stress strain curves for the diamond and interlocked lattices. The
solid portion of the curves represents the region where the effective modulus was
measured, 7.5-60% of the peak stress. The grey band behind each curve represents
the average plus and minus the standard deviation of the experimental data. ................. 213
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Figure 5.16 The purpose of Case Study 3 is to generate a unit cell topology that has an
orthotropic constitutive matrix, specifically that it is three times stiffer in the X-
direction than in the Y-direction. ..................................................................................... 214
Figure 5.17 Graph of the DfAM penalty function that penalizes elements near the
relevant cross-sectional area limit and reduces non-critical elements. Image adapted
from [361]. ....................................................................................................................... 217
Figure 5.18 Lowest optimization scores for the constitutive matching objective function.
Each image represents the best solution of 20 initial guesses for the particular
combination of penalty factor level and relevant diameter limit. Image adapted from
[361]. ................................................................................................................................ 219
Figure 5.19 Optimized unit cell topology with the area of the elements connected to the
corners set to the lower limit. Compression in X and Y is also shown, demonstrating
the target constitutive matrix. Total error in the constitutive matrix was 3x10-6 %.
Image adapted from [361]. ............................................................................................... 221
Figure 5.20: Example of a lattice structure patterned from the unit cell topology shown in
Figure 5.19. This unit cell topology was generated using a penalty function level of
one third and the relevant diameter limit set to 0.5 mm. Image adapted from [361]. ...... 221
Figure 5.21 Without DfAM restrictions and penalties to ensure manufacturability, there
were a variety of pitfalls that appear in optimization solutions. The DfAM
considerations help to reduce the need for design interpretation and prepare more
print-ready solutions. ....................................................................................................... 226
Figure 6.1 Graphic summary of the methods section including how sources were found
and refined, data collection and assumptions made, and finally incorporation into
LUCIE. ............................................................................................................................. 232
Figure 6.2 Schematic showing the inputs and outputs for LUCIE .......................................... 244
Figure 6.3 Typical stress-strain curves for stretch-dominated or bend-dominated
structures, labeled with the relevant mechanical property terminology for the data
compiled in this work ....................................................................................................... 249
Figure 6.4 Relative frequency of uncertainty in experimental data points. Uncertainty was
normalized by the nominal value to obtain the relative error. 75% of the reported
uncertainty values had a relative error of less than 10%. ................................................. 252
Figure 6.5 LUCIE Case Study 1 output comparing analytical, experimental, and FEA
data for the BCC, Diamond, and Truncated Cube unit cell topologies. From the
output you can visualize differences in analytical models for a single unit cell
topology, such as the Diamond, as well as gain an understanding of the normalized
effective modulus for different unit cell topologies. ........................................................ 253
Figure 6.6 LUCIE Case Study 2 output shows a comparison of 7 different unit cell
topologies and the variability that exists in experimental data. The error bars around
a single point represent a small “bin” of data. The point itself represents the average
xvii
value within the bin, while the error bars denote the maximum and minimum
reported values for each property. .................................................................................... 254
Figure 6.7 LUCIE Case Study 3 output from LUCIE for comparison of the lattice
structure properties of different unit cell topologies. The red box with a dashed line
highlights the region of interest (EC 0.02-0.03 and Volume Fraction 0.2-0.25) for
determining appropriate unit cell selection. ..................................................................... 256
Figure 6.8 LUCIE Case Study 3 plot comparison of normalized effective modulus and
normalized yield stress for comparison of the six unit cell topologies that were down
selected from Figure 6.7. The thumbnails of the unit cell topologies have been added
to the plot; from top to bottom they are Cubic, TPMS Gyroid,
Rhombicuboctahedron, Truncated Cuboctahedron, BCC. No yield stress data was
available for the Octet-truss at this volume fraction range. ............................................. 257
Figure 6.9 Number of sources found for each unit cell topology. The results show the
amount of lattice structure data available is limited for many common unit cell
topologies. ........................................................................................................................ 262
xviii
List of Tables
Table 1.1 A wide range of opportunistic and restrictive DfAM topics exist for which
there is ongoing research .................................................................................................. 19
Table 2.1 Initial and boundary conditions for the TAM .......................................................... 64
Table 2.2 Material properties for TAM ................................................................................... 71
Table 2.3 Case 1 and Case 2 optimization and electrode design parameters ........................... 72
Table 2.4 Electrode designs selected for comparison from the Case 1 and Case 2 Pareto-
optimal solution sets ......................................................................................................... 77
Table 2.5 Quantitative comparison of a straight electrode and optimal Case 1 and Case 2
solutions ........................................................................................................................... 79
Table 2.6: Tissue phantom material properties [357] .............................................................. 95
Table 3.1 Summary of ALTO objective, constraint, and penalty functions for unit cell
generation and their applicability to DfAM ..................................................................... 115
Table 3.2 Examples of 2D unit cells for comparing pass/fail criteria of different penalty
functions. .......................................................................................................................... 138
Table 3.3 Comparison of homogenization benchmark problems and ground structure
calculation. For these benchmark problems, the ground structure was an exact match
to the reported constitutive matrix, images adapted from [297,298] ............................... 151
Table 4.1 Case Study 1: 3D Multi-functional lattice optimization summary .......................... 155
Table 4.2 Comparison of objective function values for two designs from Case Study 1.
Design A is not manufacturable due to long unsupported overhangs in the topology. .... 160
Table 4.3 List of Pareto-optimal solutions with the modified thermal conductance
objective less than -0.10, strain energy objective less than 95, and the volume
fraction objective sorted smallest to largest. According to the requirements of the
designer, the optimal solution would be Design B shown in the first row in bold
typeface. ........................................................................................................................... 163
Table 4.4 Case Study 2: 2D Multi-functional lattice optimization summary .......................... 165
Table 5.1 Case Study 1: Improved powder removal lattice optimization summary ................ 178
Table 5.2 Summary of the objective function values for the optimization results. The
octet-truss results are shown for individual evaluation of the unit cell, separate from
the optimization. ............................................................................................................... 182
xix
Table 5.3 Experimental validation results for the effective modulus and comparison to the
ground structure model prediction in ALTO and AM-based corrections. For AM
corrections, IO = infill-only correction and SC = stair case correction. The error
relative to the experimental value is noted in the parenthesis. ......................................... 187
Table 5.4 Effective modulus comparison between experimental data and ALTO
predictions of “constrained” and “relaxed” boundary conditions. The error relative to
the experimental value is noted in the parenthesis. .......................................................... 193
Table 5.5 Case Study 2: 3D homogenized unit cells with different stiffnesses
optimization summary ...................................................................................................... 205
Table 5.6 Case Study 3: 2D orthotropic unit cell generation optimization summary .............. 215
Table 5.7 Lattice structures demonstrate a substantial difference in mechanical properties
based on the number of layers. As the number of layers increase, the effective
modulus decreases, data from [207]. ................................................................................ 223
Table 6.1 For sources without bulk mechanical properties of the as-built material,
reported within the source, these generic AM mechanical property values were used
to normalize data to create a material independent comparison of unit cell topology. .... 234
Table 6.2 Summarized list of all sources for data compiled in LUCIE. In this table the
unit cell topology, properties recorded, and data type available are sorted by source. .... 236
Table 6.3 For each unit cell topology included in LUCIE, there are 6 different properties
that can be compared. The data type available for each unit cell topology are denoted
by analytical (A), experimental (E), or FEA (F). A dash (-) signifies that no
reference was found that provided this information for any of the three data types. ....... 246
Table 6.4 Name and associated picture for each lattice structure included in LUCIE ............ 247
xx
Abbreviations
2D Two-dimensional
3D Three-dimensional
A Analytical data type for LUCIE
AM Additive manufacturing
AMI Actuated Medical Incorporated
BCC Body-centered cubic
BS Lattice structure buckling stress normalized by the solid yield stress
CAD Computer-aided design
CM Compliant mechanism
CT Computed tomography
DED Directed energy deposition
DfAM Design for additive manufacturing
E Experimental data type for LUCIE
EC Lattice structure compressive effective elastic modulus normalized by the solid
material modulus
EUS-RFA Endoscopic ultrasound-guided radiofrequency ablation
F Finite element analysis data type for LUCIE
FCC Face-centered cubic
FEA Finite element analysis
FFF Fused filament fabrication
GSTO Ground structure topology optimization
L-PBF Laser powder bed fusion
LMM Laser micromachining
LUCIE Lattice unit-cell characterization interface for engineering
MRI Magnetic resonance imaging
NSGA-II Non-dominated sorting genetic algorithm
PBF Powder bed fusion
PR Lattice structure Poisson’s ratio
xxi
PS Lattice structure plateau stress normalized by the solid yield stress
PVC Polyvinyl chloride
PRF Powder removability factor
RFA Radiofrequency ablation
RVE Representative volume element
SIMP Simplified isotropic material with penalization
STL Stereolithograhy file type, a common model file for 3D-printing
TAM Thermal ablation model
TPMS Triply periodic minimal surfaces
US Ultrasound
YS Lattice structure yield stress normalized by the solid yield stress
xxii
Nomenclature
𝐴𝑒 Cross-sectional area of each element
𝐴𝑙𝑜𝑤𝑒𝑟 Lower limit of cross-sectional area
𝐴𝑟𝑒𝑙 Relevant cross-sectional area limit
𝐴𝑢𝑝𝑝𝑒𝑟 Upper limit of cross-sectional area
𝑨 Vector of element cross-sectional areas
𝑩 Connectivity vector
𝑐𝑖 Weight values for combining optimization objectives
𝐶 Specific heat of pancreatic tissue
𝐶𝑏 Specific heat of blood
𝐶𝑙𝑖 Constraint value for relative center and outer tine length
𝑑𝑒 Stowed electrode outer diameter
𝑑𝑙𝑜𝑤𝑒𝑟 Lower diameter limit
𝑑𝑢 Distance to the utopia point
𝑑𝑢𝑝𝑝𝑒𝑟 Upper diameter limit
𝑑𝑟𝑒𝑙 Relevant diameter limit or minimum feature size
𝒅0𝑒𝑘 Induced displacement vector for element “e”, subject to test strain case “k”
𝒅𝑒𝑘 Applied displacement vector for element “e”, subject to test strain case “k”
𝐷𝑖𝑗𝑒𝑒 Distance from midpoint to midpoint between element i and element j
�̃�𝑒𝑒 Element-to-element distance matrix
𝐸 Modulus of elasticity
�̃� Material constitutive matrix
�̃�𝐻 Homogenized constitutive matrix
�̃�𝐷𝐻 Diamond unit cell homogenized constitutive matrix
�̃� Overlap penalty matrix
�̃�∗ Target constitutive matrix
F Ground structure force vector
𝑓 Overall objective function for gradient-based optimization algorithm
xxiii
𝑓𝑖 Percentage greater or smaller than 𝑡𝑖 for a grid point
𝑓𝑜𝑣𝑒𝑟 Objective function representing ablation zone outside the target region
𝑓𝑢𝑛𝑑𝑒𝑟 Objective function representing ablation zone inside the target region
ℎ𝑏 Height of insulated shaft connected to the electrode
ℎ𝑡 Height of biological tissue during simulation
𝑱 Current Density
𝑱𝑒 Externally generated current density
𝑘 Thermal conductivity
�̃�𝑒 Element stiffness matrix for element “e” (4x4 for 2D, 6x6 for 3D)
�̃� Ground structure global stiffness matrix
𝑙𝑏 Exposed electrode base length
𝑙𝑐 Electrode center tine length
𝐿𝑒 Ground structure element length
𝑙𝑜 Length of outer tines
𝑙𝑐𝑙𝑜𝑤𝑒𝑟 Lower bound of center tine length
𝑙𝑐𝑢𝑝𝑝𝑒𝑟
Upper bound of center tine length
𝑙𝑜𝑒𝑣𝑒𝑛 Length of even numbered electrode tines for Case 2
𝑙𝑜𝑖 Electrode outer tine length, 𝑖 = 1,… , 𝑛𝑡
𝑙𝑜𝑙𝑜𝑤𝑒𝑟 Lower bound of outer tine length
𝑙𝑜𝑜𝑑𝑑 Length of odd numbered electrode tines for Case 2
𝑙𝑜𝑢𝑝𝑝𝑒𝑟
Upper bound of center tine length
𝐿𝑒 Ground structure element length
𝐿𝑙𝑜𝑤𝑒𝑟 Lower limit of ground structure element length
𝐿𝑢𝑝𝑝𝑒𝑟 Upper limit of ground structure element length
𝑳 Vector of element lengths
𝑚𝑖 Minimizes (𝑚𝑖 = 1) or maximizes (𝑚𝑖 = −1) an objective function
𝑀 Total number of elements in the ground structure
𝑛𝑅𝐸 Number of active elements in the ground structure (𝐴𝑒 > 𝐴𝑟𝑒𝑙)
xxiv
𝑛𝑔𝑝 Number of grid points for computing the objective functions
𝑛𝑜𝑏𝑗 Number of objective functions in the optimization
𝑛𝑜𝑣𝑒𝑟 Number of grid point locations outside the target region
𝑛𝑝 Number of penalty functions in the optimization
𝑛𝑠𝑡𝑎𝑙𝑙 Number of stall generations for determining convergence
𝑛𝑡 Number of electrode outer tines
𝑛𝑢𝑛𝑑𝑒𝑟 Number of grid point locations inside the target region
𝒏 Surface normal vector
𝑁𝑡 Total number of nodes in the ground structure
𝑁1, 𝑁2 Node (or set of nodes) for thermal conduction
𝑁𝑑𝑜𝑓 Number of degrees of freedom
𝑶 Null vector
�̃� Null matrix
𝑝∗ Pore threshold for powder removability factor
𝑝𝐶 Connectivity penalty
𝑝𝑖 Individual penalty scores
𝑝𝑂 Overlap penalty
𝑝𝑠𝑒 Penalty for each active element (gradient-based only)
𝑝𝑈𝑁 Unsupported–N penalty
𝑝𝑈𝑆 Unsupported–S penalty
𝑃 Combined penalty value
�̃� Pore distance matrix for powder removability
𝒒 Heat flux
𝑞𝑒𝑓𝑓 Effective heat flux from 𝑁1 to 𝑁2 for thermal conduction
�̃�𝒆 Element mutual energy matrix for element “e”
𝑄𝑒𝑥𝑡 Heat source from spatial heating
𝑄𝑗 Current source
𝑄𝑚𝑒𝑡 Metabolism heat source
xxv
𝑟𝑏 Radius of insulated shaft connected to the electrode
𝑟𝑖 Radial distance from the center of the ablation zone to a grid point
𝑟𝑜 Radius of curvature of outer tines
𝑟𝑡 Radius of biological tissue during simulation
𝑟𝑖𝑒 Radius of element i in ground structure
𝑟𝑜𝑒𝑣𝑒𝑛 Radius of curvature for even numbered electrode tines for Case 2
𝑟𝑜𝑖 Electrode outer tine radius of curvature, 𝑖 = 1,… , 𝑛𝑡
𝑟𝑜𝑙𝑜𝑤𝑒𝑟 Lower bound of outer tine radius of curvature
𝑟𝑜𝑜𝑑𝑑 Radius of curvature for odd numbered electrode tines for Case 2
𝑟𝑜𝑢𝑝𝑝𝑒𝑟
Upper bound of outer tine radius of curvature
𝑅𝑒 Element thermal resistance
𝑠𝑖 Penalty function scalar to determine impact of penalty
𝑺 Unsupported elements vector
𝑺𝑁 Unsupported elements vector, augmented by neighbors
𝑆𝐸 Strain energy
𝑇 Temperature
�̅� Average temperature at heat input nodes for modified thermal conduction
𝑇1, 𝑇2 Temperature assigned to corresponding 𝑁1, 𝑁2 node(s) for thermal conduction
𝑇𝑏 Blood temperature
𝑡𝑖 Target radial distance from the center of the ablation zone to a grid point
𝑇∞ Ambient tissue temperature
𝒖 Ground structure nodal displacement vector
𝒗 Velocity vector
𝑉 Voltage potential
𝑉𝑢𝑐 Unit cell volume
𝑉𝑓 Volume fraction
𝑉𝑓∗ Target volume fraction
𝑉𝑚𝑎𝑥 Total volume of all ground structure elements at 𝐴𝑢𝑝𝑝𝑒𝑟
xxvi
𝑉0 Voltage applied to the electrode
𝑤𝑖 Individual objective score
𝑤𝑅𝐸 Relevant elements objective
𝑤𝐻 Homogenization objective function
𝑤𝑀𝑇𝐶 Modified thermal conduction objective function
𝑤𝑀𝑇𝐶̅̅ ̅̅ ̅̅ ̅ Normalized modified thermal conduction objective function
𝑤𝑃𝑅𝐹 Powder removability factor objective function
𝑤𝑆𝐸̅̅̅̅ Normalized strain energy objective function
𝑤𝑆𝐸𝑚𝑖𝑛 Strain energy for all element areas set to 𝐴𝑢𝑝𝑝𝑒𝑟
𝑤𝑆𝐸𝑚𝑎𝑥 Strain energy for all element areas set to 𝐴𝑙𝑜𝑤𝑒𝑟
𝑤𝑇𝐶 Thermal conduction objective function
𝑤𝑇𝐶̅̅̅̅ Normalized thermal conduction objective function
𝑤𝑇𝐶𝑚𝑖𝑛 Thermal conduction for all element areas set to 𝐴𝑙𝑜𝑤𝑒𝑟
𝑤𝑇𝐶𝑚𝑎𝑥 Thermal conduction for all element areas set to 𝐴𝑢𝑝𝑝𝑒𝑟
𝑤𝑉 Volume objective function
𝑤𝑉𝐹 Volume fraction objective function
𝑊𝑖 Penalized objective function score
𝑊 Combined objective function value
𝑌 Characteristic length of the representative volume element
𝛼 Ratio of 𝐸𝑥𝑥/𝐸𝑦𝑦 for 2D constitutive matrix
𝛼1, 𝛼2, 𝛼3, 𝛼4 Parameters used to define the penalty function for gradient-based algorithms
𝛽1, 𝛽2, 𝛽3 Property-volume fraction constants for power law regression model
𝛾 Penalty impact level
𝜖 Threshold or tolerance on the objective function values for Borg
𝜂 Tolerance for Pareto front distribution change during 𝑛𝑠𝑡𝑎𝑙𝑙 generations
𝜈 Poisson’s ratio
𝜙 Spherical coordinates angle from Z axis to direction vector
𝜌 Density of pancreatic tissue
xxvii
𝜌𝑏 Density of blood
𝜎 Electrical conductivity of pancreatic tissue
𝜃 Spherical coordinates angle from X axis in XY plane
𝜃𝑒 Element build angle measured from the build plate
𝜃𝑙𝑜𝑤𝑒𝑟 Lower limit for build angle constraint
𝜃𝑢𝑝𝑝𝑒𝑟 Upper limit for build angle constraint
𝜓 Mechanical property for LUCIE power law regression
𝜔𝑏 Perfusion rate of blood
xxviii
Acknowledgements
The completion of this dissertation was a process that started long before my first day at
the Pennsylvania State University and would not have been possible without so much help along
the way. First, I would like to acknowledge the mentorship of my adviser Dr. Mary Frecker. Her
continued support and guidance throughout my MS and PhD programs have been invaluable,
without which, I would not have been pushed to become a better researcher, scholar, and person.
Dr. Frecker has opened my eyes to design optimization and I’ll never be able to look at another
problem again without wondering about the tradeoffs in the design and how it could be
optimized. In addition, I am grateful for the feedback and recommendations of my committee
members, Dr. Kraft, Dr. Moore, Dr. Reutzel, and Dr. Simpson which have helped shape my
research.
I would like to acknowledge my undergraduate institution, Brigham Young University.
The education and opportunities there strengthened my drive for engineering and laid the
groundwork for my education. Working with Professor Brian Jensen and various graduate and
undergraduate students in the BioMEMS group solidified my love of research, ultimately leading
to my decision to pursue a PhD.
During my time at Penn State, I have had so much help from many different groups and
individuals such as the EDOG Lab with all of its students, past and present. I’m grateful for the
support of Dr. Simpson, through attending the additive manufacturing research group meetings
and as part of the Additive Manufacturing and Design (AMD) program. It was my first parts
printed in the EDOG lab and later at CIMP-3D that spurred my fascination for additive
manufacturing which later led me a major portion of my research and contributions to the
scientific community. I appreciate the support of Dr. Moyer who provided invaluable insight for
the development of radiofrequency ablation electrodes. I’m grateful for support from Dr. Ounaies
xxix
and Dr. vonLockette and their labs in the beginnings of my research. I would like to thank Joseph
Calogero whose support in coding and optimization saved me countless hours of learning to
program and debugging. I would like to thank the undergraduate students I mentored and
collaborated with during my research including multiple Capstone teams and more directly,
Katherine Reichert, Fariha Azhar, and Joseph Berthel. I have also had the pleasure of working
with various collaborators outside of Penn State which have made this research possible including
Kevin Snook, Ryan Clement, and Jenna Greaser from Actuated Medical Inc. and Greg Hayes and
the entire Additive Minds Team from EOS North America.
Numerous sources have also supported this work financially including the National
Science Foundation (NSF) EFRI grant #1240459, Air Force Office of Scientific Research
(AFOSR), National Institutes of Health (NIH) award #CA225169, Actuated Medical Inc., NSF
Center for Healthcare Organization (CHOT), CIMP-3D, EOS Additive Minds, Industry 4.0 Seed
Grant, Kulakowski Travel Grant, Reiss Fellowship, Marley Fellowship, and the University
Graduate Fellowship from the Pennsylvania State University. Any opinions, findings, and
conclusions or recommendations expressed in this dissertation are those of the author and do not
necessarily reflect the views of the funding agencies.
Finally, I acknowledge the never-ending support of my family. I’m grateful to my wife,
Brooke, for her encouragement, understanding, and for always listening to my successes and
challenges, even when I didn’t do a great job of explaining them. My parents, Byron and Patti
Hanks for always encouraging me to learn all I can and their support and confidence in me to
accomplish what I set my mind to. I’m thankful for my kids that help me take a break from
thinking of research 24/7 and my extended family for their endless encouragement. Finally, I’m
grateful for God, who I feel has blessed and inspired me throughout my life.
1
Chapter 1 Introduction
1.1 Background and Motivation
Recent advances in the capabilities of model simulation and manufacturing technology
are paving the way for systematic design tools. These tools enable designers to explore organic
and nonconventional designs that were previously not possible to analyze or manufacture.
Through design optimization, these nonconventional designs may be used to maximize part
performance in a variety of applications such as medical, aerospace, and defense industries. One
sector of nonconventional designs that has been relatively unexplored is that of compliant
mechanisms or components with tailored properties specific to the application.
The goal of this research is to develop systematic design optimization approaches for
compliant structures. In this dissertation, two systematic design optimization approaches are
described: a shape matching approach for a compliant deployable surgical tool for radiofrequency
ablation (RFA) of tumors and the development of specialized lattice structures for additive
manufacturing (AM). The five objectives of this research are as follows:
1. Develop systematic optimization approach for compliant deployable radiofrequency
ablation electrodes
2. Develop three-dimensional ground structure topology optimization for unit cell DfAM
3. Explore multi-functional lattices through case study on unit cell generation for thermal
conductance and minimum strain energy
4. Explore homogenized lattices through case study on generation of unit cells with tailored
compliance
5. Develop mechanical property database for metal lattice structures from current literature
containing analytical, finite element, and experimental data
2
This chapter includes relevant literature for the optimization approaches discussed in the
dissertation. There are six major sections in the chapter: compliant mechanisms, pancreatic
cancer, radiofrequency ablation, AM, lattice structures, and design optimization.
1.1.1 Compliant Mechanisms
A compliant mechanism (CM) gains its motion through the deformation of a material as
opposed to traditional hinges or joints [1]. Because the mechanisms do not use traditional hinges
for movement, they have several natural benefits such as reduced part count or assembly costs,
energy absorption in deflected members, and reduced wear due to friction. In addition, there are
some unique advantages for surgical CMs. In surgical applications, compliant mechanisms do not
require lubrication, lack wear, reduce the number of parts, and can improve mobility or dexterity
[2,3]. A number of surgical tools have been designed which exploit the unique advantages of
compliant mechanisms [4–7]. An example of a compliant surgical forceps is shown in Figure 1.1.
Figure 1.1 Compliant surgical forceps reduce the need for complex hinges on a small scale, lack
debris from wear, and do not require lubrication, adapted from [8]
3
One major challenge associated with CMs is that they often consist of complex structures
making them difficult to manufacture. For this reason, CMs were often planar mechanisms to
make them manufacturable. Recently, with the advancements of AM, there are new opportunities
to manufacture complex 3D CMs. Because AM also enables part consolidation and the ability to
manufacture more complex parts than traditional subtractive processes, it becomes a natural fit to
consider AM for building CMs.
With the ease of accessibility to polymer AM machines, polymer CMs have become very
common [9–12] , especially in the area of multi-material polymer AM [13–15]. Metal AM CMs
are much less common than polymer CMs for multiple reasons. One major challenge is that the
cost of metal AM remains high, making it less accessible than polymer AM processes. A second
reason is because much of the focus of AM is on structures designed to be as stiff and lightweight
as possible. For metal AM CMs, Merriam et al. [16–19] have developed titanium compliant
mechanisms for space applications. Other examples of AM CMs found in literature include
functionally graded superelastic compliant joints [20] and a high precision compliant parallel
mechanism [21]. The potential for CMs or other metamaterials designed specifically for metal
AM is largely unexplored with limited guidance or feedback from the manufacturing process to
direct the design. Two examples of compliant mechanisms designed for AM are shown in Figure
1.2. With the increased manufacturability of CMs, there are opportunities to improve
performance of components in a variety of different industries. One area where CMs excel is in
the medical industry because they do not require lubrication or wearing surfaces at the joints. In
this dissertation, the motivating application of CMs is a surgical tool required for treating cancers,
such as pancreatic cancer.
4
Figure 1.2 Examples of multi-material (left) and metal (right) compliant mechanisms designed for
AM, adapted from [13,19]
1.1.2 Pancreatic Cancer
Pancreatic cancer is the fourth most deadly cancer for men and women in the United
States [22]. A major cause of the poor prognosis is the difficulty of detection resulting in nearly
85-90% of patients being diagnosed at advanced stages of the disease [23]. Pancreatic cancer is
diagnosed into one of four stages based on the location of the tumor and how much it has spread.
Often the first symptoms a patient feels are pain and discomfort due to an enlarged tumor.
Depending on the location of the tumor, additional symptoms may include weight loss or
jaundice. At this point, when the symptoms become noticeable, the tumor has likely impacted
surrounding organs and vessels and possibly spread to other areas of the body. While universal
screening has been suggested to increase earlier detection, it has not yet been proven effective
[24]. In addition, pancreatic cancer is more common in older patients with poorer health than in
young healthy patients.
Due to the late diagnosis and poor health of many pancreatic cancer patients, the average
five year survival rate from the time of diagnosis is only 7% [22]. A comparison of ten of the
most common cancers and their five-year survival rates are shown in Figure 1.3. A patient
5
diagnosed in late stages of pancreatic cancer is often limited to palliative care to reduce pain and
discomfort because of the spread of the disease. The poor health of many patients also limits the
treatment options available. While earlier detection of pancreatic cancer remains a challenge,
incidental detection is increasing due to improved patient imaging. A study by Lee et al. [25]
found incidental pancreatic cysts in 13.5% of the adult population using retrospectively viewed
MR images. Another similar study determined that 60% of the pancreatic cysts in patients older
than 70 are likely to be malignant [26]. With the increase of incidental pancreatic cysts being
discovered at earlier stages, more patients will be seeking curative treatment.
Figure 1.3 Five year survival rates for the 10 most common cancers [22]. Pancreatic cancer is the
lowest with an average fiver year survival rate of 7%.
When treating cancer, interdisciplinary teams of doctors meet to review a patient’s case
and determine which forms of treatment should be used to best combat their disease. A treatment
plan is generated based on various information available to the doctors including the cancer type,
† Includes renal pelvis
‡ Includes intrahepatic bile duct
6
location, and stage, as well as the health of the patient. Medical imaging plays a critical role in
helping doctors plan an appropriate treatment by visualizing potential risks associated with the
disease and side-effects from any action they take. There are a number of treatment modes for
cancer, the most common being surgical resection, chemotherapy, radiation therapy. In addition,
Knavel et al. [27] provides a review of various other ablative technologies, such as cryoablation,
radiofrequency ablation, microwave ablation, and high intensity focused ultrasound. These teams
of doctors seek to combine the benefits of multiple forms of treatment while minimizing the
negative effects on a case by case basis.
The most effective form of treatment for non-metastatic pancreatic cancer is surgical
resection followed by an adjuvant or secondary treatment. During surgery, a doctor removes the
tumor and layers of surrounding tissue until a negative margin can be developed around the
tumor. When the full tumor cannot be removed due to the size or location, the surgeon seeks to
remove large portions to reduce the overall size. Surgery allows the doctor to visualize the tumor
and surrounding tissues as well as inspect other areas of the body where cancer may have spread.
A secondary treatment such as chemotherapy or radiation therapy is typically used to destroy any
cancer cells that may have been missed or further reduce unresected portions of the tumor. Chen
and Prinz [28] and Werner et al. [29] reviewed many studies which have discussed the effects of
various treatments on long-term survival of pancreatic cancer patients. While multiple studies
reviewed by Chen and Prinz [28] showed that resection followed by chemotherapy tends to
increase the survival of patients, the patient population for each study was often very small
resulting in unconvincing results of the benefit of chemotherapy following resection. One of the
strongest studies reviewed by Chen and Prinz was published in 2004 by Neoptolemos et al. [30].
This study included 289 patients and presented convincing results that surgical resection of
pancreatic cancer followed by chemotherapy increased survival. Werner highlights a number of
7
recent studies which show improved long term survival for patients who received an adjuvant
therapy following surgery [31–34].
While the combination of surgery with chemotherapy is the most effective form of
treatment, many patients with non-metastatic pancreatic cancer are not healthy enough to undergo
this complex surgery and recovery. Minimally invasive approaches, such as laparoscopy or
endoscopy, are available to treat many cancer patients who are ineligible for open surgery.
Laparoscopy can be used for delivery of chemotherapy, radiation therapy, and various forms of
ablation technologies. One common minimally invasive approach which has shown promise is
radiofrequency ablation (RFA).
1.1.3 Radiofrequency Ablation
Radiofrequency ablation (RFA) is a common mode of cancer treatment that has been
used since the early 1990’s. The effect of heating due to radiofrequency waves was first reported
in 1891 by D’Arsonval [35], followed by the explanation of the physical principles in 1976 by
Organ [36,37]. The radiofrequency waves being passed through tissue were shown to produce
oscillation in the ions causing frictional heating in the tissue. Later in the 1990’s McGahan et al.
[38] and Rossi et al. [39] first attempted RFA in the liver. RFA is now a common treatment
method that has been applied in various locations throughout the body including the lungs, liver,
kidneys, and pancreas [27,40].
RFA is performed by inserting one or more electrodes into a tumor and a large counter
electrode is placed on the exterior of the patient. A radiofrequency current applied to the electrode
causes frictional heating of the tissue immediately surrounding the electrode where the
concentration of the electric field is highest [41]. The frictional heating is a result of the
alternating current causing oscillation of the ions near the electrode. The primary mode of heat
8
transfer is conduction through the tissue, raising its temperature and destroying the cells. The cell
destruction is time and temperature dependent. The target tissue temperature for RFA is between
50-100°C [41]. For temperatures below 50°C, the cells require a long exposure time to be
destroyed and at temperatures greater than 100°C, the cells char, increasing their resistance and
limiting the effectiveness of the current flow. In the range of 50-60°C, reported values for the
exposure time required to cause cell death vary greatly in literature [27,40–43]. However, there is
agreement in that for cells heated to temperatures between 60-100°C, cell death is nearly
instantaneous [40,42,44–46]. The region of cell death surrounding the electrode is referred to as
the ablation zone. The study of the effects of temperature on cells is called thermal therapy. Many
studies have been reported in literature about the effects of hyperthermia on cells [42,43].
Dewhirst et al. [47] provides an summary of previous work and the focus of thermal therapy
moving forward.
RFA is primarily limited by three factors: tumor location, tumor geometry, and tracking
the treatment zone. First, location-based limitations occur when the tumor is near other vital
structures or organs, blood vessels, or in a region of the body which is difficult to access using
traditional delivery methods. Effective RFA treatment requires the entire tumor plus a small
margin around the periphery to be fully heated by the electrode. Because RFA relies on thermal
therapy to sufficiently heat and destroy tumor, if there are vital structures or organs adjacent to
the tumor then they may be negatively affected by the heating. In particular, it is difficult to
develop a negative margin around the tumor to ensure all cancerous tissue is destroyed without
damaging healthy surrounding tissue. If the tumor is located near a large blood vessel (diameter
larger than 3mm), the blood vessel counteracts the effects of RFA by acting as a significant heat
sink [48,49]. As the heating from RFA is primarily based on conduction, the blood vessel quickly
draws away the heat, preventing the tumor from heating sufficiently to receive permanent
damage. A final challenge with the tumor location is simply accessibility for surgical tools. In
9
certain regions such as the pancreas or other areas of the upper abdomen, it is difficult to find an
access point for the RFA electrode.
The second limitation is due to the geometry of the electrode and tumor. These
limitations occur when the heating profile generated by the electrode does not match the tumor
geometry. For example, if the heating profile is cylindrical and the tumor is spherical, there is a
mismatch which either damages healthy tissue or does not destroy all the cancer; see Figure 1.4.
To overcome this limitation, a few different electrode configurations are commercially available
and have been reported in literature [41,50,51]. Multiple reviews of single electrodes and multi-
electrode systems are available such as those by Hong and Georgiades [41] and Mulier et al. [51].
Hong and Georgiades [41] gave a review of current single electrodes and electrode systems. The
most basic shape of RFA electrodes is cylindrical which typically generates an ellipsoidal
ablation zone. The exposed length of the electrode can be adjusted or multiple cylindrical
electrodes can be inserted in an attempt to alter the geometry of the ablation zone. Hong and
Georgiades [41] also describe complex expanding electrodes that broaden the ablation zone
through multiple tines that deploy or expand into tissue as the electrode is inserted. Mulier et al.
[51] includes a review of current electrodes and multiple electrode systems as part of his proposal
for standardized terminology of RFA devices; see Figure 1.5 [51]. In his review of RFA for
pancreatic adenocarcinoma, D’Onofrio et al. [52] describe how straight electrodes are used for
cylindrical-like tumors and deployable electrodes are used for spherical-like tumors. Even with
several configurations of electrodes available to try to match the ablation zone and tumor
geometry, it is hard to ensure a negative margin is developed around the tumor because the
ablation zone is difficult to predict and the tumor geometry is unique to each patient.
10
Figure 1.4 (Left) Diagram of radiofrequency ablation showing the mismatch that can occur between
a tumor and ablation zone and (Right) thermal-imaging of radiofrequency ablation showing the
ellipsoidal temperature profile around a straight electrode, adapted from [41]
Figure 1.5 Basic electrode types and several commercial models of radiofrequency ablation
electrodes described by Mulier et al., images adapted from [51]
Tumor
Ablation
Zone
11
The third limitation is the ability to track and get feedback on the treatment zone shape
and size. Three primary methods of tracking of the ablation zone during radiofrequency ablation
have been proposed: ultrasound (US), computed tomography (CT), magnetic resonance imaging
(MRI). US provides real-time tracking of the ablation zone but can be limited due to gas bubble
artifacts which blur the image during ablation and it is often associated with underestimation of
the actual ablation zone [53–55]. CT imaging has been correlated with improved ablation zone
prediction but is less available than US and often is unable to provide real-time feedback [56].
MR imaging is expensive, time consuming, and cannot be performed real-time [57]. In current
practice, radiofrequency generators often monitor the impedance or temperature at the electrode
to provide automated control power delivery and ablation zone generation. Other research groups
are currently pursuing impedance- or temperature-based automated control to predict ablation
zones [58]. Because it is difficult to predict and monitor the treatment zone, recurrence or
multiple treatments are common following the initial RFA.
The result of the current limitations of RFA is incomplete tumor ablation. One reason
surgical resection is so effective is because the doctor is able to visibly remove the tumor
followed by a few layers of extra tissue around the periphery to ensure that no tumor cells have
been missed. If no cancer cells can be found at the outer edge of the tissue where the tumor was
located, the patient has a negative margin resection. With RFA it is very difficult to ensure a
negative margin is developed around the tumor because of the limitations described above. Using
a specially shaped electrode to create an ablation zone which matches the tumor geometry and
combining RFA with other treatment modalities becomes increasingly important in developing
the desired negative margin around the tumor and preventing reoccurrence.
To understand RFA in tissue, numerous finite element models have been created. RFA
models have been applied in thyroid tissue based on MRI reconstructed models and also in liver-
tissue using a hyperbolic bioheat equation and a new-voltage calibration method [59,60]. Various
12
ablation models have been used to study the effects of tissue properties such as electrical
conductivity and blood perfusion rate [61–66] or the heat sink effect from blood vessels [67–69] .
Duan et al. [64] and Huang and Chui [66] both present unique work for treatment planning that
statistically predicts the ablation zone by varying tissue properties [70]. Tungjitkusolmun et al.
[67] presented a 2D and 3D model of ablation using the Rita Medical Systems Electrode and
explored the effects of ablation near blood vessels. Other models focus on specific commercially
available electrodes [44,71–74] or automatic power delivery methods [58,60,75]. Chang [61]
focused on the effects of the blood perfusion rate and temperature dependent electrical
conductivity. Later, Chang and Nguyen [71] studied how isotherms can be used to predict the
ablation zone. In addition, Altrogge et al. [73] studied the optimization of the placement of a
commercial electrode to increase the efficiency of RFA. Another model by Lim et al. [68]
focused on improving efficiency by considering alternative input waveform patterns. While many
models have been developed to understand RFA, the use of these models within a formal design
optimization of probe tip geometry to create a specific ablation zone has not been considered.
Up to this point, models available in literature study commercially available electrodes
and have not explored the use of such models in conjunction with a systematic design
optimization tool. For a given tumor geometry, these models could be used with design tools to
direct the shape and size of an electrode to match the ablation zone and tumor geometries.
1.1.3.1 Endoscopic RFA
Due to the challenges with placing an RFA electrode percutaneously in certain regions of
the body and with the intent of making surgery even less invasive than open surgery, recently
endoscopic RFA has been explored. For example, in the case of pancreatic tumors, the tumor is
13
often inaccessible because of the surrounding organs and location of the tumor in the pancreas.
Endoscopic RFA offers a unique approach that allows access for insertion of an electrode.
Endoscopic ultrasound-guided radiofrequency ablation (EUS-RFA) is a relatively new
approach which allows for increased access to certain areas in the upper abdomen such as the
pancreas. It has been tested in several animal models and clinical trials. Varadarajulu et al. [76]
and Gaidhane et al. [77] performed EUS-RFA in pigs on the liver and pancreas respectively.
Varadarajulu found that after seven minutes the ablation was obtained and the electrode could
easily be retracted. The pigs remained stable and without significant damage to surrounding
organs after producing a 2.6cm diameter ablation zone. Gaidhane found similar promising results
in 5 Yucatan pigs causing minimal pancreatitis and concluded that EUS-RFA is a safe procedure.
Following their successful results, Pai et al. [78] conducted clinical trials on eight human patients
and found the EUS-RFA treatment safe, with minimal risks to patients. They found a range of
results from complete resolution to 50% reduction in diameter. Other recent studies and reviews
have continued to show promising results [79–84]. While the trials conducted have been effective
and safe, repositioning the electrode and repeated applications of RFA in a single session were
required to generate a large ablation zone and match the geometry of the tumor [78].
While EUS-RFA has been shown to be safe and increases accessibility to other areas of
the body, there are only two EUS-RFA electrodes which have been tested in clinical trials, the
Habib EUS-RFA catheter (Emcision Ltd, London) [77,78,85] and the cryotherm probe (ERBE
Elektromedizin GmbH, Tübingen, Germany) [86,87]. Each electrode is a straight cylinder, which
generates an elliptical ablation zone similar to that shown in Figure 1.6. There is a need for
specialized electrode designs that can reduce the geometrical mismatch that often occurs in
treatment of pancreatic cancer with RFA.
The first systematic optimization method described in the dissertation is focused on a
shape matching method RFA to design specialized electrodes. The optimization method is
14
applied to a novel EUS-RFA electrode design that is proposed in this work. The second
systematic optimization is for tailored mesostructures or lattices. This optimization includes AM
considerations and process constraints.
1.1.4 Additive Manufacturing
AM has seen significant growth in recent years due to the increased interest from a
variety of industries such as automotive, aerospace, medicine, biological systems, and food
supply chains due to several of its unique advantages [88–90]. Traditional subtractive
manufacturing processes begin with a solid piece of material and progressively remove or
subtract material to achieve the final form. AM refers to seven different manufacturing processes
that involve adding material layer-by-layer: material extrusion, powder bed fusion, vat
polymerization, material jetting, binder jetting, sheet lamination, and directed energy deposition.
Abdulhameed et al. [91] reviewed all seven processes in depth to identify advantages and
challenge with each process. The layer-by-layer approach of AM enables generation of parts that
were previously impossible or too complex to manufacture and reduces material waste compared
to subtractive manufacturing. In reality, with AM, the cost or manufacturability of a part is not
directly related to its complexity as with traditional subtractive processes, where the number of
steps to create the part or number of different tools drive the cost of the part. While AM is not
without its limitations, there is arguably an inherent “free complexity” in design based on the fact
that a part is built layer by layer with material deposited where it is needed. The concept of free
complexity combined with the increased capability to manufacture complex internal geometries
has led to a variety of research studies into AM for applications such as compliant mechanisms,
discussed in Section 1.1.1.
15
In their review of AM, Gao et al. [88] describes how AM, also known as desktop
manufacturing, rapid manufacturing, rapid prototyping, was most commonly used for quick
demonstration pieces but has progressed toward end use products. Gao et al. [88] outlines several
pros and cons associated with AM that are often lumped into two groups called opportunistic and
restrictive. From the opportunistic side, Gao highlights increased design flexibility, free
complexity, part consolidation and/or printed assemblies, and cost efficiency for low production
quantities [88]. Many examples of opportunistic AM exist in literature including non-assembly
builds [10,92], customized parts [93–95], part consolidation [96,97], and multi-functional parts
[98]. A few examples of multi-functional parts are embedded components or actuators [99–101]
and multi-material printing [102,103].
The restrictions or challenges with AM include high cost for mass production, limited
scalability due to increased build time or loss of resolution, part consistency and reliability, or
AM standardization and intellectual property [88]. Two of the biggest limitations to AM are the
cost for mass production and part reliability or consistency. With other processes such as
injection molding, after molds are produced, the cost per part decreases as the production quantity
increases. In contrast, for AM there is a much small upfront cost for machines but the cost per
part does not decrease with production quantity. As the technology is still in its infancy for end
use products, there are still many concerns over the reliability and consistency of parts. In
particular for custom parts, an important question is, how can the safety and quality of every
product be validated if each product is customized? To improve part reliability a major area of
research is in-situ monitoring methods to improve consistency and detect flaws [104–111]. While
many industries rely on driving down cost by mass production through consistent and reliable
processes, AM is continually improving to reduce build time and improve consistency and
dependability.
16
1.1.4.1 Metal Powder Bed Fusion
Several divisions in AM exist based on material, process, and energy type. For metals,
there are two major processes: powder bed fusion (PBF) and directed energy deposition (DED).
For DED there are two material feedstock methods: wire and powder. In DED, the energy source
heats the part generating a melting pool and a wire or powder is fed into melt pool. Either the part
itself or the energy source and feed material are moved generating layers of added material. These
deposited layers progressively build the part. PBF is performed by spreading a thin layer of metal
powder on a metal build plate, then using a laser or electron beam to melt the powder to the build
plate. The melted powder forms to the first layer of the final part. The build plate is then lowered
and a new layer of metal powder is spread across the previous layer. Again, the powder is melted
to the previous layer and the process is repeated, slowly building the metal part layer by layer.
Schematics of PBF, powder feed DED, and wire feed DED systems are shown in Figure 1.6.
In the review of metal AM, Frazier [112] reports that several challenges exist for metal
AM, such as the need for monitoring to improve part consistency and increased material selection
through alloy development. One challenge with consistency is the numerous thermal cycles that a
part experiences causing variations in microstructure and deformation of the part. Even with the
need to improve consistency and develop materials specific for AM, Frazier [112] reports that the
static mechanical properties are similar to those of conventionally processed parts. Specific to
PBF, the speed, accuracy, and cost need to improve and the material processing-structure-
property relationships need to be understood to improve the technology [113]. By reviewing
published works reporting microstructure and material properties, Herzog et al. [114] provides a
thorough review of metal AM processes, linking processing parameters and material properties in
various types of metal.
17
Figure 1.6 Metal AM system types include powder bed fusion and wire/powder feed directed energy
deposition, typically using either a laser or electron beam energy source, adapted from [112]
1.1.4.2 Design for Additive Manufacturing
Design for AM (DfAM) includes the theories and principles behind preparing parts to be
manufactured using AM, taking full advantage of the unique opportunities and avoiding the
restrictions. A common misconception exists in that there is complete design freedom with AM,
but the design freedom of AM is in fact not limitless. In 2016, the U.S. Department of Defense
established a roadmap for AM including four focus areas: Design, Material, Process, and Value
Chain. In the design focus area, specifically DfAM tools that “streamline [the] design process,
Powder Bed System Wire Feed System
Powder Feed System
18
reduced cycle time, and [create] higher performance products” [115] were highlighted as future
tasks. Because the manufacturing process is very distinct, the design rules for traditional
manufacturing processes are naturally different than those for AM. In addition, because the
design freedom is increased, DfAM tools are needed to help guide a designer in enhancing
product performance over traditional designs. With DfAM tools, parts can be tailored for design
requirements ranging from biologically inspired effective modulus properties to improve
orthopedic implants to multifunctional structures [116–118]. While there is a wide range of
possible applications, DfAM optimization tools to design these structures may be one in the same.
DfAM is generally separated into two categories: opportunistic and restrictive DfAM.
Describing opportunistic DfAM, Gibson et al. [119] organized it into four groups: shape
complexity, hierarchical complexity, material complexity, and functional complexity. These
groups were defined as follows:
• Shape complexity – ability to create almost any freeform shape,
• Hierarchical complexity – ability to design across multiple length scales ranging from
the grain structure and orientation, or micro-level, internal geometric complexity, or
meso-level, and the general object shape or form, or macro-level,
• Material complexity – ability to create parts one point at a time, mixing and blending
multiple materials,
• Functional complexity – ability to access the entire 3D geometry during the
manufacturing process enables integration of multiple functional parts through
printed assemblies or embedded components.
Using a slightly different method of categorization, a review by Thompson et al. [120]
provides a discussion on numerous areas of both opportunistic and restrictive DfAM; see Table
1.1 for a complete list of the topics addressed. Thompson et al. [120] covers everything from the
ability to print full color to designing the microstructure to the regulatory limitations of the
relatively new technology. Gibson and Thompson both acknowledge that many of these DfAM
topics are in their infancy and merit further study to maximize their capability of minimize their
19
limitation [119,120]. For example, on the opportunistic side, meso-level DfAM enables further
enhancement to maximize performance by reducing weight or tailoring properties, but no fixed
guidelines or design rules have been established. Much of the meso-level design remains based
on designer intuition as opposed to concrete design rules or methods. On the restrictive side,
minimum feature size and the need for support material are established as necessary
considerations in DfAM, but there is ongoing research in ways to reduce the negative impact of
these topics. In an attempt to incorporate a variety of DfAM topics, there are several DfAM
frameworks that have been proposed.
Table 1.1 A wide range of opportunistic and restrictive DfAM topics exist for which there is ongoing
research
Opportunistic DfAM
Macro-Level Meso-Level Product-Level
• Material Choice
• Full Color
• Freeform geometry for
art/aesthetics
• Internal freeform geometry
• Topology optimized macro-
structure
• Custom fit or mass
customization
• Custom material, metallurgy,
microstructure
• Custom surfaces, textures,
porosity
• Lattices, trusses, cellular
materials
• Multi-material
• Functionally graded
• Metamaterials
• Part Consolidation
• Embedded objects
• Printed Assemblies
Restrictive DfAM
CAD and Digitization Discretization, Directionality,
Build Orientation, and Supports
Process Characteristics and
Machine Capabilities
• Complete model
• Inadequate for organic shapes
• Limited meso-structure
design
• Surface roughness dependent
on orientation
• Supports for overhangs
• Boundaries between layers
• Minimum Feature Size
• Accuracy
• Build Volume
• Compatible materials
• Process specific design rules
Material properties and
processing
Metrology and quality
control
Maintenance, repair,
recycling
External and regulatory
constraints
• Need materials for
AM
• Properties change
with processing
• Voids, porosity,
grain characteristics
• Internal geometries
• Disassembly of
printed assemblies
for maintenance
• Relatively new
technology
• Inherent variability
20
DfAM frameworks are a set of guidelines or methods to aid designers in creating efficient
and successful parts for AM that take advantage of the opportunities available while accounting
for the limitations. For example, ASTM 52910 is a standard which very broadly addresses
guidelines for DfAM [121]. In literature there are many research groups trying to provide the
necessary design guidelines for DfAM [122]. After a review of 27 publications on DfAM for
decision making, Laverne et al. [123] describe that the articles could be categorized by their focus
on opportunistic DfAM, restrictive DfAM, or dual DfAM, a combination of both. To take full
advantage of AM and have a successful build both opportunistic and restrictive DfAM must be
considered, yet Laverne et al. [123] found that only 36% were dual DfAM. Yang and Zhao [124]
provide a review of AM design theory, describing several general design considerations for AM,
DfAM frameworks, systematic design tools for AM, and makes recommendations for future
research areas in DfAM. In reviewing DfAM frameworks, they were grouped into two categories:
AM-enabled structural optimization and DfAM methodologies.
AM-enabled structural optimization covers the novel shapes and structures that can be
created at both the macro and meso-levels. This includes various forms of topology optimization
to generate optimized structures that would otherwise be difficult to manufacture. Additional
detail on topology optimization is covered in Section 1.1.6.2. Of these DfAM optimization
methods, Yang and Zhao [124] describe the benefits of combining functional requirements
through objective functions to optimize functional surfaces or complex designs. While there are
clear advantages to such approaches for generating an optimal design, there are also limitations
with the ability to include other unique opportunistic features of AM. Later in their discussion of
systematic design tools for DfAM, Yang and Zhao [124] describe design tools that allow a
designer to alter the interior volume or meso-level design. While there are several commercial
tools for optimizing the interior volume such as OptiStruct [125] and Netfabb Ultimate [126],
they perform with relatively limited options, not providing full control for a designer to specify
21
various objective functions, but primarily focus on stiff and light weight solutions. In their review
of AM design theories and methodologies, Yang and Zhao [124] found that there is a need for a
design framework that “initializes design from the perspective of functionality achievement” and
a “method to better synthesize functional requirements and process knowledge simultaneously.”
Regarding DfAM, Doubrovski et al. [127] studied the relationships between structure and
performance as well as optimization approaches. Nearly half the articles reviewed by Doubrovski
et al. [127] related the structure to strength- and stiffness-based performance. Aside from
strength and stiffness, the other articles were spread throughout other performance objectives
including thermal, compliance, dynamic, and visual. With early interest in AM from the
automotive and aerospace industries, much of the focus of designing AM parts has been
concentrated around weight reduction and creating stiff light-weight structures. Little work was
reported in the area of tailored compliance or thermal properties at the mesostructural level.
Doubrovski et al. [127] conclude that “One of the promising possibilities of AM is the complexity
of geometry… combined with methods for optimization” and to take full advantage of AM in the
product development process systematic design tools have to be developed [127]. While it is
challenging to create a general set of DfAM guidelines or design tools that apply across all AM
processes, there are many researchers exploring design rules specifically for metal AM.
In 2017, a report from the Fraunhofer institute detailed the design of 7 different
components for AM. One of the purposes of detailing these 7 different industry-relevant case
studies was to highlight DfAM guidelines and limitations with current metal AM technology.
Over 20 different metal AM-relevant industry standards were identified and considered during the
different case studies. Only three of these standards deal directly with DfAM (ISO/ASTM 52910,
VDI 3405 Part 3, VDI 3405 Part 3.5). Few other general metal AM guidelines or standards exist
for metal AM though it is a rapidly evolving field [128–132].
22
A few of the primary manufacturing limitations and considerations for metal PBF are the
required support structures, overhangs, and powder removal. Metal support structures are
required in PBF for three purposes: anchoring the part to the build platform, conducting heat
away from the part, and supporting overhanging features. In metal AM, the part undergoes large
changes in temperature caused by reheating for several subsequent layers [114]. During
processing, the resulting thermal stresses cause parts to warp upward away from the build plate.
The warping occurs when hot upper layers cool and contract while lower layers do not contract,
causing the upward warping. Figure 1.7 shows a basic schematic of the upward warping. The
support structures in metal AM help to anchor parts to the build plate and also prevent warping.
Heat buildup causes increased surface roughness and residual stresses in the part. Because solid
metal structures have higher conductivity than metal powder, support structures conduct heat
away from the part better than powder. Similar to other AM processes, overhanging features
require support material. As a rule of thumb, overhangs of less than 45° require support structures
to both support the melted material and anchor it to the build plate. In recent years, the minimum
build angle has continued to decrease, including some companies, such as VELO3D, that boast
support-free designs. Specific to PBF processes for AM is the requirement that the geometry of
the design allow for powder removal, which limits enclosed cavities and other small features that
may trap powder. It occurs due to geometrically tight channels that are difficult for powder to
flow from as well as due to dross or excess powder being picked up because of poor thermal
management of the process. Trapped powder increases the weight, cost, and safety of a
component. Rather than empty cavities in a part, cavities filled with powder adds cost through
unrecoverable powder and weight without any structural integrity. Additional safety concerns
come into play with powder that could eventually find its way out of the part posing health risks
if aspirated. Processes to remove powder from a component include vacuuming, compressed air,
23
tapping the part, ultrasonic vibration, and mounting it on a multi-axis turntable for rotating the
part and allowing powder to flow out in various directions.
Figure 1.7 Example of warping due to thermal contraction
1.1.5 Lattice Structures
One major area of research for AM, with significant potential for enhanced performance,
is the ability to design at the mesoscale [133,134]. Mesoscale design, also known as metamaterial
design or design of cellular structures, is designing the functions or properties of a material as a
direct result of its mesostructure geometry as opposed to the raw material. The mesoscale is an
intermediate level between macroscale and microscale, including features at length scales of
approximately 0.1–10mm [135]. For example, in a beehive, the macroscale of a component is the
overall structure or topology of the outside, the mesoscale is the honeycomb structure within the
outer shell, and the microstructure is the makeup or material state of the individual walls of the
honeycomb. The ability to design and manufacture the macrostructure or overall topology of
complex parts is a natural benefit of the “free complexity” of AM. As the accuracy and precision
of AM processes improve, increased complexity at the mesoscale and even microscale may be
designed and manufactured without direct increase in cost of the part. Common examples of
mesostructural design with traditional manufacturing processes are honeycomb structures and
metal foams; see Figure 1.8. Honeycomb and cellular structures have been observed in nature for
24
thousands of years but it wasn’t until the early 1900’s that they were first used in structural
applications such as airplane wings and buildings. Since then, many researchers have studied and
tried to replicate cellular structures, but for many years the complexity of these microstructures
was limited [136–139]. For organized structures, such as the honeycomb, only limited designs
could be manufactured efficiently, if at all. Several examples of traditionally manufactured lattice
structures have been researched and developed by Haydn Wadley [140–157]. More variety can be
found with metal foams than organized structures but they suffer in applications where a
repeatable structure is needed, due to the stochastic nature of their fabrication. For many years,
researchers have been creating and analyzing unique cellular structures analytically or with the
finite element method. Now, there is a renewed interest in mesoscale design due to advances in
AM technology that allow for these complex structures to be manufactured in 3D.
25
Figure 1.8 Various forms of cellular solids including a (top) 2D honeycomb, (middle) 3D open-cell
foam, and (bottom) 3D closed-cell foam, adapted from [136]
Though a variety of names have been used in the past for cellular structures, lattice
structure is the terminology used throughout this dissertation. Lattice structure refers specifically
to an organized repeating mesostructure for which the most basic repeating unit is called a unit
cell [117,158,159]. Lattice structures offer improved strength to weight performance over solid
components by maintaining structural integrity while decreasing weight [136]. Helou and Kara
[160] give a comprehensive review of lattice structure literature and specify that AM processes
have an “immense impact on the form and functionality of the specimen being produced.”
Though there are other methods to create lattice structures, the focus of this dissertation and
literature review is on lattice structures fabricated with AM. Lattice structures have been used for
26
a variety of applications including light-weighting [161], functionally graded structures [162],
tissue engineering scaffolds [163–169], medical implants [170–181], actuation[182], heat transfer
[118,183–185], reinforcement [186], and energy absorption [187–193]. Figure 1.9 shows an
example of various medical implants, each with a different approach to lattice structure design.
Because lattice structures may be incorporated into such a wide variety of applications, there is a
need for further research exploring how to design or select an appropriate unit cell topology for
the application.
27
Figure 1.9 Lattice structures have been proposed for a variety of medical implants with the intent of
improving performance and functionality. Due to the variety of lattice structures that can be used in
such applications, there is a need for further research to understand how to select or design an
appropriate unit cell for the intended application. Images adapted from A) [179], B) [177], C) [181],
D) & E) [194].
Designing a lattice structure within a component is typically performed by separating the
process into two distinct steps: unit cell generation and lattice population. Unit cell generation is
commonly based on three different approaches as described by Tao and Leu [195], (1) basic
primitive shapes, (2) mathematical equation based surfaces, or (3) topology optimization. Basic
primitive unit cells are generated by combining common shapes and different Boolean operations,
A) B)
C)
E)
D)
28
often in ways that mimic common crystal structures seen in nature [196,197]; see Figure 1.10. A
few examples of common basic primitives are the body-centered cubic (BCC), face-centered
cubic (FCC), octet-truss. For modeling unit cells with mathematical equations, three-dimensional
surfaces are generated using special formulations. Examples of unit cells from mathematical
equation-based surfaces are the gyroid or another example shown by Tao and Leu [195], whose
shape and unit cell density may be adjusted by two parameters; see Figure 1.11. With infinite
possibilities for the topology of the unit cell, several naming conventions or classifications have
been proposed [198–200]. Unit cell density refers specifically to the volume fraction of the filled
space within the bounding volume of the unit cell. For example, 50% dense would be a unit cell
sized such that half of the volume within the bounding box is full and 100% dense would mean
the bounding box is filled completely. Topology optimization is a developing research area for
unit cell generation and is covered in Section 1.1.6.2.
Figure 1.10 Unit Cells are often generated based on Boolean operations with primitive shapes such as
these examples, adapted from [195]
29
Figure 1.11 Unit cells may be generated using implicit mathematical equations to represent a surface.
In this particular case, the surface is defined at F(x,y,z) = 0 with parameters "a" and "b" used to
tailor the shape of the surface; adapted from [195].
To understand the physical behavior of different unit cells and their properties, numerous
researchers are developing analytical [178,201–206] and finite element methods [202,204–210] to
predict the properties of different unit cell types and sizing, and performing experimental
validation. In current literature for metal AM, unit cell properties may be found for cube
[178,203,211], rhombic dodecahedron [178,211,212], rhombicuboctahedron [178,204,211], BCC
[178,208,213–216], FCC[208,213], diamond[211,217–219], gyroid [213,220–223], pyramidal
[190], octahedral [178], and several other variations on these. A complete set of sources is
described as part of Chapter 6. Dong et al. [116] performed a review of modeling approaches,
comparing the advantages and disadvantages of each approach. Among their primary findings
was that lattice structures are promising with their potential to be multi-functional materials in
various engineering applications.
𝐹(𝑥, 𝑦, 𝑧) = cos(2𝜋𝑥) + cos(2𝜋𝑦) + cos(2𝜋𝑧)+ 𝑎(cos(2𝜋𝑥) cos(2𝜋𝑦) + cos(2𝜋𝑦) cos(2𝜋𝑧)+ cos(2𝜋𝑧) cos(2𝜋x)) + 𝑏
30
Lattice population or patterning throughout a part is accomplished using either Boolean
operations or conformal methods. Basic Boolean operations are the most common and simplest
method for populating a unit cell within a part. The unit cell is repeated into a lattice block and
the edges of a part or volume are used to trim the lattice. Often only partial unit cells remain near
the edges of the part resulting potential weak points. Conformal lattice population methods
consist of modifying the lattice block to match the surface geometry of your part. A comparison
of conformal and non-conformal lattices is shown in Figure 1.12. Rosen and his students have
performed extensive work into conformal lattice methods and design processes for generating
lattice structures [159,224–227].
Figure 1.12 Two common methods for lattice population are (top) uniform in which the basic lattice
is not modified based on the component shape and (bottom) conformal in which each unit cell may be
deformed to follow the surface of the component, adapted from [224]
After these two steps of generating a unit cell and populating the lattice structure, often
the lattice is altered by a sizing optimization, changing thickness or strut diameters to provide
higher stiffness in certain areas than others. Several other approaches have been proposed in
Uniform lattice
Conformal lattice
31
recent years which blend together these steps: unit cell generation, population, and sizing, often
all mixed with an optimization routine [228]. For example, Brackett et al. [229] and Arabnejad et
al. [230] use a density mapping to size and position their lattice structure. Brackett et al. [229]
uses a gray-scale image of the component stress to generate the density mapping for the lattice
structure. The gray-scale stress image is generated by finite element analysis and then dithering
the image; see Figure 1.13. A lattice structure is populated through the part with small stiffer
trusses in regions of high stress or a high density of points. Arabnejad et al. [179,230–232]
generates a lattice structure inside a hip implant using the density mapping from a CT-scan. The
densities from the scan are correlated to a particular unit cell density to fill the part; see Figure
1.14. Graf et al. [233] and Chang and Rosen [234] use similar approaches for generating and
sizing a lattice. First, they begin with a basic lattice and either add, remove, or resize struts on
each unit cell based on the type of loading and amount of stress in that region of the component.
The stress is found using finite element analysis. High stress regions of the component under load
are correlated with increasing unit cell density; see Figure 1.15. While these methods have proven
useful, they lack the full customization and ability to optimize performance that is possible for
AM. While they may serve to lightweight a component and improve its performance, AM allows
for customization by tailoring the unit cells to demonstrate specific bulk properties.
Figure 1.13 One method of generating a functionally graded lattice is by computing the stress within
the part, plotting the stress in gray scale, and then dithering the image to generate a density map.
Next, small stiff unit cells are placed in dense regions, with high stress, and large unit cells are placed
in less dense regions, adapted from [229].
32
Figure 1.14 Functionally graded lattice structure populated using a relative density map based on CT
imaging, adapted from [178]
33
Figure 1.15 Lattice generation and sizing method using stress results from a finite element analysis,
adapted from [234]
Problem Definition
Base-Lattice
Topology Generation
Solid-Body
Analysis
34
In addition to these methods being researched, there is a variety of commercial software
that offer lattice structure population within a part as well as some capabilities to generate custom
lattice structures. Among the prominent commercial software is nTopology [235], Paramatters
[236], Netfabb Ultimate [126], 3DXpert [237], Discovery Spaceclaim [238], 3D CASTS
(Computer Aided System for Tissue Scaffolds) [170,175,239–241], or software developed by
Vongbunyong and Kara [242,243]. While all of this software has been developed to ease the
workflow for lattice structures in AM components, there is limited ability within the software to
provide feedback for selecting the proper unit cell topology for the application. For example, if a
company wants to populate a lattice inside their component, they can do so using Netfabb
Ultimate [126]; there are even some limited options for optimizing their lattice structure based on
the loading conditions. However, there is no way to know which of the 19 lattice structure options
is ideal for their application. Any of the 19 lattices could be populated within the component and
subsequently optimized without a knowledge of which lattices are manufacturable or provide the
desired bulk mechanical properties. In addition, most software only offers pre-generated unit cell
types, limiting the options and variations that are possible with AM.
Of these generation and population methods in research or commercial software, there is
a lack of reasoning or a systematic approach for why a particular unit cell is chosen. Essentially
any unit cell could be populated based on the density mapping or sized based on the finite
element results and there is no clear logic provided as to why a unit cell is chosen over another.
While there is data available for a variety of lattice structures, there is a need for design guides
and unit cell generation tools that may be used to understand the current unit cell structures or
customize the lattice structure for the application. AM has the capability to manufacture
multifunctional unit cells and lattice structures but there are currently limited options for
designing or optimizing unit cells based on application-specific requirements [116].
35
1.1.5.1 DfAM of Lattice Structures
For PBF, there are several challenges with printing lattice structures including four
discussed by Nazir et al. [244] in their review of lattice structures fabricated with AM: (1) support
material removal, (2) powder removal, (3) large file size and (4) high surface roughness. Similar
to large components, build angle and the requirement to anchor low build angles and overhangs to
the build plate remain a concern for lattice structures. Due to the intricate complexity and feature
size of lattice structures, support material removal quickly becomes infeasible or impossible. For
powder removal, Koizumi et al. [245] specifically mentions the challenge of removing powder
for low porosity lattice structures. Low porosity lattice structures can trap powder due to minor
manufacturing variations that close off connected pores. The speed of post-processing can also be
affected by the time required to evacuate all the loose powder through a complex maze of
interconnected pores. Regarding file size, the most common file format for 3D-printing is the
Stereolithography file (STL). An STL is simply a triangulated surface mesh of the component.
Due to the intricate surface complexity of a lattice structure, an STL is ill-suited to be able to
accurately represent a component in a compact format, due to a file size that quickly explodes,
even for small components. As such, there are a variety of research areas being explored to
improve modeling and STL (or mesh) generation for lattices compared to traditional software
[246–249]. In practical cases, file size for lattice structures can be reduced by changing the cross
section of the struts from circular to hexagonal or octagonal to make the tessellation simpler.
Depending on the diameter of the lattice struts, the file size can be significantly reduced without
impact on the structure because the simplification of circular to hexagonal or octagonal can
become blurred out by the minimum feature size of the process. In addition, some commercial
software and production systems are moving away from the traditional STL file format due to
issues with file size and an inaccurate tessellated representation of the object. While all AM
36
components are subject to poor surface roughness, the effects are compounded by the fact that the
feature sizes of lattice structures are much closer that of the surface roughness. Compared with
macroscale topology, surface variations on lattice structures have a larger impact on the expected
properties and performance. In addition to the four limitations covered by Nazir et al. [244],
Maconachie et al. [250] and others [194,223] also address the staircase effect which can cause
significant changes in cross section for thin lattice features, negatively impacting the mechanical
properties from the intended design. Others have reported on bending or deformation of intricate
or delicate structures during the printing process [251,252].
1.1.6 Design Optimization
For many years various algorithms have been proposed to optimize a design, especially
valuable for a manufacturing process with substantial design freedom. With the “free complexity”
allowed by AM, the ability to manufacture complex optimized shapes becomes a reality. While
AM remains an expensive process for large batch sizes, the geometric complexity of a one-off
part does not necessarily directly relate to the cost. This fact allows for geometrically complex
and customizable parts to be reasonably manufactured. With this reality, areas such as
customization and design optimization have seen an explosion of interest from the research
community. Optimization algorithms that have been researched for many years to maximize
performance in theoretical components. With AM, the cost and ability to manufacture these
optimized components is more reasonable than traditional subtractive manufacturing processes.
This section addresses two areas of optimization: common optimization algorithms and topology
optimization.
37
1.1.6.1 Optimization Algorithms
In design optimization, numerous algorithms have been proposed including a variety of
bio-inspired algorithms, mathematical based algorithms such as gradient search, and other
physics based methods [253–255]. Bio-inspired algorithms include several different methods
based on principles in natural such as natural selection or evolutionary algorithms, swarming of
birds, or insects. Gradient search or gradient descent is a common method with several variations
for calculation or estimation of the gradient. Other physics-based models mimic physical
behavior or laws of physics. For all optimization algorithms and throughout the dissertation, there
are several terms used to describe the optimization. The objective function refers to a measure of
the goal of the optimization, sometimes called the cost function. The objective function, by
convention, is typically minimized to find the best solution. The design variables are the set of
values which may be changed to create different possible solutions. For each variable there is
usually an upper and lower bound which define feasible solution space. In addition to upper and
lower bounds, additional constraints impose restrictions on the solution space that relate two or
more optimization variables. A penalty function is similar to a constraint but is typically applied
after evaluation of the objective function. In addition, a constraint completely restricts certain
designs, whereas penalty functions have the flexibility of discouraging certain designs by
negatively impacting the objective function, without necessarily removing the design from the
solution space.
Evolutionary algorithms are based on several principles including reproduction, mutation,
gene crossover, and natural selection. Of evolutionary algorithms, the genetic algorithm is the
most common [256]. In general, a population of possible solutions are formed randomly
throughout the solution space, this set of individual solutions is called the population in the first
generation. Each individual in the population is evaluated based on the objective function(s), also
38
known as the fitness function(s). The high performing designs according to the fitness function(s)
are selected as parents for the next generation. The parents, or dominant designs, are used to
“reproduce” and create children designs that are a mixture of these dominant designs. This
reproduction or mixture process of the parents is often referred to as crossover, consistent with
the chromosome crossover that happens in genetics. These children form the subsequent
generation, creating a new population of designs. In reality, the subsequent generations are mixes
of the dominant designs, essentially attempting to breed better individuals within each generation.
In addition to crossover, another process called mutation is also implemented for a small
percentage of each population. It allows some portion of the mixture for a child to be randomly
generated, creating a mutation for the child. The mutation process is used to help expand the
search of the design space by randomly searching other areas of the solution space. The algorithm
progresses until the convergence criterion is met, typically based on how much the population is
changing with each generation. A general diagram of a genetic algorithm is shown in Figure 1.16.
There are several genetic algorithms and variations that have been proposed, but one of the most
common algorithms is the Non-dominated Sorting Genetic Algorithm (NSGA-II) [257] for multi-
objective optimization problems, discussed in Section 2.3.2. In addition to NSGA-II, Borg is
another multi-objective evolutionary algorithm which incorporates a variety of search methods
and identification of the top performing designs [258]. Additional information on Borg is
available in Section 3.2.2.2.
39
Figure 1.16 Diagram of genetic algorithm optimization process
Gradient search algorithms numerically estimate or analytically calculate the gradients of
the objective function to find the minimum or maximum of the objective function. A cartoon
image description of a gradient search algorithm for maximization of an objective is shown in
Figure 1.17. In the cartoon, it depicts a blindfolded person trying to find the top of the hill. The
height on the hill is equivalent to the objective function. In this case, the goal is to maximize the
objective function or find the highest point on the hill. The position of the blindfolded person is
the optimization variable. The fences define constraints or bounds on the feasible solution space.
As the blindfolded person takes a step, they can feel whether they are moving up or down the hill
and adjust accordingly. Eventually the person will reach the highest point on the hill at the red
flag. Just as described the cartoon, the gradient search method follows a similar process. By
taking a small step in the solution space, the gradient of the objective function or change in
objective function is calculated determining which direction to move. There are several variations
40
of the gradient search algorithm based on different methods to calculate or estimate the gradient
and acceleration methods for faster convergence [259–261].
Figure 1.17 Cartoon graphic used to describe maximization of an objective for gradient-based
algorithms, adapted from [254]
Referring again to Figure 1.17 to compare genetic and gradient-based algorithms, a
genetic algorithm would start with a random population of people in the design space. The
individuals highest on the hill would be used to create children in attempts to find a higher
position on the hill than their current positions. The process would be repeated with an occasional
random mutation to try to search other areas until the population is no longer finding higher
locations on the hill. As a quick comparison, initially the genetic algorithm is able to get a sense
for the design space, assuming the space is well-sampled by the initial random population. Using
the parents of the population the objective value, or average height on the hill for a population
should increase quickly but it is not guaranteed that the crossover of parents will generate lower
objective values than the parents. It may take many generations of gradual improvements to
progressively increase the objective function. In contrast, the gradient-based algorithm begins
41
with only a single objective function evaluation and should be able to continuously improve the
objective function unless there is difficulty with calculating or estimating the gradient.
Genetic and gradient-based optimization algorithms have been compared for many
different applications from which a general comparison can be formed between the two
optimization methods. In general, Salomon [262] compared the two methods and highlighted the
benefit of the population that exists for genetic algorithms. The population is argued to provide a
parallel search method as opposed to the gradient method which progresses from a single
solution. However, arguably, Salomon [262] points out that the gradient-based method does
search multiple directions to find the steepest gradient. The parallel search of a population in
genetic algorithms has also has shown a greater tendency to find the global minimum than
gradient-based algorithms [262]. The small step size of gradient-based algorithms, required to
maintain stability, makes it difficult to not become stuck in local minimums. Zingg et al. [263]
describes the tradeoff in cost between genetic and gradient-based algorithms. The high cost of the
genetic algorithm is based on the number of calculations which must be performed. Zingg et al.
[263] found that the genetic algorithm required between 6 and 200 times more function
evaluations than a gradient-based algorithm. The high cost of the gradient-based approach is the
requirement that the gradient must be computed. This can be challenging for non-smooth
objective functions or where the gradient cannot easily be defined [262]. In cases where the
gradient is readily available, Zingg [263] describes the primary advantage of the gradient-based
approach being rapid convergence. In addition, Zingg [263] points out that the convergence
criteria for gradient-based approaches is straightforward. As the gradient decreases by orders of
magnitude, one can be sure that at least a local minimum has been found [263]. In contrast, for
genetic algorithms they are able to handle non-smooth objective functions because a
mathematical relation between the optimization variables and objective function is not required
[264]. Genetic algorithm convergence is generally based on how much change exists in the
42
population but it is not as straightforward. Because genetic algorithms often excel where gradient-
based algorithms do not, and vice versa, performance often varies based on application [265].
Many researchers have also attempted to combine them to get the best of both algorithms
[262,266,267].
Optimization algorithms may be applied to a variety of different problems using one or
more objectives. In the case of multi-objective optimization algorithms, there is generally no
single best solution but instead there is often a tradeoff between objectives because the objective
functions may conflict with one another. The multi-objective nature results in a set of solutions,
each being optimal depending on how important each objective function is. This set of optimal
solutions is referred to as the Pareto front and is defined by the set of solutions which are non-
dominated by any other solution. A non-dominated solution is a solution for which no other
solution is better in all objective functions. Multi-objective optimization problems present a
unique advantage because all solutions along the final Pareto front are optimal. On the contrary,
weighting factors or ratios are required when multiple objectives are combined into a single
objective. To test different weights or ratios, the optimization problem has to be run multiple
times. A multi-objective optimization enables the designer to view a set of Pareto optimal
solutions and select one based on the relative importance or limits of each objective function.
1.1.6.2 Topology Optimization
Topology Optimization is a method to optimize material placement throughout a design
space. Originally proposed in 1988 by Bendsøe and Kikuchi [268], it was based on discretizing a
design space into small elements and using optimization techniques to determine which elements
should contain materials and which should be void. After determining the size of the design space
and applying boundary and loading conditions, the design space is discretized where each
43
element contains a “density.” Using a homogenization technique to determine the bulk properties
of the overall design space based on the density of individual elements, an optimized topology
can be generated. One of the challenges with the initial homogenization approach was the slow
convergence and difficulty in finding a clear boundary around the geometry. Because the density
value for each element can range from 0-1, 0 being void and 1 being full material, there was an
inherent gray-area where it was unclear whether material should be present or not. Later, the
Solid Isotropic Material with Penalization (SIMP) method was developed to improve
convergence and clearly define the boundaries of the solution [269,270]. The SIMP method uses
a power-law approach to drive the element densities to away from the gray area range in the
middle and force them closer to 0 or 1. The power-law approach acts as a penalization factor for
element densities in the mid-range. An example of the effect of varying the penalization
parameter for a minimization of compliance benchmark is shown in Figure 1.18. Traditionally
topology optimization was based on minimization of compliance and a material volume
constraint, with the intent of creating stiff and light-weight structures. For many years, topology
optimization has been developed, now using a variety of approaches and applied to complex
situations such as compliant mechanisms [271–276], multiple load cases [277], heat transfer
[118,278], multiple materials [13,279], and constraints for different manufacturing processes
[280]. A review of topology optimization methods and applications is presented by Bendsøe and
Sigmund [281] and Sigmund and Maute [282], and a review of topology optimization in
commercial software is provided by Reddy et al. [283].
44
Figure 1.18 Minimization of compliance benchmark optimized with the SIMP method. Varying the
power-law penalization factor, p, reduces the amount of gray material in the solution, clearly
defining the boundaries of the topology. Images adapted from [270].
In the case of compliant mechanism design or tailored compliance, one method of design
synthesis is topology optimization. Often in topology optimization of structures, the objective of
the optimization is minimization of compliance. On the other hand, for tailored compliance or
compliant mechanisms, compliance is desired, requiring new objectives for optimization. Often
the objective is a coupling between deflections at an input point and output point. To determine
the input/output relationship, the standard minimization of compliance problem is altered to
account for motion or force at the output point. Ananthasuresh and Frecker [284] describe
compliant mechanism synthesis objective functions based on mutual strain energy and strain
energy. The mutual strain energy is a measure of the output displacement and the strain energy is
a measure of the inverse of stiffness. These are conflicting objectives that must be balanced
through optimization. For example, the desired output force or displacement must be balanced
with a structure that stiff enough to handle the external loads. Ananthasuresh and Frecker [284]
describe two compliant mechanism synthesis approaches using topology optimization: ground
45
structure parameterization and a continuous material density parameterization. The ground
structure parameterization method is discussed in the following paragraph and the continuous
material density parameterization was described previously, though the objective functions for the
optimization must be reconsidered as described by Ananthasuresh and Frecker [284]. Many
examples of topology optimization and compliant mechanisms exist in the literature, especially
for the classic example of a compliant displacement inverter, Figure 1.19, and compliant gripper,
Figure 1.20.
Figure 1.19 Boundary and Loading conditions for a compliant inverter as well as the evolution of the
topology during convergence, adapted from [285]
46
Figure 1.20 A variety of compliant gripper topologies generated by varying optimization parameters,
adapted from [285]
One class of topology optimization methods is a called ground structure topology
optimization (GSTO). GSTO is known by many other names such as, structural optimization,
structural topology optimization, truss optimization, truss topology optimization, and ground
structure optimization. In contrast to traditional topology optimization, GSTO methods are a
discrete approximation of the design space by using truss, beam, or frame elements connected to
nodes throughout the design space. While in traditional topology optimization the design space is
discretized using continuum elements, the GSTO method approximates the design space as a set
of nodes distributed throughout and truss elements connecting the nodes; see Figure 1.21.
Assigning a cross-sectional area value to each member, the domain may be optimized with a
volume constraint for a given set of boundary and loading conditions. The earliest references to
GSTO dates back to the 1960’s with work by Dorn et al. [286]. The methodology and formulation
47
have been described by Bendsoe and Sigmund [281] and Ohsaki [287] in their textbooks on
topology optimization. GSTO has since advanced to tackle a number of different challenges
including multi-material design [288], reliability [289], arbitrary 2D and 3D domains [290,291],
adding and removing nodes or elements [292], buckling and overlap [293], and other AM
considerations [294–296].
48
Figure 1.21 An example comparison of the continuum and ground structure approach for topology
optimization
Continuum Approach Ground Structure
Approach
Problem Definition
49
1.1.6.3 Unit Cell Topology Optimization
In 1995, Sigmund researched a method for using a GSTO to generate a representative
base cell, or unit cell topology, for mesostructure design [297,298]. Using homogenization, the
unit cell topology was optimized for tailored compliance, or desired bulk mechanical properties.
Though it was mathematically possible to solve for the unit cell topology, the ability to
manufacture complex mesostructures was severely limited. Since then, multiple uses of this
approach have been used such as the work by Guth et al. [299,300], Messner [301], Neves et al.
[302], Liu et al. [303], and Huang et al. [304]. In 2012, Guth et al. [299] reported work very
similar to that of Sigmund, showing several 2D examples of unit cells tailored to specific bulk
properties. In 2015, Guth et al. [300] then reported 3D unit cell examples while considering
matching thermal and mechanical properties while enforcing isotropy constraints. Messner [301]
reported a broader use of the approach by not limiting the bounding volume of the unit cell to a
predetermined size and shape. Neves et al. [302] reported on the use of a penalization factor for
localized buckling in slender beams to increase the stability of the unit cell. Liu et al. [303] and
Huang et al. [304] both used the approach to develop cells of minimum compliance. A variety of
other works have also focused on homogenization and comparison of different techniques [305–
309].
In addition to the ground structure and homogenization method proposed by Sigmund, a
variety of other unit cell optimization methods have been proposed. Chen et al. [133], Xia et al.
[310], and Neves et al. [311] describe the use of continuum-based topology optimization to
design novel lattice structures with tailored properties. Ground structure approaches have also
been used to explore unit cell optimization with uncertainty and robust design considerations
[118,312]. Watts et al. [313] and Watts and Tortorelli [314] have published methods for reducing
computational cost of lattice structure design through surrogate models and geometric projection
50
methods. Geometric projection relies on geometric primitives that are projected onto a continuum
mesh for finite element analysis [315]. It has been applied through various way including simple
cylindrical primitives [316], plates [317], or supershapes [318]. Other methods have also
proposed hybrid approaches that combine topology optimization of the overall geometry with
sizing the lattice structure [319–324]. While there are numerous research groups exploring
methods for unit cell optimization or metamaterial design, the direct incorporation of AM
constraints into the optimization of unit cells has not been addressed.
1.1.7 Topology Optimization and Additive Manufacturing
With the growing interest in AM discussed previously, there has been a renewed interest
in topology optimization. Topology optimization methods often produce complex geometries that
are organic in nature and tend to be difficult, if not impossible, to manufacture with traditional
methods. While there are many complications to overcome, it is often possible to directly print
topology optimized solutions because the part is manufactured directly from a computer model.
One challenge with topology optimization for AM, is how to include manufacturing constraints
and consideration, such as build time and cost [325]. Without manufacturing constraints, the
solution generated does not account for process capabilities or limitations.
In 2016, Liu et al. [280] published a review of how manufacturing constraints for
machining, injection molding, and casting are implemented in topology optimization and then
highlighted topology optimization for AM as a promising future research direction. Then, in
2018, Liu et al. [295] provided a thorough review specifically for topology optimization for AM.
In the review, he highlights the importance of support material considerations for topology
optimization and the approaches that various research groups have taken. Langelaar [326,327] has
created an AM filter for post processing topology optimized designs into an as-printed part,
51
including a 2D and 3D implementation. In addition, Langelaar [328] proposed a cost function to
understand the tradeoff between restricting the topology optimization to be overhang-free and
unrestrained topology optimization; see Figure 1.22. Brackett et al. [329] proposed an idea for
using a linear approximation of overhangs to determine the build angle and penalize designs with
low build angles. Gaynor and Guest [330] used an integrated approach to include the critical
overhang angle within the topology optimization fully self-supporting components. Another
option that Liu et al. [295] highlights for reducing support structures is support slimming. Liu et
al. [331] proposed a method for detecting and restricting closed voids to avoid trapped powder.
While various research groups continue to explore methods for incorporating DfAM into
topology optimization, there is a lack of research which explore DfAM constraints on unit cell
topology optimization for designing lattice structures.
Figure 1.22 Potential tradeoffs exist in the ability to find design that consider the amount of support
material used while simultaneously considering the performance of the component, adapted from
[328]
1.1.7.1 Combining Lattice Structure DfAM and Topology Optimization
The second area of topology optimization for AM addressed by Liu et al. [295] is porous
infill and mesoscale design. Their work highlights lattice optimization or complex mesoscale
design as an important research area of topology optimization and AM. Due to the length scale,
topology optimization should produce fully self-supporting designs because of the difficulty
required to remove supports, especially for metal AM. Regarding mesoscale design, researchers
52
have produced topology optimized structures for tissue engineering scaffolds [168,169], auxetic
structures [332], other tailored structures for biomedical applications [245,333]. Though the area
of topology optimization for tailored properties has been an area of research for many years, as
described in Section 1.1.6.3, there is a lack of research for incorporating DfAM. In particular,
DfAM and unit cell topology optimization using a ground structure approach for tailored lattice
properties has not been addressed. With much of the focus for topology optimization and AM
centered on making stiff and light-weight structures, there is still much to be explored for tailored
compliance or multifunctional structures. In particular, the free complexity of AM that allows for
mesostructural design may be blended with GSTO synthesis for a variety of objective functions
including tailored bulk properties, thermal conductance, and AM consideration such as powder
removability.
There are several examples of topology optimization used to design novel metamaterials
with unique mechanical properties [133,197,304], biomedical purposes [163,245], or other multi-
functional lattices [118]. However, many of these approaches give little regard to incorporation of
the manufacturing process constraints into the optimization. In 2018, Tamburrino et al. [334]
review the design process for lattice structures fabricated with AM. As a result of their review,
they call for further research in five areas:
1) strategies to predict mismatch between as-designed versus as-manufactured,
2) optimization strategies for material distribution for enhanced and customized
properties allowed by AM,
3) optimized models that reduce computational effort,
4) integrated design tools for a multi-disciplinary approach, simultaneously taking into
account technological, geometric, and technical requirements, and
5) development of novel functionalities or properties that go beyond simple
lightweighting.
Roadmaps and reviews of the current status and future roadmaps have come to similar
conclusions [122,295,335–337]. Liu et al. [295] further supports this call for improved
incorporation and consideration of DfAM into the optimization. Plocher et al. [336] refer to
53
DfAM constraints in optimization as a, “bottleneck for a streamlined design procedure”. In
summary, there is a need for multi-objective optimization methods for lattice structure design that
incorporate DfAM constraints and variations that may occur during printing. This dissertation
presents a systematic approach for the optimization of unit cell topology that directly incorporates
DfAM considerations and demonstrates it with multiple case studies.
1.2 Research Objectives and Tasks
Recent advances in the capabilities of model simulation and manufacturing technology
are paving the way for systematic design optimization methods. These methods enable designers
to explore organic and nonconventional designs that were previously not possible to analyze or
manufacture. Through design optimization, these nonconventional designs may be used to
maximize part performance in a variety of applications such as medical, aerospace, and defense
industries.
With the increased need for design optimization tools for applications across multiple
industries, the following research objectives and tasks are outlined below for the development of
two systematic design optimization approaches.
1. Develop systematic optimization approach for compliant deployable radiofrequency
ablation electrodes
1.1. Design deployable electrode for endoscopic radiofrequency ablation
1.2. Develop finite element model to simulate radiofrequency ablation in soft tissue
1.3. Couple finite element with optimization approach for design of electrode based on
tumor geometry
1.4. Evaluate the deployable electrode design
1.4.1. Assess deployment of early prototypes in tissue phantom
1.4.2. Validate optimized electrode design through ex-vivo experimentation
2. Develop three-dimensional ground structure topology optimization for unit cell DfAM
2.1. Develop 3D-parametric ground structure model
2.2. Establish systematic optimization method for unit cell generation
54
2.2.1. Generate set of potential objective functions and constraints
2.2.2. Quantify manufacturing considerations for metal AM
2.3. Validate optimization with benchmark problems
3. Explore multi-functional lattices through case study on unit cell generation for thermal
conductance and minimum strain energy
3.1. Define constraints and process limitations
3.2. Generate optimal lattice structure with and without DfAM
3.3. Fabricate lattice structure with DfAM considerations
3.4. Evaluate solutions and revise ground structure topology optimization
4. Explore homogenized lattices through case study on generation of unit cells with tailored
compliance
4.1. Define constraints and process limitations
4.2. Generate optimal lattice structure with tailored properties
4.3. Fabricate and test to determine mechanical properties
4.4. Characterize fabricated samples with mechanical testing
4.4.1. Compare boundary conditions of homogenization with experimental
characterization
5. Develop mechanical property database for metal lattice structures from current literature
containing analytical, finite element, and experimental data
5.1. Perform literature search for metal lattice structure data
5.2. Collect data from selected papers
5.3. Create user interface for generating mechanical property charts of collected data
1.3 Dissertation Contents and Structure
The remainder of the dissertation is structured in the following format. Chapter 2
addresses work related to objective 1. The design of a deployable compliant endoscopic RFA
electrode is detailed. Next, a finite element model to predict the treatment zone for
radiofrequency ablation is explained. Then, a systematic optimization approach for shape
matching of an RFA treatment zone with electrode shape is explained. The optimization approach
is demonstrated in treatment of a pancreatic tumor using two variations of the proposed compliant
electrode. Finally, the deployment testing of the electrode prototypes and validation of the
computational model with a thermochromic tissue phantom is described.
Chapter 3 describes a systematic optimization approach for unit cell generation using a
3D GSTO, addressing objective 2 and its tasks. The GSTO approach is called Additive Lattice
55
Topology Optimization (ALTO). DfAM optimization objective functions, constraints, and
penalty functions were applied to improve manufacturability for AM. Next, a set of potential
objective functions, constraints, and penalties for a variety of applications is presented. Finally,
comparison of the ALTO results for optimization benchmark test problems validate the approach.
Chapters 4 and 5 address the tasks related to objectives 3 and 4, detailing the case-studies
included in this work. The case studies include 2D and 3D generation of novel unit cell
topologies for (1) thermal conductance and mechanical stiffness, (2) tailoring the mechanical
properties of a unit cell while maintaining a constant volume fraction, and (3) improving powder
removability while maintaining mechanical properties. Each case study includes description of
the purpose, optimization parameters, and results. From each case study, the outcomes and insight
gained are discussed, including adjustments and acknowledgement of limitations of the ALTO
approach.
In Chapter 6, the development of Lattice Unit-cell Characterization Interface for
Engineers (LUCIE) is presented. LUCIE combines published literature surrounding analytical,
finite element, and experimental analysis of lattice structures into MATLAB-based GUI for
comparison of mechanical properties of unit cell topologies. The selection of a unit cell topology
is available in commercial software; however, the selection is essentially blind because there is no
information regarding unit cell selection for the intended application. Though still in its infancy,
the future of topology optimization and AM, such as ALTO, would allow for complete design
and optimization of a customized lattice structure within a component. LUCIE offers a bridge
between rigorous optimization and blind selection in commercial software by providing a
centralized database for mechanical characterization of unit cell topologies.
Finally, Chapter 7 is a brief overview of the entire work. The overview includes major
outcomes and conclusions regarding each objective and task. To conclude, the research
contributions and recommendations for future research directions are addressed.
56
Chapter 2 Optimization and Experimental Validation of an EUS-RFA
Electrode
The focus of this chapter is on the development of a systematic optimization approach for
a deployable EUS-RFA compliant electrode. It includes a brief summary of the relevant
background and motivation as well as descriptions of the proposed electrode design, a finite
element model to predict the RFA in tissue, the optimization approach, results to date, and
experimental validation of the electrode design. This entire work was performed in collaboration
with Dr. Moyer from Hershey Medical Center. The following objectives and sub tasks are
addressed in this chapter:
1. Develop systematic optimization approach for compliant deployable radiofrequency
ablation electrodes
1.1. Design deployable electrode for endoscopic radiofrequency ablation
1.2. Develop finite element model to simulate radiofrequency ablation in soft tissue
1.3. Couple finite element with optimization approach for design of electrode based on
tumor geometry
1.4. Evaluate the deployable electrode design
1.4.1. Assess deployment of early prototypes in tissue phantom
1.4.2. Validate optimized electrode design through ex-vivo experimentation
2.1 Background and Motivation
Radiofrequency Ablation (RFA) is an increasingly used, minimally invasive, cancer
treatment modality for patients who are unwilling or unable to undergo a surgical resection of
abdominal tumors. RFA is performed by heating the tissue with an electrode inserted into the
tumor. The shape of the treatment zone is primarily determined by the shape the electrode and the
thermal and electrical properties of the tissue. While it has shown great promise in treatment, it is
limited by tumor location, tumor geometry, and tracking the treatment zone. Endoscopic
57
ultrasound-guided radiofrequency ablation (EUS-RFA) offers access to additional regions of the
abdomen that were previously inaccessible through traditional percutaneous RFA. Current EUS-
RFA electrodes are straight needle-shaped, generating an elliptical ablation zone, and offer
limited ability to treat the entire tumor; see Figure 2.1. This requires multiple insertions and
treatments to destroy a single tumor. To understand RFA, numerous finite element models have
developed; see Section 1.1.3 for additional information. While computational models have been
developed for understanding the effect of tissue properties, heat sink effects, placement
optimization, and commercial electrodes, there have been no attempts to use optimization to
determine the design of an RFA electrode for a specific tumor shape. There is a need to develop a
systematic optimization approach for generating RFA electrode designs to effectively treat tumor
geometries. In this chapter, a systematic optimization approach for a deployable EUS-RFA
electrode is applied specifically to pancreatic cancer, however, it is important to note that the
approach is not limited to pancreatic cancer. The fundamental approach and analysis may be
applied to cancers throughout the abdomen and mediastinum.
Figure 2.1 For straight electrodes, multiple insertions are often required to overlap a series of
ablation zones to destroy tumors. The image above is for an example of a percutaneous approach,
but, for EUS-RFA, the ability to insert multiple electrodes is limited creating the need for specialized
electrodes, adapted from [51].
58
2.2 Deployable Compliant EUS-RFA Electrode Design
The compliant EUS-RFA electrode design was inspired by the tail feathers of a peacock,
which begin as a straight bundle and then expand out into a large circular display. An earlier
version of the design was proposed by Hanks et al. [338]. The peacock-inspired electrode was
designed to be introduced into the tumor endoscopically, through a hollow needle. As shown by
Figure 2.2, the endoscopic needle, or sheath for the electrode, is advanced through the working
channel of an endoscope up to the periphery of the tumor. Then, the electrode is delivered
through the needle and deployed into the tumor. The design of the electrode is such that it may be
retracted into the endoscopic needle and, if necessary, redeployed in another location.
Figure 2.2 The electrode is deployed into the tumor through an endoscopic needle, acting as a sheath,
whose tip has been positioned at the periphery of the tumor, adapted from [339]
Two variations of a generic or base design for the optimization, shown in Figure 2.3, are
based on a circular array of outer tines and a single central tine which are connected at the
exposed electrode base. It is shown in both the stowed (Figure 2.3A,D) and deployed (Figure
2.3B,E) configurations. The principle of the design was that the natural state of the device was
open. The outer tines were compressed around the central tine and stowed within a rigid sheath,
such as an endoscopic needle. When the electrode was extended out of the rigid sheath, the
59
electrode tines would deploy and cut paths radially outward into the tumor facilitated by their
beveled tips. The deployment of the electrode tines broadens the ablation zone as compared to a
traditional straight needle electrode. The deployable electrode design was parameterized in terms
of the outer diameter in the stowed configuration (𝑑𝑒), length of the center tine (𝑙𝑐), number of
outer tines (𝑛𝑡), length of each outer tine (𝑙𝑜𝑖 , 𝑖 = 1,… , 𝑛𝑡), radius of curvature of each outer tine
(𝑟𝑜𝑖, 𝑖 = 1,… , 𝑛𝑡), and length of the exposed electrode base (𝑙𝑏) which connected the outer tines to
the center tine. Figure 2.3C shows the electrode design parameters 𝑑𝑒, 𝑙𝑐, 𝑙𝑜𝑖 , 𝑟𝑜
𝑖, and 𝑙𝑏 for the 𝑖-
th outer tine of an 8 outer tine example (𝑛𝑡 = 8). Figure 2.3F shows the design parameters
associated with a 12 outer tine variation (𝑛𝑡 = 12) of the electrode which is used as a second case
for optimization. The outer tines were designed to have constant curvature because studies of
flexible needles showed that during insertion, the curvature of the needle path was essentially
constant [340–342].
60
Figure 2.3 Two variations of the proposed electrode design are pictured in the stowed (A,D) and
deployed (B,E) configurations. The design parameters for each are shown (C,F), adapted from [339]
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2.3 Systematic Optimization Approach
2.3.1 Ablation Zone Prediction with a Finite Element Model
The Electric Currents and Bioheat Transfer modules of COMSOL Multiphysics
Modeling Software Version 5.2a [343] were used to develop the thermal ablation model (TAM).
These modules coupled the thermal and electrical components of the simulation of RFA in
biological tissue through the Multiphysics module. The finite element analysis was transient and
simulated an ablation procedure to obtain the geometry of the ablation zone generated by an
electrode. During the simulation, the electrode design was held fixed, as defined by design
variables shown in Figure 2.3. The TAM used similar underlying assumptions and boundary
conditions as the finite element models by Chang [61] and Tungjitkusolmun et al. [67].
A cylindrical block with a radius of 𝑟𝑡 and height ℎ𝑡 was used to represent the tissue and
must be large enough to avoid boundary interactions with the elevated temperatures generated by
the ablation. An insulated shaft, that would deliver the power to the electrode in the actual
procedure, was represented by a cylinder of radius 𝑟𝑏 and height ℎ𝑏. It was placed concentrically
inside the cylinder extending from the bottom of the tissue up to the bottom of the exposed
electrode base. The compliant electrode design was generated in COMSOL and fixed to the
insulated shaft, centered within the tissue block. To reduce the computational size of the TAM, a
simplified design without the beveled tips was generated for the ablation modeling. RFA tissue
heating was caused by the concentrated electric field near the electrode and thus heat transfer in
the tissue remained unaffected by the simplified TAM.
The fundamental equations used to model the thermal ablation are shown in Eqn. 2.1 and
Eqn. 2.2. (the bioheat equation).
∇ ∙ (−𝜎∇𝑉 + 𝑱𝑒) = 𝑄𝑗 (2.1)
62
𝜌𝐶𝜕𝑇
𝜕𝑡+ 𝜌𝐶𝒗 ∙ ∇𝑇 + ∇ ∙ (−𝑘∇𝑇) = 𝜌𝑏𝐶𝑏𝜔𝑏(𝑇𝑏 − 𝑇) + 𝑄𝑚𝑒𝑡 + 𝑄𝑒𝑥𝑡 (2.2)
In Eqn. 2.1, 𝜎(S/m) is the electrical conductivity, 𝑉(V) is the voltage potential, 𝑱𝑒 (A/m2)
is an externally generated current density and 𝑄𝑗(A/m3) is a current source. In Eqn. 2.2, 𝜌 (kg/m3)
is the tissue density, 𝐶 (J/(kg∙K)) is the tissue specific heat, T (K) is the temperature, 𝒗 (m/s) is
the velocity vector, 𝑘 (W/(m∙K)) is the tissue thermal conductivity, 𝜌𝑏 (kg/m3) is the blood
density, 𝐶𝑏 (J/(kg∙K)) is the blood specific heat, 𝜔𝑏 (1/s) is the blood perfusion rate, 𝑇𝑏 (K) is the
arterial blood temperature, 𝑄𝑚𝑒𝑡 (W/m3) is the heat source from metabolism and 𝑄𝑒𝑥𝑡 (W/m3) is
the heat source from the spatial heating.
The Electric Currents module available in COMSOL is based on Eqn. 2.1 In this TAM,
the electric field generated by the radiofrequency current in ablation was approximated by an
applied voltage, thus 𝑱𝑒 and 𝑄𝑗 are assumed to be zero. This simplifies Eqn. 2.1 to Eqn. 2.3
below.
−𝜵 ∙ (𝝈𝜵𝑽) = 𝟎 (2.3)
Electric boundary conditions of the TAM were an applied voltage to the active portion of
the electrode design (𝑉 = 𝑉0) and grounded external surfaces of the tissue block (𝑉 = 0). The
applied voltage which generated a direct current as opposed to the radiofrequency current used in
the actual procedure was a simplification, which was also used in the finite element model by
Chang [71]. The grounded external surfaces of the cylindrical tissue block were representative of
the large grounding pads used in the procedure to disperse the current.
The Bioheat Transfer module of COMSOL, which describes the heating that occurs in
biological tissue, is based on Eqn. 2.2, the bio-heat equation. With the velocity (𝒗) and metabolic
heat source (𝑄𝑚𝑒𝑡) set to zero, Eqn. 2.2 is simplified to Eqn. 2.4.
𝜌𝐶𝜕𝑇
𝜕𝑡+ ∇ ∙ (−𝑘∇𝑇) = 𝜌𝑏𝐶𝑏𝜔𝑏(𝑇𝑏 − 𝑇) + 𝑄𝑒𝑥𝑡 (2.4)
63
The thermal boundary condition was specified as a constant temperature on the external
surfaces of the tissue block (𝑇 = 𝑇∞). Heat conduction was permitted through the active portion
of the electrode and exposed electrode base. Tetrahedral elements were chosen to mesh the tissue
and electrode. Element size limitations were imposed in three sections to accommodate the small
features of the electrode geometry without drastically increasing the computational size.
To simplify the TAM, quarter symmetry was used to reduce the computational time; see
Figure 2.4. On the symmetry planes used to divide the geometry, an insulation boundary
condition was specified. For the Electrical Currents module of COMSOL, the insulation boundary
requires that no current flows through the boundary. In other words, the electric potential was
assumed to be symmetric about the boundary. For the Bioheat Transfer module, an insulation
boundary condition requires that the heat flux through the boundary be zero, or that the
temperature was symmetric about the boundary. For a summary of the initial and boundary
conditions; see Table 2.1.
64
Figure 2.4 General setup for the TAM. Due to the symmetry of the electrode, the TAM was reduced
to quarter symmetry to decrease computation time. The grounding pads regions are the top and
bottom quarter circles as well as the exterior wall of the cylinder (not shown) , adapted from [339].
Table 2.1 Initial and boundary conditions for the TAM
Initial Condition Boundary Condition
Tissue Electrical 𝑉 = 0 --
Thermal 𝑇 = 𝑇∞ --
Electrode Electrical 𝑉 = 𝑉0 𝑉 = 𝑉0
Thermal 𝑇 = 𝑇∞ --
Grounding Pad Electrical 𝑉 = 0 𝑉 = 0
Thermal 𝑇 = 𝑇∞ 𝑇 = 𝑇∞
Symmetry Plane Electrical 𝑉 = 0 𝒏 ∙ 𝑱 = 0
Thermal 𝑇 = 𝑇∞ −𝒏 ∙ 𝒒 = 0
The ablation zone was defined by the 60°C isothermal surface surrounding the electrode.
Time and temperature dependent models for cell death vary in literature; however, at 60°C,
literature agrees that cell death is essentially instantaneous [40,42,44–46]. Chang and Nguyen
[71] found that the 60°C isothermal surface was a good representation of the ablation zone by
comparing their finite element model with experimental data.
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2.3.2 Optimization using a Genetic Algorithm
The optimization of the electrode was performed using a multi-objective genetic
algorithm in MATLAB [344] and was an extension of our previous work [345]. A genetic
algorithm was chosen because of the increased likelihood of finding a global minimum compared
to gradient-based approaches, and its ability to converge to an optimal solution without the need
to calculate numerical or analytical gradients. The multi-objective genetic algorithm is part of the
Global Optimization Toolbox in MATLAB [346] and is a variant of the Non-dominated Sorting
Genetic Algorithm (NSGA-II) algorithm [257]. The purpose of the algorithm was to find the
Pareto-Optimal set of designs, or Pareto front, which were non-dominated by any other design. A
non-dominated design is a design for which no other design has better than the current design in
all objective functions.
The NSGA-II algorithm variant in MATLAB works by evaluating a series of design sets
using the TAM. Each design set, or generation, was created based on the results of previous
generations, gradually converging to the Pareto-Optimal set of designs. MATLAB created the
initial generation randomly distributed throughout the design space. The design space was
defined by the user-defined constraints and the upper and lower bounds of each design variable.
Next, the electrode geometry, defined by the design variables, was sent to COMSOL for
simulation and the objective function data from each simulation was tabulated. Following each
COMSOL simulation, MATLAB collected the data and evaluated the objective functions for the
design. After all the designs in the generation were simulated, they were ranked based on the
objective function (fitness) evaluations. The non-dominated designs were rank one, designs
dominated by one other design were rank two, and designs dominated by two designs were given
a rank of three, etc. After all the designs of the generation were ranked, a portion of the best-
ranked designs were carried over to the next generation and other designs were generated by
66
creating crossovers and mutations according to MATLAB default settings. The crossovers and
mutations allow a genetic algorithm to sample a large design space and help avoid local minima.
The process was repeated for subsequent generations, gradually improving with each generation.
Multi-objective optimization algorithms present a unique advantage over single objective
algorithms because all designs along the final Pareto front are optimal. On the contrary, weighting
factors or ratios are required when multiple objectives are combined into a single objective. To
test different weights or ratios, the optimization problem would have to be run multiple times. A
multi-objective optimization enables the designer to view a set of Pareto optimal designs and
select a solution based on the relative importance or limits of each objective function for a
particular application. In the case of treating tumors, the multi-objective optimization allows the
designer to view the set of optimal solutions, consider the importance of each objective function,
and select the best design while considering other patient-specific factors such as surrounding
anatomy.
As the optimization progressed, the metric used to determine convergence was the
change in the distribution or spread change of the Pareto front. Once the Pareto front was
essentially stagnant, the algorithm has determined a set of optimal designs which represent the
best tradeoffs of multiple objective functions. The change in the distribution of the Pareto front
was computed as the average relative change in the spread of the optimal designs, where the
spread is a distance measurement of the distribution of designs along the Pareto front. The
algorithm was considered converged when the average relative spread change for the previous
𝑛𝑠𝑡𝑎𝑙𝑙 generations was less than 𝜂, the stall tolerance, and the current spread was less than the
average of the previous 5 generations. In other words, the distribution of the optimal designs has
had minimal changes for 𝑛𝑠𝑡𝑎𝑙𝑙 generations and the spread of designs was decreasing.
The purpose of the optimization was to match the size and shape of the ablation zone for
a particular electrode with the size and shape of target ablation zone. After all the variables and
67
objective functions are described, a formal description of the optimization is given by Eqns. 2.5-
2.9. The design variables included in the optimization are 𝑙𝑐, 𝑛𝑡, 𝑙𝑜𝑖 , 𝑟0
𝑖 as defined previously. The
upper and lower bounds for each design variable, represented by Eqn. 2.5, define the design
space.
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒
(𝑓𝑜𝑣𝑒𝑟 (𝑡𝑖 , 𝑟𝑖(𝑙𝑐 , 𝑙𝑜𝑖 , 𝑟𝑜
𝑖)) , 𝑓𝑢𝑛𝑑𝑒𝑟 (𝑡𝑖, 𝑟𝑖(𝑙𝑐 , 𝑙𝑜𝑖 , 𝑟𝑜
𝑖)))
𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜
{
𝑙𝑐𝑙𝑜𝑤𝑒𝑟
𝑙𝑜𝑙𝑜𝑤𝑒𝑟
𝑟𝑜𝑙𝑜𝑤𝑒𝑟
} ≤ {
𝑙𝑐𝑙𝑜𝑖
𝑟𝑜𝑖
} ≤ {
𝑙𝑐𝑢𝑝𝑝𝑒𝑟
𝑙𝑜𝑢𝑝𝑝𝑒𝑟
𝑟𝑜𝑢𝑝𝑝𝑒𝑟
}
𝑓𝑜𝑟 𝑖 = 1,… , 𝑛𝑡
(2.5)
−𝑙𝑐 + 𝐶𝑙𝑖𝑙0𝑖 ≤ 0 (2.6)
𝑊ℎ𝑒𝑟𝑒,
𝑓𝑖 =𝑟𝑖 − 𝑡𝑖𝑡𝑖
(2.7)
𝑓𝑜𝑣𝑒𝑟 = ∑ 𝑓𝑖𝑛𝑜𝑣𝑒𝑟1
𝑛𝑜𝑣𝑒𝑟 𝑓𝑜𝑟 𝑓𝑖 > 0 (2.8)
𝑓𝑢𝑛𝑑𝑒𝑟 = ∑ 𝑓𝑖𝑛𝑢𝑛𝑑𝑒𝑟1
𝑛𝑢𝑛𝑑𝑒𝑟 𝑓𝑜𝑟 𝑓𝑖 < 0 (2.9)
Two objective functions were used to quantify the performance of each design, each
based on a spherical coordinate system centered in the ablation zone; see Figure 2.5. As
mentioned previously, the 60°C isothermal surface determines the shape of the ablation zone. The
radial distance was measured from the center of the ablation zone to this isothermal surface for
𝑛𝑔𝑝 points. The center of the ablation zone was determined taking the average of the highest and
lowest ablation zone coordinates along the Z-axis. Each radial distance measurement, 𝑟𝑖, was
normalized in terms of the target radial distance, 𝑡𝑖, resulting in 𝑓𝑖, a percentage over or under the
target value for a specific θ, φ location; see Eqn. 2.7. The resulting 𝑓𝑖 values were grouped by
whether they were greater or less than the target distance and then averaged into two objective
functions, 𝑓𝑜𝑣𝑒𝑟 and 𝑓𝑢𝑛𝑑𝑒𝑟; see Eqns. 2.8-2.9. The objective function 𝑓𝑜𝑣𝑒𝑟 focuses on
68
minimizing the amount of tissue destroyed outside the target zone (healthy tissue beyond the
target zone). The objective function 𝑓𝑢𝑛𝑑𝑒𝑟 focuses on maximizing the amount of tissue destroyed
within the target zone. While these two objective functions were not always in direct conflict,
they result in a set of Pareto-optimal designs from which the designer can make a more informed
decision than a single objective optimization. The use of these objective functions allows for
consideration of optimal electrode designs ranging from no ablation outside the target region
(𝑓𝑜𝑣𝑒𝑟 = 0) to ablating the entire target region (𝑓𝑢𝑛𝑑𝑒𝑟 = 0) and a variety in between. The
constraint in Eqn. 2.6 allows the designer to control the length of the center tine relative to the
length of the outer tine through 𝐶𝑙𝑖.
Figure 2.5 Graphical description of the objective functions used for optimization, adapted from [339]
2.3.2.1 Objective Function Computation through Image Analysis
Several different objective functions were explored including sphericity calculations, a
ratio of the second moment of area through different cross sections, and finally the comparison of
a series of distance measurements for the target and simulated ablation zone. Ultimately, the
69
distance measurements proved to be the most robust and may be easily applied to a variety of
target shapes beyond a sphere. These objective functions make the approach applicable to a
variety of tumor shapes and even patient specific optimization based on patient tumor geometry.
To compute the objective functions two methods have been utilized. The first methods
used probes internal to COMSOL to measure the distance to the edge of the ablation zone at
different locations. The second approach uses image analysis to compute the cross sections of the
ablation zone. Image analysis tools within MATLAB determine the center of the ablation zone
and measure the distance to the edge. The MATLAB code for performing the image analysis was
developed with support from Zackary Snow and is shown in Appendix A. In both approaches,
these distances were compared against the target shape to compute the objective functions.
Compared to the first approach, the second approach enables the designer to compare a greater
number of measurement locations between the simulated and target ablations, but ultimately the
results were similar. For arbitrary shapes, where many points would be necessary to define the
shape, the second approach may show better robustness than the first approach. As the results for
both approaches were similar, the results from the first method are described in the following
section.
The use of these objective functions resulted in a set of Pareto optimal electrode designs
for which the ablation zone geometry ranges from completely inside the target to variations that
fill more of the target zone. For the purposes of further analysis and comparison of the two
electrode variations shown in Figure 2.3, a single design was selected from the set of optimal
solutions. Without patient specific anatomy to consider in narrowing down the set of optimal
solutions, the distance to the utopia point was used as a metric to select a single design. By
plotting each electrode design according to its objective functions, the design closest to the utopia
point was selected for analysis. The utopia point is an idealized location where all objective
functions are simultaneously minimized. In this case, the utopia point was defined as the location
70
where all the objective functions simultaneously equal zero. As a metric to the compare optimal
solutions from Case 1 and Case 2. The distance to the utopia point (𝑑𝑢) was defined as shown in
Eqn. 2.10.
𝑑𝑢 = √𝑓𝑜𝑣𝑒𝑟2 + 𝑓𝑢𝑛𝑑𝑒𝑟
2 (2.10)
This metric is unitless and represents a measure of how closely the ablation zone matches
the target shape.
2.3.3 Results from the Systematic Optimization Approach
For the purpose of the EUS-RFA electrode for pancreatic cancer, the materials used in
the TAM were assumed to be pancreatic tissue, polyurethane for the insulated shaft, and nickel-
titanium (Nitinol) for the exposed electrode base, outer tines, and center tine. Nitinol was chosen
for its superelasticity and biocompatibility. For Nitinol, the superelastic property means there is
minimal increase in stress for a large increase in strain during the phase transition from austenite
to martensite. The material properties used for each of the above listed materials are specified in
Table 2.2 [347–352]. The target ablation zone was assumed to be a 25 mm sphere, based on
typical tumors that would be treated with such a procedure. Each simulation was for an ablation
procedure time of six minutes. A voltage 𝑉0 = 20 Volts was applied to the electrode. An ambient
temperature 𝑇∞ = 37°C was chosen based on standard body temperature. The size of the
pancreatic tissue used during the simulation was a cylinder of radius 𝑟𝑡 = 6cm with a height ℎ𝑡 =
12cm, for which boundary effects from the proximity of grounding pads are negligible. The
convergence of the TAM was performed for a set of electrode design parameters by continuously
refining the mesh until the ablation zone volume remained essentially unchanged (within 0.5%).
The insulated shaft was represented as a cylinder of radius 𝑟𝑏 = 0.48 mm with a height ℎ𝑏 = 5 cm,
71
centering the electrode in the pancreatic tissue. The length of the exposed electrode base which
connects the outer tines to the central tine was held fixed at 1 mm.
Table 2.2 Material properties for TAM
Material
Density
[𝑘𝑔 𝑚3⁄ ] Thermal
Conductivity [𝑊 (𝑚 ∙ 𝐾)⁄ ]
Electrical
Conductivity [𝑆/𝑚]
Heat
Capacity [𝐽 (𝑘𝑔 ∙ 𝐾)⁄ ]
Ambient
Temperature [𝐾]
Perfusion
Coefficient [1 𝑠⁄ ]
Pancreatic
Tissue[347,348] 1087 0.51 0.566 3164 37 --
Blood[347,349] 1050 -- -- 3617 37 0.0064
Polyurethane[350,35
1] 70 0.026 1x10-8 1045 -- --
Nitinol[352] 6450 18 1.25x106 836.8 -- --
Two cases are presented to illustrate how the optimization technique can be useful in
systematic design of EUS-RFA electrodes. Each case is summarized in Table 2.3. Case 1, shown
in Figure 2.3A-C, presents an electrode design in which there were eight outer tines and each tine
has the same length and radius of curvature. In keeping each of the outer tines the same length
and curvature, the number of design variables was reduced to three: 𝑙𝑐, 𝑙𝑜, and 𝑟𝑜. Case 2, shown
in Figure 2.3D-F, allows more freedom to the optimization than Case 1 by considering more
complex electrode designs. The number of tines was increased to 12 and there were two
configurations of outer tines, such that every other tine had the same configuration. All odd
number tines had the same length (𝑙𝑜𝑜𝑑𝑑) and radius of curvature (𝑟𝑜
𝑜𝑑𝑑) while all even number
tines had the same length (𝑙𝑜𝑒𝑣𝑒𝑛) and radius of curvature (𝑟𝑜
𝑒𝑣𝑒𝑛). The upper and lower bounds for
each design variable are shown in Table 2.3. They were chosen such that a wide range of
electrode designs could be simulated, from a long and thin profile to a very short and round
profile.
72
Table 2.3 Case 1 and Case 2 optimization and electrode design parameters
Case 1 Case 2
Design parameters 3 5
Stowed diameter (𝑑𝑒) 0.96 mm 0.96 mm
Center length (𝑙𝑐) 𝑙𝑐 𝑙𝑐 Number of tines (𝑛𝑡) 8 12
Outer tine length (𝑙𝑜𝑖 ) 𝑙𝑜
𝑖 = 𝑙𝑜 , 𝑖 = 1, … ,8 𝑙𝑜𝑖 = {
𝑙𝑜𝑜𝑑𝑑 , 𝑖 = 1, 3, … , 11
𝑙𝑜𝑒𝑣𝑒𝑛 , 𝑖 = 2, 4, … ,12
}
Outer tine radius (𝑟𝑜𝑖) 𝑟𝑜
𝑖 = 𝑟𝑜 , 𝑖 = 1, … ,8 𝑟𝑜𝑖 = {
𝑟𝑜𝑜𝑑𝑑 , 𝑖 = 1, 3, … , 11
𝑟𝑜𝑒𝑣𝑒𝑛 , 𝑖 = 2, 4, … ,12
}
Exposed electrode base (𝑙𝑏) 1.0 mm 1.0 mm
Constraint parameter (𝐶𝑙𝑖) 0.8 0.8
Stall generations (𝑛𝑠𝑡𝑎𝑙𝑙) 5 5
Pareto change tolerance (𝜂) 0.001 0.001
Grid Points (𝑛𝑔𝑝) 23 23
Target radial distance (𝑡𝑖) 12.5 mm 12.5 mm
Optimization Bounds and
Constraints
{5 𝑚𝑚5 𝑚𝑚5 𝑚𝑚
} ≤ {𝑙𝑐𝑙𝑜𝑟𝑜
} ≤ {25 𝑚𝑚25 𝑚𝑚200 𝑚𝑚
}
−𝑙𝑐 + 0.8𝑙0𝑖 ≤ 0
{
5 𝑚𝑚5 𝑚𝑚5 𝑚𝑚5 𝑚𝑚5 𝑚𝑚}
≤
{
𝑙𝑐𝑙𝑜𝑜𝑑𝑑
𝑙𝑜𝑒𝑣𝑒𝑛
𝑟𝑜𝑜𝑑𝑑
𝑟𝑜𝑒𝑣𝑒𝑛}
≤
{
30 𝑚𝑚30 𝑚𝑚30 𝑚𝑚200 𝑚𝑚200 𝑚𝑚}
−𝑙𝑐 + 0.8𝑙0𝑒𝑣𝑒𝑛 ≤ 0
−𝑙𝑐 + 0.8𝑙0𝑜𝑑𝑑 ≤ 0
The objective functions 𝑓𝑜𝑣𝑒𝑟 and 𝑓𝑢𝑛𝑑𝑒𝑟 were based on grid points around the ablation
zone surface; see Figure 2.5. These points correspond to positions at θ = 0°, 45°, and 90°. At each
θ location (0°, 45°, and 90°), 8 positions were calculated for corresponding to φ = 0°, 22.5°, 45°,
67.5°, 90°, 112.5°, 135°, 157.5°, and 180°. Repeated grid points at φ = 0° and 180°, for θ = 0°,
45°, and 90°, were removed, resulting in a total of 23 grid points to determine the objective
functions. The target radial distance (𝑡𝑖) was equivalent for each point around the surface (𝑡𝑖 =
12.5 mm). Convergence was determined by an average relative spread change of less than 0.1%
over the final 5 generations (𝑛𝑠𝑡𝑎𝑙𝑙 = 5, 𝜂 = .001).
73
2.3.3.1 Case 1
The optimization was performed using a population of 75 designs per generation and
converged after 138 generations. Figure 2.6 shows the entire performance space, i.e., a
comparison of the objective function values for each design evaluated. Each point on the plot
represents a feasible design, the plot shows the tradeoff between objective functions which exist
for some designs. From the objective functions, one can quickly understand the relative size and
shape of the ablation zone. For example, for designs along the horizontal axis (𝑓𝑜𝑣𝑒𝑟 = 0), all 23
grid points were within the treatment zone. Other designs along the Pareto front decrease 𝑓𝑢𝑛𝑑𝑒𝑟,
or fill more of the target volume, but also treat some regions of tissue beyond the target zone.
This range of optimal designs allows for discretion of whether it is preferable to only treat tissue
within the target region or to increase volume of treated tissue within the target region at the
expense of destroying other tissue outside the target region. In this study, a single optimal
electrode design was selected for further consideration based on the design located closest to the
utopia point.
74
Figure 2.6 Each design tested during the optimization is plotted in this figure according to its
corresponding objective function values, adapted from [339]
For Case 1, the parameters for the optimal electrode design selected for comparison
against Case 2 were 𝑙𝑐, 𝑙𝑜, and 𝑟𝑜 equal to 18.33 mm, 15.57 mm, and 10.42 mm, respectively,
with 𝑓𝑜𝑣𝑒𝑟 and 𝑓𝑢𝑛𝑑𝑒𝑟 equal to 0.0 and 0.1027, respectively. The distance from the utopia point to
the optimal design was 0.1027. The temperature profile and ablation zone for the optimal design
from Case 1 are shown in Figure 2.7. The electrode design parameters for Case 1 are summarized
in Table 2.4 for comparison against Case 2.
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Figure 2.7 (Left) The side view of the temperature profile for an optimal electrode solution for Case
1. The ablation zone is marked by the white isothermal contour (60°C). (Right) The ablation zone
surface (green) is shown compared to the target shape (black outline) for the top and side views,
adapted from [339].
2.3.3.2 Case 2
The optimization was performed using a population of 75 designs per generation and
converged after 91 generations. The entire design space is shown in Figure 2.8. As with Case 1, a
wide range of optimal designs lie along the Pareto front providing a variety of options for optimal
designs. From the final Pareto front of designs, the parameters for the optimal electrode design
selected for comparison against Case 1 were 𝑙𝑐, 𝑙𝑜𝑜𝑑𝑑, 𝑟𝑜
𝑜𝑑𝑑, 𝑙𝑜𝑒𝑣𝑒𝑛, and 𝑟𝑜
𝑒𝑣𝑒𝑛 equal to 22.42 mm,
19.49 mm, 20.31 mm, 15.39 mm, and 8.49 mm, respectively, with 𝑓𝑜𝑣𝑒𝑟 and 𝑓𝑢𝑛𝑑𝑒𝑟 equal to
0.0434 and 0.0555, respectively. The distance from the utopia point to the optimal design was
0.0705. The temperature profile and ablation zone for the optimal design from Case 2 are shown
(Axis units are shown in meters)
Side View
Top View
76
in Figure 2.9. The electrode design parameters for Case 2 are summarized in Table 2.4 for
comparison against Case 1.
Figure 2.8 Each Case 2 design simulated is plotted above according to its corresponding objective
function values, adapted from [339]
77
Figure 2.9 (Left) The side view of the temperature profile for an optimal electrode solution for Case
2. The ablation zone is marked by the white isothermal contour (60°C). (Right) The ablation zone
surface (green) is shown compared to the target shape (black outline) for the top and side views,
adapted from [339].
Table 2.4 Electrode designs selected for comparison from the Case 1 and Case 2 Pareto-optimal
solution sets
Center Length Outer Length Outer Radius
Case 1 𝑙𝑐(mm) 𝑙𝑜(mm) 𝑟𝑜(mm)
Optimal 18.33 15.57 10.42
Case 2 𝑙𝑐(mm) 𝑙𝑜𝑜𝑑𝑑(mm) 𝑙𝑜
𝑒𝑣𝑒𝑛(mm) 𝑟𝑜𝑜𝑑𝑑(mm) 𝑟𝑜
𝑒𝑣𝑒𝑛(mm)
Optimal 22.42 19.49 15.39 20.31 8.49
Top View
Side View
78
2.3.4 Discussion and Comparison to Standard Straight Electrode
A systematic design optimization approach was useful in comparing electrode designs by
reducing the amount of trial and error and intuition-based experimental work. A comparison of
the two electrode designs in Cases 1 and 2 can be used to show the benefit of such a systematic
approach.
The purpose of Cases 1 and 2 was to compare two electrode designs, where the second
was a more complex design, which increases freedom of the optimization to find an electrode
which generates an ablation zone that matches the target, a 25 mm sphere. For Case 1, all the
outer tines have the same length and radius of curvature, limiting spherical ablation zones which
can be generated. In particular, as the size of the target shape increases, it becomes difficult for
the electrode to fill the target shape while all outer tines have the same configuration. This can be
seen in Figure 2.7, where the ablation zone could be improved by broadening the lower and upper
portions. With a single outer tine configuration, the electrode was unable to fill these lower and
upper portions. Increasing the number of tines and allowing multiple configurations should allow
for better shape matching than only eight outer tines with a single configuration.
For Case 2, Figure 2.9 shows that increasing the number of outer tines and allowing for
two outer tine configurations resulted in significant improvement in the ability of the electrode to
cover the target zone. The optimization improved the ablation zone by using the second set of tine
configurations. As evident from the objective function values, the optimization was able to find a
design with a decreased distance to the utopia point in Case 2. Thus, the optimization approach
was able to show that the Case 2 design has the potential for improved shape matching compared
to Case 1. Depending on the increase in complexity for manufacturing, the improved objective
function values of Case 2 may be desirable. For complex, non-symmetrical, target shapes, it is
expected that deployable electrode designs that allow even more freedom than Case 1 or Case 2
79
will be necessary. The objective functions utilized for this optimization were useful for matching
a sphere and can easily be expanded to complex or arbitrary shapes.
To compare against currently available endoscopic electrodes, the ablation zone for a
standard straight needle is shown in comparison to the Case 1 and Case 2 solutions; see Figure
2.10. The length of the straight electrode is typically adjustable up to 20 mm. In this case, the full
electrode length (20 mm) was used for simulation and comparison. By comparing the ablation
zone volumes with the target shape, one can obtain a quantitative sense of the improvements seen
with the optimal designs. The percentage of the target volume filled by the ablation zone was
computed by dividing the ablation zone volume inside the target zone by the total target volume.
The percentage of the ablation zone outside the target volume was computed by dividing the
ablation zone outside the target volume by the total ablation zone volume. The current straight
electrode ablated only 25.1% of the target volume. The optimal solutions show increased
treatment volumes of 70.9% and 87.0% for Case 1 and Case 2, respectively. The results for the
straight electrode and optimal solutions for Case 1 and Case 2 are summarized in Table 2.5.
Table 2.5 Quantitative comparison of a straight electrode and optimal Case 1 and Case 2 solutions
Single Case 1 Case 2
Percentage of target zone filled 25.1% 70.9% 87.0%
Percentage of ablation volume outside target zone 0.5% 0.4% 3.2%
80
Figure 2.10 Top and side views of the ablation zones for the straight electrode and optimal solutions
of Case 1 and Case 2. The green surface represents the 60°C isothermal surface and the black circle
represents the target zone, adapted from [339].
While this optimization approach has shown promise as a useful tool in simulating the
ablation zone and designing RFA electrodes, limitations exist in accounting for the numerous
variables of the human body. Experimental validation of the TAM using a thermochromic tissue
phantom is described in Section 2.4.3. When compared with other similar models reported in
literature, the average error between temperature profiles across the electrode was found to be
1.8% [61]. In addition, the pancreatic tissue in this TAM does not account for the inhomogeneity
which exists in all biological tissue; however, the random and constant dividing of cancer cells
produces more homogeneous tissue as compared to typical healthy tissue. The tissue properties
81
were also chosen based on an approximation of tumors, though, in reality, a variety of values are
reported in literature for various pancreatic tissue properties.
One challenge with selecting tissue properties was that the reported values in literature
vary and are often temperature or frequency dependent. For this optimization, the electrical
conductivity and blood perfusion were assumed to be constant according to reported values for
pancreatic tissue. The electrical conductivity is temperature dependent as discussed by Chang
[71,347] and has been explored in many other studies [63,348,353]. The blood perfusion rate has
also been shown to decrease with increased metabolic activity in tumors making it difficult to
determine an appropriate value [349]. Although the overall size and temperature of the ablation
zone was found to be sensitive to these parameters, the shape of ablation zone was not heavily
affected. Thus, the current TAM could be used to understand the general shape of the ablation
zone and then tissue properties can be used to tune the TAM to match experimental ex-vivo
testing for validation. It is also important to recognize that RFA may have limited effectiveness
near large blood vessels because they act as a significant heat sink, limiting the temperature to
which the tissue can be heated. While this TAM does not include the heterogeneities present in
living tissue (blood vessels, surrounding organs, or other structures), the same optimization
approach could be utilized with models of heterogeneous tissue to generate an electrode design
for patient specific treatment planning.
While this approach for design optimization of RFA ablation electrodes shows great
promise, the TAM only accounts for an electrode shape after it has been inserted in the tissue.
The deployment of the electrode, and therefore the ablation zone produced, were likely to be
affected by variations in tissue elastic modulus and other tissue properties. To consider the
robustness of the current electrode designs, the outer tine radius of curvature was varied by ±10%
for the Case 1 solution and the ablation zone volume was calculated. For an outer tine radius of
curvature of 9.38 mm, the ablation zone volume increased by 3.3%. For an outer tine radius of
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curvature of 11.46 mm, the ablation zone volume decreased by 3.7%. Therefore, for slight
variations in outer tine radius of curvature, the change in total ablation zone volume was small.
To account for these variations in deployment, it is necessary to develop modeling which predicts
the deformation and mechanical response of the electrode as it deploys while being inserted into
the tumor. This type of deployment model will be useful in relating the manufactured electrode
shape with the final electrode shape deployed in tumors.
2.3.5 Conclusions from Systematic Optimization Approach
The purpose of this study was to optimize the shape of an RFA electrode to match the
ablation zone for a target tumor geometry to improve the efficacy of treatment of pancreatic
cancer. Using a computational model based on coupling the electric and bioheat transfer
equations in COMSOL Multiphysics software, a TAM was used to predict the ablation zone for
unique electrodes. The resulting electrode shape was optimized to generate a 2.5cm spherical
ablation zone using a multi-objective genetic algorithm. For a target 2.5cm spherical tumor, the
optimal design parameters of the compliant electrode design were found for two cases. Case 1
and Case 2 optimal solutions filled 70.9% and 87.0% of the target volume as compared to only
25.1% for a standard straight electrode. While the TAM and electrode were specifically focused
on EUS-RFA of pancreatic cancer, RFA is a common procedure for tumors throughout the
abdomen. This approach of optimizing the electrode shape for a target ablation zone has the
potential to personalize treatment for many cancer patients and may be applied to a variety of
tissues where RFA is practical.
83
2.4 Testing and Validation of Deployable Electrode
2.4.1 Prototype Development
An early challenge in the development of the deployable EUS-RFA electrode was size
restrictions due to the requirement that the deployable electrode pass through an endoscopic
needle (~1 mm diameter). The number of manufacturing processes that can produce such a small-
scale device was limiting but ultimately two different processes were used for early prototypes:
metal AM with laser powder bed fusion (L-PBF) and laser micromachining (LMM).
L-PBF was used to create a 2x scale version of the electrode, printed on the 3D Systems
ProX 320 using Oerlikon MetcoADD 625A powder at Penn State’s CIMP3D facility. The parts
were removed from the build plate using wire electrical discharge machining and post treated
with hot isostatic pressing to reduce voids and stress relieve the part. Stowing the L-PBF
prototype inside the sheath, inner and outer diameter 2.16 and 2.41 mm, respectively, resulted in
some visible plastic deformation. Next the base of the part was welded onto a steel wire with a
diameter of 0.991 mm. The final part dimensions of the L-PBF prototype are shown in Figure
2.11. These prototypes were used in the deployment testing in a tissue phantom described in
Section 2.4.2. An additional version of the L-PBF prototype with only six outer tines was created
for testing the limits of the manufacturing process and resulted in a nearly 1x version; see Figure
2.12.
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Figure 2.11 L-PBF 2X scale electrode prototype
Figure 2.12 Near 1X scale L-PBF electrode prototype
The second manufacturing method, LMM, used a superelastic Nitinol tube to create the
outer compliant tines. Initial prototypes were manufactured by L4is. The Nitinol tube had an
outer diameter of 0.991 mm and an inner diameter of 0.584 mm. A nanosecond laser was used to
make 14.8 mm cuts along the length of the tube starting at the end of the tube. In total four cuts of
this length were equally spaced around the tube circumferentially creating 4 tines. The final width
of each tine was 0.635 mm. The tip of each tine was also sharpened with two cuts from the
nanosecond laser to make the tip of each tine pointed. Finally, each electrode tine was wrapped
49.8 mm
15.4 mm
85
around a mandrel to form it into its naturally deployed configuration. These outer compliant tines
were fitted onto a steel wire with a diameter of 0.584 mm which serves as the stiff central needle.
The final dimensions of the electrode fabricated with LMM are shown in Figure 2.13. These
prototypes were used in the deployment testing in a tissue phantom described in section 2.4.2.
Since that time, additional prototypes were fabricated based on the Case 1 and Case 2 optimal
configurations described previously, featuring 8 and 12 outer tines, respectively; see Figure 2.14.
These prototypes were fabricated through collaboration with Actuated Medical Inc. (AMI) with
LMM by Spectralytics from a Nitinol tube with an outer diameter of 0.030” ± 0.001” and an inner
diameter of 0.019” ± 0.001”. These later prototypes were used in the ex-vivo experimentation for
validation of the TAM.
Figure 2.13 4-tine laser micromachined prototype
19.6 mm
13.2
mm
86
Figure 2.14 8-tine and 12-tine prototypes based on optimal solution configurations that used for ex-
vivo experimentation
2.4.2 Deployment in a Tissue Phantom
The purpose of the deployment testing was to provide proof of concept for the deployable
EUS-RFA electrode and to determine how insertion speed affects the insertion force and amount
of deployment in the two electrodes when inserted into a tissue phantom. The deployment testing
was completed with the help of undergraduate students Katherine Reichert and Fariha Azhar who
assisted with the experimental setup and testing.
2.4.2.1 Methods
The test setup, shown in Figure 2.15, includes an Aerotech ECO115SL linear slide,
controlled by a Soloist MP10 controller. A Futek LSB302 single axis load cell was used to
measure the amount of force required during insertion. The electrode was positioned in line with
the load cell that was fixed to the slide. LabView software was used to control the insertion speed
and distances while acquiring data from the load cell.
87
Figure 2.15 Insertion force and deployment experimental setup for early prototypes of the EUS-RFA
electrode
A Polyvinyl Chloride (PVC)-based tissue phantom was prepared using M-F
Manufacturing Company’s regular liquid plastic and softener, mixed with a ratio of 4:1. This ratio
was chosen from literature on flexible needle insertions which reported the phantom elastic
modulus similar to porcine tissue [340]. The tissue phantom was placed on an elevated platform
to align it with the load cell and electrode. The tip of the sheath for the electrode was partially
inserted into the phantom and fixed to the platform. The electrode prototype was initially
compressed and stowed inside the distal end of the sheath and connected in line with the load cell.
As the slide was advanced, the compressed electrode slid through the sheath and deployed into
the tissue phantom. After each trial, the position of the tissue phantom was adjusted on the
platform to prevent overlap in the deployment region with previous trials.
The force and deployment of LMM and L-PBF electrodes were measured at speeds of 1,
3, 5, 7, 10, 25, 35, and 50 mm/s. To minimize noise in the force data, a Savitzky-Golay
88
smoothing filter was applied to the data in MATLAB. The filter successively fits polynomials to
the data using a method of linear least squares. The force for the L-PBF electrode was measured
along 80 mm of travel which was split into three segments. Segment 1 was 36 mm and represents
the friction force associated with the electrode sliding through the sheath. It ends when the central
needle first enters the tissue phantom. Next, segment 2 was 10 mm and ends when the outer tines
leave the sheath and pierce the tissue phantom. Finally, segment 3 was 34 mm and ends when the
electrode was fully inserted. The force for the LMM electrode was measured along 40 mm of
travel which was be split into three segments using a method similar to the L-PBF electrode.
Segment 1 was 20 mm, segment 2 was 4 mm and finally, segment 3 was 16 mm.
To determine the amount of deployment, a digital photo was taken after each trial to
record the spread, or maximum width of the electrode tines. The photos were analyzed using
digital image correlation to determine the average spread at different speeds.
2.4.2.2 Results
The force, position and amount of deployment readings collected for the L-PBF and
LMM electrodes were plotted and analyzed using MATLAB and ImageJ [354] image analysis
software.
The average spread for the L-PBF prototype at the speeds of insertion is shown in Figure
2.16 (left). Error bars denote one standard deviation from the average. The average spread at
insertion speeds of 1, 3, 5, 7, 10, 25, 35, and 50 mm/s were 12.3, 14.5, 12.1, 12.3, 14.8, 12.5,
12.4, 8.6 mm, respectively. The testing shows a maximum spread around 10 mm/s. The insertion
force as a function of the slide position for each speed was then averaged over the five trials for
each of the 8 speeds. Figure 2.16 (right) shows a comparison of the average insertion force profile
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for each of the insertion speeds for the L-PBF electrode. Here, a zero position corresponds to the
point at which the tip of the electrode first penetrates the tissue phantom.
Figure 2.16 (Left) The spread or amount of deployment is shown based on the insertion speed.
(Right) The insertion force profile is separated into three segments showing the friction sliding
through the sheath, the penetration of the central needle, and then the deployment of the outer tines.
Both graphs are for the L-PBF electrode.
Similarly, the average spread for the LMM prototype was plotted against speeds of
insertion and is shown in Figure 2.17 (left). Error bars denote one standard deviation from the
average. The average spread at insertion speeds of 1, 3, 5, 7, 10, 25, 35, and 50 mm/s were 16.7,
19.2, 17.5, 15.5, 19.9, 17.5, 14.9, 14.8 mm/s, respectively. The testing shows a maximum spread
to be around 10 mm/s which was the same as the L-PBF electrode. The insertion force as a
function of the slide position for each speed was averaged over the five trials. Figure 2.17 (right)
shows a comparison of the average insertion force profile for each of the insertion speeds. Here a
zero position corresponds to the point at which the tip of the electrode first penetrates the tissue
phantom.
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Figure 2.17 (Left) The spread or amount of deployment is shown based on the insertion speed.
(Right) The insertion force profile is separated into three segments showing the friction sliding
through the sheath, the penetration of the central needle, and then the deployment of the outer tines.
Both graphs are for the LMM electrode.
2.4.2.3 Discussion
The data collected reveals some interesting trends for future development of deployable
endoscopic electrodes. In addition, important limitations and advantages due to the manufacturing
process became apparent.
Spread was at a maximum around 10 mm/s and appeared to decrease as insertion speed
increases beyond 10 mm/s; see Figure 2.16 (left). The decrease in average spread for the L-PBF
electrode from 10 mm/s to 50 mm/s was nearly 60%. However, in the case of the LMM electrode,
a decrease in spread can be observed from 10 mm/s to 50 mm/s, though the decrease was less
than for the L-PBF electrode. This could be because of the considerable difference in size
between the two prototypes. At speeds lower than 10 mm/s the data did not show a clear trend for
either electrode. For some speeds the error bars were large and were attributed to the complexities
of surface friction and the viscoelasticity of the tissue phantom. There appears to be an ideal
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speed to maximize spread. It was likely that the viscoelasticity and the speed dependent friction
behave differently across the range of speeds tested. For example, at slow speeds the tissue
phantom was able to spring back quickly relative to the insertion speed which affects the overall
deployment.
During the insertion, the tissue phantom deforms as the central needle tip pushes on it.
After a certain point, the tip pierces and ruptures the tissue phantom. The rupture allows the tissue
phantom to spring back toward its original position. At slow speeds, the insertion force remains
low because the tissue phantom was allowed to spring back, reducing the potential energy like a
spring. This spring back behavior was visible twice along insertion force profiles. The first spring
back behavior occurred after deformation from the central needle and the second occurred after
deformation from the outer tines. The spring back is most clearly visible in segment 3 of Figure
2.16 (right) (L-PBF) and Figure 2.17 (right) (LMM), for the 1, 3, 5 and 7 mm/s force profiles. At
high speeds, the tissue phantom was not able to spring back as much before it was pushed further
and further, preventing the force from dropping as much. By segmenting the force profiles, it was
clear that the slope of the force increased as the electrode was inserted. Yet, while the slope of the
force profiles was similar across different speeds, the spring back in the tissue at slow speeds
results in low overall forces. It was interesting to note that the spring back behavior was very
pronounced in the LMM electrode, with the force dropping almost to the force before the initial
penetration. This was likely due to the size and surface roughness of the LMM electrode
compared to the L-PBF electrode. Because the L-PBF electrode had a rougher surface finish and
was larger than the LMM electrode, the friction forces were higher and did not allow the tissue to
spring back as much as for the LMM electrode.
The L-PBF manufacturing processes presents a unique opportunity for fabrication of
electrodes with complex geometry. While the testing results reported were for a 2x scale
prototype, recent work has shown that the electrode may be printed near actual size; see Figure
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2.12. The primary limitation with the L-PBF process was the material selection. Inconel is not
biocompatible due to the high nickel content and lacks the superelastic property exhibited by
Nitinol. In addition, the initial L-PBF prototype after the HIP post processing exhibited spread of
21.4 mm, requiring a significant amount of plastic deformation to be stowed in the sheath. After
being stowed in the sheath, the spread in air without resistance was decreased to 15.4 mm. Plastic
deformation negatively impacts the deployment potential by limiting the spread in air and the
amount of elastic strain energy that can be stored when compressed into the sheath. While other
biocompatible material choices exist for L-PBF, the ideal material with a superelastic behavior to
resist plastic deformation, Nitinol, is in the early stages of process parameter development on
metal printers.
For the LMM electrode, there were some clear advantages in the inherent surface
roughness as compared to the L-PBF electrode, though the forming process to bend the outer
tines following the micromachining process was challenging because of the large strains that the
Nitinol can undergo prior to plastic deformation. A special forming technique with a mold or
another method based on a mandrel would have to be devised to improve the consistency.
Additionally, the shapes of the electrodes would be difficult to customize as compared to the L-
PBF electrode in which the complexity was not a major issue.
2.4.2.4 Conclusions
While some trends can be seen in the data presented, there were several limitations that
should be taken into account. First, the tissue phantom was chosen based on matching elastic
modulus properties for healthy porcine tissue, but other properties such as friction could play a
role in deployment. Second, the tissue phantom was homogeneous. In reality, a tumor consists of
a heterogeneous fibrotic structure. Third, the L-PBF prototype tested was a 2x scale version. With
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these limitations in mind, the authors expect the trends in the spread and insertion force to remain
similar across various elastic modulus values for tissue, heterogeneous compositions, and scales.
In conclusion, it has been shown that the speed of insertion results in minor variations in
the amount of deployment. The insertion speed should be controlled for consistency during use.
Furthermore, while AM was able to produce functional metal parts with complex geometries,
there are limited materials that can be printed at such scales that allow for the compliance needed
to stow and deploy effectively.
2.4.3 Ex-vivo validation of TAM
The purpose of the ex-vivo experimentation was to validate the TAM in predicting the
ablation zone surrounding the electrode. The experimental validation of the TAM was performed
using the 8-tine prototype shown in Figure 2.18; see also Figure 2.14. The prototype was
fabricated using laser micromachining of a Nitinol tube. Next, a machined tube was fitted onto a
beveled stainless-steel wire that serves as the stiff central needle. The shape of the deployed
electrode was defined by the length of the center needle, length/curvature of the outer tines, and
length of the conductive base; see Figure 2.18.
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Figure 2.18 Geometry details for 8-tine deployable electrode prototype for TAM validation, adapted
from [355]
For validation of the TAM, the thermal and electrical material properties of the
cylindrical tissue block were selected to match the material properties of the thermochromic
tissue phantom used in the experimental setup; see Section 2.4.3.1. For the TAM, the electrode
was modeled in its deployed state and its conductive surfaces were applied a constant applied
voltage. The applied voltage levels were 26.4V and 17.5V, corresponding to the high and low
power levels in the tissue phantom model, respectively. The voltage levels were calculated using
a known power level and measured resistance. As described in more detail in Section 2.3.1, the
60°C temperature contour was used to mark the ablation zone for the TAM. The size and shape of
the 60°C contour predicted by the TAM was validated by comparison to the thermochromic
tissue phantom.
Ex-vivo experimentation was performed with assistance from our collaborators at AMI
and undergraduate student Fariha Azhar in development of the experimental setup and testing.
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2.4.3.1 Methods
Tissue phantom models were fabricated to visualize thermal tissue damage [356]. In this
work, a thermochromic tissue phantom for tumor ablation was chosen as the material for
experimental validation of the electrode [357]. The phantom was based on a polyacrylamide gel
that mimics tissue. The thermochromic agent was selected so that the phantom permanently
turned a deep shade of magenta as the temperature rises above 60°C, the same temperature that
causes instantaneous cell death (i.e. ablation or necrotic zone). The ingredients, percent volume
for each ingredient, and fabrication method for the phantom were consistent with Negussie et al.
[357], but the total volume was modified. The thermochromic tissue phantom was molded in
cylinders with a 3.65 cm diameter and a 7.62 cm height. Electrical conductivity of the tissue
phantom was measured, and the other material properties of the tissue phantom were from
Negussie et al. [357]; see Table 2.6.
Table 2.6: Tissue phantom material properties [357]
Material Property Value
Density 1033 kg/m3
Heat Capacity 3939 J/kg∙K
Thermal Conductivity 0.59 W/m∙K
Electrical Conductivity 0.587 S/m
The thermochromic tissue phantom experiments were carried out in a custom tank
containing warmed (~37°C) physiologic (0.9%) saline; see Figure 2.19. Prior to test, the phantom
sample was placed in the saline until warmed to within several degrees of tank temperature. The
outer electrode sheath (with un-deployed electrode inside) was inserted into the phantom to a
depth that ensured the entire electrode was contained in the phantom sample once deployed. After
deploying the electrode, the sheath was retracted to fully expose the electrode while avoiding
electrical contact with it. To monitor temperatures of the phantom during ablation, a custom 7-
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channel thermocouple needle array was inserted at the level of the electrode to a depth of ~1.5
cm, being careful not to contact the electrode tines. An 8th thermocouple was used to monitor tank
temperature. All thermocouples were connected to a multi-channel thermocouple reader (TC-08,
Omega) and output was displayed on a laptop during the experiment. A patient ground pad
(hydrogel coating removed) was submerged and secured to one end of the tank with adhesive
tape.
Figure 2.19 Schematic and photo of the experimental set-up for thermochromic tissue phantom
testing, adapted from [355]
A clinical RF ablation generator (model 950, Aaron) was connected to the electrode and
ground pad. The generator was operated in “blend” mode (357 kHz RF in short bursts delivered
at a rate of 30 kHz). Two different power levels (15 W and 20 W) were tested in ablation trials
lasting 15 minutes. A 2-channel digital oscilloscope (TDS1000, Tektronix) was used to monitor
voltage signals on both the electrode and ground pad (oscilloscope channel grounds connector
together, floating). Additionally, an LCR meter (7600 Model B, QuadTech) was used to measure
impedance (at 357 kHz) between the electrode and ground pad before/after ablation trials.
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Three repetitions of each power level (15W, 20W) in a different tissue phantom sample
were used to determine an average ablation zone for each power level. After conducting the
experiments, each tissue phantom sample was sliced in half, length-wise along the cylinder.
Preliminary studies showed the ablation zone to be axisymmetric so a single cross section through
the center of the cylinder, length-wise, was used to compare the experimental and TAM ablation
zones. The ablation zone, marked by the thermochromic color change, was imaged and analyzed
using ImageJ and compared to TAM results in MATLAB.
Digital images for each trial were scaled and calibrated based on the resolution of the
image and a known distance in the image. An ellipse was fit to each ablation zone to approximate
the size and shape of the ablation zone. The ellipse was fit by matching the normalized second
central moment to those of the ablation zone, using regionprops, a MATLAB function. Similar to
the experimental results, an image of the 60°C contour from the TAM was generated for the low
and high-power setting. The measured major and minor axes of the fit ellipse for each trial were
averaged for the respective power level and compared against the measured axes in the simulated
ablation zones from the TAM.
2.4.3.2 Results
Figure 2.20 and Figure 2.21 show examples of the original and processed images for the
experimental and TAM ablation zones at 15W and 20W, respectively. In these figures, the dark
pink and lime green are the ablation zones. Using image analysis, the white profile was the
ablation zone and the red ellipse was fit to approximate the size and shape by the major and minor
axes. The average major and minor axes of the elliptical fits to ablation zones for the tissue
phantom experiments (3 trials per power level) and the TAM are plotted in Figure 2.22. The error
bars denote the standard deviation for major and minor axes. See discussion below for possible
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reasons for disparities between the experimental and TAM-generated results. A second set of
simulation results are shown to demonstrate the effect of differing electrical conductivity of the
electrode base and tissue phantom as a way to model tissue damage; see “Adapted” in Figure
2.22. These varied conditions account for differences between the ex-vivo and TAM models and
are detailed in the discussion.
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Figure 2.20: Tissue phantom and TAM ablation zone and analyzed images for the 15W power level,
adapted from [355]. (Scale 15.4 px/mm)
Figure 2.21: Tissue phantom and TAM ablation zone and analyzed images for the 20W power level,
adapted from [355]. (Scale 17.4 px/mm)
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Figure 2.22: Comparison of the ablation zone size and shape by the major and minor axis of an
ellipse fitted to each ablation zone with image analysis; see examples in Figure 2.20 and Figure 2.21,
adapted from [355]
2.4.3.3 Discussion
The data collected reveals interesting differences between the tissue phantom and TAM
results. In particular it was clear that the TAM overpredicted the size of the ablation zone in
comparison to the tissue phantom, particularly for the 20W power level. Two primary reasons for
the difference in ablation zone size between the two models were the tissue damage and electrical
conductivity of the electrode.
First, the TAM does not account for tissue damage such as charring or sparking that
occurs at elevated temperatures and voltages, causing decreased electrical conductivity of the
tissue. Figure 2.21 clearly shows damage to the tissue phantom resulting in smaller major and
minor axes at 20W power level than predicted by the TAM. At the 15W power level, the damage
was less apparent than for 20W. In RFA, charring or damage increases the tissue resistance,
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reducing the efficiency of the treatment by limiting the power that can be delivered to the
surrounding tissue. In the TAM, not accounting for increased tissue resistance allowed for
elevated temperatures near the electrode, resulting in larger thermal conduction than the ex-vivo
experiment. One method to correct this in the TAM was to apply a temperature dependent
thermal conductivity, however, the computational cost was high to recompute the electrical
conductivity and electric field throughout the simulation. A simplified approach was to reduce the
constant electrical conductivity of the entire tissue. Reducing the electrical conductivity of the
tissue in the TAM by fifty percent for the high-power level and five percent for the low power
levels, the TAM closely matches the ex-vivo model.
A second reason for the difference in the ablation zone size was the heating near the
proximal end of the electrode. In the TAM, the initial heating in the tissue occurs near the tips of
the outer tines and central needle, and proximal end of electrode. In some of the experimental
trials, heating near the proximal electrode end was inconsistent, showing signs of reduced
electrical conductivity and increased heat sink effects. The inconsistencies were suspected to be
caused by saline solution entering the tissue phantom along the electrode path.
The difference in conductivity between the tissue phantom and TAM at the proximal end
of the electrode was best seen at the low power level where tissue damage was less significant
than the high power level; see Figure 2.20. In the TAM, heating at the proximal end of the
electrode causes a broader and longer ablation region near the bottom of the profile. In contrast,
the ablation zone in the tissue phantom comes to a narrow point. The result was that the minor
axis of the tissue phantom and TAM were similar, while the major axis was not; see Figure 2.22.
By reducing the electrical conductivity of the electrode base by 4 orders of magnitude, mimicking
the ex-vivo model, the major and minor axes become very similar between the tissue phantom
and TAM. By changing both the electrical conductivity of the tissue phantom and the
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conductivity of the electrode base, we see good agreement between the TAM and tissue phantom;
see Figure 2.22.
There were a few limitations that exist in the thermal ablation and tissue phantom
models. The TAM assumes constant material properties while properties such as electrical and
thermal conductivity were known to be temperature dependent. In addition, the TAM assumes the
initial temperature was 37°C and was constant at the exterior tissue surface. The TAM also used a
constant applied voltage, a simplification of the actual pulsed RF voltage, based on the root mean
square voltage. As mentioned previously, the TAM also did not directly account for tissue
damage or charring at elevated temperatures. Though some error between the experiment and
TAM was expected by not capturing the full physics of tissue ablation, the purpose of the TAM
was a quick approximation of the ablation zone for the formal optimization with a genetic
algorithm [339].
For the tissue phantom testing, the tank heater and controller helped to maintain a
consistent starting temperature and tissue phantom boundary temperature, though there were
small variations during testing. In addition, as described previously, in some cases the tissue
phantom did not change color in the tissue phantom experiment, which were not included in the
data analysis. This result was attributed to short circuiting when the saline solution entered the
tissue phantom and/or thermocouple array contacted the outer electrode tines during testing.
2.4.3.4 Conclusions
In conclusion, the tissue phantom experiment showed that the TAM overpredicted the
ablation zone size, likely due to not accounting for tissue damage and inaccurate prediction of
heating at proximal end of electrode. By adapting the conditions of the TAM to match the ex-vivo
experiment, the TAM was validated to approximate the ablation zone. Future work should
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consider improving the TAM to consider tissue charring or damage at elevated temperatures. By
considering a variety of power levels, a modified electrical conductivity factor for the TAM could
be determined based on the maximum temperature or applied voltage during the simulation.
2.5 Closing Remarks
Optimization of the electrode geometry with a TAM can be a valuable tool when
designing specially shaped ablation electrodes. The potential to develop unique ablation
electrodes, based on a specific target geometry, or a specific tumor, can help to customize patient
care. In this work, the target shape of the ablation zone was spherical, however, the desired
ablation zone may correlate to a patient specific tumor shape. In addition, more complex models
may be developed which incorporate patient specific blood vessel structures to optimize the
electrode while considering heat sink effects. With the ex-vivo validation completed, this method
of systematic design through optimization can be used to design specially shaped electrodes for
patient specific cases. These results provide evidence toward the benefits of combining
optimization with computational modeling and its potential impact on personalized medicine. In
the next chapter, a similar methodology of developing efficient computational models, paired
with optimization, for personalized or customized applications of AM is addressed.
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Chapter 3 Unit Cell Generation through Ground Structure Topology
Optimization
The focus of this chapter is the theory behind unit cell generation through a ground
structure topology optimization method. The chapter includes a brief summary of the relevant
background and motivation as well as descriptions of the approach. The approach includes
development of a 3D ground structure model, a formal description of the optimization, and
solution of several benchmark problems to validate the approach. The following objectives and
sub tasks are addressed in this chapter:
2. Develop three-dimensional ground structure topology optimization for unit cell DfAM
2.1. Develop 3D-parametric ground structure model
2.2. Establish systematic optimization method for unit cell generation
2.2.1. Generate set of potential objective functions and constraints
2.2.2. Quantify manufacturing considerations for metal AM
2.3. Validate optimization with benchmark problems
3.1 Background and Motivation
At the core, AM processes are based on a layer-by-layer approach for building up a part,
rather than subtractively removing material. With its layer-by-layer build process, AM offers
access to the entire build volume during fabrication. This unique advantage of AM over other
manufacturing technologies enables design complexity to be easily incorporated into components
and has led to increased interest and rapid development of AM processes, particularly in the past
decade [91,337].
Lattice structures are one example of design complexity enabled by AM, and are an
increasingly active area of research for academia and industry [160,228,250]. Whether for
lightweighting, improving energy absorption capabilities, adding functionality, or improving
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biocompatibility, lattice structures are being explored extensively for designing a component’s
mesostructure. A non-stochastic lattice structure is defined as a patterned structure based on a
repeated unit cell topology. Though the freedom of AM allows for complex and intricate lattice
structures to be manufactured, the challenge remains to be able to effectively design components
with such structures in mind.
One challenge with lattice structures is that their behavior is not well understood and are
often used without specific consideration for why a particular unit cell topology was chosen over
another. The addition of lattice structures complicates traditional designs because the unit cell
topology changes the effective bulk mechanical properties from the material properties of the
base material. For example, the assumption of isotropic behavior common to many metals is no
longer a valid assumption when considering the macroscale properties of a metal lattice structure.
Commercial software for AM makes it easy to lightweight a part by simply adding a lattice, but
does not provide any guidance for which lattice will be most beneficial for the component.
Similar to the need for topology optimization methods for overall geometric design of a
component, systematic design methods are needed for the appropriate design of lattice structures.
Topology optimization is the process of determining material placement within a design space to
achieve a specific goal or objective [282]. Chen et al. [133] describe the use of continuum-based
topology optimization to design novel isotropic lattice structures that exhibit high stiffness. Watts
et al. [313] and Watts and Tortorelli [314] have published methods for reducing computational
cost of lattice structure design through surrogate models and geometric projection methods.
While there are numerous research groups exploring methods for metamaterial design, or the
design of materials with properties not found in traditional materials (i.e. multi-functional,
negative Poisson’s ratio, self-healing), the direct incorporation of AM constraints into the
optimization has not been addressed. Liu et al. [295] review of the current state of topology
optimization and AM and they describe a need for further incorporation of design for AM
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(DfAM) limitations in topology design. In a review of structural optimization for AM, Plocher
and Panesar [336] cited integration of DfAM as a major bottleneck for a streamlined design
process.
DfAM constraints include overhangs and support structures, powder removal, and
minimum feature size. In AM the build angle is defined as the angle between the build plate and
the surface of the component being built. If this angle is less than 35 degrees, additional structures
are generally required to be built underneath the part to support it by anchoring it to the build
plate, otherwise, significant warping occurs. These support structures are frequently used in AM
components but become nearly impossible to remove for intricate and delicate lattice structures.
Overhangs are defined as a region where part extends out horizontally. If the horizontal region
reconnects after a short distance (~5 mm) it can form a bridge anchoring the material. Otherwise,
overhanging regions typically cause failures do to warping and overheating. For powder-based
AM processes, anywhere that material is not solidified, there is excess powder to be removed. For
lattice structures, one common challenge is avoiding trapping powder and the ability to remove
all the powder from the complex maze of channels that form in the lattice. The minimum feature
size is the smallest features that can be built by the AM process. For L-PBF, this is typically
based on the spot size of the laser. As lattice structures are typically very intricate, they quickly
push the limits of the minimum feature size or resolution of the AM machine. Without
incorporation of DfAM considerations into topology optimization methods, the optimized
structures may not be manufacturable or lose much of the enhancement gained from the
optimization after the component is built.
One method of topology optimization is ground structure topology optimization (GSTO),
a discretized approach for determining for an optimized topology. GSTO discretizes the design
space with a grid of nodes and connects the nodes using truss, beam, or frame elements. By
applying periodic boundary conditions to the ground structure model and considering
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homogenization techniques, it is proposed that GSTO can be used to design unit cell topologies
for a lattice structure with tailored properties. As compared to continuum-based topology
optimization, GSTO generally offers a reduced number of variables for lower computational cost.
It is particularly well-suited to AM unit cell topology design because many lattice structures are
based on an arrangement of trusses and nodes, and DfAM considerations can be directly
incorporated. While this approach has been used in the past for mesoscale design, there is a need
to expand such an approach to include additional objective functions and DfAM considerations.
With the need for systematic approaches for the design of unit cell topologies, this
chapter describes the additive lattice topology optimization (ALTO) approach. ALTO is a 3D
GSTO method for generation of novel unit cell topologies tailored for their application. The
GSTO serves as a systematic optimization method for generating novel lattice structures, offering
a fast simulation with fewer design variables than continuum-based approaches. ALTO includes
direct incorporation of various DfAM considerations such as overhangs and support structures,
powder removal, and minimum feature size. Through a variety of optimization objective,
constraint, and penalty functions, ALTO offers a systematic approach to unit cell topology design
that can be applied to a wide range of applications.
3.2 Additive Lattice Topology Optimization (ALTO)
The generation of novel unit cell topologies with DfAM and optimization were
accomplished through ALTO. First a ground structure model was applied to represent the unit
cell topology. Next, the unit cell topology was tailored for the application with multi-objective
optimization. After optimization, the solution was interpreted and verified for manufacturability.
Next the solution may be recreated in design software and ultimately fabricated with AM. An
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overview of the process is shown in Figure 3.1. MATLAB [344] code for the 3D version of
ALTO is provided in Appendix B.
Figure 3.1 Overview of Additive Lattice Topology Optimization (ALTO) to generate optimized unit
cell topologies for lattices fabricated with AM
3.2.1 Ground Structure Model
GSTO methods discretize a design space into a set of nodes and then connects nodes with
truss, beam, or frame elements. In a full ground structure, each node was connected to every other
node by an element, forming a highly redundant system. The ground structure is evaluated using
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standard finite element analysis. For optimization, the cross-sectional areas of each element are
the design variables, which are appropriately resized to minimize the objective function(s).
The ground structure approach for topology optimization is an alternative to the so-called
continuum-based topology optimization where the design space is discretized into finite elements
and the relative element densities are the design variables. The earliest references to GSTO dates
back to the 1960’s with work by Dorn et al. [286]. The methodology and formulation have been
described by Bendsoe and Sigmund [281] and Ohsaki [287] in their textbooks on topology
optimization. GSTO has since advanced to tackle a number of different challenges including
multi-material design [288], reliability [289], arbitrary 2D and 3D domains [290,291], adding and
removing nodes or elements [292], buckling and overlap [293], sizing non-patterned lattice
structures [159,224–227], and other AM considerations [294–296].
In particular, GSTO has been researched previously by Sigmund [297,298] for
metamaterial design of structures with homogenized material properties. Since the work by
Sigmund in the late 1990’s, others have produced similar work, generally using the approach to
develop novel materials matching a specific constitutive matrix. Guth et al.[299,300] developed
2D and 3D structures for maximizing elastic and thermal constitutive properties. Others have
expanded on the work by exploring penalization for buckling [302], not limiting the design
volume to a predetermined size shape [301], or for minimum compliance applications [303,304].
Despite extensive research in the field, for many years one of the biggest challenges was the
ability to manufacture such structures.
With the recent increased interest in AM and improved fabrication resolution, optimized
metamaterial design is seeing renewed interest. The ground structure approach is well suited for
lattice structure design because it is a good representation of common lattice structures which
consist of struts and nodes, and can reduce the number of design variables compared to the
continuum-based approach.
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In this work, the ground structure was formed using truss elements which carry only axial
loads. The use of truss elements assumes the elements were connected by pin joints at the nodes.
The ground structure formulation is valid for linear geometric deformations. The design space
was assumed to be cubic with an equal number of nodes along each edge. The design variables
for the structure were the element cross-sectional areas. By adjusting the cross-sectional area of
each element, 𝐴𝑒, the optimization algorithm identified elements that were non-critical to
improving the objective function(s). The upper and lower bounds on the design variables were
𝐴𝑢𝑝𝑝𝑒𝑟 and 𝐴𝑙𝑜𝑤𝑒𝑟, respectively. As opposed to an element deletion technique for the ground
structure model, 𝐴𝑒 was allowed to approach a minimum, 𝐴𝑙𝑜𝑤𝑒𝑟, which was set to a small value
several orders of magnitude lower than the other element areas, yielding an insignificant impact
on the structure. In other words, as the cross-sectional area approaches 𝐴𝑙𝑜𝑤𝑒𝑟, that element has
negligible impact on the performance of the structure determined by the objective function for
optimization. As opposed to simply a sizing optimization, as 𝐴𝑒 approaches 𝐴𝑙𝑜𝑤𝑒𝑟, the topology
changes because elements become negligible and can later be discarded from the final structure.
3.2.2 Optimization Algorithms
Two optimization algorithms have been used for optimization of the ground structure
model in ALTO: fmincon, and gradient-based algorithm, and Borg, an evolutionary algorithm. A
brief description of how each algorithm works is explained in the sections that follow. In these
sections, general advantages and disadvantages of each algorithm are mentioned. For additional
information on the optimization algorithms and findings from this research see the following
sections:
• Section 1.1.6.1 – general comparison of gradient-based and evolutionary
algorithms,
• Sections 4.4 and 5.5 – discussion of fmincon and Borg for specific case studies,
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• Sections 7.1 and Section 7.2 – summary and primary conclusions.
3.2.2.1 fmincon: A Gradient-based Optimization Algorithm
MATLAB’s fmincon is a gradient-based nonlinear programming solver [358]. In cases
where the gradient of the system is not given, fmincon approximates the gradient numerically by
gradually adjusting each design variable to determine an appropriate search direction. The
convergence criteria for fmincon is based on when the objective values are no longer decreasing
to within the tolerance on the design variables. As with other gradient-based algorithms, fmincon
generally requires fewer function evaluations, has a clear convergence criterion, and is a faster
method than non-gradient based algorithms because it searches in a single direction based on the
gradient approximation. The downsides to gradient-based algorithms are that they are highly
dependent on the initial values for the design variables, easily become trapped in local-minima,
and can be challenging for considering multi-objective optimizations. For gradient-based
algorithms, multi-objective problems are generally reduced to a single objective using a weighted
sum of the objectives being considered. As a result, the optimization searches for a single solution
which is highly dependent on an appropriate selection of the weight values.
For a gradient-based optimization algorithm, such as fmincon, a description of the
optimization problem is given in Eqn. (3.1)-(3.7),
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓(𝑊, 𝑃)
𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝐴𝑒 − 𝐴𝑙𝑜𝑤𝑒𝑟 ≥ 0 (3.1)
𝐴𝑒 − 𝐴𝑢𝑝𝑝𝑒𝑟 ≤ 0 (3.2)
𝐴𝑒 = 𝐴𝑙𝑜𝑤𝑒𝑟 𝑓𝑜𝑟 𝐴𝑒 < 𝐴𝑟𝑒𝑙 (3.3)
𝑊ℎ𝑒𝑟𝑒 𝑓 = (𝑊)(𝑃) (3.4)
𝑊 =∑𝑐𝑖𝑚𝑖𝑤𝑖(𝑨, 𝑳), 𝑓𝑜𝑟 𝑖 = 1…𝑛𝑜𝑏𝑗
𝑛
1
(3.5)
112
1 = ∑𝑐𝑖
𝑛
1
, 𝑓𝑜𝑟 𝑖 = 1…𝑛𝑜𝑏𝑗 (3.6)
𝑃 =∑𝑝𝑗(𝑨)
𝑛𝑝
𝑗
, 𝑓𝑜𝑟 𝑗 = 1…𝑛𝑝 (3.7)
where 𝐴𝑟𝑒𝑙 was a DfAM consideration called the relevant area limit discussed in Sections
3.2.3.1.5 and 0, the overall objective, 𝑓, was based on the combined objective value, 𝑊, and
combined penalty value, 𝑃. A weighted sum was used to calculate 𝑊, where 𝑤𝑖 were the
individual objective scores, 𝑐𝑖 were the corresponding weight values, and 𝑚𝑖 was either 1 or -1 to
minimize or maximize an objective, respectively. The number of objectives was 𝑛𝑜𝑏𝑗 and 𝑨, 𝑳
were vectors of the element cross-sectional areas and lengths, respectively. The combined penalty
value, 𝑃, was calculated by adding the individual penalty scores, 𝑝𝑖, for 𝑛𝑝 penalty functions.
Penalties are used to negatively impact the objective function score, making the design less
economical than non-penalized designs, without removing it from the solution space. Additional
information on the a DfAM penalty function for gradient-based optimization is given in Section
3.2.3.3.2.
3.2.2.2 Borg: A Multi-objective Evolutionary Algorithm
Borg is a multi-objective evolutionary algorithm (MOEA). Evolutionary algorithms are
heuristic approaches which have little or no information about the problem while iteratively
trying to improve the design. They evaluate a population of designs and then try to form new
designs by recombination and mutation of the design variables from the best performing designs.
Evolutionary algorithms tend to excel where gradient-based algorithms struggle: multi-objective
solutions, finding global minimums, and less dependence on a single initialization of the design
variables. They accomplish this by evaluating designs throughout the design space and taking the
designs that perform the best in each of the objectives into consideration. They maintain a set of
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good designs making it easier to search multiple directions and develop a set of Pareto-optimal
solutions. A Pareto-optimal solution is one for which no other design is better than the current
design in all objectives. The downside of evolutionary algorithms is that they are generally slow
because they are exploring multiple search directions and have no gradient-type information to
direct their search.
Borg was developed by Hadka and Reed [258] to address issues encountered with
traditional MOEA’s. In their analysis of MOEA’s, the pareto front is found to be typically only
developed in a very localized region of the solution space. This localized resolution of the
solution space is referred to as dominance resistance and ultimately leads to deterioration of the
pareto front. Deterioration is the case in which the best designs at the current iteration are actually
worse than designs simulated during previous iterations. Borg, based on 𝜖-MOEA [359],
incorporates an 𝜖-box dominance, 𝜖-progress, adaptive population sizing, and six different
recombination methods, where 𝜖 is a threshold or tolerance on the objective function values.
Through the incorporation of these features, Borg preserves the diversity of designs across the
pareto front and improves search efficiency compared to other MOEAs [258]. In Borg, the default
termination criterion is the number of design evaluations.
For an evolutionary algorithm , such as Borg, a description of the optimization problem is
given in Eqn. (3.8)-(3.12),
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 (𝑊1,𝑊2, … ,𝑊𝑖) 𝑓𝑜𝑟 𝑖 = 1…𝑛𝑜𝑏𝑗
𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝐴𝑒 − 𝐴𝑙𝑜𝑤𝑒𝑟 ≥ 0 (3.8)
𝐴𝑒 − 𝐴𝑢𝑝𝑝𝑒𝑟 ≤ 0 (3.9)
𝐴𝑒 = 𝐴𝑙𝑜𝑤𝑒𝑟 𝑓𝑜𝑟 𝐴𝑒 < 𝐴𝑟𝑒𝑙 (3.10)
𝑊ℎ𝑒𝑟𝑒 𝑊𝑖 = 𝑚𝑖(𝑠𝑖𝑃 + 𝑤𝑖(𝑨, 𝑳)) (3.11)
𝑃 =∑𝑝𝑗(𝑨)
𝑛𝑝
𝑗
, 𝑓𝑜𝑟 𝑗 = 1…𝑛𝑝 (3.12)
114
where, 𝑊𝑖 was the penalized objective score, 𝑤𝑖 was the raw or individual objective score, 𝐴𝑟𝑒𝑙
was a DfAM consideration called the relevant area limit discussed in Sections 3.2.3.1.5 and 0.
Without the need to combine all the objectives into a single value using the weighted sum, the
penalty was applied individually to each objective and may be scaled with 𝑠𝑖 based on the how
heavily each objective should be penalized. As mentioned previously, 𝑚𝑖 was either 1 or -1 to
minimize or maximize an objective, respectively. The number of objectives was 𝑛𝑜𝑏𝑗 and 𝑨, 𝑳
were vectors of the element cross-sectional areas and lengths, respectively. Penalty functions
were utilized to negatively impact the objective function values, making the design less
economical than non-penalized designs, without removing the design from the solution space.
Additional information on penalty functions used with Borg is given in Section 3.2.3.3.1.
3.2.3 ALTO Objectives, Constraints, and Penalties
The purpose of the optimization was to minimize the objective function(s) subject to the
constraints. First, designs were evaluated for feasibility based on the optimization constraints.
Each design that satisfies all constraints was then evaluated for each objective function. Next,
each design was evaluated against the penalty functions. Penalties were distinguished from
constraints as they negatively impact the objective function values but do not render designs
infeasible and were applied after evaluation of the objective functions. As such, a design that was
penalized may be part of the solution space, however, designs which do not meet the constraints
were not. Constraints were applied to the design variables and restrict the range of possible values
for the design variables.
As the purpose of ALTO was an application-agnostic method for design and optimization
of unit cell topologies, a library of objective, constraint, and penalty functions were developed to
include a wide variety of applications. Based on the intended application, the user selects
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appropriate functions from the library to customize the optimization. In addition, the formulation
of ALTO with an evolutionary algorithm was designed to be robust for accepting additional
objective, constraint, or penalty functions. In general, the functions are described in the context of
L-PBF but the approach can be tailored to a variety of AM processes. A summary of the current
options within ALTO is presented in Table 3.1. Each of the objective, constraint, and penalty
functions are described in detail in the following subsections.
Table 3.1 Summary of ALTO objective, constraint, and penalty functions for unit cell generation and
their applicability to DfAM
ALTO Purpose DfAM
Objectives (Maximize or Minimize…)
Strain Energy Strain energy subject to a loading condition
Thermal Conductance Thermal conductance between two sets of nodes
Homogenization Relative error from a specified target constitutive matrix
Unit Cell Volume Total volume or relative error from a specified volume fraction Yes
Relevant Elements Number of elements within the unit cell to control complexity Yes
Powder Removability Factor (PRF) Proximity of elements with respect to other elements Yes
Optimization Constraints (Constrain design space for…)
Relevant Cross- Sectional Area Minimum feature size for DfAM Yes
Cross-Sectional Area Limits Upper and lower bounds on element area
Volume Fraction Allowable material volume for design space
Ground Structure Restrictions (Convert elements to non-design elements for…)
Build Angle Elements within a specified DfAM build angle range Yes
Element Length Elements within a specified length range
Surface Elements Elements on unit cell surface
Interconnectivity Nodes Connectivity to neighboring unit cells
Symmetry Element symmetry across any/all orthogonal midplanes
Penalties (Discourage solutions …)
Overlap Containing overlapping elements Yes
Connectivity Without connection to neighboring unit cells
Unsupported‒S Not self-supported for AM Yes
Unsupported‒N Not supported for AM by neighboring unit cells Yes
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3.2.3.1 Objectives
Six different objective functions have been included for calculation in ALTO and are
described in the following paragraphs. The objective function calculations are for strain energy,
thermal conductance, homogenization, volume, the number of relevant elements, and powder
removability.
3.2.3.1.1 Strain Energy
As with most topology optimization approaches, the structural compliance, or
proportionally the strain energy, was calculated and represents the stored elastic energy in the
structure. The purpose of the minimization of strain energy was to maximize the stiffness of the
structure subject to a specific loading condition. For structural loading, the stiffness matrix, �̃�,
was a symmetric matrix formed for the structure and then loads and boundary conditions were
applied to nodes on the structure. The quasi-static displacements of each node were computed
using Eqn. (3.13),
�̃�𝒖 = 𝑭 (3.13)
where 𝑭 and 𝒖 were vectors of size [𝑁𝑑𝑜𝑓𝑥1] and �̃� was a matrix of size [𝑁𝑑𝑜𝑓𝑥𝑁𝑑𝑜𝑓] where
𝑁𝑑𝑜𝑓 was the number of degrees of freedom of the system. The vector of externally applied
forces corresponding to each of the degrees of freedom for the nodes was represented by 𝑭. The
vector of displacements for each degree of freedom was represented by 𝒖. The global stiffness
matrix, �̃�, was a function of 𝐴𝑒, where 𝑒 = 1…𝑀 and 𝑀 was the total number of elements in the
system. In the ground structure model, 𝑭 and the Young’s modulus, 𝐸, are assumed constant and
the formulation is for linear elastic and small deformations with the nodes assumed to act like pin
joints as explained previously.
117
The work done on the system by externally applied loads was proportional to the strain
energy, 𝑆𝐸. The strain energy was given by Eqn. (3.14). The strain energy objective function,
𝑤𝑆𝐸, is simplified by ignoring the 1
2 because it does not affect the optimization; see Eqn. (3.15).
SE =1
2𝒖�̃�𝒖 (3.14)
wSE = 𝒖𝑇�̃�𝒖 (3.15)
For gradient-based optimization algorithms, the strain energy was normalized to assist
with the weighted sum combination of multiple objectives. In this situation, the objective function
was modified to Eqn.(3.16),
𝑤𝑆𝐸̅̅̅̅ =𝑤𝑆𝐸𝑚𝑖𝑛 −𝑤𝑆𝐸
𝑤𝑆𝐸𝑚𝑖𝑛 −𝑤𝑆𝐸
𝑚𝑎𝑥 (3.16)
where 𝑤𝑆𝐸𝑚𝑖𝑛 and 𝑤𝑆𝐸
𝑚𝑎𝑥 are the strain energy calculated with all the element areas set to 𝐴𝑢𝑝𝑝𝑒𝑟
and 𝐴𝑙𝑜𝑤𝑒𝑟, respectively.
3.2.3.1.2 Thermal Conductance
The purpose of the thermal conductivity objective was to maximize or minimize the
thermal conductance between two sets of nodes within the structure. The thermal conductance
was defined in terms of conductance from one node, or set of nodes, 𝑁1, to another node, or set of
nodes, 𝑁2. Temperatures, 𝑇1 and 𝑇2, were assigned to the 𝑁1 and 𝑁2, respectively. The thermal
resistance of each truss element within the unit cell was calculated using Eqn. (3.17),
𝑅𝑒 =𝐿𝑒𝑘𝐴𝑒
(3.17)
118
where 𝑅𝑒 was the element thermal resistance, 𝐿𝑒 was the element length, 𝐴𝑒 was the element
cross-sectional area, and 𝑘 was the material thermal conductivity.
Next, using an analog to Kirchhoff’s voltage and current laws for solving an electrical
circuit, a linear system of equations was developed to calculate the temperature at each node and
the heat flux through each element using 𝑅𝑒, 𝑇1, 𝑇2, and the connectivity of the elements in the
unit cell. By calculating the heat flux through each element, the effective heat flux, 𝑞𝑒𝑓𝑓, across
the unit cell was calculated from the total heat flux leaving 𝑁1 or entering 𝑁2. Finally, the thermal
conductance objective was calculated using Eqn. (3.18)
𝑤𝑇𝐶 =𝑞𝑒𝑓𝑓
𝑇1 − 𝑇2 (3.18)
where 𝑤𝑇𝐶 was the thermal conductance objective function from 𝑁1 to 𝑁2. The thermal
conductance may be maximized or minimized between 𝑁1 and 𝑁2, depending on the application;
see an example 2D schematic in Figure 3.2.
Figure 3.2 Example of traditional thermal circuits and analogy to the ground structure formulation
for calculating the unit cell thermal conductance
119
In the thermal conductance calculation described above, two sets of nodes must be
defined and given specific temperatures. As a slight variation, a modified thermal conductance
formulation was available to maximize or minimize heat conduction from a set of nodes where
heat was supplied, to a heat sink node(s). In the modified formulation, heat was supplied to nodes
within the design space, similar to a current source within a circuit. As the input heat drives the
heat flux through the system, the average temperature at the heat input nodes, �̅�, determines the
effective thermal conductance. If the average nodal temperature was high, the thermal
conductance was low, because high nodal temperatures were required to drive the heat flux
through a high thermal resistance. On the other hand, if the average nodal temperature was low,
the thermal conductance was high because low temperatures were sufficient to drive the heat flux
through a low thermal resistance. The modified thermal conductance objective is shown in Eqn.
(3.19),
𝑤𝑀𝑇𝐶 =𝑞𝑒𝑓𝑓
�̅� (3.19)
where 𝑤𝑀𝑇𝐶 was the modified thermal conductance, 𝑞𝑒𝑓𝑓 was the effective heat flux through the
unit cell, and �̅� was the average node temperature of the nodes where heat was supplied.
For gradient-based optimization, the thermal conductance objective function was
normalized as shown in Eqn. (3.20),
𝑤𝑇𝐶̅̅̅̅ =𝑤𝑇𝐶𝑚𝑖𝑛 −𝑤𝑇𝐶
𝑤𝑇𝐶𝑚𝑖𝑛 −𝑤𝑇𝐶
𝑚𝑎𝑥 (3.20)
where 𝑤𝑇𝐶𝑚𝑖𝑛and 𝑤𝑇𝐶
𝑚𝑎𝑥e calculated with all the element areas set to 𝐴𝑙𝑜𝑤𝑒𝑟 and 𝐴𝑢𝑝𝑝𝑒𝑟,
respectively. Similarly, the objective function for 𝑤𝑀𝑇𝐶 may be modified to form the normalized
equivalent 𝑤𝑀𝑇𝐶̅̅ ̅̅ ̅̅ ̅.
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3.2.3.1.3 Homogenization
The purpose of the homogenization objective was to design a lattice structure with
tailored compliance or designer-specified mechanical properties. For a given material, the
relationship between stresses and strains was defined by the elastic constitutive matrix, �̃�.
Homogenization is a method for approximating �̃�, by calculating the homogenized elastic
constitutive matrix, �̃�𝐻, which is based on averaging the properties of a representative volume
element (RVE). Throughout this dissertation, the term “constitutive matrix” refers to the elastic
constitutive matrix. In the case of lattice structures, homogenization means taking the potentially
complex behavior of the unit cell topology and averaging it across an entire body or patterned
lattice structure.
The homogenization approach was based on a formulation developed by Sigmund and is
summarized as follows; for additional details; see references [297,298]. For 2D, the approach
requires three test strain cases to be applied to the unit cell (six for 3D). The test strain cases were
applied in each primary orthogonal direction (X and Y), and in the shear direction (XY). As the
2D case can be readily demonstrated in schematics, an example of the three test strain cases for
2D are shown in Figure 3.3. For expansion to 3D, three test strain cases were applied in the
primary orthogonal directions (X, Y, Z), followed by three test strain cases in the shear directions
(XY, YZ, XZ). The test strain cases were applied using periodic boundary conditions on the unit
cell to ensure edge compatibility with neighboring unit cells in the patterned lattice structure. The
periodic boundary conditions were based on work by van der Sluis et al. [360]. The approach is
valid for calculating the homogenized constitutive matrix for small deformations.
121
Figure 3.3 Schematic for three test strain cases to an RVE for calculation of the homogenized
constitutive matrix, adapted from [298]
Using the standard finite element method (see Section 3.2.3.1.1) to solve each of the test
strain cases, the resulting global displacement vector, 𝒖, was solved. Based on the deformations
and element stiffnesses, averaged mechanical properties were calculated for the unit cell from the
element mutual energies using Eqns. (3.21)-(3.22),
𝐸𝑘𝑙𝐻 =∑𝑄𝑘𝑙
𝑒
𝑀
𝑒=1
(3.21)
𝑄𝑘𝑙𝑒 =
1
𝑌(𝒅0𝑒
𝑘 − 𝒅𝑒𝑘)𝑇�̃�𝑒(𝒅0𝑒
𝑙 − 𝒅𝑒𝑙 ), 𝑓𝑜𝑟 {
𝑒 = 1,… ,𝑀 2𝐷: 𝑘, 𝑙 = 1,2,3 3𝐷: 𝑘, 𝑙 = 1,… ,6
} (3.22)
where �̃�𝐻 was the homogenized constitutive matrix, �̃�𝑒 was the element mutual energy, 𝒅0𝑒𝑘 was
the induced element displacement vector for each test strain case, 𝒅𝑒𝑘 was the applied element
displacement vector for each test strain case, �̃�𝑒 was the element stiffness matrix (4x4 for 2D, or
6x6 for 3D), 𝑀 was the number of elements, and 𝑌 was the characteristic length of the RVE. In
�̃�𝐻, the terms 𝐸11𝐻 , 𝐸22
𝐻 , 𝐸33𝐻 are the effective elastic modulus values for the homogenized
material, heretofore referenced as “effective modulus” values.
After calculating the constitutive matrix, the homogenization objective function was
calculated using Eqn. (3.23),
X
Y
122
𝑤𝐻 =∑|𝐸𝑘𝑙
∗ − 𝐸𝑘𝑙𝐻 |
𝐸𝑘𝑙∗ , 𝑓𝑜𝑟 {
2𝐷: 𝑘, 𝑙 = 1,2,3 3𝐷: 𝑘, 𝑙 = 1,… ,6
} (3.23)
where �̃�∗ was the desired or target homogenized constitutive matrix. This objective function,
Eqn. (3.23), represented the total relative error between the target and calculated homogenized
constitutive matrices.
3.2.3.1.4 Unit Cell Volume
The purpose of the unit cell volume objective was to match a target unit cell volume
fraction or minimize/maximize the total unit cell volume. For matching a target volume fraction,
𝑉𝑓∗, the objective function value was the relative error between the target and calculated volume
fraction; see Eqn. (3.24),
𝑤𝑉𝐹 =|𝑉𝑓
∗ −𝑨𝑇𝑳𝑉𝑢𝑐
|
𝑉𝑓∗
(3.24)
where 𝑤𝑉𝐹 was the volume fraction objective function and 𝑉𝑢𝑐 was the total unit cell volume. On
the other hand, instead of using a target volume fraction, the total volume of material within the
unit cell can be minimized or maximized; see the volume objective function, 𝑤𝑉, in Eqn. (3.25).
In both methods, the volume calculation does not account for material overlap from neighboring
or intersecting elements.
𝑤𝑉 = 𝑨𝑇𝑳 (3.25)
The unit cell volume objective was indirectly used in the context of DfAM because the
volume of material within the unit cell can affect manufacturability. While in general the
complexity or amount of material is not a major concern for AM, the length scale for designing
unit cell topology quickly approaches the minimum feature size, making it difficult to
123
individually print each intended strut. For example, high volume fraction unit cells are more
likely to lead to trapped powder than low volume fraction unit cells [245]. In addition,
minimizing or matching a specific volume fraction indirectly reduces complexity by encouraging
designs with low volume. In the subsequent section, a more direct approach than the volume
fraction objective for reducing complexity is addressed.
3.2.3.1.5 Number of Relevant Elements
In DfAM, component design complexity or intricacy is often less of a consideration than
with subtractive manufacturing processes because, arguably, design complexity is free. However,
when considering DfAM for unit cells, the unit cell intricacy can play a role in the
manufacturability because the unit cells are small and the elements or features of the unit cell are
often at or near the minimum feature resolution of the manufacturing process. Depending on the
laser scan strategy of the lattice structures, added complexity can also increase build time due to a
large number of short infill and contour laser passes. Asadpoure et al. [294] address the same
issue using a smooth and differentiable objective function for gradient-based optimization
approaches.
The purpose of the number of relevant elements objective was to reduce the design
complexity, encouraging fewer elements or struts within the unit cell solution. In this text,
“relevant element(s)”, means elements that have a cross-sectional area larger than 𝐴𝑟𝑒𝑙. Based on
the relevant cross-sectional area limit, 𝐴𝑟𝑒𝑙 (see Section 3.2.3.2.1), and by applying Eqn. (3.3),
any element cross-sectional areas that fell below 𝐴𝑟𝑒𝑙 were temporarily set to 𝐴𝑙𝑜𝑤𝑒𝑟 for the
current iteration. The relevant elements objective, 𝑤𝑅𝐸, was simply the current number of
elements with a cross-sectional area larger than 𝐴𝑟𝑒𝑙; see Eqn. (3.26).
𝑤𝑅𝐸 = 𝑐𝑜𝑢𝑛𝑡(𝐴𝑒 > 𝐴𝑟𝑒𝑙) (3.26)
124
For DfAM, reducing the number of relevant elements reduced the complexity of the unit
cell. Unit cells that have a large number of struts are more likely to blend or mix the struts than
unit cells with few struts due to the process tolerances and minimum feature size. As a result, the
as-printed lattices will not match the as-designed model nor perform as predicted or intended. In
addition, more struts will generally lead to small pores and cavities where powder can become
partially sintered or even sealed off sections of the unit cell, making powder removal impossible.
3.2.3.1.6 Powder Removability
The powder removability objective function was for evaluation of the risk of trapping
powder within the unit cell. The purpose of the objective was to encourage unit cells with large
pores and spacing between relevant elements. Small pores or tight spacing between elements can
trap unsintered powder in L-PBF components [244,245]. A simple schematic for how powder can
become trapped is shown in Figure 3.4, where, as the build progresses, cavities may become
entirely or partially sealed off. Whether AM lattices are applied to aerospace components where
trapped powder adds weight to the component or medical components in which any unsintered
powder that eventually escapes could be toxic, it is critical to be able to efficiently remove all
unsintered powder from lattice structures. Aside from trapping powder, small pores also increase
the post-processing time to remove powder from a maze of intricate channels.
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Figure 3.4 Schematic showing how powder can become trapped in internal cavities for powder-based
AM processes.
In this dissertation, a new metric to measure powder removability from strut and node
type lattices is proposed. To the authors’ best knowledge, this is the first objective function or
evaluation related to powder removability for AM lattice structures. The powder removability
factor (PRF) was based on evaluation of the spacing between elements. The midpoint distance
between each element to every other element in the unit cell was calculated based on the initial
full ground structure, where �̃�𝑒𝑒 was the element-to-element distance matrix. Within each
position of the matrix, 𝐷𝑖𝑗𝑒𝑒 was the distance from midpoint to midpoint between element 𝑖 and
element 𝑗. During the PRF objective evaluation of each design, the radius of each element was
subtracted; see Eqn. (3.27),
𝑃𝑖𝑗 = 𝐷𝑖𝑗
𝑒𝑒 − 𝑟𝑖𝑒 − 𝑟𝑗
𝑒 , 𝑓𝑜𝑟 {𝑖 = 1,… ,𝑀𝑗 = 1,… ,𝑀
} (3.27)
where �̃� was the pore distance matrix and 𝑟𝑖𝑒, 𝑟𝑗
𝑒 were the radii of the i-th and j-th elements. An
example of the pore size measurement is shown in Figure 3.5.
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From the resulting matrix containing all the distance measurements for relevant elements
(𝐴𝑒 > 𝐴𝑟𝑒𝑙), the PRF objective score was the number of instances where the distance fell below a
given threshold. Eqn. (3.28) shows the raw objective function for the PRF objective (𝑤𝑃𝑅𝐹),
𝑤𝑃𝑅𝐹 = 𝑐𝑜𝑢𝑛𝑡(𝑃𝑖𝑗 < 𝑝∗) (3.28)
where 𝑝∗ is the pore threshold.
The pore threshold value, 𝑝∗, may vary based on the process and intent of the objective.
For example, to simply improve powder removability during post processing, 𝑝∗ is chosen to
encourage pores larger than the specified threshold, improving flowability of the powder out of
the lattice. On the other hand, to specifically discourage designs that are likely to have closed
pores due to overmelt or dross, excess powder particles sintered to the downskin surface, 𝑝∗ may
be chosen based on the minimum gap size for which powder may escape. In this second case 𝑝∗ is
directly dependent on the AM process, powder size distribution, laser process parameters, etc.
Figure 3.5 Ground structure-based pore size measurement for the PRF. On the left part of the image,
an example of the element-to-element distance measurement is made midpoint to midpoint and then
the radius of each element is subtracted to obtain the pore size for the pore distance matrix. A few
examples of some of the pore measurements are shown on the right part of the image. In the pore
matrix, each element is measured relative to every other element.
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3.2.3.2 Optimization Constraints and Ground Structure Restrictions
Eight different constraints and restrictions can be applied to the ground structure and
optimization problem to reduce the number of design variables and constrain design space. Three
were optimization constraints that control the relevant cross-sectional area, cross-sectional area
limits, and volume fraction. The remaining five are restrictions on the ground structure model
configuration for DfAM and are not enforced as constraint equations in the optimization problem.
Each of these optimization constraints and ground structure restrictions are described in the
following sections.
3.2.3.2.1 Relevant Cross-Sectional Area
The relevant cross-sectional area constraint is for elements that were deemed not
manufacturable because they fell below the minimum feature size. The constraint was based on
the relevant cross-sectional area limit, 𝐴𝑟𝑒𝑙, which defines how large an element should be to be
printable and structurally sound. Eqn. (3.3) and Eqn. (3.10) define the relevant cross-sectional
area constraint. Elements with a cross-sectional area less than 𝐴𝑟𝑒𝑙 are temporarily set to the
lower limit, 𝐴𝑙𝑜𝑤𝑒𝑟, for the current iteration and remain in the optimization as design variables.
𝐴𝑟𝑒𝑙 was based on the AM process-specific limitations. For example, in L-PBF, the
manufacturable spot size is typically close to 150-200 microns in diameter. From this spot size,
the generally accepted minimum feature size for L-PBF printers is a diameter of 300-500 microns
so that more than a single point can be used to represent the contour or infill. In this work, where
all element cross sections were assumed to be circular, 𝐴𝑟𝑒𝑙 was defined by the relevant diameter
limit, 𝑑𝑟𝑒𝑙, as shown in Eqn. (3.29).
𝐴𝑟𝑒𝑙 =
𝜋𝑑𝑟𝑒𝑙2
4 (3.29)
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3.2.3.2.2 Cross-Sectional Area Limits
The cross-sectional area of each element in the system was constrained by upper and
lower bounds, 𝐴𝑢𝑝𝑝𝑒𝑟 and 𝐴𝑙𝑜𝑤𝑒𝑟. The purpose of the lower limit was to serve as a sufficiently
low bound that any element near or set to this value had a negligible contribution on the system
and objective function scores. The purpose of the upper limit was to help determine the allowable
topologies based on the largest feature size of the unit cell. The upper bound was chosen based on
the desired diameter upper limit, 𝑑𝑢𝑝𝑝𝑒𝑟. The lower diameter limit, 𝑑𝑙𝑜𝑤𝑒𝑟, was set such that
𝐴𝑙𝑜𝑤𝑒𝑟 ≪ 𝐴𝑢𝑝𝑝𝑒𝑟. The effect of 𝑑𝑢𝑝𝑝𝑒𝑟 depended on the objective of the optimization. For
example, for high stiffness unit cells, increasing 𝑑𝑢𝑝𝑝𝑒𝑟 may enable unit cell topologies with
thicker and fewer elements. In general, the relevant diameter limits were chosen such that 𝐴𝑙𝑜𝑤𝑒𝑟
was five or more orders of magnitude smaller than 𝐴𝑢𝑝𝑝𝑒𝑟. The exact relevant diameter limits
were chosen based on the optimization problem and were specified for each optimization case
study.
3.2.3.2.3 Volume Fraction
In cases where the volume was not used as an objective, the volume fraction can be a
constraint that limits the total volume that can be included in the unit cell. The purpose was to
restrict the allowable material volume, especially in cases where other objectives encourage high
volume fraction unit cells. The constraint was formulated using Eqn. (3.30),
𝑉𝑓∗𝑉𝑢𝑐 ≥ 𝑨
𝑇𝑳 (3.30)
where 𝑉𝑓∗ was the target volume fraction, and 𝑉𝑢𝑐 was the total volume of the unit cell. A common
approach to GSTO is also to use the total possible volume, 𝑉𝑚𝑎𝑥, which is the combined volume
of all elements at 𝐴𝑢𝑝𝑝𝑒𝑟 instead of the total enclosed volume of the unit cell.
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3.2.3.2.4 Ground Structure Restrictions
For a full ground structure, each node is connected to every other node by an element.
While a full ground structure lends the most design freedom to the optimization for a given set of
nodes, the number of elements or design variables, 𝑀, increases based on Eqn. (3.31),
𝑀 =
𝑁𝑡(𝑁𝑡 − 1)
2 (3.31)
where 𝑁𝑡 was the total number of nodes in the ground structure. As such, the number of design
variables can quickly become unwieldy for optimization. In addition, a full ground structure was
highly redundant due to overlapping elements. For example, for a unit cell defined by a 5x5x5
array of nodes, the total number of nodes is 125, leading to 7750 design variables in a full ground
structure. To reduce the number of design variables, restrictions were placed on the ground
structure configuration, permanently converting the elements to non-design elements in the
optimization problem.
There were five restrictions that could be applied to reduce the ground structure
configuration: build angle, element length, surface elements, interconnectivity nodes, and
symmetry. In each of these restrictions, elements from the initial ground structure that did not
meet the criteria were set as non-design elements. Non-design elements were present in the
ground structure model but were not design variables in the optimization. The cross-sectional
areas of non-design elements were either permanently set to a fixed value (i.e., 𝐴𝑒 = 𝐴𝑙𝑜𝑤𝑒𝑟) or
linked to another element with symmetry. As described in Section 3.2.1, as opposed to removal
from the ground structure model, non-design elements maintain model continuity and prevent
singularity of the global stiffness matrix.
In AM, the build angle is defined as the angle between horizontal build plate and the
component surface. Due to the nature of AM, built layer-by-layer, low build angles can cause
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manufacturability issues in some AM processes. The purpose of the build angle restriction was to
improve manufacturability of the unit cell by converting elements that would be difficult to build
to void non-design elements. Eqn. (3.32) defines the restriction,
𝐴𝑒 = 𝐴𝑙𝑜𝑤𝑒𝑟 𝑓𝑜𝑟 𝜃𝑙𝑜𝑤𝑒𝑟 < 𝜃𝑒 < 𝜃𝑢𝑝𝑝𝑒𝑟 (3.32)
where 𝜃𝑢𝑝𝑝𝑒𝑟 and 𝜃𝑙𝑜𝑤𝑒𝑟 are the upper and lower limits on the build angle and 𝜃𝑒 is the element
build angle, measured as smallest angle between the build plate and element direction vector.
The upper and lower limits of the build angle were chosen based on the AM process.
Generally, for L-PBF, the upper limit was set between 30-45 degrees. As processing and
manufacturing parameters continue to improve, small support-free build angles can be achieved
with reasonable surface roughness. In this work, the lower limit was often chosen as 1 degree,
allowing horizontal elements (𝜃𝑒 = 0) because at the length scale of lattice structures it is
frequently possible to span between two vertical supports. As shown in Figure 3.6, there is an
intermediate build angle range which is difficult to build due to the low build angle.
Figure 3.6 For lattice structures, horizontal elements are allowed to span between two vertical
supports because of the generally small length-scale. However, there is an intermediate range for
which the build angle is too low to accurately build.
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As reported in the author’s previous work, based on our preliminary studies of lattice
structures, the metal powder bed fusion process can create horizontal members for bridging [361].
Bridging refers to horizontal members connected at each end, forming a bridge between two
members. The term bridging is common in fused filament fabrication (FFF), where the filament is
stretched horizontally across two supports. Figure 3.7 shows several examples of horizontal
elements printed in lattice structures built with metal powder bed fusion, including 3x3x3 arrays
of 4 mm and 7 mm octet-truss unit cells, and 4 mm and 6 mm cubic unit cells. Note that the
horizontal members do print, however, for high element aspect ratios (length/diameter), or in the
case of cantilevers, distortion would occur, resulting in poor print quality and increased likelihood
of connection failures in the lattice.
Figure 3.7 Horizontal lattice members are shown in two lattice examples, 3x3x3 arrays of octet-truss
unit cells (4 mm, 7 mm) and cubic unit cells (4 mm, 6 mm). These lattices were printed in Ti64 on a
3D Systems ProX DMP 320. Image adapted from [361].
The element length restriction converted elements whose length fell within a specified
range to void non-design elements. The purpose of the restriction was to find only designs with
elements outside of the specified length range. Eqn. (3.33) defines the element length restriction,
𝐴𝑒 = 𝐴𝑙𝑜𝑤𝑒𝑟 𝑓𝑜𝑟 𝐿𝑙𝑜𝑤𝑒𝑟 < 𝐿𝑒 < 𝐿𝑢𝑝𝑝𝑒𝑟 (3.33)
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where 𝐿𝑢𝑝𝑝𝑒𝑟 and 𝐿𝑙𝑜𝑤𝑒𝑟 were the upper and lower limit, respectively, on element length, 𝐿𝑒.
Any element(s) within this length range were set to 𝐴𝑙𝑜𝑤𝑒𝑟 in the ground structure model and
became void non-design elements.
With the cubic design space, the surface elements restriction converted elements that lie
on the cubic faces of the unit cell to void non-design elements and the cross-sectional area is set
to the lower limit. The purpose of setting the cross-sectional area of the surface elements from the
unit cell to the lower limit was to reduce the impact on neighboring structures or connectivity to
other unit cells. This was accomplished by examining the starting and ending node locations of
each element to determine which elements lie solely on the cube faces.
Interconnectivity nodes were defined as nodes that lie on the surface of the unit cell cube.
These nodes determine the connectivity to neighboring unit cells or component structures. The
purpose of the interconnectivity nodes restriction was to limit how the unit cell connects with
itself or neighboring structures. For a given set of nodes, the cross-sectional area of any element
connected to one of these nodes was set to the lower limit and the element was converted to a
void non-design element. Using this restriction did not necessarily set the cross-sectional area of
all surface elements to the lower limit. Surface elements connected to two nodes that are not part
of the interconnectivity nodes restriction would remain as design variables in the optimization.
Finally, the symmetry restriction links the cross-sectional areas of symmetric elements to
a single design variable and converts the elements to non-design elements. The purpose of the
symmetry restriction was to reduce the number of design variables. Symmetry was applied based
on the three mid-planes (X, Y, Z) of the cubic design domain, individually or in any combination
of the three, resulting in seven distinct symmetry restriction options (X, Y, Z, XY, XZ, YZ,
XYZ).
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3.2.3.3 Penalties
For penalty functions, two different approaches were taken based on the type of
algorithm. To differentiate penalty functions from constraints, constraints were evaluated prior to
evaluating the objective functions of a unit cell design. Constraints restricted the design variable
values and if the design violated any of the constraints, the unit cell design was removed from the
solution space. In contrast, penalty functions were evaluated after the objective function(s) and
may negatively impact the objective function score, but the unit cell design may still be included
in the solution space.
3.2.3.3.1 Evolutionary Algorithm Penalty Functions
For the evolutionary algorithm, a set of penalties have been created. For each penalty, if
the current design being evaluated does not meet the criteria of the penalty, a flag was triggered.
As shown in Eqn. (3.12), the sum of the total number of flags was calculated as 𝑃. Using 𝑠𝑖 the
penalty 𝑃 can be scaled based on the relative order of magnitude of the corresponding objective
function or by the magnitude of the penalty impact. Next the penalty value, 𝑠𝑖𝑃, was added to
each raw objective score, 𝑤𝑖, and then multiplied by 𝑚𝑖 to determine whether an objective should
be minimized (𝑚𝑖 = 1) or maximized (𝑚𝑖 = −1); see Eqn. (3.11).
Four types of penalties may be applied to the unit cell optimization to discourage
infeasible or undesirable unit cell solutions without removing them from the solution space. The
four penalties were overlap, connectivity, and two options for verifying AM support,
unsupported‒S or unsupported-N. For each penalty, the unit cell was evaluated using a short
computational function to check for any violations. If any violations were detected during the
check, the penalty being checked was flagged. As shown in Eqn. (3.12), the penalty, 𝑃, was the
sum of the flags raised during the evaluation of the unit cell.
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3.2.3.3.1.1 Overlap Penalty
With a full ground structure, where an element connects each node to every other node in
the system, there were numerous overlapping or intersecting elements due to the highly redundant
elements. Mathematically, elements can overlap partially or entirely with no problem, but
physically two separate elements cannot occupy the same physical space. Similarly, elements can
intersect at locations that were not their endpoints mathematically, but physically these were not
desirable. The purpose of the overlap penalty was to discourage a unit cell design that had
overlapping or intersecting elements at non-end points of the elements.
Using the line vector that defined the position and direction of each element, the overlap
function determined which line vectors overlapped or intersected at non-end points of the
element. Overlapping or intersecting line vectors for elements were stored in a binary 𝑀x𝑀
matrix, where 𝑀 was the total number of elements in the full ground structure. To increase speed,
the overlap matrix was computed once for the ground structure as opposed to every iteration,
meaning that the current radius of each overlapping element was not considered.
During each iteration, the overlap matrix was reduced to only the relevant elements. If
any relevant elements for the iteration overlap or intersect according to the overlap matrix, a flag
was raised for the overlap penalty. Evaluation of the overlap penalty is shown in Eqn. (3.34)
𝑝𝑜(𝑨) = {
0, �̃� = �̃�
1, �̃� ≠ �̃� (3.34)
where 𝑝𝑜 is the overlap penalty, �̃� is the overlap matrix, and �̃� is the null matrix.
3.2.3.3.1.2 Connectivity Penalty
The connectivity penalty function was designed to prevent free hanging elements that
would not connect to any neighboring unit cells. For each patterned unit cell, there are 26
neighboring unit cells that interact with it. The simplest way to visualize this is to consider a
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3x3x3 array of unit cells, for which there is a single central cell that is connected to the other 26
unit cells surrounding it.
The connectivity computational function first determined node-neighbor pairs, or nodes
that were shared by neighboring unit cells (i.e. face-centered nodes on opposing faces). Next, it
determined which nodes were used by relevant elements within the unit cell. Finally, it checked
to ensure that if any nodes were used by relevant elements, at least one of their node-neighbor
pairs were also used by other relevant elements within the unit cell, creating a binary connectivity
vector marking whether a violation had occurred. The purpose of the connectivity penalty is to
ensure that the unit cell will connect to neighboring unit cells; see Eqn. (3.35),
𝑝𝐶(𝑨) = {0,𝑩 = 𝑶1,𝑩 ≠ 𝑶
(3.35)
where 𝑝𝑐 is the connectivity penalty, 𝑩 is the connectivity vector, and 𝑶 is the null vector. If
connectivity was flagged, that signified that there were relevant elements that would be free
hanging and not connected to relevant elements in the neighboring unit cells when patterned. In
optimization problems where homogenization was not the optimization objective, the
connectivity penalty could be used to ensure that relevant elements connect to other relevant
elements in neighboring unit cells.
3.2.3.3.1.3 Unsupported‒S Penalty
A major manufacturing limitation that is unique to most AM processes is the need for
support material. In L-PBF, support material is used to prevent distortion and ensure there is a
surface present on which to build the layer. For lattice structures fabricated with AM, the
presence of support material is generally infeasible because it is inaccessible and cannot be
removed from the intricate cavities and geometries that are created. The purpose of the
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unsupported‒S penalty (unsupported by self) was to ensure the unit cell can be printed without
support material.
With this penalty, the unit cell was checked to ensure that each relevant element was self-
supported by other elements within the unit cell. Based on the relevant elements within the
structure, each node was determined to be supported or unsupported. Layer by layer for nodes in
the XY planes of the ground structure model, each node is determined to be supported or
unsupported based on the relevant elements of the current design. A supported node is a node that
is supported by relevant elements below it and upon which other relevant elements could feasibly
build. In the current formulation, there is no max overhang or max allowable bridging distance
within a unit cell topology. All horizontal cantilevered elements (connected to a single supported
node) are considered unsupported and would trigger the penalty. All horizontal elements that
bridge between two supported nodes are considered supported and not penalized. The assumption
is that elements feasibly can bridge horizontally in the XY plane across two supported nodes
within the unit cell topology. As such, bridge distance is not a parameter for unsupported
penalties and acts under. If a relevant element was connected to an unsupported node, a violation
was marked in a binary unsupported elements vector. In addition, for a self-supported unit cell,
any nodes on the bottom cube face that were connected to relevant elements should have had their
mirrored opposite nodes also connected to relevant elements on the upper plane to ensure that for
a vertical column of patterned unit cells it continued to be self-supported. Eqn. (3.36) shows the
unsupported‒S penalty,
𝑝𝑈𝑆(𝑨) = {0, 𝑺 = 𝑶1, 𝑺 ≠ 𝑶
(3.36)
where 𝑝𝑈𝑆 is the unsupported‒S penalty, 𝑺 is the unsupported elements vector, and 𝑶 is the null
vector. If the unsupported‒S penalty was flagged, it signified that there were relevant elements
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building vertically upward from nodes that were unsupported. These were elements that would
not build properly without added support material.
3.2.3.3.1.4 Unsupported‒N Penalty
There were cases for which a relevant element could be supported by its neighboring unit
cells but it would not qualify as a self-supported unit cell on its own. The purpose of the
unsupported‒N penalty (unsupported by neighbors) was to ensure that the unit cell can be built
without added support, as long as when it was patterned it had other unit cells built around it in
the same plane.
Similar to the unsupported‒S penalty, the unsupported‒N determined which nodes were
supported by relevant elements in the unit cell. Next, considering node-neighbor pairs described
in Section 3.2.3.3.1.2, the list of supported nodes could be augmented by nodes shared by
neighboring unit cells. As such, the structures that pass the unsupported‒S penalty check would
be a subset of any structures that passed the unsupported‒N penalty. Similar rules for horizontal
cantilevered or bridging elements apply as described for the unsupported-S penalty. Finally, if a
relevant element was connected to an unsupported node, a violation was marked in a binary
unsupported elements vector. Eqn. (3.37) shows the unsupported‒N penalty,
𝑝𝑈𝑁(𝑨) = {
0, 𝑺𝑁 = 𝑶1, 𝑺𝑁 ≠ 𝑶
(3.37)
where 𝑝𝑈𝑁 is the unsupported‒N penalty, 𝑺𝑁 is the unsupported elements vector considering
neighbored support nodes, and 𝑶 is the null vector.
Similar to the connectivity penalty, unsupported‒N considers relevant elements that
would be connected to the neighboring unit cells that could provide support for what would
otherwise be free hanging structures. The unsupported‒N penalty assumes an infinite array of unit
cells in the build plane and does not consider unit cells on the boundary of a finite array.
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To aid with visualization of the connectivity, unsupported-S, and unsupported‒N
penalties, a few 2D examples are shown in Table 3.2. As any structures that pass the
unsupported‒S penalty are a subset of the unsupported‒N penalty, it is not possible to have a
structure pass unsupported‒S but not unsupported‒N.
Table 3.2 Examples of 2D unit cells for comparing pass/fail criteria of different penalty functions.
ALTO Penalties
Unit Cell Lattice Connectivity Unsupported‒S Unsupported‒N
Fail Fail Fail
Pass Fail Fail
Pass Fail Pass
Fail Pass Pass
Pass Pass Pass
Figure 3.8 shows a 2D example of determining supported and unsupported nodes for the
unsupported‒S and unsupported‒N penalties. In this example, the unit cell topology would pass
the unsupported‒N penalty but not the unsupported‒S penalty (row 3 of Table 3.2). There are
three rows of elements in the XY plane and non-relevant elements are not shown. First, the
relevant elements that start at row 1 determine which nodes are supported in rows 2 and 3. The
supported nodes are added to a list of supported nodes. Next, node-neighbor pairs are considered
and added to the list. Then, the next set of relevant elements starting from supported nodes in row
2 determines additional nodes supported in row 3. The top row determines which nodes on the
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bottom row are supported. Horizontal elements that bridge supported nodes are considered
feasible. After the list of all supported nodes in the unit cell has been compiled, it is compared
against a list of all nodes associated with relevant elements. If there are any nodes that are not in
the supported list, then the unsupported‒N penalty is triggered.
The unsupported‒S penalty check is performed similarly but without the node-neighbor
pairs check for which a node supported by a neighboring unit cell would augment the list of
supported nodes. In this example, node 4 would not have been added for the unsupported‒S
penalty and in the final check would have identified two unsupported elements that connect to
node 4. This basic example is shown in 2D for simplicity of explaining the process but the
penalties are extended to 3D for ALTO.
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Figure 3.8 Basic 2D example for determining the supported nodes for the unsupported‒N penalty.
This example comes from row three of Table 3.2 which passes the unsupported‒N penalty but would
fail the unsupported‒S penalty. The unsupported‒S penalty follows a similar process but the node-
neighbor pairs would not be considered meaning that node 4 is not self-supported, but instead
supported by a neighboring unit cell. Horizontal elements that bridge supported nodes are
considered feasible.
3.2.3.3.2 Gradient-based Algorithm Penalty
For the gradient-based optimization algorithm, a continuous penalty function was
formulated. The purpose of the penalty function was to reduce the complexity of the final
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topology by penalizing unit cell topologies with a higher number of relevant elements than other
topologies. By reducing the number of elements in a unit cell, the unit cell becomes more
manufacturable because there were fewer elements that can overlap and solidify together during
printing. The penalty function was defined by Eqns. (3.38)-(3.42),
𝑝𝑠𝑒 = 𝛼3 (
𝐴𝑟𝑒𝑙𝐴𝑒
) + 𝛼4 , 𝑓𝑜𝑟 𝐴𝑒 < 𝛼2𝐴𝑟𝑒𝑙 (3.38)
𝑝𝑠𝑒 = 1, 𝑓𝑜𝑟 𝐴𝑒 ≥ 𝛼2𝐴𝑟𝑒𝑙 (3.39)
𝛼3 =
𝛼2(𝛼1 − 1)
𝛼2 − 1 (3.40)
𝛼4 =𝛼2 − 𝛼1𝛼2 − 1
(3.41)
𝑝𝑖 = (𝑝𝑠1 ∗ 𝑝𝑠2 ∗ …∗ 𝑝𝑠𝑒)𝛾 , 𝑓𝑜𝑟 𝑒 = 1…𝑛𝑅𝐸 (3.42)
where 𝑛𝑅𝐸 was the number of relevant elements, 𝑝𝑠𝑒 was the penalty associated with each
relevant element, 𝛾 was used to change the importance of the penalty, and (𝛼1, 𝛼2, 𝛼3, 𝛼4) were
parameters to define the penalty function values and range.
By using the penalty function, elements with an area close to 𝐴𝑟𝑒𝑙 become less
economical in improving the objective function than elements not close to 𝐴𝑟𝑒𝑙, encouraging few
elements with large cross-sectional areas as opposed to many elements with small cross-sectional
areas. An example plot of the penalty function is shown in Figure 3.9. In this case (𝛼1, 𝛼2, 𝛼3,
𝛼4) were set to (2, 100,100
99,98
99), respectively. The result is that for elements at 𝐴𝑟𝑒𝑙, 𝑝𝑠𝑒 = 2. The
value of 𝑝𝑠𝑒 decreases as 𝐴𝑒 increases until 𝐴𝑒 = 𝛼2𝐴𝑟𝑒𝑙, at which point 𝑝𝑠𝑒 = 1.
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Figure 3.9 Graph of penalty function that reduces complexity by encouraging elements with a cross-
sectional area larger than the relevant cross-sectional area limit.
3.2.3.4 ALTO Summary and DfAM Considerations
As a major aspect of ALTO was to enable unit cell DfAM, the DfAM considerations are
reiterated in this section as well as noted in Table 3.1. By selectively choosing different
combinations of optimization components from Table 3.1, ALTO enables researchers and
designers to optimize lattice structures for specific applications and take advantage of the design
freedom offered by AM.
The primary AM-related criteria considered in ALTO were minimum feature size, design
complexity, and support material. The purpose of these DfAM considerations was to improve
manufacturability of the model’s optimal solution and reduce the need for design interpretation
and manipulation prior to fabrication the physical solution.
1) The minimum feature size was controlled by 𝐴𝑟𝑒𝑙. Elements with an area less than 𝐴𝑟𝑒𝑙
were considered infeasible because they would be difficult to reliably manufacture.
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2) Design complexity was reduced by including objective functions such unit cell volume
and number of relevant elements, or through an overlap penalty. Minimization of the unit
cell volume or number of relevant elements encouraged less material volume and
elements. In both cases, fewer elements or less material reduced the risk of overlapping
elements and blending or mixing of elements printed too close to each other. The overlap
penalty reduced complexity and time required for design interpretation.
3) Designs that would require support material were discouraged or eliminated through the
build angle restriction and the unsupported‒S or unsupported‒N penalties.
3.2.4 ALTO Validation
Three types of benchmark problems were used for validation of different components of
ALTO. The benchmark types were minimization of strain energy, maximization of thermal
conductance, and homogenization. In optimization, it is difficult to prove that the result is not a
false positive, a result that is not truly optimal. This would occur if the optimization is not able to
find the global minimum. The purpose of the validating ALTO against these benchmark problems
is to show that the ground structure formulation and optimization are functioning as expected and
that ALTO is able to replicate similar results to those found in literature for standard topology
optimization problems.
3.2.4.1 Minimization of Strain Energy
Minimization of compliance benchmarks were test problems for optimization of a simply
supported beam and a cantilever beam. In both test problems, the goal was to minimize the
compliance, or equivalently strain energy, subject to a volume constraint. The elastic modulus
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was set to 1 and the load applied was also 1; their magnitudes do not affect the optimal solution.
For the simply supported beam, also known as the MBB beam, a load is applied at the center of
the beam. The beam is standardized to a 6:1 ratio, or a 120 mm x 20 mm beam. In this validation,
and common to the structural optimization literature, half symmetry was used to reduce the
computational cost. For the ground structure, a 7x5 array of nodes represented the half symmetry
design space; see Figure 3.10. For the cantilever, the load is applied in the middle of the end; see
Figure 3.11. The design space is standardized to a ratio of 1.6:1 or a 40 mm x 25 mm beam. A
full 2D ground structure with 7x5 nodes represents the design space; see Figure 3.11.
Both benchmarks were solved using the minimization of strain energy objective and the
fmincon optimization algorithm in MATLAB. In the GSTO, it determines how to size each
element to minimize the compliance of the structure subject to the volume constraint. As the
solution progressed, the elements with the smallest cross-sectional areas approach the lower limit
and are later discarded from the final structure. For additional details on the benchmark problems;
see [273,336].
145
Figure 3.10 Schematics for the MBB beam benchmark problem and the initial ground structure used
to represent the design space.
Figure 3.11 Schematics for the cantilever beam benchmark problem and the initial ground structure
used to represent the design space.
146
For topology optimization, the solution is highly dependent on the original mesh and
constraints such as minimum feature size. For the ground structure model, the mesh density and
minimum feature size are comparable to the nodal array density and the relevant cross-sectional
area, respectively. The results of the GSTO of the MBB beam and cantilever benchmarks are
shown in Figure 3.12 and Figure 3.13, respectively. In the ALTO results, the line thickness in the
plot was correlated to the size of the cross-sectional area, thicker lines mean a thicker cross
section. Multiple solutions were optimized for a fixed allowable volume, while varying the
minimum allowable thickness. The minimum thickness value for the MBB beam and cantilever
benchmarks were (0.1, 0.5, 0.8) and (0.1, 0.5, 0.7), respectively. The compliance value of each
structure is noted on each image. In both test problems, the ground structure solutions were not
identical matches to those found in literature because the literature results are for a continuum-
based topology optimization as opposed to the discrete GSTO. However, they agree quite well
qualitatively; the trends in the topology of the minimum compliance structure are similar.
Figure 3.12 MBB beam comparison with results reported in literature. The different solutions are
shown for a fixed volume and changing the minimum feature size limit. The line width on the bottom
images is correlated to the cross-sectional area from the GSTO solutions. Top row of images adapted
from [273].
147
Figure 3.13 Cantilever benchmark comparison with literature results. Various solutions are shown
for a fixed allowable volume and changing the minimum feature size limit. The line width on the
bottom images is correlated to the cross-sectional area from the GSTO solutions. Top row of images
adapted from [273].
The GSTO results follow similar patterns compared to results found in literature. As
shown, as the minimum feature size increases, the compliance value also increases, meaning a
less stiff structure. By adjusting the volume constraint as well as adjusting the minimum feature
size, the structure may be further refined. In addition, the ground structure model was a more
discrete representation of the design space than the continuum models. For example, the
cantilever example from literature had a smooth curved surface flowing from the supports to the
load which was difficult for the ground structure to replicate without increasing the node density.
By increasing the node density, the ground structure solutions could more closely approach the
literature results. The similarity between the GSTO and literature results is considered validation
of the ground structure formulation and optimization.
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3.2.4.2 Thermal Conductance Benchmark
The thermal conductance formulation and corresponding objective functions have been
developed for ALTO applications requiring specific thermal conductance properties from the
lattice structure. Through combination with other objective functions, it offers the opportunity for
unique multi-functional lattices. The thermal conductance formulation is validated through a
common topology optimization benchmark for a heat sink problem.
In the benchmark, a design space is uniformly heated with thermal insulation surrounding
the design space. At one point along the edge of the design space, a heat sink is applied. The
objective of the optimization is to maximize the heat transfer through the heat sink. A schematic
of the benchmark is shown in Figure 3.14 and more details are provided by Bendsoe and
Sigmund [281] and Lohan et al. [278]. Using ALTO, the benchmark was solved using a 2D
square (10 mm x 10 mm) design space with an array of 7x7 nodes; see Figure 3.14. The gradient-
based algorithm fmincon was used to solve the benchmark. To reduce the number of design
variables, a non-full ground structure was utilized by restricting the element length to 6 mm or
less.
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Figure 3.14 Schematic for thermal conductance benchmark from literature. The continuum topology
optimization approach (left) is shown compared to the GSTO approach (right).
The solution to the benchmark problem is shown in Figure 3.15. In addition, the solutions
from two other sources using a continuum approach for topology optimization are shown in
Figure 3.15. In the ALTO solution, the line thickness represents the relative thickness of each
strut. As shown, the solutions agree qualitatively, creating tree-like structures that branch out to
fill the design space. Common to all solutions in Figure 3.15 were the branches gradually tapering
and reducing in cross section as the distance from the heat sink increases. The branching and
tapering of the solution were characteristic of benchmark solutions reported in literature. As with
the previous benchmark, the nodal array, 𝐴𝑟𝑒𝑙 parameter value, and the maximum element size
can all be adjusted to make minor modifications to the solution. The similarities between the
benchmark results and literature results validate the thermal conductance formulation and
objective function for the optimization.
150
Figure 3.15 Thermal conductance benchmark results from literature (left, middle) and from the
GSTO in ALTO show good correlation, validating the thermal conductance formulation and
optimization objective. Left and middle images adapted from [278] and [281], respectively.
3.2.4.3 Homogenization Benchmark
The final benchmark problem for validation of the ground structure model and
homogenization formulation was replication of results from the origin of the homogenization
approach [297,298]. In Sigmund’s work, he reported several 2D and 3D structures and their
constitutive matrices. For validation, these same structures were replicated in ALTO to compare
the calculated constitutive matrix against those reported by Sigmund.
Considering various examples from two sources by Sigmund [297,298], 6 different
example lattice structures were replicated as part of the validation. These structures include five
2D examples and one 3D example; see Table 3.3. In order replicate these structures in ALTO, the
same 2D square design space, discretized into a 4x4 nodal array was created. Similarly, a square
3D design space was discretized into a 4x4x4 nodal array for the 3D example problem. The exact
dimensions of the nodal array were not given and cross-sectional areas of all the elements were
not given in the source. The dimensions and assumed material properties of the elements were not
critical as the final result was normalized. However, these structures represent optimized
solutions and the exact cross-sectional areas of each individual truss element were not specified in
151
the sources. As such, some variation was expected without the ability to exactly match all the
cross-sectional areas.
Table 3.3 Comparison of homogenization benchmark problems and ground structure calculation.
For these benchmark problems, the ground structure was an exact match to the reported constitutive
matrix, images adapted from [297,298]
Literature Benchmark
Reported Constitutive Matrix
Replicated Structure
Calculated Constitutive Matrix
[𝐸] = [1.00 0.00 0.000.00 0.00 0.000.00 0.00 0.00
]
[𝐸] = [1.00 1.00 0.001.00 1.00 0.000.00 0.00 0.00
]
[𝐸] = [1.00 1.00 0.001.00 1.00 0.000.00 0.00 0.00
]
[𝐸] = [1.00 1.00 0.001.00 1.00 0.000.00 0.00 0.00
]
[𝐸] = [ 1.00 −1.00 0.00−1.00 1.00 0.00 0.00 0.00 1.00
]
[𝐸] =
[ 1.0 1.0 1.0 0.0 0.0 0.01.0 1.0 1.0 0.0 0.0 0.01.0 1.0 1.0 0.0 0.0 0.00.0 0.0 0.0 0.0 0.0 0.00.0 0.0 0.0 0.0 0.0 0.00.0 0.0 0.0 0.0 0.0 0.0]
Table 3.3 shows the structure in the original source, the constitutive matrix, and the
ALTO structure, and the calculated constitutive matrix. In all cases, the constitutive matrices
were found to be a match to the results reported by Sigmund. For 2D examples, it was relatively
simple to determine the appropriate truss connectivity and appropriate element cross-sectional
areas. Only a single 3D example was completed due to the added complexity of determining the
appropriate connectivity and cross-sectional areas of each element. Using these six examples of
152
homogenized materials, the ground structure model and homogenization formulation in ALTO
were validated.
3.3 Closing Remarks
This chapter details the basis and theory of ALTO, a systematic optimization approach
for unit cell topology optimization. As detailed in this chapter, there are a variety of objective,
constraint, and penalty functions that may be applied to tailor the optimization for specific
applications. Of particular importance is the contribution of this work toward the advancing field
of DfAM. Using various methods, AM optimization constraints and ground structure restrictions
can be directly incorporated into the optimization to improve manufacturability and reduce
interpretation steps to be able to print an optimized solution. Three examples of benchmark
problems have been demonstrated to validate various aspects of the ground structure formulation,
objective functions, and optimization. With the validation complete, the next chapters explore the
application of ALTO to several case studies including multi-functional unit cells, tailoring
compliance through homogenization, and improving powder removability.
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Chapter 4 Case Studies on Multi-functional Lattices
The focus of this chapter is demonstration of ALTO for multi-functional lattice
structures. The chapter includes a summary of the relevant background and motivation as well as
descriptions of the case studies performed. Two case studies are presented for demonstrating the
benefits of using ALTO to generate a novel lattice structures, specifically with the intent of
finding a balance between two conflicting objectives. The first case study is a 3D example with
the evolutionary optimization algorithm Borg. The second case study is a 2D example using the
gradient-based optimization algorithm fmincon. Each case study follows the same general flow of
presenting the motivation, a formal description of the optimization, and presentation of the
results. Afterward, there is a combined discussion of the results of the case studies and a few
closing remarks. The following objectives and sub tasks are addressed in this chapter:
3. Explore multi-functional lattices through case study on unit cell generation for thermal
conductance and minimum strain energy
3.1. Define constraints and process limitations
3.2. Generate optimal lattice structure with and without DfAM
3.3. Fabricate lattice structure with DfAM considerations
3.4. Evaluate solutions and revise ground structure topology optimization
4.1 Background and Motivation
Though the freedom of AM allows for complex and intricate structures to be
manufactured, the challenge remains to be able to effectively design components with such
structures in mind. In the author’s opinion, our ability to manufacture has surpassed our ability to
model and design to the fullest potential of AM. The inclusion of an appropriate lattice structure
in a component that enhances the macroscale performance becomes a metamaterial design
problem requiring novel systematic design and optimization tools such as ALTO.
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One method to approach the metamaterial design problem is through topology
optimization of the mesoscale structure. While there are numerous research groups exploring
methods for metamaterial design, the direct incorporation of AM constraints into the optimization
has not been addressed. As mentioned by Liu et al. [295], there is a need for further incorporation
of design for AM (DfAM) limitations in topology design. Plocher and Panesar [336] cited
integration of DfAM as a major bottleneck for a streamlined design process. In a review on
modeling approaches by Dong et al. [116], one of their primary findings was that lattice
structures were particularly promising with their potential to be multi-functional materials in
various engineering applications.
The purpose of these case studies is to explore ALTO for generation of multi-functional
lattice structures. Topology optimization and lattice structures have been of particular interest for
heat sinks and other heat transfer applications [118,183–185,278]. With AM, heat sinks can be
integrated into components and serve multiple purposes such as structural stiffness in addition to
including enhanced heat transfer topology. In these case studies, the focus is on using
optimization to balance two conflicting objectives: (1) minimization of strain energy and (2)
maximization of thermal conduction. The strain energy objective seeks to use all available
volume to maximize the stiffness of the structure against the loading configuration. The thermal
conduction objective seeks to use all available volume to maximize the thermal conductance of
the unit cell for the specified heat sink configuration. For additional details regarding the
objective functions, see Section 3.2.3.1.1 and Section 3.2.3.1.2. In these case studies, the loading
and heating configurations were specifically posed to be in conflict, such that the optimization
was forced to explore a balance between these two primary objectives.
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4.2 Case Study 1: Generation of a Novel 3D Multi-functional Lattice
4.2.1 Optimization Description
The purpose of Case Study 1 was a 3D example to demonstrate a multi-objective
optimization with an evolutionary algorithm and to show the effects of DfAM considerations. The
optimization algorithm for this example is Borg. A formal description of the optimization
formulation is given in Eqns. (4.1)-(4.5), with additional details available in Section 3.2.2.2. A
summary of the objectives, constraints, and penalties is shown in Table 4.1 and described in the
following paragraphs.
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 (𝑊𝑆𝐸 ,𝑊𝑀𝑇𝐶 ,𝑊𝑉𝐹 ,𝑊𝑅𝐸)
𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝐴𝑒 − 𝐴𝑙𝑜𝑤𝑒𝑟 ≥ 0 (4.1)
𝐴𝑒 − 𝐴𝑢𝑝𝑝𝑒𝑟 ≤ 0 (4.2)
𝐴𝑒 = 𝐴𝑙𝑜𝑤𝑒𝑟 𝑓𝑜𝑟 𝐴𝑒 < 𝐴𝑟𝑒𝑙 (4.3)
𝑊ℎ𝑒𝑟𝑒 𝑊𝑖 = 𝑚𝑖(𝑠𝑖𝑃 + 𝑤𝑖(𝑨, 𝑳)) (4.4)
𝑃 =∑𝑝𝑗(𝑨, 𝑳)
𝑛𝑝
𝑗
, 𝑓𝑜𝑟 𝑗 = 1…𝑛𝑝 (4.5)
Table 4.1 Case Study 1: 3D Multi-functional lattice optimization summary
Min. Strain Energy (𝑤𝑆𝐸) (𝑚𝑖, 𝑠𝑖) = (1,100)
Max. Thermal Conductance (𝑤𝑀𝑇𝐶) (𝑚𝑖, 𝑠𝑖) = (−1,100)
Min. Volume Fraction Error (𝑤𝑉𝐹) (𝑉𝑓∗, 𝑚𝑖, 𝑠𝑖) = (0.15,1,100)
Min. Relevant Elements (𝑤𝑅𝐸) (𝑚𝑖, 𝑠𝑖) = (1,100)
Symmetry YZ midplanes
𝑑𝑟𝑒𝑙 0.3mm
𝑑𝑙𝑜𝑤𝑒𝑟 , 𝑑𝑢𝑝𝑝𝑒𝑟 (9.5x10-5,1.5) mm
𝐿𝑙𝑜𝑤𝑒𝑟 , 𝐿𝑢𝑝𝑝𝑒𝑟 (2.1,15) mm
𝜃𝑙𝑜𝑤𝑒𝑟 , 𝜃𝑢𝑝𝑝𝑒𝑟 (0,35) degrees
DfAM Unsupported-S Penalty Optimized with and without
Borg Evaluations 100,000
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Objective 1 was to minimize the strain energy subject to the loading case; see Figure
4.1A and Eqn. (4.6). Objective 2 was to maximize thermal conductance out of the unit cell
through a single face-centered node; see Figure 4.1B and Eqn. (4.7). If each objective were
evaluated individually, one would expect elements to primarily be oriented vertically for
Objective 1 and horizontally for Objective 2. This example demonstrates finding a balance
between two objectives and is analogous to solving for a heat sink lattice with rigidity in one
direction. The case study considers the optimization of the unit cell individually and not global
heat transfer or stiffness across lattice structure patterned from this unit cell. Objective 3,
minimization of the volume fraction relative error, encourages a 15% volume fraction within the
unit cell; see Eqn. (4.8). Objective 4, minimization of relevant elements, was added to encourage
reduced unit cell complexity; see Eqn. (4.9).
wSE = 𝒖𝑇𝑲𝒖 (4.6)
𝑤𝑀𝑇𝐶 =𝑞𝑒𝑓𝑓
�̅� (4.7)
𝑤𝑉𝐹 =
|𝑉𝑓∗ −
𝑨𝑇𝑳𝑉𝑢𝑐
|
𝑉𝑓∗ (4.8)
𝑤𝑅𝐸 = 𝑐𝑜𝑢𝑛𝑡(𝐴𝑒 > 𝐴𝑟𝑒𝑙) (4.9)
157
Figure 4.1 Case Study 1: 3D multi-functional lattice example problem, simultaneously optimizing for
(A) minimization of strain energy and (B) maximizing the thermal conductance
Beginning with an 8mm unit cell size, using a 5x5x5 cubic array of nodes, each node was
connected to every other node in the system using a truss element, for a total of 7750 elements or
design variables. Next, the optimization constraints were limits on the relevant diameter and
bounds on the upper and lower cross-sectional area. Ground structure restrictions were applied for
an element length range, build angle range, and symmetry across the Y and Z midplanes. The
element length restriction converted elements with a length range of 2.1-15 mm to void non-
design elements. The build angle restriction converted elements with a build angle larger than 0°
and less than 35° to void non-design elements. The symmetry restriction links symmetric
elements to a single design variable. Based on the ground structure restrictions, the total number
of design variables was reduced from 7750 to 96. For comparison of the effects of the DfAM
penalty, two identical optimizations were completed with and without application of the
unsupported‒S penalty.
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4.2.2 Optimization Results
The results of the case study show a solution with vertical members at the corners that
provides stiffness for the applied loading, while the horizontal members draw heat toward the
heat sink. Using a serial version of Borg, the computation times for the optimization with and
without DfAM were 5.5 hours and 5.4 hours, respectively. The heat sink structure resembles the
branching tree-like structure that reaches throughout the unit cell as described previously by the
benchmark problem; see Section 3.2.4.2. The ground structure restriction on element length
restricts the elements to those connecting only to its closest neighboring nodes. The resulting
solutions from the optimization, Design A (without DfAM) and Design B (with DfAM), are
shown in Figure 4.2. Design A is not manufacturable due to the unsupported horizontal elements
and would require extensive design interpretation and manipulation; see Figure 4.3. On the other
hand, by including the DfAM penalty, Design B shows a similar branching structure and is fully
self-supported.
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Figure 4.2 Solutions to the optimization problem from Case Study 1. Design A and Design B are the
solutions without and with the DfAM penalty, respectively.
Figure 4.3 Design A has several long horizontal overhangs reducing the manufacturability of the unit
cell. Incorporating a DfAM penalty into the optimization, the algorithm discovered Design B which is
fully self-supported. The images depict an XZ plane through the center of the unit cell topology for
improved visibility of some of the unsupported regions.
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Table 4.2 shows the objective function values (𝑊𝑖), for the two optimizations, without
DfAM (Design A) and with DfAM (Design B). The ideal result would be the lowest objective
values in each of these cases. Comparing the objective function scores, Design A and Design B
have equivalent strain energy and thermal conductance, despite one being infeasible for
manufacturing without support material. The difference in objectives in Table 4.2 comes in the
volume fraction and the number of relevant elements, where Design B is higher in each objective
than Design A to accommodate the added material/elements needed for a self-supporting unit
cell. In general, one would expect an improved strain energy score with added material. However,
the equivalent strain energy and thermal conductance score mean that the added support material
for Design B is used for its self-supporting features.
Table 4.2 Comparison of objective function values for two designs from Case Study 1. Design A is not
manufacturable due to long unsupported overhangs in the topology.
𝑾𝑺𝑬 𝑾𝑴𝑻𝑪 𝑾𝑽𝑭 𝑾𝑹𝑬 Self-Supported
Design A 90.54 -0.100 0.544 138 No
Design B 90.54 -0.100 0.570 168 Yes
In multi-objective algorithms, the optimization finds a Pareto-optimal set of solutions.
Because there are multiple objectives there are many solutions that can be considered optimal
because of the tradeoffs in the objective functions. The Pareto-optimal solutions are non-
dominated solutions, or solutions for which no other solutions are better in all objectives. As
described in Section 3.2.2.2, Borg utilizes 𝜖-box dominance. In this problem large 𝜖 values for
each objective were chosen (𝜖𝑆𝐸 , 𝜖𝑀𝑇𝐶 , 𝜖𝑉𝐹 , 𝜖𝑅𝐸) = (20,0.05,300,300) such that the algorithm
narrowed down the Pareto-optimal solutions and returned only a single design for each instance
(Design A and Design B); see [258] for additional details on 𝜖-box dominance. With lower
epsilon values than those specified for this problem, hundreds or even thousands of optimal
solutions may be returned by Borg and populate the Pareto front.
161
For another instance of the optimization, the 𝜖 values were reduced to
(𝜖𝑆𝐸 , 𝜖𝑀𝑇𝐶 , 𝜖𝑉𝐹, 𝜖𝑅𝐸) = (0.05, 0.001,0.005, 0.5) to demonstrate optimization algorithm, resulting
in 84 different Pareto-optimal solutions. Figure 4.4 shows two graphs of the solution space for 84
Pareto-optimal designs: (A) modified thermal conductance versus strain energy and (B) modified
thermal conductance versus volume fraction. The optimization was searching for designs in the
lower left of each graph. In addition, the red point in each graph is the optimized solution, Design
B, from case study. For ease of viewing, the fourth objective is not depicted in these figures of the
solution space. As shown in the figure, a natural result of the 𝜖-box dominance is the stepped
solution space. Further reduction of the 𝜖 value for the modified thermal conductance objective
would reduce this stepped solution. From Figure 4.4A, the optimization has found a minimum
strain energy value (𝑊𝑆𝐸 ≅ 90.54), where the four corner pillars are at near the upper limit for
the cross-sectional area and there are multiple designs that can achieve this with varying levels of
thermal conductance. From Figure 4.4B, the optimization results show a clear tradeoff between
the volume fraction and thermal conductance.
162
Figure 4.4 Graphs of the solutions space for an instance of the Case Study 1 Optimization. Each point
is a Pareto-optimal solution for (A) modified thermal conductance versus strain energy objectives
and (B) modified thermal conductance versus volume fraction objectives. The red point in each
graph represents Design B shown in Figure 4.2
One advantage of a multi-objective optimization is the opportunity for a designer to
explore the tradeoffs and select a design from the optimal solution set. To select a single design,
the Pareto optimal solutions have to be narrowed based on the intent of the design. For example,
the designer could desire a solution with 𝑊𝑀𝑇𝐶 ≤ −0.10, 𝑊𝑆𝐸 < 95 , and minimum 𝑊𝑉𝐹. Table
163
4.3 shows the Pareto-optimal solutions from Figure 4.4 fitting this requirement and sorted by the
𝑊𝑉𝐹 objective, also included in this list is Design B from Figure 4.2. As shown, the design which
minimizes the volume fraction with the aforementioned requirements on the thermal conductance
and strain energy objectives is Design B.
Table 4.3 List of Pareto-optimal solutions with the modified thermal conductance objective less than
-0.10, strain energy objective less than 95, and the volume fraction objective sorted smallest to
largest. According to the requirements of the designer, the optimal solution would be Design B shown
in the first row in bold typeface.
𝑾𝑺𝑬 𝑾𝑴𝑻𝑪 𝑾𝑽𝑭 𝑾𝑹𝑬 Self-Supported
Top Design (Design B) 90.541 -0.10000 0.570345 168 Yes
90.629 -0.10002 0.745860 176 Yes
90.541 -0.10010 0.788925 168 Yes
94.458 -0.10008 0.849405 166 Yes
90.541 -0.10000 0.861188 166 Yes
90.541 -0.11048 0.991856 196 Yes
90.544 -0.11008 1.045903 180 Yes
91.109 -0.11071 1.096788 176 Yes
90.594 -0.11013 1.098463 178 Yes
90.541 -0.11009 1.111183 174 Yes
92.243 -0.12000 1.337346 198 Yes
91.077 -0.12043 1.393450 186 Yes
90.562 -0.12019 1.398234 188 Yes
90.562 -0.12010 1.416453 184 Yes
In Case Study 1, the overlap and self-supported penalties resulted in a unit cell topology
that was immediately manufacturable as opposed to another solution that would have required
extensive design interpretation. The optimized solution was formatted in a file type that is
readable using nTopology [235] to generate the full solid 3D geometric model. As demonstration
of the printability, the unit cell was printed with FFF using a single unit cell and then patterned
into a 2x2x2 array and fabricated using L-PBF. The L-PBF example was printed using a 2x scale
for visibility. Figure 4.5 shows each of the example prints that completed successfully without
any support material. The single unit cell was printed in PLA using standard processing
164
conditions on an Afinia H480 3D Printer. To meet minimum feature size requirements of the FFF
process, the model was scaled up 5x so that a single unit cell was a 40 mm cube. The L-PBF
example was printed on a 3D Systems ProX DMP 320 using recycled Inconel 625 powder with
standard processing conditions.
Figure 4.5 Printed examples of the optimized solution from the 3D multi-functional lattice case study.
The examples were printed as a single unit cell (scaled up 5x) with FFF and in a 2x2x2 array (scaled
2x) with L-PBF. By incorporating DfAM considerations into the optimization, the optimized solution
was print-ready, not requiring any design interpretation.
4.3 Case Study 2: Generation of a Novel 2D Multi-functional Lattice
4.3.1 Optimization Description
Case Study 2 demonstrates a 2D example of the benefits of multi-objective optimization
with a gradient-based algorithm by solving the problem multiple times with different weighting
factors to show the tradeoffs between the objective functions. In this example, the two objective
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functions were the minimization of strain energy (𝑤𝑆𝐸̅̅̅̅ ) and maximization of thermal conductance
(𝑤𝑇𝐶̅̅̅̅ ). A formal description of the optimization problem is given in Eqns. (4.10)-(4.15),
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓(𝑊)
𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝐴𝑒 − 𝐴𝑙𝑜𝑤𝑒𝑟 ≥ 0 (4.10)
𝐴𝑒 − 𝐴𝑢𝑝𝑝𝑒𝑟 ≤ 0 (4.11)
𝑨𝑇𝑳 − 𝑉𝑓∗𝑉𝑚𝑎𝑥 ≤ 0
𝐴𝑒 = 𝐴𝑙𝑜𝑤𝑒𝑟 𝑓𝑜𝑟 𝐴𝑒 < 𝐴𝑟𝑒𝑙 (4.12)
𝑊ℎ𝑒𝑟𝑒 𝑓 = (𝑊) (4.13)
𝑊 =∑𝑐𝑖𝑚𝑖𝑤𝑖(𝑨, 𝒍), 𝑓𝑜𝑟 𝑖 = 𝑆𝐸̅̅̅̅ , 𝑇𝐶̅̅̅̅
𝑛
1
(4.14)
1 = ∑𝑐𝑖
𝑛
1
, 𝑓𝑜𝑟 𝑖 = 𝑆𝐸̅̅̅̅ , 𝑇𝐶̅̅̅̅ (4.15)
with additional details available in Section 3.2.2.1. A summary of the objectives, constraints, and
penalties is shown in Table 4.4 and described in the following paragraphs.
Table 4.4 Case Study 2: 2D Multi-functional lattice optimization summary
Min. Strain Energy (𝑤𝑆𝐸̅̅̅̅ ) (𝑚𝑖) = (1𝑥1010)
Max. Thermal Conductance (𝑤𝑇𝐶̅̅̅̅ ) (𝑚𝑖) = (−1)
Weight values for combined objective (𝑊) 𝑐𝑆𝐸 = 0,0.01,0.02,… ,1𝑐𝑇𝐶 = 1,0.99,0.98,… ,0
Volume Fraction Constraint 𝑉𝑓∗ = 0.1
Symmetry XY midplanes
𝑑𝑟𝑒𝑙 0.3mm
𝑑𝑙𝑜𝑤𝑒𝑟 , 𝑑𝑢𝑝𝑝𝑒𝑟 (1x10-6,2) mm
𝜃𝑙𝑜𝑤𝑒𝑟 , 𝜃𝑢𝑝𝑝𝑒𝑟 (1,35) degrees
Young’s Modulus E = 200 GPa
Thermal Conductivity 54 (W/m·K)
Coefficient of thermal expansion 1.2x10-5 (m/m·K)
Structural Load 4x: 1000N
𝑇1, 𝑇2 100°C, 25°C
For the minimization of strain energy, the load case is shown in Figure 4.6. Loads were
applied vertically downward on the four nodes at the top of the design space with pin supports in
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the bottom corners. The purpose of the first objective function, 𝑤𝑆𝐸̅̅̅̅ , was to minimize strain
energy due to loading in the negative Y-direction. While these conditions do not represent
periodic boundary conditions that would be present in a unit cell, the purpose was to illustrate the
minimization of strain energy of a lattice structure under the compressive loading. These loading
and boundary conditions may easily be varied by a designer depending on the application. The
purpose of the second objective function, 𝑤𝑇𝐶̅̅̅̅ , was to maximize thermal conductance in the X-
direction, from the left-edge nodes to the right-edge nodes; see Figure 4.6. A temperature
difference between the nodes on the left and right edges of the design domain was applied. The
unit cell optimization was solved using a 4x4 array of nodes for a square 5 mm design space and
assumes a ground structure restriction of symmetry across the vertical and horizontal midplane,
shown by the dashed lines in Figure 4.6.
Figure 4.6 Case Study 2: (A) Minimize strain energy and (B) Maximize thermal conductance. Blue
dashed lines denote symmetry planes. Images adapted from [361].
Because the objective function for the gradient-based optimization was a weighted sum
of the two objectives, a variety of weight values were required to find various solutions showing
the tradeoffs in the objectives. The weighting values (𝑐1 = 0,… ,1) and (𝑐2 = 1,… ,0), with
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increments of 0.01, were applied to each objective function, resulting in a set of 101 solutions. To
normalize the strain energy objective function to be of a similar order of magnitude as the thermal
conductance objective function, a scale factor of 1x1010 was applied to 𝑤𝑆𝐸̅̅̅̅ . The individual
objective functions and combined objective are shown in Eqns. (4.16)-(4.18). No penalty
functions were applied this optimization. DfAM considerations for the lattice structure
optimization were the ground structure restrictions on build angle and the relevant diameter limit,
shown in Table 4.4. The ground structure restrictions of build angle and symmetry reduced the
number of design variables from 120 to 30.
𝑤𝑆𝐸̅̅̅̅ =
𝑤𝑆𝐸𝑚𝑖𝑛 −𝑤𝑆𝐸
𝑤𝑆𝐸𝑚𝑖𝑛 −𝑤𝑆𝐸
𝑚𝑎𝑥 (4.16)
𝑤𝑇𝐶̅̅̅̅ =
𝑤𝑇𝐶𝑚𝑖𝑛 −𝑤𝑇𝐶
𝑤𝑇𝐶𝑚𝑖𝑛 −𝑤𝑇𝐶
𝑚𝑎𝑥 (4.17)
𝑊 = (1𝑥1010)𝑐𝑆𝐸𝑤𝑆𝐸̅̅̅̅ − 𝑐𝑇𝐶𝑤𝑇𝐶̅̅̅̅ (4.18)
4.3.2 Optimization Results
The result of the optimization was a set of 101 solutions showing the tradeoffs that exist
between the two objective functions. The total computation wall time was 2-3 hours using 6
computing cores. A convergence plot for the 𝑐𝑆𝐸 = 0.5 and 𝑐𝑇𝐶= 0.5 is shown in Figure 4.7.
Figure 4.8 illustrates 11 of the solutions with weight value increments of 0.1, showing the range
solutions possible. In each plot, the line widths of the elements were scaled based on its cross-
sectional area. Elements with a diameter less than 𝑑𝑟𝑒𝑙 were excluded from the plots. From Figure
4.8, it is shown that the optimal solution for maximizing the thermal conductance was comprised
of primarily horizontal bars while the optimal solution for minimization of strain energy was
comprised of primarily vertical bars. The result was that for a unit cell that has both high thermal
conductance and low strain energy, there must be a tradeoff.
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Figure 4.7 Example convergence plot for the solution to the optimization problem with equal
weighting factors for both objectives. The result shows an increasing objective function score until
the constraints are met and then gradually decreasing until convergence.
Figure 4.8: Case Study 2 results showing the tradeoffs between minimization of strain energy and
maximization of thermal conductance. 11 different solutions are shown beginning for incrementing
the weight values by 0.1 as the solutions move clockwise. Image adapted from [361].
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A graph of the objective function values for the 101 solutions generated is shown in
Figure 4.9. Optimal solutions would simultaneously maximize the thermal conductance and
minimize the strain energy. This figure shows the reciprocal of the thermal conductance objective
so the optimization searches for designs near the lower left corner of the plot. This figure
represents the designs obtained by incrementing the weighting factors and shows the Pareto front
that develops. The Pareto front is the set of designs for which no design is better than another in
both objectives, or non-dominated designs. These non-dominated solutions formed the Pareto
front and were all considered optimal. As one non-dominant design slightly improves in one
objective, it gets slightly worse in the other. Dominated designs are worse in both objectives than
at least one other solution. In Figure 4.9, non-dominated solutions are shown in grey circles and
dominated designs are shown as black ‘+’.
Figure 4.9: Comparison of the objective function values for minimization of strain energy and
maximization of thermal conductivity (minimization of its’ reciprocal plotted here). This plot shows
the solution space, demonstrating the tradeoffs between objectives. On the plot, all 101 solutions are
shown, with the grey dots being non-dominated solutions on the Pareto-front. Image adapted from
[361].
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4.4 Discussion of Results
The two case studies demonstrated in this chapter were for generating novel unit cell
topologies based on specific strain energy and thermal conductance targets. Though the case
studies were different, 3D versus 2D and Borg vs fmincon, both were successful in generating
novel unit cells and demonstrate unique advantages and disadvantages of each optimization
approach.
Through Case Study 1, ALTO demonstrates the ability to generate a manufacturable unit
cell topology for multiple conflicting design objectives. One challenge was the large number of
design variables in a full 3D ground structure. Even after reducing the optimization problem
through ground structure restrictions by considering symmetry and DfAM, the number of
variables was large, making it difficult for the algorithm to find a solution. One advantage of the
evolutionary algorithm Borg, was a true multi-objective optimization that can handle objectives
with different ranges and orders of magnitude. They can offer the additional flexibility in
returning a set of Pareto-optimal solutions allowing the designer to explore tradeoffs and select
from a set of optimal solutions. In addition, the result of the optimization clearly shows that
without inclusion of the DfAM penalty function, the unit cell topology would require additional
design interpretation and refinement. With the included DfAM penalty, we see tradeoffs arise in
the objectives due to the need for generating a self-supporting unit cell. The optimized solutions
could also be manufactured without any additional manipulation as shown by the printed
examples.
In Case Study 2, the 2D example illustrates the tradeoffs that exist between different
objectives because the weight values were given and the solution searches in a single direction.
One challenge of the 2D approach, using the gradient-based algorithm, was appropriately scaling
each objective function before the weighted sum was calculated for the final objective value.
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Initially, each objective function was normalized based on the range of the objective function
value; however, the minimization of strain energy objective has such a large range that the
normalized value was substantially smaller than the normalized thermal conductance value. If one
objective function is significantly smaller, or larger, than the other, then a single objective
function can become dominant and overshadow another objective. To rectify this difference,
significant time was required to determine an appropriate scaling factor for the minimization of
strain energy objective. With the scale factor applied, it placed each objective function value on a
similar order of magnitude however, this scale factor selection remains case-dependent and
somewhat arbitrary. This same issue exists for all optimization where multiple objectives and
penalties are combined into a single objective optimization through a weighted sum.
In general, there are several challenges and limitations of a GSTO approach, 1) the initial
nodal array restricts the design space to truss connected to those elements, 2) the assumption that
elements are connected with hinges of zero stiffness, and 3) finding a balance between the
number of elements, or design variables, and not overly restricting the design space.
In GSTO, the final solution is limited by the node and truss locations available in the
initial configuration. The nodal configurations were 5x5x5 and 4x4 for the 3D and 2D case
studies, respectively. These configurations were chosen based on recommendations from
literature. Sigmund [297] recommended that 4 or 5 nodes per edge were sufficient for a RVEs,
representing a good balance between not restricting the available design space and a reasonable
number of design variables. As future work, other ground structure approaches with moving
nodes could be considered to reduce the impact of the node array on limiting the design space and
compare the effect on the optimized solutions.
For the pin-joint connected truss elements, the stiffness at the nodes may be assumed to
be small when the structure carries the majority of the load through the elements themselves.
Deshpande et al. [201] found that hinge or pin joints can be a reasonable assumption. In addition,
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experimental results presented in Chapter 5 also demonstrate that the pin joint assumption was
reasonable. Where these assumptions break down is for thicker elements or high volume
fractions, where the size or prevalence of nodes within the unit cell increase relative to the void or
truss element space. For other approaches that account for nodal stiffness in unit cell design, the
reader may consider geometric projection methods [313–315].
Finally, there is a tradeoff between the complexity of the initial configuration and the
number of design variables included in the optimization. Apart from the nodal locations
themselves, the number of truss elements can become intractable. For a 5x5x5 nodal array there
are 7750 possible truss elements or design variables. While this allows for a wide range of
solutions, reducing 7750 design variables to a simple set of non-intersecting and manufacturable
elements was quickly intractable for multi-objective evolutionary algorithms. In this work,
ground structure restrictions such as symmetry and element build angle were used to reduce the
number of design elements. However, in comparison to continuum element approaches, 7750
elements are approximately equal to a cube discretized into 20 elements per edge, which would
not be capable of capturing the intricacy that can be accomplished through GSTO. As a result,
after simplification for symmetry and other restrictions, there was a significant reduction in the
number of design variables required compared to continuum topology optimization approaches.
As with all topology optimization methods, a final challenge is the interpretation step to
move from the optimized solution to the final design. This design interpretation is required for the
ground structure approach as with other continuum-based approaches. In the case of ALTO, Case
Study 1 was able to remove this interpretation step and generate a unit cell topology that was
manufacturable.
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4.5 Closing Remarks
In this work, ALTO has been demonstrated as an approach to improve unit cell DfAM,
including the ability to incorporate multiple design objectives into a single lattice structure. The
case studies shown in this work represent examples of metamaterial design for novel AM lattices
by incorporating stiffness in one direction, while transferring heat in an orthogonal direction.
Particularly with Case Study 1, the two solutions show the importance of including DfAM
considerations into the optimization. Without direct integration of DfAM constraints, penalties or
ground structure restrictions, design interpretation to ensure manufacturability would be essential.
For Case Study 2, the wide range of solutions along the Pareto-front demonstrate the importance
of including multiple objectives that can allow for the tradeoffs in designs. Through continued
improvement, ALTO offers an approach to take advantage of the intricate design freedom offered
through AM, including the development of multi-functional lattices or metamaterials.
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Chapter 5 Case Studies on Homogenized Lattice Structures
The focus of this chapter is exploration of ALTO for generating unit cells with tailored
elastic properties. The chapter includes a brief summary of the relevant background and
motivation as well as descriptions of the case studies performed. Three case studies are presented
for demonstrating the benefits of using ALTO to generate a novel lattice structures, specifically
with the intent of matching a target homogenized constitutive matrix. The first case study is a 3D
example exhibiting improved powder removability. The second case study is a 3D example for
creating a unit cells with different stiffness by changing topology without decreasing the volume
fraction. The third case study is a 2D example for creating an orthotropic material that is three
times stiffer in one direction than the other. Each case study follows the same general flow of
presenting the motivation, a formal description of the optimization, and presentation of the
results. Afterward, there is a combined discussion of the results of the case studies and a few
closing remarks. The following objectives and sub tasks are addressed in this chapter:
4. Explore homogenized lattices through case study on generation of unit cells with tailored
compliance
4.1. Define constraints and process limitations
4.2. Generate optimal lattice structure with tailored properties
4.3. Fabricate and test to determine mechanical properties
4.4. Characterize fabricated samples with mechanical testing
4.4.1. Compare boundary conditions of homogenization with experimental
characterization
5.1 Background and Motivation
Lattice structures present a unique opportunity for materials with tailored
properties/behavior. Topology design of cellular materials has been ongoing for over thirty years,
dating back to the significant contributions of Bendsoe and Sigmund [268,298]. In these works,
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the mesostructure or cellular structure of a component was designed to match specific elastic
constitutive properties through homogenization. Through their work, they were able to
demonstrate mesostructures with extreme mechanical properties, such as designs with Poisson’s
ratios in the range (-1,1) [297]. At the time of this early research, the limited manufacturing
options were recognized as a primary challenge and rapid prototyping was mentioned as a
potential prospect. With recent advances in AM, there is renewed interest in design of cellular
structures, or lattice structures.
For L-PBF, there are several challenges with printing lattice structures which are
summarized here; for additional details, see Section 1.1.4.2 and Section 1.1.5.1. In their review of
lattice structures fabricated with AM, Nazir et al. [244] identified four challenges: (1) support
material removal, (2) powder removal, (3) large file size and (4) high surface roughness. In
addition to the four limitations covered by Nazir et al. [244], Maconachie et al. [250] address the
staircase effect which can cause significant changes in cross section for thin lattice features and
describe the impact of these challenges on mechanical properties. The ability to design unique
lattice structures with tailored properties, while accommodating DfAM is a metamaterial problem
requiring optimization.
The purpose of this chapter is demonstration of ALTO for generation of unit cells that
have a specific target constitutive matrix. The unit cell topology for a lattice structure is
frequently chosen from a list of common structures, without understanding of how the unit cell
topology will affect the overall properties of the component. In these case studies, the desired
lattice structure properties were assumed to be known as a target constitutive matrix and then
ALTO was used to generate novel unit cell topologies that matched that constitutive matrix.
Three unique case studies are presented to demonstrate the abilities of ALTO. Case Study 1 is a
3D example highlighting a novel powder removability objective. The optimized solution matches
the orthogonal effective elastic moduli of a common lattice structure but exhibits improved
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powder removability. Case Study 2 demonstrates the generation of three novel 3D unit cell
topologies by scaling the constitutive matrix for fixed volume fraction. Case Study 3
demonstrates generation of an orthotropic unit cell topology that is three times stiffer in one
direction than the other. Using a multi-objective optimization approach, such as ALTO, to design
unit cell topologies with a target constitutive matrix enables metamaterial design with enhanced
functionality.
5.2 Case Study 1: Generation of a 3D Unit Cell Topology with Improved Powder
Removability
The purpose of Case Study 1 was to demonstrate ALTO for generation of a new unit cell
that has the same orthogonal effective moduli as a baseline unit cell and improved powder
removal. After optimization, the optimized and baseline unit cell topologies were printed and the
mechanical properties were compared through compression testing. Powder removal was
evaluated through comparison of measured weight of the as-designed and as-printed components
and further validated through CT-scan evaluation.
The baseline unit cell chosen for the study was an octet-truss, shown in Figure 5.1. The
octet-truss is an arrangement of 36 struts that create a large number of potential trapped powder
regions. For this study, the unit cell size was 8 mm with 1 mm struts, or a volume fraction of
17.1% based on the CAD model (20.8% in ALTO). The octet-truss unit cell [362], chosen as the
baseline unit cell, is very common in literature because the effective modulus was predicted to
scale linearly with volume fraction [141,201]. This linear scaling was recently demonstrated in
[363]. Other studies have also used the octet-truss as a baseline for comparison against other unit
cells that follow similar linear scaling trends [133,301,363]. Mechanical properties that have been
studied include static effective modulus [150,191,364–366], high-strain rate compression
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[191,367], and fracture toughness [157]. Various other groups have studied it for optimization
[368], medical cellular scaffolds [176], or nanoscale applications [369].
5.2.1 Optimization Description
For the optimization, four objectives were included: 1) minimize homogenization error,
2) minimization of volume fraction, 3) minimization of the relevant elements in the unit cell, and
4) minimize the powder removability factor (PRF). The optimization constraints were upper and
lower bounds on the cross-sectional area and the relevant diameter limit for the minimum feature
size. Ground structure restrictions were the build angle limit, symmetry across ‘X’, ‘Y’, and ‘Z’
midplanes of the unit cell, and interconnectivity nodes. No penalties were applied for this
optimization. The optimization algorithm was Borg with a stopping criterion of 70,000 unit cell
design evaluations. A formal description of the optimization is shown in Eqns. (4.1)-(5.4). A
summary of the optimization problem is shown in Table 5.1. The optimization objectives and
constraints are described briefly below. For additional details, see Chapter 3.
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 (𝑊𝐻 ,𝑊𝑉𝐹 ,𝑊𝑅𝐸 ,𝑊𝑃𝑅𝐹)
𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝐴𝑒 − 𝐴𝑙𝑜𝑤𝑒𝑟 ≥ 0 (5.1)
𝐴𝑒 − 𝐴𝑢𝑝𝑝𝑒𝑟 ≤ 0 (5.2)
𝐴𝑒 = 𝐴𝑙𝑜𝑤𝑒𝑟 𝑓𝑜𝑟 𝐴𝑒 < 𝐴𝑟𝑒𝑙 (5.3)
𝑊ℎ𝑒𝑟𝑒 𝑊𝑖 = 𝑚𝑖(𝑤𝑖(𝑨, 𝑳)) (5.4)
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Table 5.1 Case Study 1: Improved powder removal lattice optimization summary
Min. Homogenization Error (𝑚𝑖) = (1)
Min. Volume Fraction Error (𝑉𝑓∗, 𝑚𝑖) = (0.01,1)
Min. Relevant Elements (𝑚𝑖) = (1)
Min. PRF (𝑝∗, 𝑚𝑖) = (4𝑚𝑚, 1)
Symmetry XYZ midplanes
𝑑𝑟𝑒𝑙 0.3 mm
𝑑𝑙𝑜𝑤𝑒𝑟, 𝑑𝑢𝑝𝑝𝑒𝑟 (9.5x10-5,3) mm
𝜃𝑙𝑜𝑤𝑒𝑟, 𝜃𝑢𝑝𝑝𝑒𝑟 (1,30) degrees
Borg Evaluations 70,000
The first objective used homogenization methods [116,297,298] to predict the
constitutive matrix and minimize the error between the calculated and target constitutive matrix.
The target constitutive matrix, based on the octet-truss baseline cell (volume fraction of 20.8%),
is shown in Eqn. (5.5), where �̃�∗ is the target constitutive matrix, and E is the material Young’s
modulus. The target constitutive matrix differs from octet-truss homogenized constitutive matrix
by a 25% reduction in each of the three orthogonal shear terms (𝐸44𝐻 , 𝐸55
𝐻 , 𝐸66𝐻 ). The objective
function is shown in Eqn. (5.6), where 𝑤𝐻 is the raw objective score and �̃�𝐻 is the calculated
homogenized constitutive matrix unit cell design.
�̃�∗ = 𝐸
[ 𝟎. 𝟎𝟑𝟒𝟕 𝟎. 𝟎𝟏𝟕𝟒 𝟎. 𝟎𝟏𝟕𝟒 𝟎 𝟎 𝟎
𝟎. 𝟎𝟑𝟒𝟕 𝟎. 𝟎𝟏𝟕𝟒 𝟎 𝟎 𝟎𝟎. 𝟎𝟑𝟒𝟕 𝟎 𝟎 𝟎
𝟎. 𝟎𝟏𝟕𝟒 𝟎 𝟎𝟎. 𝟎𝟏𝟕𝟒 𝟎
𝒔𝒚𝒎 𝟎. 𝟎𝟏𝟕𝟒]
(5.5)
𝑤𝐻 = ∑
|(𝐸𝑘𝑙∗ −𝐸𝑘𝑙
𝐻)|
𝐸𝑘𝑙∗ for 𝑘, 𝑙 = 1…6 (5.6)
The second objective minimizes the error between the target and current volume fraction.
Eqn. (5.7) shows the volume fraction objective function 𝑤𝑉𝐹, where 𝑨 is a vector of the element
cross-sectional areas, 𝑳 is vector of element lengths, and 𝑉𝑓∗ is the target volume fraction. Note
that the volume fraction calculated in ALTO overpredicts the actual volume fraction because it
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does not account for overlapping volumes near nodes. In this case study, the volume fraction was
set to 1% to effectively minimize the volume fraction. The third objective encourages simpler
unit cell designs with fewer struts; see Eqn. (5.8), where 𝐴𝑒 is the element cross-sectional area
and 𝐴𝑟𝑒𝑙 is the relevant cross-sectional area limit or minimum feature size. Though reducing
complexity is not often a focus for AM, at the mesoscale, the strut diameters and proximity of
struts within a unit cell are so close to the minimum feature size of industrial machines that
reduced complexity can improve print quality in lattice structures.
𝑤𝑉𝐹 =
|𝑉𝑓∗ − 𝑨𝑇𝑳|
𝑉𝑓∗ (5.7)
𝑤𝑅𝐸 = 𝑐𝑜𝑢𝑛𝑡(𝐴𝑒 > 𝐴𝑟𝑒𝑙) (5.8)
The fourth objective was to minimize the PRF. For each unit cell design, the pore
distance matrix, �̃�, was assembled based on the midpoint to midpoint distance between all
elements in the initial ground structure, �̃�𝒆𝒆, minus the radius of each element, 𝑟𝑖, 𝑟𝑗 ; see Eqn.
(5.9). After assembling �̃�, the objective function was calculated by counting the number of
instances that fell below the pore threshold, 𝑝∗; see Eqn. (5.10). For the fourth objective, the
threshold value for the pore size was set to 4 mm, or 50% of the unit cell size, which is much
larger than the powder particle size. For a large unit cell size and low volume fraction, such as the
baseline octet-truss, the purpose of the objective is to minimize the number of pores smaller than
4 mm in an 8 mm unit cell for improved powder flowability out of a lattice. For a small unit cell
size or high volume fraction of the optimized topology, the 𝑝∗ value will also help reduce
potential trapped regions.
𝑃𝑖𝑗 = 𝐷𝑖𝑗
𝑒𝑒 − 𝑟𝑖𝑒 − 𝑟𝑗
𝑒 , 𝑓𝑜𝑟 {𝑖 = 1,… ,𝑀𝑗 = 1,… ,𝑀
} (5.9)
𝑤𝑃𝑅𝐹 = 𝑐𝑜𝑢𝑛𝑡(𝑃𝑖𝑗 < 𝑝∗) (5.10)
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For optimization constraints, the upper and lower bounds on the cross-sectional area and
the relevant cross-sectional area limit were based on the equivalent diameter for the circular cross
section as defined in Table 5.1. Ground structure restrictions included build angle limit,
interconnectivity nodes, and symmetry. The unit cell was modeled by a full ground structure for a
5x5x5 cubic node array. Next, with the exception of 5 nodes per face (four corners and the center
of each face), all nodes on the surface of the unit cell and their corresponding elements were
converted to void non-design elements which have a fixed cross-sectional area at the lower limit.
This ground structure restriction mimics the nodes on the surface of the octet-truss unit cell; see
Figure 5.1. In addition, based on the ground structure restriction for build angle limit, any
elements between a build angle of 1 and 30 degrees (defined from the build plate) were converted
to void non-design elements. For ALTO, horizontal elements that form bridges between nodes
were allowed because the short distance they spanned across the unit cell is generally printable.
Finally, the symmetry ground structure restriction across all three orthogonal midplanes was
applied for cubic symmetry in the unit cell, linking all symmetric element cross-sectional areas to
a single design variable in the optimization. Beginning with a total of 7750 design variables, the
ground structure restrictions on interconnectivity nodes, build angle limit, and symmetry reduced
the total number of design variables for the ground structure to 120.
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Figure 5.1 The octet-truss (A) is the baseline unit cell topology for matching the constitutive matrix,
colors have been added to improve visualization of 3D topology. The initial ground structure is
shown in (B), where each line represents a truss element. With symmetry, there were a total of 120
distinct design variables.
5.2.2 Optimization Results
The optimization problem was successful in identifying multiple unit cell topologies with
𝑤𝑆𝐸 < 10%, or less than 10% total relative error between the calculated and target constitutive
matrix. From the designs with less than 10% total relative error, two unit cell topologies were
identified based on their ability to exactly match the target constitutive matrix. The two unit cells
from the optimization were the octahedral and a new cell type called the opti-octet. Figure 5.2
shows a single unit cell and a side view of the patterned structure for the two unit cells from the
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optimization on either side of the baseline octet-truss unit cell. A summary of the objective
function values is shown in Table 5.2.
Figure 5.2 The two unit cells output from the optimization are shown on either side of the octet-truss.
The top images show a single unit cell and the bottom images are side views of a 3x3 patterned array.
As a result of the PRF as an objective, both the octahedral and opti-octet open up the porous space to
improve powder removal and flowability.
Table 5.2 Summary of the objective function values for the optimization results. The octet-truss
results are shown for individual evaluation of the unit cell, separate from the optimization.
Objectives Opti-Octet Octet-Truss Octahedral
𝑤𝐻 0% -- 0%
𝑤𝑉𝐹 20.8% 20.8% 20.8%
𝑤𝑅𝐸 20 36 12
𝑤𝑃𝑅𝐹 36 252 60
For the two unit cells identified from the optimization results, minor adjustments to the
optimized solutions were made as part of the design interpretation step. One challenge with non-
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gradient-based optimization algorithms is that they lack an intelligence about the model. Rather
than progressively increasing/decreasing a design variable value based on a gradient
approximation, non-gradient-based algorithms rely on various recombination methods. These
recombination methods can make it difficult for minor variations of an optimal solution when
recombining with other designs. In the optimization results, the octahedral and opti-octet had 𝑤𝑆𝐸
values of 6.65% and 2.31%, respectively. In each case, manual adjustments to the strut diameters
were found to result in an exact match for the constitutive matrix and volume fraction as
compared to the baseline unit cell topology. The average manual adjustment to strut diameters
was 0.7% and 2.3% for the octahedral and opti-octet, respectively. This interpretation step is
common to topology optimization. The octet-truss baseline unit cell has a 75% difference from
the target constitutive matrix because of the reduction of the primary shear terms (𝐸44𝐻 , 𝐸55
𝐻 , 𝐸66𝐻 )
by 25% each, but matches exactly in all other terms of the constitutive matrix.
For multi-objective optimization, there are many designs which may be considered
optimal or said to lie on the Pareto front, where neither design dominates another in all objectives.
For example, the octahedral has fewer relevant elements but a larger number of potential trapped
regions than the opti-octet. On the other hand, the opti-octet has more relevant elements but fewer
potential trapped regions than the octahedral. With these tradeoffs in the objectives, these were
two solutions along the Pareto front.
From the side view in Figure 5.2, one can see that both the opti-octet and octahedral open
up larger pathways for powder flowability than the baseline octet-truss. The octahedral primarily
accomplishes large channels for powder flowability as a result of the minimization of relevant
elements objective. Having few elements in the unit cell structure creates wide open channels
with neighboring unit cells. However, the tight cavity at the center of the unit cell leads to the
high PRF score. On the other hand, the opti-octet accomplishes large channels for powder
flowability by spreading out the elements within the unit cell as shown by PRF. Based on the
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primary intention of improved powder removability, the opti-octet was chosen for fabrication and
further analysis in comparison with the octet-truss.
5.2.3 Ground Structure Model Validation
5.2.3.1 Fabrication and Testing
For validation of the ground structure model used in ALTO, a set of octet-truss and opti-
octet lattices were printed and evaluated using uniaxial compression testing. For both structures,
the unit cell size was scaled down to 4 mm and arrayed into a 10x10x10 lattice structure or 40
mm x 40 mm cubic samples. The intent of the 10x10x10 lattice was to reduce boundary effects
from the compression platens on the top and bottom of the sample as recommended in ISO 13314
[370]. The completed build plate is shown in Figure 5.3 with all the octet-truss and opti-octet
lattices among two other lattice types. The location of each lattice on the build plate was
randomized to minimize any effects from build location. The strut diameters for the octet-truss
were a uniform thickness of 0.5 mm. For the opti-octet, it is a unique combination of a cubic and
body-centered cubic (BCC) lattice, with strut diameters of 0.595 mm and 0.678 mm, respectively.
For testing, four replicates of each lattice were printed on an EOS M290 in Ti64 using standard
laser process parameters with infill only, no contours. The parts were heat treated at 800 C for
two hours in an argon inert atmosphere as recommended per the EOS material datasheet. The
parts were removed from the build plate using wire electrical discharge machining.
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Figure 5.3 Complete build plate with octet-truss and opti-octet lattice structures among a few other
lattice types. In the picture, one of the five samples of each lattice type are labeled. The interlocked
and diamond lattice are discussed as part of Case Study 2 in Section 5.3.
Using an MTS QTest/100 test frame, each sample was loaded in compression until the
first peak force at a crosshead speed of 1.0 mm/min, an equivalent strain rate of 4.17x10-4[1/s].
Figure 5.4 shows an example of the compressed lattice structures. From the load displacement
data, the global stress was calculated as the force divided by the original cross section of the
sample (1600 mm2). The global strain was calculated based on the crosshead displacement with
compensation for the compliance of the test frame.
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Figure 5.4 Examples of the octet-truss (left) and opti-octet (right) crushed after compression loading.
Prior to compression, the samples were 40 x 40 mm cubes or a unit cell size of 5mm as noted by the
scale bars.
5.2.3.2 Experimental Results
The average stress-strain curves from four samples each for the octet-truss and opti-octet
lattice structures are shown in Figure 5.5. The grey band behind each curve in the figure
represents the average plus and minus the standard deviation of the experimental data. The
effective modulus of each was computed by a linear least squares fit to the data between 10-50%
of the peak load. In Figure 5.5, the effective modulus region for each lattice is denoted by the
solid portion overlaid on the stress strain curves. From this region, the effective modulus was
calculated to be 2420.6 ± 65.4 MPa and 2152.4 ± 73.2 MPa for the octet-truss and opti-octet
lattices, respectively. The peak stress was determined to be 50.7 ± 1.5 MPa and 40.8 ± 0.9 MPa
for the octet-truss and opti-octet lattices, respectively. Table 5.3 shows a comparison of the
experimental, ALTO, and ALTO+corrections results of the effective modulus.
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Figure 5.5 Average stress strain curves for the octet-truss and opti-octet truss. The solid portion of
the curves represents the region where the effective modulus was measured, 10-50% of the peak
stress. The grey band behind each curve represents the average plus and minus the standard
deviation of the experimental data.
Table 5.3 Experimental validation results for the effective modulus and comparison to the ground
structure model prediction in ALTO and AM-based corrections. For AM corrections, IO = infill-only
correction and SC = stair case correction. The error relative to the experimental value is noted in the
parenthesis.
Octet-truss (MPa) Opti-octet (MPa)
Experimental 2420.6 ± 65.4 2152.4 ± 73.2
ALTO Prediction 2661.1 (+9.9 %)
2660.4 (+23.6 %)
ALTO + BOC 2279.4 (-5.9 %)
2276.4 (+5.8 %)
ALTO + BOC + SCC 2133.1 (-11.9 %)
2260.1 (+5.0 %)
From the experimental data, the ground structure model was found to slightly overpredict
the effective modulus and was attributed to differences between the as-designed and as-printed
models. There were several assumptions in the ground structure model and variations in the as-
printed part that can explain the discrepancy. For the ground structure, the model assumed perfect
geometry without deviations due to surface roughness or the scan strategy, that all nodes were
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representative of pin-joints, and that the stiffness of a unit cell was solely attributed to axial
loading of the printed struts. Deshpande et al. [201] showed that the pin-jointed truss element
assumption was a valid assumption, meaning that the axial stiffness of a strut plays a much
greater role in the effective modulus than the joint or bending stiffness at the nodes. In addition,
previous studies have shown that the number of layers of lattice changes the effective modulus of
the lattice due to the constrained, non-sliding, boundary conditions created by friction at the
surface of the compression platens [207]. For the results presented in Table 5.3, the experimental
data is compared against a homogenization prediction using “relaxed” boundary conditions that
represent the least constrained layers of the lattice sample, centrally located between compression
the platens. Additional detail on a comparison of the boundary conditions is given in Section
5.2.3.3.
For printing variations, the AM process is susceptible to deviations in geometry due to
the STL file conversion, the stair stepping effect, variable surface roughness depending on the
build angle, and the laser scan strategy or parameters to recreate the intended geometry. It is
assumed that powder removal does not affect the failure mechanisms in the lattice structures.
Two AM-based model corrections were considered for implementation in the ground structure: 1)
the Beam Offset Correction (BOC) and 2) the Stair Case Correction (SCC). The BOC is an offset
correction based on the laser processing parameters for the infill of each region to be scanned
with the laser in a layer. Lattice structures have a large surface to volume ratio, meaning that the
build time is a more evenly split between laser scan time for infill versus laser scan time for
contours as compared to a non-lattice part. In addition, scanning both infill and contours tends to
cause overheating in the thin lattice struts. In this case, the build time was decreased by nearly
50% by using an infill-only scan strategy as compared to including both the infill and contour
laser scans. As a result of using infill-only laser scans to build the lattice, the laser processing
parameters for Ti64 on an EOS M290 introduced a 0.025 mm beam offset from the as-designed
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cross-section in the XY build plane due to the laser scan strategy. The beam offset in the XY
plane caused a reduction in the cross-section which was dependent on the build angle of the strut.
Similarly, the BOC reduces the cross-sectional area of each strut in the lattice to match the actual
geometry scanned by the laser during the build process. Figure 5.6 shows a schematic of the
reduction caused by the beam offset and the adjusted cross-sectional area calculation.
The SCC is an additional offset correction to account for the staircase effect that is
frequently described for the high surface roughness on angled features. For lattice structures, it
has been suggested that the staircase effect can have a large impact on the stiffness of the lattice
struts because the roughness is large relative to the diameter of the struts [194,223,250,365].
Figure 5.6 shows how the staircase effect can negatively impact the cross-sectional area, by
reducing the continuous cross-sectional area over which a load is transferred. In the illustration,
underneath the overhang from each layer shift, there is dross buildup but the effectiveness of the
dross is limited, causing the reduction in cross section. Though the staircase effect has been
mentioned in several studies, a build angle dependent formula for approximating the effective
cross-sectional area of struts has not been used as a correction factor. The SCC was only applied
to angled struts, meaning struts with a non-vertical and non-horizontal build angle. The geometry
change for these two corrections was applied first as just the BOC, based on scan parameters, and
second, BOC combined with the SCC.
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Figure 5.6 AM-based corrections applied to ground structure prediction. As a result of either or both
corrections, the cross-section of the strut is effectively reduced.
The variation in percent error relative to the experimental values stemmed from the
nature of the unit cell types. First, consider the impact of the AM corrections based on strut angle.
The BOC correction affects all struts, with the biggest impact on vertical struts. On the other
hand, the stair case effect impacts only angled struts, with no impact on vertical or horizontal
struts. The BOC correction was first applied to account for the beam offset for all cross-sections
in the build plane. Next, the SCC correction was applied to adjust the effective cross section
based on the stair case effect of AM. For the opti-octet, it was apparent that the vertical and
horizontal struts carry the majority of the load because the BOC had a large impact on the
effective modulus while the SCC had very little impact on the effective modulus. On the other
hand, for the octet-truss, BOC decreased the effective modulus and underpredicts the
experimental result and then the added SCC reduces the effective modulus even further.
It is difficult to generalize an AM correction factor to account for multiple unit cell
topologies because there are many factors at play in the assumptions of the ground structure and
the AM geometric variations are geometry dependent. However, comparing the total error from
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Table 5.3, or combined error for both unit cell types, the BOC was found to be a sufficient
correction. While the SCC applied to the opti-octet effective modulus prediction was shown to
slightly reduce the relative error, in other cases, such as the octet-truss, the correction may lead to
an overcorrection. As such, the SCC was found to be an unnecessary correction for ground
structure models. In summary, the ground structure model, combined with the BOC was shown to
be a validated model that accurately predicted the effective modulus of two unique unit cells to
within 6%.
5.2.3.3 Comparison of Boundary Conditions
During compression, lattice layers near the upper or lower compression platens will
experience greater constraint than the centrally located layers due to the edge effects from friction
between the sample and platens. Figure 5.7 shows a schematic of a compression sample, with the
color gradient representative of the impact of edge effects and two examples of representative
boundary conditions for the unit cells in the “constrained” or “relaxed” regions of the
compression sample. Because globally the experimental testing measures the least stiff region, or
“relaxed” middle layers of the sample, the homogenization should predict the stiffness at the
central or “relaxed” region of the lattice.
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Figure 5.7 Schematic of a compression sample where the color gradient represents the impact of edge
effects from the friction with compression platens. Near the compression platens a unit cell
experiences “constrained” boundary conditions as compared to “relaxed” boundary conditions for a
unit cell near the middle of the sample.
Based on the homogenization method from Sigmund [297,298] the original boundary
conditions that were established in the homogenization objective function for the optimization
were found to be highly constrained over the experimental results. In the original boundary
conditions, the unit cell was constrained such that during the test strain cases, edges and faces
cannot contract as would be expected due to a Poisson’s ratio-like effect. From Section 3.2.3.1.3,
Figure 3.3 shows the boundary conditions for the 2D test strain cases. Similarly, in 3D the
boundary conditions restrict motion of the unit cell during the test strain cases such that the
effective modulus of the homogenized constitutive matrix may be overpredicted. For compression
testing a single layer of lattices, the unit cells would be highly constrained and expected to more
closely match the original boundary conditions as described in Section 3.2.3.1.3. Testing with a
minimum of 10 lattice layers, as suggested by ISO 13314 [370], produce a “relaxed” boundary
condition near the center of the sample. By relaxing the boundary conditions to mimic those
shown for the “relaxed” unit cell from Figure 5.7, the homogenized properties are significantly
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reduced. Table 5.4 shows the comparison of the effective modulus calculated from experimental
data and for two different boundary conditions in ALTO.
Table 5.4 Effective modulus comparison between experimental data and ALTO predictions of
“constrained” and “relaxed” boundary conditions. The error relative to the experimental value is
noted in the parenthesis.
Octet-truss (MPa) Opti-octet (MPa)
ALTO – “Constrained” Boundary Conditions
3991.7 (+64.9 %)
3991.8 (+85.5 %)
ALTO – “Relaxed” Boundary Conditions
2661.1 (+9.9 %)
2660.4 (+23.6 %)
Experimental 2420.6 ± 65.4 2152.4 ± 73.2
For matching the experimental data in Table 5.3 and validating the ground structure
model, ALTO predictions of the effective modulus were calculated using “relaxed” boundary
conditions that allow for expansion of the sides of the unit cell, similar to a Poisson’s ratio effect
in a traditional material. Using “relaxed” boundary conditions, and applying AM corrections
ultimately provides a very accurate prediction of the effective modulus of the compression
samples. Additional discussion of the issues that arise due to the choice of boundary conditions
for homogenization is given in Section 5.5.
5.2.4 Powder Removability
For comparison of powder removal between the octet-truss and the opti-octet, samples
fabricated with AM were weighed to compare the as-designed versus as-printed weight. The
ground structure model was used to compare the effective modulus prediction of the two unit
cells. If all powder was removed from the structure, the relative error between as-designed and
as-printed weight should be small and increase as trapped powder or partially sintered powder
exists inside the lattice structure. The purpose of the experiment was to demonstrate that the PRF
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objective function generated a unit cell with the same orthogonal effective moduli properties as
the octet-truss unit cell topology, but with improved powder removability.
5.2.4.1 Fabrication of Powder Removal Samples
For each unit cell type, 12 different test specimens were created by changing the unit cell
size while holding the volume fraction constant. This provided a set of 12 samples where each
sample was predicted to have the same effective modulus as its paired size in the other unit cell
type and the spacing between struts where powder can escape gradually decreased as the unit cell
size was scaled down. The largest unit cell size was 7 mm decremented by 0.5 mm until the
smallest unit cell size of 1.5 mm. For the octet-truss in the largest unit cell size, the diameter of
the struts was 2.391 mm, or a volume fraction of 52.6%. For the opti-octet in the largest unit cell
size, the diameters of the struts corresponding to the cubic and BCC struts were 2.843 and 3.242,
respectively. The range of unit cell sizes was intended for the largest to be able to evacuate all
powder down to the smallest where the channels for powder to escape the lattice were expected to
be too small for any powder removal. Each lattice structure sample was a 5x5x5 array of unit
cells. In summary, 72 total samples were created based on the 2 unit cell topologies, 12 unit cell
sizes with the same volume fraction, and 3 replicates of each for repeatability. Figure 5.8 shows a
lineup of 12 as-printed specimens for the octet-truss unit cell topology.
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Figure 5.8 As-printed lineup of octet-truss powder removal specimens. The largest specimen is a
5x5x5 array of 7 mm unit cells, approximately 35 mm in width. The smallest specimen is a 5x5x5
array of 1.5 mm unit cells, approximately 7.5 mm in width.
The powder removal specimens where fabricated on an EOS M290 in SS 17-4. Prior to
removal from the build plate, vacuuming, blowing compressed air, bead blasting, and tapping the
plate were all used to attempt to extract all unsintered powder from the lattices. Next, the
specimens were cut off from the build plate using wire electrical discharge machining. After
being removed from the plate, specimens were again individually processed using similar
methods to attempt to remove all unsintered powder.
5.2.4.2 Mass Comparison
For the as-printed mass, the samples were weighed using an A&D GR-202 scale. The
three largest unit cell sizes fell outside of the 210g capacity of the GR-202 scale and were
weighed using an Ohaus Scout II SC4010. Using a density value of 7.77 g/cm3 [371], the as-
designed mass of each specimen was calculated using the as-designed build volume. The relative
error was calculated as the mass difference between the as-printed and as-designed, normalized
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by the as-designed value. With the relative error defined in this way, a positive error value
demonstrates excess mass exists within the lattice structure, while a negative error value
demonstrates less mass than the as-designed value.
The comparison of relative error between the octet-truss and the opti-octet is shown in
Figure 5.9. The error bars represent the standard deviation for the three replicates. As a general
trend, as the unit cell size increases, the relative error decreases because as the unit cell size scales
up, the pore size is also scaled up, making it easier for powder to be removed. For the largest unit
cell sizes, there were a few cases where the relative error was negative. The negative relative
error was attributed to the cutoff height from the build plate, where the bottom of each sample
was slightly trimmed, making the mass slightly less than the as-designed model. Also, of note
was that the octet-truss always had a larger relative error than the opti-octet suggesting that more
powder was trapped within the octet-truss.
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Figure 5.9 Comparison of the relative error between the as-designed and as-printed mass for the
powder removal specimens. In (B) three regions are predicted based on the relative error (1) no
powder removed, (2) partial removal, and (3) all powder removed, shown with the red ellipses
overlaid.
From the relative error plot in Figure 5.9B, there were three distinct regions that develop
on the plot, (1) a plateau in relative error for the three smallest unit cell sizes, (2) a sloped relative
error region in the middle, and (3) a plateau in relative error for the largest 4-5 unit cell sizes.
This trend is consistent with the explanation that below a certain threshold no powder can be
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removed and above a certain threshold all the powder can be removed. For region 1, the relative
error was consistent because all the specimens were essentially solid blocks. For region 3, the
relative error was also similar because there was no powder trapped inside, only slight variations
based on surface roughness or other AM defects. Region 2 represents the transition zone where
the powder can be removed in some cases but is inconsistent. The error bars are the standard
deviation of the averaged data and represent the amount of variation. The largest error bars, or
largest variation, in Region 2 further support the claim in that in the transition region some
powder could be removed but it was inconsistent at the threshold of when powder can be
removed.
Based on the results, the opti-octet demonstrated better powder removal than the octet,
particularly in the transition region. At the extremes of Region 1 and Region 3 the relative error
was expected to be similar between both unit cell topologies because either all powder was
removed or all powder was trapped. However, in the transition region was where the opti-octet
was expected to differentiate itself from the octet-truss. While the octet-truss consistently has a
larger relative error than the opti-octet, the lower and statistically significant relative error of the
opti-octet for unit cell sizes 4.5, 5, 6, 6.5 mm demonstrated improved powder removal compared
to its octet-truss equivalents. One potential confounding effect not taken into account in the mass
measurement comparison was the surface roughness and dross on downskin surfaces. The impact
of surface roughness relative to the amount of total volume could conceivably also follow a
similar trend, where the amount of dross relative to the total volume would be higher for the
smallest lattices and decrease with increasing unit cell size. As such further validation of the opti-
octet was explored through computed tomography (CT) scans to visualize the internal feature of
the lattice structures.
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5.2.4.3 Computed Tomography Validation
As further validation of the powder removability study, CT scans were taken of the 1.5,
4.5, and 7 mm unit cell sizes for the octet-truss and opti-octet. Due to the size and density, the
largest sizes were sectioned into 4 vertical columns to obtain a higher scan quality than scanning
the entire specimen as a single piece. The CT scan resolution for the 1.5 mm unit cell size was 15
µm. For the 4.5 mm and 7 mm unit cell sizes the CT scan resolution was 35 µm. Qualitative
comparison of a central slice for each unit cell size and the as-designed model are shown in
Figure 5.10. Though these single slices represent only a fraction of the data, these central slices
were representative of features seen throughout the sample for their respective size and unit cell
topology.
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Figure 5.10 Qualitative CT scan comparison of a central slice for the octet-truss and opti-octet unit
cell sizes. The dashed line squares represent a single unit cell within each image. Comparing the 4.5
mm unit cell sizes against the computer generated as-designed cross-section, the octet-truss shows
more trapped powder or partially sintered regions as compared to the opti-octet, validating that the
opti-octet demonstrates improved powder removability.
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In comparing each of the three unit cell sizes against the as-designed cross section, the
CT scans confirmed the findings from the mass measurement of the samples shown in Figure 5.9.
In the context of describing the CT scan slices shown in Figure 5.10, the term void refers to an
empty region as detected by the CT scan. The term cavity refers to an internal empty space that
should exist according to the as-designed model. The term channel refers to pathways of
continuous voids in the as-designed model through which unsintered powder would pass to be
evacuated from the sample.
For the 1.5 mm unit cell sizes, the finding from the mass comparison was indicative of no
powder being removed. In nearly every cavity, the CT scans show sintered or trapped powder.
The CT scan also shows small voids located in many of cavity regions, suggesting that some
powder was removed from the sample. However, using Avizo [372] for analysis of the pore
network, or void space mapping, there were no channels large enough to be able to evacuate
unsintered powder from the majority of these internal cavities. This finding suggests that these
voids were not formed by evacuated powder, but occurred during the build, possibly during the
recoat stage or blown out by the air flow across the powder bed. The sintered or trapped powder
in every internal cavity confirms the finding that at the smallest size, all trapped powder either
became sintered or could not be evacuated.
For the 4.5 mm unit cell sizes, the finding from the mass comparison suggested that some
powder was removed from internal cavities but not all, and that the octet-truss had more trapped
powder in its internal cavities and channels than the opti-octet. The CT scans show that for the
octet-truss, many of the cavities were filled with sintered or trapped powder. Cavities near the
outer surface had a greater tendency to be void than cavities near the center of the specimen. This
suggests that it was easier for the powder to find a way out of the specimen from outer cavities
than inner cavities due to the maze of channels to navigate and less risk of lightly sintered powder
blocking the channels. Though only a single slice, this finding was representative of cross
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sections throughout the 4.5 mm octet-truss sample. Unfortunately, due to the quality of the scan,
complete segmentation for pore network analysis was not feasible. For the opti-octet, there were a
few cavities that showed deviations from the as-designed cross-section due to sintered powder,
however these were significantly less than the octet-truss. Using pore network analysis in Avizo
for the opti-octet, all void space in the cavities was shown to map to the outside of the unit cell,
demonstrating that powder could be evacuated throughout the unit cell. The sintered or trapped
regions within the octet-truss and lack thereof in the opti-octet, confirm the findings that the opti-
octet shows improved powder removability.
For the 7 mm unit cell size the finding from the mass measurement suggested that these
samples matched very closely with the as-designed model. In some cases, the relative error was
also negative. For the opti-octet cross section, the circles at the bottom of the image are visibly
cutoff, showing the loss in material that accounts for the negative relative error. In addition, due
to the need to section these samples into four vertical columns each, the CT slice shown in Figure
5.10 was slightly offset from the center. Overall, the CT scans show clean cross sections with
channels and cavities clear of unsintered powder, validating the findings that the largest size for
both unit cells were successful in removing all powder from the internal cavities and channels.
Finally, using the as-designed models for the 4.5 mm unit cell size, the tortuosity of each
unit cell topology was computed using pore network analysis in Avizo. Tortuosity is a measure of
how tortuous a path is, in other words, a quantitative measure reflecting how complex the maze is
to remove powder from a structure. In its most basic form, tortuosity is defined as the ratio of
curve length to chord length, meaning a value of 1 is ideal and the more tortuous the path, the
higher the ratio. In the pore network analysis calculates the hydraulic tortuosity. Using a pressure
difference applied across the sample, the absolute permeability of the structure is calculated. With
the flow velocities through each channel known, hydraulic tortuosity is calculated by summing
the length of all velocities and dividing by the sum of the projection of the velocities along the
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flow direction. From the pore network analysis, the octet-truss tortuosity ratio was 1.736. On the
other hand, the opti-octet had a tortuosity ratio of 1.419, an 18.3% improvement.
Using mass measurement comparison of the as-designed and as-printed samples,
qualitative CT analysis, and quantitative comparison of the hydraulic tortuosity, the opti-octet
demonstrates improved powder removability over the octet-truss. These results are consistent
with the PRF factors calculated by ALTO which predicted better powder removability for the
opti-octet than the octet-truss. This case study demonstrates an example of using ALTO to create
a novel lattice structure matching the orthogonal effective moduli of a baseline lattice but
exhibiting improved powder removability over the baseline unit cell.
5.3 Case Study 2: Generation of a 3D Unit Cell Topologies with a Fixed Volume Fraction
The purpose of Case Study 2 was to generate several novel unit cells with different
constitutive matrices but similar interconnectivity to a baseline unit cell design. Assuming an
application of a lattice structure with graded stiffness, the intent was to create different stiffness
without changing the volume fraction and while maintaining connectivity between unit cells. For
a given unit cell size and volume fraction, the homogenized constitutive matrix may be
calculated. To adjust the stiffness of a lattice, one typically increases or decreases the volume
fraction, changes the unit cell topology, or a combination of both. Increasing or decreasing the
volume fraction changes the unit cell weight. As a result, a lattice with different unit cells of
varying stiffness generally has a non-uniform weight distribution which may be undesirable for
the application. Changing the unit cell topology within a lattice may lead to connectivity issues
between unit cells. ALTO can be used to generate unit cells with different constitutive matrices
but similar connectivity and volume fraction. In this case study, the goal was to generate three
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different unit cell topologies that all have the same volume fraction and similar interconnectivity
but distinct stiffness properties from their homogenized constitutive matrix.
5.3.1 Optimization Description
For this optimization three different objective functions were included: 1) minimization
of error between the target and calculated homogenized constitutive matrix, 2) minimization of
error between the target and calculated volume fraction, and 3) minimization of relevant
elements. This optimization problem also utilized the overlay and DfAM unsupported–N
penalties. A formal description of the optimization problem is given in Eqns. (5.11)-(5.15) with
additional details in Section 3.2.2.2. A summary of the objectives, constraints, and penalties is
shown in Table 5.5 and described in the following paragraphs.
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 (𝑊𝐻 ,𝑊𝑉𝐹 ,𝑊𝑅𝐸)
𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝐴𝑒 − 𝐴𝑙𝑜𝑤𝑒𝑟 ≥ 0 (5.11)
𝐴𝑒 − 𝐴𝑢𝑝𝑝𝑒𝑟 ≤ 0 (5.12)
𝐴𝑒 = 𝐴𝑙𝑜𝑤𝑒𝑟 𝑓𝑜𝑟 𝐴𝑒 < 𝐴𝑟𝑒𝑙 (5.13)
𝑊ℎ𝑒𝑟𝑒 𝑊𝑖 = 𝑚𝑖(𝑠𝑖𝑃 + 𝑤𝑖(𝑨, 𝑳)) (5.14)
𝑃 =∑𝑝𝑗(𝑨)
𝑛𝑝
𝑗
, 𝑓𝑜𝑟 𝑗 = 1…𝑛𝑝 (5.15)
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Table 5.5 Case Study 2: 3D homogenized unit cells with different stiffnesses optimization summary
Min. Homogenization Error (𝑚𝑖, 𝑠𝑖) = (1,100)
Min. Volume Fraction Error (𝑚𝑖 , 𝑠𝑖, 𝑉𝑓∗) = (1,100, .085)
Min. Relevant Elements (𝑚𝑖, 𝑠𝑖) = (1,100)
Constitutive Scalar (𝑏𝑚) 1
4,1
2, 1
Interconnectivity Nodes See Figure 5.11(E) and (F)
Symmetry XYZ midplanes
𝜃𝑙𝑜𝑤𝑒𝑟, 𝜃𝑢𝑝𝑝𝑒𝑟 (−1, 35) 𝑑𝑒𝑔𝑟𝑒𝑒𝑠
𝑑𝑟𝑒𝑙 0.5𝑚𝑚 𝑑𝑙𝑜𝑤𝑒𝑟, 𝑑𝑢𝑝𝑝𝑒𝑟 (9.5𝑥10−5, 1.5) 𝑚𝑚
Penalties Overlap, Unsupported-N
Borg Evaluations 90,000
Objective 1 was to minimize the total relative error from the target constitutive matrix.
For additional details regarding the homogenization method, see Section 3.2.3.1.3 or
[297,298,361]. The baseline unit cell (Dbase) used to determine the target constitutive matrices
was a diamond unit cell, unit cell size 8 mm and strut diameters 1 mm; see Figure 5.11A and
Figure 5.11B. The diamond lattice structure was chosen for its common use in other lattice
structure research. The graph representation showing the locations of the unit cell struts is shown
in Figure 5.11C and Figure 5.11D. After calculating the Dbase homogenized constitutive matrix,
�̃�𝐷𝐻, shown in Eqn. (5.16), the target constitutive matrices were calculated as scalar multiples
using Eqn. (5.17), where 𝐸 is the material Young’s modulus and the 𝑚-th target constitutive
matrix is scaled by 𝑏𝑚. For this case study, three optimizations were set up 𝑚 = 1,2,3 for 𝑏𝑚 =
1
4,1
2, 1, respectively. Finally, the objective function for objective 1 is shown in Eqn. (5.18).
�̃�𝑫𝑯 = 𝐸
[ 𝟗. 𝟓 𝟗. 𝟓 𝟗. 𝟓 𝟎 𝟎 𝟎
𝟗. 𝟓 𝟗. 𝟓 𝟎 𝟎 𝟎𝟗. 𝟓 𝟎 𝟎 𝟎
𝟎 𝟎 𝟎𝟎 𝟎
𝒔𝒚𝒎 𝟎]
𝑥10−3 (5.16)
�̃�𝑚∗ = 𝑏𝑚�̃�𝐷
𝐻 (5.17)
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𝑤𝐻 = ∑
(𝐸𝑘𝑙∗ − 𝐸𝑘𝑙
𝐻)
𝐸𝑘𝑙∗ 𝑓𝑜𝑟 𝑘, 𝑙 = 1…6 (5.18)
Objective 2 was to minimize the relative error between the calculated and target volume
fraction; see Eqn. (5.19). The target volume fraction was 8.5%, the same as the volume fraction
of Dbase. Objective 3 was to minimize the number of relevant elements to reduce complexity of
the unit cell; see Eqn. (5.20).
𝑤𝑉𝐹 =|𝑉𝑓
∗ −𝑨𝑇𝑳𝑉𝑢𝑐
|
𝑉𝑓∗
(5.19)
𝑤𝐴𝐸 = 𝑐𝑜𝑢𝑛𝑡(𝐴𝑒 > 𝐴𝑟𝑒𝑙) (5.20)
Beginning with a 5x5x5 cubic array of nodes, each node was connected to every other
node in the structure, resulting in 7750 elements or design variables. Next, the optimization
constraints were the relevant diameter limit and bounds on the upper and lower limit of the cross-
sectional area. Ground structure restrictions were limits on the interconnectivity nodes based on
Dbase (see Figure 5.11), surface elements, build angle, and symmetry. Cubic symmetry across
the XYZ mid-planes was applied such that the interconnectivity restriction was maintained. The
optimization penalties were overlap and unsupported‒N. After application of the ground structure
restrictions, the total number of design variables was reduced from 7750 to 67. Figure 5.11E and
Figure 5.11F show the side and isometric views of the initial ground structure for the
optimization, where each line represents a truss element.
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Figure 5.11 Case Study 2 is a multi-objective optimization for creating a lattice structure that mimics
the homogenized constitutive matrix of a Diamond unit cell (A) and (B). The unit cell graph is shown
in (C) and (D). To ensure similar interconnectivity to the Diamond unit cell, the full ground structure
is reduced to the initial configuration shown in (E) and (F).
5.3.2 Optimization Results
The results of the case study were three unique cells with homogenization matrices
matching those of the target constitutive matrices. Examples of these unit cells are shown in
Figure 5.12. The unit cells in each solution were found to be based on a 4-element pattern, with
the best shown in Figure 5.12C. This four-element configuration is referred to as a tetrahedra.
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Figure 5.12 Case Study 2 resulted in three unit cells which match the scaled target constitutive
matrix of the diamond unit cell. In each solution, a fundamental tetrahedra (grey) shape was found
with decreasing diameter. For the middle and top solutions, several non-load bearing elements (blue)
form a mechanism around the tetrahedra shape, based on the pin-joint assumption, to
simultaneously match the volume fraction and homogenization targets.
As shown in Figure 5.12, each optimization found a variation of a simple tetrahedra
pattern that exactly mimics the diamond unit cell constitutive matrix properties. This tetrahedra
fulfills two of the three objectives by matching the target constitutive matrix and minimizing the
number of elements to only four. To achieve the final objective, matching the volume fraction of
the diamond unit cell, the optimization added elements that contribute to the weight but not the
209
stiffness, shown in a blue color in Figure 5.12. That is, these elements do not change the
homogenized constitutive matrix because they rely on the pin joint assumptions at the nodes,
meaning that the elements act like a mechanism and do not increase the stiffness of the unit cell.
From here, each design could be further refined through designer interpretation to more
accurately represent a hinged mechanism. It is interesting to note that the solution shown in
Figure 5.12B has a ring of elements that surround the tetrahedra, never connecting. This ring was
formed as additional weight to match the volume fraction and by design interpretation could be
made to bear very little load if the elements were reduced in size near the upper and lower node.
In the end, both the 𝑏𝑚 =1
4,1
2 solutions represent a load bearing tetrahedra with mechanism-like
elements for adding weight without stiffness.
Regarding the 1x stiffness cell result, Figure 5.12C, the solution is referred to as the
tetrahedra unit cell, which when patterned forms an interlocked lattice. As compared to the
diamond unit cell, the tetrahedra represents a more scalable solution because the smallest feature
size is larger than that of the diamond unit cell. In addition, powder removal from the interlocked
lattice is expected to be easier due to the larger spacing between lattice struts than the diamond
unit cell. On the other hand, the longer struts are likely to make the tetrahedra unit cell more
prone to buckling.
By comparison of both the tetrahedra and diamond unit cells, the tetrahedra shape is
actually a small section of the original diamond structure. Upon further inspection of the diamond
unit cell, one can see that the tetrahedra is repeated 4 times within the unit cell; see Figure 5.13.
From this finding, it may be argued that both structures are the same. However, the diamond and
interlocked lattice structures are indeed different because the repetition or patterning of this unit
cell results in a different structure; see Figure 5.13. In the diamond lattice, the tetrahedra is
repeated in a checkerboard pattern as shown in the patterned lattice schematic. For the interlocked
lattice, the repetition is continuous; see Figure 5.13. The continuous patterning allows for the
210
same predicted constitutive matrix while using larger element diameters and fewer elements than
the diamond lattice. Due to objective 4, the minimization of relevant elements, fewer and thicker
elements mean that the structure is better for powder removal than the diamond lattice and can be
more scalable for printing than the diamond lattice because the smallest diameters of the
tetrahedra were larger than the diameters for the diamond unit cell.
Figure 5.13 The interlocked lattice is a unique patterning of the tetrahedra base that results in a
scaled and interlocked set of two diamond lattice. In the final patterned interlocked lattice, the unit
cells only connect to diagonally neighboring unit cells.
One unique result was that the interlocked lattice was named such because after
patterning, it formed two interlocking lattices that move independent of each other, much like
interconnected chain mail. Figure 5.14 shows the interlocked lattice, with the blue lattice being
independent from the grey. The unit cells connect to diagonally neighboring unit cells, as opposed
to connecting to other unit cells neighboring the 6 cubic faces.
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Figure 5.14 The Interlocked lattice is a set of two interconnected lattices. The upper lattices were
printed in Alumide® on an EOS P110. The lower left image shows the interlocking lattices,
distinguished by the blue and grey colors.
Case Study 2 demonstrates the ability to create novel unit cell topologies with specified
properties and interconnectivity to neighboring unit cells. For a graded lattice stiffness, changing
unit cell topology or reducing the volume fraction within a lattice can lead to issues of
interconnectivity between unit cells or varying weight distribution. By fixing the volume fraction
and interconnectivity between lattice structures, ALTO manipulates the unit cell topology to vary
the lattice stiffness while enabling a uniform weight distribution. The three solutions from Case
Study 2 represent a novel approach to tailoring lattice stiffness while simplifying
interconnectivity for the same weight. The optimization accomplishes this by combining
mechanism-like behavior into novel lattice structure designs. Using the DfAM objective function
minimization of relevant elements, a novel tetrahedra unit cell was discovered which offers the
same theoretical constitutive matrix as the diamond lattice and larger spacing than the diamond
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lattice for powder removability. In addition, the tetrahedra unit cell has greater scalability because
the diameter of the struts is larger than the diameter of the diamond unit cell struts.
5.3.3 Fabrication and Compression Testing
For demonstration of printing the diamond and interlocked lattice structures, a set of
diamond and interlocked lattices were and evaluated using uniaxial compression testing. The test
specimens were built on the same build plate and follow the same specimen format and testing
procedure as in Section 5.2.3.1. The strut diameters for the diamond were a uniform thickness of
0.5 mm. For the interlocked lattice, the 1x stiffness optimized result was printed using a uniform
diameter 0.707 mm.
The average stress-strain curves from four samples each for the diamond and interlocked
lattices are shown in Figure 5.15. The grey band for each curve represents the average plus and
minus one standard deviation of the experimental data. The peak load for each lattice was
determined to be 9.2 ± 0.03 MPa and 5.9 ± 0.10 MPa for the diamond and interlocked lattices,
respectively. Based on the stress strain curves, the linear region was determined to be between
7.5-60% of the peak load. In Figure 5.15, the effective modulus region for each lattice is denoted
by the solid portion of the stress strain curves. From this region, the effective modulus of the
lattice was calculated using a linear least squares fit. The diamond and interlocked lattice
effective moduli were calculated to be 305 ± 4.3 MPa and 162 ± 6.8 MPa, respectively.
213
Figure 5.15 Average stress strain curves for the diamond and interlocked lattices. The solid portion
of the curves represents the region where the effective modulus was measured, 7.5-60% of the peak
stress. The grey band behind each curve represents the average plus and minus the standard
deviation of the experimental data.
Based on the need to relax the boundary condition discussed for the previous case study,
the ground structure formulation in ALTO is not capable of accurately capturing the effective
moduli of the diamond and interlocked lattices. With the “relaxed” boundary conditions as
described in Section 5.2.3.3, the diamond and interlocked lattices become mechanisms. For
mechanism-like behavior, the effective modulus captured during experimental compression tests
is related directly to the stiffness at the nodes as opposed to axial stiffness in the struts. With a
significantly higher number of nodes in the diamond structure than the interlocked, it is not
surprising that the effective modulus is higher in the diamond lattice than the interlocked lattice
when comparing the results for “relaxed” boundary conditions measured experimentally. For a
ground structure model composed of truss elements, the pin joints provide no stiffness and a zero-
stiffness result is calculated in the homogenized constitutive matrix from ALTO. However, with a
raw material Young’s modulus of 115 GPa, the experimental results further support the need to
214
relax the boundary conditions, with the extremely low calculated effective modulus values, only
0.27% and 0.14% when normalized by the raw material Young’s modulus.
5.4 Case Study 3: Generation of a 2D Orthotropic Unit Cell Topology
Up to this point, the case studies have been for isotropic materials, materials which
exhibit the same effective mechanical properties in the three primary orthogonal directions.
ALTO theoretically has the ability to match any constitutive matrix. Case Study 3 is a 2D
example of generating a unit cell to match an orthotropic target constitutive matrix. The target
constitutive matrix was selected so that the effective modulus of unit cell was three times larger in
the X-direction than the Y-direction; see Figure 5.16.
Figure 5.16 The purpose of Case Study 3 is to generate a unit cell topology that has an orthotropic
constitutive matrix, specifically that it is three times stiffer in the X-direction than in the Y-direction.
5.4.1 Optimization Description
For the optimization, a single objective function was utilized, minimization of error
between the target and calculated homogenized constitutive matrix. This case study used the
gradient-based DfAM penalty for minimizing complexity and the DfAM relevant diameter limit
constraint. The optimization was performed using the gradient-based algorithm fmincon with a
volume fraction constraint on the total volume. A formal description of the optimization is shown
215
in Eqns. (5.21)-(5.27). A summary of the optimization problem is shown in Table 5.6. For more
details on the optimization formulation or objective, constraint, and penalty functions see Chapter
3.
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓(𝑊, 𝑃)
𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝐴𝑒 − 𝐴𝑙𝑜𝑤𝑒𝑟 ≥ 0 (5.21)
𝐴𝑒 − 𝐴𝑢𝑝𝑝𝑒𝑟 ≤ 0 (5.22)
𝑨𝑇𝑳 − 𝑉𝑓∗𝑉𝑚𝑎𝑥 ≤ 0 (5.23)
𝐴𝑒 = 𝐴𝑙𝑜𝑤𝑒𝑟 𝑓𝑜𝑟 𝐴𝑒 < 𝐴𝑟𝑒𝑙 (5.24)
𝑊ℎ𝑒𝑟𝑒 𝑓 = (𝑊)(𝑃) (5.25)
𝑊 = 𝑐𝐻𝑚𝐻𝑤𝐻(𝑨, 𝑳) (5.26)
𝑃 = 𝑝(𝑨, 𝑳) (5.27)
Table 5.6 Case Study 3: 2D orthotropic unit cell generation optimization summary
Min. Homogenization Error 𝑚𝑖 = 1
Volume Fraction Constraint 𝑉𝑓∗ = 0.15
Material Modulus Ratio (𝛼) 3
Poisson’s Ratio (𝜈12) 0.6
Shear Modulus (𝐺12) 𝐸𝑥𝑥
3(1 + 𝜈𝑥𝑦)
Target Constitutive Matrix (�̃�∗) 𝐸𝑥𝑥 [1.136 0.227 0
0.379 0𝑠𝑦𝑚 0.208
]
DfAM penalty function parameters (𝛼1, 𝛼2, 𝛼3, 𝛼4) = (2, 100,100
99,98
99)
DfAM penalty function levels 𝛾 = 0,1
6,1
3,1
2,2
3
𝑑𝑟𝑒𝑙 0.3, 0.5, 1.0 mm
𝑑𝑙𝑜𝑤𝑒𝑟, 𝑑𝑢𝑝𝑝𝑒𝑟 (1𝑥10−6, 2) mm
𝜃𝑙𝑜𝑤𝑒𝑟, 𝜃𝑢𝑝𝑝𝑒𝑟 (1, 35) degrees
Symmetry XY midplanes
The design space for the optimization problem was a 5 mm square unit cell discretized by
a 4x4 array of nodes. The objective for the optimization was to minimize the error between the
target constitutive matrix and the calculated homogenized constitutive matrix. In this formulation,
the solution was restricted to 2D orthotropic materials, resulting in a [3x3] symmetric constitutive
matrix defined by two elastic moduli, two Poisson’s ratios, and the shear modulus shown in Eqn.
216
(5.28), where �̃� is the constitutive matrix, 𝐸𝑥𝑥 and 𝐸𝑦𝑦 are elastic moduli, 𝜈𝑥𝑦 and 𝜈𝑦𝑥 are
Poission’s ratios, and 𝐺𝑥𝑦 is the shear modulus for a plane stress condition. Using symmetry, the
terms in the 𝐸12 and 𝐸21 positions are equal, resulting in Eqn. (5.29), where 𝛼 is the ratio of the
moduli (Exx
Eyy) and Poisson’s ratios (
𝜈𝑥𝑦
𝜈𝑦𝑥), resulting in a constitutive matrix defined by four
parameters (𝐸𝑥𝑥 , 𝜈𝑥𝑦 , 𝐺𝑥𝑦, 𝛼). According to the problem statement 𝑎 = 3 so that the effective
modulus in the X-direction is three times larger than in the Y-direction. In this case, the focus of
the constitutive matrix was the effective modulus in the X- and Y-directions so the value of 𝜈𝑥𝑦
was chosen arbitrarily to be 0.6. In summary, by setting (𝜈𝑥𝑦, 𝐺𝑥𝑦, 𝛼) = (. 6,𝐸𝑥𝑥
3(1+𝜈𝑥𝑦), 3) the final
target constitutive matrix was determined, normalized by 𝐸𝑥𝑥; see Eqn. (5.30). The objective
function for the optimization is shown in Eqn. (5.31).
�̃� =
[
𝐸𝑥𝑥1 − 𝜈𝑥𝑦𝜈𝑦𝑥
𝜈𝑦𝑥𝐸𝑥𝑥
1 − 𝜈𝑥𝑦𝜈𝑦𝑥0
𝜈𝑥𝑦𝐸𝑦𝑦
1 − 𝜈𝑥𝑦𝜈𝑦𝑥
𝐸𝑦𝑦
1 − 𝜈𝑥𝑦𝜈𝑦𝑥0
0 0 𝐺𝑥𝑦]
(5.28)
�̃� =1
𝛼 − 𝜈𝑥𝑦2 [
𝛼𝐸𝑥𝑥 𝜈𝑥𝑦𝐸𝑥𝑥 0
𝐸𝑥𝑥 0
𝑠𝑦𝑚 (𝛼 − 𝜈𝑥𝑦2 )𝐺𝑥𝑦
] (5.29)
�̃�∗ = 𝐸𝑥𝑥 [
1.136 0.227 00.379 0
𝑠𝑦𝑚 0.208] (5.30)
𝑤𝐻 =
∑|𝐸𝑘𝑙∗ − 𝐸𝑘𝑙
𝐻 |
𝐸𝑘𝑙∗ 𝑓𝑜𝑟 𝑘, 𝑙 = 1,2,3 (5.31)
For optimization constraints, the case study has a limit on the upper and lower bounds of
the element cross-sectional areas, a relevant diameter limit, and a total volume fraction limit.
Ground structure restrictions include the build angle limit and symmetry across the horizontal and
217
vertical midplanes. The restrictions reduced the number of design variables from 120 to 30.
Depending on the overall size of the unit cell, high complexity within the unit cell could result in
trapped powder or fused members. A DfAM penalty function to reduce unit complexity was
applied for the gradient-based algorithm. The function penalizes elements near the relevant
diameter limit, encouraging fewer and thicker elements; see additional details in Section
3.2.3.3.2. The penalty function was controlled by four parameters and then scaled by an exponent,
the penalty function level, to change the level of impact from the penalty function. The
parameters chosen for this optimization are given in Table 5.6 and an image for the resulting
penalty function is shown in Figure 5.17. In addition to the penalty function to reduce complexity,
increasing the relevant diameter limit forces fewer members due to the volume constraint. This
effectively reduces the total number of elements that can be used to form the unit cell. Because
the gradient-based algorithms are prone to find local minima, multiple initial guesses were used
to search the design space. DfAM considerations applied to this case study were the build angle
limit, relevant diameter limit, and penalty function to reduce overall unit cell complexity.
Figure 5.17 Graph of the DfAM penalty function that penalizes elements near the relevant cross-
sectional area limit and reduces non-critical elements. Image adapted from [361].
218
5.4.2 Optimization Results
To explore the penalty function and relevant diameter limit, a range of values were used
for each. For each penalty function level (𝛾 = 0,1
6,1
3,1
2,2
3) and relevant diameter limit
(𝑑𝑟𝑒𝑙 = 0.3,0.5,1.0), the optimization was initialized from 20 different randomly generated initial
guesses. The range of penalty function levels allows for varied impact from the penalty function.
For 𝛾 = 0, there was no penalty factor. With each increasing penalty factor level, the impact of
the penalty increases, providing a range of solutions. From 20 different initial guesses for each
combination of penalty function level and relevant diameter limit, the results represent a total of
300 different instances of the optimization problem. The total computation wall time was
approximately 3 hours using 6 cores. Figure 5.18 shows the best solutions from 20 randomized
initial guesses, each for a different penalty factor level and relevant diameter limit. Though there
were similar features between many of these designs, the figure shows that there were a variety of
solutions that closely matched the target constitutive matrix. The penalty function and relevant
diameter limit may be used to further refine the solution set and promote or penalize designs with
characteristics that are beneficial or challenging for AM processes.
219
Figure 5.18 Lowest optimization scores for the constitutive matching objective function. Each image
represents the best solution of 20 initial guesses for the particular combination of penalty factor level
and relevant diameter limit. Image adapted from [361].
In this case study, both the penalty function and relevant diameter limit were used to
encourage simpler solutions by encouraging elements cross-sectional area larger than the relevant
diameter limit, thereby reducing the total number of relevant elements. In general, viewing Figure
5.18, as you look from left to right (increasing relevant diameter limit) or from top to bottom
(increasing penalty factor level), the complexity of the solution decreases, showing fewer and
thicker members within the unit cell. While both the penalty and relevant diameter were
successful in reducing the complexity of the unit cell topology, this comes at a cost of error in the
homogenized constitutive matrix because there were fewer elements in a unit cell and tighter
220
limits on the diameters of each element. For example, of the 20 random initial guesses for 𝛾 =
0 , 𝑑𝑟𝑒𝑙 = 0.3, 95% of the designs had a total error in the constitutive matrix of less than 10%.
However, for 𝛾 =2
3, 𝑑𝑟𝑒𝑙 = 0.3 only 25% of designs had an error of less than 10%. For 𝑑𝑟𝑒𝑙=1.0,
only 3 solutions out of out all 5 penalty function levels with 20 initial guesses each (100
optimizations in total) were able to achieve a total error of less than 10%. Therefore, reducing the
complexity of the unit cell was desirable but it may come at the expense of increased error for
matching the target constitutive matrix.
Another method to reduce cell complexity was to reduce the number of elements based
on the node configuration or the interconnectivity ground structure restriction. For example,
setting the cross-sectional area of elements connected to the corners to 𝐴𝑙𝑜𝑤𝑒𝑟 forces those
elements to be uneconomical and reduces the number of design variables to only 18. Figure 5.19
shows an example of a unit cell generated with 𝑑𝑟𝑒𝑙 = 0.5 and 𝛾 =1
3 and no elements connected
to the corners. The deformed shape of the unit cell is shown under the same compression load
applied horizontally Figure 5.19A and vertically Figure 5.19B. As shown in the figure, the
deformation was three times larger in the vertical direction than it was in the horizontal direction,
demonstrating the target constitutive matrix. Figure 5.20 shows the patterned lattice structure
based on the unit cell topology from Figure 5.19.
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Figure 5.19 Optimized unit cell topology with the area of the elements connected to the corners set to
the lower limit. Compression in X and Y is also shown, demonstrating the target constitutive matrix.
Total error in the constitutive matrix was 3x10-6 %. Image adapted from [361].
Figure 5.20: Example of a lattice structure patterned from the unit cell topology shown in Figure
5.19. This unit cell topology was generated using a penalty function level of one third and the relevant
diameter limit set to 0.5 mm. Image adapted from [361].
A) X-Compression
B) Y-Compression
Y
X
�̃�𝑯 = 𝐸𝑥𝑥 [1.136 0.227 0⬚ 0.379 0𝑠𝑦𝑚 ⬚ 0.208
]
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Case Study 3 demonstrates the ability of ALTO to successfully generate unit cell
topologies that match an orthotropic constitutive matrix. The DfAM considerations and penalty
function reduced complexity and improved manufacturability of the lattice structure. With this
approach, novel metamaterials may be developed and customized for specific applications
through optimization.
5.5 Discussion of Results
While the results for each case study show generation of novel unit cell topologies,
topology optimization through an approach like ALTO is not without its limitations or challenges.
The focus of this discussion section is on results found from each of the case studies. Excluded
from this discussion are many of the general limitations and challenges associated with ground
structures, which were covered previously in Section 4.4.
Case Study 1 demonstrates a novel powder removability objective function for improved
manufacturability. Using the PRF, the opti-octet unit cell topology was generated and
demonstrated improved powder removal over the baseline octet-truss unit cell. Aside from
generation of the novel unit cell, this case study also included experimental validation of ALTO
through compression testing of the two lattice structures. Two AM modifications were considered
and discussed in this section including a novel build angle dependent effective cross-sectional
area to account for the staircase effect. The ALTO homogenization calculation was found to be an
accurate prediction of the compressive effective modulus when appropriate boundary conditions
were considered. One limitation of the homogenization approach was the boundary conditions, as
discussed in Section 5.2.3.2. For lattice structures, it has been shown in other research that the
number of layers can significantly change the experimentally calculated effective properties
[207]. In the work by Lei et al. [207], experimental data and finite element analysis are from
223
compression testing of lattice samples with 1, 3, 5, and 7 layers, the theoretical values are
calculated based on single unit cell for which the number of layers is cannot be taken into
account. Table 5.7 shows data from Lei et al. [207] for two different lattice types that further
supports the changing boundary conditions for lattices with different numbers of layers.
Table 5.7 Lattice structures demonstrate a substantial difference in mechanical properties based on
the number of layers. As the number of layers increase, the effective modulus decreases, data from
[207].
Lattice Data Type 1 Layer 3 Layers 5 Layers 7 Layers
BCC Theoretical 225.00 225.00 225.00 225.00
Experimental 198.62 ± 2.36 87.11 ± 1.24 39.16 ± 1.07 21.71 ± 0.86
FEA 210.79 109.68 41.79 18.11
BCCZ Theoretical 554.90 554.90 554.90 554.90
Experimental 490.22 ± 3.41 378.32 ± 2.87 330.31 ± 2.66 317.07 ± 2.05
FEA 557.87 443.26 391.40 366.70
A challenge with lattice structures is that depending on how they are applied within a
structure they may exhibit very different performance. For example, consider a component that
has an embedded lattice for lightweighting and is surrounded by solid material. In this case, the
lattice would be highly constrained by the solid material surrounding it and exhibit a high
stiffness. However, consider a second component in which the lattice is not surrounded by solid
material. In this second case, if the lattice is left unconstrained by a surrounding structure, the
stiffness of the lattice would be significantly less than in the first component. While other
traditional materials may exhibit a similar behavior, the result may be exaggerated for lattices
because the stiffness of bending at the nodes is significantly less than the axial stiffness along the
truss element. In unconstrained cases, the bending or deformation near the nodes significantly
reduces its stiffness. Despite this challenge, the effective mechanical properties can be accurately
predicted if the boundary conditions are carefully considered. As with the example of Case Study
1, the effective modulus predicted in ALTO was found to match within 6% to the experiment
data.
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One other limitation demonstrated with ALTO was the ability to capture geometric
variations based on the AM manufacturing. In this case, AM manufacturing modifications were
considered for correcting the ground structure prediction to match the as-printed geometry of the
lattice structure. These corrections were for the staircasing effect and reduction in the cross
section based on the laser scan strategy. The two corrections showed promise in improving the
predicted result but other factors such as the 1) dross on horizontal elements or 2) geometric
anisotropy due to the build process were not accounted for. For example, on horizontal elements
it is known that a circular cross-section becomes ellipsoidal due to the extra material that is
picked up on the downskin surface. At the bottommost extreme of this ellipse, the powder is
likely lightly sintered and would not significantly contribute to the stiffness, meaning the entire
ellipse cross section is not contributing to the stiffness. However, some of the added material is
likely to make a contribution to the stiffness. The question then arises, should this correction and
prediction be performed as part of the topology optimization step or corrected as the part is sliced
model in preparation for printing to more accurately represented the as-designed geometry.
Likely there is a balance of both steps that needs to occur.
A second issue mentioned is the anisotropic behavior of printed lattice structures,
specifically due to geometric discrepancies in the as-designed versus as-printed topology, as
opposed to material anisotropy due to AM. The dross on horizontal elements, mentioned earlier,
means that for elements that were printed vertically, the cross-section should circular compared to
the elliptical cross-section of horizontal elements. As a result, the AM process geometrically
induces anisotropy when compared to the original as-designed model. For example, in this work,
the primary load path for compression testing of the opti-octet would be along the four vertical
columns with the circular cross-sections. However, if tested in X- or Y-axis compression, the
result is likely to vary due to the primary load path now along the ellipsoidal cross-sections.
While these geometric variations due to surface roughness or dross occur in large non-lattice parts
225
as well, the issue is likely to be exaggerated for lattices because the surface roughness is much
closer to the lattice feature sizes than in a non-lattice part.
In Case Study 2, the optimization was successful in finding solutions that met the desired
criteria. However, early work on this case study highlighted several potential issues that required
additions or corrections in ALTO for 3D unit cell topologies. Two examples of these were
overlapping and disconnected material. Overlapping or intersecting elements at non-nodal
locations was common in many cases because there were so many elements crossing in the unit
cell. Due to the discrete nature of the optimization, it determines the effective mechanical
properties by beams connected to the array of nodes. If elements intersect at non-nodal locations,
ALTO sees these as separate elements, essentially sliding past each other, as opposed the
inevitable fusion that would occur during printing. As a result, this gave rise to the penalty
function which discourages designs with overlapping elements in the solution, resulting in more
print-ready solutions and less design interpretation.
Disconnected material was an issue that arose in the 𝑏𝑚 =1
4,1
2 examples. In these cases,
the optimization was trying to increase the volume fraction without changing the constitutive
matrix. One of the simplest ways to accomplish this was to include elements that never connect to
the load bearing structure, thereby increasing volume fraction without stiffening the structure. As
opposed to simply requiring each individual element to meet the build angle requirements, the
support structure penalties incorporate a more rigorous approach to ensure each element is
buildable.
Without the DfAM build angle restriction or unsupported‒N penalty applied to Case
Study 2, the solutions would not be feasible to print. Figure 5.21 shows a series of results that
occur without DfAM restrictions and penalties. Due to the addition of DfAM restrictions, such as
the build angle, overlap penalty, and unsupported penalties, the optimization resulted in more
print-ready solutions requiring less design interpretation.
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Figure 5.21 Without DfAM restrictions and penalties to ensure manufacturability, there were a
variety of pitfalls that appear in optimization solutions. The DfAM considerations help to reduce the
need for design interpretation and prepare more print-ready solutions.
Finally, for Case Study 3, the result demonstrated a method to reduce unit cell complexity
for DfAM using a gradient-based penalty function; however, a major challenge continues to be
the ability to reduce complexity and match a target constitutive matrix. With such a variety of
solutions that were able to closely match the constitutive matrix, there were numerous local
minima where the optimization became trapped. The penalty function was effective in reducing
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complexity of the unit cell, however, only to an extent due to the tradeoff with the constitutive
matrix error.
One of the primary challenges or limitations with Case Study 3 was the repeatability of
gradient-based algorithms. In this type of problem there were many different solutions that could
match the target constitutive matrix. For a given starting ground structure, the algorithm was
consistent in finding the same results; however, it was clear that numerous local minima existed
in the solution space because the 20 different randomized starting guesses frequently resulted in a
variety of different solutions. This result was caused by the single search direction nature of
gradient-based algorithms versus the multiple search paths of evolutionary algorithms. In general,
gradient-based optimization case studies were found to require significantly more time upfront
tweaking parameters and adjusting the formulation. The parameters and tweaking were problem
specific and have to be repeated with each type of optimization. If ALTO were consistently
solving the same types of problems, the formulation could likely be tuned in for a specific set of
objectives and constraints and take advantage of the faster speed of a single search direction for
gradient based-optimization algorithms as compared to evolutionary algorithms. In contrast, case
studies utilizing evolutionary algorithms were found to be more robust to varying applications
and less affected by tweaking different parameters, at the expense of potentially increased
optimization time. As the intent of ALTO is a systematic optimization method for a variety of
unit cell topologies and an application agnostic purpose, evolutionary algorithms are
recommended. The ability of evolutionary algorithms to handle multiple objectives and explore
multiple search directions was found to be amendable to unit cell topology optimization.
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5.6 Closing Remarks
The focus of this chapter is the use of ALTO for generating unit cell topologies that
match a target constitutive matrix. Three different case studies were presented showing multiple
examples of uses for ALTO in generating novel and customized unit cell topologies. Case Study
1 demonstrates matching the orthogonal effective moduli of a baseline unit cell topology while
showing improved powder removability. In addition, this case study shows compression testing
data to validate the ground structure model. Case Study 2 shows a unique exploration of
generating unit cell topologies with decreasing stiffness while maintaining the same target
volume fraction, forcing the volume to be rearranged, not removed, from the unit cell. Finally,
Case Study 3 shows generation of a unique 2D unit cell that exhibits an orthotropic material
constitutive matrix. In all cases, DfAM considerations increase manufacturability of the solution
to ensure that the optimized solutions can be fabricated with minimal design interpretation and no
support material. AM is a powerful manufacturing process that enables advanced design and
complexity to be incorporated through lattice structures and it requires development of systematic
topology optimization approaches like ALTO to generate novel unit cell topologies and ensure
manufacturability.
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Chapter 6 LUCIE: Lattice Unit-cell Characterization Interface for Engineers
The focus of this chapter will be on the development of a comprehensive compilation of
mechanical properties for lattice structures with regular unit cell topologies through an interactive
interface. The chapter includes a brief summary of the relevant background and motivation, the
methods for gathering sources and extracting data, developing a graphical user interface LUCIE
(Lattice Unit-cell Characterization Interface for Engineers), and finally three examples of how
LUCIE can be used to understand lattice structures. This project was completed with help from
undergraduate student Joseph Berthel. From this work, LUCIE contains 69 sources which
resulted in over 1400 experimental data points, over 200 finite element data points, and over 45
analytical models for visualization of lattice structure. It is the largest known database for
mechanical property data for AM lattice structures [373]. The following objectives and sub tasks
are addressed in this chapter:
5. Develop mechanical property database for metal lattice structures from current literature
containing analytical, finite element, and experimental data
5.1. Perform literature search for metal lattice structure data
5.2. Collect data from selected papers
5.3. Create user interface for generating mechanical property charts of collected data
6.1 Background and Motivation
Lattice structures are frequently designed into a component through standard computer-
aided design (CAD) modeling software (e.g., SolidWorks [374], Autodesk Inventor [375], PTC
Creo [376]) or AM-specific software (e.g., Autodesk Netfabb Ultimate [126], ANSYS
SpaceClaim [238], nTopology nTop Platform [235], 3D Systems 3DXpert [237], Materialise 3-
matic [377]). Within the AM-specific software, a library of unit cell topologies to generate
230
different lattice structures or infill patterns are offered; however, their implications on the bulk
(i.e. macroscale) mechanical properties of the component are unknown in many cases (e.g., the
unit cell topology has inherent mechanical properties in compression and shear that will affect the
overall component performance).
Various methodologies have been recommended regarding the design or inclusion of
lattice structures in a component [228,334]. Tamburrino et al. [334] discuss unit cell selection
based on the choice of either a stretched-dominated or a bending-dominated lattice structure
[334]. However, grouping all truss-based lattice structures into these two categories without
distinguishing them further essentially leaves the decision wide open. In recent work by Bhate
[228], the first of four questions regarding cellular material design is, “What is (are) the optimum
unit cell(s)?” [228]. Bhate describes several methods that have been explored in the literature,
namely, Maxwell’s stability criterion, Gibson-Ashby models of cellular materials, computational
analysis, and experimental testing.
In a recent publication, Maconachie et al. [250] provide a review of topics surrounding L-
PBF lattice structures. The topics include types of lattice structures and their applications,
processing parameters, microstructure and mechanical properties, numerical modeling,
experimental approaches, and outstanding challenges. As discussed in the review, with all the
current research into lattice structures, there is a need to compile and provide a comprehensive
analysis of the data available. This review does compile some experimental data; nonetheless, a
comprehensive compilation of metal AM lattice structure performance data in literature does not
exist.
To expand the current understanding of lattice structure properties and to improve unit
cell selection, this work seeks to compile reported mechanical properties for AM lattice structures
of regular unit cell topologies, gathered through analytical modeling, finite element analysis
(FEA), and experimental testing. The gathered data was compiled and illustrated in Ashby-style
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plots through the Lattice Unit-cell Characterization Interface for Engineering (LUCIE), a
graphical user interface that allows researchers to easily compare mechanical properties of metal
lattice structures based on different unit cell topologies made with AM. Similar to traditional
material selection through Ashby plots [378], this work seeks to make lattice structure selection
available through scatter plots with axes chosen based on lattice structure properties. The primary
intent was not to provide exact mechanical property values, but instead to provide the designer
with visualization of performance trends based on the unit cell topology.
6.2 Methods
In this section, the methods for identifying relevant research articles, gathering and
storing data, visualizing the data, as well as assumptions made during data collection are
described. A graphic summary of the methods section is shown in Figure 6.1
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Figure 6.1 Graphic summary of the methods section including how sources were found and refined,
data collection and assumptions made, and finally incorporation into LUCIE.
6.2.1 Literature Compilation
To identify research articles containing relevant lattice structure data, literature searches
were conducted in multiple databases such as Compendex. The primary goal of the search was to
find research articles from theses, dissertations, conference, and refereed journal articles
containing mechanical performance information for metal lattice structures made with AM
powder bed fusion (laser or electron beam).
With the recent increase in research in AM, there are a wide variety of terms to describe
similar if not the same concepts. In recent years, the term “lattice structure” has become common,
though additional terms such as “metamaterial”, “cellular structure”, or “mesoscale design” are
233
also used, often interchangeably. In addition to these search terms regarding the lattice structure,
there are a variety of search terms for the manufacturing process including “additive
manufacturing”, “rapid prototyping”, “3D printing”, etc. Specific to this paper, the search was for
metal AM such as, “powder bed fusion”, “selective laser melting”, “direct metal laser sintering”,
and “electron beam melting”. To narrow the wealth of papers that these search terms can produce,
other search terms such as “mechanical performance”, “mechanical characterization”,
“mechanical properties”, etc. were included. No restrictions were placed on publication year of
the source. While the intent was to gather all published literature articles containing mechanical
characterization of metal lattice structures made with AM, it became difficult to find all current
and relevant literature due to a wide variety of terminology included in today’s research articles.
In various combinations, the aforementioned keywords were searched using Google Scholar and
Compendex. Additional references were also found through snowballing, a technique for
expanding literature reviews through exploration of relevant sources cited within the compiled
literature findings.
From each paper, three main data types were considered for inclusion: (1) analytical
models, (2) FEA, and (3) experimental. These three data types were defined as follows.
Analytical data was defined as mechanical performance or properties derived from
mathematical equations based on structural shear/normal forces or bending moments. For
example, models that were grouped into the analytical category were often derived using beam
theory, considering each individual strut within the unit cell topology. The result of the papers in
the analytical category was a continuous equation, generally relying on unit cell size and volume
fraction as inputs or independent variables and the predicted mechanical property being the
dependent variable.
Papers grouped in the FEA category were defined in terms of mechanical performance or
properties gathered through modeling and simulation of individual or arrayed unit cells. This data
234
was available in research articles as either data points or extracted using a power law regression
fit as described in Section 6.2.2. Key features of this data type were the requirement to have full
information regarding unit cell topology, volume fraction or strut diameter, and material. To
normalize all results, raw data was divided by the corresponding bulk, or solid, mechanical
property. In cases where the reported data was not normalized, or where the bulk mechanical
property was not given in the source, the raw data was normalized using the mechanical property
values shown in Table 6.1.
Finally, papers in the experimental category were defined in terms of mechanical
performance or properties determined based on experimental testing of metal AM-fabricated
lattice structures. Experimental data was available in published literature as data points in tables
and on graphs. In some sources these data points represented an average from multiple tests;
however, in other sources, data points for all replicates in the experiment were reported as
opposed to a single average value. Similar to the FEA category, a clear description of the
geometry, volume fraction, and, where possible, as-built mechanical properties of solid material
specimens were gathered. In some cases, as-built strut diameter or volume fraction were reported;
however, nominal, or as-designed, values were also collected for consistency. As with the FEA
data type, if the reported data was not normalized, then the mechanical properties from Table 6.1
were used to normalize the data.
Table 6.1 For sources without bulk mechanical properties of the as-built material, reported within
the source, these generic AM mechanical property values were used to normalize data to create a
material independent comparison of unit cell topology.
AM Material Young’s Modulus (GPa) Yield Stress (MPa)
Co-Cr [371] 200 1060
AlSi10Mg [371] 75 270
Ti64 [379] 129 1007
SS 316L [371] 185 530
Hastelloy X [371] 195 630
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After gathering 117 research articles from the literature search, the total number of
articles was reduced to 69 using the following criteria:
• Paper must contain mechanical properties for quasi-static compressive behavior
• Paper must contain a clear depiction of unit cell topology through figures,
pictures, and/or textual description
• Paper must contain a clear description of unit cell volume fraction (or porosity)
• Paper must contain data for a standard/common unit cell topology (available in at
least one additional source)
A detailed breakdown of how the research articles were reduced is as follows. The total
number of articles was reduced from 117 to 109 by first removing those which were duplicates,
or reported data repeated in other sources. Next the total number of articles was reduced to 92 by
considering only those containing lattice structure mechanical performance and properties for
quasi-static compressive behavior. Though a few instances of lattice structure test data for tensile,
shear, bending, or combined loading conditions were found in the literature, the data was
extremely limited, and as such, only quasi-static compressive characterization data was included
in this work [213,217,323,380,381]. For other types of loading conditions, information regarding
lattice structure fatigue performance or high-strain deformation was recently becoming available
(c.f., [188,189,382–385]) but was not included in this work. The selection of papers was further
reduced to 69 due to insufficient information such as an ambiguous depiction of the unit cell
topology or volume fraction and non-standard unit cell topologies for which little data was
available [190,386,387]. Similarly, in some cases, part of the data was excluded based on the
limited data available on a unique topic, such as how the aspect ratio or rotation of the unit cell
topology affects mechanical properties [201,216,219,388]. In general, data was collected that
236
considered regularly shaped unit cells, often cubic, that could be scaled by adjusting the unit cell
size or densified by increasing the thickness of struts or walls.
After the literature search and filtering, metal AM lattice structure data was obtained
from a total of 69 papers that were published between 1997-2019. Table 6.2 shows the references
and a summary of the unit cell topology, material, and the data type gathered from the paper. In
the table, the properties were abbreviated as normalized compressive effective modulus (EC),
yield stress (YS), Poisson’s ratio (PR), plateau stress (PS), and buckling stress (BS). Each of
these properties are defined in the context of this work in the results section (see Section 6.3.1).
The data types were abbreviated as analytical (A), FEA (F), and experimental (E).
Table 6.2 Summarized list of all sources for data compiled in LUCIE. In this table the unit cell
topology, properties recorded, and data type available are sorted by source.
Author Unit Cell Topology Property Data Type Material
Ahmadi [209] Diamond EC, YS, PR,
BS, PS A, E, F Ti-6Al-4V
Ahmadi [211]
Cubic, Diamond, Truncated Cube, Truncated Cuboctahedron, Rhombic
Dodecahedron, Rhombicuboctahedron
EC, YS, PS E Ti-6Al-4V
Arabnejad [364] Octet-truss EC, YS E Ti-6Al-4V
Ataee [389] TPMS Gyroid EC, YS, PS E CP-Ti
Beyer [390] BCC YS E AlSi10Mg
Bobbert [391] TPMS Gyroid, TPMS Diamond EC, YS E Ti-6Al-4V
Borleffs [392] Cubic, Diamond, Rhombic Dodecahedron, Truncated
Octahedron EC, PR A, F Ti-6Al-4V
Campanelli [393] FCCZ YS, PS E Ti-6Al-4V
Cheng [212] Rhombic Dodecahedron EC, YS E Ti-6Al-4V
Choy [388] Cubic EC, PS E Ti-6Al-4V
de Formanoir [366]
Octet-truss EC E Ti-6AL-4V
Deshpande [201] Octet-truss EC, YS A n/a
Gümrük [394] BCC EC, YS E, F SS 316L
Gümrük [380] BCC, BCCZ, F2BCC EC, YS E SS 316L
237
Han [208] Cubic, BCC EC, YS E, F Co-Cr
Harrysson [395] Rhombic Dodecahedron EC, YS E, F Ti-6Al-4V
Hasib [396] Octahedron, Rhombic
Dodecahedron EC, YS E Ti-6AL-4V
He [369] Octet-truss EC, YS A n/a
Hedayati [381] Cubic, Truncated Octahedron,
Rhombic Dodecahedron, Diamond EC, YS, PR F Ti-6Al-4V
Hedayati [204] Rhombicuboctahedron EC, YS, PR A, E, F Ti-6Al-4V
Hedayati [203] Truncated Cube EC, YS, PR,
BS A, E, F Ti-6Al-4V
Hedayati [202] Cubic, Rhombic Dodecahedron,
Diamond, Truncated Octahedron EC, YS, PR A, F Ti-6Al-4V
Hedayati [206] Truncated Cuboctahedron EC, YS, PR A, F Ti-6Al-4V
Hedayati [205] Octahedron EC, PR A, F Ti-6Al-4V
Heinl [397] Diamond EC, YS E Ti-6Al-4V
Heinl [398] Diamond EC, YS E Ti-6Al-4V
Helou [192] TPMS Gyroid, TPMS Diamond EC E, F SS GP1
Hussein [220] TPMS Gyroid, TPMS Diamond EC, YS E Ti-6Al-4V,
AlSi10Mg, SS 316L
Leary [399] BCC, BCCZ, FCC, FCCZ, F2BCC EC, YS E AlSi12Mg
Leary [400] BCC, BCCZ, FCC, FCCZ EC, YS E Inconel 625
Lei [207] BCC, BCCZ EC, YS E, F AlSi10Mg
Li [401] BCCZ EC A AlSi10Mg
Li [402] Cubic, Rhombic Dodecahedron, G7 EC, PS E Ti-6Al-4V
Liu [403] Rhombic Dodecahedron EC E Ti2448
Liu [404] Rhombic Dodecahedron EC E Ti2448
Mazur [405] BCC, FCC, F2BCCZ EC, YS E Ti-6Al-4V
McKown [385] BCC, BCCZ EC, YS, PR E SS 316L
Mines [406] BCC EC, YS E Ti-6Al-4V
Mullen [407] BCC YS E CP-Ti
Murr [408] Rhombic Dodecahedron EC E Co-Cr
Murr [409] Cubic, Rhombic Dodecahedron, G7 EC E Ti-6Al-4V
Murr [410] Rhombic Dodecahedron EC E Inconel 625
Ozdemir [188] Diamond EC, YS E Ti-6Al-4V
Parthasarathy [411]
Cubic EC, YS E Ti-6Al-4V
Ptochos [412] BCC EC, PR A n/a
Rehme [413] FCC, FCCZ, BCC, BCCZ, F2BCC,
F2BCCZ YS E SS 316L
Sallica-Leva [414] Cubic EC, YS E Ti-6Al-4V
Shen [415] BCC, BCCZ YS E SS 316L
Smith [214] BCC, BCCZ EC, YS, PS E, F SS 316L
238
Suard [365] Octet-truss EC E, F Ti-6Al-4V
Tancogne-Dejean [191]
Octet-truss EC, PR F SS 316L
Tsopanos [416] BCC EC, YS E SS 316L
Ushijima [215] BCC EC, PR A, F SS 316L
Ushijima [216] BCC EC, PR, YS A SS 316L
Van Grunsven [417]
Diamond EC, YS E Ti-6Al-4V
Wauthle [418] Diamond EC, YS E Ti-6Al-4V
Xiao [419] Rhombic Dodecahedron EC, YS, PS E Ti-6Al-4V
Yan [221] TPMS Gyroid YS E AlSi10Mg
Yan [218] TPMS Diamond EC, YS E AlSi10Mg
Yan [222] TPMS Gyroid EC, YS E SS 316L
Yan [223] TPMS Gyroid EC, YS E SS 316L
Yan [420] TPMS Gyroid, TPMS Diamond EC, YS E Ti-6Al-4V
Yanez [421] TPMS Gyroid EC, YS E, F Ti-6Al-4V
Yang [422] TPMS Gyroid EC E, F Ti-6Al-4V
Zadpoor [178]
Cubic, Diamond, Truncated Cube, Truncated Cuboctahedron, Rhombic
Dodecahedron, Rhombicuboctahedron, Truncated Octahedron, Octahedron, BCC, FCC
EC, PR, YS, BS
A n/a
Zaharin [423] Cubic, TPMS Gyroid EC, YS E, F Ti-6Al-4V
Zhao [424] Cubic, Rhombic Dodecahedron, G7 EC, PS E Ti-6Al-4V
Zhao [425] BCC EC, YS E Ti-6Al-4V
Zhu [426] Truncated Octahedron PR A n/a
6.2.2 Data Collection
Data was collected from the 69 papers that provided all of the required mechanical
property information for lattice structure characterization. The volume fraction of the tested unit
cell topology was required to compare data from different sources. Additionally, for experimental
and FEA data, the material needed to be given to normalize all the data. As noted earlier, data
was normalized by dividing the reported property value of the lattice structure by the bulk
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mechanical property for the as-built material. Normalization allowed comparison of data taken
from sources using different materials.
Data collection depended on how each paper presented their findings. If data was
explicitly reported as numerical values, then those values were simply catalogued along with the
other required information. In some cases, numerical data was not provided, and plotted graphs
were given instead. To extract the relevant data points, WebPlotDigitizer [427] was used to find
the approximate values of the plotted data. WebPlotDigitizer is a web-based image analysis tool
specifically designed to extract data from images of graphs and plots. The accuracy of the
extracted data was dependent on the figure image quality within the cited source and on the user
error in precisely selecting data points from the extracted image. WebPlotDigitizer has been
recognized as a viable method for data collection when the raw data is unavailable. Several
studies have been performed to verify and validate the use of such digital image data collection
tools [428–430]. Though digital image extraction of data points introduces uncertainty, one study
[429] reported that WebPlotDigitizer resulted in a Lin’s concordance value of 0.982, for which
0.99-1 is considered a near-perfect match [431]. From these studies, the extracted data was found
to be reliable, valid, have little absolute difference from the true values, and have excellent
consistency.
Meanwhile, in eight FEA data type sources [202–206,365,381,394], instead of reporting
specific simulation data points, the data was reported as points connected by line segments, i.e., a
segmented curve. Though in some cases a slight and sudden change in slope indicated the
likelihood of a data point, it was not possible to identify each of these data points on all plots.
Because individual points of the FEA simulation could not be distinguished, a power law
regression fit was used to match the segmented curve as best as possible and extract the data from
the graph. The regression fit of these segmented curves introduced some additional error;
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however, the primary intent of LUCIE was not to provide an exact mechanical property value, but
to provide property trends based on the unit cell type.
The power law regression was based on a popular property-volume fraction model for
cellular structures by Gibson and Ashby [136], where 𝜓 is the property, 𝑉𝑓 is the volume fraction,
and 𝛽1, 𝛽2 are constants determined by the fit of the model to the experimental data; see Eqn.
(3.13).
𝜓 = 𝛽1(𝑉𝑓)𝛽2
(6.1)
A modified power regression equation, with an additional constant, 𝛽3, was used
specifically for the Poisson’s ratio property, ν, to match data that did not intersect the origin (see
Eqn. (6.2)). For these eight sources with segmented curves, instead of individual data points,
WebPlotDigitizer was first used to extract data points along the curve. Second, the power law
regression fit coefficients for the extracted points were solved using Desmos [432]. Desmos is an
online graphing calculator that can fit a regression model to a set of data points.
𝜈 = 𝛽1(𝑉𝑓)𝛽2 + 𝛽3 (6.2)
6.2.3 Assumptions for Comparing Data
To compare data from different papers, many assumptions had to be made. Data was
collected only from papers that tested metal prints made with laser or electron beam powder bed
fusion processes. It was assumed that all types of metallic materials and the printing process
parameters were valid. We also assumed that data from different metallic powder feedstock was
comparable when normalized (e.g., different suppliers provide equivalent grades of Ti-6Al-4V
feedstock). Where possible, the normalized value was calculated using the mechanical property of
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the as-built material reported in the paper. In cases where the as-built material was not
experimentally characterized or given in the paper, bulk values were chosen from published data
sheets. The mechanical property values of different materials built with AM are given in Table
6.1.
Additionally, we assumed that all methods of testing, both experimental and FEA, as well
as analytical derivations can be compared. For FEA and analytical data, it was assumed that unit
cell size does not influence mechanical properties. This may seem like a limiting assumption, but
in many cases, papers did not report unit cell size for analytical models or FEA simulations. For
experimental data, unit cell size was recorded if it was given in the source. For analytical data
specifically, it was assumed that the derivations for volume fraction consider strut overlap and
were accurate. For many papers, data was extracted from plots, and it was assumed that data
taken from plots were valid within the resolution available. Though some papers measured the
actual volume fraction, or relative density, of the AM lattice structures, the theoretical volume
fraction, or as-designed value, was used in this work because it was reported more consistently in
literature than the as-printed volume fraction.
The potential sources of error include: experimental error within the reported results, low
image quality in figures used to extract data points with WebPlotDigitizer, human error in
precisely selecting data points in an image when gathering data, and the regression fits of the
segmented curves for some of the FEA data type data. While there were several potential sources
of error, the total error was assumed to be minimal when combining data from so many different
models and experiments across the literature. With all the variations in data due to testing
methods, AM process parameters, and materials, the purpose of the assumptions was to isolate
the effect of unit cell topology in mechanical property trends.
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6.2.4 LUCIE
The Lattice Unit-cell Characterization Interface for Engineering (LUCIE) was developed
in MATALB [344] as a design tool to compare and visualize mechanical properties of lattice
structures based on various unit cell topologies. LUCIE presents a large plot area along with
several adjustable settings to compare the lattice structure data that was collected from the
literature. The data can be sorted by the user adjusting the range of volume fractions and unit cell
sizes that are plotted. Few data points were available for a single volume fraction; so, a range
allows for more data to be viewed. Drop-down menus labeled “X Property” and “Y Property”
specify which lattice structure properties of interest are to be plotted on the X and Y axis,
respectively. Check boxes can then be used to select the type of data (Analytical, FEA, and/or
Experimental) and the unit cell topologies that are plotted. Multiple data types and unit cell
topologies can be selected to aid in comparison.
The FEA and experimental data collected from the literature search were stored in
spreadsheets. Numerical data catalogued the data’s source, unit cell topology, mechanical
property, volume fraction, unit cell size, numerical property value, units, bulk mechanical
property value, and the normalized property value. Regression data cataloged the data source, unit
cell topology, mechanical property, volume fraction min/max, unit cell size min/max, and
constants (𝑎, 𝑏, 𝑐) for the power regression equation; see Eqn. (6.2).
Equations for the analytical data were stored explicitly in a MATLAB [344] function. It
was typical for these equations to have either the volume fraction or aspect ratio (ratio of strut
radius to strut length) as the independent variable. In many cases, algebraic manipulations of the
analytical equations were required to use volume fraction or aspect ratio as the independent
variable.
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An example of inputs and outputs for LUCIE is shown in Figure 6.2. To use LUCIE, a
user must first set the desired data range values and unit cell topologies to be displayed along
with data types. Volume fraction and unit cell size are set either with the sliders or the text edit
boxes. For both volume fraction, a minimum and maximum value need to be specified so that
data can be selected from a range. For unit cell size, either “All” data can be plotted or a range
may be selected. For sources that did not directly specify the unit cell size of the test data, the
data points are included when “All” unit cell size data is selected. However, if a unit cell size
range is specified, only data with a known unit cell size is plotted. The desired properties for the
X and Y axes need to be specified along with the types of data and unit cell topologies are to be
plotted. Once all settings have been chosen, the MATLAB code filters all of the data so that only
data that satisfies the user’s specifications are plotted.
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Figure 6.2 Schematic showing the inputs and outputs for LUCIE
In the case where one of the properties selected was volume fraction, the data could be
plotted directly, because all the data types were reported, or calculated, as a function of volume
fraction. In the case where volume fraction was not selected as one of the properties to be plotted,
the properties were calculated by the specified range of volume fraction and unit cell size and
then parametrically represented.
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In the case of the analytical data type, data was represented by continuous functions. For
each specified lattice structure, the two functions that pertain to the selected properties of interest
were plotted parametrically with volume fraction as the input parameter.
In the case of the experimental and FEA data types, the data was represented by discrete
points, which were first combined into bins by volume fraction. Each bin contained a range of 5%
volume fraction, combining any data pertaining to that range. After the data from each property
type was sorted into bins, the data could be represented parametrically for each property by
volume fraction. In these cases, the data was plotted using two error bars that show the
maximums and minimums of the data in that bin. The location where two error bars intersect
denoted the average value. In the case of only a single data point within the bin range error bars
were not shown.
6.3 Results and Case Studies
6.3.1 Results of Literature Compilation
A summary of the type of data and properties for each unit cell topology is shown in
Table 6.3, and the unit cell topologies are illustrated in Table 6.4. Compiling all the data from 69
papers, 1,439 experimental data points were collected across 18 different unit cell topologies. For
the experimental data points, 41% were collected from tabulated values, and the remaining 59%
were extracted from digital images in the original sources using WebPlotDigitizer. In addition,
209 data points for finite element simulations were also collected. The majority of data was
available for the effective modulus (compressive) and yield stress of lattice structures; however,
there was also some data available in literature for Poisson’s ratio, buckling stress, and plateau
stress.
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Table 6.3 For each unit cell topology included in LUCIE, there are 6 different properties that can be
compared. The data type available for each unit cell topology are denoted by analytical (A),
experimental (E), or FEA (F). A dash (-) signifies that no reference was found that provided this
information for any of the three data types.
Unit Cell Topology Volume Fraction
EC YS PR BS PS
Cubic A A, E, F A, E, F A, F A E
Diamond A A, E, F A, E, F A, F A A, E
Truncated Cube A A, E, F A, E, F A, F A E
Truncated Cuboctahedron A A, E, F E, F A, F A E
Rhombic Dodecahedron A A, E, F E, F A, F - E
Rhombicuboctahedron A A, E, F A, E, F A, F - E
Truncated Octahedron A A, F F A, F - -
Octahedron A A, E, F E, F A, F - -
Octet-truss A A, E, F A, E F - -
BCC A A, E, F A, E, F A, E, F - E, F
BCCZ A A, E, F E, F E - E, F
FCC A A, E E A - -
FCCZ A E E - - E
F2BCC A E E - - -
F2BCCZ A E E - - -
TPMS Gyroid - E, F E, F - - E
TPMS Diamond - E, F E - - -
G7 - E - - - E
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Table 6.4 Name and associated picture for each lattice structure included in LUCIE
Unit Cell Topology Image Unit Cell Topology Image
Cubic
BCC
Diamond
BCCZ
Truncated Cube
FCC
Truncated Cuboctahedron
FCCZ
Rhombic Dodecahedron
F2BCC
Rhombicuboctahedron
F2BCCZ
Truncated Octahedron
TPMS Gyroid
Octahedron
TPMS Diamond
Octet-truss
G7
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As shown in Table 6.3, the mechanical properties for compiled data include normalized
compressive effective modulus (EC), yield stress (YS), Poisson’s ratio (PS), buckling stress (BS),
and plateau stress (PS). In Table 6.3, a dash (-) signifies that no reference was found that provided
this information for any of the three data types. One limitation of this work, as mentioned
previously, was the challenge with inconsistent terminology and methods for computing and
reporting mechanical properties. As such, a brief description of each property as we have defined
it follows.
Compressive effective modulus was defined as the slope of the linear portion of the
stress-strain curve during the initial loading or the linear portion of the unloading curve during
testing, although Ashby et al. [137] suggest that unloading effective modulus may be a more
accurate representation of the structure’s performance than the loading effective modulus. Yield
stress was defined as the first peak stress for stretch-dominated lattice structures, and was
generally the same as the plateau stress for bend-dominated lattice structures [250,400,405]. In a
few cases, sources may have instead reported yield stress based on a strain offset as opposed to a
first peak or plateau stress. Plateau stress was defined as the average stress value during the
cyclical collapse of the bending-dominated lattice structures or during the relatively level stress
that occurs in stretch-dominated structures [400,405]. Buckling stress was defined only in the
analytical data type because, in general, changes in slope on the stress-strain curve due to yielding
or buckling are difficult to distinguish. As such, the buckling stress of lattice structures was not
reported experimentally. The distinction or classification of stretch- or bend-dominated unit cells
was not categorized or recorded within LUCIE, but it was useful to understand what the yield or
plateau stress represents in different unit cell topologies.
Figure 6.3 illustrates each property and the differences between loading curves for
stretch-dominated and bending-dominated truss-like lattice structures. Bend-dominated structures
tend to have a peak stress at yield, and then the stress decreases as layers begin to collapse.
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Stretch-dominated structures tend to maintain a plateau in their stress strain curve as the struts
yield in tension and collapse prior to densification. Stretch-dominated and bend-dominated lattice
structures are determined based on their Maxwell number, which suggests how deformation
occurs in the structure [138,433].
Figure 6.3 Typical stress-strain curves for stretch-dominated or bend-dominated structures, labeled
with the relevant mechanical property terminology for the data compiled in this work
Regarding experimental data collected, the data may be grouped in four categories: (1)
“non-averaged”, (2) “non-averaged+uncertainty”, (3) “averaged”, and (4)
“averaged+uncertainty”.
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• “Non-averaged” data points were defined as data points that did not represent an
average from experimental replicates and did not include uncertainty reported in
the source. These data points were from sources that reported all of the data from
their experimental replicates as individual points as opposed to only the average
of the data, or from sources that did not have any experimental replicates. In the
LUCIE database for experimental data, “non-averaged” accounts for 65.8% of
the data points we collected.
• “Non-averaged+uncertainty” data points were defined as data points that did not
represent an average from experimental replicates but did include uncertainty
reported in the source. In a single source, replicates were not reported, but the
source estimated the uncertainty based on the testing equipment. In the LUCIE
database for experimental data, “non-averaged+uncertainty” accounts for only
0.7% of the data points we collected.
• “Averaged” data points were defined as data points that represent an average of
the experimental replicates but for which an uncertainty value was unavailable.
The unavailability of the uncertainty associated with the averaged data points
was for one of two reasons: (i) the uncertainty was not reported or (ii) it was not
visible for digital image extraction. For example, if uncertainty was reported as
error bars on a plot, then the error bars were occasionally hidden by the data
point marker they were attached to or other nearby markers, making it impossible
to accurately extract the uncertainty. In the LUCIE database for experimental
data, “averaged” accounts for 15.8% of the data points we collected.
• “Averaged+uncertainty” data points were defined as data points that represent an
average of the experimental replicates and included an uncertainty value. These
data points were often reported as tabulated values (e.g., 10 ± 1 MPa) or on plots
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that included error bars. In the LUCIE database for experimental data,
“averaged+uncertainty” accounts for 17.7% of the data points we collected.
From the collection of experimental uncertainty, 264 of 1439 (18.4%) data points were
reported with uncertainty. A histogram has been created to show the frequency of the relative
error for the data points reported with uncertainty; see Figure 6.4. The relative error was defined
as the uncertainty divided by the nominal value. For example, for a data point 10 ± 1, the relative
error was ± 10%. In cases where the error bars extracted from digital images were different on the
high or low side, the conservative, or larger, error value was taken. From the histogram of the
relative error, we find that when the experimental uncertainty was reported, 75% of the data had
an uncertainty of less than 10%. As there was only a select portion of the data, 18.4%, for which
experimental uncertainty was quantified in the literature, the uncertainty value was not directly
accounted for within the LUCIE database. As described in Section 6.2.4, error bars shown on a
plot within LUCIE denote the maximum and minimum values averaged from the bin of data for
that point. As more data becomes available with quantified uncertainty, future revisions allow the
user to restrict data to a specific cutoff threshold.
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Figure 6.4 Relative frequency of uncertainty in experimental data points. Uncertainty was
normalized by the nominal value to obtain the relative error. 75% of the reported uncertainty values
had a relative error of less than 10%.
6.3.2 LUCIE Case Study 1
Case Study 1 was a demonstration of the output from LUCIE. Figure 6.5 shows a
comparison of volume fraction and the normalized effective modulus using analytical,
experimental, and FEA data. The unit cell topologies plotted here are the BCC, Diamond, and
Truncated Cube for a volume fraction range and unit cell size of 0-50% and 0-10mm,
respectively, to capture a large range of data. It can be seen that for the analytical Diamond cell
data (red), three curves are plotted to reflect analytical results from different sources. In general,
these analytical curves represent the experimental and FEA data fairly well at low volume
fractions and at high volume fractions the range of data is large. In contrast, other analytical
models tend to underpredict or overpredict the experimental and FEA data. For example, in
Figure 6.5 the analytical BCC curve (blue) underpredicts the experimental data beyond a volume
fraction of 0.15. However, the analytical curve for the Truncated Cube overpredicts the
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experimental data reported for that unit cell topology and diverges from the FEA simulation
results.
Figure 6.5 LUCIE Case Study 1 output comparing analytical, experimental, and FEA data for the
BCC, Diamond, and Truncated Cube unit cell topologies. From the output you can visualize
differences in analytical models for a single unit cell topology, such as the Diamond, as well as gain
an understanding of the normalized effective modulus for different unit cell topologies.
6.3.3 LUCIE Case Study 2
Case Study 2 contains only experimental data from LUCIE; see Figure 6.6. This figure
plots the volume fraction and normalized yield stress for Diamond, Truncated Cube, Truncated
Cuboctahedron, Rhombic Dodecahedron, Rhombicuboctahedron, BCCZ, and TPMS Gyroid unit
cell topologies. The ranges of volume fraction and unit cell size were set to 5-40% and 1-5mm,
respectively. The plot indicates a general area where a particular unit cell topology’s data exists
and how it compares to other unit cell topologies and types of data. For example, the Diamond
unit cell topology was reported to have a lower normalized yield stress than the Truncated Cube
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or TPMS Gyroid; however, it was on par with reported data for the Rhombic Dodecahedron. By
making these comparisons, a user can easily update the settings in the interface to narrow the
search for a particular unit cell topology.
Figure 6.6 LUCIE Case Study 2 output shows a comparison of 7 different unit cell topologies and the
variability that exists in experimental data. The error bars around a single point represent a small
“bin” of data. The point itself represents the average value within the bin, while the error bars
denote the maximum and minimum reported values for each property.
LUCIE can benefit designers by allowing them to easily compare unit cell topologies. For
example, based on the plot shown in Figure 6.6, a designer who was previously considering a
diamond unit cell topology would see that by switching to a TPMS gyroid unit cell topology there
are opportunities for further reducing weight or increasing the normalized yield stress value.
Specifically, starting from a diamond unit cell topology with a volume fraction of 0.35-0.4, a
designer can achieve the same normalized yield stress with nearly 50% weight savings by
switching to a TPMS gyroid unit cell topology with a volume fraction of 0.2. Similarly, by
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switching from the previously mentioned diamond unit cell to a TPMS gyroid unit cell topology
with volume fraction between 0.35-0.4, the normalized yield stress value is increased by 76%
with no change in weight.
6.3.4 LUCIE Case Study 3
Case Study 3 illustrates how to use LUCIE as a design tool to aid unit cell selection by
providing information to differentiate unit cell topologies and their performance. Consider the
design requirements for a lattice structure with a target density, normalized effective modulus,
and normalized yield stress of 0.2-0.25, 0.02-0.03, and 0.05 or greater, respectively. LUCIE was
used to plot experimental data for volume fraction and normalized effective modulus of the 18
different available unit cell topologies; see Figure 6.7. Next, the user locates the region of interest
on the plot according to the design constraints on volume fraction and normalized effective
modulus. The region of interest is highlighted in Figure 6.7 by a red box with a dashed line. From
this plot we can identify six potential unit cell topologies, whose average normalized effective
modulus falls within the region of interest. This reduces our selection from 18 to 6 unit cell
topologies: Cubic, BCC, Truncated Cuboctahedron, and Rhombicuboctahedron, TPMS Gyroid,
and Octet-truss.
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Figure 6.7 LUCIE Case Study 3 output from LUCIE for comparison of the lattice structure
properties of different unit cell topologies. The red box with a dashed line highlights the region of
interest (EC 0.02-0.03 and Volume Fraction 0.2-0.25) for determining appropriate unit cell selection.
A new plot was generated by selecting only the six unit cell topologies identified from
Figure 6.7, with a comparison of normalized effective modulus and normalized yield stress with a
volume fraction restriction of 0.2-0.25; see Figure 6.8. It can be seen that there was a clearer
distinction between the possible cells. The Cubic cell has the highest average reported normalized
yield stress followed by the TPMS Gyroid, Rhombicuboctahedron, Truncated Cuboctahedron,
and finally the BCC. No normalized yield stress data was available in this range for the Octet-
truss. With a stipulation that the normalized yield stress be greater than 0.05, the designer could
move forward with either a Cubic, TPMS Gyroid, or Rhombicuboctahedron. One consideration
could be the large range of normalized yield stress that has been reported for the TPMS Gyroid,
directing a user to further explore how the normalized yield stress can be high or low for similar
volume fractions. Further distinction can be considered based on desired print tolerances, unit-cell
pore sizes, etc. For example, the Rhombicuboctahedron has more struts than the Cubic unit cell,
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meaning that, for the same volume fraction and unit cell size, the strut diameters and unit cell
topology pore sizes are smaller than those of the Cubic cell. If a small unit cell topology pore size
is critical to the application, the Rhombicuboctahedron would likely be a better choice than the
Cubic unit cell topology; however, if minimum feature size is a concern, the Cubic unit cell
topology has larger strut sizes and pores than the Rhombicuboctahedron making it easier to print
remove powder. For an AM designer, rather than an expensive experimental trial with 18
different unit cell topologies, LUCIE was able to quickly narrow the search to only three unit cell
topologies that could be compared or prepared for further testing, resulting in valuable cost and
time savings.
Figure 6.8 LUCIE Case Study 3 plot comparison of normalized effective modulus and normalized
yield stress for comparison of the six unit cell topologies that were down selected from Figure 6.7.
The thumbnails of the unit cell topologies have been added to the plot; from top to bottom they are
Cubic, TPMS Gyroid, Rhombicuboctahedron, Truncated Cuboctahedron, BCC. No yield stress data
was available for the Octet-truss at this volume fraction range.
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6.4 Discussion
While the data presented in the interface can be used by designers to compare the lattice
structure properties to improve unit cell selection, there were a number of challenges that remain
and limitations that exist in the current approach. AM remains a continuously changing and
evolving process with a wide range of process parameters, materials, and build strategies still
being developed. This work compiled data by unit cell topology, but not by the material or
processing conditions. The intended result was not to say that all data for a unit cell topology was
processed with a single set of process parameters; rather, it shows the range of mechanical
properties that can be achieved with a single unit cell topology. Even the analytical models that
have been developed differ based on their underlying assumptions and the experimental data
differs even further; see the diamond cell analytical models shown in Figure 6.5 as an example.
In cited sources, several reasons for differences between analytical, finite element, and
experimental data were suggested. Analytical models tend to overestimate effective modulus
when compared to finite element and experimental data and deviate more as volume fraction
increases. Analytical models derived using the Euler-Bernoulli method ignore shear and
rotational inertia which decrease the effective modulus [189,203,204,206]. Shear and rotational
inertia become more influential as volume fraction increases where more unit cell volume exists
at strut intersections. Analytical and FEA values for yield stress can overlap if critical struts
experience no bending and therefore were unaffected by shear and rotational inertia [204]. Finite
element models using the Timoshenko method account for shear and rotational inertia.
As for experimental data compared to analytical and FEA data, it was stated in the
literature that imperfections in the printing process were responsible for the biggest discrepancies.
For example, surface roughness as a result of unmelted or semi-melted powder can cause
variations in strut radius and cause stress concentrations [203–206,208,392,394,395,423].
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Additionally, quality of print for vertical, inclined, and horizontal struts vary, and this was not
captured by finite element or analytical models, which assume perfectly cylindrical and
homogeneous struts [206]. Meanwhile, strut alignment often differs between experimental and
finite element models as well. Analytical and finite element models do not account for slight
misalignments among struts and loading conditions [192,392]. Other causes for disagreements
may be due to the number of unit cells tested (finite or infinite) or lack of accounting for strut
interference during densification [203,214].
Different approaches have been proposed to increase the accuracy of analytical and finite
element models. Several sources try to account for imperfections in their models by changing the
as-designed strut radius to match a geometric average of a physical radius or by applying a
randomly varying strut radius [365,392]. One study improved their finite element model by using
reconstructed struts based on CT-scans which had greater accuracy than as-designed and average
radii models [207]. Having misaligned struts in finite element simulations has also been proposed
to improve the model [392].
As another example of the variation that was present in the literature, consider the gyroid
data presented in Figure 6.6. The range of data that was available from 0.1-0.15 volume fraction
had a large range in the reported yield stress as shown by the large vertical error bars. This
finding suggests that a difference in testing conditions, build quality, definition of the gyroid, or
processing parameters had major effects on the yield stress of the gyroid unit cell topology. The
variation seen in the gyroid could lead to several outcomes. First, one researcher could see the
data and take it as an argument for standardization of process parameters and testing procedures.
While standardization is critical, by combining all the results into a single interface, the range of
capabilities of the gyroid are represented. A second researcher could take the result and wish to
understand how the gyroid can be tailored to meet a specific yield stress criterion. By reviewing
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the original literature sources compiled in LUCIE, the ability for researchers to explore the
capabilities of the gyroid lattice structure is simplified.
Another factor in the variability of data could be the variations in testing conditions such
as the number of unit cells included in the array, boundary conditions in the compression test,
strain rate, etc. For example, ISO 13314:2011, the standard that addresses mechanical testing of
porous and cellular metal structures, recommends a periodicity of 10x in each direction to
minimize the effects from boundary conditions. However, as mechanical testing of lattice
structures has not been formalized, many authors do not strictly follow this guideline and test
samples with fewer than ten unit cells in any of the three directions. In this work, test samples
with fewer than ten unit cells in each direction were not excluded, resulting in data that may be
skewed by differences in the impact of boundary conditions on testing results. For additional
detail on the impact of boundary conditions on testing, see Section 5.2.3.3 and Section 5.5. As
with standard materials and testing, consistent standards applicable to lattice structures are needed
to ensure that the scientific community—and eventually designers and engineers—can regularize
the testing methods to achieve consistent results.
Another inherent challenge with AM is the design freedom that is available. Because
there is so much design freedom, there are infinite minor variations on a single unit cell topology,
such as rounding or fillets at node locations, making it impossible to name and track each
individual unit cell topology. The purpose of this work was not to say that the optimal unit cell
topology is found within the 18 topologies included in LUCIE. In our opinion, there is a large gap
between what can be designed and quantified for analysis and what can be printed in metal using
current laser and electron beam powder bed fusion technology. AM processes can print
complexity that designers would have never dreamed of or imagined previously. However, for the
designer to feasibly characterize such infinite possibilities and complexity into a functional and
reliable component is currently not possible. The intention of LUCIE was to help bridge this gap
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by providing a database to allow researchers and designers an understanding of regular unit cells
that have been characterized through analytical, experimental, and finite element analysis. In this
work, we have grouped similar unit cell topologies where possible, though minor variations in
geometric models do exist. As such, the purpose of LUCIE was visualization of performance
trends based on the unit cell type.
Related to the challenge imposed by unit cell topology design freedom, comes the
challenge of naming convention. Several authors have tried to address this issue for lattice
structures and unit cell topologies and it remains an ongoing battle [198–200]. When compiling
our data, the unit cell topology naming conventions were similar; however, variations were seen
across different research articles and software libraries. In this work, we have attempted to use the
most common name found in literature for describing the unit cell topology.
The amount of data available is increasing all the time but remains limited, especially for
some unit cell topologies. In the unit cell selection example, Case Study 3, represented by Figure
6.7 and Figure 6.8, there was lattice structure data available that fit the design requirements;
however, in some cases data may not be available because there are many gaps in the published
literature. For example, the Octet-truss was initially identified as a potential candidate in Case
Study 3, but no data was available for yield stress in the specified volume fraction range. The
focus of this initial work was common unit cell topologies, or those for which more than a single
source could be found. As a result, less common and modified versions of common lattice
structures were excluded for lack of sufficient data. Figure 6.9 shows the number of sources used
for each unit cell topology. With the amount of published data increasing all the time, we expect
to be able to continue expanding the database for LUCIE and the research community’s
understanding of the range of mechanical properties that may be achieved with AM lattice
structures.
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Figure 6.9 Number of sources found for each unit cell topology. The results show the amount of
lattice structure data available is limited for many common unit cell topologies.
As the focus of this work was identifying the material independent mechanical property
trends of lattices structures based on the unit cell topology, all materials were combined through
normalization. At present, sorting by material is not available to a front-end user due to the
limited amount of data available for each material. With continued expansion of the database for
LUCIE, future work may include sorting or categorizing data by material as an option to further
refine the results in LUCIE.
Another limitation of LUCIE is that the exact testing conditions (testing machine type,
crosshead rate, boundary conditions, etc.) and AM processing parameters are not stored within
the current LUCIE database or available to the user on the front-end. For the testing conditions,
the information is available in the cited sources. In contrast, in terms of fabrication, few papers
report the specific details of the AM process parameters, often due to the proprietary information
of the AM machine manufacturer. Of those papers that do report AM process parameters, the
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amount and type of information that was reported is inconsistent across different sources. The
ability to incorporate testing conditions and AM process parameters for each data point in LUCIE
in a useful manner was thus limited because each data point represents numerous papers, each
with their own testing methods and AM process parameters. Future work for LUCIE could also
address how to make the sources for each combined data point accessible to the user, along with
information regarding testing methods and AM process parameters for comparison of how these
factors affect the mechanical properties.
Based on the research articles which have been compiled into the LUCIE database, a
number of future directions have been identified for short-term and long-term lattice structure
research. In the short-term, testing methods, naming conventions, and common practices should
be developed and standardized to allow researchers to collaborate easily and compare
experimental data and analytical and simulation results. The issue of lack of standardization
makes it very difficult for researchers to understand the current state when published works are
obfuscated by unique naming conventions. Standardization across naming conventions and
testing appears to be the low hanging fruit that can help accelerate lattice structure research for
AM. In the long-term, as the interest in lattice structures grows, research initiatives should
include exploration of:
1) additional experimental loading conditions such as tensile, shear, and fatigue,
2) novel modeling approaches to account for hierarchical complexity,
3) incorporation of manufacturing variability in modeling and simulation,
4) novel unit cell topologies through optimization,
5) post-processing opportunities and effects, and
6) industrial-proven case studies of lattice structure advantages.
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Additional methods to model and predict the effect of lattice structures within the full
component will be an enabling factor for driving forward industrial applications. As discussed
previously, the majority of the present literature was focused on compressive behavior, although
most components undergo a variety of loading conditions. Non-compressive behavior, especially
cyclic and fatigue loading, is a major concern among researchers for the applications of lattice
structures. Finally, learning how to further improve lattice structures with novel topologies or
post-processing methods continue to advance the opportunities available for lattice-enabled
components.
There are ongoing efforts to make LUCIE publicly available for all researchers and
engineers. Currently, LUCIE can be downloaded as a standalone application for installation on a
Windows operating system (see https://sites.psu.edu/edog/lucie/). This standalone version
requires the MATLAB Compiler Runtime which will be downloaded and installed for free
automatically during the installation process, meaning that a MATLAB license is not required to
use the application. In addition, ongoing efforts include development of a web-based version of
LUCIE that will allow researchers to utilize the application without having to download and
maintain an up-to-date version of the application. The intent of the web-based tool was to
simplify access to LUCIE with additional plans to allow researchers to be able to submit their
own data for incorporation into the database.
6.5 Closing remarks
Through review of over 69 papers, the largest known mechanical property database for
lattice structures fabricated with metal AM was compiled and the Lattice Unit-cell
Characterization Interface for Engineering (LUCIE) was developed to provide Ashby-style plots
for unit cell selection of AM lattice structures. Researchers and AM designers stand to benefit
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from simplified methods to observe and understand unit cell topology effects on lattice structure
mechanical properties through LUCIE and similar interfaces and databases. Though the amount
of data and lack of testing standards were limiting, ongoing efforts to characterize lattice
structures will continue to improve the models and experimental data available.
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Chapter 7 Conclusions and Future Work
In this dissertation, two systematic design optimization approaches were presented: a
shape matching approach for a compliant deployable surgical tool for radiofrequency ablation
(RFA) of tumors and the development of specialized lattice structures for additive manufacturing
(AM). This final chapter of the dissertation summarizes the work completed regarding each
objective. Next, major conclusions drawn from work are given. Following the major conclusions,
research contributions to the scientific community and broader impacts of the research are stated.
Finally, recommended future research directions are addressed.
7.1 Summary of Objectives and Tasks
7.1.1 Objective 1: Develop systematic optimization approach for compliant deployable
radiofrequency ablation electrodes
1.1 Design deployable electrode for endoscopic radiofrequency ablation
1.2 Develop finite element model to simulate radiofrequency ablation in soft tissue
1.3 Couple finite element with optimization approach for design of electrode based on
tumor geometry
1.4 Evaluate the deployable electrode design
1.4.1 Assess deployment of early prototypes in tissue phantom
1.4.2 Validate optimized electrode design through ex-vivo experimentation
To improve treatment for endoscopic RFA of tumors, a novel compliant deployable RFA
electrode was presented and optimized to match the treatment zone with the tumor geometry. The
deployable electrode design is based on an array of compliant tines which compress and stow
inside an endoscopic needle. The endoscopic needle serves as a sheath and then the electrode is
advanced out of the sheath and a circular array of tines deploy into the cancerous tissue. A finite
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element model which combines the electrical and thermal interactions of an electrode heating the
surrounding tissue has been developed in COMSOL Multiphysics version 5.2a [343]. The finite
element model predicts the ablation zone that develops around a deployed electrode geometry.
Coupling the finite element model with an optimization algorithm, the deployable electrode
design was optimized for treatment of a 2.5 cm spherical tumor. Comparing computational results
of a baseline electrode with 25% treatment efficiency, the deployable electrode design was shown
to increase treatment efficiency to 71-87%. Prototypes of the electrode were fabricated using AM
and laser micromachining. These prototypes were evaluated for deployment and later used for
validation in a thermochromic tissue phantom. Validation of the computational models showed a
strong dependence on temperature dependent properties that can be averaged over the simulation
time to improve computational efficiency. With corrections to the computational model to
account for tissue damage at elevated temperatures, the computational model was shown to
accurately replicate the ex-vivo testing results. At the time of prototyping, AM was not capable of
producing a functional deployable endoscopic electrode due to the minimum feature size and
material selection. However, the materials and processes available are constantly evolving and
this approach should be reconsidered for mass customization of electrodes possibly through AM
of Nitinol or metal binder jet processes. For additional details regarding Objective 1 and its
associated tasks, see Chapter 2.
7.1.2 Objective 2: Develop three-dimensional ground structure topology optimization for
unit cell DfAM
2.1 Develop three-dimensional ground structure topology optimization for unit cell DfAM
2.2 Establish systematic optimization method for unit cell generation
2.2.1 Generate set of potential objective functions and constraints
2.2.2 Quantify manufacturing considerations for metal AM
2.3 Validate optimization with benchmark problems
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With the increasing interest of AM for fabrication topology optimized structures, there is
a need for systematic approaches for the design of lattice structures. For non-stochastic lattice
structures, the most basic repeating pattern is the called the unit cell. In current practice, there is
little known about how the choice of unit cell topology affects the performance of a lattice
structure. Using commercial software, the choice of unit cell topology is typically based on trial
and error with little thought to how the choice will influence the structure. AM enables
customization and optimized lattice structures, but there is a lack of design optimization methods
that incorporate DfAM restrictions to ensure manufacturability.
Additive Lattice Topology Optimization (ALTO) is an application-agnostic systematic
approach for generating unit cell topologies. The foundation of this work relies on ground
structure topology optimization. Ground structures are particularly well suited to the design of
lattice structures because of their computational efficiency and they are a good representation of
traditional strut and node type unit cell topologies. The ground structure representation of the unit
cell has been developed in MATLAB and is paired with either a gradient-based or evolutionary
optimization algorithm. The gradient-based algorithm was utilized for 2D optimization problems
with two or fewer objectives. It was found to be very sensitive to the initial guess and the order of
magnitude of the objectives, requiring problem specific scale factors to equilibrate different
objective functions. The evolutionary algorithm was used for 3D optimization problems with
three or more objectives. It was found to be very robust to including different objective and
penalty functions for various types of optimization problems was able to successfully solve
optimization problems with more than one hundred design variables.
For the optimization, a library of objective, constraint, and penalty functions have been
developed. DfAM is incorporated throughout ALTO in various objective functions (i.e., unit cell
volume, minimization of relevant elements, minimization the powder removability factor),
constraint functions (i.e., minimum feature size), ground structure restrictions (i.e., build angle),
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and penalty functions (i.e., overlap, unsupported–S, unsupported–N). The DfAM considerations
reduce the need for design interpretation prior to printing the optimized solution. Specific
considerations to ensure manufacturability include bridging of horizontal elements, a no-build
range for the build angle, and minimum feature size for printing. Validation of the ground
structure formulation and optimization were performed using several traditional topology
optimization benchmark problems. Benchmark problems were solved for minimization of strain
energy, maximization of thermal conductance, and predicting macroscale properties from a
representative volume element through homogenization. In each benchmark, ALTO was able to
replicate results similar to the benchmarks from literature, validating its formulation for DfAM
cases studies. For additional detail regarding Objective 2 and its associated tasks, see Chapter 3.
7.1.3 Objective 3: Explore multi-functional lattices through case study on unit cell
generation for thermal conductance and minimum strain energy
3.1 Define constraints and process limitations
3.2 Generate optimal lattice structure with and without DfAM
3.3 Fabricate lattice structure with DfAM considerations
3.4 Evaluate solutions and revise ground structure topology optimization
Multi-objective design optimization offers a unique opportunity to enhance performance
through multi-functional lattices. Two case studies were presented for multi-functional lattices
that combine minimization of strain energy and maximization of thermal conductance. Using
various constraints and ground structure restrictions the case studies each produce designs that
exhibit the tradeoffs required to balance two conflicting objectives. Case Study 1 from Chapter 4
clearly illustrated the benefits of DfAM considerations. With DfAM, the optimized solution was
immediately manufacturable and was fabricated using two different AM processes. Without
DfAM, the optimized result would require significant design interpretation prior to fabrication.
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Case Study 2 from Chapter 4, demonstrated the Pareto-optimal solutions from which a designer
can select a design. Gradient-based algorithms were found to require significant time upfront to
determine appropriate weighting values and optimization parameters to balance the two
objectives. On the other hand, evolutionary algorithms were found to be robust in solving multi-
objective optimizations with less control over the development of the Pareto front. For additional
details regarding Objective 3 and it associated tasks, see Chapter 4.
7.1.4 Objective 4: Explore homogenized lattices through case study on generation of unit
cells with tailored compliance
4.1 Define constraints and process limitations
4.2 Generate optimal lattice structure with tailored properties
4.3 Fabricate and test to determine mechanical properties
4.4 Characterize fabricated samples with mechanical testing
4.4.1 Compare boundary conditions of homogenization with experimental
characterization
Lattice structures are of particular interest to the topology optimization community for
the design of materials with tailored compliance. Three case studies were presented for finding
structures that match a specific target constitutive matrix. Using a variety of optimization
objective, constraint, and penalty functions, these case studies add to the range of applications for
ALTO. Case Study 1 from Chapter 5 demonstrated a novel powder removability factor as an
objective function. The powder removability factor encouraged solutions with large spacing in
the unit cell topology to ease powder removal. After fabrication of the optimized and baseline
unit cell topologies, the optimized solution demonstrated improved powder removability
compared to the baseline unit cell through mass measurement and CT scan analysis. Experimental
compression testing was also used to validate the homogenization calculation within ALTO. With
corrections for the boundary conditions and differences between the as-designed and as-printed
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models, the homogenization predicted the effective modulus of two different unit cell topologies
to within 6%. Case Study 2 from Chapter 5 generated multiple optimized solutions, each for a
scaled version of the target constitutive matrix. The target volume fraction was constant,
requiring the optimization to rearrange, as oppose to simply remove, material. The result was
three different stiffness unit cells that all had the same weight by embedding mechanism-like
behavior into the unit cell topology. This enables lattice structures with a non-uniform stiffness
and uniform weight distribution. Fabricated samples were also experimentally tested and the
results further support boundary conditions corrections discussed with Case Study 1. Case Study
3 from Chapter 5 demonstrated generation of an orthotropic material with three times the
effective modulus in one direction as compared to the other. In addition, the DfAM penalty
function and relevant diameter limit were explored for their impact on the optimization problem.
For additional details regarding Objective 4 and it associated tasks, see Chapter 5.
7.1.5 Objective 5: Develop mechanical property database for metal lattice structures from
current literature containing analytical, finite element, and experimental data
5.1 Perform literature search for metal lattice structure data
5.2 Collect data from selected papers
5.3 Create user interface for generating mechanical property charts of collected data
While AM enables fabrication of optimized unit cell topologies, there is a need to be able
to characterize and understand standard unit cell topologies. Researchers around the world are
characterizing lattice structure through analytical, finite element, and experimental data.
Compiling data from 69 sources, the Lattice Unit-cell Characterization Interface for Engineers
enabled visualization of mechanical properties of metal lattice structure fabricated with AM. As a
result of the compilation, LUCIE contains data for 18 different unit cell topologies. The Ashby-
style material property charts are generated from over 1400 experimental data points, 200 FEA
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data points, and 45 analytical models. To the author’s knowledge, LUCIE is the largest publicly
available database of its kind. LUCIE can be downloaded as a standalone application for
installation on a Windows operating system (see https://sites.psu.edu/edog/lucie/). This
standalone version requires the MATLAB Compiler Runtime which will be downloaded and
installed automatically during the installation process; a MATLAB license is not required to use
the application. The interface enables researchers and scientists to understand the trends for
mechanical property data sorted by the data type, unit cell topology, unit cell size, and volume
fraction. It serves to help bridge the gap between formal optimization of the unit cell topology
with ALTO and the current practice of selecting blindly from a unit cell library. For additional
details regarding Objective 5 and it associated tasks, see Chapter 6.
7.2 Primary Conclusions
From the objectives and tasks in this dissertation, fifteen conclusions were drawn
regarding the work completed:
1. Deployable endoscopic electrodes are feasible for expanding RFA treatment zones and
have the potential to significantly impact the size of tumors that can be treated with a
single insertion of the electrode. The deployable electrode designs presented increased
treatment efficiency from 25% to 71-87%.
2. Temperature dependent properties such as electrical conductivity play a significant role
in the development of the ablation zone surrounding an electrode and must be addressed
in computational models.
3. The degree of deployment for a deployable electrode demonstrated no statistical
difference based on insertion speed, however, a minor trend appeared to show decreasing
deployment with increasing speed above 10 mm/s.
4. At the time of initial investigation, mass customization of endoscopic electrodes with L-
PBF was limited due to the minimum feature size of large-scale production machines and
material selection but is worth reconsidering as available material and processes continue
to improve.
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5. ALTOs unit cell topology optimization framework is capable of handling a wide variety
of applications as demonstrated through multi-functional lattices and various tailored
compliance lattice structures in the case studies presented.
6. The gradient-based optimization algorithm fmincon was capable of solving multi-
objective optimization problems with 2 objectives and 30 design variables.
7. The evolutionary algorithm Borg was capable of solving multi-objective optimization
problems with 4 objectives and 120 design variables.
8. The ground structure formulation and optimization of ALTO were validated through
common topology optimization benchmarks for minimization of compliance,
maximization of thermal conductance, and homogenization.
9. The novel powder removability factor was found to be a sufficient comparison metric for
manufacturability of a lattice structure.
10. Appropriate homogenization boundary conditions for the test strain cases applied to the
unit cell are strongly dependent on the number of layers and overall size of the intended
lattice structure.
11. AM-based correction factors for differences between as-printed and as-designed
structures, such as the BOC or SCC, can be incorporated directly into ALTO for
mechanical property prediction.
12. Through compression testing of printed lattice structures, the ground structure model and
homogenization method were validated experimentally.
13. DfAM optimization of unit cell topology simultaneously improves our ability to design
novel lattice structures and their manufacturability using AM.
14. Current experimental characterization of metal lattice structures fabricated with AM is
primarily limited to static compression testing without a standardized protocol for details
such as specimen size and strain rate.
15. Discrepancies between predicted mechanical properties from analytical and finite
element analysis as compared to experimental characterization remains a challenge for
advancing lattice structures to practical components.
7.3 Research Contributions
The research contributions of this work have been summarized into a list of seven key
points below. Following the list of research contributions, the intellectual merit for the scientific
community and broader impacts to the general community are addressed. With regards to the
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intellectual merit and broader impacts, the discussion is divided based on the two systematic
optimization methods developed: a shape matching approach for a compliant deployable surgical
tool for radiofrequency ablation of tumors and the development of specialized lattice structures
for additive manufacturing.
1. Validated systematic design approach for finite element-based optimization of
radiofrequency ablation electrodes
2. Developed an application-agnostic DfAM framework for lattice unit cell design and
optimization
3. Demonstrated first known DfAM powder removability optimization objective for lattice
structures
4. Demonstrated unit cell DfAM framework through multiple 2D and 3D examples
5. Generated novel unit cell topology with improved manufacturability and the same
orthogonal effective modulus properties as traditional unit cell
6. Developed build angle dependent staircase correction factor for the reduced effective
cross-sectional area of loading for unit cell struts
7. Established largest known database for analytical, finite element, and experimental
mechanical characterization of metal lattice structures into an interactive interface.
7.3.1 Electrode Optimization for Radiofrequency Ablation
Research of RFA has included numerous modeling methods for understanding how tissue
properties, power settings, and proximity to surrounding structures affect the treatment efficiency.
A few methods have also considered using these models for optimizing placement of the
electrode to improve treatment efficiency. While these models have improved our understanding
of RFA, there is a lack of systematic design approaches which take full advantage of these
models, optimization, and the increased manufacturing capabilities available.
For the scientific community, this work provides an experimentally validated novel
systematic design method for finite element-based optimization of the electrode geometry. The
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electrode geometry was optimized such that the ablation zone generated matched a target
geometry. In ablation technology research, the completed research fills the gap of using ablation
zone modeling as a design tool by providing validated models which predict ablation zones for
RFA electrodes as a methodology for optimizing them for a target shape. Though a very specific
case was considered as an example, EUS-RFA for a 2.5cm spherical tumor in pancreatic tissue,
the approach is applicable to tumors of any size/shape and to other locations where RFA may be
applied such as the lungs, liver, and kidneys. This systematic approach opens the doors for future
research into personalized medical care as it can easily be extended to customized non-spherical
and non-symmetric target shapes.
For a broader impact on the community, these models will benefit both doctors and
patients. This research moves surgeons one step closer to being able to provide truly customized
surgical treatment by enabling more advanced surgical tools. With optimized surgical electrodes,
surgeons can efficiently destroy cancerous tissue, preserving the surrounding healthy tissue. With
increased CT-imaging for medical diagnosis, many patients are learning of tumors in earlier
stages and seeking curative treatment options. While surgical resection is currently the most
effective option, a complete surgical resection of the tumor is often not possible due to
complications associated with the location of the tumor or the negative impact that an open
surgery would have on a patient in poor health. Improving less invasive treatment options than
open surgery through EUS-RFA, while simultaneously increasing efficiency, increases the odds
of survival for these rising numbers of cancer patients.
7.3.2 Additive Lattice Topology Optimization
With the advancing capabilities of AM, our ability to manufacture has exceeded our
current ability to design. AM processes enable access to the entire design envelope of a
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component and can fabricate designs that are difficult or impossible for traditional modeling tools
to analyze. While AM offers extensive freedom, there are additional design rules and process
limitations that have to be considered. There is a lack of systematic design tools and methods for
optimizing performance while accounting for AM process constraints, evident by numerous
research groups exploring topology optimization for AM, development of DfAM frameworks,
and a push from industry to be able to maximize performance of every component. In particular,
mesostructure design for lattice structures have broad applications; however, they are not well
understood or characterized and often used without consideration for why one lattice type is
chosen over another.
For the scientific community researching AM and DfAM, novel contributions of the
work completed are an application-agnostic systematic optimization method for unit cell topology
optimization, the powder removability factor (PRF) for improving manufacturability, and LUCIE,
the largest known database for mechanical characterization of lattice structures fabricated with
metal AM. The research combines ground structure topology optimization and DfAM to enable
generation of novel unit cell topologies with application-specific properties. Using an
evolutionary algorithm, the approach fills the gap of a robust methodology to incorporate a
variety of objective, constraint, and penalty functions that improve manufacturability for multi-
functional lattices. For example, in L-PBF, powder removal from lattice structures is a continuing
concern. The PRF developed as part of this work offers the first powder removal metric for strut
and node type lattice structures, assessing the manufacturability and post-processing of the lattice.
Generation of a lattice structure with the PRF as an objective has been shown to improve
manufacturability. For common lattice structure types, numerous research groups around the
world are generating much needed analytical, finite element, and experimental data for
characterization of lattice structures. LUCIE offers the largest compilation of this data in a format
that allows researchers to gain an understanding of the tradeoffs and trends for different unit cell
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topologies. In addition, while the focus of this work was for L-PBF, ALTO enables consideration
of various AM processes by changing the minimum feature size, build angle restriction, and other
parameters. These contributions and those of others in the community advance our ability to
apply lattice structures to everyday components.
For impact on the broader community, lattice structures form the basis for development
of novel materials and structures with applications across a variety of industries. For example,
consider lattice structures in medical implants. Metal implants are used for their durability and
strength; however, they are often too stiff causing stress shielding and allow limited bone
integration. As referenced in Section 1.1.5, numerous research groups are exploring lattice
structures because they offer the same traditional metallic properties with the opportunity for
reduced stiffness and enhanced bone integration. This research may serve as a stepping stone for
patients to receive custom multi-functional implants, with properties tailored to integrate into
their anatomy and enhance performance. Apart from the medical applications, lattice structures
also offer increased freedom over traditional 3D foams and materials, such as honeycomb
structures for impact absorbing applications. Other applications that benefit the broader
community could include examples such as lightweighting for improving fuel efficiency and
impact absorption for safer cars and planes. Lattice structures offer additional design freedom for
novel material development; systematic design and optimization tools are what is needed to take
advantage of them.
7.4 Recommended Future Research Directions
Based on this work, five recommended research direction are proposed: 1) improving
boundary conditions for homogenization, 2) understanding the impact of as-designed versus as-
printed topology, 3) porosity development in small unit cell sizes (< 2 mm), 4) expansion of
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LUCIE, 5) powder removal from AM structures, and 6) continued development of optimization
for additional applications.
As discussed in Chapter 5, the experimental data initially showed significant deviations
from the predicted homogenization properties. The discrepancy is associated with the boundary
conditions for homogenization over constraining and artificially increasing the predicted effective
modulus. Lattice structures have been shown to be sensitive to the number of layers in the testing
configuration. For similar reasons, test coupons are standardized to minimize the edge effects
during testing. Experimental and finite element analysis of large lattice structures to reduce the
boundary effects are expensive to physically and computationally perform, due to the time /cost
of printing and traditional modeling techniques utilized. Further exploration of homogenization
methods, and comparison with other approaches, will benefit DfAM methods such as ALTO so
that they can provide accurate predictions at reduced computational cost.
On the AM side, the practical use of lattice structures has remained limited due to lack of
understanding in performance and build quality. For practical use there are concerns regarding the
reliability of the structures, especially under cyclic or fatigue loading. The majority of
experimental data is for static loading and dynamic loading remains a continuing area of interest
before application of lattice structures can be widespread. Part of the challenge with developing
lattice structure for practical applications is the unknowns of how the as-designed part varies from
the as-printed part. While much is known about surface roughness and anisotropy in non-lattice
components, significantly less information is available for intricate lattice structures. In addition,
these traditional discrepancies between as-designed and as-printed parts are compounded in
lattice structures because the feature size of the discrepancies is often near or similar to the
feature sizes of the lattice. Issues such as surface roughness, sharp corners, anisotropy, dross, and
other geometric variations have a much bigger impact than in non-lattice parts. The focus of this
dissertation regarding unit cell design and optimization was to ensure the manufacturability of the
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topology. Future research should address how as-designed parts differ from as-printed parts, the
implications on performance, how to mitigate those discrepancies, microstructures of lattice
struts, qualification strategies, and finishing or post-processing strategies. This practical
understanding of lattice structure performance, combined with systematic optimization
approaches like ALTO, will enable lattice structures in advanced designs.
One example of an issue that arose during CT scanning of powder removability samples
described in Section 5.2.4, was the presence of void regions within the smallest unit cell size. For
the lattice structures with a large unit cell size, these void regions would be expected. After
building the component, the powder would be removed and evacuated from these non-solidified
regions of the lattice through void channels. However, after analysis of the void regions, there
were no channels or pathways from which powder could escape. The CT scan resolution was 15
microns, much smaller than many of the powder particles, and with the exception of a few void
regions near the exterior, there is currently no confirmed explanation of how powder could be
evacuated from these regions. The regularity and spherical nature of the voids is of interest for
understanding of what is causing these void regions as it may relate to other porosity or flaws that
develop in other components. Due to the lack of channels through which powder could be
removed, an assumption is made that these void regions formed during the build process. A few
theories for the void formation are powder particles being drawn away from the region and pulled
into the melt pool on either side or being blown away due to the air flow across the build plate. In
either case, the ability to understand the void formation could lead to improved build reliability in
AM components and lattice structures.
For LUCIE, there is new experimental data available all the time and multiple features
could improve its accessibility and usability. First, as more data becomes available, the database
should be expanded because as additional data will improve the quality of the trends. To support
this effort, a web-based version where researchers can add their own data to the public database,
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would improve the sustainability and accessibility of LUCIE. Second, the user interface can
continue to be updated to increase functionality and usability. Third, the ability to directly access
the sources of the data for each data point would provide the user a direct connection between the
data and its source. For example, adding direct access to key process parameters, PBF system,
heat treatment, and finishing processes for each data point or group of data points would helpful
to the user. Fourth, standardization of unit cell names and testing specifications should be
established to make the data reliable and easier to communicate between research groups. Lastly,
in its current state, LUCIE is sufficient to provide mechanical property trends based on the unit
cell topology. Through standardized testing methods, continued additions to the database of key
performance parameters, and incorporation of experimental error or uncertainty into LUCIE, it
will advance beyond simply providing trends to providing exact data based on the specified
volume fraction.
The ability to remove powder from AM components remains is a concern for lattice
structures as well as larger components such as heat sinks. Future research in this area could
include development of predictive tools for powder flow from an AM built component. As a part
of this simulation, thermal simulation to anticipate how cavities or channels will change the
printing process could be used to inform the designer about area way small regions could trap
powder. Generalization of the powder removability factor for larger components and a standard
approach to powder removal would help researchers be able to compare powder removal
techniques and processes to further improve the ability to remove powder. While it remains a
challenge to use full-scale predictive tools in optimization, simplified versions of these full-scale
tools could be paired with ALTO or other topology optimization processes to improve powder
removability.
Finally, the expansion of lattice structures for advanced applications relies on design and
optimization. Regarding expansion of ALTO, there are various research directions that could be
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considered such as build orientation consideration, exploration of optimization limits, combined
material and topology optimization, and robust optimization. For this work, the build orientation
is assumed to be vertical according to the z-axis of the ground structure model, however,
changing the build orientation of the lattice would impact the build angle ground structure
restriction, unsupported penalties, and AM-based geometry correction factors. For optimization
limits, additional work could be completed to consider what are the primary limiting factors for
being able to increase the number of design variables that can be optimized in a reasonable time.
For example, the current version of Borg was serial, though a parallelized version of Borg paired
with ALTO could substantially reduce computation time and increase the number of design
variables that can be optimized. Another area where ALTO could be expanded is through
microscale material optimization such as through multi-material unit cell topology optimization,
or simultaneous optimization of the material properties and topology. In this work the microscale
properties are assumed to be fixed and known, however, AM offers opportunities for multiple
materials or changing material properties based on different process parameters which could offer
additional payoffs for unit cell topology optimization. Finally, the field of robust optimization
considers process variability in the optimization problem. Including robust optimization in ALTO
would help the optimal solutions are less susceptible to common flaws or process variation in
AM.
In this work, various optimization objective, constraint, and penalty functions have been
combined to demonstrate a few examples of unit cell generation for differing applications. There
are endless possibilities of other objective functions that could be used to design unit cell
topologies or how the current objectives could be expanded to incorporate other scenarios. For
the powder removability factor, it acts solely on the unit cell topology as opposed to the lattice
structure as a whole. Exploration of how to expand the powder removability factor to account for
the entire lattice would increase the robustness and more accurately represent the actual scenario.
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In addition, tortuosity is difficult to solve analytically because the ground structure model has no
consideration for empty space. Therefore, in the ground structure approach there is no model of
the void cavities or channels which gave rise to the method for the powder removability factor.
However, one could consider a correlation between the tortuosity ratio for a lattice structure and
powder removability factor for a unit cell topology in order to understand their correlation.
Another correlation of interest would be to explore if there is any relation between common unit
cell parameters (unit cell size, lattice size, strut diameter, volume fraction, etc.) and the tortuosity
ratio. This could provide insight into powder removal without requiring the powder removability
factor calculation. Another future research direction would be additional exploration of
calculating the tortuosity ratio analytically, possibly through geometric projection methods. As
another example, current objective functions utilize thermal conduction for heat transfer
applications, however, convection plays a critical role in many applications. Another example for
an objective function could be the dynamic response of a lattice structure for impact absorption
scenarios. By expanding the objective functions available, the application of lattice structures
may continue to grow through design and optimization.
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References
[1] Howell, L. L., 2001, Compliant Mechanism, Wiley-Interscience.
[2] Kota, S., Lu, K. J., Kreiner, Z., Trease, B., Arenas, J., and Geiger, J., 2005, “Design and
Application of Compliant Mechanisms for Surgical Tools,” J. Biomech. Eng., 127(6), pp.
981–989.
[3] Liu, J., 2012, “Compliant Mechanisms Using Superelastic Nitinol,” Pennsylvania State
University.
[4] Edmondson, B. J., Bowen, L. A., Grames, C. L., Magleby, S. P., Howell, L. L., and
Bateman, T. C., 2016, “Oriceps: Origami-Inspired Forceps,” Proc. ASME SMASIS 2013,
pp. 1–6.
[5] Cronin, J. a., Frecker, M. I., and Mathew, A., 2008, “Design of a Compliant Endoscopic
Suturing Instrument,” J. Med. Device., 2(2), p. 025002.
[6] Aguirre, M. E., 2010, “Design and Optimization of Hybrid Compliant Narrow-Gauge
Surgical Forceps,” Proceedings of the ASME 2010 Conf on SMASIS.
[7] Frecker, M. I., Powell, K. M., and Haluck, R., 2005, “Design of a Multifunctional
Compliant Instrument for Minimally Invasive Surgery.,” J. Biomech. Eng., 127(6), pp.
990–993.
[8] Aguirre, M. E., 2011, “Design and Optimization of Narrow-Gauge Contact-Aided
Compliant Mechanisms for Advanced Minimally Invasive Surgery,” The Pennsylvania
State University.
[9] Lee, C., and Tarbutton, J. A., 2015, “Compliance and Control Characteristics of an
Additive Manufactured-Flexure Stage,” Rev. Sci. Instrum., 86(4).
[10] Cuellar, J. S., Smit, G., Plettenburg, D., and Zadpoor, A., 2018, “Additive Manufacturing
of Non-Assembly Mechanisms,” Addit. Manuf., 21(January), pp. 150–158.
[11] Blanes, C., Mellado, M., and Beltran, P., 2014, “Novel Additive Manufacturing Pneumatic
Actuators and Mechanisms for Food Handling Grippers,” Actuators, 3(3), pp. 205–225.
[12] Mirth, J. A., 2013, “Preliminary Investigations into the Design and Manufacturing of Fully
Compliant Layered Mechanisms,” ASME 2013 International Design Engineering
Technical Conferences & Computers and Information in Engineering Conference.
[13] Gaynor, A. T., Meisel, N. A., Williams, C. B., and Guest, J. K., 2014, “Multiple-Material
Topology Optimization of Compliant Mechanisms Created via Polyjet 3D Printing,” J.
Manuf. Sci. Eng., 136(6), pp. 980–997.
284
[14] Sakhaei, A. H., Kaijima, S., Lee, T. L., Tan, Y. Y., and Dunn, M. L., 2018, “Design and
Investigation of a Multi-Material Compliant Ratchet-like Mechanism,” Mech. Mach.
Theory, 121, pp. 184–197.
[15] Mirth, J. A., 2016, “The Design and Prototyping of Complex Compliant Mechanisms Via
Multi-Material Additive Manufacturing Techniques,” ASME 2016 International Design
Engineering Technical Conferences and Computers and Information in Engineering
Conference, pp. 1–6.
[16] Merriam, E. G., Jones, J. E., Magleby, S. P., and Howell, L. L., 2013, “Monolithic 2 DOF
Fully Compliant Space Pointing Mechanism,” Mech. Sci., 4(2), pp. 381–390.
[17] Merriam, E. G., Lund, J. M., and Howell, L. L., 2015, “Compound Joints: Behavior and
Benefits of Flexure Arrays,” Precis. Eng., 45, pp. 79–89.
[18] Merriam, E. G., Tolman, K. A., and Howell, L. L., 2016, “Integration of Advanced
Stiffness-Reduction Techniques Demonstrated in a 3D-Printable Joint,” Mech. Mach.
Theory, 105, pp. 260–271.
[19] Merriam, E. G., Jones, J. E., and Howell, L. L., 2014, “Design of 3D-Printed Titanium
Compliant Mechanisms,” Proceedings of the 42nd Aerospace Mechanisms Symposium,
pp. 169–174.
[20] Jovanova, J., and Frecker, M., 2017, “Two Stage Design of Compliant Mechanisms With
Superelastic Compliant,” ASME 2017 Conference on Smar Materials, Adaptive Structures
and Intelligent Systems, pp. 1–9.
[21] Pham, M. T., Teo, T. J., Yeo, S. H., Wang, P., and Nai, M. L. S., 2017, “A 3-D Printed Ti-
6Al-4V 3-DOF Compliant Parallel Mechanism for High Precision Manipulation,”
IEEE/ASME Trans. Mechatronics, 22(5), pp. 2359–2368.
[22] American Cancer Society, 2016, Cancer Facts & Figures 2013, Atlanta, Georgia.
[23] 2015, “Pancreatic Cancer: Treatment Options,” Cancer.Net [Online]. Available:
http://www.cancer.net/cancer-types/pancreatic-cancer/treatment-options.
[24] Greenhalf, W., Grocock, C., Harcus, M., and Neoptolemos, J., 2009, “Screening of High-
Risk Families for Pancreatic Cancer,” Pancreatology, 9(3), pp. 215–222.
[25] Lee, K. S., Sekhar, A., Rofsky, N. M., and Pedrosa, I., 2010, “Prevalence of Incidental
Pancreatic Cysts in the Adult Population on MR Imaging.,” Am. J. Gastroenterol., 105(9),
pp. 2079–84.
[26] Spinelli, K. S., Fromwiller, T. E., Daniel, R. A., Kiely, J. M., Nakeeb, A., Komorowski, R.
A., Wilson, S. D., and Pitt, H. A., 2004, “Cystic Pancreatic Neoplasms: Observe or
Operate.,” Ann. Surg., 239(5), pp. 651–7; discussion 657-9.
[27] Knavel, E. M., and Brace, C. L., 2013, “Tumor Ablation: Common Modalities and
General Practices,” Tech. Vasc. Interv. Radiol., 16(4), pp. 192–200.
285
[28] Chen, E. L., and Prinz, R. A., 2007, “Long-Term Survival after Pancreatic Cancer
Treatment,” Am. J. Surg., 194(4A), pp. S127–S130.
[29] Werner, J., Combs, S. E., Springfeld, C., Hartwig, W., Hackert, T., and Buchler, M. W.,
2013, “Advanced-Stage Pancreatic Cancer: Therapy Options,” Nat Rev Clin Oncol, 10(6),
pp. 323–333.
[30] Neoptolemos, J. P., Stocken, D. D., Friess, H., Bassi, C., Dunn, J. A., Hickey, H., Beger,
H., Fernandez-Cruz, L., Dervenis, C., Lacaine, F., Falconi, M., Pederzoli, P., Pap, A.,
Spooner, D., Kerr, D. J., and Buchler, M. W., 2004, “A Randomized Trial of
Chemoradiotherapy and Chemotherapy after Resection of Pancreatic Cancer,” N Engl J
Med, 350(12), pp. 1200–1210.
[31] Neoptolemos, J. P., Stocken, D. D., Bassi, C., Ghaneh, P., Cunningham, D., Goldstein, D.,
Padbury, R., Moore, M. J., Gallinger, S., Mariette, C., Wente, M. N., Izbicki, J. R., Friess,
H., Lerch, M. M., Dervenis, C., Olah, A., Butturini, G., Doi, R., Lind, P. A., Smith, D.,
Valle, J. W., Palmer, D. H., Buckels, J. a, Thompson, J., Mckay, C. J., Rawcliffe, C. L.,
and Büchler, M. W., 2010, “Adjuvant Chemotherapy With Fluorouracil Plus Folinic Acid
vs Gemcitabine Following Pancreatic Cancer Resection,” J. Am. Med. Assoc., 304(10),
pp. 1073–1081.
[32] Oettle, H., Post, S., Neuhaus, P., Gellert, K., Langrehr, J., Ridwelski, K., Schramm, H.,
Fahlke, J., Zuelke, C., Burkart, C., Gutberlet, K., Kettner, E., Schmalenberg, H., Weigang-
Koehler, K., Bechstein, W.-O., Niedergethmann, M., Schmidt-Wolf, I., Roll, L., Doerken,
B., and Riess, H., 2007, “Adjuvant Chemotherapy With Gemcitabine vs Observation in
Patients Undergoing Curative-Intent Resection of Pancreatic Cancer,” J. Am. Med.
Assoc., 297(3), pp. 267–277.
[33] Ueno, H., Kosuge, T., Matsuyama, Y., Yamamoto, J., Nakao, A., Egawa, S., Doi, R.,
Monden, M., Hatori, T., Tanaka, M., Shimada, M., and Kanemitsu, K., 2009, “A
Randomised Phase III Trial Comparing Gemcitabine with Surgery-Only in Patients with
Resected Pancreatic Cancer: Japanese Study Group of Adjuvant Therapy for Pancreatic
Cancer.,” Br. J. Cancer, 101(6), pp. 908–15.
[34] Schmidt, J., Abel, U., Debus, J., Harig, S., Hoffmann, K., Herrmann, T., Bartsch, D.,
Klein, J., Mansmann, U., Jäger, D., Capussotti, L., Kunz, R., and Büchler, M. W., 2012,
“Open-Label, Multicenter, Randomized Phase III Trial of Adjuvant Chemoradiation plus
Interferon Alfa-2b versus Fluorouracil and Folinic Acid for Patients with Resected
Pancreatic Adenocarcinoma,” J. Clin. Oncol., 30(33), pp. 4077–4083.
[35] D’Arsonval, A., 1891, “Action Physiologique Des Courants Alternatifs a Grande
Frequence,” Arch Physiol Norm Pathol, 5, pp. 401–408.
[36] Organ, L., 1976, “Electrophysiologic Principles of Radiofrequency Lesion Making,” Appl.
Neurophysiol., 39(2), pp. 69–76.
[37] Krishnamurthy, V. N., Casillas, V. J., and Latorre, L., 2003, “Radiofrequency Ablation of
Hepatic Tumors,” Appl. R, 32(10).
[38] McGahan, J. P., Browning, P. D., Brock, J. M., and Tesluk, H., 1990, “Hepatic Ablation
286
Using Radiofrequency Electrocautery,” Investig. Radiol., 25(3), pp. 267–270.
[39] Rossi, S., Fornari, F., Pathies, C., and Buscarini, L., 1990, “Thermal Lesions Induced by
480 KHz Localized Current Field in Guinea Pig and Pig Liver,” Tumori, 76(1), pp. 54–57.
[40] VanSonnenberg, E., McMullen, W., and Solbiati, L., 2010, Tumor Ablation, Springer New
York.
[41] Hong, K., and Georgiades, C., 2010, “Radiofrequency Ablation: Mechanism of Action
and Devices,” J. Vasc. Interv. Radiol., 21(SUPPL. 8), pp. S179–S186.
[42] Dewhirst, M. W., Viglianti, B. L., Lora-Michiels, M., Hanson, M., and Hoopes, P. J.,
2003, “Basic Principles of Thermal Dosimetry and Thermal Thresholds for Tissue
Damage from Hyperthermia,” Int. J. Hyperth., 19(3), pp. 267–294.
[43] Sapareto, S. A., and Dewey, W. C., 1984, “Thermal Dose Determination in Cancer
Therapy,” Int. J. Radiat. Oncol. Biol. Phys., 10(6), pp. 787–800.
[44] Ryan, T. P., Kwok, J., and Beetel, R., 2003, “Simulations of Percutaneous RF Ablation
Systems,” SPIE Conference Proceedings, pp. 71–88.
[45] Brace, C. L., 2009, “Radiofrequency and Microwave Ablation of the Liver, Lung, Kidney
and Bone: What Are the Differences,” Curr. Probl. Diagn. Radiol., 38(3), pp. 135–143.
[46] Feldman, L. S., Fuchshuber, P. R., and Editors, D. B. J., 2012, The SAGES Manual on the
Fundamental Use of Surgical Energy (FUSE), Springer-Verlag, New York.
[47] Dewhirst, M. W., Vujaskovic, Z., Jones, E., and Thrall, D., 2005, “Re-Setting the Biologic
Rationale for Thermal Therapy.,” Int. J. Hyperthermia, 21(8), pp. 779–90.
[48] Nemcek, A. a., 2006, “Complications of Radiofrequency Ablation of Neoplasms,” Semin.
Intervent. Radiol., 23(2), pp. 177–187.
[49] Lu, D. S. K., Raman, S. S., Limanond, P., Aziz, D., Economou, J., Busuttil, R., and Sayre,
J., 2003, “Influence of Large Peritumoral Vessels on Outcome of Radiofrequency
Ablation of Liver Tumors,” J. Vasc. Interv. Radiol., 14(10), pp. 1267–1274.
[50] Saldanha, D. F., Khiatani, V. L., Carrillo, T. C., Yap, F. Y., Bui, J. T., Knuttinen, M. G.,
Owens, C. A., and Gaba, R. C., 2010, “Current Tumor Ablation Technologies: Basic
Science and Device Review,” Semin. Intervent. Radiol., 27(3), pp. 247–254.
[51] Mulier, S., Miao, Y., Mulier, P., Dupas, B., Pereira, P., De Baere, T., Lencioni, R.,
Leveillee, R., Marchal, G., Michel, L., and Ni, Y., 2006, “Electrodes and Multiple
Electrode Systems for Radio Frequency Ablation: A Proposal for Updated Terminology,”
Eur. J. Radiol., 15, pp. 57–73.
[52] D’Onofrio, M., Barbi, E., Girelli, R., Martone, E., Gallotti, A., Salvia, R., Martini, P.-T.,
Bassi, C., Pederzoli, P., and Pozzi Mucelli, R., 2010, “Radiofrequency Ablation of Locally
Advanced Pancreatic Adenocarcinoma: An Overview.,” World J. Gastroenterol., 16(28),
pp. 3478–3483.
287
[53] Varghese, T., Techavipoo, U., Zagzebski, J. a, and Lee, F. T., 2004, “Impact of Gas
Bubbles Generated during Interstitial Ablation on Elastographic Depiction of in Vitro
Thermal Lesions.,” J. Ultrasound Med., 23(4), pp. 535–544; quiz 545–546.
[54] Raman, S. S., Lu, D. S. K. K., Vodopich, D. J., Sayre, J., and Lassman, C., 2000,
“Creation of Radiofrequency Lesions in a Porcine Model: Correlation with Sonography,
CT, and Histopathology,” Am. J. Roentgenol., 175(5), pp. 1253–1258.
[55] Leyendecker, J. R., Dodd, Gerald D, I., Halff, G. A., Mccoy, V. A., Napier, D. H.,
Hubbard, L. G., Chintapalli, K. N., Chopra, S., Washburn, W. K., Esterl, R. M., Cigarroa,
F. G., Kohlmeier, R. E., and Sharkey, F. E., 2002, “Sonographically Observed Echogenic
Response During Intraoperative Radiofrequency Ablation of Cirrhotic Livers : Pathologic
Correlation,” Am. J. Roentgenol., 178(May), pp. 1147–1151.
[56] Cha, C. H., Lee, F. T., Gurne, J. M., Markhardt, B. K., Warner, T. F., Kelcz, F., and
Mahvi, D. M., 2000, “CT Versus Monitoring Ablation Sonography for Radiofrequency in
a Porcine Liver,” Imaging, 175(3), pp. 705–711.
[57] Wu, H., Wilkins, L. R., Ziats, N. P., Haaga, J. R., and Exner, A. A., 2014, “Real-Time
Monitoring of Radiofrequency Ablation and Postablation Assessment: Accuracy of
Contrast-Enhanced US in Experimental Rat Liver Model.,” Radiology, 270(1), pp. 107–
16.
[58] Isobe, Y., Watanabe, H., Yamazaki, N., Lu, X., Kobayashi, Y., Miyashita, T., Hashizume,
M., and Fujie, M. G., 2013, “Real-Time Temperature Control System Based on the Finite
Element Method for Liver Radiofrequency Ablation: Effect of the Time Interval on
Control.,” International Conference of the IEEE Engineering in Medicine and Biology
Society, pp. 392–6.
[59] Jin, C., He, Z., and Liu, J., 2014, “MRI-Based Finite Element Simulation on
Radiofrequency Ablation of Thyroid Cancer,” Comput. Methods Programs Biomed.,
113(2), pp. 529–538.
[60] Zhang, M., Zhou, Z., Wu, S., Lin, L., Gao, H., and Feng, Y., 2015, “Simulation of
Temperature Field for Temperature-Controlled Radio Frequency Ablation Using a
Hyperbolic Bioheat Equation and Temperature-Varied Voltage Calibration: A Liver-
Mimicking Phantom Study,” Phys. Med. Biol., 60(24), pp. 9455–9471.
[61] Chang, I., 2003, “Finite Element Analysis of Hepatic Radiofrequency Ablation Probes
Using Temperature-Dependent Electrical Conductivity.,” Biomed. Eng. Online, 2, p. 12.
[62] Schutt, D. J., and Haemmerich, D., 2008, “Effects of Variation in Perfusion Rates and of
Perfusion Models in Computational Models of Radio Frequency Tumor Ablation.,” Med.
Phys., 35(8), pp. 3462–3470.
[63] Solazzo, S. a, Liu, Z., Lobo, S. M., Ahmed, M., Hines-Peralta, A. U., Lenkinski, R. E.,
and Goldberg, S. N., 2005, “Radiofrequency Ablation: Importance of Background Tissue
Electrical Conductivity--an Agar Phantom and Computer Modeling Study.,” Radiology,
236(2), pp. 495–502.
288
[64] Duan, B., Wen, R., Fu, Y., Chua, K. J., and Chui, C. K., 2016, “Probabilistic Finite
Element Method for Large Tumor Radiofrequency Ablation Simulation and Planning,”
Med. Eng. Phys., 38(11), pp. 1360–1368.
[65] Subramanian, S., and Mast, T. D., 2015, “Optimization of Tissue Physical Parameters for
Accurate Temperature Estimation from Finite-Element Simulation of Radiofrequency
Ablation,” Phys. Med. Biol., 60(19), pp. N345–N355.
[66] Huang, W., and Chui, C., 2012, “A Radio-Frequency Ablation Planning System Using
Stochastic Finite Element Method,” IEEE/SCIE.
[67] Tungjitkusolmun, S., Staelin, S. T., Haemmerich, D., Tsai, J. Z., Cao, H., Webster, J. G.,
Lee, F. T., Mahvi, D. M., and Vorperian, V. R., 2002, “Three-Dimensional Finite-Element
Analyses for Radio-Frequency Hepatic Tumor Ablation,” IEEE Trans. Biomed. Eng.,
49(1), pp. 3–9.
[68] Lim, D., Namgung, B., Woo, D. G., Choi, J. S., Kim, H. S., and Tack, G. R., 2010, “Effect
of Input Waveform Pattern and Large Blood Vessel Existence on Destruction of Liver
Tumor Using Radiofrequency Ablation: Finite Element Analysis.,” J. Biomech. Eng.,
132(6), p. 061003.
[69] Haemmerich, D., Wright, A. W., Mahvi, D. M., Lee, F. T., and Webster, J. G., 2003,
“Hepatic Bipolar Radiofrequency Ablation Creates Coagulation Zones Close to Blood
Vessels: A Finite Element Study,” Med. Biol. Eng. Comput., 41(3), pp. 317–323.
[70] Dos Santos, I., Haemmerich, D., Schutt, D., Da Rocha, A. F., and Menezes, L. R., 2009,
“Probabilistic Finite Element Analysis of Radiofrequency Liver Ablation Using the
Unscented Transform.,” Phys. Med. Biol., 54(3), pp. 627–640.
[71] Chang, I. a, and Nguyen, U. D., 2004, “Thermal Modeling of Lesion Growth with
Radiofrequency Ablation Devices.,” Biomed. Eng. Online, 3(27).
[72] Barauskas, R., Gulbinas, A., Vanagas, T., and Barauskas, G., 2008, “Finite Element
Modeling of Cooled-Tip Probe Radiofrequency Ablation Processes in Liver Tissue,”
Comput. Biol. Med., 38(6), pp. 694–708.
[73] Altrogge, I., Preusser, T., Kröger, T., Büskens, C., Pereira, P. L., Schmidt, D., and
Peitgen, H. O., 2007, “Multiscale Optimization of the Probe Placement for
Radiofrequency Ablation,” Acad. Radiol., 14(11), pp. 1310–1324.
[74] Haemmerich, D., Tungjitkusolmun, S., Staelin, S. T., Lee, F. T., Mahvi, D. M., and
Webster, J. G., 2002, “Finite-Element Analysis of Hepatic Multiple Probe Radio-
Frequency Ablation,” IEEE Trans. Biomed. Eng., 49(8), pp. 836–842.
[75] Haemmerich, D., and Webster, J. G., 2005, “Automatic Control of Finite Element Models
for Temperature-Controlled Radiofrequency Ablation,” Biomed. Eng. Online, 4, pp. 1–8.
[76] Varadarajulu, S., Jhala, N. C., and Drelichman, E. R., 2009, “EUS-Guided
Radiofrequency Ablation with a Prototype Electrode Array System in an Animal Model
(with Video),” Gastrointest. Endosc., 70(2), pp. 372–376.
289
[77] Gaidhane, M., Smith, I., Ellen, K., Gatesman, J., Habib, N., Foley, P., Moskaluk, C., and
Kahaleh, M., 2012, “Endoscopic Ultrasound-Guided Radiofrequency Ablation (EUS-
RFA) of the Pancreas in a Porcine Model,” Gastroenterol. Res. Pract., 2012, pp. 3–8.
[78] Pai, M., Habib, N., Senturk, H., Lakhtakia, S., Reddy, N., Cicinnati, V. R., Kaba, I.,
Beckebaum, S., Drymousis, P., Kahaleh, M., and Brugge, W., 2015, “Endoscopic
Ultrasound Guided Radiofrequency Ablation, for Pancreatic Cystic Neoplasms and
Neuroendocrine Tumors,” World J. Gastrointest. Surg., 7(4), pp. 52–59.
[79] Bas-Cutrina, F., Bargalló, D., and Gornals, J. B., 2017, “Small Pancreatic Insulinoma:
Successful Endoscopic Ultrasound-Guided Radiofrequency Ablation in a Single Session
Using a 22-G Fine Needle,” Dig. Endosc., 29(5), pp. 636–638.
[80] Chaudhary, S., and Sun, S. Y., 2017, “Endoscopic Ultrasound-Guided Radiofrequency
Ablation in Gastroenterology: New Horizons in Search,” World J. Gastroenterol., 23(27),
pp. 4892–4896.
[81] Choi, J. H., Seo, D. W., Song, T. J., Park, D. H., Lee, S. S., Lee, S. K., and Kim, M. H.,
2018, “Endoscopic Ultrasound-Guided Radiofrequency Ablation for Management of
Benign Solid Pancreatic Tumors,” Endoscopy, 50(11), pp. 1099–1104.
[82] Barthet, M., Giovannini, M., Lesavre, N., Boustiere, C., Napoleon, B., Koch, S., Gasmi,
M., Vanbiervliet, G., and Gonzalez, J. M., 2019, “Endoscopic Ultrasound-Guided
Radiofrequency Ablation for Pancreatic Neuroendocrine Tumors and Pancreatic Cystic
Neoplasms: A Prospective Multicenter Study,” Endoscopy, 51(9), pp. 836–842.
[83] Oleinikov, K., Dancour, A., Epshtein, J., Benson, A., Mazeh, H., Tal, I., Matalon, S.,
Benbassat, C. A., Livovsky, D. M., Goldin, E., Gross, D. J., Jacob, H., and Grozinsky-
Glasberg, S., 2019, “Endoscopic Ultrasound-Guided Radiofrequency Ablation: A New
Therapeutic Approach for Pancreatic Neuroendocrine Tumors,” J. Clin. Endocrinol.
Metab., 104(7), pp. 2637–2647.
[84] Ofosu, A., Ramai, D., and Adler, D. G., 2019, “Endoscopic Ultrasound-Guided Ablation
of Pancreatic Cystic Neoplasms: Ready for Prime Time?,” Ann. Gastroenterol., 32(1), pp.
39–45.
[85] Waung, J. A., Todd, J. F., Keane, M. G., and Pereira, S. P., 2016, “Successful
Management of a Sporadic Pancreatic Insulinoma by Endoscopic Ultrasound-Guided
Radio-Frequency Ablation,” Endoscopy, 48, pp. E144–E145.
[86] Carrara, S., Arcidiacono, P. G., Albarello, L., Addis, A., Enderle, M. D., Boemo, C.,
Campagnol, M., Ambrosi, A., Doglioni, C., and Testoni, P. A., 2008, “Endoscopic
Ultrasound-Guided Application of a New Hybrid Cryotherm Probe in Porcine Pancreas: A
Preliminary Study,” Endoscopy, 40, pp. 321–326.
[87] Arcidiacono, P. G., Carrara, S., Reni, M., Petrone, M. C., Cappio, S., Balzano, G., Boemo,
C., Cereda, S., Nicoletti, R., Enderle, M. D., Neugebauer, A., Von Renteln, D., Eickhoff,
A., and Testoni, P. A., 2012, “Feasibility and Safety of EUS-Guided Cryothermal
Ablation in Patients with Locally Advanced Pancreatic Cancer,” Gastrointest. Endosc.,
76(6), pp. 1142–1151.
290
[88] Gao, W., Zhang, Y., Ramanujan, D., Ramani, K., Chen, Y., Williams, C. B., Wang, C. C.
L., Shin, Y. C., Zhang, S., and Zavattieri, P. D., 2015, “The Status, Challenges, and Future
of Additive Manufacturing in Engineering,” Comput. Des., 69, pp. 65–89.
[89] Wong, K. V., and Hernandez, A., 2012, “A Review of Additive Manufacturing,” ISRN
Mech. Eng., 2012, pp. 1–10.
[90] Guo, N., and Leu, M. C., 2013, “Additive Manufacturing: Technology, Applications and
Research Needs,” Front. Mech. Eng., 8(3), pp. 215–243.
[91] Abdulhameed, O., Al-Ahmari, A., Ameen, W., and Mian, S. H., 2019, “Additive
Manufacturing: Challenges, Trends, and Applications,” Adv. Mech. Eng., 11(2), pp. 1–27.
[92] Cali, J., Calian, D. A., Amati, C., Kleinberger, R., Steed, A., Kautz, J., and Weyrich, T.,
2012, “3D-Printing of Non-Assembly, Articulated Models,” ACM Trans. Graph., 31(6).
[93] Mohammed, M. I., P. Fitzpatrick, A., and Gibson, I., 2017, “Customised Design of a
Patient Specific 3D Printed Whole Mandible Implant,” KnE Eng., 2(2), p. 104.
[94] Tuck, C. J., Hague, R. J. M., Ruffo, M., Ransley, M., and Adams, P., 2008, “Rapid
Manufacturing Facilitated Customization,” Int. J. Comput. Integr. Manuf., 21(3), pp. 245–
258.
[95] Maji, P. K., Banerjee, A. J., Banerjee, P. S., and Karmakar, S., 2014, “Additive
Manufacturing in Prosthesis Development – a Case Study,” Rapid Prototyp. J., 20(6), pp.
480–489.
[96] Schmelzle, J., Kline, E. V., Dickman, C. J., Reutzel, E. W., Jones, G., and Simpson, T.
W., 2015, “(Re)Designing for Part Consolidation: Understanding the Challenges of Metal
Additive Manufacturing,” J. Mech. Des., 137(11), p. 111404.
[97] Kellner, T., 2017, “An Epiphany Of Disruption: GE Additive Chief Explains How 3D
Printing Will Upend Manufacturing,” GE Reports [Online]. Available:
https://www.ge.com/reports/epiphany-disruption-ge-additive-chief-explains-3d-printing-
will-upend-manufacturing/. [Accessed: 24-May-2018].
[98] MacDonald, E., and Wicker, R., 2016, “Multiprocess 3D Printing for Increasing
Component Functionality,” Science (80-. )., 353(6307).
[99] Kataria, A., and Rosen, D. W., 2001, “Building around Inserts: Methods for Fabricating
Complex Devices in Stereolithography,” Rapid Prototyp. J., 7(5), pp. 253–262.
[100] Meisel, N. A., Elliott, A. M., and Williams, C. B., 2015, “A Procedure for Creating
Actuated Joints via Embedding Shape Memory Alloys in PolyJet 3D Printing,” J. Intell.
Mater. Syst. Struct., 26(12), pp. 1498–1512.
[101] Ota, H., Emaminejad, S., Gao, Y., Zhao, A., Wu, E., Challa, S., Chen, K., Fahad, H. M.,
Jha, A. K., Kiriya, D., Gao, W., Shiraki, H., Morioka, K., Ferguson, A. R., Healy, K. E.,
Davis, R. W., and Javey, A., 2016, “Application of 3D Printing for Smart Objects with
Embedded Electronic Sensors and Systems,” Adv. Mater. Technol., 1(1).
291
[102] Doubrovski, E. L., Tsai, E. Y., Dikovsky, D., Geraedts, J. M. P., Herr, H., and Oxman, N.,
2015, “Voxel-Based Fabrication through Material Property Mapping: A Design Method
for Bitmap Printing,” CAD Comput. Aided Des., 60, pp. 3–13.
[103] Garland, A., and Fadel, G., 2015, “Design and Manufacturing Functionally Gradient
Material Objects With an Off the Shelf Three-Dimensional Printer: Challenges and
Solutions,” J. Mech. Des., 137(11), p. 111407.
[104] Everton, S. K., Hirsch, M., Stravroulakis, P., Leach, R., and Clare, A. T., 2016, “Review
of in Situ Process Monitoring and in Situ Metrology for Metal Additive Manufacturing,”
Mater. Des., 95, pp. 431–445.
[105] Chua, Z. Y., Ahn, I. H., and Moon, S. K., 2017, “Process Monitoring and Inspection
Systems in Metal Additive Manufacturing: Status and Applications,” Int. J. Precis. Eng.
Manuf. - Green Technol., 4(2), pp. 235–245.
[106] Grasso, M., and Colosimo, B. M., 2017, “Process Defects and in Situ Monitoring Methods
in Metal Powder Bed Fusion: A Review,” Meas. Sci. Technol., 28(4).
[107] Montazeri, M., and Rao, P., 2017, “In-Process Condition Monitoring in Laser Powder Bed
Fusion (LPBF),” pp. 1264–1278.
[108] Repossini, G., Laguzza, V., Grasso, M., and Colosimo, B. M., 2017, “On the Use of
Spatter Signature for In-Situ Monitoring of Laser Powder Bed Fusion,” Addit. Manuf., 16,
pp. 35–48.
[109] Zhang, B., Land, W. S., Ziegert, J., Davies, A., Ii, W. S. L., Ziegert, J., and Davies, A.,
2015, “In Situ Monitoring of Laser Powder Bed Fusion Additive Manufacturing Using
Digital Fringe Projection Technique,” Proc. ASPE 2015 Spring Top. Meet., (January), pp.
47–52.
[110] Zhao, C., Fezzaa, K., Cunningham, R. W., Wen, H., De Carlo, F., Chen, L., Rollett, A. D.,
and Sun, T., 2017, “Real-Time Monitoring of Laser Powder Bed Fusion Process Using
High-Speed X-Ray Imaging and Diffraction,” Sci. Rep., 7(1), pp. 1–11.
[111] Yadroitsev, I., Krakhmalev, P., and Yadroitsava, I., 2014, “Selective Laser Melting of
Ti6Al4V Alloy for Biomedical Applications: Temperature Monitoring and
Microstructural Evolution,” J. Alloys Compd., 583, pp. 404–409.
[112] Frazier, W. E., 2014, “Metal Additive Manufacturing: A Review,” J. Mater. Eng.
Perform., 23(6), pp. 1917–1928.
[113] Bhavar, V., Kattire, P., Patil, V., Khot, S., Gujar, K., and Singh, R., 2014, “A Review on
Powder Bed Fusion Technology of Metal Additive Manufacturing,” Mater. Sci. Technol.
2014.
[114] Herzog, D., Seyda, V., Wycisk, E., and Emmelmann, C., 2016, “Additive Manufacturing
of Metals,” Acta Mater., 117, pp. 371–392.
[115] Fielding, J. D., Gorham, R., Davis, A., Bouffard, B., and Morris, K., 2016, Department of
292
Defense Joint Additive Manufacturing Roadmap.
[116] Dong, G., Tang, Y., and Zhao, Y. F., 2017, “A Survey of Modeling of Lattice Structures
Fabricated by Additive Manufacturing,” J. Mech. Des., 139(10), pp. 1–13.
[117] Chu, C., Graf, G., and Rosen, D. W., 2008, “Design for Additive Manufacturing of
Cellular Structures,” Comput. Aided. Des. Appl., 5(5), pp. 686–696.
[118] Seepersad, C. C., Allen, J. K., McDowell, D. L., and Mistree, F., 2008, “Multifunctional
Topology Design of Cellular Material Structures,” J. Mech. Des., 130.
[119] Gibson, I., Rosen, D. W., and Stucker, B., 2015, “Directed Energy Deposition,” Additive
Manufacturing Technologies: 3D Printing, Rapid Prototyping, and Direct Digital
Manufacturing, pp. 245–268.
[120] Thompson, M. K., Moroni, G., Vaneker, T., Fadel, G., Campbell, R. I., Gibson, I.,
Bernard, A., Schulz, J., Graf, P., Ahuja, B., and Martina, F., 2016, “Design for Additive
Manufacturing: Trends, Opportunities, Considerations, and Constraints,” CIRP Ann. -
Manuf. Technol., 65(2), pp. 737–760.
[121] ASTM International, 2017, ISO/ASTM 52910:2017 Standard Guidelines for Design for
Additive Manufacturing.
[122] Wiberg, A., Persson, J., and Ölvander, J., 2019, “Design for Additive Manufacturing – a
Review of Available Design Methods and Software,” Rapid Prototyp. J., 25(6), pp. 1080–
1094.
[123] Laverne, F., Segonds, F., Anwer, N., and Le Coq, M., 2015, “Assembly Based Methods to
Support Product Innovation in Design for Additive Manufacturing: An Exploratory Case
Study,” J. Mech. Des., 137(12), p. 121701.
[124] Yang, S., and Zhao, Y. F., 2015, “Additive Manufacturing-Enabled Design Theory and
Methodology: A Critical Review,” Int. J. Adv. Manuf. Technol., 80(1–4), pp. 327–342.
[125] Altair, “Optistruct” [Online]. Available: https://www.altair.com/optistruct/.
[126] Autodesk, 2019, “Netfabb Ultimate 2019” [Online]. Available:
https://www.autodesk.com/products/netfabb/overview.
[127] Doubrovski, Z., Verlinden, J. C., and Geraedts, J. M. P., 2011, “OPTIMAL DESIGN FOR
ADDITIVE MANUFACTURING: OPPORTUNITIES AND CHALLENGES,” ASME
2011 International Design Engineering Technical Conferences & Computers and
Information in Engineering Conference, pp. 1–12.
[128] Thomas, D., 2009, “The Development of Design Rules for Selective Laser Melting The
Development of Design Rules for Selective Laser Melting.”
[129] Vayre, B., Vignat, F., and Villeneuve, F., 2012, “Designing for Additive Manufacturing,”
Procedia CIRP, 3(1), pp. 632–637.
293
[130] ASTM International, 2019, Additive Manufacturing — Design — Part 1 : Laser-Based
Powder Bed Fusion of Metals 1 (ISO/ASTM 52911-1).
[131] ASTM International, 2019, Additive Manufacturing — Technical Design Guideline for
Powder Bed Fusion Part 2: Laser-Based Powder Bed Fusion of Polymers (ISO/ASTM
52911-2).
[132] ASTM International, 2019, Additive Manufacturing - Design - Requirements, Guidelines
and Recommendations (ISO/ASTM52910).
[133] Chen, W., Watts, S., Jackson, J. A., Smith, W. L., Tortorelli, D. A., and Spadaccini, C. M.,
2019, “Stiff Isotropic Lattices beyond the Maxwell Criterion,” Sci. Adv., 5(9), pp. 1–7.
[134] Pham, M.-S., Liu, C., Todd, I., and Lertthanasarn, J., 2019, “Damage-Tolerant Architected
Materials Inspired by Crystal Microstructure,” Nature.
[135] Tang, Y., and Zhao, Y. F., 2016, “A Survey of the Design Methods for Additive
Manufacturing to Improve Functional Performance,” Rapid Prototyp. J., 22(3), pp. 569–
590.
[136] Gibson, L. J., and Ashby, M. F., 1997, Cellular Solids, Cambridge University Press,
Cambridge, New York.
[137] Ashby, M. F., Evans, A. G., Fleck, N. A., Gibson, L. J., Hutchinson, J. W., and Wadley,
H., 2000, Metal Foams: A Design Guide, Butterworth-Heinemann, Boston, MA.
[138] Ashby, M. ., 2006, “The Properties of Foams and Lattices,” Philos. Trans. R. Soc. A
Math. Phys. Eng. Sci., 364(1838), pp. 15–30.
[139] Gibson, L. J., and Ashby, M. F., 1982, “The Mechanics of Three-Dimensional Cellular
Materials,” Proc. R. Soc. London, 382(1782), pp. 305–331.
[140] Wadley, H., Fleck, N. A., and Evans, A. G., 2003, “Fabrication and Structural
Performance of Periodic Cellular Metal Sandwich Structures,” Compos. Sci. Technol.,
63(16), pp. 2331–2343.
[141] Wadley, H. N. ., 2006, “Multifunctional Periodic Cellular Metals,” Philos. Trans. R. Soc.
A Math. Phys. Eng. Sci., 364(1838), pp. 31–68.
[142] Queheillalt, D. T., and Wadley, H., 2011, “Hollow Pyramidal Lattice Truss Structures,”
Int. J. Mater. Res., 102(4).
[143] Queheillalt, D. T., Murty, Y., and Wadley, H., 2008, “Mechanical Properties of an
Extruded Pyramidal Lattice Truss Sandwich Structure,” Scr. Mater., 58(1), pp. 76–79.
[144] Queheillalt, D. T., and Wadley, H., 2005, “Pyramidal Lattice Truss Structures with
Hollow Trusses,” Mater. Sci. Eng. A, 397(1–2), pp. 132–137.
[145] Queheillalt, D. T., and Wadley, H., 2005, “Cellular Metal Lattices with Hollow Trusses,”
Acta Mater., 53(2), pp. 303–313.
294
[146] Queheillalt, D. T., and Wadley, H., 2009, “Titanium Alloy Lattice Truss Structures,”
Mater. Des., 30(6), pp. 1966–1975.
[147] Wadley, H., Masta, M. R. O., Dharmasena, K. P., Compton, B. G., Gamble, E. A., and
Zok, F. W., 2013, “Effect of Core Topology on Projectile Penetration in Hybrid
Aluminum / Alumina Sandwich Structures Q,” Int. J. Impact Eng., 62, pp. 99–113.
[148] Wadley, H., Dharmasena, K. P., O’Masta, M. R., and Wetzel, J. J., 2013, “Impact
Response of Aluminum Corrugated Core Sandwich Panels,” Int. J. Impact Eng., 62, pp.
114–128.
[149] Yungwirth, C. J., Radford, D. D., Aronson, M., and Wadley, H., 2008, “Experiment
Assessment of the Ballistic Response of Composite Pyramidal Lattice Truss Structures,”
Compos. Part B Eng., 39(3), pp. 556–569.
[150] Dong, L., Deshpande, V., and Wadley, H., 2015, “Mechanical Response of Ti–6Al–4V
Octet-Truss Lattice Structures,” Int. J. Solids Struct., 60–61, pp. 107–124.
[151] Dong, L., King, W. P., Raleigh, M., and Wadley, H., 2018, “A Microfabrication Approach
for Making Metallic Mechanical Metamaterials,” Mater. Des., 160, pp. 147–168.
[152] Holloman, R. L., Deshpande, V., Hanssen, A. G., Fleming, K. M., Scully, J. R., and
Wadley, H., 2013, “Tubular Aluminum Cellular Structures: Fabrication and Mechanical
Response,” J. Mech. Mater. Struct., 8(1), pp. 65–94.
[153] Kooistra, G. W., Deshpande, V. S., and Wadley, H., 2004, “Compressive Behavior of Age
Hardenable Tetrahedral Lattice Truss Structures Made from Aluminium,” Acta Mater.,
52(14), pp. 4229–4237.
[154] Kooistra, G. W., and Wadley, H., 2007, “Lattice Truss Structures from Expanded Metal
Sheet,” Mater. Des., 28(2), pp. 507–514.
[155] Moongkhamklang, P., Deshpande, V. S., and Wadley, H., 2010, “The Compressive and
Shear Response of Titanium Matrix Composite Lattice Structures,” Acta Mater., 58(8),
pp. 2822–2835.
[156] Moongkhamklang, P., and Wadley, H., 2010, “Titanium Alloy Lattice Structures with
Millimeter Scale Cell Sizes,” Adv. Eng. Mater., 12(11), pp. 1111–1116.
[157] O’Masta, M. R., Dong, L., St-Pierre, L., Wadley, H., and Deshpande, V. S., 2017, “The
Fracture Toughness of Octet-Truss Lattices,” J. Mech. Phys. Solids, 98, pp. 271–289.
[158] Rosen, D. W., Johnston, S., and Reed, M., 2006, “Design of General Lattice Structures for
Lightweight and Compliance Applications,” Rapid Manuf. Conf.
[159] Rosen, D. W., 2007, “Computer-Aided Design for Additive Manufacturing of Cellular
Structures,” Comput. Aided. Des. Appl., 4(1–6), pp. 585–594.
[160] Helou, M., and Kara, S., 2017, “Design, Analysis and Manufacturing of Lattice
Structures: An Overview,” Int. J. Comput. Integr. Manuf., 31(3), pp. 243–261.
295
[161] Nguyen, J., Park, S., and Rosen, D. W., 2013, “Heuristic Optimization Method for
Cellular Structure Design of Light Weight Components,” Int. J. Precis. Eng. Manuf.,
14(6), pp. 1071–1078.
[162] Cheng, L., Bai, J., and To, A. C., 2019, “Functionally Graded Lattice Structure Topology
Optimization for the Design of Additive Manufactured Components with Stress
Constraints,” Comput. Methods Appl. Mech. Eng., 344, pp. 334–359.
[163] Boccaccio, A., Uva, A. E., Fiorentino, M., Mori, G., and Monno, G., 2016, “Geometry
Design Optimization of Functionally Graded Scaffolds for Bone Tissue Engineering: A
Mechanobiological Approach,” PLoS One, 11(1).
[164] An, J., Teoh, J. E. M., Suntornnond, R., and Chua, C. K., 2015, “Design and 3D Printing
of Scaffolds and Tissues,” Engineering, 1(2), pp. 261–268.
[165] Yeong, W.-Y., Chua, C.-K., Leong, K.-F., and Chandrasekaran, M., 2004, “Rapid
Prototyping in Tissue Engineering: Challenges and Potential,” Trends Biotechnol., 22(12),
pp. 643–652.
[166] Warnke, P. H., Douglas, T., Wollny, P., Sherry, E., Steiner, M., Galonska, S., Becker, S.
T., Springer, I. N., Wiltfang, J., and Sivananthan, S., 2009, “Rapid Prototyping: Porous
Titanium Alloy Scaffolds Produced by Selective Laser Melting for Bone Tissue
Engineering,” Tissue Eng. Part C Methods, 15(2), pp. 115–124.
[167] Leong, K. F., Chua, C. K., Sudarmadji, N., and Yeong, W. Y., 2008, “Engineering
Functionally Graded Tissue Engineering Scaffolds,” J. Mech. Behav. Biomed. Mater.,
1(2), pp. 140–152.
[168] Hollister, S. J., 2005, “Porous Scaffold Design for Tissue Engineering,” Nat. Mater., 4(7),
pp. 518–524.
[169] Dias, M. R., Guedes, J. M., Flanagan, C. L., Hollister, S. J., and Fernandes, P. R., 2014,
“Optimization of Scaffold Design for Bone Tissue Engineering: A Computational and
Experimental Study,” Med. Eng. Phys., 36(4), pp. 448–457.
[170] Chua, C. K., Leong, K. F., Cheah, C. M., and Chua, S. W., 2003, “Development of a
Tissue Engineering Scaffold Structure Library for Rapid Prototyping. Part 1: Investigation
and Classification,” Int. J. Adv. Manuf. Technol., 21(4), pp. 291–301.
[171] Taniguchi, N., Fujibayashi, S., Takemoto, M., Sasaki, K., Otsuki, B., Nakamura, T.,
Matsushita, T., Kokubo, T., and Matsuda, S., 2016, “Effect of Pore Size on Bone Ingrowth
into Porous Titanium Implants Fabricated by Additive Manufacturing: An in Vivo
Experiment,” Mater. Sci. Eng. C, 59, pp. 690–701.
[172] Hanks, B., Dinda, S., and Joshi, S., 2018, “Redesign of the Femoral Stem for a Total Hip
Arthroplasty for Additive Manufacturing,” ASME International Design Engineering
Technical Conferences and Computers and Information in Engineering Conference, pp.
1–13.
[173] Burton, H. E., Eisenstein, N. M., Lawless, B. M., Jamshidi, P., Segarra, M. A., Addison,
296
O., Shepherd, D. E. T., Attallah, M. M., Grover, L. M., and Cox, S. C., 2019, “The Design
of Additively Manufactured Lattices to Increase the Functionality of Medical Implants,”
Mater. Sci. Eng. C, 94(September 2018), pp. 901–908.
[174] Jonitz-Heincke, A., Wieding, J., Schulze, C., Hansmann, D., and Bader, R., 2013,
“Comparative Analysis of the Oxygen Supply and Viability of Human Osteoblasts in
Three-Dimensional Titanium Scaffolds Produced by Laser-Beam or Electron-Beam
Melting,” Materials (Basel)., 6(11), pp. 5398–5409.
[175] Chua, C. K., Leong, K. F., Cheah, C. M., and Chua, S. W., 2003, “Development of a
Tissue Engineering Scaffold Structure Library for Rapid Prototyping. Part 2: Parametric
Library and Assembly Program,” Int. J. Adv. Manuf. Technol., 21(4), pp. 302–312.
[176] Tan, X. P., Tan, Y. J., Chow, C. S. L., Tor, S. B., and Yeong, W. Y., 2017, “Metallic
Powder-Bed Based 3D Printing of Cellular Scaffolds for Orthopaedic Implants: A State-
of-the-Art Review on Manufacturing, Topological Design, Mechanical Properties and
Biocompatibility,” Mater. Sci. Eng. C, 76, pp. 1328–1343.
[177] Sing, S. L., An, J., Yeong, W. Y., and Wiria, F. E., 2016, “Laser and Electron-Beam
Powder-Bed Additive Manufacturing of Metallic Implants: A Review on Processes,
Materials and Designs,” J. Orthop. Res., 34(3), pp. 369–385.
[178] Zadpoor, A. A., and Hedayati, R., 2016, “Analytical Relationships for Prediction of the
Mechanical Properties of Additively Manufactured Porous Biomaterials,” J. Biomed.
Mater. Res. - Part A, 104(12), pp. 3164–3174.
[179] Arabnejad, S., Johnston, B., Tanzer, M., and Pasini, D., 2017, “Fully Porous 3D Printed
Titanium Femoral Stem to Reduce Stress-Shielding Following Total Hip Arthroplasty,” J.
Orthop. Res., 35(8), pp. 1774–1783.
[180] De Wild, M., Ghayor, C., Zimmermann, S., Rüegg, J., Nicholls, F., Schuler, F., Chen, T.-
H., and Weber, F. E., 2019, “Osteoconductive Lattice Microarchitecture for Optimized
Bone Regeneration,” 3D Print. Addit. Manuf., 6(1), pp. 40–49.
[181] He, Y., Burkhalter, D., Durocher, D., and Gilbert, J. M., 2018, “Solid-Lattice Hip
Prosthesis Design: Applying Topology and Lattice Optimization to Reduce Stress
Shielding From Hip Implants,” 2018 Design of Medical Devices Conference, American
Society of Mechanical Engineers.
[182] Pronk, T. N., Ayas, C., and Tekõglu, C., 2017, “A Quest for 2D Lattice Materials for
Actuation,” J. Mech. Phys. Solids, 105, pp. 199–216.
[183] Moustafa, A. R., Dinwiddie, R. B., Pawlowski, A. E., Splitter, D. A., Shyam, A., and
Cordero, Z. C., 2018, “Mesostructure and Porosity Effects on the Thermal Conductivity of
Additively Manufactured Interpenetrating Phase Composites,” Addit. Manuf.,
22(February), pp. 223–229.
[184] Takezawa, A., Zhang, X., and Kitamura, M., 2019, “Optimization of an Additively
Manufactured Functionally Graded Lattice Structure with Liquid Cooling Considering
Structural Performances,” Int. J. Heat Mass Transf., 143, p. 118564.
297
[185] Yu, J., Song, Q. F., Ma, X. L., Zhao, Y. K., Cao, J. H., and Chen, X., 2015, “Study of Heat
Transfer of Composite Lattice Structure for Active Cooling Used in the Scramjet
Combustor,” Mater. Res. Innov., 19, pp. S5843–S5849.
[186] Pawlowski, A. E., Cordero, Z. C., French, M. R., Muth, T. R., Keith Carver, J., Dinwiddie,
R. B., Elliott, A. M., Shyam, A., and Splitter, D. A., 2017, “Damage-Tolerant Metallic
Composites via Melt Infiltration of Additively Manufactured Preforms,” Mater. Des.,
127(March), pp. 346–351.
[187] Brennan-Craddock, J., Brackett, D., Wildman, R., and Hague, R., 2012, “The Design of
Impact Absorbing Structures for Additive Manufacture,” J. Phys. Conf. Ser., 382(1), pp.
2–9.
[188] Ozdemir, Z., Hernandez-Nava, E., Tyas, A., Warren, J. A., Fay, S. D., Goodall, R., Todd,
I., and Askes, H., 2016, “Energy Absorption in Lattice Structures in Dynamics:
Experiments,” Int. J. Impact Eng., 89, pp. 49–61.
[189] Ozdemir, Z., Tyas, A., Goodall, R., and Askes, H., 2017, “Energy Absorption in Lattice
Structures in Dynamics: Nonlinear FE Simulations,” Int. J. Impact Eng., 102.
[190] Hammetter, C. I., Rinaldi, R. G., and Zok, F. W., 2013, “Pyramidal Lattice Structures for
High Strength and Energy Absorption,” J. Appl. Mech., 80(4), p. 041015.
[191] Tancogne-Dejean, T., Spierings, A. B., and Mohr, D., 2016, “Additively-Manufactured
Metallic Micro-Lattice Materials for High Specific Energy Absorption under Static and
Dynamic Loading,” Acta Mater., 116, pp. 14–28.
[192] Helou, M., Vongbunyong, S., and Kara, S., 2016, “Finite Element Analysis and Validation
of Cellular Structures,” Procedia CIRP, 50, pp. 94–99.
[193] Khurana, J., Hanks, B., and Frecker, M., 2018, “Design for Additive Manufacturing of
Cellular Compliant Mechanism Using Thermal History Feedback,” ASME International
Design Engineering Technical Conferences and Comp, pp. 1–15.
[194] Zhang, X. Z., Leary, M., Tang, H. P., Song, T., and Qian, M., 2018, “Selective Electron
Beam Manufactured Ti-6Al-4V Lattice Structures for Orthopedic Implant Applications:
Current Status and Outstanding Challenges,” Curr. Opin. Solid State Mater. Sci., 22(3),
pp. 75–99.
[195] Tao, W., and Leu, M. C., 2016, “Design of Lattice Structure for Additive Manufacturing,”
2016 International Symposium on Flexible Automation (ISFA), IEEE, pp. 325–332.
[196] Wettergreen, M. A., Bucklen, B. S., Starly, B., Yuksel, E., Sun, W., and Liebschner, M. A.
K., 2005, “Creation of a Unit Block Library of Architectures for Use in Assembled
Scaffold Engineering,” CAD Comput. Aided Des., 37(11), pp. 1141–1149.
[197] Satterfield, Z., Kulkarni, N., Fadel, G., Li, G., Coutris, N., and Castanier, M. P., 2017,
“Unit Cell Synthesis for Design of Materials With Targeted Nonlinear Deformation
Response,” J. Mech. Des., 139(12), p. 121401.
298
[198] Fazelpour, M., Summers, J. D., and Bluoin, V. Y., 2016, “A Taxonomy For Representing
Prismatic Cellular Materials,” ASME 2016 Int. Des. Eng. Tech. Conf. Comput. Inf. Eng.
Conf., 2, p. V02BT03A001.
[199] Zok, F. W., Latture, R. M., and Begley, M. R., 2016, “Periodic Truss Structures,” J. Mech.
Phys. Solids, 96, pp. 184–203.
[200] Bhate, D., Penick, C., Ferry, L., and Lee, C., 2018, “Classification and Selection of
Cellular Materials in Mechanical Design : Engineering and Biomimetic Approaches,” pp.
1–27.
[201] Deshpande, V. S., Fleck, N. A., and Ashby, M. F., 2001, “Effective Properties of the
Octet-Truss Lattice Material,” J. Mech. Phys. Solids, 49(8), pp. 1747–1769.
[202] Hedayati, R., Sadighi, M., Mohammadi-Aghdam, M., and Zadpoor, A. A., 2016, “Effect
of Mass Multiple Counting on the Elastic Properties of Open-Cell Regular Porous
Biomaterials,” Mater. Des., 89, pp. 9–20.
[203] Hedayati, R., Sadighi, M., Mohammadi-Aghdam, M., and Zadpoor, A. A., 2016,
“Mechanical Properties of Regular Porous Biomaterials Made from Truncated Cube
Repeating Unit Cells: Analytical Solutions and Computational Models,” Mater. Sci. Eng.
C, 60, pp. 163–183.
[204] Hedayati, R., Sadighi, M., Mohammadi-Aghdam, M., and Zadpoor, A. A., 2016,
“Mechanics of Additively Manufactured Porous Biomaterials Based on the
Rhombicuboctahedron Unit Cell,” J. Mech. Behav. Biomed. Mater., 53, pp. 272–294.
[205] Hedayati, R., Sadighi, M., Mohammadi-Aghdam, M., and Zadpoor, A. A., 2017,
“Analytical Relationships for the Mechanical Properties of Additively Manufactured
Porous Biomaterials Based on Octahedral Unit Cells,” Appl. Math. Model., 46, pp. 408–
422.
[206] Hedayati, R., Sadighi, M., Mohammadi-Aghdam, M., and Zadpoor, A. A., 2016,
“Mechanical Behavior of Additively Manufactured Porous Biomaterials Made from
Truncated Cuboctahedron Unit Cells,” Int. J. Mech. Sci., 106, pp. 19–38.
[207] Lei, H., Li, C., Meng, J., Zhou, H., Liu, Y., Zhang, X., Wang, P., and Fang, D., 2019,
“Evaluation of Compressive Properties of SLM-Fabricated Multi-Layer Lattice Structures
by Experimental Test and μ-CT-Based Finite Element Analysis,” Mater. Des., 169, p.
107685.
[208] Han, C., Yan, C., Wen, S., Xu, T., Li, S., Liu, J., Wei, Q., and Shi, Y., 2017, “Effects of
the Unit Cell Topology on the Compression Properties of Porous Co-Cr Scaffolds
Fabricated via Selective Laser Melting,” Rapid Prototyp. J., 23(1), pp. 16–27.
[209] Ahmadi, S. M., Campoli, G., Amin Yavari, S., Sajadi, B., Wauthle, R., Schrooten, J.,
Weinans, H., and Zadpoor, A. A., 2014, “Mechanical Behavior of Regular Open-Cell
Porous Biomaterials Made of Diamond Lattice Unit Cells,” J. Mech. Behav. Biomed.
Mater., 34, pp. 106–115.
299
[210] Tang, T. L. E., Liu, Y., Lu, D., Arisoy, E. B., and Musuvathy, S., 2017, “Lattice Structure
Design Advisor for Additive Manufacturing Using Gaussian Process,” ASME
International Design Engineering Technical Conferences and Computers and Information
in Engineering Conference, pp. 1–10.
[211] Ahmadi, S. M., Yavari, S. A., Wauthle, R., Pouran, B., Schrooten, J., Weinans, H., and
Zadpoor, A. A., 2015, “Additively Manufactured Open-Cell Porous Biomaterials Made
from Six Different Space-Filling Unit Cells: The Mechanical and Morphological
Properties,” Materials (Basel)., 8(4), pp. 1871–1896.
[212] Cheng, X. Y., Li, S. J., Murr, L. E., Zhang, Z. B., Hao, Y. L., Yang, R., Medina, F., and
Wicker, R. B., 2012, “Compression Deformation Behavior of Ti-6Al-4V Alloy with
Cellular Structures Fabricated by Electron Beam Melting,” J. Mech. Behav. Biomed.
Mater., 16(1), pp. 153–162.
[213] Aremu, A. O., Maskery, I., Tuck, C., Ashcroft, I. A., Wildman, R. D., and Hague, R. I. .,
2014, “A Comparative Finite Element Study of Cubic Unit Cells for Selective Laser
Melting,” Int. Solid Free. Fabr. Symp., pp. 1238–1249.
[214] Smith, M., Guan, Z., and Cantwell, W. J., 2013, “Finite Element Modelling of the
Compressive Response of Lattice Structures Manufactured Using the Selective Laser
Melting Technique,” Int. J. Mech. Sci., 67, pp. 28–41.
[215] Ushijima, K., Cantwell, W. J., and Chen, D. H., 2013, “Prediction of the Mechanical
Properties of Micro-Lattice Structures Subjected to Multi-Axial Loading,” Int. J. Mech.
Sci., 68, pp. 47–55.
[216] Ushijima, K., Cantwell, W. J., Mines, R. A. W., Tsopanos, S., and Smith, M., 2011, “An
Investigation into the Compressive Properties of Stainless Steel Micro-Lattice Structures,”
J. Sandw. Struct. Mater., 13(3), pp. 303–329.
[217] Brodin, H., and Saarimäki, J., 2013, “Mechanical Properties of Lattice Truss Structures
Made of a Selective Laser Melted Superalloy,” 13th International Conference on Fracture
(ICF13), pp. 1–10.
[218] Yan, C., Hao, L., Hussein, A., Bubb, S. L., Young, P., and Raymont, D., 2014,
“Evaluation of Light-Weight AlSi10Mg Periodic Cellular Lattice Structures Fabricated via
Direct Metal Laser Sintering,” J. Mater. Process. Technol., 214(4), pp. 856–864.
[219] Park, S. I., Rosen, D. W., Choi, S. kyum, and Duty, C. E., 2014, “Effective Mechanical
Properties of Lattice Material Fabricated by Material Extrusion Additive Manufacturing,”
Addit. Manuf., 1, pp. 12–23.
[220] Hussein, A. Y., 2013, “The Development of Lightweight Cellular Structures for Metal
Additive Manufacturing,” University of Exeter.
[221] Yan, C., Hao, L., Hussein, A., Young, P., Huang, J., and Zhu, W., 2015, “Microstructure
and Mechanical Properties of Aluminium Alloy Cellular Lattice Structures Manufactured
by Direct Metal Laser Sintering,” Mater. Sci. Eng. A, 628, pp. 238–246.
300
[222] Yan, C., Hao, L., Hussein, A., and Raymont, D., 2012, “Evaluations of Cellular Lattice
Structures Manufactured Using Selective Laser Melting,” Int. J. Mach. Tools Manuf., 62,
pp. 32–38.
[223] Yan, C., Hao, L., Hussein, A., Young, P., and Raymont, D., 2014, “Advanced Lightweight
316L Stainless Steel Cellular Lattice Structures Fabricated via Selective Laser Melting,”
Mater. Des., 55, pp. 533–541.
[224] Nguyen, J., Park, S., Rosen, D. W., Folgar, L., and Williams, J., 2012, “Conformal Lattice
Structure Design and Fabrication,” Solid Freeform Fabrication Symposium, pp. 138–161.
[225] Wang, H., Chen, Y., and Rosen, D. W., 2005, “A Hybrid Geometric Modeling Method for
Large Scale Conformal Cellular Structures,” 25th Comput. Inf. Eng. Conf. Parts A B, 3,
pp. 421–427.
[226] Wang, H. V., and Rosen, D. W., 2002, “Computer-Aided Design Methods for the
Additive Fabrication of Truss Structures,” Int. Conf. Manuf. Autom., (August), pp. 181–
198.
[227] Engelbrecht, S., Luis Folgar, Rosen, D. W., Schulberger, G., and Williams, J., 2009,
“Cellular Structures for Optimal Performance,” Proc. SFF …, pp. 831–842.
[228] Bhate, D., 2019, “Four Questions in Cellular Material Design,” Materials (Basel)., 12(7).
[229] Brackett, D., Ashcroft, I., and Hague, R., 2011, “A Dithering Based Method to Generate
Variable Volume Lattice Cells for Additive Manufacturing,” 22nd Annual International
Solid Freeform Fabrication Symposium, pp. 671–679.
[230] Arabnejad Khanoki, S., and Pasini, D., 2012, “Multiscale Design and Multiobjective
Optimization of Orthopedic Hip Implants with Functionally Graded Cellular Material,” J.
Biomech. Eng., 134(3), p. 031004.
[231] Arabnejad Khanoki, S., and Pasini, D., 2013, “The Fatigue Design of a Bone Preserving
Hip Implant With Functionally Graded Cellular Material,” J. Med. Device., 7(2), pp. 1–2.
[232] Arabnejad Khanoki, S., and Pasini, D., 2013, “Fatigue Design of a Mechanically
Biocompatible Lattice for a Proof-of-Concept Femoral Stem,” J. Mech. Behav. Biomed.
Mater., 22, pp. 65–83.
[233] Graf, G. C., Chu, J., Engelbrecht, S., and Rosen, D. W., 2009, “Synthesis Methods for
Lightweight Lattice Structures,” ASME International Design Engineering Technical
Conferences and Computers and Information in Engineering Conference, pp. 1–11.
[234] Chang, P. S., and Rosen, D. W., 2013, “The Size Matching and Scaling Method: A
Synthesis Method for the Design of Mesoscale Cellular Structures,” Int. J. Comput. Integr.
Manuf., 26(10), pp. 907–927.
[235] nTopology, “NTop Platform” [Online]. Available: https://ntopology.com/.
[236] Paramatters Inc, “Paramatters” [Online]. Available: https://paramatters.com/.
301
[237] 3D Systems, “3DXpert” [Online]. Available:
https://www.3dsystems.com/software/3dxpert.
[238] ANSYS Inc, 2020, “Discovery SpaceClaim” [Online]. Available:
https://www.ansys.com/products/3d-design/ansys-spaceclaim.
[239] Naing, M. W., Chua, C. K., Leong, K. F., and Wang, Y., 2005, “Fabrication of
Customised Scaffolds Using Computer‐aided Design and Rapid Prototyping Techniques,”
Rapid Prototyp. J., 11(4), pp. 249–259.
[240] Cheah, C., Chua, C., Leong, K., Cheong, C.-H., and Naing, M.-W., 2004, “Automatic
Algorithm for Generating Complex Polyhedral Scaffold Structures for Tissue
Engineering,” Tissue Eng., 10(3–4), pp. 595–610.
[241] Sudarmadji, N., Tan, J. Y., Leong, K. F., Chua, C. K., and Loh, Y. T., 2011,
“Investigation of the Mechanical Properties and Porosity Relationships in Selective Laser-
Sintered Polyhedral for Functionally Graded Scaffolds,” Acta Biomater., 7(2), pp. 530–
537.
[242] Vongbunyong, S., and Kara, S., 2016, “Development of Software Tool for Cellular
Structure Integration for Additive Manufacturing,” Procedia CIRP, 48, pp. 489–494.
[243] Vongbunyong, S., and Kara, S., 2017, “Rapid Generation of Uniform Cellular Structure
by Using Prefabricated Unit Cells,” Int. J. Comput. Integr. Manuf., 30(8), pp. 792–804.
[244] Nazir, A., Abate, K. M., Kumar, A., and Jeng, J. Y., 2019, “A State-of-the-Art Review on
Types, Design, Optimization, and Additive Manufacturing of Cellular Structures,” Int. J.
Adv. Manuf. Technol., 104(9–12), pp. 3489–3510.
[245] Koizumi, Y., Okazaki, A., Chiba, A., Kato, T., and Takezawa, A., 2016, “Cellular Lattices
of Biomedical Co-Cr-Mo-Alloy Fabricated by Electron Beam Melting with the Aid of
Shape Optimization,” Addit. Manuf., 12, pp. 305–313.
[246] Savio, G., Meneghello, R., and Concheri, G., 2018, “Geometric Modeling of Lattice
Structures for Additive Manufacturing,” Rapid Prototyp. J., 24(2), pp. 351–360.
[247] McMillan, M. L., Jurg, M., Leary, M., and Brandt, M., 2017, “Programmatic Generation
of Computationally Efficient Lattice Structures for Additive Manufacture,” Rapid
Prototyp. J., 23(3), pp. 486–494.
[248] Gupta, A., Kurzeja, K., Rossignac, J., Allen, G., Kumar, P. S., and Musuvathy, S., 2019,
“Programmed-Lattice Editor and Accelerated Processing of Parametric Program-
Representations of Steady Lattices,” CAD Comput. Aided Des., 113, pp. 35–47.
[249] Goel, A., and Anand, S., 2019, “Design of Functionally Graded Lattice Structures Using
B-Splines for Additive Manufacturing,” Procedia Manuf., 34, pp. 655–665.
[250] Maconachie, T., Leary, M., Lozanovski, B., Zhang, X., Qian, M., Faruque, O., and Brandt,
M., 2019, “SLM Lattice Structures: Properties, Performance, Applications and
Challenges,” Mater. Des., p. 108137.
302
[251] Hanzl, P., Zetkova, I., and Dana, M., 2017, “Issues of Lattice Structures Production via
Metal Additive Manufacturing,” Manuf. Technol., 17(6).
[252] Tang, Y., Dong, G., Zhou, Q., and Zhao, Y. F., 2018, “Lattice Structure Design and
Optimization with Additive Manufacturing Constraints,” IEEE Trans. Autom. Sci. Eng.,
15(4), pp. 1546–1562.
[253] Behera, S., Sahoo, S., and Pati, B. B., 2015, “A Review on Optimization Algorithms and
Application to Wind Energy Integration to Grid,” Renew. Sustain. Energy Rev., 48, pp.
214–227.
[254] Venter, G., 2010, “Review of Optimization Techniques,” Encycl. Aerosp. Eng., pp. 1–12.
[255] Marthaler, D. E., 2013, “An Overview of Mathematical Methods for Numerical
Optimization,” Numerical Methods for Metamaterial Design, pp. 31–54.
[256] Vikhar, P. A., 2016, “Evolutionary Algorithms : A Critical Review and Its Future
Prospects,” pp. 261–265.
[257] Deb, K., Pratap, A., Agarwal, S., and Meyarivan, T., 2002, “A Fast and Elitist
Multiobjective Genetic Algorithm: NSGA-II,” IEEE Trans. Evol. Comput., 6(2), pp. 182–
197.
[258] Hadka, D., and Reed, P., 2013, “Borg: An Auto-Adaptive Many-Objective Evolutionary
Computing Framework,” Evol. Comput., 21(2), pp. 231–259.
[259] Kim, S., 2006, “Gradient-Based Simulation Optimization,” IEEE Winter Simulation
Conference, pp. 159–167.
[260] Ruder, S., 2016, “An Overview of Gradient Descent Optimization Algorithms,” CoRR2,
arXiv:1609.
[261] Golfetto, W. A., and Fernandes, S. da S., 2012, “A Review of Gradient Algorithms for
Numerical Computation of Optimal Trajectories,” J. Aerosp. Technol. Manag., 4(2), pp.
131–143.
[262] Salomon, R., 1998, “Evolutionary Algorithms and Gradient Search: Similarities and
Differences,” IEEE Trans. Evol. Comput., 2(2), pp. 45–55.
[263] Zingg, D. W., Nemec, M., and Pulliam, T. H., 2008, “A Comparative Evaluation of
Genetic and Gradient-Based Algorithms Applied to Aerodynamic Optimization,” Rev.
Eur. mécanique numérique, 17, pp. 103–126.
[264] Haupt, R., 1995, “Comparison between Genetic and Gradient-Based Optimization
Algorithms for Solving Electromagnetics Problems,” IEEE Trans. Magn., 31(3), pp.
1932–1935.
[265] Ahmad, F., Isa, N. A. M., Osman, M. K., and Hussain, Z., 2010, “Performance
Comparison of Gradient Descent and Genetic Algorithm Based Artificial Neural
Networks Training,” IEEE Intelligent Systems Design and Applications, ISDA, Cairo, pp.
303
604–609.
[266] Song, Y., and Xi, P., 2009, “Genetic Algorithm and Gradient-Based Algorithm
Optimization of Vehicle Turning Mechanism,” IEEE International Conference on
Computer Science and Information Technology, ICCSIT, pp. 81–84.
[267] Montastruc, L., Azzaro-Pantel, C., Pibouleau, L., and Domenech, S., 2004, “Use of
Genetic Algorithms and Gradient Based Optimization Techniques for Calcium Phosphate
Precipitation,” Chem. Eng. Process. Process Intensif., 43(10), pp. 1289–1298.
[268] Bendsøe, M. P., and Kikuchi, N., 1988, “Generating Optimal Topologies in Structural
Design Using a Homogenization Method,” Comput. Methods Appl. Mech. Eng., 71(2),
pp. 197–224.
[269] Bendsøe, M. P., 1989, “Optimal Shape Design as a Material Distribution Problem,” Struct.
Optim., 1(4), pp. 193–202.
[270] Bendsøe, M. P., and Sigmund, O., 1999, “Material Interpolation Schemes in Topology
Optimization,” Arch. Appl. Mech., 69(9–10), pp. 635–654.
[271] Frecker, M. I., Ananthasuresh, G. K., Njshiwaki, S., Kikuchi, N., and Kota, S., 1997,
“Topological Synthesis of Compliant Mechanisms Using Multi-Criteria Optimization,” J.
Mech. Des., 119(June).
[272] Yin, L., and Ananthasuresh, G. K., 2001, “Topology Optimization of Compliant
Mechanisms with Multiple Materials Using a Peak Function Material Interpolation
Scheme,” Struct. Multidiscip. Optim., 23(1), pp. 49–62.
[273] Guest, J. K., Asadpoure, A., and Ha, S. H., 2011, “Eliminating Beta-Continuation from
Heaviside Projection and Density Filter Algorithms,” Struct. Multidiscip. Optim., 44(4),
pp. 443–453.
[274] Mankame, N. D., and Ananthasuresh, G. K., 2004, “Topology Optimization for Synthesis
of Contact-Aided Compliant Mechanisms Using Regularized Contact Modeling,”
Comput. Struct., 82(15–16), pp. 1267–1290.
[275] De Leon, D. M., Alexandersen, J., Jun, J. S., and Sigmund, O., 2015, “Stress-Constrained
Topology Optimization for Compliant Mechanism Design,” Struct. Multidiscip. Optim.,
52(5), pp. 929–943.
[276] Frecker, M., Kikuchi, N., and Kota, S., 1999, “Topology Optimization of Compliant
Mechanisms with Multiple Outputs,” Struct. Optim., 17(4), pp. 269–278.
[277] Santos Iwamura, R., and Rocha De Faria, A., 2013, “Topology Optimization of Multiple
Load Case Structures,” International Symposium on Solid Mechanics, Porto Alegre, pp. 1–
11.
[278] Lohan, D. J., Dede, E. M., and Allison, J. T., 2017, “Topology Optimization for Heat
Conduction Using Generative Design Algorithms,” Struct. Multidiscip. Optim., 55(3), pp.
1063–1077.
304
[279] Wang, N., and Zhang, X., 2014, “Multi-Material Topology Optimization of Compliant
Mechanism Using Ground Structure Approach,” 2014 Int. Conf. Manip. Manuf. Meas.
Nanoscale, (October), pp. 249–254.
[280] Liu, J., and Ma, Y., 2016, “A Survey of Manufacturing Oriented Topology Optimization
Methods,” Adv. Eng. Softw., 100, pp. 161–175.
[281] Bendsøe, M. P., and Sigmund, O., 2004, Topology Optimization, Springer Berlin
Heidelberg, Berlin, Heidelberg.
[282] Sigmund, O., and Maute, K., 2013, “Topology Optimization Approaches: A Comparative
Review,” Struct. Multidiscip. Optim., 48(6), pp. 1031–1055.
[283] Reddy K., S. N., Ferguson, I., Frecker, M., and Simpson, T. W., 2016, “Topology
Optimization Software for Additive Manufacturing: A Review of Current Capabilities and
a Real-World Example,” Proc. ASME 2016 Int. Des. Eng. Tech. Conf. Comput. Inf. Eng.
Conf.
[284] Ananthasuresh, G. K., and Frecker, M. I., 2001, “Optimal Synthesis with Continuum
Models,” Compliant Mechanisms, L.L. Howell, ed., John Wiley & Sons.
[285] Huang, X., Li, Y., Zhou, S. W., and Xie, Y. M., 2014, “Topology Optimization of
Compliant Mechanisms with Desired Structural Stiffness,” Eng. Struct., 79(May 2018),
pp. 13–21.
[286] Dorn, W. S., Gomory, R. E., and Greenberg, H. G., 1964, “Automatic Design of Optimal
Structures,” J. Mec., 3(1).
[287] Ohsaki, M., 2016, Optimization of Finite Dimensional Structures, CRC Press.
[288] Zhang, X. S., Paulino, G. H., and Ramos, A. S., 2018, “Multi-Material Topology
Optimization with Multiple Volume Constraints: A General Approach Applied to Ground
Structures with Material Nonlinearity,” Struct. Multidiscip. Optim., 57(1), pp. 161–182.
[289] Liu, K., Paulino, G. H., and Gardoni, P., 2016, “Reliability-Based Topology Optimization
Using a New Method for Sensitivity Approximation - Application to Ground Structures,”
Struct. Multidiscip. Optim., 54(3), pp. 553–571.
[290] Zegard, T., and Paulino, G. H., 2014, “GRAND — Ground Structure Based Topology
Optimization for Arbitrary 2D Domains Using MATLAB,” Struct. Multidiscip. Optim.,
50(5), pp. 861–882.
[291] Zegard, T., and Paulino, G. H., 2015, “GRAND3 — Ground Structure Based Topology
Optimization for Arbitrary 3D Domains Using MATLAB,” Struct. Multidiscip. Optim.,
52(6), pp. 1161–1184.
[292] Hagishita, T., and Ohsaki, M., 2009, “Topology Optimization of Trusses by Growing
Ground Structure Method,” Struct. Multidiscip. Optim., 37(4), pp. 377–393.
[293] Cui, H., An, H., and Huang, H., 2018, “Truss Topology Optimization Considering Local
305
Buckling Constraints and Restrictions on Intersection and Overlap of Bar Members,”
Struct. Multidiscip. Optim., pp. 1–20.
[294] Asadpoure, A., Guest, J. K., and Valdevit, L., 2015, “Incorporating Fabrication Cost into
Topology Optimization of Discrete Structures and Lattices,” Struct. Multidiscip. Optim.,
51(2), pp. 385–396.
[295] Liu, J., Gaynor, A. T., Chen, S., Kang, Z., Suresh, K., Takezawa, A., Li, L., Kato, J.,
Tang, J., Wang, C. C. L., Cheng, L., Liang, X., and To, A. C., 2018, “Current and Future
Trends in Topology Optimization for Additive Manufacturing,” Struct. Multidiscip.
Optim., pp. 2457–2483.
[296] Zegard, T., and Paulino, G. H., 2016, “Bridging Topology Optimization and Additive
Manufacturing,” Struct. Multidiscip. Optim., 53, pp. 175–192.
[297] Sigmund, O., 1995, “Tailoring Materials with Prescribed Elastic Properties,” Mech.
Mater., 20(4), pp. 351–368.
[298] Sigmund, O., 1994, “Materials With Prescribed Constitutive Parameters: An Inverse
Homogenization Problem,” Int. J. Solids Struct., 31(17), pp. 2313–2329.
[299] Guth, D. C., Luersen, M. A., and Muñoz-Rojas, P. A., 2012, “Optimization of Periodic
Truss Materials Including Constitutive Symmetry Constraints,” Materwiss. Werksttech.,
43(5), pp. 447–456.
[300] Guth, D. C., Luersen, M. A., and Muñoz-Rojas, P. A., 2015, “Optimization of Three-
Dimensional Truss-like Periodic Materials Considering Isotropy Constraints,” Struct.
Multidiscip. Optim., 52(5), pp. 889–901.
[301] Messner, M. C., 2016, “Optimal Lattice-Structured Materials,” J. Mech. Phys. Solids, 96,
pp. 162–183.
[302] Neves, M. M., Sigmund, O., and Bendsøe, M. P., 2002, “Topology Optimization of
Periodic Microstructures with a Penalization of Highly Localized Buckling Modes,” Int. J.
Numer. Methods Eng., 54(6), pp. 809–834.
[303] Liu, L., Yan, J., and Cheng, G., 2008, “Optimum Structure with Homogeneous Optimum
Truss-like Material,” Comput. Struct., 86(13–14), pp. 1417–1425.
[304] Huang, X., Zhou, S. W., Xie, Y. M., and Li, Q., 2013, “Topology Optimization of
Microstructures of Cellular Materials and Composites for Macrostructures,” Comput.
Mater. Sci., 67, pp. 397–407.
[305] De Pasquale, G., Montemurro, M., Catapano, A., Bertolino, G., and Revelli, L., 2018,
“Cellular Structures from Additive Processes: Design, Homogenization and Experimental
Validation,” Procedia Struct. Integr., 8, pp. 75–82.
[306] Cheng, G. D., Cai, Y. W., and Xu, L., 2013, “Novel Implementation of Homogenization
Method to Predict Effective Properties of Periodic Materials,” Acta Mech. Sin. Xuebao,
29(4), pp. 550–556.
306
[307] Park, S., and Rosen, D. W., 2016, “Homogenization of Mechanical Properties for
Additively,” ASME International Design Engineering Technical Conferences, pp. 1–10.
[308] Arabnejad, S., and Pasini, D., 2013, “Mechanical Properties of Lattice Materials via
Asymptotic Homogenization and Comparison with Alternative Homogenization
Methods,” Int. J. Mech. Sci., 77, pp. 249–262.
[309] Vigliotti, A., and Pasini, D., 2012, “Stiffness and Strength of Tridimensional Periodic
Lattices,” Comput. Methods Appl. Mech. Eng., 229–232, pp. 27–43.
[310] Xia, L., and Breitkopf, P., 2015, “Design of Materials Using Topology Optimization and
Energy-Based Homogenization Approach in Matlab,” Struct. Multidiscip. Optim., 52(6),
pp. 1229–1241.
[311] Neves, M. M., Rodrigues, H., and Guedes, J. M., 2000, “Optimal Design of Periodic
Linear Elastic Microstructures,” Comput. Struct., 76.
[312] Seepersad, C. C., Allen, J. K., McDowell, D. L., and Mistree, F., 2006, “Robust Design of
Cellular Materials With Topological and Dimensional Imperfections,” J. Mech. Des.,
128(6), p. 1285.
[313] Watts, S., Arrighi, W., Kudo, J., Tortorelli, D. A., and White, D. A., 2019, “Simple,
Accurate Surrogate Models of the Elastic Response of Three-Dimensional Open Truss
Micro-Architectures with Applications to Multiscale Topology Design,” Struct.
Multidiscip. Optim., 3, pp. 1887–1920.
[314] Watts, S., and Tortorelli, D. A., 2017, “A Geometric Projection Method for Designing
Three-Dimensional Open Lattices with Inverse Homogenization,” Int. J. Numer. Methods
Eng., 112(11), pp. 1564–1588.
[315] Norato, J. A., Bell, B. K., and Tortorelli, D. A., 2015, “A Geometry Projection Method for
Continuum-Based Topology Optimization with Discrete Elements,” Comput. Methods
Appl. Mech. Eng., 293, pp. 306–327.
[316] Sharpe, C., Seepersad, C. C., Watts, S., and Tortorelli, D., 2018, “Design of Mechanical
Metamaterials via Constrained Bayesian Optimization,” Volume 2A: 44th Design
Automation Conference, American Society of Mechanical Engineers, pp. 1–11.
[317] Zhang, S., Norato, J. A., Gain, A. L., and Lyu, N., 2016, “A Geometry Projection Method
for the Topology Optimization of Plate Structures,” Struct. Multidiscip. Optim., 54(5), pp.
1173–1190.
[318] Norato, J. A., 2018, “Topology Optimization with Supershapes,” Struct. Multidiscip.
Optim., 58(2), pp. 415–434.
[319] Wang, Y., Zhang, L., Daynes, S., Zhang, H., Feih, S., and Wang, M. Y., 2018, “Design of
Graded Lattice Structure with Optimized Mesostructures for Additive Manufacturing,”
Mater. Des., 142, pp. 114–123.
[320] Cheng, L., Zhang, P., Biyikli, E., Bai, J., Robbins, J., and To, A., 2017, “Efficient Design
307
Optimization of Variable-Density Cellular Structures for Additive Manufacturing: Theory
and Experimental Validation,” Rapid Prototyp. J., 23(4), pp. 660–677.
[321] Han, Y., and Lu, W. F., 2018, “Optimization Design of Nonuniform Cellular Structures
for Additive Manufacturing,” Vol. 1 Addit. Manuf. Bio Sustain. Manuf., p.
V001T01A033.
[322] Panesar, A., Abdi, M., Hickman, D., and Ashcroft, I., 2018, “Strategies for Functionally
Graded Lattice Structures Derived Using Topology Optimisation for Additive
Manufacturing,” Addit. Manuf., 19, pp. 81–94.
[323] Kang, D., Park, S., Son, Y., Yeon, S., Kim, S. H., and Kim, I., 2019, “Multi-Lattice Inner
Structures for High-Strength and Light-Weight in Metal Selective Laser Melting Process,”
Mater. Des., 175, p. 107786.
[324] Steuben, J. C., Iliopoulos, A. P., and Michopoulos, J. G., 2018, “Multiscale Topology
Optimization for Additively Manufactured Objects,” J. Comput. Inf. Sci. Eng., 18(3), pp.
1–9.
[325] Sabiston, G., Ryan, L., and Kim, I. Y., 2018, “Minimization of Cost and Print Time of
Additive Manufacturing Via Topology,” ASME International Design Engineering
Technical Conferences & Computers and Information in Engineering, pp. 1–10.
[326] Langelaar, M., 2017, “An Additive Manufacturing Filter for Topology Optimization of
Print-Ready Designs,” Struct. Multidiscip. Optim., 55(3), pp. 871–883.
[327] Langelaar, M., 2016, “Topology Optimization of 3D Self-Supporting Structures for
Additive Manufacturing,” Addit. Manuf., 12, pp. 60–70.
[328] Langelaar, M., 2016, “Topology Optimization for Additive Manufacturing With
Controllable Support Structures,” ECCOMAS Congress.
[329] Brackett, D., Ashcroft, I., and Hague, R., 2011, “Topology Optimization for Additive
Manufacturing,” Solid Freeform Fabrication Symposium, pp. 348–362.
[330] Gaynor, A. T., and Guest, J. K., 2016, “Topology Optimization Considering Overhang
Constraints: Eliminating Sacrificial Support Material in Additive Manufacturing through
Design,” Struct. Multidiscip. Optim., 54(5), pp. 1157–1172.
[331] Liu, S., Li, Q., Chen, W., Tong, L., and Cheng, G., 2015, “An Identification Method for
Enclosed Voids Restriction in Manufacturability Design for Additive Manufacturing
Structures,” Front. Mech. Eng., 10(2), pp. 126–137.
[332] Schwerdtfeger, J., Wein, F., Leugering, G., Singer, R. F., Körner, C., Stingl, M., and
Schury, F., 2011, “Design of Auxetic Structures via Mathematical Optimization,” Adv.
Mater., 23(22–23), pp. 2650–2654.
[333] Lin, C.-Y., Wirtz, T., LaMarca, F., and Hollister, S. J., 2015, “Structural and Mechanical
Evaluations of a Topology Optimized Titanium Interbody Fusion Cage Fabricated by
Selective Laser Melting Process,” Clin. Exp. Rheumatol., 33, pp. 97–103.
308
[334] Tamburrino, F., Graziosi, S., and Bordegoni, M., 2018, “The Design Process of Additively
Manufactured Mesoscale Lattice Structures: A Review,” J. Comput. Inf. Sci. Eng., 18(4),
pp. 1–16.
[335] Meng, L., Zhang, W., Quan, D., Shi, G., Tang, L., Hou, Y., Breitkopf, P., Zhu, J., and
Gao, T., 2019, “From Topology Optimization Design to Additive Manufacturing: Today’s
Success and Tomorrow’s Roadmap,” Arch. Comput. Methods Eng., (0123456789).
[336] Plocher, J., and Panesar, A., 2019, “Review on Design and Structural Optimisation in
Additive Manufacturing: Towards next-Generation Lightweight Structures,” Mater. Des.,
183, p. 108164.
[337] Bacciaglia, A., Ceruti, A., and Liverani, A., 2020, “Additive Manufacturing Challenges
and Future Developments in the Next Ten Years,” Design Tools and Methods in Industrial
Engineering, Springer International Publishing, pp. 891–902.
[338] Hanks, B., Frecker, M., and Moyer, M., 2016, “Design of a Compliant Endoscopic
Ultrasound-Guided Radiofrequency Ablation Probe,” ASME International Design
Engineering Technical Conferences and Computers and Information in Engineering
Conference, pp. 1–9.
[339] Hanks, B., Frecker, M., and Moyer, M., 2018, “Optimization of an Endoscopic
Radiofrequency Ablation Electrode,” ASME J. Med. Devices, 12(3), pp. 1–11.
[340] Misra, S., Reed, K. B., Schafer, B. W., Ramesh, K. T., and Okamura, A. M., 2010,
“Mechanics of Flexible Needles Robotically Steered through Soft Tissue,” Int. J. Rob.
Res., 29(13), pp. 1640–1660.
[341] Webster III, R. J., Memisevic, J., and Okamura, a M., 2005, “Design Considerations for
Robotic Needle Steering,” Proc. IEEE Int. Conf. Robot. Autom., (April), pp. 3599–3605.
[342] Webster III, R. J., Cowan, N. J., Chirikjian, G. S., and Okamura, a M., 2006,
“Nonholonomic Modelling of Needle Steering,” Int. J. Rob. Res., 25(5/6), pp. 509–526.
[343] COMSOL Inc, “COMSOL Multiphysics” [Online]. Available: https://www.comsol.com/.
[344] MathWorks, “MATLAB” [Online]. Available:
https://www.mathworks.com/products/matlab.html.
[345] Hanks, B., Frecker, M., and Moyer, M., 2017, “Optimization of a Compliant Endoscopic
Radiofrequency Ablation Electrode,” ASME International Design Engineering Technical
Conferences and Computers and Information in Engineering Conference, DETC2017-
673576, Cleveland.
[346] Mathworks, 2016, Global Optimization Toolbox.
[347] Hasgall, P., Di Gennaro, F., Baumgartner, C., Neufeld, E., Gosselin, M., Payne, D.,
Klingenbock, A., and Kuster, N., 2015, “IT’IS Database for Thermal and Electromagnetic
Parameters of Biological Tissues,” Version 3.0 [Online]. Available:
www.itis.ethz.ch/database.
309
[348] Gabriel, C., 1996, Compilation of the Dielectric Properties of the Body Tissues at RF and
Microwave Frequencies, London.
[349] Komar, G., Kauhanen, S., Liukko, K., Seppänen, M., Kajander, S., Ovaska, J., Nuutila, P.,
and Minn, H., 2009, “Decreased Blood Flow with Increased Metabolic Activity: A Novel
Sign of Pancreatic Tumor Aggressiveness,” Clin. Cancer Res., 15(17), pp. 5511–5517.
[350] Bergman, T. L., Lavine, A. S., Incropera, F. P., and Dewitt, D. P., 2011, Fundamentals of
Heat and Mass Transfer, Wiley.
[351] Committee, A. I. H., 1995, “Electrical Testing of Polymers,” Engineered Materials
Handbook Desk Edition, ASM International, Materials Park, Ohio, pp. 423–438.
[352] 2015, “Nitinol Technical Properties,” Johnson Matthey Med. Components, pp. 1–2
[Online]. Available: http://jmmedical.com/resources/221/Nitinol-Technical-
Properties.html.
[353] Rossman, C., and Haemmerich, D., 2014, “Review of Temperature Dependence of
Thermal Properties, Dielectric Properties, and Perfusion of Biological Tissues at
Hyperthermic and Ablation Temperatures,” Crit Rev Biomed Eng, 42(6), pp. 467–492.
[354] Rasband, W. S. W., 2012, “ImageJ,” U. S. Natl. Institutes Heal. Bethesda, Maryland,
USA.
[355] Hanks, B., Azhar, F., Frecker, M., Clement, R., Greaser, J., and Snook, K., 2019, “A
Deployable Multi-Tine Endoscopic Radiofrequency Ablation Electrode: Simulation
Validation in a Thermochromic Tissue Phantom,” Design of Medical Devices Conference,
DMD2019-3214.
[356] Datla, N. V., Konh, B., Koo, J. J. Y., Choi, D. J. W., Yu, Y., Dicker, A. P., Podder, T. K.,
Darvish, K., and Hutapea, P., 2014, “Polyacrylamide Phantom for Self-Actuating Needle-
Tissue Interaction Studies,” Med. Eng. Phys., 36(1), pp. 140–145.
[357] Negussie, A. H., Partanen, A., Mikhail, A. S., Xu, S., Abi-Jaoudeh, N., Maruvada, S., and
Wood, B. J., 2016, “Thermochromic Tissue-Mimicking Phantom for Optimisation of
Thermal Tumour Ablation,” Int. J. Hyperth., 32(3), pp. 239–243.
[358] Mathworks, “Fmincon” [Online]. Available:
https://www.mathworks.com/help/optim/ug/fmincon.html.
[359] Deb, K., Mohan, M., and Mishra, S., 2003, A Fast Multi-Objective Evolutionary
Algorithm for Finding Well-Spread Pareto-Optimal Solutions.
[360] van der Sluis, O., Schreurs, P. J. G., Brekelmans, W. A. M., and Meijer, H. E. H., 2000,
“Overall Behaviour of Heterogeneous Elastoviscoplastic Materials: Effect of
Microstructural Modelling,” Mech. Mater., 32, pp. 449–462.
[361] Hanks, B., and Frecker, M., 2019, “Lattice Structure Design for Additive Manufacturing:
Unit Cell Topology Optimization,” ASME International Design Engineering Technical
Conferences & Computers and Information in Engineering, DETC2019-97863.
310
[362] Fuller, R. B., 1961, “Synergetic Building Construction,” pp. 1–11.
[363] Zheng, X., Lee, H., Weisgraber, T. H., Shusteff, M., DeOtte, J., Duoss, E. B., Kuntz, J. D.,
Biener, M. M., Ge, Q., Jackson, J. A., Kucheyev, S. O., Fang, N. X., and Spadaccini, C.
M., 2014, “Ultralight, Ultrastiff Mechanical Metamaterials,” Science (80-. )., 344(6190),
pp. 1373–1377.
[364] Arabnejad, S., Burnett Johnston, R., Pura, J. A., Singh, B., Tanzer, M., and Pasini, D.,
2016, “High-Strength Porous Biomaterials for Bone Replacement: A Strategy to Assess
the Interplay between Cell Morphology, Mechanical Properties, Bone Ingrowth and
Manufacturing Constraints,” Acta Biomater., 30, pp. 345–356.
[365] Suard, M., 2015, “Characterization and Optimization of Lattice Structures Made by
Electron Beam Melting,” Universite De Grenoble.
[366] de Formanoir, C., Suard, M., Dendievel, R., Martin, G., and Godet, S., 2016, “Improving
the Mechanical Efficiency of Electron Beam Melted Titanium Lattice Structures by
Chemical Etching,” Addit. Manuf., 11, pp. 71–76.
[367] Gangireddy, S., Komarasamy, M., Faierson, E. J., and Mishra, R. S., 2019, “High Strain
Rate Mechanical Behavior of Ti-6Al-4V Octet Lattice Structures Additively
Manufactured by Selective Laser Melting (SLM),” Mater. Sci. Eng. A, 745, pp. 231–239.
[368] Stankovic, T., Mueller, J., and Shea, K., 2017, “The Effect of Anisotropy on the
Optimization of Additively Manufactured Lattice Structures,” Addit. Manuf., 17, pp. 67–
76.
[369] He, Z. Z., Wang, F. C., Zhu, Y. B., Wu, H. A., and Park, H. S., 2017, “Mechanical
Properties of Copper Octet-Truss Nanolattices,” J. Mech. Phys. Solids, 101, pp. 133–149.
[370] International Organization for Standardization, 2011, Mechanical Testing of Metals -
Ductility Testing - Compression Test for Porous and Cellular Metals (ISO 13314:2011.
[371] EOS, “Materials for Metal Additive Manufacturing” [Online]. Available:
https://www.eos.info/material-m. [Accessed: 29-Aug-2019].
[372] Thermo Fisher Scientific, “Avizo” [Online]. Available:
https://www.thermofisher.com/us/en/home/industrial/electron-microscopy/electron-
microscopy-instruments-workflow-solutions/3d-visualization-analysis-software.html.
[373] Hanks, B., Berthel, J., Frecker, M., and Simpson, T., 2020, “Mechanical Properties of
Additively Manufactured Metal Lattice Struturs: Data Review and Design Interface,”
Addit. Manuf., (Accepted for publication).
[374] Dessault Systems, “SolidWorks” [Online]. Available: https://www.solidworks.com/.
[375] Autodesk, “Inventor” [Online]. Available:
https://www.autodesk.com/products/inventor/overview.
[376] PTC, “Creo” [Online]. Available: https://www.ptc.com/en/products/cad/creo.
311
[377] Materialise, “3-Matic” [Online]. Available: https://www.materialise.com/en/software/3-
matic.
[378] Ashby, M. F., 2017, Materials Selection in Mechanical Design, Butterworth-Heinemann,
Oxford, United Kingdom.
[379] Renishaw, “Data Sheets - Additive Manufacturing” [Online]. Available:
https://www.renishaw.com/en/data-sheets-additive-manufacturing--17862. [Accessed: 29-
Aug-2019].
[380] Gümrük, R., Mines, R. A. W., and Karadeniz, S., 2013, “Static Mechanical Behaviours of
Stainless Steel Micro-Lattice Structures under Different Loading Conditions,” Mater. Sci.
Eng. A, 586, pp. 392–406.
[381] Hedayati, R., Sadighi, M., Mohammadi-Aghdam, M., and Hosseini-Toudeshky, H., 2018,
“Comparison of Elastic Properties of Open-Cell Metallic Biomaterials with Different Unit
Cell Types,” J. Biomed. Mater. Res. - Part B Appl. Biomater., 106(1), pp. 386–398.
[382] Li, S. J., Murr, L. E., Cheng, X. Y., Zhang, Z. B., Hao, Y. L., Yang, R., Medina, F., and
Wicker, R. B., 2012, “Compression Fatigue Behavior of Ti-6Al-4V Mesh Arrays
Fabricated by Electron Beam Melting,” Acta Mater., 60(3), pp. 793–802.
[383] Yavari, S. A., Wauthle, R., Van Der Stok, J., Riemslag, A. C., Janssen, M., Mulier, M.,
Kruth, J. P., Schrooten, J., Weinans, H., and Zadpoor, A. A., 2013, “Fatigue Behavior of
Porous Biomaterials Manufactured Using Selective Laser Melting,” Mater. Sci. Eng. C,
33(8), pp. 4849–4858.
[384] Yavari, S. A., Ahmadi, S. M., Wauthle, R., Pouran, B., Schrooten, J., Weinans, H., and
Zadpoor, A. A., 2015, “Relationship between Unit Cell Type and Porosity and the Fatigue
Behavior of Selective Laser Melted Meta-Biomaterials,” J. Mech. Behav. Biomed. Mater.,
43(March), pp. 91–100.
[385] McKown, S., Shen, Y., Brookes, W. K., Sutcliffe, C. J., Cantwell, W. J., Langdon, G. S.,
Nurick, G. N., and Theobald, M. D., 2008, “The Quasi-Static and Blast Loading Response
of Lattice Structures,” Int. J. Impact Eng., 35(8), pp. 795–810.
[386] Niu, J., Choo, H. L., Sun, W., and Mok, S. H., 2018, “Analytical Solution and
Experimental Study Young’s Modulus of SLM-Fabricated Lattice Structure with
Triangular Unit Cells,” J. Manuf. Sci. Eng.
[387] Al-Saedi, D. S. J., Masood, S. H., Faizan-Ur-Rab, M., Alomarah, A., and Ponnusamy, P.,
2018, “Mechanical Properties and Energy Absorption Capability of Functionally Graded
F2BCC Lattice Fabricated by SLM,” Mater. Des., 144, pp. 32–44.
[388] Choy, S. Y., Sun, C.-N., Leong, K. F., and Wei, J., 2017, “Compressive Properties of Ti-
6Al-4V Lattice Structures Fabricated by Selective Laser Melting: Design, Orientation and
Density,” Addit. Manuf., 16, pp. 213–224.
[389] Ataee, A., Li, Y., Brandt, M., and Wen, C., 2018, “Ultrahigh-Strength Titanium Gyroid
Scaffolds Manufactured by Selective Laser Melting (SLM) for Bone Implant
312
Applications,” Acta Mater., 158, pp. 354–368.
[390] Beyer, C., and Figueroa, D., 2016, “Design and Analysis of Lattice Structures for Additive
Manufacturing,” J. Manuf. Sci. Eng., 138(12), p. 121014.
[391] Bobbert, F. S. L., Lietaert, K., Eftekhari, A. A., Pouran, B., Ahmadi, S. M., Weinans, H.,
and Zadpoor, A. A., 2017, “Additively Manufactured Metallic Porous Biomaterials Based
on Minimal Surfaces: A Unique Combination of Topological, Mechanical, and Mass
Transport Properties,” Acta Biomater., 53, pp. 572–584.
[392] Borleffs, M. S., 2012, “Finite Element Modeling To Predict Bulk Mechanical Properties
of 3D Printed Metal Foams,” TUDelft.
[393] Campanelli, S. L., Contuzzi, N., Ludovico, A. D., Caiazzo, F., Cardaropoli, F., and Sergi,
V., 2014, “Manufacturing and Characterization of Ti6Al4V Lattice Components
Manufactured by Selective Laser Melting,” Materials (Basel)., 7(6), pp. 4803–4822.
[394] Gümrük, R., and Mines, R. A. W., 2013, “Compressive Behaviour of Stainless Steel
Micro-Lattice Structures,” Int. J. Mech. Sci., 68, pp. 125–139.
[395] Harrysson, O. L. A., Cansizoglu, O., Marcellin-Little, D. J., Cormier, D. R., and West, H.
A., 2008, “Direct Metal Fabrication of Titanium Implants with Tailored Materials and
Mechanical Properties Using Electron Beam Melting Technology,” Mater. Sci. Eng. C,
28(3), pp. 366–373.
[396] Hasib, H. Bin, 2011, “Mechanical Behavior of Non-Stochastic Ti-6Al-4V Cellular
Structures Produced via Electron Beam Melting (EBM),” North Carolina State University.
[397] Heinl, P., Müller, L., Körner, C., Singer, R. F., and Müller, F. A., 2008, “Cellular Ti-6Al-
4V Structures with Interconnected Macro Porosity for Bone Implants Fabricated by
Selective Electron Beam Melting,” Acta Biomater., 4(5), pp. 1536–1544.
[398] Heinl, P., Körner, C., and Singer, R. F., 2008, “Selective Electron Beam Melting of
Cellular Titanium: Mechanical Properties,” Adv. Eng. Mater., 10(9), pp. 882–888.
[399] Leary, M., Maxur, M., Elambasseril, J., Matthew, M., Chirent, T., Sun, Y., Qian, M.,
Easton, M., and Brandt, M., 2016, “Selective Laser Melting (SLM) of AlSi12Mg Lattice
Structures,” Mater. Des., 98, pp. 119–161.
[400] Leary, M., Mazur, M., Williams, H., Yang, E., Alghamdi, A., Lozanovski, B., Zhang, X.,
Shidid, D., Farahbod-Sternahl, L., Witt, G., Kelbassa, I., Choong, P., Qian, M., and
Brandt, M., 2018, “Inconel 625 Lattice Structures Manufactured by Selective Laser
Melting (SLM): Mechanical Properties, Deformation and Failure Modes,” Mater. Des.,
157, pp. 179–199.
[401] Li, C., Lei, H., Liu, Y., Zhang, X., Xiong, J., Zhou, H., and Fang, D., 2018, “Crushing
Behavior of Multi-Layer Metal Lattice Panel Fabricated by Selective Laser Melting,” Int.
J. Mech. Sci., 145(June), pp. 389–399.
[402] Li, S. J., Xu, Q. S., Wang, Z., Hou, W. T., Hao, Y. L., Yang, R., and Murr, L. E., 2014,
313
“Influence of Cell Shape on Mechanical Properties of Ti-6Al-4V Meshes Fabricated by
Electron Beam Melting Method,” Acta Biomater., 10(10), pp. 4537–4547.
[403] Liu, Y. J., Wang, H. L., Li, S. J., Wang, S. G., Wang, W. J., Hou, W. T., Hao, Y. L.,
Yang, R., and Zhang, L. C., 2017, “Compressive and Fatigue Behavior of Beta-Type
Titanium Porous Structures Fabricated by Electron Beam Melting,” Acta Mater., 126, pp.
58–66.
[404] Liu, Y. J., Li, S. J., Wang, H. L., Hou, W. T., Hao, Y. L., Yang, R., Sercombe, T. B., and
Zhang, L. C., 2016, “Microstructure, Defects and Mechanical Behavior of Beta-Type
Titanium Porous Structures Manufactured by Electron Beam Melting and Selective Laser
Melting,” Acta Mater., 113, pp. 56–67.
[405] Mazur, M., Leary, M., Sun, S., Vcelka, M., Shidid, D., and Brandt, M., 2016,
“Deformation and Failure Behaviour of Ti-6Al-4V Lattice Structures Manufactured by
Selective Laser Melting (SLM),” Int. J. Adv. Manuf. Technol., 84(5–8), pp. 1391–1411.
[406] Mines, R. A. W., Tsopanos, S., Shen, Y., Hasan, R., and McKown, S. T., 2013, “Drop
Weight Impact Behaviour of Sandwich Panels with Metallic Micro Lattice Cores,” Int. J.
Impact Eng., 60, pp. 120–132.
[407] Mullen, L., Stamp, R. C., Brooks, W. K., Jones, E., and Sutcliffe, C. J., 2009, “Selective
Laser Melting: A Regular Unit Cell Approach for the Manufacture of Porous, Titanium,
Bone in-Growth Constructs, Suitable for Orthopedic Applications,” J. Biomed. Mater.
Res. - Part B Appl. Biomater., 89(2), pp. 325–334.
[408] Murr, L. E., Amato, K. N., Li, S. J., Tian, Y. X., Cheng, X. Y., Gaytan, S. M., Martinez,
E., Shindo, P. W., Medina, F., and Wicker, R. B., 2011, “Microstructure and Mechanical
Properties of Open-Cellular Biomaterials Prototypes for Total Knee Replacement Implants
Fabricated by Electron Beam Melting,” J. Mech. Behav. Biomed. Mater., 4(7), pp. 1396–
1411.
[409] Murr, L. E., Gaytan, S. M., Medina, F., Lopez, H., Martinez, E., Machado, B. I.,
Hernandez, D. H., Martinez, L., Lopez, M. I., Wicker, R. B., and Bracke, J., 2010, “Next-
Generation Biomedical Implants Using Additive Manufacturing of Complex, Cellular and
Functional Mesh Arrays,” Philos. Trans. R. Soc. A Math. Phys. Eng. Sci., 368(1917), pp.
1999–2032.
[410] Murr, L., Li, S., Tian, Y., Amato, K., Martinez, E., and Medina, F., 2010, “Open-Cellular
Co-Base and Ni-Base Superalloys Fabricated by Electron Beam Melting,” Materials
(Basel)., 4(4), pp. 782–790.
[411] Parthasarathy, J., Starly, B., Raman, S., and Christensen, A., 2010, “Mechanical
Evaluation of Porous Titanium (Ti6Al4V) Structures with Electron Beam Melting
(EBM),” J. Mech. Behav. Biomed. Mater., 3(3), pp. 249–259.
[412] Ptochos, E., and Labeas, G., 2012, “Elastic Modulus and Poisson ’ s Ratio Determination
of Micro-Lattice Cellular Structures by Analytical , Numerical and Homogenisation
Methods,” J. Sandw. Struct. Mater., 14(5).
314
[413] Rehme, O., and Emmelmann, C., 2006, “Rapid Manufacturing of Lattice Structures with
Selective Laser Melting,” Laser-based Micropackaging, 6107(February 2006), p. 61070K.
[414] Sallica-Leva, E., Jardini, A. L., and Fogagnolo, J. B., 2013, “Microstructure and
Mechanical Behavior of Porous Ti-6Al-4V Parts Obtained by Selective Laser Melting,” J.
Mech. Behav. Biomed. Mater., 26, pp. 98–108.
[415] Shen, Y., McKown, S., Tsopanos, S., Sutcliffe, C. J., Mines, R. A. W., and Cantwell, W.
J., 2010, “The Mechanical Properties of Sandwich Structures Based on Metal Lattice
Architectures,” J. Sandw. Struct. Mater., 12(2), pp. 159–180.
[416] Tsopanos, S., Mines, R. A. W., McKown, S., Shen, Y., Cantwell, W. J., Brooks, W., and
Sutcliffe, C. J., 2010, “The Influence of Processing Parameters on the Mechanical
Properties of Selectively Laser Melted Stainless Steel Microlattice Structures,” J. Manuf.
Sci. Eng., 132, pp. 1–12.
[417] van Grunsven, W., Hernandez-Nava, E., Reilly, G., and Goodall, R., 2014, “Fabrication
and Mechanical Characterisation of Titanium Lattices with Graded Porosity,” Metals
(Basel)., 4(3), pp. 401–409.
[418] Wauthle, R., Vrancken, B., Beynaerts, B., Jorissen, K., Schrooten, J., Kruth, J.-P., and
Van Humbeeck, J., 2015, “Effects of Build Orientation and Heat Treatment on the
Microstructure and Mechanical Properties of Selective Laser Melted Ti6Al4V Lattice
Structures,” Addit. Manuf., 5, pp. 77–84.
[419] Xiao, L., Song, W., Wang, C., Liu, H., Tang, H., and Wang, J., 2015, “Mechanical
Behavior of Open-Cell Rhombic Dodecahedron Ti–6Al–4V Lattice Structure,” Mater. Sci.
Eng. A, 640, pp. 375–384.
[420] Yan, C., Hao, L., Hussein, A., and Young, P., 2015, “Ti-6Al-4V Triply Periodic Minimal
Surface Structures for Bone Implants Fabricated via Selective Laser Melting,” J. Mech.
Behav. Biomed. Mater., 51, pp. 61–73.
[421] Yánez, A., Cuadrado, A., Martel, O., Afonso, H., and Monopoli, D., 2018, “Gyroid Porous
Titanium Structures: A Versatile Solution to Be Used as Scaffolds in Bone Defect
Reconstruction,” Mater. Des., 140, pp. 21–29.
[422] Yang, E., Leary, M., Lozanovski, B., Downing, D., Mazur, M., Sarker, A., Khorasani, A.,
Jones, A., Maconachie, T., Bateman, S., Easton, M., Qian, M., Choong, P., and Brandt,
M., 2019, “Effect of Geometry on the Mechanical Properties of Ti-6Al-4V Gyroid
Structures Fabricated via SLM: A Numerical Study,” Mater. Des., 184, p. 108165.
[423] Zaharin, H. A., Rani, A. M. A., Azam, F. I., Ginta, T. L., Sallih, N., Ahmad, A., Yunus, N.
A., and Zulkifli, T. Z. A., 2018, “Effect of Unit Cell Type and Pore Size on Porosity and
Mechanical Behavior of Additively Manufactured Ti6Al4V Scaffolds,” Materials (Basel).,
11(12).
[424] Zhao, S., Li, S. J., Hou, W. T., Hao, Y. L., Yang, R., and Misra, R. D. K., 2016, “The
Influence of Cell Morphology on the Compressive Fatigue Behavior of Ti-6Al-4V Meshes
Fabricated by Electron Beam Melting,” J. Mech. Behav. Biomed. Mater., 59, pp. 251–264.
315
[425] Zhao, M., Liu, F., Fu, G., Zhang, D. Z., Zhang, T., and Zhou, H., 2018, “Improved
Mechanical Properties and Energy Absorption of BCC Lattice Structures with Triply
Periodic Minimal Surfaces Fabricated by SLM,” Materials (Basel)., 11(12).
[426] Zhu, H. X., Knott, J. F., and Mills, N. J., 1997, “Analysis of the Elastic Properties of
Open-Cell Foams with Tetrakaidecahedral Cells,” J. Mech. Phys. Solids, 45(3), pp. 319–
343.
[427] Rohatgi, A., 2019, “WebPlotDigitizer” [Online]. Available:
https://automeris.io/WebPlotDigitizer.
[428] Drevon, D., Fursa, S. R., and Malcolm, A. L., 2017, “Intercoder Reliability and Validity
of WebPlotDigitizer in Extracting Graphed Data,” Behav. Modif., 41(2), pp. 323–339.
[429] Moeyaert, M., Maggin, D., and Verkuilen, J., 2016, “Reliability, Validity, and Usability of
Data Extraction Programs for Single-Case Research Designs,” Behav. Modif., 40(6), pp.
874–900.
[430] Burda, B. U., O’Connor, E. A., Webber, E. M., Redmond, N., and Perdue, L. A., 2017,
“Estimating Data from Figures with a Web-Based Program: Considerations for a
Systematic Review,” Res. Synth. Methods, 8(3), pp. 258–262.
[431] McBride, G. B., 2005, “A Proposal for Strength-of-Agreement Criteria for Lins
Concordance Correlation Coefficient,” NIWA Client Rep., HAM2005-06.
[432] 2019, “Desmos Online Calculator” [Online]. Available:
https://www.desmos.com/calculator/7yuojxowmx. [Accessed: 10-Jan-2019].
[433] Maxwell, J. C., 1864, “XLV. On Reciprocal Figures and Diagrams of Forces,” London,
Edinburgh, Dublin Philos. Mag. J. Sci., 27(182), pp. 250–261.
316
Appendix A: MATLAB Image Analysis
For image analysis in MATLAB, this file is set up to read in multiple images and
calculate the f_over and f_under objective functions from thermal ablation model. Special thanks
to Zackary Snow for his support in recommending and providing this direction of image analysis
for computing the objective functions.
% This file computes the f_over and f_under objective function for an
% ablation zone. The image analysis is performed on three different cross
% sections of the ablation that are output from COMSOL.
%Wipes previously stored variables, cleans workspace, and closes open figures
clear; %clc;
files = char('image2_45_600.tiff', 'image2_xz_600.tiff', 'image2_yz_600.tiff');
pixels_to_mm = 1/36;
abl_rad = 12.5; % Target ablation radius (mm)
angle_res = .15; % Take measurements every 'angle_res' degrees
angle_tol = .15; % Take measurements at each angle + or - the tolerance
%% DATA ANALYSIS %%
%For loop to analyze each image file
nfiles = size(files,1);
fover = zeros(nfiles,2);
funder = zeros(nfiles,2);
for k = 1:nfiles
% Imports an image file and converts to grayscale and inverts colors
filename = files(k,:);
[A,map] = imread(filename);
grayscale_version = imcomplement(rgb2gray(A));
% Converts grayscale image to black and white image
thresholded_image = imbinarize(grayscale_version);
% Analyzes the properties of the ablation zone in the image
image_stats = regionprops(thresholded_image,'BoundingBox');
bounding_box = cat(1,image_stats.BoundingBox);
%% UNGULATION COUNTING AND QUANTIFICATION ALGORITHM %%
% Establishes the bounding box around the ablation zone
x_lower = int16(round(bounding_box(1)));
x_upper = int16(fix(bounding_box(1)))+int16(round(bounding_box(3)));
x_mid = double((x_upper+x_lower)/2);
x_range = x_lower:x_upper;
y_lower = int16(round(bounding_box(2)));
y_upper = int16(fix(bounding_box(2)))+int16(round(bounding_box(4)));
y_mid = double((y_upper+y_lower)/2);
317
y_range = y_lower:y_upper;
% Generates smaller image of the bounding box around each ablation zone
bb_BW = thresholded_image( y_range(1)-1:y_range(end), x_range(1)-
1:x_range(end)+1);
% Calculates the distance from and angle relative to the center of
% area of the particle for each pixel contained within the boundning
% box of the particle.
distance = zeros(length(y_range),length(x_range));
phi = zeros(length(y_range),length(x_range));
for j = 1:length(y_range)
for i = 1:length(x_range)
distance(j,i) = ((double(i-1+x_lower-x_mid))^2 + (double(j-
1+y_lower-y_mid))^2)^(0.5);
phi(j,i) = -1*atan2d(double(j-1+y_lower-y_mid),double(i-
1+x_lower-x_mid));
if phi(j,i)<0
phi(j,i) = phi(j,i)+360;
else
phi(j,i) = phi(j,i);
end
if bb_BW(j,i) == 0 % If pixel is black, set distance to zero
distance(j,i) = 0;
else
distance(j,i) = distance(j,i);
end
end
end
%'edge_distance' is the distance from the edge of the particle from
%the center of area of the particle.
edge_distance = zeros(1,round(360/angle_res));
count = 0;
for angle = 0:angle_res:(360-angle_res)
count = count + 1;
positions = find(phi > (angle - angle_tol) & phi < (angle +
angle_tol));
if isempty(positions)
edge_distance(1,count) = 0;
else
edge_distance(1,count) = max(distance(positions))*pixels_to_mm;
end
end
% Computes the differences between the edge distance (from center
% to edge) and the target ablation radius
Distance_Difference = edge_distance - abl_rad;
ind_over = Distance_Difference > 0;
ind_under = Distance_Difference < 0;
fover(k,:) = [sum(Distance_Difference(ind_over))/abl_rad,
length(Distance_Difference(ind_over))];
funder(k,:) = [abs(sum(Distance_Difference(ind_under))/abl_rad),
length(Distance_Difference(ind_under))];
end
318
f_o = sum(fover(:,1))/sum(fover(:,2))
f_u = sum(funder(:,1))/sum(funder(:,2))
if isnan(f_o)
f_o = 0;
end
if isnan(f_u)
f_u = 0;
end
319
Appendix B: ALTO 3D MATLAB Files
Main File
The main file is used to set up the optimization problem and contains essentially all the
parameters to determine what type of optimization will be run. It calls the execute function to
execute the optimization problem.
ALTO3D_main.m
% Multi-objective unit cell generation
% Written by Brad Hanks
%
% This file contains all of the preliminary design and input information
% required from the user for the development of the optimization.
clear; close all; clc; % Begin with a fresh slate
% Specify the size of the ground structure for optimization
opt.ucs = [60,20,20]; % Unit cell size in mm
opt.gssize = [ 7, 2, 5]; % Number of nodes in [ x, y, z]
% Specify the objective functions and constraints
% opt.objs = [ TYPE, MINIMIZE, WEIGHT, TARGET]
% TYPE: 1 - calculated the strain energy
% 2 - calculates the thermal conductivity between a set of nodes
% 3 - calculates the homogenized unit cell properties
% 4 - calculates the total volume of the material
% 5 - Number of relevant members
% 6 - Powder Removability (Maxi Avg Dist, Maxi Min Dist, Mini #
trapped regions
% MINIMIZE: 1 to minimize the objective
% -1 to maximize the objective
% Should be 1 for objectives 3 and 4
% WEIGHT: specify weight value or set of weight values
% leave blank for general pareto sweep
% TARGET: Specify target value for objective
% For powder removability, if maximize:
% 1 = maximize the avg distance of between relevant members
% 2 = maximize the min distance between relevant members
% For powder removability, if minimize, provide tolerance here
% NOTE: For coupled strain energy and conductivity, the conductivity
% objective function must follow immediately after the strain energy
% objective function.
% Homogenization Target Options
% Isotropic
% E = 0.8*2e5;
% nu = 0.3;
% target = E/(1-nu^2)*[1, nu, 0;
320
% nu, 1, 0;
% 0, 0, (1-nu)/2];
% % Orthotropic
% factor = 3;
% opt.E1 = 0.8*2e5;
% opt.E2 = opt.E1/factor;
% opt.nu12 = 0.6;
% opt.nu21 = opt.nu12/factor;
% opt.G12 = opt.E1/factor/(1+opt.nu12);
% homog_target = [ opt.E1/(1-opt.nu12*opt.nu21), opt.nu21*opt.E1/(1-
opt.nu12*opt.nu21), 0;
% opt.nu12*opt.E2/(1-opt.nu12*opt.nu21), opt.E2/(1-
opt.nu12*opt.nu21), 0;
% 0, 0, opt.G12];
% % Match Diamond UCS 8mm, Diam 1mm
% val = (1/2)*1.8944e3;
% homog_target = zeros(6,6);
% homog_target([1,2,3,7,8,9,13,14,15]) = val;
% Match Octet Truss UCS 8mm, Diam 1mm
val = 6.9420e3;
homog_target = zeros(6,6);
homog_target([1,8,15]) = val;
homog_target([2,3,7,9,13,14]) = val/2;
homog_target([22,29,36]) = 0.625*val;
opt.objs(1) = struct('type', 1,'minimize',
1,'scale',[1],'weight',[1.0],'target',[]);
% opt.objs(1) = struct('type', 2,'minimize',-
1,'scale',[1],'weight',[1.0],'target',[0]);
% opt.objs(1) = struct('type',
3,'minimize',1,'scale',[1],'weight',[1.0],'target',homog_target);
% opt.objs(2) = struct('type',
4,'minimize',1,'scale',[1],'weight',[1.0],'target',[0.01]); % The target is a
volume fraction and will OVERRIDE opt.volfrac
% opt.objs(3) = struct('type',
5,'minimize',1,'scale',[1],'weight',[1.0],'target',[]);
% opt.objs(4) = struct('type',
6,'minimize',1,'scale',[1],'weight',[1.0],'target',4);
% Specify Optimization constraint parameters
opt.volfrac = [.58]; % Volume Fraction constraint, leave empty to remove
constraints
opt.dlim = [9.5e-5, 7.5]; % Element diameter limits [min,max] in mm
opt.setsym = ['y']; % Set symmetry mid planes { X, Y, Z, XY, XZ, XYZ,
NONE }
opt.buildang = []; % Build Angle Limit measured from horizontal
opt.removenode = []; % Remove elements connected to these nodes if row
vector, column vector removes only the element between the two nodes;
% ~~ 4x4x4 nodes for common structures ~~
% ~~ 5x5x5 nodes for common structures ~~
% Diamond List -
[2,4,6:10,12,14,16:20,22,24,26:31,35,36,40,41,45:50,52,54,56,60,66,70,72,74,76:
81,85,86,90,91,95,96:100,102,104,106:110,112,114,116:120,122,124,1,3,11,15,23,2
5,51,55,71,75,103,105,111,115,121,123]
% Diamond List2-
[2,4,6:10,12,14,16:20,22,24,26:31,35,36,40,41,45:50,52,54,56,60,66,70,72,74,76:
321
81,85,86,90,91,95,96:100,102,104,106:110,112,114,116:120,122,124,3,5,11,15,21,2
3,51,55,71,75,101,103,111,115,123,125]
% Octet List -
[2,3,4,6:10,11,12,14,15,16:20,22,23,24,26:31,35,36,40,41,45:50,51,52,54,55,56,6
0,66,70,71,72,74,75,76:81,85,86,90,91,95,96:100,102,103,104,106:110,111,112,114
,115,116:120,122,123,124]
% 9pts per face -
[2,4,6:10,12,14,16:20,22,24,26:31,35,36,40,41,45:50,52,54,56,60,66,70,72,74,76:
81,85,86,90,91,95,96:100,102,104,106:110,112,114,116:120,122,124]
% Interior List - [32:34,37:39,42:44,57:59,62:64,67:69,82:84,87:89,92:94]
% 2,4 Edge -
[1,3,5,7:9,11,12,14,15,17:19,21,23,25,27:29,31,35,36,40,41,45,47,48,49,51,52,54
,55,56,60,66,70,71,72,74,75,77:79,81,85,86,90,91,95,97,98,99,101,103,105,107:10
9,111,112,114,115,117:119,121,123,125]
% Edge Only -
[7:9,12:14,17:19,27:29,31,35,36,40,41,45,47:49,52:54,56,60,61,65,66,70,72:74,77
:79,81,85,86,90,91,95,97:99,107:109,112:114,117:119];
opt.removesurface = false; % Remove surface elements
opt.removelength = []; % Remove members whose length falls between this range
% Penalty functions
opt.drel = [1]; % Relevant diameter (mm) - Used to determine which members
are relevant
opt.Pcount = []; % '+' few members, '-' more members
opt.Pdmid = []; % '+' few members between drel and dlim(1)
opt.Ptype = 1; %1 - None, 2 - overlap, 3 - connectivity, 4 - both
opt.SPtype = 1; %1 - None, 2 - neighbor, 3 - self Support Structure
Requirements
% Specify material properties
opt.mod = 2e5; % Modulus of elasticity N/mm^2
opt.kcond = 0.054; % Thermal conductivity W/mm-K
opt.alpha = 12e-6; % Coefficient of thermal expansion mm/(mm-K)
% Specify optimization options
opt.algorithm = 'fmincon'; % Specify algorithm type, fmincon or borg or NSGA-
II
switch lower(opt.algorithm)
case {'borg'}
opt.epsilon = [1]; % Specify objective tolerances
opt.FuncCount = 1000; % Specify max function evaluations for borg
case {'fmincon'}
opt.parallel = false; % Run in parallel
opt.MaxIterations = 500; % Maximum number of iterations, enter
1 for single run and not optimization
opt.minstep = 1e-6; % Step size tolerance, default 1e-10
opt.saveobj = 0; % Save all objective function values
(EXPENSIVE)
case {'nsga-ii'}
opt.FunctionTolerance = 1e-4; % Tolerance for relative spread change
for convergence over MaxStall Generations, default 1e-4
opt.MaxIterations = 2449; % Maximum Generations, default is
200*numvars, To define by FuncCount-- ((FuncCount/Population) - 1)
opt.MaxStallGenerations = 100; % Number of stall generations
considered in convergence, default is 100
opt.ParetoFraction = 0.35; % Number of individuals to keep from
first Pareto Front, default is 0.35
opt.PopulationSize = 200; % Population size, default 50 (numvars
<= 5), 200 (numvars > 5)
322
opt.UseParallel = true; % Use Parallel computing when available
opt.CrossoverFraction = 0.80;
end
% Initial Population options
opt.init = []; % Populate initial condition *ROW vector, will
randomize if blank
% opt.init = pi/4*(opt.dlim(1)^2)*ones(1,2016); % Populate initial
condition at A_lower
% opt.init = pi/4*(opt.drel^2)*ones(1,2016); % Populate initial
condition at A_rel
% opt.init = pi/4*(opt.dlim(2)^2)*ones(1,7750); % Populate initial
condition at A_upper
% opt.init([5,45,66,69,79,100,102,110])=1.0;
% Specify the type of loading and boundary conditions
% Required for strain energy objective function
% opt.load syntax: [ node#, fx, fy, fz;
% node#, fx, fy, fz];
opt.loads = [ 13, 0, 0, 1000;
14, 0, 0, 1000];
% opt.bc syntax: [ node#, ux, uy, uz;
% node#, ux, uy, uz];
% ux = 1 is free, ux = 0 is constrained
opt.bcs = [ 1, 0, 0, 0;
2, 0, 0, 0;
15, 0, 0, 0;
16, 0, 0, 0;
29, 0, 0, 0;
30, 0, 0, 0];
% Specify temperature and nodes for thermal conduction objective function
% Required for thermal conduction objective function
% in = [1:12,14:31,35,36,40,41,45:56,60,61,65,66,70:81,85,86,90,91,95:125]; %
Nodes at temperature T1
% out = [13]; % Nodes at temperature T2
in = [];
out = [53];
Iin = 5*ones(124,1);
opt.temps = [ 100, 25, 25]; % [T1, T2, T-reference] [degC]
opt.T1nodes = in;
opt.T2nodes = out;
opt.Iin = Iin;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%Single Optimization
modopt = []; % Leave empty for non-parallel runs
filename = 'SE_bench_R4_dr_10_5';
ALTO3D_exec(opt,filename,modopt)
% % Parallel Runs
% nprocs = 3;
% parfor(modopt = 1:3,nprocs)
323
% filename = ['C1_R5_AM_eps2_M',num2str(modopt)]; % Filename prefix, do not
include ".mat"
% ALTO3D_exec(opt,filename,modopt)
% end
Execution Function
The execution file takes all the parameters from the problem set in the main file and the
initializes the ground structure, applies ground structure restrictions, and prepares variables for
the optimization. It then sends the problem to the specified optimization algorithm.
ALTO3D_exec.m
% Multi-objective unit cell generation
% Written by Brad Hanks
%
% This file reads all of the input information from the main file and
% initializes the ground structure accordingly. After initializing the
% ground structure, this file executes the optimization.
function [] = ALTO3D_exec(opt,filename,modopt)
% Setup filename
opt.filename = filename;
% Setup different modifications for parfor loop here
if modopt == 1
disp('Running configuration 1 - (1/2x) no DfAM')
else
disp('Running main file config')
end
% Initialize Ground Structure Geometry
gs = ALTO3D_initgs(opt);
A = gs.Ainitial;
% Initialize
% Determine objective functions
optim.numobjs = length([opt.objs.type]);
for i = 1:optim.numobjs
if opt.objs(i).type == 1
disp('Initializing Strain Energy ObjFunc')
optim.compl.loads = opt.loads;
optim.compl.bcs = opt.bcs;
optim.compl.objnum = i;
% Calculate range of objective function
opt.type = 1;
optim.compl.range(1) =
nonzeros(ALTO3D_compltherm(gs.Aupper,opt,optim,gs));
optim.compl.range(2) =
nonzeros(ALTO3D_compltherm(gs.Alower,opt,optim,gs));
elseif opt.objs(i).type == 2
disp('Initializing Thermal Conductivity ObjFunc')
324
optim.therm.T1 = opt.temps(1);
optim.therm.T2 = opt.temps(2);
optim.therm.Tref = opt.temps(3);
optim.therm.in = opt.T1nodes;
optim.therm.out = opt.T2nodes;
% Calculate range of objective function
opt.type = 2;
optim.therm.range(1) =
nonzeros(ALTO3D_compltherm(gs.Alower,opt,optim,gs));
optim.therm.range(2) =
nonzeros(ALTO3D_compltherm(gs.Aupper,opt,optim,gs));
elseif opt.objs(i).type == 3
disp('Initializing Homogenization Properties')
optim.homog.target = opt.objs(i).target;
elseif opt.objs(i).type == 4
disp('Initializing Volume ObjFunc')
optim.volume.range(1) = gs.Alower*gs.L;
optim.volume.range(2) = gs.Aupper*gs.L;
optim.volume.vftarget = opt.objs(i).target;
elseif opt.objs(i).type == 5
disp('Initializing Member Count ObjFunc')
elseif opt.objs(i).type == 6
if opt.objs(i).minimize == 1
optim.powder.typename = 'Minimize Trapped Regions';
optim.powder.type = 3;
optim.powder.tol = opt.objs(i).target;
else
if opt.objs(i).target == 1
optim.powder.typename = 'Maximize Avg Distance';
optim.powder.type = 1;
elseif opt.objs(i).target == 2
optim.powder.typename = 'Maximize Min Distance';
optim.powder.type = 2;
else
error('Unknown Powder Removability Target')
end
end
disp(['Initializing Powder Removability ObjFunc:
',[optim.powder.typename]])
optim.powder.target = opt.objs(i).target;
else
error(['Unknown objective function type, TYPE:
',num2str(opt.objs(i).type)])
end
end
% % Check Code
% gs.Aupper = 0.04*ones(1,gs.M);
% opt.type = 12;
% ALTO3D_compltherm(gs.Aupper,opt,optim,gs)
% Objective Function Weights
try
optim.weight = vertcat(opt.objs.weight)';
catch
error('Weights are not the same length for each objective')
end
if isempty(optim.weight)
if optim.numobjs == 2
optim.weight = [ 1.0:-0.1:0.0; 0.0:0.1:1.0]';
325
elseif optim.numobjs == 3
optim.weight = [ 1, 0, 0, 0.5, 0.5, 0, (1/3), 0.4, 0.4, 0.2;
0, 1, 0, 0.5, 0, 0.5, (1/3), 0.4, 0.2, 0.4;
0, 0, 1, 0, 0.5, 0.5, (1/3), 0.2, 0.4, 0.4]';
else
error('Requires weight values to be defined')
end
end
%% Prepare Optimization
% Optimization variables
optim.Ainitial = gs.Ainitial;
optim.L = gs.L;
optim.IEN = gs.IEN;
optim.Alower = gs.Alower;
optim.Aupper = gs.Aupper;
optim.theta = gs.theta;
optim.thetab = gs.thetab;
if ~isempty(opt.volfrac)
% optim.Vstar = opt.volfrac*(optim.Aupper*gs.L);
optim.Vstar = opt.volfrac*(opt.ucs(1)*opt.ucs(2)*opt.ucs(3));
else
optim.Vstar = [];
end
optim.wght = [];
optim.en_symmap = zeros(gs.M,1);
optim.en_count = zeros(gs.M,1);
optim.symm = [];
optim.keep = [];
optim.remove = [];
%% Reduce number of elements
% Remove elements by symmetry
for i = 1:gs.N
symnodes(1,i) = find(abs(gs.coord(1,:)-(opt.ucs(1)-gs.coord(1,i)))<.00001 &
gs.coord(2,:) == gs.coord(2,i) & gs.coord(3,:) == gs.coord(3,i));
symnodes(2,i) = find(abs(gs.coord(2,:)-(opt.ucs(2)-gs.coord(2,i)))<.00001 &
gs.coord(3,:) == gs.coord(3,i) & gs.coord(1,:) == gs.coord(1,i));
symnodes(3,i) = find(abs(gs.coord(3,:)-(opt.ucs(3)-gs.coord(3,i)))<.00001 &
gs.coord(1,:) == gs.coord(1,i) & gs.coord(2,:) == gs.coord(2,i));
end
IEN1 = zeros(2,gs.M); % Flip across X-mid
IEN2 = zeros(2,gs.M); % Flip across Y-mid
IEN3 = zeros(2,gs.M); % Flip across Z-mid
IEN4 = zeros(2,gs.M); % Flip across X-mid then Y-mid
IEN5 = zeros(2,gs.M); % Flip across Z-mid then X-mid
IEN6 = zeros(2,gs.M); % Flip across Z-mid then Y-mid
IEN7 = zeros(2,gs.M); % Flip across Z-mid then X-mid then Y-mid
for i = 1:gs.N
map1 = gs.IEN == i;
IEN1(map1) = symnodes(1,i);
IEN2(map1) = symnodes(2,i);
IEN3(map1) = symnodes(3,i);
end
for i = 1:gs.N
map2 = IEN1 == i;
326
IEN4(map2) = symnodes(2,i);
end
for i = 1:gs.N
map3 = IEN3 == i;
IEN5(map3) = symnodes(1,i);
IEN6(map3) = symnodes(2,i);
end
for i = 1:gs.N
map4 = IEN5 == i;
IEN7(map4) = symnodes(2,i);
end
IEN1 = sort(IEN1);
IEN2 = sort(IEN2);
IEN3 = sort(IEN3);
IEN4 = sort(IEN4);
IEN5 = sort(IEN5);
IEN6 = sort(IEN6);
IEN7 = sort(IEN7);
switch lower(opt.setsym)
case{'x'}
IEN_f = [gs.IEN;IEN1];
case{'y'}
IEN_f = [gs.IEN;IEN2];
case{'z'}
IEN_f = [gs.IEN;IEN3];
case{'xy'}
IEN_f = [gs.IEN;IEN1;IEN2;IEN4];
case{'xz'}
IEN_f = [gs.IEN;IEN1;IEN3;IEN5];
case{'yz'}
IEN_f = [gs.IEN;IEN2;IEN3;IEN6];
case{'xyz'}
IEN_f = [gs.IEN;IEN1;IEN2;IEN3;IEN4;IEN5;IEN6;IEN7];
case{'none'}
IEN_f = [gs.IEN];
end
gs.IEN_sym = zeros(2,gs.M);
for i = 1:gs.M
row = find(IEN_f(:,i)==min(IEN_f(1:2:(size(IEN_f,1)-1),i)));
if length(row)>1
min_pos = find(IEN_f(row+1,i)==min(IEN_f(row+1,i)));
row = row(min_pos(1));
end
gs.IEN_sym(:,i) = IEN_f([row,row+1],i);
end
elem = 1;
for i = 1:gs.M
if optim.en_symmap(i) <1
nodeA = gs.IEN_sym(1,i);
nodeB = gs.IEN_sym(2,i);
e_pos = find(gs.IEN_sym(1,:) == nodeA & gs.IEN_sym(2,:) == nodeB);
optim.en_symmap(e_pos) = elem;
optim.en_count(e_pos) = length(e_pos); %Add count of number of elements
repeated
327
elem = elem + 1;
if length(e_pos) > 1
optim.symm = [optim.symm,e_pos(2:end)];
end
end
end
optim.Ainitial(optim.symm) = [];
optim.en_count(optim.symm) = [];
optim.L(optim.symm) = [];
optim.Alower(optim.symm) = [];
optim.Aupper(optim.symm) = [];
optim.theta(optim.symm,:) = [];
optim.thetab(optim.symm) = [];
optim.IEN(:,optim.symm) = [];
% Remove elements from optimization that will be set to lower limit
for i = 1:length(optim.L)
% Remove based on build angle
if ~isempty(opt.buildang)
if ((abs(optim.thetab(i)) > pi*opt.buildang(1)/180) &&
(abs(optim.thetab(i)) < pi*opt.buildang(2)/180))
optim.remove = [optim.remove;i];
continue;
end
end
% Remove based on nodal connections
if ~isempty(opt.removenode)
% if any(optim.IEN(:,i) == opt.removenode, 'all') % MIN VERSION 2018b
if any(any(optim.IEN(:,i) == opt.removenode))
optim.remove = [optim.remove;i];
continue;
end
end
% Remove surface elements
if opt.removesurface
coordA = gs.coord(:,optim.IEN(1,i));
coordB = gs.coord(:,optim.IEN(2,i));
surface = [coordA,coordB]==[min(gs.coord,[],2),min(gs.coord,[],2)];
surface2 = [coordA,coordB]==[max(gs.coord,[],2),max(gs.coord,[],2)];
if any(and(surface(:,1),surface(:,2))) ||
any(and(surface2(:,1),surface2(:,2)))
optim.remove = [optim.remove;i];
continue;
end
end
% Remove based on element length
if ~isempty(opt.removelength)
if ((optim.L(i) > opt.removelength(1)) && (optim.L(i) <
opt.removelength(2)))
optim.remove = [optim.remove;i];
continue;
end
end
optim.keep = [optim.keep;i];
end
328
optim.Ainitial(optim.remove) = [];
optim.en_count(optim.remove) = [];
optim.L(optim.remove) = [];
optim.Alower(optim.remove) = [];
optim.Aupper(optim.remove) = [];
optim.theta(optim.remove,:) = [];
optim.thetab(optim.remove) = [];
optim.IEN(:,optim.remove) = [];
% %% Visualize Prior to Optimization
% Ain = optim.Ainitial;
% Areduced_ck = optim.Ainitial;
% Areduced_ck([optim.keep]) = Areduced_ck;
% Areduced_ck([optim.remove]) = gs.Alower(1);
%
% A_ck = zeros(1,gs.M);
% if isfield(optim,'en_symmap')
% for i = 1:gs.M
% A_ck(i) = Areduced_ck(optim.en_symmap(i));
% if ~isempty(opt.removenode)
% if any(any(gs.IEN(:,i) == opt.removenode))
% A_ck(i) = gs.Alower(1);
% end
% end
% end
% end
% ALTO3D_plot(A_ck,gs.IEN,gs.coord,.5)
% return
%%% Execute Optimization %%%
switch lower(opt.algorithm)
case{'fmincon'}
Results = zeros(size(optim.weight,1),size(optim.weight,2)+1);
for run = 1:size(optim.weight,1)
optim.wght = optim.weight(run,:);
%% Save current
if ~isempty(opt.filename)
opt.tempname = [opt.filename,'-wghts',sprintf('_%.2f',
optim.wght),'.mat'];
if exist(opt.tempname,'file')
error('Change filename to avoid overwiting')
end
outvars =
struct('Iteration',[1],'FuncCount',[],'A',[],'vol',[],'fval',[],'objs',[],'P',[
]);
save(opt.tempname,'opt','gs','optim','outvars')
end
f = @(A)ALTO3D_ucsim(A,opt,optim,gs);
optout = @(x,optimValues,state)
ALTO3D_out(x,optimValues,state,opt); % Pass vars to output fcn
options =
optimoptions(@fmincon,'StepTolerance',opt.minstep,'MaxIterations',opt.MaxIterat
ions,'PlotFcns',@optimplotfval,'MaxFunctionEvaluations',2*opt.MaxIterations*len
gth(optim.Ainitial),'OutputFcn',
{optout},'UseParallel',opt.parallel,'OptimalityTolerance',1e-6);
329
fprintf('Initializing Optimization at %s \n',datestr(now)); tic;
fprintf('Time \t\t Iter \t Fval \n')
[A,fval,exitflag,output] =
fmincon(f,optim.Aupper,((optim.L).*optim.en_count)',optim.Vstar,[],[],optim.Alo
wer,optim.Aupper,[],options);
% [A,fval,exitflag,output] =
fmincon(f,optim.Ainitial,[],[],[],[],optim.Alower,optim.Aupper,[],options);
fprintf('Optimization Completed at %s \n',datestr(now))
Results(run,:) = [optim.wght, fval];
time = toc;
save(opt.tempname,'time','-append')
end
case{'borg'}
optim.wght = ones(1,optim.numobjs);
if ~isempty(opt.filename)
if exist([opt.filename,'.mat'],'file')
warning(['Tried to overwrite ',opt.filename,'.mat'])
warning('Change Filename to avoid overwrite')
return
elseif exist([opt.filename],'file')
warning('Change Filename to avoid overwrite')
disp(['Tried to overwrite ',opt.filename,'.mat'])
return
else
save(opt.filename,'opt','gs','optim')
end
end
f = @(A)ALTO3D_ucsim(A,opt,optim,gs);
disp(['ModOpt ',num2str(modopt),' Vars -
',num2str(length(optim.Ainitial))])
fprintf('Initialilzing BORG Optimization at %s \n',datestr(now)); tic;
[vars, objs] =
borg(length(optim.Ainitial),optim.numobjs,1,f,opt.FuncCount,opt.epsilon,optim.A
lower,optim.Aupper)
fprintf('Optimization Terminated at %s \n',datestr(now))
time = toc;
save(opt.filename,'vars','objs','time','-append')
case{'nsga-ii'}
optim.wght = ones(1,optim.numobjs);
if ~isempty(opt.filename)
if exist([opt.filename,'.mat'],'file')
error('Change Filename to avoid overwrite')
elseif exist([opt.filename],'file')
error('Change Filename to avoid overwrite')
else
save(opt.filename,'opt','gs','optim')
end
end
f = @(A)ALTO3D_ucsim(A,opt,optim,gs);
optout = @(options,state,flag)ALTO3D_gaout(options,state,flag,opt); %
Pass vars to output fcn
options =
optimoptions(@gamultiobj,'FunctionTolerance',opt.FunctionTolerance,'MaxGenerati
330
ons',opt.MaxIterations,'MaxStallGenerations',opt.MaxStallGenerations,'ParetoFra
ction',opt.ParetoFraction,'PopulationSize',opt.PopulationSize,'UseParallel',opt
.UseParallel,'OutputFcn',{optout},'CrossoverFraction',opt.CrossoverFraction,'Us
eParallel',opt.UseParallel);
fprintf('Intializing NSGA-II Optimization at %s \n',datestr(now)); tic;
if isempty(optim.Vstar)
[x, fval, exitflag, output, population, scores] =
gamultiobj(f,length(optim.Ainitial),[],[],[],[],optim.Alower,optim.Aupper,optio
ns);
else
[x, fval, exitflag, output, population, scores] =
gamultiobj(f,length(optim.Ainitial),((optim.L).*optim.en_count)',optim.Vstar,[]
,[],optim.Alower,optim.Aupper,options);
end
fprintf('Optimization Completed at %s \n',datestr(now));
time=toc;
save(opt.filename,'x','fval','exitflag','output','population','scores','time','
-append')
end
end
Ground Structure Initialization Function
The ground structure initialization function initializes the entire ground structure
according to the optimization problem specifications. It also sets up the node-neighbor pairs for
neighboring unit cells and reads in the overlapping elements matrix.
ALTO3D_initgs.m
% Multi-objective unit cell generation
% Written by Brad Hanks
%
% This file reads all of the input information from the main file and
% initializes the ground structure accordingly. After initializing the
% ground structure, this file executes the optimization.
function [gs] = ALTO3D_initgs(opt)
% Initialize Ground Structure Geometry
ucs = opt.ucs;
gs.b = opt.gssize(1) - 1;
gs.h = opt.gssize(2) - 1;
gs.d = opt.gssize(3) - 1;
% set up nodes and coordinates
% length of truss member on a side = 1.0
nb = gs.b+1;
nh = gs.h+1;
nd = gs.d+1;
% total # of nodes
gs.N=nb*nh*nd;
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% total # of dof
gs.Ndof = 3*gs.N;
% total # of truss members = M
% connectivity array = IEN
gs.M = gs.N*(gs.N-1)/2;
gs.IEN = zeros(2,gs.M);
member=0;
for i=1:gs.N
for j=i+1:gs.N
member=member+1;
gs.IEN(1,member)=i;
gs.IEN(2,member)=j;
end
end
% coordinate array (coordinates of node numbers)
% nodes are numbered starting at the bottom left corner and go up the
% column first and then move to the next column (see example below)
%
% *3 *6 *9 *12
%
% *2 *5 *8 *11
%
% *1 *4 *7 *10
gs.coord=zeros(3,gs.N);
number=1;
for z=0:ucs(3)/gs.d:ucs(3)
for x=0:ucs(1)/gs.b:ucs(1)
for y=0:ucs(2)/gs.h:ucs(2)
gs.coord(3,number)=z;
gs.coord(2,number)=y;
gs.coord(1,number)=x;
number=number+1;
end
end
end
% calc member lengths
gs.L = zeros(gs.M,1);
for i=1:gs.M
nodeA=gs.IEN(1,i);
nodeB=gs.IEN(2,i);
xA=gs.coord(1,nodeA);
yA=gs.coord(2,nodeA);
zA=gs.coord(3,nodeA);
xB=gs.coord(1,nodeB);
yB=gs.coord(2,nodeB);
zB=gs.coord(3,nodeB);
gs.L(i,1)=sqrt((xA-xB)^2 + (yA-yB)^2 + (zA-zB)^2);
end
% ID array (maps global node numbers to equation and DOF numbers)
gs.ID=zeros(3,gs.N);
count=1;
for i=1:gs.N
for j=1:3
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gs.ID(j,i) = count;
count = count + 1;
end
end
% member orientation
gs.theta=zeros(gs.M,3);
gs.thetab=zeros(gs.M,1);
for i=1:gs.M
nodeA=gs.IEN(1,i);
nodeB=gs.IEN(2,i);
xcoordA=gs.coord(1,nodeA);
ycoordA=gs.coord(2,nodeA);
zcoordA=gs.coord(3,nodeA);
xcoordB=gs.coord(1,nodeB);
ycoordB=gs.coord(2,nodeB);
zcoordB=gs.coord(3,nodeB);
gs.theta(i,:) = [(xcoordB - xcoordA)/gs.L(i),
(ycoordB - ycoordA)/gs.L(i),
(zcoordB - zcoordA)/gs.L(i)];
gs.thetab(i) = atan2((zcoordB-zcoordA),sqrt((xcoordB - xcoordA)^2 +
(ycoordB - ycoordA)^2));
end
% LM array (location matrix) for part 1
gs.LM=zeros(6,gs.M);
for e=1:gs.M
for a=1:2
for i=1:3
p=3*(a-1)+i;
gs.LM(p,e)=gs.ID(i,gs.IEN(a,e));
end
end
end
if any([opt.objs.type] == 6)
% Calculate midpoints of all members and calculate mid2mid distances
gs.midpts = zeros(3,gs.M);
for k = 1:gs.M
gs.midpts(:,k) = 0.5*(gs.coord(:,gs.IEN(1,k)) +
gs.coord(:,gs.IEN(2,k)));
end
% Calculate distance between all midpoints
gs.middist = 10*ones(gs.M,gs.M);
for k = 1:gs.M
for l = (k+1):gs.M
gs.middist(k,l) = norm(gs.midpts(:,l)-gs.midpts(:,k));
end
end
end
% Calculate surface element area reduction
gs.surfred = ones(1,gs.M);
for i = 1:length(gs.surfred)
coordA = gs.coord(:,gs.IEN(1,i));
coordB = gs.coord(:,gs.IEN(2,i));
surface = [coordA,coordB]==[min(gs.coord,[],2),min(gs.coord,[],2)];
surface2 = [coordA,coordB]==[max(gs.coord,[],2),max(gs.coord,[],2)];
if any(and(surface(:,1),surface(:,2))) ||
any(and(surface2(:,1),surface2(:,2)))
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sizefactor = nnz(and(surface(:,1),surface(:,2))) +
nnz(and(surface2(:,1),surface2(:,2)));
gs.surfred(i) = 1/(2*sizefactor);
continue;
end
end
% Setup initial areas and limits
gs.Ainitial = opt.init;
gs.Alower = pi/4*(opt.dlim(1)^2)*ones(1,gs.M);
gs.Aupper = pi/4*(opt.dlim(2)^2)*ones(1,gs.M);
% Initialize element area
if isempty(gs.Ainitial)
gs.Ainitial = gs.Alower;
% % Short Only
% index = gs.L <=4.25;
% gs.Ainitial(index) = gs.Aupper(1);
% 4x4x4 arrangement
%
gs.Ainitial([12,75,137,198,900,947,993,1038,1532,1563,1593,1622,1908,1923,1937,
1950])=gs.Aupper(1); % ALL X
%
gs.Ainitial([3,249,479,693,891,1073,1239,1389,1523,1641,1743,1829,1899,1953,199
1,2013])=gs.Aupper(1); % ALL Y
%
gs.Ainitial([48,111,173,234,294,353,411,468,524,579,633,686,738,789,839,888])=g
s.Aupper(1); % ALL Z
% gs.Ainitial([195,1035,1619,1947]) = gs.Aupper(1); % XY Shear 1
% gs.Ainitial([15,903,1535,1911]) = gs.Aupper(1); % XY Shear 2
% gs.Ainitial([51,297,527,741]) = gs.Aupper(1); % YZ Shear 1
% gs.Ainitial([231,465,683,885]) = gs.Aupper(1); % YZ Shear 2
% gs.Ainitial([726,777,827,876]) = gs.Aupper(1); % ZX Shear 1
% gs.Ainitial([60,123,185,246]) = gs.Aupper(1); % ZX Shear 2
% 5x5x5 arrangement
% gs.Ainitial([20,510,5142,7470,7560])=gs.Aupper(1); % ALL X
% gs.Ainitial([4,2294,5674,7454,7744])=gs.Aupper(1); % ALL Y
% gs.Ainitial([100,590,1522,2390,2800])=gs.Aupper(1); % ALL Z
% gs.Ainitial([506,3206,5281,6731,7556])=gs.Aupper(1); % ALL XY
% gs.Ainitial([104,714,1299,1859,2394])=gs.Aupper(1); % ALL YZ
% gs.Ainitial([2370,2474,2577,2679,2780])=gs.Aupper(1); % ALL ZX
% gs.Ainitial = gs.Alower + (gs.Aupper(1)-gs.Alower(1)).*rand(1,gs.M);
% gs.Ainitial = pi/4*(opt.drel^2) + (gs.Aupper(1) -
pi/4*(opt.drel^2)).*rand(1,gs.M);
end
%% 4x4x4 and 5x5x5 specifics %%
% Load appropriate overlap file
if all([opt.gssize] == [4,4,4])
% Tiers
gs.tiers = [16,32,48,64];
% load appropriate overlap file
load('4x4x4overlap.mat');
gs.overlap = overlaps;
% Set up edg_cor and faces for connectivity
334
gs.edg_cor = [ 1, 4,13,16,49,52,61,64;
0, 0, 0, 0, 2,14,50,62;
0, 0, 0, 0, 3,15,51,63;
0, 0, 0, 0, 5, 8,53,56;
0, 0, 0, 0, 9,12,57,60;
0, 0, 0, 0,17,20,29,32;
0, 0, 0, 0,33,36,45,48];
gs.faces = [ 6,
7,10,11,18,19,21,24,25,28,30,31,34,35,37,40,41,44,46,47,54,55,58,59
54,55,58,59,30,31,24,21,28,25,18,19,46,47,40,37,44,41,34,35, 6,
7,10,11];
% Support neighbors Row represents the node and the values in each row
% are potential neighbor supports for that node
gs.supneighbor = [49,52,61,64;
0, 0,50,62;
0, 0,51,63;
49,52,61,64;
0, 0,53,56;
0, 0, 0,54;
0, 0, 0,55;
0, 0,53,56;
0, 0,57,60;
0, 0, 0,58;
0, 0, 0,59;
0, 0,57,60;
49,52,61,64;
0, 0,50,62;
0, 0,51,63;
49,52,61,64;
0,20,29,32;
0, 0, 0,30;
0, 0, 0,31;
0,17,29,32;
0, 0, 0,24;
0, 0, 0, 0;
0, 0, 0, 0;
0, 0, 0,21;
0, 0, 0,28;
0, 0, 0, 0;
0, 0, 0, 0;
0, 0, 0,25;
0,17,20,32;
0, 0, 0,18;
0, 0, 0,19;
0,17,20,29;
0,36,45,48;
0, 0, 0,46;
0, 0, 0,47;
0,33,45,48;
0, 0, 0,40;
0, 0, 0, 0;
0, 0, 0, 0;
0, 0, 0,37;
0, 0, 0,44;
0, 0, 0, 0;
0, 0, 0, 0;
0, 0, 0,41;
0,33,36,48;
0, 0, 0,34;
0, 0, 0,35;
335
0,33,36,45;
1, 4,13,16;
0, 0, 2,14;
0, 0, 3,15;
1, 4,13,16;
0, 0, 5, 8;
0, 0, 0, 6;
0, 0, 0, 7;
0, 0, 5, 8;
0, 0, 9,12;
0, 0, 0,10;
0, 0, 0,11;
0, 0, 9,12;
1, 4,13,16;
0, 0, 2,14;
0, 0, 3,15;
1, 4,13,16];
elseif all([opt.gssize] == [5,5,5])
% Tiers
gs.tiers = [25,50,75,100,125];
% load appropriate overlap file
load('5x5x5overlap.mat');
gs.overlap = overlaps;
% Set up edg_cor and faces for connectivity
gs.edg_cor = [1,5,21,25,101,105,121,125; % Edges and corners
0,0, 0, 0, 2,102, 22,122;
0,0, 0, 0, 3,103, 23,123;
0,0, 0, 0, 4,104, 24,124;
0,0, 0, 0, 6, 10,106,110;
0,0, 0, 0, 11, 15,111,115;
0,0, 0, 0, 16, 20,116,120;
0,0, 0, 0, 26, 30, 46, 50;
0,0, 0, 0, 51, 55, 71, 75;
0,0, 0, 0, 76, 80, 96,100];
gs.faces = [ 7:9, 12:14,
17:19,27:29,31,35,36,40,41,45,47:49,52:54,56,60,61,65,66,70,72:74,77:79,81,85,8
6,90,91,95,97:99,107:109,112:114,117:119;
107:109,112:114,117:119,47:49,35,31,40,36,45,41,27:29,72:74,60,56,65,61,70,66,5
2:54,97:99,85,81,90,86,95,91,77:79, 7:9, 12:14, 17:19]';
% Support neighbors Row represents the node and the values in each row
% are potential neighbor supports for that node
gs.supneighbor = [101,105,121,125;
0, 0,102,122;
0, 0,103,123;
0, 0,104,124;
101,105,121,125;
0, 0,106,110;
0, 0, 0,107;
0, 0, 0,108;
0, 0, 0,109;
0, 0,106,110;
0, 0,111,115;
0, 0, 0,112;
0, 0, 0,113;
0, 0, 0,114;
0, 0,111,115;
336
0, 0,116,120;
0, 0, 0,117;
0, 0, 0,118;
0, 0, 0,119;
0, 0,116,120;
101,105,121,125;
0, 0,102,122;
0, 0,103,123;
0, 0,104,124;
101,105,121,125;
0, 30, 46, 50;
0, 0, 0, 47;
0, 0, 0, 48;
0, 0, 0, 49;
0, 26, 46, 50;
0, 0, 0, 35;
0, 0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 31;
0, 0, 0, 40;
0, 0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 36;
0, 0, 0, 45;
0, 0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 41;
0, 26, 30, 50;
0, 0, 0, 27;
0, 0, 0, 28;
0, 0, 0, 29;
0, 26, 30, 46;
0, 55, 71, 75;
0, 0, 0, 72;
0, 0, 0, 73;
0, 0, 0, 74;
0, 51, 71, 75;
0, 0, 0, 60;
0, 0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 56;
0, 0, 0, 65;
0, 0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 61;
0, 0, 0, 70;
0, 0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 66;
0, 51, 55, 75;
0, 0, 0, 52;
0, 0, 0, 53;
0, 0, 0, 54;
0, 51, 55, 71;
0, 80, 96,100;
337
0, 0, 0, 97;
0, 0, 0, 98;
0, 0, 0, 99;
0, 76, 96,100;
0, 0, 0, 85;
0, 0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 81;
0, 0, 0, 90;
0, 0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 86;
0, 0, 0, 95;
0, 0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 0;
0, 0, 0, 91;
0, 76, 80,100;
0, 0, 0, 77;
0, 0, 0, 78;
0, 0, 0, 79;
0, 76, 80, 96;
1, 5, 21, 25;
0, 0, 2, 22;
0, 0, 3, 23;
0, 0, 4, 24;
1, 5, 21, 25;
0, 0, 6, 10;
0, 0, 0, 7;
0, 0, 0, 8;
0, 0, 0, 9;
0, 0, 6, 10;
0, 0, 11, 15;
0, 0, 0, 12;
0, 0, 0, 13;
0, 0, 0, 14;
0, 0, 11, 15;
0, 0, 16, 20;
0, 0, 0, 17;
0, 0, 0, 18;
0, 0, 0, 19;
0, 0, 16, 20;
1, 5, 21, 25;
0, 0, 2, 22;
0, 0, 3, 23;
0, 0, 4, 24;
1, 5, 21, 25];
else
warning('Non-standard opt.gssize , several options may be unavailable')
end
gs.removednode = false(gs.M,1);
if ~isempty(opt.removenode)
for i = 1:gs.M
if any(any(gs.IEN(:,i) == [opt.removenode]))
gs.removednode(i,1) = true;
end
end
338
end
end
Overlap Function
The overlap function is a standalone function that is run prior to the optimization
problem. It can be used to generate the overlapping elements matrix for different size cubic arrays
of elements. In general, it was run once for 4x4x4 and 5x5x5 node arrays. Running the function
creates a file that is read by the ground structure initialization function for the overlap penalty.
ALTO_overlap.m
% This code generates the binary overlap matrix offline so that it does
% need to be calculated in real time. The same 4x4x4 or 5x5x5 binary matrix
% applies for every UCS as long as the full ground structure is considered.
% Most of the options included below are arbitrary but required for
% ALTO3D_initgs.m to run
% The overlaps matrix should be saved to a file that can be loaded in
% ALTO3D_initgs.m
close all;
clear;
clc;
array = 5; % Array Size (5x5x5 or 4x4x4, etc)
opt.ucs = 2; % Unit cell size in mm
opt.diam = 1; % Beam Diameter
opt.mod = 2e5; % Modulus of elasticity N/mm^2
opt.dlim = [9.5e-5, 9.5]; % Element diameter limits [min,max] in mm
opt.drel = [0.3]; % Relevant diameter (mm) - Used to determine which
members are relevant
opt.gssize = [ array, array, array];
opt.init = pi/4*(opt.dlim(1)^2)*ones(1,((array^3)*(array^3-1)/2));
gs = ALTO3D_initgs(opt);
overlaps = false(gs.M,gs.M);
pairs = [];
solutions = [];
for i = 1:gs.M
for j = (i+1):gs.M
ptL1 = gs.coord(:,gs.IEN(1,i)); % L1 starting point
ptL2 = gs.coord(:,gs.IEN(1,j)); % L2 starting point
L1 = gs.L(i); % L1 length
L2 = gs.L(j); % L2 length
dir1 = gs.theta(i,:)'; % L1 direction unit vector
dir2 = gs.theta(j,:)'; % L2 direction unit vector
% Ax = b, solve for positions along lines where they intersect
b = ptL2 - ptL1;
339
A = [L1*dir1(1) -L2*dir2(1)
L1*dir1(2) -L2*dir2(2)
L1*dir1(3) -L2*dir2(3)];
options = optimoptions(@lsqlin,'Display','None');
x = lsqlin(A,b,[],[],[],[],[0.01,0.01],[0.99,0.99],[],options);
err = b-A*x;
% Check if result is an intersection point
if (all(abs(err) < .001))
% disp(['Found overlap',num2str(size(solutions,2)+1)])
overlaps(i,j) = true;
pairs = [pairs,[gs.IEN(:,i);gs.IEN(:,j)]];
solutions = [solutions,x];
else
continue
end
end
end
%% This section shows an example of how to evaluate whether the relevant
% members of the current state are overlapping.
A = gs.Ainitial;
A([8,20,24,55,67,75,104,106,108,110,153,161,192,196,217,219,227,250,256,275,281
,302,323,334]) = 1;
ALTO3D_plot(A,gs.IEN,gs.coord,.1)
rel_overlaps = overlaps((A>.1),(A>.1));
nnz(rel_overlaps)
Unit Cell Simulation Function
The unit cell simulation file is the function given to the optimization algorithm by the
executive function to simulate a unit cell design. It contains calls to any of the objective functions
and code to evaluate penalty functions for a unit cell design. The result of this file is the objective
function values for the optimization algorithm.
ALTO3D_ucsim.m
function [obj,constrs] = ALTO3D_ucsim(Areduced,opt,optim,gs)
% This function executes the appropriate simulations and combines the
% resulting objective functions to a single objective function value.
objtypes = [opt.objs.type];
Ain = Areduced;
340
% Rebuild A
Areduced([optim.keep]) = Areduced;
Areduced([optim.remove]) = gs.Alower(1);
A = zeros(1,gs.M);
if isfield(optim,'en_symmap')
for i = 1:gs.M
A(i) = Areduced(optim.en_symmap(i));
end
end
A(gs.removednode) = gs.Alower(1);
A_relevant = (pi/4*(opt.drel)^2);
A_rel = A>=A_relevant;
Azero = (A < A_relevant);
A(Azero) = gs.Alower(1);
% Reduce areas for shared surface elements (ONLY USE FOR HOMOGENIZATION
% PROPERTIES)
A_surfred = A;
A_surfred(A_rel) = gs.surfred(A_rel).*(A(A_rel));
score = zeros(1,length(objtypes));
objs = zeros(1,length(objtypes));
% Apply constraint if borg
switch lower(opt.algorithm)
case{'borg'}
if ~isempty(optim.Vstar)
constrs = (A_surfred*gs.L) - optim.Vstar;
% constrs = Ain*(optim.L.*optim.en_count) - optim.Vstar;
pass = (constrs<=0);
constrs(pass) = 0;
if constrs ~= 0
obj = 1e15*ones(1,length(objtypes));
obj(3) = abs((A_surfred*gs.L)/((opt.ucs)^3) -
optim.volume.vftarget)/optim.volume.vftarget;
% constrs = 0;
return
end
else
constrs = 0;
end
end
switch lower(opt.algorithm)
case{'borg','nsga-ii'}
i = 1;
while (i <= length(objtypes))
if objtypes(i) == 1
if (i+1 <= length(objtypes))
if objtypes(i+1) == 2
opt.type = 12;
objs(i:(i+1)) = ALTO3D_compltherm(A,opt,optim,gs)';
i = i + 2;
else
opt.type = 1;
objs(i) = nonzeros(ALTO3D_compltherm(A,opt,optim,gs));
i = i + 1;
341
end
else
opt.type = 1;
objs(i) = nonzeros(ALTO3D_compltherm(A,opt,optim,gs));
i = i + 1;
end
elseif objtypes(i) == 2
if (i+1 <= length(objtypes))
if objtypes(i+1) == 1
opt.type = 12;
objs(i:(i+1)) = ALTO3D_compltherm(A,opt,optim,gs)';
i = i + 2;
else
opt.type = 2;
objs(i) = nonzeros(ALTO3D_compltherm(A,opt,optim,gs));
i = i + 1;
end
else
opt.type = 2;
objs(i) = nonzeros(ALTO3D_compltherm(A,opt,optim,gs));
i = i + 1;
end
elseif objtypes(i) == 3
if all([opt.gssize]==[4,4,4])
Eh = ALTO3D_homog(A_surfred,A,opt,optim,gs);
elseif all([opt.gssize]==[5,5,5])
Eh = ALTO3D_homog5(A_surfred,A,opt,optim,gs);
else
error('Homogenization cannot be performed with this array
size')
end
% objs(i) = sqrt(sumsqr((optim.homog.target -
Eh)./optim.homog.target)); % Normalized
% objs(i) = sqrt(sumsqr((optim.homog.target - Eh))); % NOT
normalized
a = abs(((optim.homog.target - Eh)./optim.homog.target));
objs(i) = sum(a(a<inf));
i = i + 1;
elseif objtypes(i) == 4
if optim.volume.vftarget > 0
objs(i) = abs((A_surfred*gs.L)/((opt.ucs)^3) -
optim.volume.vftarget)/optim.volume.vftarget;
i = i + 1;
elseif abs(optim.volume.vftarget) <= 1e-10
objs(i) = A_surfred*gs.L;
i = i + 1;
end
elseif objtypes(i) == 5
objs(i) = nnz(A_rel);
i = i + 1;
elseif objtypes(i) == 6
A_red = A(A_rel); %
r_rel = (A_red/pi).^(.5);
middist_red = gs.middist((A_rel),(A_rel));
middist_adj = middist_red;
for k = 1:length(A_red)
for l = (k+1):length(A_red)
middist_adj(k,l) = middist_red(k,l)-
(r_rel(k)+r_rel(l));
end
end
342
if optim.powder.type == 1
objs(i) = mean(mean(middist_adj(middist_adj<900)));
elseif optim.powder.type == 2
objs(i) = min(min(middist_adj(middist_adj<900)));
elseif optim.powder.type == 3
objs(i) = nnz(middist_adj < optim.powder.tol);
end
i = i + 1;
end
end
% Penalties
P = [0];
if opt.Ptype == 1
elseif opt.Ptype == 2
% Overlap Penalty
rel_overlaps = gs.overlap((A_rel),(A_rel));
if nnz(rel_overlaps) > 0
P = [P,100];
end
elseif opt.Ptype == 3
% Connectivity
rel_nodes = unique(gs.IEN(:,(A_rel)'));
check1 = sum(ismember(gs.edg_cor,rel_nodes),2);
check2 = sum(ismember(gs.faces,rel_nodes),2);
if any([check1;check2]==1)
flaws = nnz([check1;check2]==1);
P = [P,100];
end
else
% Overlap Penalty
rel_overlaps = gs.overlap((A_rel),(A_rel));
if nnz(rel_overlaps) > 0
P = [P,100];
end
% Connectivity
rel_nodes = unique(gs.IEN(:,(A_rel)'));
check1 = sum(ismember(gs.edg_cor,rel_nodes),2);
check2 = sum(ismember(gs.faces,rel_nodes),2);
if any([check1;check2]==1)
flaws = nnz([check1;check2]==1);
P = [P,100];
end
end
% Support Structure Penalties
if any(opt.SPtype == [2,3])
gs.IENred = gs.IEN(:,A_rel);
horz = gs.thetab(A_rel) == 0;
343
if all([opt.gssize] == [4,4,4])
t1 = unique(gs.IENred(gs.IENred<=gs.tiers(1)))';
grounded = [];
neighbored = [];
new = unique(gs.IENred(2,(ismember(gs.IENred(1,:),t1) &
gs.IENred(2,:) > gs.tiers(1))));
if opt.SPtype == 2
for i = new
neighbored = [neighbored,gs.supneighbor(i,:)];
end
end
grounded = unique([grounded,new,neighbored(neighbored >0)]);
t2 = unique(grounded(grounded>gs.tiers(1) &
grounded<=gs.tiers(2)));
new = unique(gs.IENred(2,(ismember(gs.IENred(1,:),t2) &
gs.IENred(2,:) > gs.tiers(2))));
grounded = [grounded,new];
if opt.SPtype == 2
for i = new
neighbored = [neighbored,gs.supneighbor(i,:)];
end
end
grounded = unique([grounded,neighbored(neighbored >0)]);
t3 = unique(grounded(grounded>gs.tiers(2) &
grounded<=gs.tiers(3)));
new = unique(gs.IENred(2,(ismember(gs.IENred(1,:),t3) &
gs.IENred(2,:) > gs.tiers(3))));
grounded = [grounded,new];
if opt.SPtype == 2
for i = new
neighbored = [neighbored,gs.supneighbor(i,:)];
end
end
grounded = unique([grounded,neighbored(neighbored >0)]);
t4 = unique(grounded(grounded>gs.tiers(3) &
grounded<=gs.tiers(4)));
elseif all([opt.gssize] == [5,5,5])
t1 = unique(gs.IENred(gs.IENred<=gs.tiers(1)))';
grounded = [];
neighbored = [];
new = unique(gs.IENred(2,(ismember(gs.IENred(1,:),t1) &
gs.IENred(2,:) > gs.tiers(1))));
if opt.SPtype == 2
for i = new
neighbored = [neighbored,gs.supneighbor(i,:)];
end
end
grounded = unique([grounded,new,neighbored(neighbored >0)]);
t2 = unique(grounded(grounded>gs.tiers(1) &
grounded<=gs.tiers(2)));
new = unique(gs.IENred(2,(ismember(gs.IENred(1,:),t2) &
gs.IENred(2,:) > gs.tiers(2))));
grounded = [grounded,new];
if opt.SPtype == 2
for i = new
neighbored = [neighbored,gs.supneighbor(i,:)];
end
344
end
grounded = unique([grounded,neighbored(neighbored >0)]);
t3 = unique(grounded(grounded>gs.tiers(2) &
grounded<=gs.tiers(3)));
new = unique(gs.IENred(2,(ismember(gs.IENred(1,:),t3) &
gs.IENred(2,:) > gs.tiers(3))));
grounded = [grounded,new];
if opt.SPtype == 2
for i = new
neighbored = [neighbored,gs.supneighbor(i,:)];
end
end
grounded = unique([grounded,neighbored(neighbored >0)]);
t4 = unique(grounded(grounded>gs.tiers(3) &
grounded<=gs.tiers(4)));
new = unique(gs.IENred(2,(ismember(gs.IENred(1,:),t4) &
gs.IENred(2,:) > gs.tiers(4))));
grounded = [grounded,new];
if opt.SPtype == 2
for i = new
neighbored = [neighbored,gs.supneighbor(i,:)];
end
end
grounded = unique([grounded,neighbored(neighbored >0)]);
t5 = unique(grounded(grounded>gs.tiers(4) &
grounded<=gs.tiers(5)));
end
if opt.SPtype == 2 % Neighbored
horz = gs.thetab(A_rel) == 0;
flaws = nnz(~ismember(gs.IENred(1,:),grounded)) +
nnz(~ismember(gs.IENred(2,horz),grounded));
elseif opt.SPtype == 3
if all([opt.gssize] == [4,4,4])
grounded = unique([grounded,t4-48]);
elseif all([opt.gssize] == [5,5,5])
grounded = unique([grounded,t5-100]);
end
horz = gs.thetab(A_rel) == 0;
flaws = nnz(~ismember(gs.IENred(1,:),grounded)) +
nnz(~ismember(gs.IENred(2,horz),grounded));
else
error('Unknown SPtype')
end
if flaws > 0
P = [P,100];
end
end
obj = sum(P)*ones(1,length(objtypes)) +
([opt.objs.minimize].*[opt.objs.scale].*[optim.wght]).*objs;
case{'fmincon'}
i = 1;
while (i <= length(objtypes))
if objtypes(i) == 1
Smin = optim.compl.range(1);
Smax = optim.compl.range(2);
if (i+1 <= length(objtypes))
345
if objtypes(i+1) == 2
opt.type = 12;
Condmin = optim.therm.range(1);
Condmax = optim.therm.range(2);
score(i:(i+1)) = ALTO3D_compltherm(A,opt,optim,gs)';
objs(i:(i+1)) = [(Smin - score(i))/(Smin - Smax),
(Condmin - score(i+1))/(Condmin - Condmax)];
i = i + 2;
end
else
opt.type = 1;
score(i) = nonzeros(ALTO3D_compltherm(A,opt,optim,gs));
% objs(i) = (Smin - score(i))/(Smin - Smax);
objs(i) = score(i);
i = i + 1;
end
elseif objtypes(i) == 2
Condmin = optim.therm.range(1);
Condmax = optim.therm.range(2);
if (i+1 <= length(objtypes))
if objtypes(i+1) == 1
Smin = optim.compl.range(1);
Smax = optim.compl.range(2);
opt.type = 12;
score(i:(i+1)) = ALTO3D_compltherm(A,opt,optim,gs)';
objs(i:(i+1)) = [(Smin - score(i))/(Smin - Smax),
(Condmin - score(i+1))/(Condmin - Condmax)];
i = i + 2;
end
else
opt.type = 2;
score(i) = nonzeros(ALTO3D_compltherm(A,opt,optim,gs));
objs(i) = (Condmin - score(i))/(Condmin - Condmax);
i = i + 1;
end
elseif objtypes(i) == 3
Eh = ALTO3D_homog(A_surfred,opt,optim,gs);
% objs(i) = sqrt(sumsqr((optim.homog.target -
Eh)./optim.homog.target)); % Normalized
% objs(i) = sqrt(sumsqr((optim.homog.target - Eh))); % NOT
normalized
a = abs(((optim.homog.target - Eh)./optim.homog.target));
objs(i) = sum(a(a<inf),'all');
i = i + 1;
elseif objtypes(i) == 4
target = (optim.volume.vftarget)*optim.volume.range(2);
score(i) = A*gs.L;
% objs(i) = abs(target - score(i))/target; % Normalized
% objs(i) = 1e6*abs(target - score(i))/target; % NOT Normalized
% objs(i) = abs(target - score(i))/optim.volume.range(2); %
Zero volume
objs(i) = score(i)/optim.volume.range(2);
i = i + 1;
end
end
% Calculate Penalty functions
P = 1;
if ~isempty(opt.drel)
if ~isempty(opt.Pcount)
346
ne_rel = nnz(A >= A_relevant);
P = [P,(ne_rel/gs.M)^(opt.Pcount)];
end
if ~isempty(opt.Pdmid)
P2 = ones(1,length(A));
num = gs.M;
for i = 1:length(A)
if A(i) > A_relevant/10
P2(i) = (A(i)/A_relevant);
else
P2(i) = .1;
end
end
P2 = prod(P2)^(opt.Pdmid/num);
P = [P,P2];
end
end
obj =
prod(P)*([opt.objs.minimize].*[opt.objs.scale].*[optim.wght])*objs';
if (opt.saveobj == 1)
load(opt.tempname,'outvars')
iter = size(outvars,2);
if ~isempty(outvars(iter).vol)
iter = iter + 1;
outvars(iter).Iteration = iter;
end
outvars(iter).objs = [[outvars(iter).objs];objs];
outvars(iter).P = [[outvars(iter).P];P];
save(opt.tempname,'outvars','-append')
end
end
end
Compliance and Thermal Conduction Function
The compliance and thermal conduction function computes the strain energy objective
function and the thermal conduction objective function.
ALTO3D_compltherm.m
function [objs] = ALTO3D_compltherm(A,opt,optim,gs)
del_strain = 1;
L = gs.L;
L_orig = gs.L;
objs = [0;0];
if (opt.type == 2 || opt.type == 12)
in = optim.therm.in;
out = optim.therm.out;
int = [];
for i = 1:gs.M
for j = 1:2
347
n = gs.IEN(j,i);
if ~any(n==[in,out,int])
int = [int,n];
end
end
end
countint = length(int);
num_nodes = max([int,in,out]);
rmindex = zeros(gs.M,1);
% if ~isempty(opt.removelength)
% rmindex = rmindex | ((gs.L > opt.removelength(1) & gs.L <
opt.removelength(2)));
% end
% if ~isempty(opt.buildang)
% if isfield(optim,'thetab')
% rmindex = rmindex | ((abs(gs.thetab) > pi*opt.buildang(1)/180) &
(abs(gs.thetab) < pi*opt.buildang(2)/180));
% end
% end
% Try this instead to remove any zeroed elements
if ~all(A'==gs.Alower(1))
rmindex = A'==gs.Alower(1);
end
num_eqs = countint + nnz(~rmindex);
Tfixed = zeros(num_nodes,1);
Tfixed(in) = optim.therm.T1;
Tfixed(out) = optim.therm.T2;
% map interior node number to equation number
for i = 1:countint
nn(int(i)) = i;
end
end
iter = 1;
% For coupled thermo mechanical response, adjust number of iterations to allow
iterative solution
% DO NOT USE coupled thermo mechanical response with heat source (current
source) input
while ((abs(del_strain(end)) > 1e-4) && (iter < 2))
fa = zeros(gs.Ndof,1);
if (opt.type == 2 || opt.type == 12)
% Mem contains all the member information for the thermal circuit
% [ Node1, Node2, Length, Area, Resistance]
Mem(:,1:5) = [gs.IEN',L,A',L./(opt.kcond*A')];
Mem_red = Mem(~rmindex,:);
% Initialize b vector and K matrix for a = K\b
b = zeros(num_eqs,1);
if ~isempty(opt.Iin)
b(end-(length(opt.Iin)-1):end) = opt.Iin;
end
K = zeros(num_eqs); % N2 N3 i1 i2 i3 i4 i5
348
% Populate resistor network matrix
for i = 1:nnz(~rmindex)
n1 = Mem_red(i,1);
n2 = Mem_red(i,2);
if ismember(n1,[in,out]) && ismember(n2,[in,out])
b(i) = Tfixed(n1) - Tfixed(n2);
elseif ismember(n1,[in,out]) && ismember(n2,int)
b(i) = Tfixed(n1);
K(i,nn(n2)) = 1;
elseif ismember(n1,int) && ismember(n2,int)
K(i,nn([n1,n2])) = [-1,1];
elseif ismember(n1,int) && ismember(n2,[in,out])
b(i) = -Tfixed(n2);
K(i,nn(n1)) = -1;
else
disp('Warning, unexpected node locations')
disp(['Node 1 = ',num2str(n1),' Node 2 = ',num2str(n2)])
end
for j = 1:countint
if n1 == int(j)
K(nnz(~rmindex)+j,countint+i) = 1;
end
if n2 == int(j)
K(nnz(~rmindex)+j,countint+i) = -1;
end
end
K(i,(countint+i)) = Mem_red(i,5);
end
scale = 1;
Ks = sparse(K);
% Determines how to treat singular matrix
if condest(Ks) == inf
if opt.type == 2
objs(:,iter) = [0;-1e20];
else
objs(:,iter) = [1e20;-1e20];
end
return
else
a = (1/scale)*(Ks\(scale*b));
end
[warnmsg, msgid] = lastwarn;
% Determines how to treat singular or near singular matrix
if strcmp(msgid,'MATLAB:singularMatrix') || strcmp(msgid,
'MATLAB:nearlySingularMatrix')
if opt.type == 2
objs(:,iter) = [0;-1e20];
else
objs(:,iter) = [1e20;-1e20];
end
lastwarn('')
return
end
% Populate Vector of Voltages at each Node
V = zeros(num_nodes,1);
349
for i=1:length(nn)
if nn(i) ~= 0
V(i) = a(nn(i));
end
end
V = V + Tfixed;
% Increase size of Mem to be N1, N2, Length, Area, Resistance, Temp
Drop,
% Temp Avg, Current
for i = 1:nnz(~rmindex)
Mem_red(i,6) = V(Mem_red(i,1)) - V(Mem_red(i,2));
Mem_red(i,7) = (V(Mem_red(i,1)) + V(Mem_red(i,2)))/2;
end
Mem_red(:,8) = a(countint+1:end);
Mem(~rmindex,6:8) = Mem_red(:,6:8);
total_i = 0;
for i = 1:nnz(~rmindex)
% if ismember(Mem_red(i,1),[in,int]) ||
ismember(Mem_red(i,2),[in,int])
if ismember(Mem_red(i,1),[out]) || ismember(Mem_red(i,2),[out])
total_i = total_i + abs(a(countint+i));
end
end
% Calculation of effective conductivity with in and out voltages
% Reff(iter) = (optim.therm.T1-optim.therm.T2)/(total_i);
% objs(2,iter) = 1/Reff(iter);
% Calculation of effective conductivity with input current
Reff(iter) = (mean(V)/(total_i));
objs(2,iter) = 1/Reff(iter);
%% For coupled thermo mechanical response, uncomment this section
% % Thermal Applied Forces
% fa_th = zeros(gs.Ndof,1);
% for i = 1:gs.M
% fth = opt.mod*Mem(i,4)*opt.alpha*(Mem(i,7)-optim.therm.Tref);
% dof_i = gs.LM(:,i);
% fa_th(dof_i) = fa_th(dof_i) + [-fth*gs.theta(i,:)';
% fth*gs.theta(i,:)'];
% end
% fa = fa + fa_th;
% if iter > 1
% del_Reff(iter) = (Reff(end) - Reff(end-1))/Reff(end-1)*100;
% end
end
if (opt.type == 1 || opt.type == 12)
% Applied Forces
for i = 1:size(optim.compl.loads,1)
node = optim.compl.loads(i,1);
fa((3*node-2):3*node) = fa(3*node-2:3*node) +
optim.compl.loads(i,2:4)';
end
350
% Fixed BC's
bcf = [];
for i = 1:size(optim.compl.bcs,1)
for j = 1:3
if optim.compl.bcs(i,j+1) == 0
bcf = [bcf,3*(optim.compl.bcs(i,1)-1) + j];
end
end
end
% local member stiffness = AE/L
% element stiffness matrix = k
% global stiffness matrix for part 1 = K1
K1=zeros(gs.Ndof,gs.Ndof);
dk1v=zeros(36,gs.M);
for i=1:gs.M
Cx = gs.theta(i,1);
Cy = gs.theta(i,2);
Cz = gs.theta(i,3);
lambda = [ Cx*Cx Cx*Cy Cx*Cz;
Cx*Cy Cy*Cy Cy*Cz;
Cx*Cz Cy*Cz Cz*Cz];
k = [ lambda, -lambda;
-lambda, lambda];
dk = opt.mod/L_orig(i)*k;
k = A(i)*opt.mod/L_orig(i)*k;
dk1v(:,i) = reshape(dk',[36,1]);
q = gs.LM(:,i);
K1(q,q) = K1(q,q) + k;
end
index = true(1,gs.Ndof);
index(bcf) = false;
nn2 = 1:gs.Ndof;
nn2_r = nn2(index);
fa_r = fa(index);
K1_r = K1(index,:);
K1_r = K1_r(:,index);
% solve for ua
ua_r = K1_r\fa_r;
strain(iter) = ua_r'*K1_r*ua_r;
objs(1,iter) = strain(iter);
% Update member lengths
ua = zeros(gs.Ndof,1);
ua(nn2_r) = ua_r;
ua_xy = reshape(ua,[3,gs.Ndof/3]);
coord_disp = gs.coord + ua_xy;
for i=1:gs.M
nodeA=gs.IEN(1,i);
nodeB=gs.IEN(2,i);
xA=coord_disp(1,nodeA);
yA=coord_disp(2,nodeA);
351
zA=coord_disp(3,nodeA);
xB=coord_disp(1,nodeB);
yB=coord_disp(2,nodeB);
zB=coord_disp(3,nodeB);
L(i,1)=sqrt((xA-xB)^2+(yA-yB)^2+(zA-zB)^2);
end
% % Plot deformed truss
% ALTO3D_plotdeformed(A,gs,coord_disp)
if iter > 1
del_strain(iter) = abs((strain(end) - strain(end-1))/strain(end-
1))*100;
end
end
if (opt.type == 1 || opt.type == 2)
break
end
iter = iter + 1;
end
objs = objs(:,end);
end
Homogenization Functions
Three different homogenization functions are included. The first “ALTO3D_homog.m”
is for a 4x4x4 nodal array, the second is for a 5x5x5 nodal array “ALTO3D_homog5.m”, and the
third is for updated boundary conditions, “ALTO3D_homog5v2.m”, that represent the relaxed
modification discussed in Chapter 5. In all cases, the homogenization function is called by the
unit cell simulation function and computes the homogenized constitutive matrix.
ALTO3D_homog.m
function [Eh,min2buckle] = ALTO3D_homog(A,Abuck,opt,optim,gs)
A_relevant = (pi/4*(opt.drel)^2);
A_rel = A>=A_relevant;
% local member stiffness = AE/L
% element stiffness matrix = k
% global stiffness matrix for part 1 = K1
K1=zeros(gs.Ndof,gs.Ndof);
k_all=zeros(6,6*gs.M);
dk1v=zeros(36,gs.M);
for i=1:gs.M
Cx = gs.theta(i,1);
Cy = gs.theta(i,2);
352
Cz = gs.theta(i,3);
lambda = [ Cx*Cx Cx*Cy Cx*Cz;
Cx*Cy Cy*Cy Cy*Cz;
Cx*Cz Cy*Cz Cz*Cz];
k = [ lambda, -lambda;
-lambda, lambda];
dk = opt.mod/gs.L(i)*k;
k = A(i)*opt.mod/gs.L(i)*k;
dk1v(:,i) = reshape(dk',[36,1]);
k_all(1:6,(6*i-5):(6*i)) = k;
q = gs.LM(:,i);
K1(q,q) = K1(q,q) + k;
end
%% Modify stiffness matrix for periodic BC's for each strain test
%%% Unit test strain cases 1 and 2 (axial strain), 3 (shear strain)
K1_1 = K1;
%%%% Boundary Conditons %%%%
% This code is for reducing K1 using the following equations where the
% numbers are the corresponding degree of freedom. These equations come
% from the following general periodicity equations (van der Sluis 2000):
% u_11 = u_12 - u_v4
% u_22 = u_21 - u_v1
% u_v3 = u_v2 + u_v4 - u_v1
%
% Using the general form
% a = b - c + d -or- a = b - c + d + e1 - e2
% Symmetry face nodes % CAREFUL WHICH NODES ARE CALCULATED FIRST, 14/15
% must be calculated before 2/3 can be calculated
xnodes = [18,19,20,34,35,36,50,51;
30,31,32,46,47,48,62,63;
13,13,13,13,13,13,13,13;
1, 1, 1, 1, 1, 1, 1, 1];
ynodes = [21,25,29,37,41,45,53,57;
24,28,32,40,44,48,56,60;
4, 4, 4, 4, 4, 4, 4, 4;
1, 1, 1, 1, 1, 1, 1, 1];
znodes = [ 6, 7, 8,10,11,12,14,15;
54,55,56,58,59,60,62,63;
49,49,49,49,49,49,49,49;
1, 1, 1, 1, 1, 1, 1, 1];
others = [ 2, 3, 5, 9,17,33;
14,15,53,57,20,36;
13,13,49,49, 4, 4;
1, 1, 1, 1, 1, 1];
nodes = [ xnodes, ynodes, znodes, others];
a1 = zeros(1,30);
b1 = zeros(1,30);
353
c1 = zeros(1,30);
d1 = zeros(1,30);
for i=1:length(nodes(1,:))
a1((3*i-2):3*i) = (3*nodes(1,i)-2):3*nodes(1,i);
b1((3*i-2):3*i) = (3*nodes(2,i)-2):3*nodes(2,i);
c1((3*i-2):3*i) = (3*nodes(3,i)-2):3*nodes(3,i);
d1((3*i-2):3*i) = (3*nodes(4,i)-2):3*nodes(4,i);
end
for i = 1:length(a1)
K1_1(:,b1(i)) = K1_1(:,b1(i)) + K1_1(:,a1(i));
K1_1(:,c1(i)) = K1_1(:,c1(i)) - K1_1(:,a1(i));
K1_1(:,d1(i)) = K1_1(:,d1(i)) + K1_1(:,a1(i));
end
% for i = 1:length(a1)
% K1_1(b1(i),:) = K1_1(b1(i),:) + K1_1(a1(i),:);
% end
K1_2 = K1_1;
K1_3 = K1_1;
K1_4 = K1_1;
K1_5 = K1_1;
K1_6 = K1_1;
% % Prescribed Displacements
pd1 = zeros(gs.Ndof,1);
pd2 = zeros(gs.Ndof,1);
pd3 = zeros(gs.Ndof,1);
pd4 = zeros(gs.Ndof,1);
pd5 = zeros(gs.Ndof,1);
pd6 = zeros(gs.Ndof,1);
nn = (1:gs.Ndof)';
INDEX = true(6,gs.Ndof); % All dof that will be removed
INDEX(:,a1) = false; % Set periodic dofs to be removed
PRDOF = zeros(6,gs.Ndof); % All dof with strain
% Strain (e); Free (f); Fixed (0);
e = 1;
f = 2;
%% Test 1
% Prescribed BC's
prenodes = f*ones(3,gs.N); % Start with all dof free
prenodes(:,13) = [e;0;0]; % Strain in x, else fix
prenodes(:,[1,4,49]) = repmat([0;0;0],[1,3]); % All fixed
prenodes(:,[24,28,40,44]) = repmat([f;0;f],[1,4]); % Fix y
prenodes(:,[30,31,46,47]) = repmat([e;f;f],[1,4]); % Strain x
prenodes(:,[54,55,58,59]) = repmat([f;f;0],[1,4]); % Fix z
prenodes(:,[32,48]) = repmat([e;0;f],[1,2]); % Strain x, fix y
prenodes(:,[56,60]) = repmat([f;0;0],[1,2]); % Fix y,z
prenodes(:,[62,63]) = repmat([e;f;0],[1,2]); % Strain x, fix z
% Calculated Corners
prenodes(:,16) = prenodes(:,13) + prenodes(:, 4) - prenodes(:, 1);
prenodes(:,52) = prenodes(:,49) + prenodes(:, 4) - prenodes(:, 1);
prenodes(:,61) = prenodes(:,49) + prenodes(:,13) - prenodes(:, 1);
prenodes(:,64) = prenodes(:,49) + prenodes(:,13) + prenodes(:, 4) -
2*prenodes(:,1);
354
% Generate PRDOF showing strains and index for removing prescribed dofs
for i = 1:size(prenodes,2)
for j = 1:3
dof = (3*(i-1) + j);
bc = prenodes(j,i);
if (bc == 0 || bc == e)
INDEX(1,dof) = false;
PRDOF(1,dof) = bc;
end
end
end
%Prepare Force Vector
F1 = -K1_1*(PRDOF(1,:)');
% for i = 1:length(a1)
% F1(b1(i)) = F1(b1(i)) + F1(a1(i));
% end
% Reduce vectors and matrices based on prescribed and periodic
nn1 = nn(INDEX(1,:));
F1_r = F1(INDEX(1,:)); % Reduced Force vector
K1_1r = K1_1(INDEX(1,:),:);
K1_1r = K1_1r(:,INDEX(1,:));
%% Test 2
% Prescribed BC's
prenodes = f*ones(3,gs.N); % Start with all dof free
prenodes(:,4) = [0;e;0]; % Strain in y, else fix
prenodes(:,[1,13,49]) = repmat([0;0;0],[1,3]); % Fix all
prenodes(:,[24,28,40,44]) = repmat([f;e;f],[1,4]); % Free x,z Strain y
prenodes(:,[30,31,46,47]) = repmat([0;f;f],[1,4]); % Fix x Free y,z
prenodes(:,[54,55,58,59]) = repmat([f;f;0],[1,4]); % Free x,y Fix z
prenodes(:,[32,48]) = repmat([0;e;f],[1,2]); % Fix x Strain y Free z
prenodes(:,[56,60]) = repmat([f;e;0],[1,2]); % Free x Strain y Fix z
prenodes(:,[62,63]) = repmat([0;f;0],[1,2]); % Fix x Free y Fix z
% Calculated Corners
prenodes(:,16) = prenodes(:,13) + prenodes(:, 4) - prenodes(:, 1);
prenodes(:,52) = prenodes(:,49) + prenodes(:, 4) - prenodes(:, 1);
prenodes(:,61) = prenodes(:,49) + prenodes(:,13) - prenodes(:, 1);
prenodes(:,64) = prenodes(:,49) + prenodes(:,13) + prenodes(:, 4) -
2*prenodes(:,1);
% Generate PRDOF showing strains and index for removing prescribed dofs
for i = 1:size(prenodes,2)
for j = 1:3
dof = (3*(i-1) + j);
bc = prenodes(j,i);
if (bc == 0 || bc == e)
INDEX(2,dof) = false;
PRDOF(2,dof) = bc;
end
end
end
%Prepare Force Vector
F2 = -K1_2*(PRDOF(2,:)');
% for i = 1:length(a1)
% F2(b1(i)) = F2(b1(i)) + F2(a1(i));
% end
355
% Reduce vectors and matrices based on prescribed and periodic
nn2 = nn(INDEX(2,:));
F2_r = F2(INDEX(2,:)); % Reduced Force vector
K1_2r = K1_2(INDEX(2,:),:);
K1_2r = K1_2r(:,INDEX(2,:));
%% Test 3
% Prescribed BC's
prenodes = f*ones(3,gs.N); % Start with all dof free
prenodes(:,49) = [0;0;e]; % Strain in x, else fix
prenodes(:,[1,4,13]) = repmat([0;0;0],[1,3]); % All fixed
prenodes(:,[24,28,40,44]) = repmat([f;0;f],[1,4]); % Fix y
prenodes(:,[30,31,46,47]) = repmat([0;f;f],[1,4]); % Strain x
prenodes(:,[54,55,58,59]) = repmat([f;f;e],[1,4]); % Fix z
prenodes(:,[32,48]) = repmat([0;0;f],[1,2]); % Strain x, fix y
prenodes(:,[56,60]) = repmat([f;0;e],[1,2]); % Fix y,z
prenodes(:,[62,63]) = repmat([0;f;e],[1,2]); % Strain x, fix z
% Calculated Corners
prenodes(:,16) = prenodes(:,13) + prenodes(:, 4) - prenodes(:, 1);
prenodes(:,52) = prenodes(:,49) + prenodes(:, 4) - prenodes(:, 1);
prenodes(:,61) = prenodes(:,49) + prenodes(:,13) - prenodes(:, 1);
prenodes(:,64) = prenodes(:,49) + prenodes(:,13) + prenodes(:, 4) -
2*prenodes(:,1);
% Generate PRDOF showing strains and index for removing prescribed dofs
for i = 1:size(prenodes,2)
for j = 1:3
dof = (3*(i-1) + j);
bc = prenodes(j,i);
if (bc == 0 || bc == e)
INDEX(3,dof) = false;
PRDOF(3,dof) = bc;
end
end
end
%Prepare Force Vector
F3 = -K1_3*(PRDOF(3,:)');
% for i = 1:length(a1)
% F3(b1(i)) = F3(b1(i)) + F3(a1(i));
% end
% Reduce vectors and matrices based on prescribed and periodic
nn3 = nn(INDEX(3,:));
F3_r = F3(INDEX(3,:)); % Reduced Force vector
K1_3r = K1_3(INDEX(3,:),:);
K1_3r = K1_3r(:,INDEX(3,:));
%% Test 4
% Prescribed BC's
prenodes = f*ones(3,gs.N); % Start with all dof free
prenodes(:,4) = [0;0;e]; % Strain in x, else fix
prenodes(:,[1,13,49]) = repmat([0;0;0],[1,3]); % All fixed
prenodes(:,[24,28,40,44]) = repmat([f;0;e],[1,4]); % Fix y
prenodes(:,[30,31,46,47]) = repmat([0;f;f],[1,4]); % Strain x
prenodes(:,[54,55,58,59]) = repmat([f;f;f],[1,4]); % Fix z
prenodes(:,[32,48]) = repmat([0;0;e],[1,2]); % Strain x, fix y
prenodes(:,[56,60]) = repmat([f;0;e],[1,2]); % Fix y,z
prenodes(:,[62,63]) = repmat([0;f;f],[1,2]); % Strain x, fix z
356
% Calculated Corners
prenodes(:,16) = prenodes(:,13) + prenodes(:, 4) - prenodes(:, 1);
prenodes(:,52) = prenodes(:,49) + prenodes(:, 4) - prenodes(:, 1);
prenodes(:,61) = prenodes(:,49) + prenodes(:,13) - prenodes(:, 1);
prenodes(:,64) = prenodes(:,49) + prenodes(:,13) + prenodes(:, 4) -
2*prenodes(:,1);
% Generate PRDOF showing strains and index for removing prescribed dofs
for i = 1:size(prenodes,2)
for j = 1:3
dof = (3*(i-1) + j);
bc = prenodes(j,i);
if (bc == 0 || bc == e)
INDEX(4,dof) = false;
PRDOF(4,dof) = bc;
end
end
end
%Prepare Force Vector
F4 = -K1_4*(PRDOF(4,:)');
% for i = 1:length(a1)
% F4(b1(i)) = F4(b1(i)) + F4(a1(i));
% end
% Reduce vectors and matrices based on prescribed and periodic
nn4 = nn(INDEX(4,:));
F4_r = F4(INDEX(4,:)); % Reduced Force vector
K1_4r = K1_4(INDEX(4,:),:);
K1_4r = K1_4r(:,INDEX(4,:));
%% Test 5
% Prescribed BC's
prenodes = f*ones(3,gs.N); % Start with all dof free
prenodes(:,49) = [e;0;0]; % Strain in x, else fix
prenodes(:,[1,4,13]) = repmat([0;0;0],[1,3]); % All fixed
prenodes(:,[24,28,40,44]) = repmat([f;0;f],[1,4]); % Fix y
prenodes(:,[30,31,46,47]) = repmat([f;f;f],[1,4]); % Strain x
prenodes(:,[54,55,58,59]) = repmat([e;f;0],[1,4]); % Fix z
prenodes(:,[32,48]) = repmat([f;0;f],[1,2]); % Strain x, fix y
prenodes(:,[56,60]) = repmat([e;0;0],[1,2]); % Fix y,z
prenodes(:,[62,63]) = repmat([e;f;0],[1,2]); % Strain x, fix z
% Calculated Corners
prenodes(:,16) = prenodes(:,13) + prenodes(:, 4) - prenodes(:, 1);
prenodes(:,52) = prenodes(:,49) + prenodes(:, 4) - prenodes(:, 1);
prenodes(:,61) = prenodes(:,49) + prenodes(:,13) - prenodes(:, 1);
prenodes(:,64) = prenodes(:,49) + prenodes(:,13) + prenodes(:, 4) -
2*prenodes(:,1);
% Generate PRDOF showing strains and index for removing prescribed dofs
for i = 1:size(prenodes,2)
for j = 1:3
dof = (3*(i-1) + j);
bc = prenodes(j,i);
if (bc == 0 || bc == e)
INDEX(5,dof) = false;
PRDOF(5,dof) = bc;
end
end
357
end
%Prepare Force Vector
F5 = -K1_5*(PRDOF(5,:)');
% for i = 1:length(a1)
% F5(b1(i)) = F5(b1(i)) + F5(a1(i));
% end
% Reduce vectors and matrices based on prescribed and periodic
nn5 = nn(INDEX(5,:));
F5_r = F5(INDEX(5,:)); % Reduced Force vector
K1_5r = K1_5(INDEX(5,:),:);
K1_5r = K1_5r(:,INDEX(5,:));
%% Test 6
% Prescribed BC's
prenodes = f*ones(3,gs.N); % Start with all dof free
prenodes(:,13) = [0;e;0]; % Strain in x, else fix
prenodes(:,[1,4,49]) = repmat([0;0;0],[1,3]); % All fixed
prenodes(:,[24,28,40,44]) = repmat([f;f;f],[1,4]); % Fix y
prenodes(:,[30,31,46,47]) = repmat([0;e;f],[1,4]); % Strain x
prenodes(:,[54,55,58,59]) = repmat([f;f;0],[1,4]); % Fix z
prenodes(:,[32,48]) = repmat([0;e;f],[1,2]); % Strain x, fix y
prenodes(:,[56,60]) = repmat([f;f;0],[1,2]); % Fix y,z
prenodes(:,[62,63]) = repmat([0;e;0],[1,2]); % Strain x, fix z
% Calculated Corners
prenodes(:,16) = prenodes(:,13) + prenodes(:, 4) - prenodes(:, 1);
prenodes(:,52) = prenodes(:,49) + prenodes(:, 4) - prenodes(:, 1);
prenodes(:,61) = prenodes(:,49) + prenodes(:,13) - prenodes(:, 1);
prenodes(:,64) = prenodes(:,49) + prenodes(:,13) + prenodes(:, 4) -
2*prenodes(:,1);
% Generate PRDOF showing strains and index for removing prescribed dofs
for i = 1:size(prenodes,2)
for j = 1:3
dof = (3*(i-1) + j);
bc = prenodes(j,i);
if (bc == 0 || bc == e)
INDEX(6,dof) = false;
PRDOF(6,dof) = bc;
end
end
end
%Prepare Force Vector
F6 = -K1_6*(PRDOF(6,:)');
% for i = 1:length(a1)
% F6(b1(i)) = F6(b1(i)) + F6(a1(i));
% end
% Reduce vectors and matrices based on prescribed and periodic
nn6 = nn(INDEX(6,:));
F6_r = F6(INDEX(6,:)); % Reduced Force vector
K1_6r = K1_6(INDEX(6,:),:);
K1_6r = K1_6r(:,INDEX(6,:));
%% Solve all test cases and rebuild displacement vectors
% Solve FEA
ua1_r = K1_1r\F1_r;
358
ua2_r = K1_2r\F2_r;
ua3_r = K1_3r\F3_r;
ua4_r = K1_4r\F4_r;
ua5_r = K1_5r\F5_r;
ua6_r = K1_6r\F6_r;
% Fill in prescribed/calculated dof
UA = [PRDOF(1,:);
PRDOF(2,:);
PRDOF(3,:);
PRDOF(4,:);
PRDOF(5,:);
PRDOF(6,:)];
% Fill in solved dof
UA(1,nn1) = ua1_r;
UA(2,nn2) = ua2_r;
UA(3,nn3) = ua3_r;
UA(4,nn4) = ua4_r;
UA(5,nn5) = ua5_r;
UA(6,nn6) = ua6_r;
% Calculate and fill in periodic displacements
for i = 1:length(a1)
UA(:,a1(i)) = UA(:,b1(i)) - UA(:,c1(i)) + UA(:,d1(i));
end
% Reshape to xy solutions
UA_XY = [reshape(UA(1,:),[3,gs.Ndof/3]);
reshape(UA(2,:),[3,gs.Ndof/3]);
reshape(UA(3,:),[3,gs.Ndof/3]);
reshape(UA(4,:),[3,gs.Ndof/3]);
reshape(UA(5,:),[3,gs.Ndof/3]);
reshape(UA(6,:),[3,gs.Ndof/3])];
%% Calculate Element Mutual Energies
D = zeros(36,gs.M);
% Prepare induced displacement matrices
for i = 1:gs.M
nodeA = gs.IEN(1,i);
nodeB = gs.IEN(2,i);
D( 1:6 ,i) = [UA_XY(1:3,nodeA); UA_XY(1:3,nodeB)];
D( 7:12,i) = [UA_XY(4:6,nodeA); UA_XY(4:6,nodeB)];
D(13:18,i) = [UA_XY(7:9,nodeA); UA_XY(7:9,nodeB)];
D(19:24,i) = [UA_XY(10:12,nodeA); UA_XY(10:12,nodeB)];
D(25:30,i) = [UA_XY(13:15,nodeA); UA_XY(13:15,nodeB)];
D(31:36,i) = [UA_XY(16:18,nodeA); UA_XY(16:18,nodeB)];
end
%%%% Compute Elasticity Matrix
A_min = min(A);
A_rel = find(A>A_min);
Y = opt.ucs(1);
q = zeros(6);
for h = 1:gs.M
for i = 1:6
for j = 1:6
val = (1/Y)*(D((6*i-5):(6*i),h))'*...
k_all(:,(6*h-5):(6*h))*(D((6*j-5):(6*j),h));
359
q(i,j) = q(i,j) + val;
end
end
end
Eh = q;
%%% Calculate Buckling Load
Kbuck = 1;
F_crit = -
pi*pi*opt.mod*(Abuck(A_rel)'.*Abuck(A_rel)'/4/pi)./(Kbuck*gs.L(A_rel)).^2;
F_crit = gs.surfred(A_rel)'.*F_crit; % Account for surface
L_chng = (sum((D(16:18,A_rel)-D(13:15,A_rel)).^2).^(.5))'; % Deformed length
F_axi = Abuck(A_rel)'*opt.mod.*(L_chng)./gs.L(A_rel); % Axial force in each
member
factor2crit = -F_crit./F_axi;
F_eff = (opt.ucs(1)^2)*Eh(3,3)*(e)/opt.ucs(1);
min2buckle = F_eff*min(factor2crit(factor2crit>0));
% ALTO3D_plotdeformed(gs.Ainitial,gs,UA_XY(1:3,:)+gs.coord,A_relevant)
% ALTO3D_plotdeformed(gs.Ainitial,gs,UA_XY(4:6,:)+gs.coord,A_relevant)
% ALTO3D_plotdeformed(gs.Ainitial,gs,UA_XY(7:9,:)+gs.coord,A_relevant)
% ALTO3D_plotdeformed(gs.Ainitial,gs,UA_XY(10:12,:)+gs.coord,A_relevant)
% ALTO3D_plotdeformed(gs.Ainitial,gs,UA_XY(13:15,:)+gs.coord,A_relevant)
% ALTO3D_plotdeformed(gs.Ainitial,gs,UA_XY(16:18,:)+gs.coord,A_relevant)
end
ALTO3D_homog5.m
function [Eh,min2buckle] = ALTO3D_homog5(A,Abuck,opt,optim,gs)
A_relevant = (pi/4*(opt.drel)^2);
A_rel = A>=A_relevant;
% local member stiffness = AE/L
% element stiffness matrix = k
% global stiffness matrix for part 1 = K1
K1=zeros(gs.Ndof,gs.Ndof);
k_all=zeros(6,6*gs.M);
dk1v=zeros(36,gs.M);
for i=1:gs.M
Cx = gs.theta(i,1);
Cy = gs.theta(i,2);
Cz = gs.theta(i,3);
lambda = [ Cx*Cx Cx*Cy Cx*Cz;
Cx*Cy Cy*Cy Cy*Cz;
Cx*Cz Cy*Cz Cz*Cz];
k = [ lambda, -lambda;
-lambda, lambda];
dk = opt.mod/gs.L(i)*k;
k = A(i)*opt.mod/gs.L(i)*k;
dk1v(:,i) = reshape(dk',[36,1]);
k_all(1:6,(6*i-5):(6*i)) = k;
q = gs.LM(:,i);
360
K1(q,q) = K1(q,q) + k;
end
%% Modify stiffness matrix for periodic BC's for each strain test
%%% Unit test strain cases 1 and 2 (axial strain), 3 (shear strain)
K1_1 = K1;
%%%% Boundary Conditons %%%%
% This code is for reducing K1 using the following equations where the
% numbers are the corresponding degree of freedom. These equations come
% from the following general periodicity equations (van der Sluis 2000):
% u_11 = u_12 - u_v4
% u_22 = u_21 - u_v1
% u_v3 = u_v2 + u_v4 - u_v1
%
% Using the general form
% a = b - c + d -or- a = b - c + d + e1 - e2
% Symmetry face nodes % CAREFUL WHICH NODES ARE CALCULATED FIRST, 14/15
% must be calculated before 2/3 can be calculated
xnodes = [27,28,29,30,52,53,54,55,77,78,79, 80,102,103,104;
47,48,49,50,72,73,74,75,97,98,99,100,122,123,124;
21,21,21,21,21,21,21,21,21,21,21, 21, 21, 21, 21;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1];
ynodes = [31,36,41,46,56,61,66,71,81,86,91, 96,106,111,116;
35,40,45,50,60,65,70,75,85,90,95,100,110,115,120;
5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1];
znodes = [ 7, 8, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24;
107,108,109,110,112,113,114,115,117,118,119,120,122,123,124;
101,101,101,101,101,101,101,101,101,101,101,101,101,101,101;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1];
others = [ 2, 3, 4, 6, 11, 16,26,51,76;
22,23,24,106,111,116,30,55,80;
21,21,21,101,101,101, 5, 5, 5;
1, 1, 1, 1, 1, 1, 1, 1, 1];
nodes = [ xnodes, ynodes, znodes, others];
a1 = zeros(1,54);
b1 = zeros(1,54);
c1 = zeros(1,54);
d1 = zeros(1,54);
for i=1:length(nodes(1,:))
a1((3*i-2):3*i) = (3*nodes(1,i)-2):3*nodes(1,i);
b1((3*i-2):3*i) = (3*nodes(2,i)-2):3*nodes(2,i);
c1((3*i-2):3*i) = (3*nodes(3,i)-2):3*nodes(3,i);
d1((3*i-2):3*i) = (3*nodes(4,i)-2):3*nodes(4,i);
end
for i = 1:length(a1)
K1_1(:,b1(i)) = K1_1(:,b1(i)) + K1_1(:,a1(i));
K1_1(:,c1(i)) = K1_1(:,c1(i)) - K1_1(:,a1(i));
361
K1_1(:,d1(i)) = K1_1(:,d1(i)) + K1_1(:,a1(i));
end
% for i = 1:length(a1)
% K1_1(b1(i),:) = K1_1(b1(i),:) + K1_1(a1(i),:); % MISTAKE? %
% end
K1_2 = K1_1;
K1_3 = K1_1;
K1_4 = K1_1;
K1_5 = K1_1;
K1_6 = K1_1;
% % Prescribed Displacements
pd1 = zeros(gs.Ndof,1);
pd2 = zeros(gs.Ndof,1);
pd3 = zeros(gs.Ndof,1);
pd4 = zeros(gs.Ndof,1);
pd5 = zeros(gs.Ndof,1);
pd6 = zeros(gs.Ndof,1);
nn = (1:gs.Ndof)';
INDEX = true(6,gs.Ndof); % All dof that will be removed
INDEX(:,a1) = false; % Set periodic dofs to be removed
PRDOF = zeros(6,gs.Ndof); % All dof with strain
% Strain (e); Free (f); Fixed (0);
e = 1;
f = 2;
bctype = 1; % 1 is original, 2 is modified but not periodic corners
%% Test 1
% Prescribed BC's
if bctype == 1
prenodes = f*ones(3,gs.N); % Start with all dof free
prenodes(:,21) = [e;0;0]; % Strain in x, else fix
prenodes(:,[1,5,101]) = repmat([0;0;0],[1,3]); % All fixed
prenodes(:,[ 35, 40, 45, 60, 65, 70, 85, 90, 95]) = repmat([f;0;f],[1,9]); %
Fix y
prenodes(:,[ 47, 48, 49, 72, 73, 74, 97, 98, 99]) = repmat([e;f;f],[1,9]); %
Strain x
prenodes(:,[107,108,109,112,113,114,117,118,119]) = repmat([f;f;0],[1,9]); %
Fix z
prenodes(:,[ 50, 75,100]) = repmat([e;0;f],[1,3]); % Strain x, fix y
prenodes(:,[110,115,120]) = repmat([f;0;0],[1,3]); % Fix y,z
prenodes(:,[122,123,124]) = repmat([e;f;0],[1,3]); % Strain x, fix z
else
% Prescribed BC's NEW
prenodes = f*ones(3,gs.N); % Start with all dof free
prenodes(:,21) = [e;0;0]; % Strain in x, else fix
prenodes(:,1) = [0;0;0]; % All fixed
prenodes(:,5) = [0;f;0]; % All fixed
prenodes(:,101) = [0;0;f]; % All fixed
prenodes(:,[ 35, 40, 45, 60, 65, 70, 85, 90, 95]) = repmat([f;f;f],[1,9]); %
Fix y
prenodes(:,[ 47, 48, 49, 72, 73, 74, 97, 98, 99]) = repmat([e;f;f],[1,9]); %
Strain x
prenodes(:,[107,108,109,112,113,114,117,118,119]) = repmat([f;f;f],[1,9]); %
Fix z
prenodes(:,[ 50, 75,100]) = repmat([e;f;f],[1,3]); % Strain x, fix y
prenodes(:,[110,115,120]) = repmat([f;f;f],[1,3]); % Fix y,z
prenodes(:,[122,123,124]) = repmat([e;f;f],[1,3]); % Strain x, fix z
362
end
% Calculated Corners
prenodes(:,25) = prenodes(:, 21) + prenodes(:, 5) - prenodes(:, 1);
prenodes(:,105) = prenodes(:,101) + prenodes(:, 5) - prenodes(:, 1);
prenodes(:,121) = prenodes(:,101) + prenodes(:,21) - prenodes(:, 1);
prenodes(:,125) = prenodes(:,101) + prenodes(:,21) + prenodes(:, 5) -
2*prenodes(:,1);
% Generate PRDOF showing strains and index for removing prescribed dofs
for i = 1:size(prenodes,2)
for j = 1:3
dof = (3*(i-1) + j);
bc = prenodes(j,i);
if (bc == 0 || bc == e)
INDEX(1,dof) = false;
PRDOF(1,dof) = bc;
end
end
end
%Prepare Force Vector
F1 = -K1_1*(PRDOF(1,:)');
% for i = 1:length(a1) % MISTAKE? %
% F1(b1(i)) = F1(b1(i)) + F1(a1(i));
% end
% Reduce vectors and matrices based on prescribed and periodic
nn1 = nn(INDEX(1,:));
F1_r = F1(INDEX(1,:)); % Reduced Force vector
K1_1r = K1_1(INDEX(1,:),:);
K1_1r = K1_1r(:,INDEX(1,:));
%% Test 2
if bctype ==1
% Prescribed BC's
prenodes = f*ones(3,gs.N); % Start with all dof free
prenodes(:,5) = [0;e;0]; % Strain in y, else fix
prenodes(:,[1,21,101]) = repmat([0;0;0],[1,3]); % Fix all
prenodes(:,[ 35, 40, 45, 60, 65, 70, 85, 90, 95]) = repmat([f;e;f],[1,9]); %
Free x,z Strain y
prenodes(:,[ 47, 48, 49, 72, 73, 74, 97, 98, 99]) = repmat([0;f;f],[1,9]); %
Fix x Free y,z
prenodes(:,[107,108,109,112,113,114,117,118,119]) = repmat([f;f;0],[1,9]); %
Free x,y Fix z
prenodes(:,[ 50, 75,100]) = repmat([0;e;f],[1,3]); % Fix x Strain y Free
z
prenodes(:,[110,115,120]) = repmat([f;e;0],[1,3]); % Free x Strain y Fix
z
prenodes(:,[122,123,124]) = repmat([0;f;0],[1,3]); % Fix x Free y Fix z
else
% Prescribed BC's
prenodes = f*ones(3,gs.N); % Start with all dof free
prenodes(:,5) = [0;e;0]; % Strain in y, else fix
prenodes(:,1) = [0;0;0]; % All fixed
prenodes(:,21) = [f;0;0]; % All fixed
prenodes(:,101) = [0;0;f]; % All fixed
prenodes(:,[ 35, 40, 45, 60, 65, 70, 85, 90, 95]) = repmat([f;e;f],[1,9]); %
Free x,z Strain y
prenodes(:,[ 47, 48, 49, 72, 73, 74, 97, 98, 99]) = repmat([f;f;f],[1,9]); %
Fix x Free y,z
363
prenodes(:,[107,108,109,112,113,114,117,118,119]) = repmat([f;f;f],[1,9]); %
Free x,y Fix z
prenodes(:,[ 50, 75,100]) = repmat([f;e;f],[1,3]); % Fix x Strain y Free
z
prenodes(:,[110,115,120]) = repmat([f;e;f],[1,3]); % Free x Strain y Fix
z
prenodes(:,[122,123,124]) = repmat([f;f;f],[1,3]); % Fix x Free y Fix z
end
% Calculated Corners
prenodes(:,25) = prenodes(:, 21) + prenodes(:, 5) - prenodes(:, 1);
prenodes(:,105) = prenodes(:,101) + prenodes(:, 5) - prenodes(:, 1);
prenodes(:,121) = prenodes(:,101) + prenodes(:,21) - prenodes(:, 1);
prenodes(:,125) = prenodes(:,101) + prenodes(:,21) + prenodes(:, 5) -
2*prenodes(:,1);
% Generate PRDOF showing strains and index for removing prescribed dofs
for i = 1:size(prenodes,2)
for j = 1:3
dof = (3*(i-1) + j);
bc = prenodes(j,i);
if (bc == 0 || bc == e)
INDEX(2,dof) = false;
PRDOF(2,dof) = bc;
end
end
end
%Prepare Force Vector
F2 = -K1_2*(PRDOF(2,:)');
% for i = 1:length(a1) % MISTAKE? %
% F2(b1(i)) = F2(b1(i)) + F2(a1(i));
% end
% Reduce vectors and matrices based on prescribed and periodic
nn2 = nn(INDEX(2,:));
F2_r = F2(INDEX(2,:)); % Reduced Force vector
K1_2r = K1_2(INDEX(2,:),:);
K1_2r = K1_2r(:,INDEX(2,:));
%% Test 3
if bctype ==1
% Prescribed BC's
prenodes = f*ones(3,gs.N); % Start with all dof free
prenodes(:,101) = [0;0;e]; % Strain in x, else fix
prenodes(:,[1,5,21]) = repmat([0;0;0],[1,3]); % All fixed
prenodes(:,[ 35, 40, 45, 60, 65, 70, 85, 90, 95]) = repmat([f;0;f],[1,9]); %
Fix y
prenodes(:,[ 47, 48, 49, 72, 73, 74, 97, 98, 99]) = repmat([0;f;f],[1,9]); %
Strain x
prenodes(:,[107,108,109,112,113,114,117,118,119]) = repmat([f;f;e],[1,9]); %
Fix z
prenodes(:,[ 50, 75,100]) = repmat([0;0;f],[1,3]); % Strain x, fix y
prenodes(:,[110,115,120]) = repmat([f;0;e],[1,3]); % Fix y,z
prenodes(:,[122,123,124]) = repmat([0;f;e],[1,3]); % Strain x, fix z
else
% Prescribed BC's
prenodes = f*ones(3,gs.N); % Start with all dof free
prenodes(:,101) = [0;0;e]; % Strain in x, else fix
prenodes(:,1) = [0;0;0]; % All fixed
prenodes(:,5) = [0;f;0]; % All fixed
364
prenodes(:,21) = [f;0;0]; % All fixed
prenodes(:,[ 35, 40, 45, 60, 65, 70, 85, 90, 95]) = repmat([f;f;f],[1,9]); %
Fix y
prenodes(:,[ 47, 48, 49, 72, 73, 74, 97, 98, 99]) = repmat([f;f;f],[1,9]); %
Strain x
prenodes(:,[107,108,109,112,113,114,117,118,119]) = repmat([f;f;e],[1,9]); %
Fix z
prenodes(:,[ 50, 75,100]) = repmat([f;f;f],[1,3]); % Strain x, fix y
prenodes(:,[110,115,120]) = repmat([f;f;e],[1,3]); % Fix y,z
prenodes(:,[122,123,124]) = repmat([f;f;e],[1,3]); % Strain x, fix z
end
% Calculated Corners
prenodes(:,25) = prenodes(:, 21) + prenodes(:, 5) - prenodes(:, 1);
prenodes(:,105) = prenodes(:,101) + prenodes(:, 5) - prenodes(:, 1);
prenodes(:,121) = prenodes(:,101) + prenodes(:,21) - prenodes(:, 1);
prenodes(:,125) = prenodes(:,101) + prenodes(:,21) + prenodes(:, 5) -
2*prenodes(:,1);
% Generate PRDOF showing strains and index for removing prescribed dofs
for i = 1:size(prenodes,2)
for j = 1:3
dof = (3*(i-1) + j);
bc = prenodes(j,i);
if (bc == 0 || bc == e)
INDEX(3,dof) = false;
PRDOF(3,dof) = bc;
end
end
end
%Prepare Force Vector
F3 = -K1_3*(PRDOF(3,:)');
% for i = 1:length(a1) % MISTAKE? %
% F3(b1(i)) = F3(b1(i)) + F3(a1(i));
% end
% Reduce vectors and matrices based on prescribed and periodic
nn3 = nn(INDEX(3,:));
F3_r = F3(INDEX(3,:)); % Reduced Force vector
K1_3r = K1_3(INDEX(3,:),:);
K1_3r = K1_3r(:,INDEX(3,:));
%% Test 4
if bctype == 1
% Prescribed BC's
prenodes = f*ones(3,gs.N); % Start with all dof free
prenodes(:,5) = [0;0;e]; % Strain in x, else fix
prenodes(:,[1,21,101]) = repmat([0;0;0],[1,3]); % All fixed
prenodes(:,[ 35, 40, 45, 60, 65, 70, 85, 90, 95]) = repmat([f;0;e],[1,9]); %
Fix y
prenodes(:,[ 47, 48, 49, 72, 73, 74, 97, 98, 99]) = repmat([0;f;f],[1,9]); %
Strain x
prenodes(:,[107,108,109,112,113,114,117,118,119]) = repmat([f;f;f],[1,9]); %
Fix z
prenodes(:,[ 50, 75,100]) = repmat([0;0;e],[1,3]); % Strain x, fix y
prenodes(:,[110,115,120]) = repmat([f;0;e],[1,3]); % Fix y,z
prenodes(:,[122,123,124]) = repmat([0;f;f],[1,3]); % Strain x, fix z
else
% Prescribed BC's
prenodes = f*ones(3,gs.N); % Start with all dof free
365
prenodes(:,5) = [0;0;e]; % Strain in x, else fix
prenodes(:,[1,21,101]) = repmat([0;0;0],[1,3]); % All fixed
prenodes(:,[ 35, 40, 45, 60, 65, 70, 85, 90, 95]) = repmat([f;f;e],[1,9]); %
Fix y
prenodes(:,[ 47, 48, 49, 72, 73, 74, 97, 98, 99]) = repmat([f;f;f],[1,9]); %
Strain x
prenodes(:,[107,108,109,112,113,114,117,118,119]) = repmat([f;f;f],[1,9]); %
Fix z
prenodes(:,[ 50, 75,100]) = repmat([f;f;e],[1,3]); % Strain x, fix y
prenodes(:,[110,115,120]) = repmat([f;f;e],[1,3]); % Fix y,z
prenodes(:,[122,123,124]) = repmat([f;f;f],[1,3]); % Strain x, fix z
end
% Calculated Corners
prenodes(:,25) = prenodes(:, 21) + prenodes(:, 5) - prenodes(:, 1);
prenodes(:,105) = prenodes(:,101) + prenodes(:, 5) - prenodes(:, 1);
prenodes(:,121) = prenodes(:,101) + prenodes(:,21) - prenodes(:, 1);
prenodes(:,125) = prenodes(:,101) + prenodes(:,21) + prenodes(:, 5) -
2*prenodes(:,1);
% Generate PRDOF showing strains and index for removing prescribed dofs
for i = 1:size(prenodes,2)
for j = 1:3
dof = (3*(i-1) + j);
bc = prenodes(j,i);
if (bc == 0 || bc == e)
INDEX(4,dof) = false;
PRDOF(4,dof) = bc;
end
end
end
%Prepare Force Vector
F4 = -K1_4*(PRDOF(4,:)');
% for i = 1:length(a1) % MISTAKE? %
% F4(b1(i)) = F4(b1(i)) + F4(a1(i));
% end
% Reduce vectors and matrices based on prescribed and periodic
nn4 = nn(INDEX(4,:));
F4_r = F4(INDEX(4,:)); % Reduced Force vector
K1_4r = K1_4(INDEX(4,:),:);
K1_4r = K1_4r(:,INDEX(4,:));
%% Test 5
if bctype == 1
% Prescribed BC's
prenodes = f*ones(3,gs.N); % Start with all dof free
prenodes(:,101) = [e;0;0]; % Strain in x, else fix
prenodes(:,[1,5,21]) = repmat([0;0;0],[1,3]); % All fixed
prenodes(:,[ 35, 40, 45, 60, 65, 70, 85, 90, 95]) = repmat([f;0;f],[1,9]); %
Fix y
prenodes(:,[ 47, 48, 49, 72, 73, 74, 97, 98, 99]) = repmat([f;f;f],[1,9]); %
Strain x
prenodes(:,[107,108,109,112,113,114,117,118,119]) = repmat([e;f;0],[1,9]); %
Fix z
prenodes(:,[ 50, 75,100]) = repmat([f;0;f],[1,3]); % Strain x, fix y
prenodes(:,[110,115,120]) = repmat([e;0;0],[1,3]); % Fix y,z
prenodes(:,[122,123,124]) = repmat([e;f;0],[1,3]); % Strain x, fix z
else
% Prescribed BC's
366
prenodes = f*ones(3,gs.N); % Start with all dof free
prenodes(:,101) = [e;0;0]; % Strain in x, else fix
prenodes(:,[1,5,21]) = repmat([0;0;0],[1,3]); % All fixed
prenodes(:,[ 35, 40, 45, 60, 65, 70, 85, 90, 95]) = repmat([f;f;f],[1,9]); %
Fix y
prenodes(:,[ 47, 48, 49, 72, 73, 74, 97, 98, 99]) = repmat([f;f;f],[1,9]); %
Strain x
prenodes(:,[107,108,109,112,113,114,117,118,119]) = repmat([e;f;f],[1,9]); %
Fix z
prenodes(:,[ 50, 75,100]) = repmat([f;f;f],[1,3]); % Strain x, fix y
prenodes(:,[110,115,120]) = repmat([e;f;f],[1,3]); % Fix y,z
prenodes(:,[122,123,124]) = repmat([e;f;f],[1,3]); % Strain x, fix z
end
% Calculated Corners
prenodes(:,25) = prenodes(:, 21) + prenodes(:, 5) - prenodes(:, 1);
prenodes(:,105) = prenodes(:,101) + prenodes(:, 5) - prenodes(:, 1);
prenodes(:,121) = prenodes(:,101) + prenodes(:,21) - prenodes(:, 1);
prenodes(:,125) = prenodes(:,101) + prenodes(:,21) + prenodes(:, 5) -
2*prenodes(:,1);
% Generate PRDOF showing strains and index for removing prescribed dofs
for i = 1:size(prenodes,2)
for j = 1:3
dof = (3*(i-1) + j);
bc = prenodes(j,i);
if (bc == 0 || bc == e)
INDEX(5,dof) = false;
PRDOF(5,dof) = bc;
end
end
end
%Prepare Force Vector
F5 = -K1_5*(PRDOF(5,:)');
% for i = 1:length(a1) % MISTAKE? %
% F5(b1(i)) = F5(b1(i)) + F5(a1(i));
% end
% Reduce vectors and matrices based on prescribed and periodic
nn5 = nn(INDEX(5,:));
F5_r = F5(INDEX(5,:)); % Reduced Force vector
K1_5r = K1_5(INDEX(5,:),:);
K1_5r = K1_5r(:,INDEX(5,:));
%% Test 6
if bctype == 1
% Prescribed BC's
prenodes = f*ones(3,gs.N); % Start with all dof free
prenodes(:,21) = [0;e;0]; % Strain in x, else fix
prenodes(:,[1,5,101]) = repmat([0;0;0],[1,3]); % All fixed
prenodes(:,[ 35, 40, 45, 60, 65, 70, 85, 90, 95]) = repmat([f;f;f],[1,9]); %
Fix y
prenodes(:,[ 47, 48, 49, 72, 73, 74, 97, 98, 99]) = repmat([0;e;f],[1,9]); %
Strain x
prenodes(:,[107,108,109,112,113,114,117,118,119]) = repmat([f;f;0],[1,9]); %
Fix z
prenodes(:,[ 50, 75,100]) = repmat([0;e;f],[1,3]); % Strain x, fix y
prenodes(:,[110,115,120]) = repmat([f;f;0],[1,3]); % Fix y,z
prenodes(:,[122,123,124]) = repmat([0;e;0],[1,3]); % Strain x, fix z
else
367
% Prescribed BC's
prenodes = f*ones(3,gs.N); % Start with all dof free
prenodes(:,21) = [0;e;0]; % Strain in x, else fix
prenodes(:,[1,5,101]) = repmat([0;0;0],[1,3]); % All fixed
prenodes(:,[ 35, 40, 45, 60, 65, 70, 85, 90, 95]) = repmat([f;f;f],[1,9]); %
Fix y
prenodes(:,[ 47, 48, 49, 72, 73, 74, 97, 98, 99]) = repmat([f;e;f],[1,9]); %
Strain x
prenodes(:,[107,108,109,112,113,114,117,118,119]) = repmat([f;f;f],[1,9]); %
Fix z
prenodes(:,[ 50, 75,100]) = repmat([f;e;f],[1,3]); % Strain x, fix y
prenodes(:,[110,115,120]) = repmat([f;f;f],[1,3]); % Fix y,z
prenodes(:,[122,123,124]) = repmat([f;e;f],[1,3]); % Strain x, fix z
end
% Calculated Corners
prenodes(:,25) = prenodes(:, 21) + prenodes(:, 5) - prenodes(:, 1);
prenodes(:,105) = prenodes(:,101) + prenodes(:, 5) - prenodes(:, 1);
prenodes(:,121) = prenodes(:,101) + prenodes(:,21) - prenodes(:, 1);
prenodes(:,125) = prenodes(:,101) + prenodes(:,21) + prenodes(:, 5) -
2*prenodes(:,1);
% Generate PRDOF showing strains and index for removing prescribed dofs
for i = 1:size(prenodes,2)
for j = 1:3
dof = (3*(i-1) + j);
bc = prenodes(j,i);
if (bc == 0 || bc == e)
INDEX(6,dof) = false;
PRDOF(6,dof) = bc;
end
end
end
%Prepare Force Vector
F6 = -K1_6*(PRDOF(6,:)');
% for i = 1:length(a1) % MISTAKE? %
% F6(b1(i)) = F6(b1(i)) + F6(a1(i));
% end
% Reduce vectors and matrices based on prescribed and periodic
nn6 = nn(INDEX(6,:));
F6_r = F6(INDEX(6,:)); % Reduced Force vector
K1_6r = K1_6(INDEX(6,:),:);
K1_6r = K1_6r(:,INDEX(6,:));
%% Solve all test cases and rebuild displacement vectors
% Solve FEA
ua1_r = K1_1r\F1_r;
ua2_r = K1_2r\F2_r;
ua3_r = K1_3r\F3_r;
ua4_r = K1_4r\F4_r;
ua5_r = K1_5r\F5_r;
ua6_r = K1_6r\F6_r;
% Fill in prescribed/calculated dof
UA = [PRDOF(1,:);
PRDOF(2,:);
PRDOF(3,:);
PRDOF(4,:);
368
PRDOF(5,:);
PRDOF(6,:)];
% Fill in solved dof
UA(1,nn1) = ua1_r;
UA(2,nn2) = ua2_r;
UA(3,nn3) = ua3_r;
UA(4,nn4) = ua4_r;
UA(5,nn5) = ua5_r;
UA(6,nn6) = ua6_r;
% Calculate and fill in periodic displacements
for i = 1:length(a1)
UA(:,a1(i)) = UA(:,b1(i)) - UA(:,c1(i)) + UA(:,d1(i));
end
% Reshape to xy solutions
UA_XY = [reshape(UA(1,:),[3,gs.Ndof/3]);
reshape(UA(2,:),[3,gs.Ndof/3]);
reshape(UA(3,:),[3,gs.Ndof/3]);
reshape(UA(4,:),[3,gs.Ndof/3]);
reshape(UA(5,:),[3,gs.Ndof/3]);
reshape(UA(6,:),[3,gs.Ndof/3])];
% Calculate new lengths
newcoord = repmat(gs.coord,6,1) + UA_XY;
for i = 1:gs.M
nodeA = gs.IEN(1,i);
nodeB = gs.IEN(2,i);
pos1 = newcoord(1:3,nodeA);
pos2 = newcoord(1:3,nodeB);
gs.Ldef(i,:) = sqrt(sum((pos2-pos1).^2));
end
%% Calculate Element Mutual Energies
D = zeros(36,gs.M);
% Prepare induced displacement matrices
for i = 1:gs.M
nodeA = gs.IEN(1,i);
nodeB = gs.IEN(2,i);
D( 1:6 ,i) = [UA_XY(1:3,nodeA); UA_XY(1:3,nodeB)];
D( 7:12,i) = [UA_XY(4:6,nodeA); UA_XY(4:6,nodeB)];
D(13:18,i) = [UA_XY(7:9,nodeA); UA_XY(7:9,nodeB)];
D(19:24,i) = [UA_XY(10:12,nodeA); UA_XY(10:12,nodeB)];
D(25:30,i) = [UA_XY(13:15,nodeA); UA_XY(13:15,nodeB)];
D(31:36,i) = [UA_XY(16:18,nodeA); UA_XY(16:18,nodeB)];
end
%%%% Compute Elasticity Matrix
A_min = min(A);
A_rel = find(A>A_min);
Y = opt.ucs(1);
q = zeros(6);
for h = 1:gs.M
for i = 1:6
for j = 1:6
val = (1/Y)*(D((6*i-5):(6*i),h))'*...
369
k_all(:,(6*h-5):(6*h))*(D((6*j-5):(6*j),h));
q(i,j) = q(i,j) + val;
end
end
end
Eh = q;
%%% Calculate Buckling Load
Kbuck = 1;
F_crit = -
pi*pi*opt.mod*(Abuck(A_rel)'.*Abuck(A_rel)'/4/pi)./(Kbuck*gs.L(A_rel)).^2;
F_crit = gs.surfred(A_rel)'.*F_crit; % Account for surface
L_chng = (sum((D(16:18,A_rel)-D(13:15,A_rel)).^2).^(.5))'; % Deformed length
F_axi = Abuck(A_rel)'*opt.mod.*(L_chng)./gs.L(A_rel); % Axial force in each
member
factor2crit = -F_crit./F_axi;
F_eff = (opt.ucs(1)^2)*Eh(3,3)*(e)/opt.ucs(1);
min2buckle = F_eff*min(factor2crit(factor2crit>0));
% ALTO3D_plotdeformed(gs.Ainitial,gs,UA_XY(1:3,:)+gs.coord,A_relevant)
% ALTO3D_plotdeformed(gs.Ainitial,gs,UA_XY(4:6,:)+gs.coord,A_relevant)
% ALTO3D_plotdeformed(gs.Ainitial,gs,UA_XY(7:9,:)+gs.coord,A_relevant)
% ALTO3D_plotdeformed(gs.Ainitial,gs,UA_XY(10:12,:)+gs.coord,A_relevant)
% ALTO3D_plotdeformed(gs.Ainitial,gs,UA_XY(13:15,:)+gs.coord,A_relevant)
% ALTO3D_plotdeformed(gs.Ainitial,gs,UA_XY(16:18,:)+gs.coord,A_relevant)
end
ALTO3D_homog5v2.m
function [Eh,min2buckle] = ALTO3D_homog5v2(A,Abuck,opt,optim,gs)
A_relevant = (pi/4*(opt.drel)^2);
A_rel = A>=A_relevant;
% local member stiffness = AE/L
% element stiffness matrix = k
% global stiffness matrix for part 1 = K1
K1=zeros(gs.Ndof,gs.Ndof);
k_all=zeros(6,6*gs.M);
dk1v=zeros(36,gs.M);
for i=1:gs.M
Cx = gs.theta(i,1);
Cy = gs.theta(i,2);
Cz = gs.theta(i,3);
lambda = [ Cx*Cx Cx*Cy Cx*Cz;
Cx*Cy Cy*Cy Cy*Cz;
Cx*Cz Cy*Cz Cz*Cz];
k = [ lambda, -lambda;
-lambda, lambda];
dk = opt.mod/gs.L(i)*k;
k = A(i)*opt.mod/gs.L(i)*k;
dk1v(:,i) = reshape(dk',[36,1]);
k_all(1:6,(6*i-5):(6*i)) = k;
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q = gs.LM(:,i);
K1(q,q) = K1(q,q) + k;
end
%% Unit Cell Periodicity constraints
K1_1 = K1;
% All node to node relations are based on this format
% NodeA - NodeB = NodeC - NodeD
% NA = NB + NC - ND
% For the stiffness matrix, in order to replace nodeA DOFs,
% col_B = col_B + col_A
% col_C = col_C + col_A
% col_D = col_D - col_A
corners = [ 25,105,121,125;
5, 5, 21,101;
21,101,101, 25;
1, 1, 1, 1];
Xedges = [ 6, 11, 16, 10, 15, 20,106,111,116;
1, 1, 1, 5, 5, 5,101,101,101;
110,115,120,110,115,120,110,115,120;
105,105,105,105,105,105,105,105,105];
Yedges = [ 2, 3, 4, 22, 23, 24,102,103,104;
1, 1, 1, 21, 21, 21,101,101,101;
122,123,124,122,123,124,122,123,124;
121,121,121,121,121,121,121,121,121];
Zedges = [ 26, 51, 76, 30, 55, 80, 46, 71, 96;
1, 1, 1, 5, 5, 5, 21, 21, 21;
50, 75,100, 50, 75,100, 50, 75,100;
25, 25, 25, 25, 25, 25, 25, 25, 25];
Xfaces = [ 27, 28, 29, 52, 53, 54, 77, 78, 79;
1, 1, 1, 1, 1, 1, 1, 1, 1;
47, 48, 49, 72, 73, 74, 97, 98, 99;
21, 21, 21, 21, 21, 21, 21, 21, 21];
Yfaces = [ 31, 36, 41, 56, 61, 66, 81, 86, 91;
1, 1, 1, 1, 1, 1, 1, 1, 1;
35, 40, 45, 60, 65, 70, 85, 90, 95;
5, 5, 5, 5, 5, 5, 5, 5, 5];
Zfaces = [ 7, 8, 9, 12, 13, 14, 17, 18, 19;
1, 1, 1, 1, 1, 1, 1, 1, 1;
107,108,109,112,113,114,117,118,119;
101,101,101,101,101,101,101,101,101];
nodes = [corners,Xedges,Yedges,Zedges,Xfaces,Yfaces,Zfaces];
for i = 1:length(nodes)
NA((3*i-2):3*i) = (3*nodes(1,i)-2):3*nodes(1,i);
NB((3*i-2):3*i) = (3*nodes(2,i)-2):3*nodes(2,i);
NC((3*i-2):3*i) = (3*nodes(3,i)-2):3*nodes(3,i);
ND((3*i-2):3*i) = (3*nodes(4,i)-2):3*nodes(4,i);
end
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for i = length(NA):-1:1
K1_1(:,NB(i)) = K1_1(:,NB(i)) + K1_1(:,NA(i));
K1_1(:,NC(i)) = K1_1(:,NC(i)) + K1_1(:,NA(i));
K1_1(:,ND(i)) = K1_1(:,ND(i)) - K1_1(:,NA(i));
end
% % Prescribed Displacements
pd1 = zeros(gs.Ndof,1);
pd2 = zeros(gs.Ndof,1);
pd3 = zeros(gs.Ndof,1);
pd4 = zeros(gs.Ndof,1);
pd5 = zeros(gs.Ndof,1);
pd6 = zeros(gs.Ndof,1);
nn = (1:gs.Ndof)';
INDEX = true(6,gs.Ndof); % All dof that will be removed
INDEX(:,NA) = false; % Set periodic dofs to be removed
PRDOF = zeros(6,gs.Ndof); % All dof with strain
% For prescribed dof's: displacement (e); Free (f); Fixed (0);
e = 1;
f = 2;
%% Case 1 - Axial X
% Prescribed DOFs
prenodes = f*ones(3,gs.N); % Start with all dof free
prenodes(:,1) = [0;0;0]; % Grounded node
prenodes(:,5) = [0;f;0]; % Free in y
prenodes(:,21) = [e;0;0]; % Disp in x
prenodes(:,101) = [0;0;f]; % Free in z
prenodes(:,[ 47, 48, 49, 72, 73, 74, 97, 98, 99]) = repmat([e;f;f],[1,9]);
prenodes(:,[ 35, 40, 45, 60, 65, 70, 85, 90, 95]) = repmat([f;f;f],[1,9]);
prenodes(:,[107,108,109,112,113,114,117,118,119]) = repmat([f;f;f],[1,9]);
prenodes(:,[110,115,120]) = repmat([f;f;f],[1,3]);
prenodes(:,[ 50, 75,100]) = repmat([e;f;f],[1,3]);
prenodes(:,[122,123,124]) = repmat([e;f;f],[1,3]);
% Generate PRDOF showing strains and index for removing prescribed dofs
for i = 1:gs.N
for j = 1:3
dof = 3*(i-1)+j;
bc = prenodes(j,i);
if (bc == 0 || bc == e)
INDEX(1,dof) = false;
PRDOF(1,dof) = bc;
end
end
end
% Prepare Force Vector
F1 = -K1_1*PRDOF(1,:)';
% Reduce vectors and matrices based on prescribed and periodic
nn1 = nn(INDEX(1,:));
F1_r = F1(INDEX(1,:)); % Reduced Force vector
K1_1r = K1_1(INDEX(1,:),:);
K1_1r = K1_1r(:,INDEX(1,:));
%% Case 2 - Axial Y
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% Prescribed BC's
prenodes = f*ones(3,gs.N);
prenodes(:,5) = [0;e;0];
prenodes(:,1) = [0;0;0];
prenodes(:,21) = [f;0;0];
prenodes(:,101) = [0;0;f];
prenodes(:,[ 35, 40, 45, 60, 65, 70, 85, 90, 95]) = repmat([f;e;f],[1,9]);
prenodes(:,[ 47, 48, 49, 72, 73, 74, 97, 98, 99]) = repmat([f;f;f],[1,9]);
prenodes(:,[107,108,109,112,113,114,117,118,119]) = repmat([f;f;f],[1,9]);
prenodes(:,[ 50, 75,100]) = repmat([f;e;f],[1,3]);
prenodes(:,[110,115,120]) = repmat([f;e;f],[1,3]);
prenodes(:,[122,123,124]) = repmat([f;f;f],[1,3]);
% Generate PRDOF showing strains and index for removing prescribed dofs
for i = 1:size(prenodes,2)
for j = 1:3
dof = (3*(i-1) + j);
bc = prenodes(j,i);
if (bc == 0 || bc == e)
INDEX(2,dof) = false;
PRDOF(2,dof) = bc;
end
end
end
%Prepare Force Vector
F2 = -K1_1*(PRDOF(2,:)');
% Reduce vectors and matrices based on prescribed and periodic
nn2 = nn(INDEX(2,:));
F2_r = F2(INDEX(2,:)); % Reduced Force vector
K1_2r = K1_1(INDEX(2,:),:);
K1_2r = K1_2r(:,INDEX(2,:));
%% Case 3 - Axial Z
% Prescribed BC's
prenodes = f*ones(3,gs.N); % Start with all dof free
prenodes(:,101) = [0;0;e]; % Strain in x, else fix
prenodes(:,1) = [0;0;0]; % All fixed
prenodes(:,5) = [0;f;0]; % All fixed
prenodes(:,21) = [f;0;0]; % All fixed
prenodes(:,[ 35, 40, 45, 60, 65, 70, 85, 90, 95]) = repmat([f;f;f],[1,9]); %
Fix y
prenodes(:,[ 47, 48, 49, 72, 73, 74, 97, 98, 99]) = repmat([f;f;f],[1,9]); %
Strain x
prenodes(:,[107,108,109,112,113,114,117,118,119]) = repmat([f;f;e],[1,9]); %
Fix z
prenodes(:,[ 50, 75,100]) = repmat([f;f;f],[1,3]); % Strain x, fix y
prenodes(:,[110,115,120]) = repmat([f;f;e],[1,3]); % Fix y,z
prenodes(:,[122,123,124]) = repmat([f;f;e],[1,3]); % Strain x, fix z
% Generate PRDOF showing strains and index for removing prescribed dofs
for i = 1:size(prenodes,2)
for j = 1:3
dof = (3*(i-1) + j);
bc = prenodes(j,i);
if (bc == 0 || bc == e)
INDEX(3,dof) = false;
PRDOF(3,dof) = bc;
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end
end
end
%Prepare Force Vector
F3 = -K1_1*(PRDOF(3,:)');
% Reduce vectors and matrices based on prescribed and periodic
nn3 = nn(INDEX(3,:));
F3_r = F3(INDEX(3,:)); % Reduced Force vector
K1_3r = K1_1(INDEX(3,:),:);
K1_3r = K1_3r(:,INDEX(3,:));
%% Case 4 - Shear YZ
% Prescribed BC's
prenodes = f*ones(3,gs.N); % Start with all dof free
prenodes(:,5) = [0;0;e];
prenodes(:,1) = [0;0;0];
prenodes(:,101) = [0;0;f];
prenodes(:,21) = [f;0;0];
prenodes(:,[ 35, 40, 45, 60, 65, 70, 85, 90, 95]) = repmat([f;f;e],[1,9]);
prenodes(:,[ 47, 48, 49, 72, 73, 74, 97, 98, 99]) = repmat([f;f;f],[1,9]);
prenodes(:,[107,108,109,112,113,114,117,118,119]) = repmat([f;f;f],[1,9]);
prenodes(:,[ 50, 75,100]) = repmat([f;f;e],[1,3]);
prenodes(:,[110,115,120]) = repmat([f;f;e],[1,3]);
prenodes(:,[122,123,124]) = repmat([f;f;f],[1,3]);
% Generate PRDOF showing strains and index for removing prescribed dofs
for i = 1:size(prenodes,2)
for j = 1:3
dof = (3*(i-1) + j);
bc = prenodes(j,i);
if (bc == 0 || bc == e)
INDEX(4,dof) = false;
PRDOF(4,dof) = bc;
end
end
end
%Prepare Force Vector
F4 = -K1_1*(PRDOF(4,:)');
% Reduce vectors and matrices based on prescribed and periodic
nn4 = nn(INDEX(4,:));
F4_r = F4(INDEX(4,:)); % Reduced Force vector
K1_4r = K1_1(INDEX(4,:),:);
K1_4r = K1_4r(:,INDEX(4,:));
%% Case 5 - Shear
% Prescribed BC's
prenodes = f*ones(3,gs.N); % Start with all dof free
prenodes(:,101) = [e;0;f]; % Strain in x, else fix
prenodes(:,1) = [0;0;0]; % All fixed
prenodes(:,5) = [0;f;0]; % All fixed
prenodes(:,21) = [f;0;0]; % All fixed
prenodes(:,[ 35, 40, 45, 60, 65, 70, 85, 90, 95]) = repmat([f;f;f],[1,9]); %
Fix y
prenodes(:,[ 47, 48, 49, 72, 73, 74, 97, 98, 99]) = repmat([f;f;f],[1,9]); %
Strain x
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prenodes(:,[107,108,109,112,113,114,117,118,119]) = repmat([e;f;f],[1,9]); %
Fix z
prenodes(:,[ 50, 75,100]) = repmat([f;f;f],[1,3]); % Strain x, fix y
prenodes(:,[110,115,120]) = repmat([e;f;f],[1,3]); % Fix y,z
prenodes(:,[122,123,124]) = repmat([e;f;f],[1,3]); % Strain x, fix z
% Generate PRDOF showing strains and index for removing prescribed dofs
for i = 1:size(prenodes,2)
for j = 1:3
dof = (3*(i-1) + j);
bc = prenodes(j,i);
if (bc == 0 || bc == e)
INDEX(5,dof) = false;
PRDOF(5,dof) = bc;
end
end
end
%Prepare Force Vector
F5 = -K1_1*(PRDOF(5,:)');
% Reduce vectors and matrices based on prescribed and periodic
nn5 = nn(INDEX(5,:));
F5_r = F5(INDEX(5,:)); % Reduced Force vector
K1_5r = K1_1(INDEX(5,:),:);
K1_5r = K1_5r(:,INDEX(5,:));
%% Case 6 - Shear
% Prescribed BC's
prenodes = f*ones(3,gs.N); % Start with all dof free
prenodes(:,21) = [f;e;0]; % Strain in x, else fix
prenodes(:,1) = [0;0;0]; % All fixed
prenodes(:,101) = [0;0;f]; % All fixed
prenodes(:,5) = [0;f;0]; % All fixed
prenodes(:,[ 35, 40, 45, 60, 65, 70, 85, 90, 95]) = repmat([f;f;f],[1,9]); %
Fix y
prenodes(:,[ 47, 48, 49, 72, 73, 74, 97, 98, 99]) = repmat([f;e;f],[1,9]); %
Strain x
prenodes(:,[107,108,109,112,113,114,117,118,119]) = repmat([f;f;f],[1,9]); %
Fix z
prenodes(:,[ 50, 75,100]) = repmat([f;e;f],[1,3]); % Strain x, fix y
prenodes(:,[110,115,120]) = repmat([f;f;f],[1,3]); % Fix y,z
prenodes(:,[122,123,124]) = repmat([f;e;f],[1,3]); % Strain x, fix z
% Generate PRDOF showing strains and index for removing prescribed dofs
for i = 1:size(prenodes,2)
for j = 1:3
dof = (3*(i-1) + j);
bc = prenodes(j,i);
if (bc == 0 || bc == e)
INDEX(6,dof) = false;
PRDOF(6,dof) = bc;
end
end
end
%Prepare Force Vector
F6 = -K1_1*(PRDOF(6,:)');
% Reduce vectors and matrices based on prescribed and periodic
375
nn6 = nn(INDEX(6,:));
F6_r = F6(INDEX(6,:)); % Reduced Force vector
K1_6r = K1_1(INDEX(6,:),:);
K1_6r = K1_6r(:,INDEX(6,:));
%% Solve all test cases and rebuild displacement vectors
% Solve FEA
ua1_r = K1_1r\F1_r;
ua2_r = K1_2r\F2_r;
ua3_r = K1_3r\F3_r;
ua4_r = K1_4r\F4_r;
ua5_r = K1_5r\F5_r;
ua6_r = K1_6r\F6_r;
% Fill in prescribed/calculated dof
UA = [PRDOF(1,:);
PRDOF(2,:);
PRDOF(3,:);
PRDOF(4,:);
PRDOF(5,:);
PRDOF(6,:)];
% Fill in solved dof
UA(1,nn1) = ua1_r;
UA(2,nn2) = ua2_r;
UA(3,nn3) = ua3_r;
UA(4,nn4) = ua4_r;
UA(5,nn5) = ua5_r;
UA(6,nn6) = ua6_r;
% Calculate and fill in periodic displacements
for i = 1:length(NA)
UA(:,NA(i)) = UA(:,NB(i)) + UA(:,NC(i)) - UA(:,ND(i));
end
% Reshape to xy solutions
UA_XY = [reshape(UA(1,:),[3,gs.Ndof/3]);
reshape(UA(2,:),[3,gs.Ndof/3]);
reshape(UA(3,:),[3,gs.Ndof/3]);
reshape(UA(4,:),[3,gs.Ndof/3]);
reshape(UA(5,:),[3,gs.Ndof/3]);
reshape(UA(6,:),[3,gs.Ndof/3])];
% % Calculate new lengths
% newcoord = repmat(gs.coord,6,1) + UA_XY;
% for i = 1:gs.M
% nodeA = gs.IEN(1,i);
% nodeB = gs.IEN(2,i);
% pos1 = newcoord(1:3,nodeA);
% pos2 = newcoord(1:3,nodeB);
% gs.Ldef(i,:) = sqrt(sum((pos2-pos1).^2));
% end
%% Calculate Element Mutual Energies
D = zeros(36,gs.M);
% Prepare induced displacement matrices
for i = 1:gs.M
nodeA = gs.IEN(1,i);
nodeB = gs.IEN(2,i);
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D( 1:6 ,i) = [UA_XY(1:3,nodeA); UA_XY(1:3,nodeB)];
D( 7:12,i) = [UA_XY(4:6,nodeA); UA_XY(4:6,nodeB)];
D(13:18,i) = [UA_XY(7:9,nodeA); UA_XY(7:9,nodeB)];
D(19:24,i) = [UA_XY(10:12,nodeA); UA_XY(10:12,nodeB)];
D(25:30,i) = [UA_XY(13:15,nodeA); UA_XY(13:15,nodeB)];
D(31:36,i) = [UA_XY(16:18,nodeA); UA_XY(16:18,nodeB)];
end
%%%% Compute Elasticity Matrix
A_min = min(A);
A_rel = find(A>A_min);
Y = opt.ucs(1);
q = zeros(6);
for h = 1:gs.M
for i = 1:6
for j = 1:6
val = (1/Y)*(D((6*i-5):(6*i),h))'*...
k_all(:,(6*h-5):(6*h))*(D((6*j-5):(6*j),h));
q(i,j) = q(i,j) + val;
end
end
end
Eh = q;
%%% Calculate Buckling Load
Kbuck = 1;
F_crit = -
pi*pi*opt.mod*(Abuck(A_rel)'.*Abuck(A_rel)'/4/pi)./(Kbuck*gs.L(A_rel)).^2;
F_crit = gs.surfred(A_rel)'.*F_crit; % Account for surface
L_chng = (sum((D(16:18,A_rel)-D(13:15,A_rel)).^2).^(.5))'; % Deformed length
F_axi = Abuck(A_rel)'*opt.mod.*(L_chng)./gs.L(A_rel); % Axial force in each
member
factor2crit = -F_crit./F_axi;
F_eff = (opt.ucs(1)^2)*Eh(3,3)*(e)/opt.ucs(1);
min2buckle = F_eff*min(factor2crit(factor2crit>0));
% ALTO3D_plotdeformed(gs.Ainitial,gs,UA_XY(1:3,:)+gs.coord,A_relevant)
% ALTO3D_plotdeformed(gs.Ainitial,gs,UA_XY(4:6,:)+gs.coord,A_relevant)
% ALTO3D_plotdeformed(gs.Ainitial,gs,UA_XY(7:9,:)+gs.coord,A_relevant)
% ALTO3D_plotdeformed(gs.Ainitial,gs,UA_XY(10:12,:)+gs.coord,A_relevant)
% ALTO3D_plotdeformed(gs.Ainitial,gs,UA_XY(13:15,:)+gs.coord,A_relevant)
% ALTO3D_plotdeformed(gs.Ainitial,gs,UA_XY(16:18,:)+gs.coord,A_relevant)
end
Output Function
The output function is for the gradient-based optimization algorithm fmincon. It stores the
output information after each iteration such as the current unit cell design, the objective function
score, constraint violations, etc.
377
ALTO3D_out.m
function stop = ALTO3D_out(xreduced,optimValues,state,opt)
stop = false;
iter = optimValues.iteration;
if ~isempty(opt.tempname)
load(opt.tempname)
end
switch state
case 'init'
hold on
case 'iter'
iter = iter + 1;
xreduced([optim.keep]) = xreduced;
xreduced([optim.remove]) = gs.Alower(1);
x_all = xreduced;
if ~isempty(optim.en_symmap)
for i = 1:gs.M
x_all(i) = xreduced(optim.en_symmap(i));
end
end
outvars(iter).FuncCount = optimValues.funccount;
outvars(iter).A = x_all';
outvars(iter).vol = x_all*gs.L;
outvars(iter).fval = optimValues.fval;
outvars(iter).Amid = nnz(x_all < ((pi/4*(opt.drel)^2)*ones(1,gs.M))
& x_all > 1e6*gs.Alower);
fprintf('%s \t %-7d %.8f \n',datestr(now,13),iter,optimValues.fval)
% Save output to file
if ~isempty(opt.tempname)
save(opt.tempname,'optim','outvars','-append')
end
case 'done'
iter = iter + 2;
xreduced([optim.keep]) = xreduced;
xreduced([optim.remove]) = gs.Alower(1);
x_all = xreduced;
if ~isempty(optim.en_symmap)
for i = 1:gs.M
x_all(i) = xreduced(optim.en_symmap(i));
end
end
outvars(iter).A = x_all';
outvars(iter).vol = x_all*gs.L;
outvars(iter).fval = optimValues.fval;
if opt.saveobj == 1
outvars(iter).objs = outvars(iter-1).objs(end,:);
outvars(iter).P = outvars(iter-1).P(end,:);
end
fprintf('%s \t %-7d %.8f \n',datestr(now,13),iter,optimValues.fval)
% Save output to file
if ~isempty(opt.tempname)
378
save(opt.tempname,'optim','outvars','-append')
end
hold off
otherwise
end
end
GA Output Function
The genetic algorithm output function stores all the output information for the
evolutionary algorithm NSGA-II. At the end of each generation, it stores all the designs simulated
and their objective function scores. It also tracks the convergence or spread change.
ALTO3D_gaout.m
function [state,options,optchanged] = ALTO3D_gaout(options,state,flag,opt)
optchanged = false;
load(opt.filename)
switch flag
case {'init'}
disp(['Generation ',num2str(state.Generation),' Complete @
',datestr(now)])
outvars(state.Generation+1) = state;
save(opt.filename,'options','outvars','-append')
case {'iter'}
disp(['Generation ',num2str(state.Generation),' Complete @
',datestr(now)])
% outvars(state.Generation+1) = state;
outvars(2) = state;
save(opt.filename,'outvars','-append')
case {'done'}
end
end
nTopology Unit Cell Function
The nTopology unit cell function is for generating a file which can be directly imported
into nTop Platform as a lattice or thickened lattice. Using this function, the results of the
379
optimized can be transferred into a representation of the 3D model that is created in nTopology
software.
ALTO3D_nToplat.m
function []=ALTO3D_nToplat(A,gs,tol,name)
filename = [name,'.ltcx'];
save(filename)
fileID = fopen(filename,'w');
fprintf(fileID,'<?xml version="1.0" encoding="UTF-8"?> \n<!--comment--> \n');
fprintf(fileID,['<graph id="0" name="',name,'" units="mm" type="rnd">\n']);
fprintf(fileID,'\t<nodegroup>\n');
offset = max(max(gs.coord)); % Used to place the centroid of the unit cell at
(0,0)
scale = 1;
nodeid = 0;
for i = 1:length(A)
if A(i)>tol
r = sqrt(A(i)/pi)/scale;
nodeA = gs.IEN(1,i);
nodeB = gs.IEN(2,i);
x1 = (gs.coord(1,nodeA) - 0.5*offset)/scale;
y1 = (gs.coord(2,nodeA) - 0.5*offset)/scale;
z1 = (gs.coord(3,nodeA) - 0.5*offset)/scale;
x2 = (gs.coord(1,nodeB) - 0.5*offset)/scale;
y2 = (gs.coord(2,nodeB) - 0.5*offset)/scale;
z2 = (gs.coord(3,nodeB) - 0.5*offset)/scale;
fprintf(fileID,'\t\t<node id="%d" x="%f" y="%f" z="%f"
r="%f"/>\n',nodeid,x1,y1,z1,r);
fprintf(fileID,'\t\t<node id="%d" x="%f" y="%f" z="%f"
r="%f"/>\n',(nodeid+1),x2,y2,z2,r);
% fprintf(fileID,'\t\t<node id="%d" x="%f" y="%f"
z="%f"/>\n',nodeid,x1,y1,z1);
% fprintf(fileID,'\t\t<node id="%d" x="%f" y="%f"
z="%f"/>\n',(nodeid+1),x2,y2,z2);
nodeid = nodeid + 2;
end
end
fprintf(fileID,'\t</nodegroup>\n');
fprintf(fileID,'\t<beamgroup>\n');
beamid = 0;
nodeid = 0;
for i = 1:length(A)
if A(i)>tol
fprintf(fileID,'\t\t<beam id="%d" n1="%d"
n2="%d"/>\n',beamid,nodeid,(nodeid+1));
beamid = beamid + 1;
nodeid = nodeid + 2;
end
end
fprintf(fileID,'\t</beamgroup>\n');
380
fprintf(fileID,'</graph>\n');
fclose(fileID);
Plot Function
The plot function generates a 3D plot of the ground structure for analyzing results or
debugging.
ALTO3D_plot.m
% Plot Undeformed Truss
function ALTO3D_plot(A,IEN,coord,Arel)
% Plot Initial Truss
Amax = max(A);
Amax = 3.14;
Amin = min(A);
dim = size(coord,1);
if dim == 2
gap = 0;
x_lim = [min(coord(1,:))-gap, max(coord(1,:))+gap];
y_lim = [min(coord(2,:))-gap, max(coord(2,:))+gap];
figure
axis equal off tight
axis([x_lim y_lim])
xlabel('X Axis')
ylabel('Y Axis')
for i=1:length(A)
if (A(i) >= Arel)
w(i) = A(i)/(Amax)*10;
nodeA=IEN(1,i);
nodeB=IEN(2,i);
xA = coord(1,nodeA);
yA = coord(2,nodeA);
xB = coord(1,nodeB);
yB = coord(2,nodeB);
line([xA,xB],[yA,yB],'LineWidth',w(i))
end
end
elseif dim == 3
gap = 1;
x_lim = [min(coord(1,:))-gap, max(coord(1,:))+gap];
y_lim = [min(coord(2,:))-gap, max(coord(2,:))+gap];
z_lim = [min(coord(3,:))-gap, max(coord(3,:))+gap];
figure
axis manual equal
axis([x_lim y_lim z_lim])
381
xlabel('X Axis')
ylabel('Y Axis')
zlabel('Z Axis')
for i=1:length(A)
if (A(i) >= Arel)
w(i) = A(i)/(44.2)*10;
nodeA=IEN(1,i);
nodeB=IEN(2,i);
xA = coord(1,nodeA);
yA = coord(2,nodeA);
zA = coord(3,nodeA);
xB = coord(1,nodeB);
yB = coord(2,nodeB);
zB = coord(3,nodeB);
line([xA,xB],[yA,yB],[zA,zB],'LineWidth',w(i))
end
end
else
error(['BH:Dimension is ',dim])
end
end
Plot Deformed Function
The plot deformed function is used to generate a 3D plot of the deformed ground
structure. The inputs are the initial and deformed positions of the nodes and then it recreates all
the elements.
ALTO3D_plotdeformed.m
% Plot Deformed Truss
function ALTO3D_plotdeformed(A,gs,coord_disp,A_relevant)
dim = size(coord_disp,1);
if dim == 2
% Plot Deformed Truss
figure
axis equal off tight
xlabel('X Axis')
ylabel('Y Axis')
for j=1:2
for i=1:gs.M
if A(i)>= A_relevant
if j==1
nodeA=gs.IEN(1,i);
nodeB=gs.IEN(2,i);
xA=gs.coord(1,nodeA);
yA=gs.coord(2,nodeA);
382
xB=gs.coord(1,nodeB);
yB=gs.coord(2,nodeB);
line([xA,xB],[yA,yB],'Color','blue','LineWidth',3)
else
nodeA=gs.IEN(1,i);
nodeB=gs.IEN(2,i);
xA=coord_disp(1,nodeA);
yA=coord_disp(2,nodeA);
xB=coord_disp(1,nodeB);
yB=coord_disp(2,nodeB);
line([xA,xB],[yA,yB],'Color','red','LineWidth',3)
end
end
end
end
hold on
% plot(gs.coord(1,:),gs.coord(2,:),'*')
% plot(coord_disp(1,:),coord_disp(2,:),'*')
% % X Compression Arrows
% quiver(coord_disp(1,14)+.5,coord_disp(2,14),-
0.5,0,'Linewidth',3,'MaxHeadSize',2,'Color','red')
% quiver(coord_disp(1,15)+.5,coord_disp(2,15),-
0.5,0,'Linewidth',3,'MaxHeadSize',2,'Color','red')
% % Y Compression Arrows
% quiver(coord_disp(1, 8),coord_disp(2, 8)+0.5,0,-
0.5,'Linewidth',3,'MaxHeadSize',2,'Color','red')
% quiver(coord_disp(1,12),coord_disp(2,12)+0.5,0,-
0.5,'Linewidth',3,'MaxHeadSize',2,'Color','red')
elseif dim == 3
% Plot Deformed Truss
gap = 2;
x_lim = [min(gs.coord(1,:))-gap, max(gs.coord(1,:))+gap];
y_lim = [min(gs.coord(2,:))-gap, max(gs.coord(2,:))+gap];
z_lim = [min(gs.coord(3,:))-gap, max(gs.coord(3,:))+gap];
figure
axis manual equal
axis([x_lim y_lim z_lim])
xlabel('X Axis')
ylabel('Y Axis')
zlabel('Z Axis')
for j=1:2
for i=1:gs.M
if A(i)>= A_relevant
if j==1
nodeA=gs.IEN(1,i);
nodeB=gs.IEN(2,i);
xA=gs.coord(1,nodeA);
yA=gs.coord(2,nodeA);
zA=gs.coord(3,nodeA);
xB=gs.coord(1,nodeB);
yB=gs.coord(2,nodeB);
zB=gs.coord(3,nodeB);
line([xA,xB],[yA,yB],[zA,zB],'Color','blue','LineWidth',1)
else
nodeA=gs.IEN(1,i);
nodeB=gs.IEN(2,i);
383
xA=coord_disp(1,nodeA);
yA=coord_disp(2,nodeA);
zA=coord_disp(3,nodeA);
xB=coord_disp(1,nodeB);
yB=coord_disp(2,nodeB);
zB=coord_disp(3,nodeB);
line([xA,xB],[yA,yB],[zA,zB],'Color','red','LineWidth',1)
end
end
end
end
grid ON
hold on
plot3(gs.coord(1,:),gs.coord(2,:),gs.coord(3,:),'*')
plot3(coord_disp(1,:),coord_disp(2,:),coord_disp(3,:),'r*')
% % X Compression Arrows
% quiver(coord_disp(1,14)+.5,coord_disp(2,14),-
0.5,0,'Linewidth',3,'MaxHeadSize',2,'Color','red')
% quiver(coord_disp(1,15)+.5,coord_disp(2,15),-
0.5,0,'Linewidth',3,'MaxHeadSize',2,'Color','red')
%
% % Y Compression Arrows
% quiver(coord_disp(1, 8),coord_disp(2, 8)+0.5,0,-
0.5,'Linewidth',3,'MaxHeadSize',2,'Color','red')
% quiver(coord_disp(1,12),coord_disp(2,12)+0.5,0,-
0.5,'Linewidth',3,'MaxHeadSize',2,'Color','red')
end
Standardized Unit Cell Simulation File
The standardized unit cell simulation file is for evaluating or creating models of generic
unit cells such as the BCC, FCC, etc. It can be used to generate and check specific unit cell types
for comparison against optimized results.
CellSim.m
close all;
clear;
% clc;
opt.type = 'interlocked'; % Lattice Type
opt.ucs = [8, 8, 8]; % Unit cell size in mm
opt.diam = [sqrt(2),1.75, 2, 2.25, 2.5, 3]; % Beam Diameter
opt.mod = 1.15e5; % Modulus of elasticity N/mm^2
opt.XYtol = []; % XY plane hatch offset for no contours
opt.layer = []; % Layer height accounts for stair step by creating area
to be ellipse
opt.new = 0; % 0 homog5, 1 homog5new, 2 homog5v2
opt.dlim = [9.5e-5, 9.5]; % Element diameter limits [min,max] in mm
opt.drel = [0.1]; % Relevant diameter (mm) - Used to determine which members
are relevant
384
opt.adjarea = false; % Adjust cross section area to account for periodicity
opt.objs.type = 1;
opt.removenode = [];
prf = false; % Compute prf's?
opt.prfrel = 4;
A_relevant = (pi/4*(opt.drel)^2);
optim = struct('blank',[]);
switch lower(opt.type)
case{'sig8a'} % Correct
array = 4;
elems =
[21,205,703,851,1114,1117,1129,1159,1171,1276,1291,1329,1666,1669,1676,1695,170
4,1764,1782,1806];
case{'sig8d'} % Incorrect
array = 4;
elems =
[64,82,146,742,752,804,1074,1086,1200,1212,1252,1366,1642,1720,1761,1651,1727,1
825,1912,2014];
case{'cubic5all'}
array = 5;
elems =
[1,5,25,125,129,149,248,252,272,370,374,394,495,515,611,615,635,730,734,754,848
,852,872,965,969,989,1085,1105,1196,1200,1220,1310,1314,1334,1423,1427,1447,153
5,1539,1559,1650,1670,1756,1760,1780,1865,1869,1889,1973,1977,1997,2080,2084,21
04,2190,2210,2291,2315,2395,2419,2498,2522,2600,2624,2725,2801,2805,2825,2900,2
904,2924,2998,3002,3022,3095,3099,3119,3195,3215,3286,3290,3310,3380,3384,3404,
3473,3477,3497,3565,3569,3589,3660,3680,3746,3750,3770,3835,3839,3859,3923,3927
,3947,4010,4014,4034,4100,4120,4181,4185,4205,4265,4269,4289,4348,4352,4372,443
0,4434,4454,4515,4535,4591,4615,4670,4694,4748,4772,4825,4849,4925,4976,4980,50
00,5050,5054,5074,5123,5127,5147,5195,5199,5219,5270,5290,5336,5340,5360,5405,5
409,5429,5473,5477,5497,5540,5544,5564,5610,5630,5671,5675,5695,5735,5739,5759,
5798,5802,5822,5860,5864,5884,5925,5945,5981,5985,6005,6040,6044,6064,6098,6102
,6122,6155,6159,6179,6215,6235,6266,6290,6320,6344,6373,6397,6425,6449,6500,652
6,6530,6550,6575,6579,6599,6623,6627,6647,6670,6674,6694,6720,6740,6761,6765,67
85,6805,6809,6829,6848,6852,6872,6890,6894,6914,6935,6955,6971,6975,6995,7010,7
014,7034,7048,7052,7072,7085,7089,7109,7125,7145,7156,7160,7180,7190,7194,7214,
7223,7227,7247,7255,7259,7279,7290,7310,7316,7340,7345,7369,7373,7397,7400,7424
,7450,7451,7455,7475,7479,7498,7502,7520,7524,7545,7561,7565,7580,7584,7598,760
2,7615,7619,7635,7646,7650,7660,7664,7673,7677,7685,7689,7700,7706,7710,7715,77
19,7723,7727,7730,7734,7740,7741,7745,7748,7750];
case{'homogerr'}
array = 5;
elems = [4987,5799,5805]; % Middle
% elems = [ 12,1424,1430]; %10,1205 % 1197
case{'homogerr2x2'}
array = 5;
% elems =
[2806,3003,3380,3383,3565,3568,3751,3928,4265,4268,4430,4433,6531,6628,6805,680
8,6890,6893,6976,7053,7190,7193,7255,7258]; % Middle
elems =
[6,253,730,733,965,968,1201,1428,1865,1868,2080,2083,4981,5128,5405,5408,5540,5
543,5676,5803,6040,6043,6155,6158];
case{'homogerr20'}
array = 5;
elems =
[6,253,730,733,965,968,1201,1428,1865,1868,2080,2083,2806,3003,3380,3383,3565,3
568,3751,3928,4265,4268,4430,4433,4981,5128,5405,5408,5540,5543,5676,5803,6040,
6043,6155,6158,6531,6628,6805,6808,6890,6893,6976,7053,7190,7193,7255,7258,7456
,7503,7580,7583,7615,7618,7651,7678,7715,7718,7730,7733];
385
case{'cubic'}
array = 4;
elems = [ 3,12,48,198,234,693,738,888,1899,1908,1950,2013];
case{'plus'}
array = 5;
elems = [ 1472,5132,5672,5799,5807,5847];
case{'plus2x2'}
array = 5;
elems =
[754,989,1889,2104,2904,3099,3286,3380,3384,3404,3473,3565,3569,3589,3839,4014,
4181,4265,4269,4289,4348,4430,4434,4454,5429,5564,6064,6179,6579,6674,6761,6805
,6809,6829,6848,6890,6894,6914,7014,7089,7156,7190,7194,7214,7223,7255,7259,727
9];
case{'cubic5'}
array = 5;
elems = [ 4,20,100,510,590,2294,2390,2800,7454,7470,7560,7744];
case{'cubic52x2'}
array = 5;
elems =
[2,10,50,249,257,297,500,540,1197,1205,1245,1424,1432,1472,1655,1695,2292,2340,
2499,2547,2750,4977,4985,5025,5124,5132,5172,5275,5315,5672,5680,5720,5799,5807
,5847,5930,5970,6267,6315,6374,6422,6525,7452,7460,7499,7507,7550,7647,7655,767
4,7682,7705,7742,7749];
case{'bcc'}
array = 5;
elems = [62,548,2332,2738,5835,5839,5855,5859];
case{'bcc2x2'}
array = 5;
elems =
[31,276,278,519,1216,1226,1441,1443,1451,1453,1664,1674,2311,2516,2518,2719,339
8,3400,3408,3410,3583,3585,3593,3595,4283,4285,4293,4295,4448,4450,4458,4460,50
06,5151,5153,5294,5691,5701,5816,5818,5826,5828,5939,5949,6286,6391,6393,6494,6
823,6825,6833,6835,6908,6910,6918,6920,7208,7210,7218,7220,7273,7275,7283,7285]
;
case{'bccz'}
array = 5;
elems = [62,100,548,590,2332,2390,2738,2800,5835,5839,5855,5859];
case{'bccz2x2'}
array = 5;
elems =
[31,50,276,278,297,519,540,1216,1226,1245,1441,1443,1451,1453,1472,1664,1674,16
95,2311,2340,2516,2518,2547,2719,2750,3398,3400,3408,3410,3583,3585,3593,3595,4
283,4285,4293,4295,4448,4450,4458,4460,5006,5025,5151,5153,5172,5294,5315,5691,
5701,5720,5816,5818,5826,5828,5847,5939,5949,5970,6286,6315,6391,6393,6422,6494
,6525,6823,6825,6833,6835,6908,6910,6918,6920,7208,7210,7218,7220,7273,7275,728
3,7285];
case{'fcc'}
array = 5;
elems =
[12,52,60,498,538,550,1430,1434,2330,2342,2740,2748,5170,5174,5710,5730,5960,59
80,6420,6424,7462,7548,7680,7684];
case{'fccnohrz'}
array = 5;
elems =
[52,60,538,550,2330,2342,2740,2748,5170,5174,5710,5730,5960,5980,6420,6424];
case{'fccz'}
array = 5;
elems =
[12,52,60,100,498,538,550,590,1430,1434,2330,2342,2390,2740,2748,2800,5170,5174
,5710,5730,5960,5980,6420,6424,7462,7548,7680,7684];
case{'fccznohrz'}
386
array = 5;
elems =
[52,60,100,538,550,590,2330,2342,2390,2740,2748,2800,5170,5174,5710,5730,5960,5
980,6420,6424];
case{'f2bcc'}
array = 5;
elems =
[12,52,60,498,538,550,1430,1434,2330,2342,2740,2748,5170,5174,5710,5730,5960,59
80,6420,6424,7462,7548,7680,7684,62,548,2332,2738,5835,5839,5855,5859];
case{'f2bccz'}
array = 5;
elems =
[62,100,548,590,2332,2390,2738,2800,5835,5839,5855,5859,12,52,60,498,538,550,14
30,1434,2330,2342,2740,2748,5170,5174,5710,5730,5960,5980,6420,6424,7462,7548,7
680,7684];
case{'f2bccnohrz'}
array = 5;
elems =
[62,548,2332,2738,5835,5839,5855,5859,52,60,538,550,2330,2342,2740,2748,5170,51
74,5710,5730,5960,5980,6420,6424];
case{'f2bccznohrz'}
array = 5;
elems =
[62,548,2332,2738,5835,5839,5855,5859,52,60,100,538,550,590,2330,2342,2390,2740
,2748,2800,5170,5174,5710,5730,5960,5980,6420,6424];
case{'column'}
array = 5;
elems = [1522];
case{'columns'}
array = 5;
elems = [100,590,1522,2390,2800];
case{'diamond'}
array = 5;
% Connected to 1
% elems =
[31,1441,1453,2719,3400,3408,4450,4458,5153,5701,5939,6391,6910,6918,7210,7218]
;
% Connected to 5
elems =
[519,1443,1451,2311,3583,3595,4283,4295,5151,5691,5949,6393,6823,6835,7273,7285
];
case{'interlocked'} % Same as 8-1 Diamond with sqrt(2) as diameter
array = 5;
% Connected to 1
% elems = [62,2738,5839,5855];
% Connected to 5
elems = [548,2332,5835,5859];
case{'interlocked2x2'} % Interlocked 2x2 homogenization
array = 5;
% All
elems =
[31,278,1226,1441,1453,1664,2516,2719,3400,3408,3585,3593,4285,4293,4450,4458,5
006,5153,5701,5816,5828,5939,6391,6494,6825,6833,6910,6918,7210,7218,7275,7283]
;
% One set
% elems =
[31,1441,1453,2719,3400,3408,4450,4458,5153,5701,5939,6391,6910,6918,7210,7218]
;
% Second set
387
% elems =
[278,1226,1664,2516,3585,3593,4285,4293,5006,5816,5828,6494,6825,6833,7275,7283
];
case{'fluorite'}
array = 5;
elems =
[31,519,1441,1443,1451,1453,2311,2719,3400,3408,3583,3595,4283,4295,4450,4458,5
151,5153,5691,5701,5939,5949,6391,6393,6823,6835,6910,6918,7210,7218,7273,7285]
;
case{'octet'}
array = 5;
elems =
[12,52,60,498,538,550,1430,1434,1462,1470,1474,1482,2330,2342,2740,2748,5130,51
34,5170,5174,5182,5682,5710,5722,5730,5928,5960,5968,5980,6412,6420,6424,7462,7
548,7680,7684];
case{'octet2x2'}
array = 5;
elems =
[6,26,30,251,253,271,273,277,494,514,520,733,735,749,753,755,759,968,970,984,98
8,990,994,1201,1215,1221,1225,1426,1428,1442,1446,1448,1452,1649,1665,1669,1675
,1868,1870,1884,1888,1890,1894,2083,2085,2099,2103,2105,2109,2310,2316,2517,252
1,2523,2720,2724,2903,2905,2923,2925,2929,3098,3100,3118,3120,3124,3291,3305,33
11,3315,3476,3478,3492,3496,3498,3502,3659,3675,3679,3685,3838,3840,3854,3858,3
860,3864,4013,4015,4029,4033,4035,4039,4186,4200,4206,4210,4351,4353,4367,4371,
4373,4377,4514,4530,4534,4540,4689,4693,4695,4844,4848,4850,4981,5001,5005,5126
,5128,5146,5148,5152,5269,5289,5295,5408,5410,5424,5428,5430,5434,5543,5545,555
9,5563,5565,5569,5676,5690,5696,5700,5801,5803,5817,5821,5823,5827,5924,5940,59
44,5950,6043,6045,6059,6063,6065,6069,6158,6160,6174,6178,6180,6184,6285,6291,6
392,6396,6398,6495,6499,6578,6580,6598,6600,6604,6673,6675,6693,6695,6699,6766,
6780,6786,6790,6851,6853,6867,6871,6873,6877,6934,6950,6954,6960,7013,7015,7029
,7033,7035,7039,7088,7090,7104,7108,7110,7114,7161,7175,7181,7185,7226,7228,724
2,7246,7248,7252,7289,7305,7309,7315,7364,7368,7370,7419,7423,7425,7456,7501,75
03,7544,7583,7585,7618,7620,7651,7676,7678,7699,7718,7720,7733,7735];
case{'truncated cube'}
array = 4;
elems =
[64,66,78,130,142,250,258,424,432,481,496,641,658,742,756,808,904,1042,1402,150
4,1537,1540,1625,1630,1838,1843,1892,1895,1912,1914,1930,1954,1984,1993,2009,20
14];
case{'kelvin'}
array = 5;
elems =
[851,853,867,1315,1333,1538,1560,2002,3021,3023,3765,3775,4115,4125,4771,4773,5
053,5075,5200,5218,5365,5635,5986,6000,6214,6230,6345,6448,6652,6996,7144,7392,
7601,7603,7665,7688];
case{'isotruss'}
array = 5;
elems =
[4,20,62,100,510,548,590,1472,2294,2332,2390,2738,2800,5132,5672,5799,5807,5835
,5839,5847,5855,5859,7454,7470,7560,7744];
case{'4cols'}
array = 5;
elems = [829,1064,1964,2179];
case{'octahedral'} % Same as Octet(8,1) with diam = 1.41421
array = 5;
elems = [1462,1470,1474,1482,5130,5134,5182,5682,5722,5928,5968,6412];
case{'octahedral2'}
array = 5;
elems =
[255,259,295,299,1207,1235,1255,1653,1685,1705,2545,2549,5027,5035,5313,5325,63
05,6317,6515,6523,7505,7509,7657,7703];
388
case{'octet2_15'} % Optioctet % Same as Octet(8,1) with perpendiculars =
1.189, diags = 1.3*Area_perps
array = 5;
elems =
[62,548,1472,2332,2738,5132,5672,5799,5807,5835,5839,5847,5855,5859];
case{'opti'} % Optioctet v2, Same as Octet(8,1) with perpendiculars =
1.189, diags = 1.3*Area_perps
array = 5;
elems =
[ 4,20,100,510,590,2294,2390,2800,7454,7470,7560,7744,62,548,2332,2738,5835,583
9,5855,5859];
case{'opti2x2'} % Optioctet 2x2 homogenization
array = 5;
elems =
[2,10,50,249,257,297,500,540,1197,1205,1245,1424,1432,1472,1655,1695,2292,2340,
2499,2547,2750,4977,4985,5025,5124,5132,5172,5275,5315,5672,5680,5720,5799,5807
,5847,5930,5970,6267,6315,6374,6422,6525,7452,7460,7499,7507,7550,7647,7655,767
4,7682,7705,7742,7749];
elems2=
[31,276,278,519,1216,1226,1441,1443,1451,1453,1664,1674,2311,2516,2518,2719,339
8,3400,3408,3410,3583,3585,3593,3595,4283,4285,4293,4295,4448,4450,4458,4460,50
06,5151,5153,5294,5691,5701,5816,5818,5826,5828,5939,5949,6286,6391,6393,6494,6
823,6825,6833,6835,6908,6910,6918,6920,7208,7210,7218,7220,7273,7275,7283,7285]
;
case{'column2x2'}
array = 5;
elems = [779,1014,1914,2129,5454,5589,6089,6204];
otherwise
error('Unknown element type')
end
%%
opt.gssize = [ array, array, array];
index = 1;
for i = [opt.diam]
for j = [opt.ucs(1)]
opt.diam = i;
opt.ucs = opt.ucs;
opt.init = pi/4*(opt.dlim(1)^2)*ones(1,((array^3)*(array^3-1)/2));
opt.init(elems) = pi/4*(opt.diam^2);
% Change some elems
if opt.type == "octet2_15"
% opt.init(elems([1,2,4,5,10,11,13,14])) = 1.3*pi/4*(opt.diam^2); %
For octet2_15
opt.init(elems([1,2,4,5,10,11,13,14])) =
pi/4*(sqrt(1.3)*opt.diam)^2; % For octet2_15
elseif opt.type == "opti"
opt.init([62,548,2332,2738,5835,5839,5855,5859]) =
1.3*pi/4*(opt.diam^2); % For cubic5bcc
% opt.init([62,548,2332,2738,5835,5839,5855,5859]) =
pi/4*(sqrt(1.3)*1.189/2-.05)^2; % For manufacturing error
elseif opt.type == "opti2x2"
opt.init(elems2) = 1.3*pi/4*(opt.diam^2);
end
gs = ALTO3D_initgs(opt);
A = gs.Ainitial;
if ~isempty(opt.XYtol) || ~isempty(opt.layer)
rel = (A>=A_relevant);
389
R1 = sqrt(A(rel)/pi); % Always the radius of the cross section
parallel to build plate
R2 = R1; % R2 angle relative to build plate changes with build
angle
% A(rel) = pi/4*((sqrt((4/pi)*A(rel))-
(2*opt.XYtol)*sin(gs.thetab(rel)')).^2);
if ~isempty(opt.XYtol) % Contour Offset
R1 = R1 - opt.XYtol;
R2 = R2 - opt.XYtol*sin(gs.thetab(rel)');
end
if ~isempty(opt.layer) % Staircase Effect
BArel = gs.thetab(rel);
SCBArel = (BArel' > 0);
R2(SCBArel) = R2(SCBArel) - (opt.layer/2)*cos(BArel(SCBArel)');
end
% if ~isempty(opt.layer)
% R2 = R2 - opt.layer/2*cos(gs.thetab(rel)');
% end
A(rel) = pi*R1.*R2;
gs.Ainitial = A;
end
% if ~isempty(opt.layer)
% rel = (A>=A_relevant & gs.thetab'>0);
% A(rel) = pi*(sqrt(A(rel)/pi)).*(sqrt(A(rel)/pi)-
opt.layer/2*cos(gs.thetab(rel)')); %Ellipsoidal
% % A(rel) = pi/4*((sqrt((4/pi)*A(rel))-
(opt.layer*cos(gs.thetab(rel)'))).^2); % Circular
% gs.Ainitial = A;
% end
if ~opt.adjarea
A_notred = A;
A_surfred = A;
A_surfred(A>=A_relevant) =
gs.surfred(A>=A_relevant).*(A(A>=A_relevant));
Volfrac = (A_surfred*gs.L)/(opt.ucs(1)*opt.ucs(2)*opt.ucs(3))
if array == 4
[Eh,min2buckle] = ALTO3D_homog(A_surfred,A,opt,optim,gs);
elseif array == 5
if opt.new == 2
[Eh,min2buckle] =
ALTO3D_homog5v2(A_surfred,A,opt,optim,gs);
elseif opt.new == 1
[Eh,min2buckle] =
ALTO3D_homog5new(A_surfred,A,opt,optim,gs);
elseif opt.new == 0
[Eh,min2buckle] = ALTO3D_homog5(A_surfred,A,opt,optim,gs);
[Eherr,min2buckleerr] =
ALTO3D_homog5err(A_surfred,A,opt,optim,gs);
Eherr
val = 3.9917e3;
target = [val , val/2, val/2, 0, 0, 0;
val/2, val , val/2, 0, 0, 0;
val/2, val/2, val, 0, 0, 0;
0, 0, 0,val/2, 0, 0;
0, 0, 0, 0,val/2, 0;
0, 0, 0, 0, 0,val/2];
390
aerr = abs(((target - Eherr)./target));
a = abs(((target - Eh)./target));
objerr = sum(aerr(aerr<inf));
obj = sum(a(a<inf));
else
error('Specify opt.new')
end
end
Eh
E33(index,1) = Eh(3,3)
M2B(index,1) = min2buckle;
GXY(index,1) = Eh(6,6)/2;
end
% PRF scores
if prf
if ~exist('middist','var')
% Perform in GS to find all midpoints and distances
% Find all midpoints
midpts = zeros(3,gs.M);
for k = 1:gs.M
midpts(:,k) =
0.5*gs.coord(:,gs.IEN(1,k))+0.5*gs.coord(:,gs.IEN(2,k));
end
% Calculate distance between all midpoints
middist = 1000*ones(gs.M,gs.M);
for k = 1:gs.M
for l = (k+1):gs.M
middist(k,l) = norm(midpts(:,l)-midpts(:,k));
end
end
end
% Realtime computation required to reduce and adjust distances for
% areas, make sure this does not apply to surface elements that
% have been reduced or it will be inaccurate.
A_rel = A(A>(pi/4*opt.drel^2));
d_rel = (4*A_rel/pi).^.5;
middist_red = middist(A>=(pi/4*opt.drel^2),A>=(pi/4*opt.drel^2));
middist_adj = middist_red;
for k = 1:length(A_rel)
for l = (k+1):length(A_rel)
middist_adj(k,l) = middist_red(k,l)-
0.5*(d_rel(k)+d_rel(l));
end
end
prfmin(index,1) = min(min(middist_adj(middist_adj<1000)))
prfavg(index,1) = mean(mean(middist_adj(middist_adj<1000)))
prfcnt = nnz(middist_adj < opt.prfrel)
end
% Connectivity
P=[];
A_rel = A>=A_relevant;
rel_nodes = unique(gs.IEN(:,(A_rel)'));
391
check1 = sum(ismember(gs.edg_cor,rel_nodes),2);
check2 = sum(ismember(gs.faces,rel_nodes),2);
if any([check1;check2]==1)
flaws = nnz([check1;check2]==1)
P = [P,100];
end
index = index + 1;
end
end
ALTO3D_plot(gs.Ainitial,gs.IEN,gs.coord,A_relevant)
VITA
BRADLEY HANKS
EDUCATION The Pennsylvania State University Doctor of Philosophy, Mechanical Engineering
Dissertation: “Design Optimization of Compliant Structures for Radiofrequency Ablation and Additive Manufacturing”
Minor in Additive Manufacturing and Design Master of Science, Mechanical Engineering
2020
2020 2018
Brigham Young University Bachelor of Science, Mechanical Engineering
2015
JOURNAL PUBLICATIONS Hanks, B., Berthel, J., Frecker, M., Simpson, T., 2020, “Mechanical Properties of Additively Manufactured Metal Lattice Structures: Data Review and Design Interface,” Additive Manufacturing, 35, p. 101301
2020
Hanks, B., Frecker, M., and Moyer, M., 2018, “Optimization of an Endoscopic Radiofrequency Ablation Electrode,” ASME J. Med. Devices, 12(3), pp. 1–11
2018
Sessions, J. W., Lindstrom, D. L., Hanks, B. W., Hope, S., and Jensen, B. D., 2016, “The Effect of Lance Geometry and Carbon Coating of Silicon Lances on Propidium Iodide Uptake in Lance Array Nanoinjection of HeLa 229 Cells,” J. Micromechanics Microengineering, 26(4).
2016
Sessions, J. W., Hanks, B. W., Lindstrom, D. L., Hope, S., and Jensen, B. D., 2016, “Transient Low Temperature Effects on Propidium Iodide Uptake in Lance Array Nanoinjected HeLa Cells,” J. Nanotechnol. Eng. Med., 6(4).
2016
Sessions, J. W., Skousen, C. S., Price, K. D., Hanks, B. W., Hope, S., Alder, J. K., and Jensen, B. D., 2016, “CRISPR-Cas9 Directed Knock-out of a Constitutively Expressed Gene Using Lance Array Nanoinjection,” Springerplus, 5(1), p. 1521.
2016
SELECTED CONFERENCE PUBLICATIONS Hanks, B., Frecker, M., 2020, “3D Additive Lattice Topology Optimization: A Unit Cell Design Approach”, ASME IDETC & CIE, DETC2020-19687, Accepted for publication
2020
Hanks, B., and Frecker, M., 2019, “Lattice Structure Design for Additive Manufacturing: Unit Cell Topology Optimization,” ASME IDETC & CIE, DETC2019-97863.
2019
Hanks, B., Azhar, F., Frecker, M., Clement, R., Greaser, J., and Snook, K., 2019, “A Deployable Multi-Tine Endoscopic Radiofrequency Ablation Electrode: Simulation Validation in a Thermochromic Tissue Phantom,” Design of Medical Devices Conference, DMD2019-3214.
2019
Hanks, B., Frecker, M., and Moyer, M., 2017, “Optimization of an Endoscopic Radiofrequency Ablation Electrode,” ASME IDETC & CIE, DETC2017- 673576
2017
HONORS AND AWARDS Marley Graduate Fellowship in Engineering – The Pennsylvania State University 2019 Reiss Fellowship in Engineering – The Pennsylvania State University 2019 1st Place Millennium Science Café Pitch Competition 2017 University Graduate Fellowship – The Pennsylvania State University 2015
