OPTIMIZATION OF ADDITIVELY MANUFACTURED MULTI-MATERIAL LATTICE STRUCTURES USING GENERALIZED OPTIMALITY CRITERIA

Tino Stankovic Engineering Design and Computing Laboratory Dept. of Mechanical and Process Engineering

ETH Zurich, Switzerland tinos@ethz.ch

Jochen Mueller Engineering Design and Computing Laboratory Dept. of Mechanical and Process Engineering

ETH Zurich, Switzerland jm@ethz.ch

Paul Egan Engineering Design and Computing Laboratory Dept. of Mechanical and Process Engineering

ETH Zurich, Switzerland pegan@ethz.ch

Kristina Shea Engineering Design and Computing Laboratory Dept. of Mechanical and Process Engineering

ETH Zurich, Switzerland kshea@ethz.ch

ABSTRACT Recent progress in additive manufacturing allows for printing

customized products with multiple materials and complex

geometries. Effectively designing such complex products for

optimal performance within the confines of additive

manufacturing constraints is challenging, due to the large

number of variables in the search space and uncertainties about

how the manufacturing processes affect fabricated materials

and structures. In this study, characteristics of materials, i.e.

Young’s modulus (E), ultimate tensile strength (UTS) and

density (𝜌), for a multi-material inkjet-based 3D-printer are

measured experimentally in order to generate data curves for a

computational optimization process in configuring multi-

material lattice structures. An optimality criteria method is

developed for computationally searching for optimal solutions

of a multi-material lattice with fixed topology and truss cross-

section sizes using the empirically obtained material

measurements. Results demonstrate the feasibility of the

approach for optimizing multi-material, lightweight truss

structures subject to displacement constraints.

1.0 INTRODUCTION AND MOTIVATION Additive manufacturing (AM) enables the automated and

continuous multi-material fabrication of three-dimensional (3D)

objects that are difficult to create with conventional

manufacturing technologies [1-3]. When additive

manufacturing technologies are utilized in combination with

pixel- and voxel-based optimization methods, it is possible to

control the distribution of materials within objects with great

precision. Such precision in creating products has the potential

to dramatically improve structural performance, and even

enable new functionalities when multiple materials are utilized

[4]. However, this freedom additionally leads to an increase in

complexity during product configuration, which can be a great

challenge for products that are already challenging to design

when only single materials are considered, such as lattice

structures [5]. Lattice structure design requires consideration of

member cross-sectional area, shape, and topology variables and

the search of a large design space that is often approached using

automated optimization approaches [6, 7]. Additionally,

solutions found by computational optimization must remain

printable according to Design for Additive Manufacturing

(DfAM) constraints such as the limitations in tolerances,

printable dimensions, and quality issues incurred by the

manufacturing process [8-11]. In this paper, we address these

DfAM challenges by developing an optimization approach for

configuring multi-material lattices that have manufacturing

constraints stemming from the limitations of an AM fabrication

process.

The development of high-performance, multi-material

lattices for complex shapes has particular applications in, e.g.,

developing customized helmets for impact resistance [12, 13].

Lattice structures are growing in interest due to their improved

strength-to-weight ratio compared with stochastic foams alone.

When combined with stochastic foams, hybrid lattice-foam

structures may show superior performance for both strength

and energy absorption under certain conditions [14]. A

complete DfAM process for designing a customized product,

such as a sports helmet with multi-material lattices, includes

phases for characterizing AM materials and processes,

Proceedings of the ASME 2015 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference

IDETC/CIE 2015 August 2-5, 2015, Boston, Massachusetts, USA

DETC2015-47403

1 Copyright © 2015 by ASME

developing quantitative and simulation models based on this

characterization, optimizing designs with computational

methods, and then fabricating and testing the final multi-

material product (Fig. 1). Once testing is completed, findings

are informative for future product design cycles such as the

characterization of different AM materials or refinement of

quantitative models.

Due to the large scope of the DfAM methodology, we

focus on the computational design and optimization portion of

Figure 1 in this paper. However, considerations of the whole

approach inform the development of the optimization method,

such as how it utilizes information from AM characterized

materials and processes. Since the total volume of the helmet is

constrained based on consumer needs, design decisions

concerning material changes in the lattice are beneficial

because they can alter the product’s performance capabilities

without altering its dimensions. The use of AM for producing

the helmet is particularly beneficial, because it enables

customized helmet shapes on a per-consumer basis such that

the helmet conforms to a particular customer’s head shape and

has performance requirements that are tailored to particular

applications [13, 15, 16]. Because of these considerations for

the helmet, multi-material AM could greatly facilitate its design

in comparison with conventional manufacturing approaches.

There is a great need for effective optimization algorithms

for multi-material AM, since such optimization problems have

only been considered for a limited set of products and

applications [6]. Particularly with lattice approaches,

evolutionary based and other meta-heuristic methods have

difficulty with single material structures because they do not

scale well as the resolution and number of structural members

in an application increases. In the helmet example, the design

consists of hundreds to hundreds of thousands of members.

Other algorithms that are well-suited for structural

optimization, such as the discrete optimality criteria method

[7], can be used for large discrete- and lattice-based structures,

but have not been formulated to solve multi-material structures.

Most often, the problem formulation for optimality criteria

methods are purely mass-based and consider dominant

displacement constraints only [7]. However, other more

constraint rich applications exist [17, 18] in addition to the

possibility for extending the method to solve multi-material

lattice problems explored here.

In this paper, we focus on developing a generalized

optimality criteria method [19] for the optimization of discrete,

multi-material mesoscale lattice structures for maximum

performance in a constrained volume. Through considering the

formulation of the optimization algorithm within the context of

a DfAM methodology (Fig. 1), the optimization approach is

developed to accept information about AM characteristics of

materials and processes that ensure that the final design found

by the algorithm can be directly fabricated with AM without the

need for post-processes. Experimental testing is conducted to

determine the properties of two AM materials that the

optimization algorithm may combine in different mixtures to

design lattices with varied properties. The optimization

algorithm is then used to search for optimal lattice designs

through altering material properties of elements within the

lattice. The paper concludes with initial results and a discussion

of future work, including fabricating and testing the optimized

AM structures.

2.0 BACKGROUND 2.1 Lattice Optimization Past studies using the size matching and scaling method

[20] have implemented an approach to design and optimize

meso-scale cellular lightweight structures. The method couples

a library of predefined unit cells, i.e. lattice configurations, with

a solid body finite element analysis that is applied over a design

domain to provide design response as a guide for unit cell

selection, scaling, and placement. The size matching and

scaling method has demonstrated better performance than

particle swarm based lattice optimization [21] and least squares

fitting method [21] when applied for predefined topology or

ground structure optimization.

The approach from Ning and Pellegrino [22] aims to

design and optimize the microstructure of lightweight sandwich

beam designs within a size distribution field using multiple

optimization steps. Starting with Delaunay triangulation, the 2D

topology search is reduced to a limited number of continuous

control variables. The first step in the optimization approach is

to use a genetic algorithm to optimize the topology and then

using the best found topology, lattice truss cross-section sizes

are optimized.

In generative design and optimization of macro scale

lattice structures a somewhat different approach to the previous

one is explored in previous work by one of the authors, the

generative synthesis of transmission towers [23]. The search for

the optimal topology, shape and member sizes is achieved

through a combination of graph grammars and simulated

annealing optimization to create spatially novel designs.

Figure 1.DfAM product design methodology

2 Copyright © 2015 by ASME

2.2 Multi-material Additive Manufacturing Unlike conventional manufacturing, additive

manufacturing (AM) is based on a fabrication process that joins

materials layer by layer to build a product based on CAD model

data. Effective AM design requires consideration of time-

required for product fabrication in addition to materials used

[24-26]. Multi-material applications are particularly relevant to

AM because they can enable a better optimization of the

mechanical properties of parts or provide additional

functionality [2] such as fabricated materials with desired

deformation behavior [27]. There is also the possibility of

including gradients of different materials in a part, which has

been investigated with optimization methods for the

manufacturing of heterogeneous 3D objects with consideration

to mechanics-manufacturing trade-offs [28]. Multiple-material

topology optimization has also been investigated in the context

of AM for 2D material structures [4]. These studies demonstrate

the potential in utilizing multi-material AM applications and the

need for further developing optimization methods in solving the

resulting complex design problems.

2.3 Optimality Criteria Methods In addition to already established lattice optimization

approaches, there is the opportunity to tailor optimization

approaches from other applications in discrete structural

optimization for lattice design, with optimality criteria being a

potentially highly effective approach. In the late 1960s’,

optimality criteria methods were developed to tackle large-scale

structural optimization of discrete member structures for the

aerospace industry [7, 17, 29, 30]. Early applications of

optimality criteria considered static load cases with

displacement constraints only, but the method was soon

extended to consider stress and buckling constraints in addition

to stability and for impact and frequency responses [31, 32]. To

balance computational efficiency with solution quality for

application to large-scale problems, a fully constrained design

method was developed [33]. This method is not gradient-based

and overcomes the requirement that the algorithm must be

customized for each unique problem formulation. However, the

method is only applied to displacement constraints. Another

successful application is the optimization of tall steel buildings

extending the optimality criteria by including commercial

standard sections in the optimization [34].

The primary motivation for the development of the

optimality criteria approach was to address the inability of

conventional mathematical programming methods to handle

optimization problems that exceed 200-300 design variables for

a generic case of a statically indeterminate structure with a non-

linear constraint set [7, 17]. Before optimality criteria was

developed as an approach, numerous alternative search

strategies and problem simplifications were applied to handle

these issues, but were not highly effective approaches. For

example, variable linking and problem order decrease

approaches are hard to implement due to the difficulty in

assessing the implications of these actions on the overall design

solution. The direct consequence of using optimality criteria is

a reduction of cost in required iteration steps to obtain a

significant improvement in the objective function, with respect

to the size of the variable vector [17]. The optimality criteria

approach additionally requires the calculation of a pseudo

energy function that is obtained when a finite element method

is applied to calculate a design response [35].

3.0 MOTIVATION FOR USING OPTIMALITY CRITERIA Findings from the literature suggest that the development

of a highly effective lattice structure optimizer that handles

many complex design cases is still an open question. Genetic

algorithms and similar approaches handle non-gradient

problems, but experience problems when the variable string

becomes too long. Simulating annealing in its simplest form

has a proof of convergence in theory if it runs for an infinite

duration of time, but requires many heuristics to be effective,

e.g. an appropriate cooling schedule definition. Least squares

fitting methods, e.g. design of experiments based responsive

surface methods, are another possible approach that alleviate

computational costs by using a surrogate approach, but they are

also limited in finding high quality solutions for large scale

structures of 10,000 or more members. With respect to the

complexity of discrete lattice structures that are produced using

AM technologies, the application of optimality criteria for

parametric optimization offers the benefit of decoupling the

variable vector size and required iteration steps which is

suggestive of its potential for effectively optimizing large-scale

lattice structures with multiple materials.

In order to assess the optimality criteria’s feasibility in

comparison to other optimization methods for complex lattice

structures, it is benchmarked against the canonical genetic

algorithm (GA) from our own genetic algorithm framework and

interior point (IP) optimization from the MATLAB

optimization toolbox. These methods are compared using a test-

case lattice with 74 members that occupies a space of 50x50x50

mm3 and is optimized for minimum mass when taking cross-

sectional area 𝐴𝑖 as the design variable. A recursive resizing

formula that is a special simplified case of optimality criteria is

applied and referred to as the fully stressed design (FSD)

approach. FSD is easily extendible for the lattice application,

fast in convergence, but is an oversimplification of optimality

criteria because it limits the possibilities to express multiple

constraints, load conditions, and the application of different

materials. Therefore, FSD is a viable approach for

benchmarking the feasibility of optimality criteria in

comparison to other methods, but is not suitable for complex

multi-material lattice structures subjected to different

constraints and loading conditions. The recursive sizing

formula for the cross-section area 𝐴𝑖 of a steel lattice structure

in the test case is as follows [7]:

𝐴𝑖𝜈+1 = (

𝑈𝑖𝐴𝑖𝜈

휀𝑖2𝜌𝑖𝑙𝑖

)

1 2⁄

(1)

where ν denotes the iteration step, 𝑈𝑖 the current strain energy

stored in the ith element, 𝜌𝑖 = 7860 kg/m3

the density, 𝑙𝑖 the

length of the ith element, and 휀𝑖2 the allowed strain per element

calculated for 𝐸 = 206 GPa and allowed stress |𝜎| ≤ 50

N/mm2. Buckling and displacement constraints are not taken

into account. The load F = 1000 N is distributed to all nodes of

the lattice top face in the direction of the negative z-axis, while

the bottom face nodes are fully constrained. The variable range

is defined as 0.01 ≤ 𝐴𝑖 ≤ 10 mm2. The results taking that

3 Copyright © 2015 by ASME

𝐴𝑖 = 𝐴 = 10 mm2 shown in Figure 2 indicate the benefits of

the optimality criteria method, as demonstrated by the FSD

approach having found the lightest structure (11.81 g), with less

computational time (less than 10 s) and iterations (20) in

comparison to both other methods.

Although there are many benefits in using optimality

criteria approaches with respect to computational time and

resources required to optimize large scale structures, its context

oriented basis lacks generality that hinders its efficient use [19].

The optimality criteria method is most commonly expressed to

meet the specific features of a design problem for a particular

application. In an effort to mitigate this, a generalized

optimality criteria was developed [19] by extending the

optimality criteria application to general multi-disciplinary

optimization problems by offering a methodological approach

in both recursive algorithm derivation as well as in the

application of compound scaling algorithm. The search strategy

is then based on a recursive sweep through the search space

while applying scaling procedures to estimate the location of

constraints boundaries [19]. This generalized formulation of the

optimality criteria is considered in this paper to be tested and

extended for multi-material lattice structure optimization

problems.

4.0 CHARACTERIZATION OF AM MATERIALS 4.1 Materials Testing Methods In order to optimize lattice structures according to AM

constraints, empirical studies are performed to characterize AM

materials that are input into the optimization model. To

facilitate multi-material lattice design, Young’s modulus (E),

ultimate tensile strength (UTS) and density (𝜌) are measured for

two different AM materials. The measurements are obtained

using the Stratasys Objet500 Connex3 due to the printer’s

capabilities for printing up to three different model materials at

the same time, thus enabling the fabrication of multi-material

lattices. The printer’s support material is a mixture of SUP705

support and a generic model material. The liquid material is

jetted onto the surface and immediately cured with UV light

positioned on the print head. A pinch roll ensures a smooth and

clean surface when jetting the next layers.

In the printer’s digital material mode, model materials are

mixed to create combinations of properties such as a hard and a

soft material with mixed properties in between. The potential to

mix materials is highly beneficial for lattice applications,

because with just two base material choices there is a broad

range of intermediate mixes of materials with unique

properties. In this study, materials of low and high strength are

tested that are referred to as VeroWhitePlus and

TangoBlackPlus by the manufacturer and chosen based on the

mechanical properties provided by the manufacturer. The

manufacturer reported that VeroWhitePlus has the following

mechanical properties, E = 2000-3000 MPa, UTS = 50-65 MPa,

𝜌 = 1.17-1.18 g/cm3, and that TangoBlackPlus has the

following mechanical properties, UTS = 0.8-1.5 MPa and 𝜌 =

1.12-1.13 g/cm3 [36]. Testing is conducted to verify these

values, in addition to measuring values for the mixed material

properties that are not provided by the manufacturer.

Testing samples are printed for each of the 14 possible

material gradings available for printing based on constraints

from the manufacturer and are inclusive of the two base

materials. Testing commenced on an Instron ElectroPuls E3000

tensile testing machine that enables the testing of the

mechanical properties according to the ASTM D638-10

standard. Additionally, materials are weighed with a Metler

Toledo XS205 DualRange scale. For each material grading,

only one measurement is collected due to negligible variations

in measurements of parts produced on the same machine in

controlled conditions [37]. 4.2 Materials Testing Results For all printed materials, the density of the material is

calculated as an independent variable to form plots with

dependent variables of the measured Young’s modulus (Fig. 3a)

and ultimate tensile strength (Fig. 3b). The plots in Figure 3

demonstrate that the densities of materials have an uneven

distribution over the range of 1.09 g/cm3 to 1.175 g/cm

3, with

most materials being close to the extremes. The plots also

include the manufacturer’s reported data as a basis of

comparison. The experimental measurements for the Young’s

modulus obtained are higher than those provided by the

manufacturer [36] (Fig. 3a). Generally, materials are bi-modally

distributed around Young’s modulus values of about 50 MPa

and 2800 MPa, and there is a lack of testable material mixtures

that may have fallen in intermediate ranges. The data suggests

that material properties are generally below 500 MPa for

densities below 1.14g/cm3, with a sharp increase to about 3000

𝑚 = 41.3 g

Genetic Algorithm

1600 iterations

Running time: ~1 min 𝑚 = 16.1 g

Interior Point

260 iterations

Running time: ~10 min 𝑚 = 11.81 g

Fully Stressed Design

20 iterations

Running time: ~ 4 s

Figure 2. Results of benchmarking algorithms with steel lattice test-case

4 Copyright © 2015 by ASME

MPa for materials with density above 1.14g/cm3. A curve was

fit to the data, since the optimization algorithm requires

continuous functions to calculate possible material properties.

For the optimization, a sigmoidal curve of shape

𝐸(𝜌) = 𝐴1 +𝐴2 − 𝐴1

1 + 10(𝐿𝑜𝑔(𝑥0)−𝜌).𝑝 (2)

is fit to the values (line, black), where A1 and A2 represent the

lower and upper asymptotes, p the hill slope, and x0 the center

point. Solving the equation for 𝜌, which is required for the

model-input, yields Equation 18 with an adjusted R-square of

0.99794.

𝜌(𝐸) = log 𝑥0 −log (

𝐴2 − 𝐴1

𝐸 − 𝐴1− 1)

𝑝 (3)

For the tensile strength, values associated to densities

below 1.14 g/cm3 span over a wider range in an almost linear

behavior (Fig. 3b). The tensile strength data provided by the

manufacturer agrees with the measured values at the densities

below 1.14 g/cm3. Between 1.14 g/cm

3 and 1.17 g/cm

3 no

strength values are found, whereas the ones above 1.17 g/cm3

tend to cumulate between 60 and 70 MPa. An exponential

function of shape:

𝑈𝑇𝑆(𝜌) = 𝐴𝐵𝜌 (4)

is fitted (adjusted R-square 0.99631), where A and B reflect

positions on the curve, respectively. Solving for 𝜌 yields

Equation (5):

𝜌(𝑈𝑇𝑆) = 𝑙𝑜𝑔𝑏 (𝑈𝑇𝑆

𝑎)

(5)

These material characterizations demonstrate the properties

and relations that can be utilized by the computational search

algorithm to only search designs with properties that are

possible to fabricate with this AM technology. Testing of these

materials also suggests that the printer can consistently

fabricate lattice elements of 1 mm in diameter and 10 mm in

length, which is an additional DfAM constraint for the

algorithm.

5.0 OPTIMALITY CRITERIA

The method derived in this section aims to extend the

optimality criteria method to solve multi-material lattice

optimization problems. In the context of additive

manufacturing, this allows for structures optimized for different

performance considerations subjected to conflicting

requirements, e.g. prescribed displacement of the lattice within

a helmet (Fig 1.), which allows customization, absorbs required

energy, and is preferably lightweight. Thus, the optimization

model aims to achieve a lightweight design subject to

displacement constraints given a lattice topology, shape and

member sizes and a range of available materials subjected to a

set of loads and boundary conditions.

5.1 Generalized Optimality Criteria Using Multi-materials with Fixed Truss Cross-section Size and Displacement Constraints

Assuming a mass minimization problem with 𝐴 being a

constant cross-sectional area of a round strut, then the objective

function is stated as:

minimize 𝐹(𝐱) = ∑ 𝐴𝑙𝑖𝜌𝑖(𝑥𝑖)

𝑛

𝑖=1

(6)

where, 𝑙𝑖 is the length of each of the corresponding members,

and 𝜌𝑖 is the material density that is material dependent over

Young’s moduli 𝑥𝑖 as:

𝜌𝑖 = 𝜌𝑖(𝑥𝑖) (7)

where the dependency is obtained through material testing.

Although equation (6) is general with respect to the type of

variable 𝑥𝑖, the rest of the expressions are derived taking the

material Young’s modulus as a continuous variable and using a

linear elastic model of the lattice structure behavior, as a

simplification. In the context of engineering applications of AM

lattice structures, e.g. the helmet in Figure 1 that contains

~100,000 truss members, the motivation for this simplification

in a first instance is justified due to the design’s complexity.

(a)

(b)

Figure 3. Empirical AM Data for (a) Young’s modulus and (b) ultimate tensile strength as a function of density.

1.08 1.10 1.12 1.14 1.16 1.18

0

500

1000

1500

2000

2500

3000 Young's modulus

Young's modulus (Objet)

Fit of E

Young's

modulu

s (

MP

a)

Density [g/cm3]

1.08 1.10 1.12 1.14 1.16 1.18

0

10

20

30

40

50

60

70 UTS

UTS (Objet)

Fit of UTS

UT

S (

MP

a)

Density [g/cm3]

5 Copyright © 2015 by ASME

Thus, to express the constraints as global constraints, e.g. as

functions of the whole lattice, a flexibility coefficient 𝐸𝑖𝑗

associated with the ith member and jth constraint is defined

with respect to the virtual energy of the system [7] as:

𝐸𝑖𝑗 = 𝑥𝑖{𝑟}𝑖𝑇[𝑘]𝑖{𝑠𝑗}𝑖 (8)

where {𝑟}𝑖𝑇 is a displacement vector, [𝑘]𝑖 is the ith truss element

stiffness matrix and {𝑠𝑗}𝑖 is the displacement vector

corresponding to constraint dependent virtual load vector. As

both 𝐴 and 𝐸 participate linearly in the stiffness of truss

members for a linear elastic material model, then the

displacement constraint with respect to the flexibility

coefficient 𝐸𝑖𝑗 , constraint boundary 𝑧�̅� and Young’s modulo 𝑥𝑖

is expressed as:

𝑧𝑗 =𝐸𝑖𝑗

𝑥𝑖

≤ 𝑧�̅� (9)

The displacement constraint gradient matrix can then follow

from (9) as:

𝑁𝑖𝑗 =𝜕𝑧𝑗

𝜕𝑥𝑖

=𝐸𝑖𝑗

−𝑥𝑖2

(10)

with 𝑁𝑖𝑗 being the derivative of the jth constraint 𝑧𝑗 with

respect to the ith variable 𝑥𝑖. Given Equation (6), the gradient

of the objective function ∇𝐹𝑖 is expressed as

∇𝐹𝑖 =𝜕𝐹

𝜕𝑥𝑖

= 𝐴𝑙𝑖

𝜕𝜌𝑖

𝜕𝑥𝑖

(11).

According to [7, 17, 19], the optimality criteria is defined

by the optimality conditions that are satisfied at the optimum

point providing a system of n equations with m active

constraints:

∑ 𝑒𝑖𝑗𝜆𝑗

𝑚

𝑗=1

= 1 (12)

where 𝑒𝑖𝑗 is the ratio of sensitivity derivates of the constraints

and the objective function and 𝜆𝑗 is the Lagrangian multiplier

corresponding to the jth constraint. The ratio of sensitivity

derivates defined with respect to (6-11) is given as:

𝑒𝑖𝑗 =𝑁𝑖𝑗

∇𝐹𝑖

= −

𝐸𝑖𝑗

𝑥𝑖2

𝐴𝑙𝑖𝜕𝜌𝑖

𝜕𝑥𝑖

(13)

However, the solution of Equation (12) requires an

independent set of active m constraints such that 𝑒𝑖𝑗𝜆𝑗 is of full

rank. Thus, according to [19], Equation (12) is expanded by

multiplication with 𝑒𝑖𝑗T 𝐴𝑖𝑗 where 𝐴𝑖𝑖 = ∇𝐹𝑖𝑥𝑖 resulting in the

following expressions written in vector notation:

𝐇𝛌 = 𝐖

(14)

with

𝐇 = 𝐞T𝐀 𝐞 (15)

𝐖 = 𝐞T𝐀 𝟏

Equations (12) and (14) yield a set of Lagrangian

multipliers that if all negative indicate an independent set of

constraints. The necessary condition is that 𝐇 is non-singular

and that 𝐀 is positive definite. If Equation (14) yields both

positive and negative Lagrangian multipliers, then the positive

multipliers indicate constraint dependency requiring their

deletion from the active constraint set and recalculation of (14)

repeating the procedure until multipliers are all negative.

5.1.1 Resizing

After the multipliers are obtained by solving Equation (14),

the algorithm either resizes if the active set of constraints are

satisfied with at least one of the constraints being at the

constraint boundary, or applies a compound scaling [19] until

the condition for resizing is met. Each resizing step completes

an optimization cycle requiring a number of iteration steps to

bring constraints to the feasible region and at least one of them

to the constraint boundary. The resizing formula derived from

the optimality condition (12) is given with respect to cycle 𝜈

and a cycle step size factor 𝛼 = 2 as:

𝑥𝑖𝜈+1 = 𝑥𝑖

𝜈 [∑ 𝑒𝑖𝑗λ𝑗

𝑚

𝑗=1

]

1/𝛼

(16)

Equation (16) takes only positive sums into the

consideration. If there are no positive members then the

corresponding variable is set to its lowest possible value as

defined by the boundary �̅�𝑖.

5.1.2 Scaling

The purpose of scaling is to move constraints to the

feasible search space in such a way that at least one constraint

is on the constraint boundary. Scaling identifies the most

critical constraint and calculates the scaling factor Λ and by

applying the equation

𝑥𝑖′ = Λ𝑥𝑖 (17)

moves the constraint to its boundary [7].

The assumption that has to hold is that all of the other

constraints are also moved to the feasible space. However, this

is not guaranteed for a nonlinear constraint set. Compound

scaling as defined in [19], on the other hand, is more robust as

it scales the variables by analyzing the first order Taylor

approximation in the constraint response due to change in the

variable vector. Each of the variables are classified as active or

passive contributors to the constraint response change that

enables the calculation of scaling factor vector Λ. For details on

compound scaling please refer to [38].

6 Copyright © 2015 by ASME

5.1.3 Optimization Stopping Conditions The overall progress of scaling versus resizing is in the

direction of objective function reduction, otherwise a stopping

condition should halt the optimization progress as the Kuhn-

Tucker conditions provide only the necessary conditions for a

local optimum. the stopping conditions are most often empirical

rather than mathematically defined [19]. However, as the

optimality criteria assumes the location of an optimum is

somewhere on the constraint boundaries, then for purposes of

this study we define the stopping condition when Equation (12)

holds over all of the constraints with a margin of error given as

휀 ≤ 0.05. This effectively allows specifying certain behavior of

the lattice structure as the goal of the optimization process.

6.0 RESULTS 6.1 Optimization Example

The illustrative example to test the method is a cubic lattice

having an individual cubic cell 10x10x10 mm3 in size and

cross-section truss diameter of 1 mm for each of the members

as shown in Figure 4. The idea is to achieve a defined

compliance of the top face middle node 1 while not allowing

significant movement of nodes 2-5. The number of cells

equates 64 with 604 truss members in total:

Figure 4. Lattice topology

Boundary conditions for this example are specified

according to Figure 5, the load is applied to nine middle nodes

on the top face in the opposite direction of z-axis. The Node 1

has displacement constraint 𝛿1 ≤ 2 mm, the rim nodes 2-5 have

prescribed displacement constraint 0.25 mm per node, in

total 𝛿2 ≤ 1 mm, which is calculated as a linear combination of

four points taken with an equal weight factor. Finally, all of the

bottom face nodes are fully constrained. All of the displacement

constraints are calculated in the direction of the applied loads.

Figure 5. Optimization boundary conditions and loads

Taking equations (2) and (3), it is possible to express the mass

𝐹(x), as a function of Young’s moduli:

𝐹(𝐱) = ∑ 𝐴𝑙𝑖

𝑛

𝑖=1

[𝑐 −log (

�̿�𝑖 − �̅�𝑖

𝑥𝑖 − �̅�𝑖− 1)

𝑝] (18)

as well as define the objective function’s gradient ∇𝐹𝑖:

∇𝐹𝑖 = −�̅�𝑖 − �̿�𝑖

𝑝 ln (10)(�̅�𝑖 − 𝑥𝑖)2(

�̅�𝑖 − �̿�𝑖

�̅�𝑖 − 𝑥𝑖− 1)

𝐴𝑙𝑖 (19)

where 𝐴 = 0.7854 mm2, the cross-sectional area of each of the

truss members, 𝑙𝑖 is member length, and �̿�𝑖 = 3250 MPa and

�̅�𝑖 = 8.3 MPa are the upper and the lower variable boundaries,

with 𝑐 = 1.16 and 𝑝 = 57.46 as the curve fitting coefficients

obtained from the material properties testing; see Section 4.2.

Finally, based on the Equations (13), (18) and (19), the

sensitivity derivates of the constraints and the objective

function are given as:

𝑒𝑖𝑗 =𝐸𝑖𝑗

𝑥𝑖2

�̅�𝑖 − �̿�𝑖

𝑝 ln (10)(�̅�𝑖 − 𝑥𝑖)2(

�̅�𝑖 − �̿�𝑖

�̅�𝑖 − 𝑥𝑖− 1)

𝐴𝑙𝑖⁄ (20)

The exit criteria for the optimization is set to find a

constraint intersection point with 5% accuracy based on their

target response ratios and is given as follows:

|𝛿1̅

𝛿1

−𝛿2̅

𝛿2

| ≤ 0.05 (21)

The starting point is selected in the upper region of the

Young’s moduli variable span as 𝑥0𝑖= 3000 MPa. Finally, the

optimization problem used as an illustrative example is

formulated as follows:

minimize 𝐹(𝐱) = ∑ 𝐴𝐿𝑖𝜌𝑖(𝑥𝑖)

𝑛

𝑖=1

(22) Subject to:

𝛿1(𝑥𝑖) ≤ 2 mm

𝛿2(𝑥𝑖) ≤ 1 mm

8.3 ≤ 𝑥𝑖 ≤ 3250 MPa 6.2 Optimization Example Results

The method itself was implemented in MATLAB together

with the FEM analysis. The running time was ~7.5 minutes on

a laptop with 8GB of RAM and an INTEL i7 CPU. The results

of the optimization as specified by the equation (22) and are

summarized as follows:

a) The constraint intersection point satisfying Equations

(12) and (21) is identified with 5% accuracy in the 27th

resizing cycle (167th

step).

10x10x10 mm3

64 cellsDstrutt = 1 mm

1

2

5

4

3

1

2

5

4

3 Fixed nodes

Load = 10 N

z - axis

7 Copyright © 2015 by ASME

b) The values of displacement constraints are: for the

middle point 𝛿1(𝑥𝑖) = 1.923 mm while the cumulative

displacements of four rim points equals 𝛿2(𝑥𝑖) = 1.0

mm. The individual displacements per node are given

as follows:

Node 2 δ22 = 0.25 mm,

Node 3 δ23 = 0.283 mm,

Node 4 δ24 = 0.25 mm,

Node 5 δ25 = 0.216 mm.

c) The minimum mass obtained is 𝐹(𝐱) = 5.886 g

The optimization of the mass over the iteration step history

is depicted according to two images in Figure 6. The left hand

side image shows the first three optimization cycles shown with

the line connected points for which the ×-symbols denote the

outcome of the resizing formula application according to

Equation (16). The o-symbols show the compound scaling

progression for each of the cycles leading to the condition at

which at least one of the constraints is an active constraint

while the other is in the feasible region. The right image in

Figure 6 depicts the overall optimization history.

The resulting structure is shown in Figure 7 where the

material is shown graduated from TangoBlackPlus (black) to

VeroWhitePlus (white). The image shows the impact of the

constraints on the overall distribution of the material, e.g. less

compliant vertical side edges of the lattice structure. The

overall structure also exhibits a symmetry over the spatial

diagonals of each of the cells, which was not constrained. The

horizontal members that are connected to the two support

points are excluded from the optimization as they introduce

singularities in Equation (13), resulting in total 548 structural

members. The structure is comprised of 366 elements with E =

8.3 MPa, 85 elements with 8.3 < 𝐸 ≤ 10 MPa, 73 elements in

the region of 10 < 𝐸 ≤ 544 MPa and finally 24 elements above

2900 MPa. Figure 8 shows the results of the optimization using

the same objective function in Equation (22) but with a

different starting point, 𝑥0i= 10 MPa. The resulting mass,

𝐹(𝐱) = 5.847 g, is slightly less than the previous result,

however, with a different layout of material, i.e. 417 elements

have E = 8.3 MPa, 112 elements in total have 8.3 < 𝐸 ≤ 10

MPa, and 19 elements are in the region of 10 < 𝐸 ≤ 324 MPa.

Figure 7. Optimization results - starting point E0 = 3000 MPa

Figure 8. Optimization results - starting point E0 = 10 MPa 7.0 DISCUSSION Both Young’s modulus and tensile strength values found in this

study are considerably higher than the manufacturer provided

values, which can be linked to a large number of in- and out-of-

process parameters available for the PolyJet 3D-printing

process [37]. No information, besides the use of the ASTM

D638-10 standard, is available on the testing procedure of the

manufacturer. A higher Young’s modulus and shorter elongation

at break makes VeroWhitePlus fragile compared to

TangoBlackPlus. In testing these materials, a strut length of 10

mm and print diameter of 1 mm is utilized for test samples,

Figure 6. The optimization mass over iteration steps history for starting point E0 = 3000 MPa

6.1

6.2

6.3

6.4

0 5 10 15 20 25 30

F[g

]

Iterration steps

5.8

5.9

6

6.1

6.2

6.3

6.4

0 50 100 150 200

F[g

]

Iterration steps

8 Copyright © 2015 by ASME

which is indicative of the minimum lattice element size the

printer can fabricate consistently. Further empirical studies

could investigate what other variables affect the minimum

lattice element diameter and length a printer may fabricate.

A limitation in material characterization occurred due to

the manufacturer limiting the possible densities for printing

because the mixing ratios provided by the manufacturer are

purely based on equal color gradients [36]. It is possible that

adding a small amount of black material to a white base

material has a large visual effect, whereas adding the same

amount of white material to a black base material has a

negligible effect. These considerations could be a potential

reason why the values of the high-density white base material

are closer together in terms of density than the ones with a

black base material and why there are few density

measurements available at densities that are representative of

near equal mixes of materials. As densities correlate with

mechanical strength, a mix of materials spread over a wider

range of densities would be preferable and is theoretically

possible. For this study, however, exact mechanical properties

of other mixing ratios other than the ones presented are not

available and therefore estimated by the fit curve used in the

optimization model.

7.2 Optimality Criteria Method Application The optimality criteria method extended to multi-material

optimization of a lattice structure under displacement

constraints achieved a mass reduction just under 10% compared

to the initial starting mass. This result can be rendered as

acceptable as the overall material density range is itself narrow

spanning from approximately 1.10-1.18 g/cm3, as shown in

Figure 7. This result is achieved considering the following

model simplifications:

a) Young’s modulus is considered as a continuous

variable. While this is theoretically possible on the

Polyjet machine, the current machine only allows

discrete combinations. Either including Young’s

modulus as a discrete variable in the optimization or

applying a second optimization to map the continuous

variable values back to the discrete available values

will be considered in future work.

b) The substitution of cross-sectional area for Young’s

modulus is applied for displacement constraints only.

Future extensions to stress constraints require the

consideration of the stress constraint boundary as

𝜎 = 𝜎(𝐸) since they change with the material. It is

difficult to assess how this will affect the scaling

procedure. Additionally, the stress constraints are

calculated individually per member, which puts a

heavy load on computational resources requiring

additional strategies for constraint management [30].

c) Optimality criteria methods require some knowledge

of the constraint behavior with respect to their activity

and optimum location to be successfully implemented.

This will be investigated in the future to develop a

generalized method for complex-shaped, multi-

material lattices for fabrication with AM.

8.0 CONCLUSION In this paper, an optimization approach using optimality

criteria is developed to facilitate effective design of complex,

multi-material products for optimal performance within the

confines of additive manufacturing constraints. The

optimization approach is developed in the context of designing

multi-material lattice structures as part of a larger DfAM

methodology and product design cycle. An optimality criteria

approach is utilized due to its effective use for optimizing large-

scale discrete structures and the possibility to extend it for

multi-material applications. In order to adhere to DfAM

considerations, empirical measurements are conducted to

characterize the properties of two AM printable materials. The

materials consisted of a low strength and high strength material

that can be combined to form mixtures of materials with

intermediate properties. These measured material properties are

used to develop a curve fit for Young’s modulus that is used by

the optimization algorithm for configuring multi-material

lattices.

The successful application of the extended optimality

criteria method reduced the mass of a multi-material lattice up

to 10% in spite of the limited density range of materials. The

optimization time for a 604 element lattice on a laptop required

only ~7.5 minutes of running time to complete 27 optimization

cycles in 167 steps. These results and the study as a whole

demonstrate the feasibility of optimality criteria in developing

specific design optimization algorithms that adhere to DfAM

constraints and enable the design of highly complex structures

for multi-material AM. Extending the method beyond

displacement constraints requires handling of dynamic stress

constraints and the introduction of a discrete material selection

procedure to adhere to current constraints of the AM machine.

These two points, extension to topology optimization and the

printing and empirical testing of the optimized designs will be

addressed in future work.

ACKNOWLEDGMENTS Partial funding for this research is provided by ETH Zurich

through the Seed Project, SP-MaP 02-14, "Additive Manufacturing of

Complex-Shaped Parts with Locally Tunable Materials".

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