26
Biomechanics
Reliability of Multiphysical Systems Set coordinated by
Abdelkhalak El Hami
Volume 5
Biomechanics
Optimization Uncertainties and Reliability
Ghias Kharmanda Abdelkhalak El Hami
First published 2017 in Great Britain and the United States by ISTE Ltd and John Wiley amp Sons Inc
Apart from any fair dealing for the purposes of research or private study or criticism or review as permitted under the Copyright Designs and Patents Act 1988 this publication may only be reproduced stored or transmitted in any form or by any means with the prior permission in writing of the publishers or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address
ISTE Ltd John Wiley amp Sons Inc 27-37 St Georgersquos Road 111 River Street London SW19 4EU Hoboken NJ 07030 UK USA
wwwistecouk wwwwileycom
copy ISTE Ltd 2017 The rights of Ghias Kharmanda and Abdelkhalak El Hami to be identified as the authors of this work have been asserted by them in accordance with the Copyright Designs and Patents Act 1988
Library of Congress Control Number 2016952066 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-025-6
Contents
Preface xi
Introduction xiii
List of Abbreviations xvii
Chapter 1 Introduction to Structural Optimization 1
11 Introduction 1 12 History of structural optimization 2 13 Sizing optimization 4
131 Definition 4 132 First works in sizing optimization 4 133 Numerical application 5
14 Shape optimization 10 141 Definition 10 142 First works in shape optimization 11 143 Numerical application 12
15 Topology optimization 16 151 Definition 16 152 First works in topology optimization 17 153 Numerical application 18
16 Conclusion 21
Chapter 2 Integration of Structural Optimization into Biomechanics 23
21 Introduction 23 22 Integration of structural optimization into orthopedic prosthesis design 23
vi Biomechanics
221 Structural optimization of the hip prosthesis 24 222 Sizing optimization of a 3D intervertebral disk prosthesis 42
23 Integration of structural optimization into orthodontic prosthesis design 47
231 Sizing optimization of a dental implant 47 232 Shape optimization of a mini-plate 49
24 Advanced integration of structural optimization into drilling surgery 52
241 Case of treatment of a crack with a single hole 53 242 Case of treatment of a crack with two holes 54
25 Conclusion 56
Chapter 3 Integration of Reliability into Structural Optimization 57
31 Introduction 57 32 Literature review of reliability-based optimization 58 33 Comparison between deterministic and reliability-based optimization 60
331 Deterministic optimization 61 332 Reliability-based optimization 63
34 Numerical application 64 341 Description and modeling of the studied problem 64 342 Numerical results 65
35 Approaches and strategies for reliability-based optimization 68
351 Mono-level approaches 68 352 Double-level approaches 68 353 Sequential decoupled approaches 68
36 Two points of view for developments of reliability-based optimization 69
361 Point of view of ldquoReliabilityrdquo 69 362 Point of view of ldquoOptimizationrdquo 70 363 Method efficiency 70
37 Philosophy of integration of the concept of reliability into structural optimization groups 72 38 Conclusion 73
Chapter 4 Reliability-based Design Optimization Model 75
41 Introduction 75 42 Classic method 76
Contents vii
421 Formulations 76 422 Optimality conditions 77 423 Algorithm 77 424 Advantages and disadvantages 79
43 Hybrid method 79 431 Formulation 79 432 Optimality conditions 82 433 Algorithm 84 434 Advantages and disadvantages 85
44 Improved hybrid method 86 441 Formulations 86 442 Optimality conditions 86 443 Algorithm 89 444 Advantages and disadvantages 90
45 Optimum safety factor method 91 451 Safety factor concept 91 452 Developments and optimality conditions 92 453 Algorithm 97 454 Advantages and disadvantages 98
46 Safest point method 98 461 Formulations 98 462 Algorithm 102 463 Advantages and disadvantages 104
47 Numerical applications 105 471 RBDO of a hook CM and HM 105 472 RBDO of a triangular plate HM amp IHM 107 473 RBDO of a console beam (sandwich beam) HM and OSF 110 474 RBDO of an aircraft wing HM amp SP 113
48 Classification of the methods developed 115 481 Numerical methods 115 482 Semi-numerical methods 116 483 Comparison between the numerical- and semi-numerical methods 118
49 Conclusion 119
Chapter 5 Reliability-based Topology Optimization Model 121
51 Introduction 121 52 Formulation and algorithm for the RBTO model 122
521 Formulation 122
viii Biomechanics
522 Algorithm 123 523 Validation of the RBTO code developed 125
53 Validation of the RBTO model 126 531 Analytical validation 126 532 Numerical validation 128
54 Variability of the reliability index 134 541 Example 1 MBB beam 136 542 Example 2 Cantilever beam 136 543 Example 3 Cantilever beam with double loads 136 544 Example 4 Cantilever beam with a transversal hole 136
55 Numerical applications for the RBTO model 137 551 Static analysis 138 552 Modal analysis 139 553 Fatigue analysis 141
56 Two points of view for integration of reliability into topology optimization 142
561 Point of view of ldquotopologyrdquo 144 562 Point of view of ldquoreliabilityrdquo 144 563 Numerical applications for the two points of view 146
57 Conclusion 152
Chapter 6 Integration of Reliability and Structural Optimization into Prosthesis Design 153
61 Introduction 153 62 Prosthesis design 154 63 Integration of topology optimization into prosthesis design 154
631 Importance of topology optimization in prosthesis design 155 632 Place of topology optimization in the prosthesis design chain 156
64 Integration of reliability and structural optimization into hip prosthesis design 157
641 Numerical application of the deterministic approach 158 642 Numerical application of the reliability-based approach 167
65 Integration of reliability and structural optimization into the design of mini-plate systems used to treat fractured mandibles 174
651 Numerical application of the deterministic approach 174 652 Numerical application of the reliability-based approach 181
Contents ix
66 Integration of reliability and structural optimization into dental implant design 184
661 Description and modeling of the problem 184 662 Numerical results 186
67 Conclusion 188
Appendices 189
Appendix 1 ANSYS Code for Stem Geometry 191
Appendix 2 ANSYS Code for Mini-Plate Geometry 197
Appendix 3 ANSYS Code for Dental Implant Geometry 201
Appendix 4 ANSYS Code for Geometry of Dental Implant with Bone 207
Bibliography 213
Index 229
Preface
The integration of structural optimization into biomechanics is a truly vast domain In this book we first focus on the integration of structural optimization into the design of orthopedic and orthodontic prostheses and also into drilling surgery Next we present the integration of reliability and structural optimization into the design of these prostheses which may be considered as a novel aspect introduced in this book The applications are made in 2D and in 3D considering the three major families of structural optimization sizing- shape- and topology optimization
In all domains of structural mechanics good design of a part is very important for its strength its lifetime and its use in service This is a challenge faced daily in sectors such as space research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve the techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile engines providing substantial energy savings The development of computer-aided design (CAD) techniques and optimization strategies is part of this context
Applying structural optimization is still somewhat complicated in certain domains Furthermore in deterministic structural optimization all parameters which are uncertain in nature are described by unfavorable characteristic values associated with safety coefficients The deterministic approach uses a pessimistic margin determined as a function of the consequences of a probable failure This approach often leads to unnecessary oversizing ndash particularly for sensitive structures
On the other hand researchers have developed a different approach which is better suited to uncertain physical phenomena In this approach the structure is deemed to have failed if the probability of failure is greater than a fixed threshold This is known as the ldquoprobabilistic approachrdquo The probabilistic approach is
xii Biomechanics
increasingly widely used in engineering as evidenced by the different applications in industry It is applied to check that the probability is sufficient when the structurersquos geometry is known or to optimize the sizing of the structure so as to respect certain fixed objectives such as a target cost or a required level of probability
Furthermore reliability analysis is an important tool in decision-making for establishing a maintenance- and inspection plan In addition it can be used in the validation of standards and regulations To perform reliability analysis various methods can be used to effectively and simply find the probability of failure Reliability analysis is a strategy used to evaluate the level of reliability without being able to control the design for a required reliability level For this reliability has become an important tool to be integrated into the process of structural optimization
This book also focuses on the necessary tools for the integration of reliability and structural optimization into biomechanics fields First the deterministic strategies of structural optimization are presented so we can implement them in structural design These deterministic strategies are applied in various domains in biomechanics including the design of orthopedic and orthodontic prostheses and drilling surgery Next reliability-based approaches pertaining to the integration of reliability into structural optimization are presented in detail with mechanical applications These reliability-based strategies are also applied in the design of orthopedic and orthodontic prostheses taking account of uncertainty in terms of geometry materials and load Finally system reliability strategies are also taken into account considering several failure scenarios
The book will provide invaluable support to teaching staff and researchers It is also intended for engineering students practising engineers and Masters students
Acknowledgements
We would like to thank all of those people who have in some way great or small contributed to the writing of this book ndash in particular Sophie Le Cann a researcher at the Biomedical Centre (BMC) at Lund University for her contribution in terms of biological language Heartfelt thanks go to our families to our students and to our colleagues for their massive moral support during the writing of this book
Ghias KHARMANDA Abdelkhalak EL-HAMI
October 2016
Introduction
This book begins with an introductory look at the fundamental principles of structural optimization before applying them to the field of biomechanics Then we present the different strategies for integrating reliability into structural optimization followed by the application of those strategies in biomechanics ndash particularly in terms of the design of orthopedic and orthodontic prostheses
In terms of structural optimization to illustrate the different techniques we can classify structural optimization into three main families
1) Sizing Optimization this model aims to improve a structural model whilst respecting the available resources (known as constraints or limitations) Sizing optimization is the particular case where we can only modify the cross section or transverse thickness of the components of a structure whose shape and topology are fixed There can be no modification of the geometric features andor models
2) Shape Optimization with this model it is possible to make changes to the shape provided they are compatible with a predefined topology This type of optimization modifies the parametric representation of the boundaries of the domain By moving those boundaries we try to find the best possible solution out of the set of all the configurations obtained by homeomorphic transformation of the original structure
3) Topology Optimization this model enables us to make more profound modifications to the shape of the structure Here the geometry of the part is examined with no prerequisites as to the connections of the domains or the structural elements present in the solution In order to optimize the topology we determine the structurersquos shape or transverse dimensions so some authors call topology optimization ldquogeneralized shape optimizationrdquo
xiv Biomechanics
In terms of reliability this concept can be integrated into all three families of structural optimization so we obtain a design that should be both optimal and reliable Sizing optimization shape optimization and topology optimization are generally classed as geometry optimization However the nature of the topology is non-quantitative in relation to shape and size For this reason this integration is divided into two models
1) RBDO Reliability-Based Design Optimization this model couples reliability analysis with sizing optimization (Reliability-Based Sizing Optimization) and also with shape optimization (Reliability-Based Shape Optimization) This coupling is a complex task requiring a lengthy computation time which seriously limits its applicability In addition during the process of reliability-based shape optimization the geometry of the structure is forced to change This coupling integrates different disciplines such as geometric modeling numerical simulation reliability analysis and optimization Thus the optimization problem becomes more complex The major difficulty lies in evaluating the reliability of the structure which in itself requires a specific optimization procedure The typical integration of reliability analysis into optimization methods is carried out in two spaces the normalized space of random variables and the physical space of design variables which requires a very significant computation time To solve this problem as we shall see there are a number of efficient methods
2) RBTO Reliability-Based Topology Optimization in the deterministic case we obtain a single optimal topology while the new RBTO model is able to generate several topologies as a function of a required reliability level However the coupling between reliability (which is quantitative) and topology (which is non-quantitative) requires the use of different methodologies relative to sizing optimization and shape optimization
In terms of biomechanics modeling is generally complicated on several different levels geometric description material properties and boundary conditions When performing the optimization process such modeling is needed with a new set of parameters being entered at each iteration In addition the problem becomes more complex when integrating reliability analysis which is itself performed by way of a particular optimization procedure In this case we couple several different software packages in order to integrate optimization and reliability into biomechanical applications Various strategies are presented in this book to simplify the coupling problem
This book is made up of six chapters presenting the different aspects of integration of reliability and structural optimization into biomechanics
Introduction xv
The first chapter offers an introduction to structural optimization and its main groups sizing optimization shape optimization and topology optimization The principles underpinning the three main types of structural optimization are presented with numerical applications for each type In structural design we have two different phases the conceptual phase and the detailed phase In the conceptual phase we use topology optimization to obtain an idea of the silhouette of the structure The detailed phase consists of shape optimization to obtain smooth geometry followed by sizing optimization to find the areas and thicknesses of the studied structure
The second chapter is devoted to the integration of structural optimization into biomechanics The integration is performed in this chapter for prosthesis design and drilling surgery Different 2D and 3D applications for orthopedic and orthodontic prostheses are presented in order to demonstrate the new strategies presented in this book and also to prove the advantages of those strategies Finally the integration of structural optimization into drilling surgery can be considered a novel aspect in relation to classic prosthesis design
The third chapter is given over to the integration of reliability into structural optimization Such integration can be viewed as a difficult task The difficulties arise in terms of coupling convergence and computation time In this chapter we present the basic principles of that integration and a literature review based on early attempts in this area A numerical application is presented to show the difference between deterministic- and reliability-based optimization Next two viewpoints are presented with the corresponding developed methods considering those two points of view to illustrate the advantages of the methods
The fourth chapter is devoted to the reliability-based design optimization (RBDO) model The developed methods are classified into two categories numerical and semi-numerical methods Numerical methods are generally applicable but require a high computation time Semi-numerical methods on the other hand are efficient in terms of computation time but can only be used in certain cases
The fifth chapter details reliability-based topology optimization (RBTO) This model enables designers and manufacturers to select the solution which is at once economical and reliable Two points of view are presented in the discussion of this model A number of applications in mechanics are presented in order to demonstrate the advantages of the RBTO model
The sixth chapter discusses the integration of reliability and structural optimization into orthopedic and orthodontic prostheses design considering two
xvi Biomechanics
approaches the deterministic approach and the reliability-based approach In orthopedic prostheses the models are presented using results drawn from the existing body of literature The aspect of materials characterization is also integrated into the design process In addition in orthodontics models are presented to test the stability of the prostheses in light of the constraints of dentistry The integration of reliability and structural optimization into the design of prostheses is performed very simply in order to aid understanding and implementation of the different strategies presented in this book
List of Abbreviations
AMA Approximate Moments Approach
AMV Advanced Mean Value
APDL ANSYS Parametric Design Language
AT Anterior Temporal (or Temporalis)
CAD Computer-Aided Design
CM Classical Method
CMV Conjugate Mean Value
CT Computed Tomography
DDO Deterministic Design Optimization
DLA Double-Level Approach
DM Deep Masseter
DO Deterministic Optimization
DTO Deterministic Topology Optimization
FE Finite Elements
FEM Finite Element Method
FORM First Order Reliability Method
HCA Hybrid Cellular Automata
HDS Hybrid Design Space
HM Hybrid Method
xviii Biomechanics
HMV Hybrid Mean Value
IAM Improved Austin-Moore
IHM Improved Hybrid Method
KKT Karush-Kuhn-Tucker
MBB Messerschmitt-Boumllkow-Blohm
MLA Mono-Level Approach
MP Medial Pterygoid
MPP Most Probable failure Point
MT Medial Temporal (or Temporalis)
OC Optimality Criteria
OSF Optimum Safety Factor
PDF Probability Density Function
PMA Performance Measure Approach
PT Posterior Temporal (or Temporalis)
RBDO Reliability-Based Design Optimization
RBO Reliability-Based Optimization
RBSO Reliability-Based Structural Optimization
RBTO Reliability-Based Topology Optimization
RIA Reliability Index Approach
SAP Sequential Approximate Programming
SDA Sequential Decoupled Approach
SED Strain Energy Density
SIMP Solid Isotropic Microstructure with Penalty
SM Superficial Masseter
SORA Sequential Optimization and Reliability Assessment
SORM Second Order Reliability Method
SP Safest Point
1
Introduction to Structural Optimization
11 Introduction
Structural optimization is a topic which affects many different physical domains ndash particularly solid mechanics ndash and which it is tricky to characterize as its formulations may have a number of different aspects Firstly a distinction regarding the way in which geometries are parameterized is presented and secondly a distribution pertaining to the intrinsic nature of optimization algorithms is established
Determining the appropriate shape for structural components is a crucially important problem for engineers In all areas of structural mechanics the impact of proper design of a part is very significant in terms of its strength its lifetime and its usage This is a challenge faced on a daily basis in the sectors of spatial research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile machinery providing substantial energy savings The different development of computer-aided design (CAD) techniques and optimization strategies is part of this context There has been keen interest in structural optimization for over thirty years Whilst it is still too infrequently applied in the conventional techniques used by research centers it is becoming more widely used as its reliability improves Having begun with the simplest of problems the field of application of structural optimization today extends to new and ever more interesting challenges
To illustrate the evolution of structural optimization techniques we can arbitrarily split structural optimization into three major groups (or families) In
Biomechanics Optimization Uncertainties and Reliability First Edition Ghias Kharmanda and Abdelkhalak El Hami copy ISTE Ltd 2017 Published by ISTE Ltd and John Wiley amp Sons Inc
2 Biomechanics
historical terms each of them has been addressed in order of increasing difficulty and generality
With sizing optimization we are only able to modify the dimensions of an object whose shape and topology are fixed There can be no modification of the geometric model We speak of a homeomorphic transformation
Shape optimization involves making changes of shape which are compatible with a predetermined topology Typical shape optimization modifies the parametric representation of the boundaries of the domain By moving the boundaries of the domains we can seek the best solution out of all the structures obtained by homeomorphic transformation of the original object In this case it is clear that we can make a change to the transverse dimensions as well as a modification to the objectrsquos configuration but it is certainly not acceptable to modify its connectivity or its nature ndash in particular the number of components that it has The optimal object exhibits the same topology as the original object
With topology optimization we can fundamentally change the nature of the object The ldquotopologyrdquo refers to the number and position of the components of the domains Here the objectrsquos geometry is presented with no prerequisites as to the connectivity of the domains or the components present in the solution We take no initial information about the topology of the optimal shape
12 History of structural optimization
It was in the early 1960s that Schmit [SCH 60] and Fox [FOX 65] laid the foundations for a modern theory of structural optimization based on the concepts of mathematical programming and sensitivity analysis Paradoxically at the time ldquofully stressed designrdquo was the only widely used technique in practice although it lacked any theoretical justification other than empiricism and the engineersrsquo intuition It was Prager and Taylor [PRA 68] who set out variational methods and Lagrangian optimality conditions to justify the criteria of fully stressed design for a class of structural optimization problems The optimality conditions of the optimization problem were then used directly to construct an iterative resolution algorithm known as the ldquooptimality criteria methodrdquo
Originally structural optimization was mainly limited to the sizing optimization of trusses or gantries Thus sizing optimization of structures was the first field of application for optimality criteria When dealing with the problem of sizing we look at the transverse sections of the structural elements though their length and the
Introduction to Structural Optimization 3
location of their joints remain fixed During the late 1960s and early 1970s optimality criteria were soon adapted to large structures modeled using the finite-element method (FEM) [VEN 73] The optimality criteria method produced a few interesting results and a number of extensions have been presented since the 1970s including Venkayyarsquos generalized criteria method [VEN 73] or Rozvany and Zhoursquos [ROZ 91] discretized optimality criteria
Although since the 1970s attention has mainly been focused on sizing the problem of truss topology has also been studied by Prager [PRA 74] on a very restricted class of structures based on the concept of a truss put forward by Michell in 1904 [MIC 04] The problem lies in finding the best possible configuration so that the truss or gantry can convey forces to the foundations whilst minimizing a given performance and satisfying the design constraints Michellrsquos theory [MIC 04] is related to the topology of trusses made of bars of minimal mass The optimal solution from Michellrsquos point of view is composed solely of perpendicular bars which form a structure whose configuration is optimal for the maximum tensile- and compressive stresses All the configuration problems studied by Prager later on were solved analytically so the practical application of topology was very limited To remedy this shortcoming Rozvany [ROZ 76] invested a great deal of effort in developing new approaches to solve these configuration problems as automatically as possible
After that optimal truss topology was studied in greater depth by Kirsch [KIR 90] If we impose a very small minimum value for the cross-section then the ldquolayoutrdquo optimization of trusses and gantries can be approached as a conventional sizing problem on a very large scale The solution is then obtained by application of generalized optimality criteria for a variety of objective functions compliance movements tensions and eigenvalues [ZHO 91]
Finally the crux of the problem of truss topology optimization seems to have been identified by Bendsoslashe et al [BEN 91] The problem of truss topology is examined with an integral approach combining analysis and design simultaneously The problem of minimum compliance is transformed into a problem of non-differentiable optimization and then in the case of a truss of bars into a linear problem In this form it can be solved on very large structures using non-differentiable optimization methods [BEN 93a] an interior point method [BEN 93b] a dual method [BEC 94] or a penaltybarrier multiplier method [BEN 97]
The three main groups of structural optimization cannot be considered recent concepts but their integration into biomechanics is ndash especially topology optimization [FRA 10]
4 Biomechanics
13 Sizing optimization
131 Definition
With sizing optimization we can only modify the cross-section or transverse thickness of the components of structure whose shape and topology are fixed There can be no modification of the geometric model and its features Figure 11 shows a truss made of 17 bars with different cross-sections (circular rectangular and I-shaped)
Figure 11 Changing the dimensions whilst preserving the same topology of the section
Sizing optimization can be performed by considering the same topology to produce various dimensions For example when the cross-section is circular we merely need to vary the diameters to minimize an objective or several objectives under certain constraints
132 First works in sizing optimization
The problem of sizing optimization has benefited the most from this research so optimization of the transverse dimensions is today a reliable tool This problem was also extended to that of flexural elements [FLE 83] to improving performance when subjected to vibration and to the stability of balance Besides the transverse dimensions of the structural elements it is possible to vary their shape To the best of our knowledge few works have been devoted to the study of the optimal shape for a truss In this type of problem only the location of the structural joints is altered whilst the topology remains unchanged Svanberg [SVA 81] was one of the only people to carry out truss shape optimization on the basis of FEM analyses and solving using mathematical programming techniques
Introduction to Structural Optimization 5
133 Numerical application
1331 Description and modeling of the studied problem
Figure 12a shows a cantilever beam and its I-shaped cross-section in Figure 12b This beam is embedded at one end and subjected to free vibration The material from which the beam is made is structural steel which has a Youngrsquos modulus 200000E MPa= and Poissonrsquos ratio of 03ν = The density of the material is 6 37854 10 Kg mmρ minus= times The material exhibits linear elastic isotropic behavior The length of that beam is 300L mm= and the dimensions of the cross-section are 60 B mm= 100H mm= and 20T mm= (Figure 12b)
a) b)
Figure 12 Cantilever beam subject to free vibration
The objective of this study is to calculate the first four modes of resonance and then perform sizing optimization on the dimensions of the cross-section The problem of optimization therefore is to minimize the structural volume under the constraint of the first resonance frequency This problem can be formulated as follows
1
min ( ) ( ) 0
40 100 80 16010 30
w
Volume B H Ts t f B H T f
BHT
minus lele lele lele le
[11]
where 20wf Hz= is the maximum value of the first resonance frequency
1332 Numerical results
To perform sizing optimization on the ANSYS software for instance we consider the dimensions of the cross-section as optimization variables so as to obtain a parameterized model Then a direct simulation is performed as the heart of the optimization loop
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
Biomechanics
Reliability of Multiphysical Systems Set coordinated by
Abdelkhalak El Hami
Volume 5
Biomechanics
Optimization Uncertainties and Reliability
Ghias Kharmanda Abdelkhalak El Hami
First published 2017 in Great Britain and the United States by ISTE Ltd and John Wiley amp Sons Inc
Apart from any fair dealing for the purposes of research or private study or criticism or review as permitted under the Copyright Designs and Patents Act 1988 this publication may only be reproduced stored or transmitted in any form or by any means with the prior permission in writing of the publishers or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address
ISTE Ltd John Wiley amp Sons Inc 27-37 St Georgersquos Road 111 River Street London SW19 4EU Hoboken NJ 07030 UK USA
wwwistecouk wwwwileycom
copy ISTE Ltd 2017 The rights of Ghias Kharmanda and Abdelkhalak El Hami to be identified as the authors of this work have been asserted by them in accordance with the Copyright Designs and Patents Act 1988
Library of Congress Control Number 2016952066 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-025-6
Contents
Preface xi
Introduction xiii
List of Abbreviations xvii
Chapter 1 Introduction to Structural Optimization 1
11 Introduction 1 12 History of structural optimization 2 13 Sizing optimization 4
131 Definition 4 132 First works in sizing optimization 4 133 Numerical application 5
14 Shape optimization 10 141 Definition 10 142 First works in shape optimization 11 143 Numerical application 12
15 Topology optimization 16 151 Definition 16 152 First works in topology optimization 17 153 Numerical application 18
16 Conclusion 21
Chapter 2 Integration of Structural Optimization into Biomechanics 23
21 Introduction 23 22 Integration of structural optimization into orthopedic prosthesis design 23
vi Biomechanics
221 Structural optimization of the hip prosthesis 24 222 Sizing optimization of a 3D intervertebral disk prosthesis 42
23 Integration of structural optimization into orthodontic prosthesis design 47
231 Sizing optimization of a dental implant 47 232 Shape optimization of a mini-plate 49
24 Advanced integration of structural optimization into drilling surgery 52
241 Case of treatment of a crack with a single hole 53 242 Case of treatment of a crack with two holes 54
25 Conclusion 56
Chapter 3 Integration of Reliability into Structural Optimization 57
31 Introduction 57 32 Literature review of reliability-based optimization 58 33 Comparison between deterministic and reliability-based optimization 60
331 Deterministic optimization 61 332 Reliability-based optimization 63
34 Numerical application 64 341 Description and modeling of the studied problem 64 342 Numerical results 65
35 Approaches and strategies for reliability-based optimization 68
351 Mono-level approaches 68 352 Double-level approaches 68 353 Sequential decoupled approaches 68
36 Two points of view for developments of reliability-based optimization 69
361 Point of view of ldquoReliabilityrdquo 69 362 Point of view of ldquoOptimizationrdquo 70 363 Method efficiency 70
37 Philosophy of integration of the concept of reliability into structural optimization groups 72 38 Conclusion 73
Chapter 4 Reliability-based Design Optimization Model 75
41 Introduction 75 42 Classic method 76
Contents vii
421 Formulations 76 422 Optimality conditions 77 423 Algorithm 77 424 Advantages and disadvantages 79
43 Hybrid method 79 431 Formulation 79 432 Optimality conditions 82 433 Algorithm 84 434 Advantages and disadvantages 85
44 Improved hybrid method 86 441 Formulations 86 442 Optimality conditions 86 443 Algorithm 89 444 Advantages and disadvantages 90
45 Optimum safety factor method 91 451 Safety factor concept 91 452 Developments and optimality conditions 92 453 Algorithm 97 454 Advantages and disadvantages 98
46 Safest point method 98 461 Formulations 98 462 Algorithm 102 463 Advantages and disadvantages 104
47 Numerical applications 105 471 RBDO of a hook CM and HM 105 472 RBDO of a triangular plate HM amp IHM 107 473 RBDO of a console beam (sandwich beam) HM and OSF 110 474 RBDO of an aircraft wing HM amp SP 113
48 Classification of the methods developed 115 481 Numerical methods 115 482 Semi-numerical methods 116 483 Comparison between the numerical- and semi-numerical methods 118
49 Conclusion 119
Chapter 5 Reliability-based Topology Optimization Model 121
51 Introduction 121 52 Formulation and algorithm for the RBTO model 122
521 Formulation 122
viii Biomechanics
522 Algorithm 123 523 Validation of the RBTO code developed 125
53 Validation of the RBTO model 126 531 Analytical validation 126 532 Numerical validation 128
54 Variability of the reliability index 134 541 Example 1 MBB beam 136 542 Example 2 Cantilever beam 136 543 Example 3 Cantilever beam with double loads 136 544 Example 4 Cantilever beam with a transversal hole 136
55 Numerical applications for the RBTO model 137 551 Static analysis 138 552 Modal analysis 139 553 Fatigue analysis 141
56 Two points of view for integration of reliability into topology optimization 142
561 Point of view of ldquotopologyrdquo 144 562 Point of view of ldquoreliabilityrdquo 144 563 Numerical applications for the two points of view 146
57 Conclusion 152
Chapter 6 Integration of Reliability and Structural Optimization into Prosthesis Design 153
61 Introduction 153 62 Prosthesis design 154 63 Integration of topology optimization into prosthesis design 154
631 Importance of topology optimization in prosthesis design 155 632 Place of topology optimization in the prosthesis design chain 156
64 Integration of reliability and structural optimization into hip prosthesis design 157
641 Numerical application of the deterministic approach 158 642 Numerical application of the reliability-based approach 167
65 Integration of reliability and structural optimization into the design of mini-plate systems used to treat fractured mandibles 174
651 Numerical application of the deterministic approach 174 652 Numerical application of the reliability-based approach 181
Contents ix
66 Integration of reliability and structural optimization into dental implant design 184
661 Description and modeling of the problem 184 662 Numerical results 186
67 Conclusion 188
Appendices 189
Appendix 1 ANSYS Code for Stem Geometry 191
Appendix 2 ANSYS Code for Mini-Plate Geometry 197
Appendix 3 ANSYS Code for Dental Implant Geometry 201
Appendix 4 ANSYS Code for Geometry of Dental Implant with Bone 207
Bibliography 213
Index 229
Preface
The integration of structural optimization into biomechanics is a truly vast domain In this book we first focus on the integration of structural optimization into the design of orthopedic and orthodontic prostheses and also into drilling surgery Next we present the integration of reliability and structural optimization into the design of these prostheses which may be considered as a novel aspect introduced in this book The applications are made in 2D and in 3D considering the three major families of structural optimization sizing- shape- and topology optimization
In all domains of structural mechanics good design of a part is very important for its strength its lifetime and its use in service This is a challenge faced daily in sectors such as space research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve the techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile engines providing substantial energy savings The development of computer-aided design (CAD) techniques and optimization strategies is part of this context
Applying structural optimization is still somewhat complicated in certain domains Furthermore in deterministic structural optimization all parameters which are uncertain in nature are described by unfavorable characteristic values associated with safety coefficients The deterministic approach uses a pessimistic margin determined as a function of the consequences of a probable failure This approach often leads to unnecessary oversizing ndash particularly for sensitive structures
On the other hand researchers have developed a different approach which is better suited to uncertain physical phenomena In this approach the structure is deemed to have failed if the probability of failure is greater than a fixed threshold This is known as the ldquoprobabilistic approachrdquo The probabilistic approach is
xii Biomechanics
increasingly widely used in engineering as evidenced by the different applications in industry It is applied to check that the probability is sufficient when the structurersquos geometry is known or to optimize the sizing of the structure so as to respect certain fixed objectives such as a target cost or a required level of probability
Furthermore reliability analysis is an important tool in decision-making for establishing a maintenance- and inspection plan In addition it can be used in the validation of standards and regulations To perform reliability analysis various methods can be used to effectively and simply find the probability of failure Reliability analysis is a strategy used to evaluate the level of reliability without being able to control the design for a required reliability level For this reliability has become an important tool to be integrated into the process of structural optimization
This book also focuses on the necessary tools for the integration of reliability and structural optimization into biomechanics fields First the deterministic strategies of structural optimization are presented so we can implement them in structural design These deterministic strategies are applied in various domains in biomechanics including the design of orthopedic and orthodontic prostheses and drilling surgery Next reliability-based approaches pertaining to the integration of reliability into structural optimization are presented in detail with mechanical applications These reliability-based strategies are also applied in the design of orthopedic and orthodontic prostheses taking account of uncertainty in terms of geometry materials and load Finally system reliability strategies are also taken into account considering several failure scenarios
The book will provide invaluable support to teaching staff and researchers It is also intended for engineering students practising engineers and Masters students
Acknowledgements
We would like to thank all of those people who have in some way great or small contributed to the writing of this book ndash in particular Sophie Le Cann a researcher at the Biomedical Centre (BMC) at Lund University for her contribution in terms of biological language Heartfelt thanks go to our families to our students and to our colleagues for their massive moral support during the writing of this book
Ghias KHARMANDA Abdelkhalak EL-HAMI
October 2016
Introduction
This book begins with an introductory look at the fundamental principles of structural optimization before applying them to the field of biomechanics Then we present the different strategies for integrating reliability into structural optimization followed by the application of those strategies in biomechanics ndash particularly in terms of the design of orthopedic and orthodontic prostheses
In terms of structural optimization to illustrate the different techniques we can classify structural optimization into three main families
1) Sizing Optimization this model aims to improve a structural model whilst respecting the available resources (known as constraints or limitations) Sizing optimization is the particular case where we can only modify the cross section or transverse thickness of the components of a structure whose shape and topology are fixed There can be no modification of the geometric features andor models
2) Shape Optimization with this model it is possible to make changes to the shape provided they are compatible with a predefined topology This type of optimization modifies the parametric representation of the boundaries of the domain By moving those boundaries we try to find the best possible solution out of the set of all the configurations obtained by homeomorphic transformation of the original structure
3) Topology Optimization this model enables us to make more profound modifications to the shape of the structure Here the geometry of the part is examined with no prerequisites as to the connections of the domains or the structural elements present in the solution In order to optimize the topology we determine the structurersquos shape or transverse dimensions so some authors call topology optimization ldquogeneralized shape optimizationrdquo
xiv Biomechanics
In terms of reliability this concept can be integrated into all three families of structural optimization so we obtain a design that should be both optimal and reliable Sizing optimization shape optimization and topology optimization are generally classed as geometry optimization However the nature of the topology is non-quantitative in relation to shape and size For this reason this integration is divided into two models
1) RBDO Reliability-Based Design Optimization this model couples reliability analysis with sizing optimization (Reliability-Based Sizing Optimization) and also with shape optimization (Reliability-Based Shape Optimization) This coupling is a complex task requiring a lengthy computation time which seriously limits its applicability In addition during the process of reliability-based shape optimization the geometry of the structure is forced to change This coupling integrates different disciplines such as geometric modeling numerical simulation reliability analysis and optimization Thus the optimization problem becomes more complex The major difficulty lies in evaluating the reliability of the structure which in itself requires a specific optimization procedure The typical integration of reliability analysis into optimization methods is carried out in two spaces the normalized space of random variables and the physical space of design variables which requires a very significant computation time To solve this problem as we shall see there are a number of efficient methods
2) RBTO Reliability-Based Topology Optimization in the deterministic case we obtain a single optimal topology while the new RBTO model is able to generate several topologies as a function of a required reliability level However the coupling between reliability (which is quantitative) and topology (which is non-quantitative) requires the use of different methodologies relative to sizing optimization and shape optimization
In terms of biomechanics modeling is generally complicated on several different levels geometric description material properties and boundary conditions When performing the optimization process such modeling is needed with a new set of parameters being entered at each iteration In addition the problem becomes more complex when integrating reliability analysis which is itself performed by way of a particular optimization procedure In this case we couple several different software packages in order to integrate optimization and reliability into biomechanical applications Various strategies are presented in this book to simplify the coupling problem
This book is made up of six chapters presenting the different aspects of integration of reliability and structural optimization into biomechanics
Introduction xv
The first chapter offers an introduction to structural optimization and its main groups sizing optimization shape optimization and topology optimization The principles underpinning the three main types of structural optimization are presented with numerical applications for each type In structural design we have two different phases the conceptual phase and the detailed phase In the conceptual phase we use topology optimization to obtain an idea of the silhouette of the structure The detailed phase consists of shape optimization to obtain smooth geometry followed by sizing optimization to find the areas and thicknesses of the studied structure
The second chapter is devoted to the integration of structural optimization into biomechanics The integration is performed in this chapter for prosthesis design and drilling surgery Different 2D and 3D applications for orthopedic and orthodontic prostheses are presented in order to demonstrate the new strategies presented in this book and also to prove the advantages of those strategies Finally the integration of structural optimization into drilling surgery can be considered a novel aspect in relation to classic prosthesis design
The third chapter is given over to the integration of reliability into structural optimization Such integration can be viewed as a difficult task The difficulties arise in terms of coupling convergence and computation time In this chapter we present the basic principles of that integration and a literature review based on early attempts in this area A numerical application is presented to show the difference between deterministic- and reliability-based optimization Next two viewpoints are presented with the corresponding developed methods considering those two points of view to illustrate the advantages of the methods
The fourth chapter is devoted to the reliability-based design optimization (RBDO) model The developed methods are classified into two categories numerical and semi-numerical methods Numerical methods are generally applicable but require a high computation time Semi-numerical methods on the other hand are efficient in terms of computation time but can only be used in certain cases
The fifth chapter details reliability-based topology optimization (RBTO) This model enables designers and manufacturers to select the solution which is at once economical and reliable Two points of view are presented in the discussion of this model A number of applications in mechanics are presented in order to demonstrate the advantages of the RBTO model
The sixth chapter discusses the integration of reliability and structural optimization into orthopedic and orthodontic prostheses design considering two
xvi Biomechanics
approaches the deterministic approach and the reliability-based approach In orthopedic prostheses the models are presented using results drawn from the existing body of literature The aspect of materials characterization is also integrated into the design process In addition in orthodontics models are presented to test the stability of the prostheses in light of the constraints of dentistry The integration of reliability and structural optimization into the design of prostheses is performed very simply in order to aid understanding and implementation of the different strategies presented in this book
List of Abbreviations
AMA Approximate Moments Approach
AMV Advanced Mean Value
APDL ANSYS Parametric Design Language
AT Anterior Temporal (or Temporalis)
CAD Computer-Aided Design
CM Classical Method
CMV Conjugate Mean Value
CT Computed Tomography
DDO Deterministic Design Optimization
DLA Double-Level Approach
DM Deep Masseter
DO Deterministic Optimization
DTO Deterministic Topology Optimization
FE Finite Elements
FEM Finite Element Method
FORM First Order Reliability Method
HCA Hybrid Cellular Automata
HDS Hybrid Design Space
HM Hybrid Method
xviii Biomechanics
HMV Hybrid Mean Value
IAM Improved Austin-Moore
IHM Improved Hybrid Method
KKT Karush-Kuhn-Tucker
MBB Messerschmitt-Boumllkow-Blohm
MLA Mono-Level Approach
MP Medial Pterygoid
MPP Most Probable failure Point
MT Medial Temporal (or Temporalis)
OC Optimality Criteria
OSF Optimum Safety Factor
PDF Probability Density Function
PMA Performance Measure Approach
PT Posterior Temporal (or Temporalis)
RBDO Reliability-Based Design Optimization
RBO Reliability-Based Optimization
RBSO Reliability-Based Structural Optimization
RBTO Reliability-Based Topology Optimization
RIA Reliability Index Approach
SAP Sequential Approximate Programming
SDA Sequential Decoupled Approach
SED Strain Energy Density
SIMP Solid Isotropic Microstructure with Penalty
SM Superficial Masseter
SORA Sequential Optimization and Reliability Assessment
SORM Second Order Reliability Method
SP Safest Point
1
Introduction to Structural Optimization
11 Introduction
Structural optimization is a topic which affects many different physical domains ndash particularly solid mechanics ndash and which it is tricky to characterize as its formulations may have a number of different aspects Firstly a distinction regarding the way in which geometries are parameterized is presented and secondly a distribution pertaining to the intrinsic nature of optimization algorithms is established
Determining the appropriate shape for structural components is a crucially important problem for engineers In all areas of structural mechanics the impact of proper design of a part is very significant in terms of its strength its lifetime and its usage This is a challenge faced on a daily basis in the sectors of spatial research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile machinery providing substantial energy savings The different development of computer-aided design (CAD) techniques and optimization strategies is part of this context There has been keen interest in structural optimization for over thirty years Whilst it is still too infrequently applied in the conventional techniques used by research centers it is becoming more widely used as its reliability improves Having begun with the simplest of problems the field of application of structural optimization today extends to new and ever more interesting challenges
To illustrate the evolution of structural optimization techniques we can arbitrarily split structural optimization into three major groups (or families) In
Biomechanics Optimization Uncertainties and Reliability First Edition Ghias Kharmanda and Abdelkhalak El Hami copy ISTE Ltd 2017 Published by ISTE Ltd and John Wiley amp Sons Inc
2 Biomechanics
historical terms each of them has been addressed in order of increasing difficulty and generality
With sizing optimization we are only able to modify the dimensions of an object whose shape and topology are fixed There can be no modification of the geometric model We speak of a homeomorphic transformation
Shape optimization involves making changes of shape which are compatible with a predetermined topology Typical shape optimization modifies the parametric representation of the boundaries of the domain By moving the boundaries of the domains we can seek the best solution out of all the structures obtained by homeomorphic transformation of the original object In this case it is clear that we can make a change to the transverse dimensions as well as a modification to the objectrsquos configuration but it is certainly not acceptable to modify its connectivity or its nature ndash in particular the number of components that it has The optimal object exhibits the same topology as the original object
With topology optimization we can fundamentally change the nature of the object The ldquotopologyrdquo refers to the number and position of the components of the domains Here the objectrsquos geometry is presented with no prerequisites as to the connectivity of the domains or the components present in the solution We take no initial information about the topology of the optimal shape
12 History of structural optimization
It was in the early 1960s that Schmit [SCH 60] and Fox [FOX 65] laid the foundations for a modern theory of structural optimization based on the concepts of mathematical programming and sensitivity analysis Paradoxically at the time ldquofully stressed designrdquo was the only widely used technique in practice although it lacked any theoretical justification other than empiricism and the engineersrsquo intuition It was Prager and Taylor [PRA 68] who set out variational methods and Lagrangian optimality conditions to justify the criteria of fully stressed design for a class of structural optimization problems The optimality conditions of the optimization problem were then used directly to construct an iterative resolution algorithm known as the ldquooptimality criteria methodrdquo
Originally structural optimization was mainly limited to the sizing optimization of trusses or gantries Thus sizing optimization of structures was the first field of application for optimality criteria When dealing with the problem of sizing we look at the transverse sections of the structural elements though their length and the
Introduction to Structural Optimization 3
location of their joints remain fixed During the late 1960s and early 1970s optimality criteria were soon adapted to large structures modeled using the finite-element method (FEM) [VEN 73] The optimality criteria method produced a few interesting results and a number of extensions have been presented since the 1970s including Venkayyarsquos generalized criteria method [VEN 73] or Rozvany and Zhoursquos [ROZ 91] discretized optimality criteria
Although since the 1970s attention has mainly been focused on sizing the problem of truss topology has also been studied by Prager [PRA 74] on a very restricted class of structures based on the concept of a truss put forward by Michell in 1904 [MIC 04] The problem lies in finding the best possible configuration so that the truss or gantry can convey forces to the foundations whilst minimizing a given performance and satisfying the design constraints Michellrsquos theory [MIC 04] is related to the topology of trusses made of bars of minimal mass The optimal solution from Michellrsquos point of view is composed solely of perpendicular bars which form a structure whose configuration is optimal for the maximum tensile- and compressive stresses All the configuration problems studied by Prager later on were solved analytically so the practical application of topology was very limited To remedy this shortcoming Rozvany [ROZ 76] invested a great deal of effort in developing new approaches to solve these configuration problems as automatically as possible
After that optimal truss topology was studied in greater depth by Kirsch [KIR 90] If we impose a very small minimum value for the cross-section then the ldquolayoutrdquo optimization of trusses and gantries can be approached as a conventional sizing problem on a very large scale The solution is then obtained by application of generalized optimality criteria for a variety of objective functions compliance movements tensions and eigenvalues [ZHO 91]
Finally the crux of the problem of truss topology optimization seems to have been identified by Bendsoslashe et al [BEN 91] The problem of truss topology is examined with an integral approach combining analysis and design simultaneously The problem of minimum compliance is transformed into a problem of non-differentiable optimization and then in the case of a truss of bars into a linear problem In this form it can be solved on very large structures using non-differentiable optimization methods [BEN 93a] an interior point method [BEN 93b] a dual method [BEC 94] or a penaltybarrier multiplier method [BEN 97]
The three main groups of structural optimization cannot be considered recent concepts but their integration into biomechanics is ndash especially topology optimization [FRA 10]
4 Biomechanics
13 Sizing optimization
131 Definition
With sizing optimization we can only modify the cross-section or transverse thickness of the components of structure whose shape and topology are fixed There can be no modification of the geometric model and its features Figure 11 shows a truss made of 17 bars with different cross-sections (circular rectangular and I-shaped)
Figure 11 Changing the dimensions whilst preserving the same topology of the section
Sizing optimization can be performed by considering the same topology to produce various dimensions For example when the cross-section is circular we merely need to vary the diameters to minimize an objective or several objectives under certain constraints
132 First works in sizing optimization
The problem of sizing optimization has benefited the most from this research so optimization of the transverse dimensions is today a reliable tool This problem was also extended to that of flexural elements [FLE 83] to improving performance when subjected to vibration and to the stability of balance Besides the transverse dimensions of the structural elements it is possible to vary their shape To the best of our knowledge few works have been devoted to the study of the optimal shape for a truss In this type of problem only the location of the structural joints is altered whilst the topology remains unchanged Svanberg [SVA 81] was one of the only people to carry out truss shape optimization on the basis of FEM analyses and solving using mathematical programming techniques
Introduction to Structural Optimization 5
133 Numerical application
1331 Description and modeling of the studied problem
Figure 12a shows a cantilever beam and its I-shaped cross-section in Figure 12b This beam is embedded at one end and subjected to free vibration The material from which the beam is made is structural steel which has a Youngrsquos modulus 200000E MPa= and Poissonrsquos ratio of 03ν = The density of the material is 6 37854 10 Kg mmρ minus= times The material exhibits linear elastic isotropic behavior The length of that beam is 300L mm= and the dimensions of the cross-section are 60 B mm= 100H mm= and 20T mm= (Figure 12b)
a) b)
Figure 12 Cantilever beam subject to free vibration
The objective of this study is to calculate the first four modes of resonance and then perform sizing optimization on the dimensions of the cross-section The problem of optimization therefore is to minimize the structural volume under the constraint of the first resonance frequency This problem can be formulated as follows
1
min ( ) ( ) 0
40 100 80 16010 30
w
Volume B H Ts t f B H T f
BHT
minus lele lele lele le
[11]
where 20wf Hz= is the maximum value of the first resonance frequency
1332 Numerical results
To perform sizing optimization on the ANSYS software for instance we consider the dimensions of the cross-section as optimization variables so as to obtain a parameterized model Then a direct simulation is performed as the heart of the optimization loop
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
Reliability of Multiphysical Systems Set coordinated by
Abdelkhalak El Hami
Volume 5
Biomechanics
Optimization Uncertainties and Reliability
Ghias Kharmanda Abdelkhalak El Hami
First published 2017 in Great Britain and the United States by ISTE Ltd and John Wiley amp Sons Inc
Apart from any fair dealing for the purposes of research or private study or criticism or review as permitted under the Copyright Designs and Patents Act 1988 this publication may only be reproduced stored or transmitted in any form or by any means with the prior permission in writing of the publishers or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address
ISTE Ltd John Wiley amp Sons Inc 27-37 St Georgersquos Road 111 River Street London SW19 4EU Hoboken NJ 07030 UK USA
wwwistecouk wwwwileycom
copy ISTE Ltd 2017 The rights of Ghias Kharmanda and Abdelkhalak El Hami to be identified as the authors of this work have been asserted by them in accordance with the Copyright Designs and Patents Act 1988
Library of Congress Control Number 2016952066 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-025-6
Contents
Preface xi
Introduction xiii
List of Abbreviations xvii
Chapter 1 Introduction to Structural Optimization 1
11 Introduction 1 12 History of structural optimization 2 13 Sizing optimization 4
131 Definition 4 132 First works in sizing optimization 4 133 Numerical application 5
14 Shape optimization 10 141 Definition 10 142 First works in shape optimization 11 143 Numerical application 12
15 Topology optimization 16 151 Definition 16 152 First works in topology optimization 17 153 Numerical application 18
16 Conclusion 21
Chapter 2 Integration of Structural Optimization into Biomechanics 23
21 Introduction 23 22 Integration of structural optimization into orthopedic prosthesis design 23
vi Biomechanics
221 Structural optimization of the hip prosthesis 24 222 Sizing optimization of a 3D intervertebral disk prosthesis 42
23 Integration of structural optimization into orthodontic prosthesis design 47
231 Sizing optimization of a dental implant 47 232 Shape optimization of a mini-plate 49
24 Advanced integration of structural optimization into drilling surgery 52
241 Case of treatment of a crack with a single hole 53 242 Case of treatment of a crack with two holes 54
25 Conclusion 56
Chapter 3 Integration of Reliability into Structural Optimization 57
31 Introduction 57 32 Literature review of reliability-based optimization 58 33 Comparison between deterministic and reliability-based optimization 60
331 Deterministic optimization 61 332 Reliability-based optimization 63
34 Numerical application 64 341 Description and modeling of the studied problem 64 342 Numerical results 65
35 Approaches and strategies for reliability-based optimization 68
351 Mono-level approaches 68 352 Double-level approaches 68 353 Sequential decoupled approaches 68
36 Two points of view for developments of reliability-based optimization 69
361 Point of view of ldquoReliabilityrdquo 69 362 Point of view of ldquoOptimizationrdquo 70 363 Method efficiency 70
37 Philosophy of integration of the concept of reliability into structural optimization groups 72 38 Conclusion 73
Chapter 4 Reliability-based Design Optimization Model 75
41 Introduction 75 42 Classic method 76
Contents vii
421 Formulations 76 422 Optimality conditions 77 423 Algorithm 77 424 Advantages and disadvantages 79
43 Hybrid method 79 431 Formulation 79 432 Optimality conditions 82 433 Algorithm 84 434 Advantages and disadvantages 85
44 Improved hybrid method 86 441 Formulations 86 442 Optimality conditions 86 443 Algorithm 89 444 Advantages and disadvantages 90
45 Optimum safety factor method 91 451 Safety factor concept 91 452 Developments and optimality conditions 92 453 Algorithm 97 454 Advantages and disadvantages 98
46 Safest point method 98 461 Formulations 98 462 Algorithm 102 463 Advantages and disadvantages 104
47 Numerical applications 105 471 RBDO of a hook CM and HM 105 472 RBDO of a triangular plate HM amp IHM 107 473 RBDO of a console beam (sandwich beam) HM and OSF 110 474 RBDO of an aircraft wing HM amp SP 113
48 Classification of the methods developed 115 481 Numerical methods 115 482 Semi-numerical methods 116 483 Comparison between the numerical- and semi-numerical methods 118
49 Conclusion 119
Chapter 5 Reliability-based Topology Optimization Model 121
51 Introduction 121 52 Formulation and algorithm for the RBTO model 122
521 Formulation 122
viii Biomechanics
522 Algorithm 123 523 Validation of the RBTO code developed 125
53 Validation of the RBTO model 126 531 Analytical validation 126 532 Numerical validation 128
54 Variability of the reliability index 134 541 Example 1 MBB beam 136 542 Example 2 Cantilever beam 136 543 Example 3 Cantilever beam with double loads 136 544 Example 4 Cantilever beam with a transversal hole 136
55 Numerical applications for the RBTO model 137 551 Static analysis 138 552 Modal analysis 139 553 Fatigue analysis 141
56 Two points of view for integration of reliability into topology optimization 142
561 Point of view of ldquotopologyrdquo 144 562 Point of view of ldquoreliabilityrdquo 144 563 Numerical applications for the two points of view 146
57 Conclusion 152
Chapter 6 Integration of Reliability and Structural Optimization into Prosthesis Design 153
61 Introduction 153 62 Prosthesis design 154 63 Integration of topology optimization into prosthesis design 154
631 Importance of topology optimization in prosthesis design 155 632 Place of topology optimization in the prosthesis design chain 156
64 Integration of reliability and structural optimization into hip prosthesis design 157
641 Numerical application of the deterministic approach 158 642 Numerical application of the reliability-based approach 167
65 Integration of reliability and structural optimization into the design of mini-plate systems used to treat fractured mandibles 174
651 Numerical application of the deterministic approach 174 652 Numerical application of the reliability-based approach 181
Contents ix
66 Integration of reliability and structural optimization into dental implant design 184
661 Description and modeling of the problem 184 662 Numerical results 186
67 Conclusion 188
Appendices 189
Appendix 1 ANSYS Code for Stem Geometry 191
Appendix 2 ANSYS Code for Mini-Plate Geometry 197
Appendix 3 ANSYS Code for Dental Implant Geometry 201
Appendix 4 ANSYS Code for Geometry of Dental Implant with Bone 207
Bibliography 213
Index 229
Preface
The integration of structural optimization into biomechanics is a truly vast domain In this book we first focus on the integration of structural optimization into the design of orthopedic and orthodontic prostheses and also into drilling surgery Next we present the integration of reliability and structural optimization into the design of these prostheses which may be considered as a novel aspect introduced in this book The applications are made in 2D and in 3D considering the three major families of structural optimization sizing- shape- and topology optimization
In all domains of structural mechanics good design of a part is very important for its strength its lifetime and its use in service This is a challenge faced daily in sectors such as space research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve the techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile engines providing substantial energy savings The development of computer-aided design (CAD) techniques and optimization strategies is part of this context
Applying structural optimization is still somewhat complicated in certain domains Furthermore in deterministic structural optimization all parameters which are uncertain in nature are described by unfavorable characteristic values associated with safety coefficients The deterministic approach uses a pessimistic margin determined as a function of the consequences of a probable failure This approach often leads to unnecessary oversizing ndash particularly for sensitive structures
On the other hand researchers have developed a different approach which is better suited to uncertain physical phenomena In this approach the structure is deemed to have failed if the probability of failure is greater than a fixed threshold This is known as the ldquoprobabilistic approachrdquo The probabilistic approach is
xii Biomechanics
increasingly widely used in engineering as evidenced by the different applications in industry It is applied to check that the probability is sufficient when the structurersquos geometry is known or to optimize the sizing of the structure so as to respect certain fixed objectives such as a target cost or a required level of probability
Furthermore reliability analysis is an important tool in decision-making for establishing a maintenance- and inspection plan In addition it can be used in the validation of standards and regulations To perform reliability analysis various methods can be used to effectively and simply find the probability of failure Reliability analysis is a strategy used to evaluate the level of reliability without being able to control the design for a required reliability level For this reliability has become an important tool to be integrated into the process of structural optimization
This book also focuses on the necessary tools for the integration of reliability and structural optimization into biomechanics fields First the deterministic strategies of structural optimization are presented so we can implement them in structural design These deterministic strategies are applied in various domains in biomechanics including the design of orthopedic and orthodontic prostheses and drilling surgery Next reliability-based approaches pertaining to the integration of reliability into structural optimization are presented in detail with mechanical applications These reliability-based strategies are also applied in the design of orthopedic and orthodontic prostheses taking account of uncertainty in terms of geometry materials and load Finally system reliability strategies are also taken into account considering several failure scenarios
The book will provide invaluable support to teaching staff and researchers It is also intended for engineering students practising engineers and Masters students
Acknowledgements
We would like to thank all of those people who have in some way great or small contributed to the writing of this book ndash in particular Sophie Le Cann a researcher at the Biomedical Centre (BMC) at Lund University for her contribution in terms of biological language Heartfelt thanks go to our families to our students and to our colleagues for their massive moral support during the writing of this book
Ghias KHARMANDA Abdelkhalak EL-HAMI
October 2016
Introduction
This book begins with an introductory look at the fundamental principles of structural optimization before applying them to the field of biomechanics Then we present the different strategies for integrating reliability into structural optimization followed by the application of those strategies in biomechanics ndash particularly in terms of the design of orthopedic and orthodontic prostheses
In terms of structural optimization to illustrate the different techniques we can classify structural optimization into three main families
1) Sizing Optimization this model aims to improve a structural model whilst respecting the available resources (known as constraints or limitations) Sizing optimization is the particular case where we can only modify the cross section or transverse thickness of the components of a structure whose shape and topology are fixed There can be no modification of the geometric features andor models
2) Shape Optimization with this model it is possible to make changes to the shape provided they are compatible with a predefined topology This type of optimization modifies the parametric representation of the boundaries of the domain By moving those boundaries we try to find the best possible solution out of the set of all the configurations obtained by homeomorphic transformation of the original structure
3) Topology Optimization this model enables us to make more profound modifications to the shape of the structure Here the geometry of the part is examined with no prerequisites as to the connections of the domains or the structural elements present in the solution In order to optimize the topology we determine the structurersquos shape or transverse dimensions so some authors call topology optimization ldquogeneralized shape optimizationrdquo
xiv Biomechanics
In terms of reliability this concept can be integrated into all three families of structural optimization so we obtain a design that should be both optimal and reliable Sizing optimization shape optimization and topology optimization are generally classed as geometry optimization However the nature of the topology is non-quantitative in relation to shape and size For this reason this integration is divided into two models
1) RBDO Reliability-Based Design Optimization this model couples reliability analysis with sizing optimization (Reliability-Based Sizing Optimization) and also with shape optimization (Reliability-Based Shape Optimization) This coupling is a complex task requiring a lengthy computation time which seriously limits its applicability In addition during the process of reliability-based shape optimization the geometry of the structure is forced to change This coupling integrates different disciplines such as geometric modeling numerical simulation reliability analysis and optimization Thus the optimization problem becomes more complex The major difficulty lies in evaluating the reliability of the structure which in itself requires a specific optimization procedure The typical integration of reliability analysis into optimization methods is carried out in two spaces the normalized space of random variables and the physical space of design variables which requires a very significant computation time To solve this problem as we shall see there are a number of efficient methods
2) RBTO Reliability-Based Topology Optimization in the deterministic case we obtain a single optimal topology while the new RBTO model is able to generate several topologies as a function of a required reliability level However the coupling between reliability (which is quantitative) and topology (which is non-quantitative) requires the use of different methodologies relative to sizing optimization and shape optimization
In terms of biomechanics modeling is generally complicated on several different levels geometric description material properties and boundary conditions When performing the optimization process such modeling is needed with a new set of parameters being entered at each iteration In addition the problem becomes more complex when integrating reliability analysis which is itself performed by way of a particular optimization procedure In this case we couple several different software packages in order to integrate optimization and reliability into biomechanical applications Various strategies are presented in this book to simplify the coupling problem
This book is made up of six chapters presenting the different aspects of integration of reliability and structural optimization into biomechanics
Introduction xv
The first chapter offers an introduction to structural optimization and its main groups sizing optimization shape optimization and topology optimization The principles underpinning the three main types of structural optimization are presented with numerical applications for each type In structural design we have two different phases the conceptual phase and the detailed phase In the conceptual phase we use topology optimization to obtain an idea of the silhouette of the structure The detailed phase consists of shape optimization to obtain smooth geometry followed by sizing optimization to find the areas and thicknesses of the studied structure
The second chapter is devoted to the integration of structural optimization into biomechanics The integration is performed in this chapter for prosthesis design and drilling surgery Different 2D and 3D applications for orthopedic and orthodontic prostheses are presented in order to demonstrate the new strategies presented in this book and also to prove the advantages of those strategies Finally the integration of structural optimization into drilling surgery can be considered a novel aspect in relation to classic prosthesis design
The third chapter is given over to the integration of reliability into structural optimization Such integration can be viewed as a difficult task The difficulties arise in terms of coupling convergence and computation time In this chapter we present the basic principles of that integration and a literature review based on early attempts in this area A numerical application is presented to show the difference between deterministic- and reliability-based optimization Next two viewpoints are presented with the corresponding developed methods considering those two points of view to illustrate the advantages of the methods
The fourth chapter is devoted to the reliability-based design optimization (RBDO) model The developed methods are classified into two categories numerical and semi-numerical methods Numerical methods are generally applicable but require a high computation time Semi-numerical methods on the other hand are efficient in terms of computation time but can only be used in certain cases
The fifth chapter details reliability-based topology optimization (RBTO) This model enables designers and manufacturers to select the solution which is at once economical and reliable Two points of view are presented in the discussion of this model A number of applications in mechanics are presented in order to demonstrate the advantages of the RBTO model
The sixth chapter discusses the integration of reliability and structural optimization into orthopedic and orthodontic prostheses design considering two
xvi Biomechanics
approaches the deterministic approach and the reliability-based approach In orthopedic prostheses the models are presented using results drawn from the existing body of literature The aspect of materials characterization is also integrated into the design process In addition in orthodontics models are presented to test the stability of the prostheses in light of the constraints of dentistry The integration of reliability and structural optimization into the design of prostheses is performed very simply in order to aid understanding and implementation of the different strategies presented in this book
List of Abbreviations
AMA Approximate Moments Approach
AMV Advanced Mean Value
APDL ANSYS Parametric Design Language
AT Anterior Temporal (or Temporalis)
CAD Computer-Aided Design
CM Classical Method
CMV Conjugate Mean Value
CT Computed Tomography
DDO Deterministic Design Optimization
DLA Double-Level Approach
DM Deep Masseter
DO Deterministic Optimization
DTO Deterministic Topology Optimization
FE Finite Elements
FEM Finite Element Method
FORM First Order Reliability Method
HCA Hybrid Cellular Automata
HDS Hybrid Design Space
HM Hybrid Method
xviii Biomechanics
HMV Hybrid Mean Value
IAM Improved Austin-Moore
IHM Improved Hybrid Method
KKT Karush-Kuhn-Tucker
MBB Messerschmitt-Boumllkow-Blohm
MLA Mono-Level Approach
MP Medial Pterygoid
MPP Most Probable failure Point
MT Medial Temporal (or Temporalis)
OC Optimality Criteria
OSF Optimum Safety Factor
PDF Probability Density Function
PMA Performance Measure Approach
PT Posterior Temporal (or Temporalis)
RBDO Reliability-Based Design Optimization
RBO Reliability-Based Optimization
RBSO Reliability-Based Structural Optimization
RBTO Reliability-Based Topology Optimization
RIA Reliability Index Approach
SAP Sequential Approximate Programming
SDA Sequential Decoupled Approach
SED Strain Energy Density
SIMP Solid Isotropic Microstructure with Penalty
SM Superficial Masseter
SORA Sequential Optimization and Reliability Assessment
SORM Second Order Reliability Method
SP Safest Point
1
Introduction to Structural Optimization
11 Introduction
Structural optimization is a topic which affects many different physical domains ndash particularly solid mechanics ndash and which it is tricky to characterize as its formulations may have a number of different aspects Firstly a distinction regarding the way in which geometries are parameterized is presented and secondly a distribution pertaining to the intrinsic nature of optimization algorithms is established
Determining the appropriate shape for structural components is a crucially important problem for engineers In all areas of structural mechanics the impact of proper design of a part is very significant in terms of its strength its lifetime and its usage This is a challenge faced on a daily basis in the sectors of spatial research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile machinery providing substantial energy savings The different development of computer-aided design (CAD) techniques and optimization strategies is part of this context There has been keen interest in structural optimization for over thirty years Whilst it is still too infrequently applied in the conventional techniques used by research centers it is becoming more widely used as its reliability improves Having begun with the simplest of problems the field of application of structural optimization today extends to new and ever more interesting challenges
To illustrate the evolution of structural optimization techniques we can arbitrarily split structural optimization into three major groups (or families) In
Biomechanics Optimization Uncertainties and Reliability First Edition Ghias Kharmanda and Abdelkhalak El Hami copy ISTE Ltd 2017 Published by ISTE Ltd and John Wiley amp Sons Inc
2 Biomechanics
historical terms each of them has been addressed in order of increasing difficulty and generality
With sizing optimization we are only able to modify the dimensions of an object whose shape and topology are fixed There can be no modification of the geometric model We speak of a homeomorphic transformation
Shape optimization involves making changes of shape which are compatible with a predetermined topology Typical shape optimization modifies the parametric representation of the boundaries of the domain By moving the boundaries of the domains we can seek the best solution out of all the structures obtained by homeomorphic transformation of the original object In this case it is clear that we can make a change to the transverse dimensions as well as a modification to the objectrsquos configuration but it is certainly not acceptable to modify its connectivity or its nature ndash in particular the number of components that it has The optimal object exhibits the same topology as the original object
With topology optimization we can fundamentally change the nature of the object The ldquotopologyrdquo refers to the number and position of the components of the domains Here the objectrsquos geometry is presented with no prerequisites as to the connectivity of the domains or the components present in the solution We take no initial information about the topology of the optimal shape
12 History of structural optimization
It was in the early 1960s that Schmit [SCH 60] and Fox [FOX 65] laid the foundations for a modern theory of structural optimization based on the concepts of mathematical programming and sensitivity analysis Paradoxically at the time ldquofully stressed designrdquo was the only widely used technique in practice although it lacked any theoretical justification other than empiricism and the engineersrsquo intuition It was Prager and Taylor [PRA 68] who set out variational methods and Lagrangian optimality conditions to justify the criteria of fully stressed design for a class of structural optimization problems The optimality conditions of the optimization problem were then used directly to construct an iterative resolution algorithm known as the ldquooptimality criteria methodrdquo
Originally structural optimization was mainly limited to the sizing optimization of trusses or gantries Thus sizing optimization of structures was the first field of application for optimality criteria When dealing with the problem of sizing we look at the transverse sections of the structural elements though their length and the
Introduction to Structural Optimization 3
location of their joints remain fixed During the late 1960s and early 1970s optimality criteria were soon adapted to large structures modeled using the finite-element method (FEM) [VEN 73] The optimality criteria method produced a few interesting results and a number of extensions have been presented since the 1970s including Venkayyarsquos generalized criteria method [VEN 73] or Rozvany and Zhoursquos [ROZ 91] discretized optimality criteria
Although since the 1970s attention has mainly been focused on sizing the problem of truss topology has also been studied by Prager [PRA 74] on a very restricted class of structures based on the concept of a truss put forward by Michell in 1904 [MIC 04] The problem lies in finding the best possible configuration so that the truss or gantry can convey forces to the foundations whilst minimizing a given performance and satisfying the design constraints Michellrsquos theory [MIC 04] is related to the topology of trusses made of bars of minimal mass The optimal solution from Michellrsquos point of view is composed solely of perpendicular bars which form a structure whose configuration is optimal for the maximum tensile- and compressive stresses All the configuration problems studied by Prager later on were solved analytically so the practical application of topology was very limited To remedy this shortcoming Rozvany [ROZ 76] invested a great deal of effort in developing new approaches to solve these configuration problems as automatically as possible
After that optimal truss topology was studied in greater depth by Kirsch [KIR 90] If we impose a very small minimum value for the cross-section then the ldquolayoutrdquo optimization of trusses and gantries can be approached as a conventional sizing problem on a very large scale The solution is then obtained by application of generalized optimality criteria for a variety of objective functions compliance movements tensions and eigenvalues [ZHO 91]
Finally the crux of the problem of truss topology optimization seems to have been identified by Bendsoslashe et al [BEN 91] The problem of truss topology is examined with an integral approach combining analysis and design simultaneously The problem of minimum compliance is transformed into a problem of non-differentiable optimization and then in the case of a truss of bars into a linear problem In this form it can be solved on very large structures using non-differentiable optimization methods [BEN 93a] an interior point method [BEN 93b] a dual method [BEC 94] or a penaltybarrier multiplier method [BEN 97]
The three main groups of structural optimization cannot be considered recent concepts but their integration into biomechanics is ndash especially topology optimization [FRA 10]
4 Biomechanics
13 Sizing optimization
131 Definition
With sizing optimization we can only modify the cross-section or transverse thickness of the components of structure whose shape and topology are fixed There can be no modification of the geometric model and its features Figure 11 shows a truss made of 17 bars with different cross-sections (circular rectangular and I-shaped)
Figure 11 Changing the dimensions whilst preserving the same topology of the section
Sizing optimization can be performed by considering the same topology to produce various dimensions For example when the cross-section is circular we merely need to vary the diameters to minimize an objective or several objectives under certain constraints
132 First works in sizing optimization
The problem of sizing optimization has benefited the most from this research so optimization of the transverse dimensions is today a reliable tool This problem was also extended to that of flexural elements [FLE 83] to improving performance when subjected to vibration and to the stability of balance Besides the transverse dimensions of the structural elements it is possible to vary their shape To the best of our knowledge few works have been devoted to the study of the optimal shape for a truss In this type of problem only the location of the structural joints is altered whilst the topology remains unchanged Svanberg [SVA 81] was one of the only people to carry out truss shape optimization on the basis of FEM analyses and solving using mathematical programming techniques
Introduction to Structural Optimization 5
133 Numerical application
1331 Description and modeling of the studied problem
Figure 12a shows a cantilever beam and its I-shaped cross-section in Figure 12b This beam is embedded at one end and subjected to free vibration The material from which the beam is made is structural steel which has a Youngrsquos modulus 200000E MPa= and Poissonrsquos ratio of 03ν = The density of the material is 6 37854 10 Kg mmρ minus= times The material exhibits linear elastic isotropic behavior The length of that beam is 300L mm= and the dimensions of the cross-section are 60 B mm= 100H mm= and 20T mm= (Figure 12b)
a) b)
Figure 12 Cantilever beam subject to free vibration
The objective of this study is to calculate the first four modes of resonance and then perform sizing optimization on the dimensions of the cross-section The problem of optimization therefore is to minimize the structural volume under the constraint of the first resonance frequency This problem can be formulated as follows
1
min ( ) ( ) 0
40 100 80 16010 30
w
Volume B H Ts t f B H T f
BHT
minus lele lele lele le
[11]
where 20wf Hz= is the maximum value of the first resonance frequency
1332 Numerical results
To perform sizing optimization on the ANSYS software for instance we consider the dimensions of the cross-section as optimization variables so as to obtain a parameterized model Then a direct simulation is performed as the heart of the optimization loop
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
First published 2017 in Great Britain and the United States by ISTE Ltd and John Wiley amp Sons Inc
Apart from any fair dealing for the purposes of research or private study or criticism or review as permitted under the Copyright Designs and Patents Act 1988 this publication may only be reproduced stored or transmitted in any form or by any means with the prior permission in writing of the publishers or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address
ISTE Ltd John Wiley amp Sons Inc 27-37 St Georgersquos Road 111 River Street London SW19 4EU Hoboken NJ 07030 UK USA
wwwistecouk wwwwileycom
copy ISTE Ltd 2017 The rights of Ghias Kharmanda and Abdelkhalak El Hami to be identified as the authors of this work have been asserted by them in accordance with the Copyright Designs and Patents Act 1988
Library of Congress Control Number 2016952066 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-025-6
Contents
Preface xi
Introduction xiii
List of Abbreviations xvii
Chapter 1 Introduction to Structural Optimization 1
11 Introduction 1 12 History of structural optimization 2 13 Sizing optimization 4
131 Definition 4 132 First works in sizing optimization 4 133 Numerical application 5
14 Shape optimization 10 141 Definition 10 142 First works in shape optimization 11 143 Numerical application 12
15 Topology optimization 16 151 Definition 16 152 First works in topology optimization 17 153 Numerical application 18
16 Conclusion 21
Chapter 2 Integration of Structural Optimization into Biomechanics 23
21 Introduction 23 22 Integration of structural optimization into orthopedic prosthesis design 23
vi Biomechanics
221 Structural optimization of the hip prosthesis 24 222 Sizing optimization of a 3D intervertebral disk prosthesis 42
23 Integration of structural optimization into orthodontic prosthesis design 47
231 Sizing optimization of a dental implant 47 232 Shape optimization of a mini-plate 49
24 Advanced integration of structural optimization into drilling surgery 52
241 Case of treatment of a crack with a single hole 53 242 Case of treatment of a crack with two holes 54
25 Conclusion 56
Chapter 3 Integration of Reliability into Structural Optimization 57
31 Introduction 57 32 Literature review of reliability-based optimization 58 33 Comparison between deterministic and reliability-based optimization 60
331 Deterministic optimization 61 332 Reliability-based optimization 63
34 Numerical application 64 341 Description and modeling of the studied problem 64 342 Numerical results 65
35 Approaches and strategies for reliability-based optimization 68
351 Mono-level approaches 68 352 Double-level approaches 68 353 Sequential decoupled approaches 68
36 Two points of view for developments of reliability-based optimization 69
361 Point of view of ldquoReliabilityrdquo 69 362 Point of view of ldquoOptimizationrdquo 70 363 Method efficiency 70
37 Philosophy of integration of the concept of reliability into structural optimization groups 72 38 Conclusion 73
Chapter 4 Reliability-based Design Optimization Model 75
41 Introduction 75 42 Classic method 76
Contents vii
421 Formulations 76 422 Optimality conditions 77 423 Algorithm 77 424 Advantages and disadvantages 79
43 Hybrid method 79 431 Formulation 79 432 Optimality conditions 82 433 Algorithm 84 434 Advantages and disadvantages 85
44 Improved hybrid method 86 441 Formulations 86 442 Optimality conditions 86 443 Algorithm 89 444 Advantages and disadvantages 90
45 Optimum safety factor method 91 451 Safety factor concept 91 452 Developments and optimality conditions 92 453 Algorithm 97 454 Advantages and disadvantages 98
46 Safest point method 98 461 Formulations 98 462 Algorithm 102 463 Advantages and disadvantages 104
47 Numerical applications 105 471 RBDO of a hook CM and HM 105 472 RBDO of a triangular plate HM amp IHM 107 473 RBDO of a console beam (sandwich beam) HM and OSF 110 474 RBDO of an aircraft wing HM amp SP 113
48 Classification of the methods developed 115 481 Numerical methods 115 482 Semi-numerical methods 116 483 Comparison between the numerical- and semi-numerical methods 118
49 Conclusion 119
Chapter 5 Reliability-based Topology Optimization Model 121
51 Introduction 121 52 Formulation and algorithm for the RBTO model 122
521 Formulation 122
viii Biomechanics
522 Algorithm 123 523 Validation of the RBTO code developed 125
53 Validation of the RBTO model 126 531 Analytical validation 126 532 Numerical validation 128
54 Variability of the reliability index 134 541 Example 1 MBB beam 136 542 Example 2 Cantilever beam 136 543 Example 3 Cantilever beam with double loads 136 544 Example 4 Cantilever beam with a transversal hole 136
55 Numerical applications for the RBTO model 137 551 Static analysis 138 552 Modal analysis 139 553 Fatigue analysis 141
56 Two points of view for integration of reliability into topology optimization 142
561 Point of view of ldquotopologyrdquo 144 562 Point of view of ldquoreliabilityrdquo 144 563 Numerical applications for the two points of view 146
57 Conclusion 152
Chapter 6 Integration of Reliability and Structural Optimization into Prosthesis Design 153
61 Introduction 153 62 Prosthesis design 154 63 Integration of topology optimization into prosthesis design 154
631 Importance of topology optimization in prosthesis design 155 632 Place of topology optimization in the prosthesis design chain 156
64 Integration of reliability and structural optimization into hip prosthesis design 157
641 Numerical application of the deterministic approach 158 642 Numerical application of the reliability-based approach 167
65 Integration of reliability and structural optimization into the design of mini-plate systems used to treat fractured mandibles 174
651 Numerical application of the deterministic approach 174 652 Numerical application of the reliability-based approach 181
Contents ix
66 Integration of reliability and structural optimization into dental implant design 184
661 Description and modeling of the problem 184 662 Numerical results 186
67 Conclusion 188
Appendices 189
Appendix 1 ANSYS Code for Stem Geometry 191
Appendix 2 ANSYS Code for Mini-Plate Geometry 197
Appendix 3 ANSYS Code for Dental Implant Geometry 201
Appendix 4 ANSYS Code for Geometry of Dental Implant with Bone 207
Bibliography 213
Index 229
Preface
The integration of structural optimization into biomechanics is a truly vast domain In this book we first focus on the integration of structural optimization into the design of orthopedic and orthodontic prostheses and also into drilling surgery Next we present the integration of reliability and structural optimization into the design of these prostheses which may be considered as a novel aspect introduced in this book The applications are made in 2D and in 3D considering the three major families of structural optimization sizing- shape- and topology optimization
In all domains of structural mechanics good design of a part is very important for its strength its lifetime and its use in service This is a challenge faced daily in sectors such as space research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve the techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile engines providing substantial energy savings The development of computer-aided design (CAD) techniques and optimization strategies is part of this context
Applying structural optimization is still somewhat complicated in certain domains Furthermore in deterministic structural optimization all parameters which are uncertain in nature are described by unfavorable characteristic values associated with safety coefficients The deterministic approach uses a pessimistic margin determined as a function of the consequences of a probable failure This approach often leads to unnecessary oversizing ndash particularly for sensitive structures
On the other hand researchers have developed a different approach which is better suited to uncertain physical phenomena In this approach the structure is deemed to have failed if the probability of failure is greater than a fixed threshold This is known as the ldquoprobabilistic approachrdquo The probabilistic approach is
xii Biomechanics
increasingly widely used in engineering as evidenced by the different applications in industry It is applied to check that the probability is sufficient when the structurersquos geometry is known or to optimize the sizing of the structure so as to respect certain fixed objectives such as a target cost or a required level of probability
Furthermore reliability analysis is an important tool in decision-making for establishing a maintenance- and inspection plan In addition it can be used in the validation of standards and regulations To perform reliability analysis various methods can be used to effectively and simply find the probability of failure Reliability analysis is a strategy used to evaluate the level of reliability without being able to control the design for a required reliability level For this reliability has become an important tool to be integrated into the process of structural optimization
This book also focuses on the necessary tools for the integration of reliability and structural optimization into biomechanics fields First the deterministic strategies of structural optimization are presented so we can implement them in structural design These deterministic strategies are applied in various domains in biomechanics including the design of orthopedic and orthodontic prostheses and drilling surgery Next reliability-based approaches pertaining to the integration of reliability into structural optimization are presented in detail with mechanical applications These reliability-based strategies are also applied in the design of orthopedic and orthodontic prostheses taking account of uncertainty in terms of geometry materials and load Finally system reliability strategies are also taken into account considering several failure scenarios
The book will provide invaluable support to teaching staff and researchers It is also intended for engineering students practising engineers and Masters students
Acknowledgements
We would like to thank all of those people who have in some way great or small contributed to the writing of this book ndash in particular Sophie Le Cann a researcher at the Biomedical Centre (BMC) at Lund University for her contribution in terms of biological language Heartfelt thanks go to our families to our students and to our colleagues for their massive moral support during the writing of this book
Ghias KHARMANDA Abdelkhalak EL-HAMI
October 2016
Introduction
This book begins with an introductory look at the fundamental principles of structural optimization before applying them to the field of biomechanics Then we present the different strategies for integrating reliability into structural optimization followed by the application of those strategies in biomechanics ndash particularly in terms of the design of orthopedic and orthodontic prostheses
In terms of structural optimization to illustrate the different techniques we can classify structural optimization into three main families
1) Sizing Optimization this model aims to improve a structural model whilst respecting the available resources (known as constraints or limitations) Sizing optimization is the particular case where we can only modify the cross section or transverse thickness of the components of a structure whose shape and topology are fixed There can be no modification of the geometric features andor models
2) Shape Optimization with this model it is possible to make changes to the shape provided they are compatible with a predefined topology This type of optimization modifies the parametric representation of the boundaries of the domain By moving those boundaries we try to find the best possible solution out of the set of all the configurations obtained by homeomorphic transformation of the original structure
3) Topology Optimization this model enables us to make more profound modifications to the shape of the structure Here the geometry of the part is examined with no prerequisites as to the connections of the domains or the structural elements present in the solution In order to optimize the topology we determine the structurersquos shape or transverse dimensions so some authors call topology optimization ldquogeneralized shape optimizationrdquo
xiv Biomechanics
In terms of reliability this concept can be integrated into all three families of structural optimization so we obtain a design that should be both optimal and reliable Sizing optimization shape optimization and topology optimization are generally classed as geometry optimization However the nature of the topology is non-quantitative in relation to shape and size For this reason this integration is divided into two models
1) RBDO Reliability-Based Design Optimization this model couples reliability analysis with sizing optimization (Reliability-Based Sizing Optimization) and also with shape optimization (Reliability-Based Shape Optimization) This coupling is a complex task requiring a lengthy computation time which seriously limits its applicability In addition during the process of reliability-based shape optimization the geometry of the structure is forced to change This coupling integrates different disciplines such as geometric modeling numerical simulation reliability analysis and optimization Thus the optimization problem becomes more complex The major difficulty lies in evaluating the reliability of the structure which in itself requires a specific optimization procedure The typical integration of reliability analysis into optimization methods is carried out in two spaces the normalized space of random variables and the physical space of design variables which requires a very significant computation time To solve this problem as we shall see there are a number of efficient methods
2) RBTO Reliability-Based Topology Optimization in the deterministic case we obtain a single optimal topology while the new RBTO model is able to generate several topologies as a function of a required reliability level However the coupling between reliability (which is quantitative) and topology (which is non-quantitative) requires the use of different methodologies relative to sizing optimization and shape optimization
In terms of biomechanics modeling is generally complicated on several different levels geometric description material properties and boundary conditions When performing the optimization process such modeling is needed with a new set of parameters being entered at each iteration In addition the problem becomes more complex when integrating reliability analysis which is itself performed by way of a particular optimization procedure In this case we couple several different software packages in order to integrate optimization and reliability into biomechanical applications Various strategies are presented in this book to simplify the coupling problem
This book is made up of six chapters presenting the different aspects of integration of reliability and structural optimization into biomechanics
Introduction xv
The first chapter offers an introduction to structural optimization and its main groups sizing optimization shape optimization and topology optimization The principles underpinning the three main types of structural optimization are presented with numerical applications for each type In structural design we have two different phases the conceptual phase and the detailed phase In the conceptual phase we use topology optimization to obtain an idea of the silhouette of the structure The detailed phase consists of shape optimization to obtain smooth geometry followed by sizing optimization to find the areas and thicknesses of the studied structure
The second chapter is devoted to the integration of structural optimization into biomechanics The integration is performed in this chapter for prosthesis design and drilling surgery Different 2D and 3D applications for orthopedic and orthodontic prostheses are presented in order to demonstrate the new strategies presented in this book and also to prove the advantages of those strategies Finally the integration of structural optimization into drilling surgery can be considered a novel aspect in relation to classic prosthesis design
The third chapter is given over to the integration of reliability into structural optimization Such integration can be viewed as a difficult task The difficulties arise in terms of coupling convergence and computation time In this chapter we present the basic principles of that integration and a literature review based on early attempts in this area A numerical application is presented to show the difference between deterministic- and reliability-based optimization Next two viewpoints are presented with the corresponding developed methods considering those two points of view to illustrate the advantages of the methods
The fourth chapter is devoted to the reliability-based design optimization (RBDO) model The developed methods are classified into two categories numerical and semi-numerical methods Numerical methods are generally applicable but require a high computation time Semi-numerical methods on the other hand are efficient in terms of computation time but can only be used in certain cases
The fifth chapter details reliability-based topology optimization (RBTO) This model enables designers and manufacturers to select the solution which is at once economical and reliable Two points of view are presented in the discussion of this model A number of applications in mechanics are presented in order to demonstrate the advantages of the RBTO model
The sixth chapter discusses the integration of reliability and structural optimization into orthopedic and orthodontic prostheses design considering two
xvi Biomechanics
approaches the deterministic approach and the reliability-based approach In orthopedic prostheses the models are presented using results drawn from the existing body of literature The aspect of materials characterization is also integrated into the design process In addition in orthodontics models are presented to test the stability of the prostheses in light of the constraints of dentistry The integration of reliability and structural optimization into the design of prostheses is performed very simply in order to aid understanding and implementation of the different strategies presented in this book
List of Abbreviations
AMA Approximate Moments Approach
AMV Advanced Mean Value
APDL ANSYS Parametric Design Language
AT Anterior Temporal (or Temporalis)
CAD Computer-Aided Design
CM Classical Method
CMV Conjugate Mean Value
CT Computed Tomography
DDO Deterministic Design Optimization
DLA Double-Level Approach
DM Deep Masseter
DO Deterministic Optimization
DTO Deterministic Topology Optimization
FE Finite Elements
FEM Finite Element Method
FORM First Order Reliability Method
HCA Hybrid Cellular Automata
HDS Hybrid Design Space
HM Hybrid Method
xviii Biomechanics
HMV Hybrid Mean Value
IAM Improved Austin-Moore
IHM Improved Hybrid Method
KKT Karush-Kuhn-Tucker
MBB Messerschmitt-Boumllkow-Blohm
MLA Mono-Level Approach
MP Medial Pterygoid
MPP Most Probable failure Point
MT Medial Temporal (or Temporalis)
OC Optimality Criteria
OSF Optimum Safety Factor
PDF Probability Density Function
PMA Performance Measure Approach
PT Posterior Temporal (or Temporalis)
RBDO Reliability-Based Design Optimization
RBO Reliability-Based Optimization
RBSO Reliability-Based Structural Optimization
RBTO Reliability-Based Topology Optimization
RIA Reliability Index Approach
SAP Sequential Approximate Programming
SDA Sequential Decoupled Approach
SED Strain Energy Density
SIMP Solid Isotropic Microstructure with Penalty
SM Superficial Masseter
SORA Sequential Optimization and Reliability Assessment
SORM Second Order Reliability Method
SP Safest Point
1
Introduction to Structural Optimization
11 Introduction
Structural optimization is a topic which affects many different physical domains ndash particularly solid mechanics ndash and which it is tricky to characterize as its formulations may have a number of different aspects Firstly a distinction regarding the way in which geometries are parameterized is presented and secondly a distribution pertaining to the intrinsic nature of optimization algorithms is established
Determining the appropriate shape for structural components is a crucially important problem for engineers In all areas of structural mechanics the impact of proper design of a part is very significant in terms of its strength its lifetime and its usage This is a challenge faced on a daily basis in the sectors of spatial research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile machinery providing substantial energy savings The different development of computer-aided design (CAD) techniques and optimization strategies is part of this context There has been keen interest in structural optimization for over thirty years Whilst it is still too infrequently applied in the conventional techniques used by research centers it is becoming more widely used as its reliability improves Having begun with the simplest of problems the field of application of structural optimization today extends to new and ever more interesting challenges
To illustrate the evolution of structural optimization techniques we can arbitrarily split structural optimization into three major groups (or families) In
Biomechanics Optimization Uncertainties and Reliability First Edition Ghias Kharmanda and Abdelkhalak El Hami copy ISTE Ltd 2017 Published by ISTE Ltd and John Wiley amp Sons Inc
2 Biomechanics
historical terms each of them has been addressed in order of increasing difficulty and generality
With sizing optimization we are only able to modify the dimensions of an object whose shape and topology are fixed There can be no modification of the geometric model We speak of a homeomorphic transformation
Shape optimization involves making changes of shape which are compatible with a predetermined topology Typical shape optimization modifies the parametric representation of the boundaries of the domain By moving the boundaries of the domains we can seek the best solution out of all the structures obtained by homeomorphic transformation of the original object In this case it is clear that we can make a change to the transverse dimensions as well as a modification to the objectrsquos configuration but it is certainly not acceptable to modify its connectivity or its nature ndash in particular the number of components that it has The optimal object exhibits the same topology as the original object
With topology optimization we can fundamentally change the nature of the object The ldquotopologyrdquo refers to the number and position of the components of the domains Here the objectrsquos geometry is presented with no prerequisites as to the connectivity of the domains or the components present in the solution We take no initial information about the topology of the optimal shape
12 History of structural optimization
It was in the early 1960s that Schmit [SCH 60] and Fox [FOX 65] laid the foundations for a modern theory of structural optimization based on the concepts of mathematical programming and sensitivity analysis Paradoxically at the time ldquofully stressed designrdquo was the only widely used technique in practice although it lacked any theoretical justification other than empiricism and the engineersrsquo intuition It was Prager and Taylor [PRA 68] who set out variational methods and Lagrangian optimality conditions to justify the criteria of fully stressed design for a class of structural optimization problems The optimality conditions of the optimization problem were then used directly to construct an iterative resolution algorithm known as the ldquooptimality criteria methodrdquo
Originally structural optimization was mainly limited to the sizing optimization of trusses or gantries Thus sizing optimization of structures was the first field of application for optimality criteria When dealing with the problem of sizing we look at the transverse sections of the structural elements though their length and the
Introduction to Structural Optimization 3
location of their joints remain fixed During the late 1960s and early 1970s optimality criteria were soon adapted to large structures modeled using the finite-element method (FEM) [VEN 73] The optimality criteria method produced a few interesting results and a number of extensions have been presented since the 1970s including Venkayyarsquos generalized criteria method [VEN 73] or Rozvany and Zhoursquos [ROZ 91] discretized optimality criteria
Although since the 1970s attention has mainly been focused on sizing the problem of truss topology has also been studied by Prager [PRA 74] on a very restricted class of structures based on the concept of a truss put forward by Michell in 1904 [MIC 04] The problem lies in finding the best possible configuration so that the truss or gantry can convey forces to the foundations whilst minimizing a given performance and satisfying the design constraints Michellrsquos theory [MIC 04] is related to the topology of trusses made of bars of minimal mass The optimal solution from Michellrsquos point of view is composed solely of perpendicular bars which form a structure whose configuration is optimal for the maximum tensile- and compressive stresses All the configuration problems studied by Prager later on were solved analytically so the practical application of topology was very limited To remedy this shortcoming Rozvany [ROZ 76] invested a great deal of effort in developing new approaches to solve these configuration problems as automatically as possible
After that optimal truss topology was studied in greater depth by Kirsch [KIR 90] If we impose a very small minimum value for the cross-section then the ldquolayoutrdquo optimization of trusses and gantries can be approached as a conventional sizing problem on a very large scale The solution is then obtained by application of generalized optimality criteria for a variety of objective functions compliance movements tensions and eigenvalues [ZHO 91]
Finally the crux of the problem of truss topology optimization seems to have been identified by Bendsoslashe et al [BEN 91] The problem of truss topology is examined with an integral approach combining analysis and design simultaneously The problem of minimum compliance is transformed into a problem of non-differentiable optimization and then in the case of a truss of bars into a linear problem In this form it can be solved on very large structures using non-differentiable optimization methods [BEN 93a] an interior point method [BEN 93b] a dual method [BEC 94] or a penaltybarrier multiplier method [BEN 97]
The three main groups of structural optimization cannot be considered recent concepts but their integration into biomechanics is ndash especially topology optimization [FRA 10]
4 Biomechanics
13 Sizing optimization
131 Definition
With sizing optimization we can only modify the cross-section or transverse thickness of the components of structure whose shape and topology are fixed There can be no modification of the geometric model and its features Figure 11 shows a truss made of 17 bars with different cross-sections (circular rectangular and I-shaped)
Figure 11 Changing the dimensions whilst preserving the same topology of the section
Sizing optimization can be performed by considering the same topology to produce various dimensions For example when the cross-section is circular we merely need to vary the diameters to minimize an objective or several objectives under certain constraints
132 First works in sizing optimization
The problem of sizing optimization has benefited the most from this research so optimization of the transverse dimensions is today a reliable tool This problem was also extended to that of flexural elements [FLE 83] to improving performance when subjected to vibration and to the stability of balance Besides the transverse dimensions of the structural elements it is possible to vary their shape To the best of our knowledge few works have been devoted to the study of the optimal shape for a truss In this type of problem only the location of the structural joints is altered whilst the topology remains unchanged Svanberg [SVA 81] was one of the only people to carry out truss shape optimization on the basis of FEM analyses and solving using mathematical programming techniques
Introduction to Structural Optimization 5
133 Numerical application
1331 Description and modeling of the studied problem
Figure 12a shows a cantilever beam and its I-shaped cross-section in Figure 12b This beam is embedded at one end and subjected to free vibration The material from which the beam is made is structural steel which has a Youngrsquos modulus 200000E MPa= and Poissonrsquos ratio of 03ν = The density of the material is 6 37854 10 Kg mmρ minus= times The material exhibits linear elastic isotropic behavior The length of that beam is 300L mm= and the dimensions of the cross-section are 60 B mm= 100H mm= and 20T mm= (Figure 12b)
a) b)
Figure 12 Cantilever beam subject to free vibration
The objective of this study is to calculate the first four modes of resonance and then perform sizing optimization on the dimensions of the cross-section The problem of optimization therefore is to minimize the structural volume under the constraint of the first resonance frequency This problem can be formulated as follows
1
min ( ) ( ) 0
40 100 80 16010 30
w
Volume B H Ts t f B H T f
BHT
minus lele lele lele le
[11]
where 20wf Hz= is the maximum value of the first resonance frequency
1332 Numerical results
To perform sizing optimization on the ANSYS software for instance we consider the dimensions of the cross-section as optimization variables so as to obtain a parameterized model Then a direct simulation is performed as the heart of the optimization loop
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
Contents
Preface xi
Introduction xiii
List of Abbreviations xvii
Chapter 1 Introduction to Structural Optimization 1
11 Introduction 1 12 History of structural optimization 2 13 Sizing optimization 4
131 Definition 4 132 First works in sizing optimization 4 133 Numerical application 5
14 Shape optimization 10 141 Definition 10 142 First works in shape optimization 11 143 Numerical application 12
15 Topology optimization 16 151 Definition 16 152 First works in topology optimization 17 153 Numerical application 18
16 Conclusion 21
Chapter 2 Integration of Structural Optimization into Biomechanics 23
21 Introduction 23 22 Integration of structural optimization into orthopedic prosthesis design 23
vi Biomechanics
221 Structural optimization of the hip prosthesis 24 222 Sizing optimization of a 3D intervertebral disk prosthesis 42
23 Integration of structural optimization into orthodontic prosthesis design 47
231 Sizing optimization of a dental implant 47 232 Shape optimization of a mini-plate 49
24 Advanced integration of structural optimization into drilling surgery 52
241 Case of treatment of a crack with a single hole 53 242 Case of treatment of a crack with two holes 54
25 Conclusion 56
Chapter 3 Integration of Reliability into Structural Optimization 57
31 Introduction 57 32 Literature review of reliability-based optimization 58 33 Comparison between deterministic and reliability-based optimization 60
331 Deterministic optimization 61 332 Reliability-based optimization 63
34 Numerical application 64 341 Description and modeling of the studied problem 64 342 Numerical results 65
35 Approaches and strategies for reliability-based optimization 68
351 Mono-level approaches 68 352 Double-level approaches 68 353 Sequential decoupled approaches 68
36 Two points of view for developments of reliability-based optimization 69
361 Point of view of ldquoReliabilityrdquo 69 362 Point of view of ldquoOptimizationrdquo 70 363 Method efficiency 70
37 Philosophy of integration of the concept of reliability into structural optimization groups 72 38 Conclusion 73
Chapter 4 Reliability-based Design Optimization Model 75
41 Introduction 75 42 Classic method 76
Contents vii
421 Formulations 76 422 Optimality conditions 77 423 Algorithm 77 424 Advantages and disadvantages 79
43 Hybrid method 79 431 Formulation 79 432 Optimality conditions 82 433 Algorithm 84 434 Advantages and disadvantages 85
44 Improved hybrid method 86 441 Formulations 86 442 Optimality conditions 86 443 Algorithm 89 444 Advantages and disadvantages 90
45 Optimum safety factor method 91 451 Safety factor concept 91 452 Developments and optimality conditions 92 453 Algorithm 97 454 Advantages and disadvantages 98
46 Safest point method 98 461 Formulations 98 462 Algorithm 102 463 Advantages and disadvantages 104
47 Numerical applications 105 471 RBDO of a hook CM and HM 105 472 RBDO of a triangular plate HM amp IHM 107 473 RBDO of a console beam (sandwich beam) HM and OSF 110 474 RBDO of an aircraft wing HM amp SP 113
48 Classification of the methods developed 115 481 Numerical methods 115 482 Semi-numerical methods 116 483 Comparison between the numerical- and semi-numerical methods 118
49 Conclusion 119
Chapter 5 Reliability-based Topology Optimization Model 121
51 Introduction 121 52 Formulation and algorithm for the RBTO model 122
521 Formulation 122
viii Biomechanics
522 Algorithm 123 523 Validation of the RBTO code developed 125
53 Validation of the RBTO model 126 531 Analytical validation 126 532 Numerical validation 128
54 Variability of the reliability index 134 541 Example 1 MBB beam 136 542 Example 2 Cantilever beam 136 543 Example 3 Cantilever beam with double loads 136 544 Example 4 Cantilever beam with a transversal hole 136
55 Numerical applications for the RBTO model 137 551 Static analysis 138 552 Modal analysis 139 553 Fatigue analysis 141
56 Two points of view for integration of reliability into topology optimization 142
561 Point of view of ldquotopologyrdquo 144 562 Point of view of ldquoreliabilityrdquo 144 563 Numerical applications for the two points of view 146
57 Conclusion 152
Chapter 6 Integration of Reliability and Structural Optimization into Prosthesis Design 153
61 Introduction 153 62 Prosthesis design 154 63 Integration of topology optimization into prosthesis design 154
631 Importance of topology optimization in prosthesis design 155 632 Place of topology optimization in the prosthesis design chain 156
64 Integration of reliability and structural optimization into hip prosthesis design 157
641 Numerical application of the deterministic approach 158 642 Numerical application of the reliability-based approach 167
65 Integration of reliability and structural optimization into the design of mini-plate systems used to treat fractured mandibles 174
651 Numerical application of the deterministic approach 174 652 Numerical application of the reliability-based approach 181
Contents ix
66 Integration of reliability and structural optimization into dental implant design 184
661 Description and modeling of the problem 184 662 Numerical results 186
67 Conclusion 188
Appendices 189
Appendix 1 ANSYS Code for Stem Geometry 191
Appendix 2 ANSYS Code for Mini-Plate Geometry 197
Appendix 3 ANSYS Code for Dental Implant Geometry 201
Appendix 4 ANSYS Code for Geometry of Dental Implant with Bone 207
Bibliography 213
Index 229
Preface
The integration of structural optimization into biomechanics is a truly vast domain In this book we first focus on the integration of structural optimization into the design of orthopedic and orthodontic prostheses and also into drilling surgery Next we present the integration of reliability and structural optimization into the design of these prostheses which may be considered as a novel aspect introduced in this book The applications are made in 2D and in 3D considering the three major families of structural optimization sizing- shape- and topology optimization
In all domains of structural mechanics good design of a part is very important for its strength its lifetime and its use in service This is a challenge faced daily in sectors such as space research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve the techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile engines providing substantial energy savings The development of computer-aided design (CAD) techniques and optimization strategies is part of this context
Applying structural optimization is still somewhat complicated in certain domains Furthermore in deterministic structural optimization all parameters which are uncertain in nature are described by unfavorable characteristic values associated with safety coefficients The deterministic approach uses a pessimistic margin determined as a function of the consequences of a probable failure This approach often leads to unnecessary oversizing ndash particularly for sensitive structures
On the other hand researchers have developed a different approach which is better suited to uncertain physical phenomena In this approach the structure is deemed to have failed if the probability of failure is greater than a fixed threshold This is known as the ldquoprobabilistic approachrdquo The probabilistic approach is
xii Biomechanics
increasingly widely used in engineering as evidenced by the different applications in industry It is applied to check that the probability is sufficient when the structurersquos geometry is known or to optimize the sizing of the structure so as to respect certain fixed objectives such as a target cost or a required level of probability
Furthermore reliability analysis is an important tool in decision-making for establishing a maintenance- and inspection plan In addition it can be used in the validation of standards and regulations To perform reliability analysis various methods can be used to effectively and simply find the probability of failure Reliability analysis is a strategy used to evaluate the level of reliability without being able to control the design for a required reliability level For this reliability has become an important tool to be integrated into the process of structural optimization
This book also focuses on the necessary tools for the integration of reliability and structural optimization into biomechanics fields First the deterministic strategies of structural optimization are presented so we can implement them in structural design These deterministic strategies are applied in various domains in biomechanics including the design of orthopedic and orthodontic prostheses and drilling surgery Next reliability-based approaches pertaining to the integration of reliability into structural optimization are presented in detail with mechanical applications These reliability-based strategies are also applied in the design of orthopedic and orthodontic prostheses taking account of uncertainty in terms of geometry materials and load Finally system reliability strategies are also taken into account considering several failure scenarios
The book will provide invaluable support to teaching staff and researchers It is also intended for engineering students practising engineers and Masters students
Acknowledgements
We would like to thank all of those people who have in some way great or small contributed to the writing of this book ndash in particular Sophie Le Cann a researcher at the Biomedical Centre (BMC) at Lund University for her contribution in terms of biological language Heartfelt thanks go to our families to our students and to our colleagues for their massive moral support during the writing of this book
Ghias KHARMANDA Abdelkhalak EL-HAMI
October 2016
Introduction
This book begins with an introductory look at the fundamental principles of structural optimization before applying them to the field of biomechanics Then we present the different strategies for integrating reliability into structural optimization followed by the application of those strategies in biomechanics ndash particularly in terms of the design of orthopedic and orthodontic prostheses
In terms of structural optimization to illustrate the different techniques we can classify structural optimization into three main families
1) Sizing Optimization this model aims to improve a structural model whilst respecting the available resources (known as constraints or limitations) Sizing optimization is the particular case where we can only modify the cross section or transverse thickness of the components of a structure whose shape and topology are fixed There can be no modification of the geometric features andor models
2) Shape Optimization with this model it is possible to make changes to the shape provided they are compatible with a predefined topology This type of optimization modifies the parametric representation of the boundaries of the domain By moving those boundaries we try to find the best possible solution out of the set of all the configurations obtained by homeomorphic transformation of the original structure
3) Topology Optimization this model enables us to make more profound modifications to the shape of the structure Here the geometry of the part is examined with no prerequisites as to the connections of the domains or the structural elements present in the solution In order to optimize the topology we determine the structurersquos shape or transverse dimensions so some authors call topology optimization ldquogeneralized shape optimizationrdquo
xiv Biomechanics
In terms of reliability this concept can be integrated into all three families of structural optimization so we obtain a design that should be both optimal and reliable Sizing optimization shape optimization and topology optimization are generally classed as geometry optimization However the nature of the topology is non-quantitative in relation to shape and size For this reason this integration is divided into two models
1) RBDO Reliability-Based Design Optimization this model couples reliability analysis with sizing optimization (Reliability-Based Sizing Optimization) and also with shape optimization (Reliability-Based Shape Optimization) This coupling is a complex task requiring a lengthy computation time which seriously limits its applicability In addition during the process of reliability-based shape optimization the geometry of the structure is forced to change This coupling integrates different disciplines such as geometric modeling numerical simulation reliability analysis and optimization Thus the optimization problem becomes more complex The major difficulty lies in evaluating the reliability of the structure which in itself requires a specific optimization procedure The typical integration of reliability analysis into optimization methods is carried out in two spaces the normalized space of random variables and the physical space of design variables which requires a very significant computation time To solve this problem as we shall see there are a number of efficient methods
2) RBTO Reliability-Based Topology Optimization in the deterministic case we obtain a single optimal topology while the new RBTO model is able to generate several topologies as a function of a required reliability level However the coupling between reliability (which is quantitative) and topology (which is non-quantitative) requires the use of different methodologies relative to sizing optimization and shape optimization
In terms of biomechanics modeling is generally complicated on several different levels geometric description material properties and boundary conditions When performing the optimization process such modeling is needed with a new set of parameters being entered at each iteration In addition the problem becomes more complex when integrating reliability analysis which is itself performed by way of a particular optimization procedure In this case we couple several different software packages in order to integrate optimization and reliability into biomechanical applications Various strategies are presented in this book to simplify the coupling problem
This book is made up of six chapters presenting the different aspects of integration of reliability and structural optimization into biomechanics
Introduction xv
The first chapter offers an introduction to structural optimization and its main groups sizing optimization shape optimization and topology optimization The principles underpinning the three main types of structural optimization are presented with numerical applications for each type In structural design we have two different phases the conceptual phase and the detailed phase In the conceptual phase we use topology optimization to obtain an idea of the silhouette of the structure The detailed phase consists of shape optimization to obtain smooth geometry followed by sizing optimization to find the areas and thicknesses of the studied structure
The second chapter is devoted to the integration of structural optimization into biomechanics The integration is performed in this chapter for prosthesis design and drilling surgery Different 2D and 3D applications for orthopedic and orthodontic prostheses are presented in order to demonstrate the new strategies presented in this book and also to prove the advantages of those strategies Finally the integration of structural optimization into drilling surgery can be considered a novel aspect in relation to classic prosthesis design
The third chapter is given over to the integration of reliability into structural optimization Such integration can be viewed as a difficult task The difficulties arise in terms of coupling convergence and computation time In this chapter we present the basic principles of that integration and a literature review based on early attempts in this area A numerical application is presented to show the difference between deterministic- and reliability-based optimization Next two viewpoints are presented with the corresponding developed methods considering those two points of view to illustrate the advantages of the methods
The fourth chapter is devoted to the reliability-based design optimization (RBDO) model The developed methods are classified into two categories numerical and semi-numerical methods Numerical methods are generally applicable but require a high computation time Semi-numerical methods on the other hand are efficient in terms of computation time but can only be used in certain cases
The fifth chapter details reliability-based topology optimization (RBTO) This model enables designers and manufacturers to select the solution which is at once economical and reliable Two points of view are presented in the discussion of this model A number of applications in mechanics are presented in order to demonstrate the advantages of the RBTO model
The sixth chapter discusses the integration of reliability and structural optimization into orthopedic and orthodontic prostheses design considering two
xvi Biomechanics
approaches the deterministic approach and the reliability-based approach In orthopedic prostheses the models are presented using results drawn from the existing body of literature The aspect of materials characterization is also integrated into the design process In addition in orthodontics models are presented to test the stability of the prostheses in light of the constraints of dentistry The integration of reliability and structural optimization into the design of prostheses is performed very simply in order to aid understanding and implementation of the different strategies presented in this book
List of Abbreviations
AMA Approximate Moments Approach
AMV Advanced Mean Value
APDL ANSYS Parametric Design Language
AT Anterior Temporal (or Temporalis)
CAD Computer-Aided Design
CM Classical Method
CMV Conjugate Mean Value
CT Computed Tomography
DDO Deterministic Design Optimization
DLA Double-Level Approach
DM Deep Masseter
DO Deterministic Optimization
DTO Deterministic Topology Optimization
FE Finite Elements
FEM Finite Element Method
FORM First Order Reliability Method
HCA Hybrid Cellular Automata
HDS Hybrid Design Space
HM Hybrid Method
xviii Biomechanics
HMV Hybrid Mean Value
IAM Improved Austin-Moore
IHM Improved Hybrid Method
KKT Karush-Kuhn-Tucker
MBB Messerschmitt-Boumllkow-Blohm
MLA Mono-Level Approach
MP Medial Pterygoid
MPP Most Probable failure Point
MT Medial Temporal (or Temporalis)
OC Optimality Criteria
OSF Optimum Safety Factor
PDF Probability Density Function
PMA Performance Measure Approach
PT Posterior Temporal (or Temporalis)
RBDO Reliability-Based Design Optimization
RBO Reliability-Based Optimization
RBSO Reliability-Based Structural Optimization
RBTO Reliability-Based Topology Optimization
RIA Reliability Index Approach
SAP Sequential Approximate Programming
SDA Sequential Decoupled Approach
SED Strain Energy Density
SIMP Solid Isotropic Microstructure with Penalty
SM Superficial Masseter
SORA Sequential Optimization and Reliability Assessment
SORM Second Order Reliability Method
SP Safest Point
1
Introduction to Structural Optimization
11 Introduction
Structural optimization is a topic which affects many different physical domains ndash particularly solid mechanics ndash and which it is tricky to characterize as its formulations may have a number of different aspects Firstly a distinction regarding the way in which geometries are parameterized is presented and secondly a distribution pertaining to the intrinsic nature of optimization algorithms is established
Determining the appropriate shape for structural components is a crucially important problem for engineers In all areas of structural mechanics the impact of proper design of a part is very significant in terms of its strength its lifetime and its usage This is a challenge faced on a daily basis in the sectors of spatial research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile machinery providing substantial energy savings The different development of computer-aided design (CAD) techniques and optimization strategies is part of this context There has been keen interest in structural optimization for over thirty years Whilst it is still too infrequently applied in the conventional techniques used by research centers it is becoming more widely used as its reliability improves Having begun with the simplest of problems the field of application of structural optimization today extends to new and ever more interesting challenges
To illustrate the evolution of structural optimization techniques we can arbitrarily split structural optimization into three major groups (or families) In
Biomechanics Optimization Uncertainties and Reliability First Edition Ghias Kharmanda and Abdelkhalak El Hami copy ISTE Ltd 2017 Published by ISTE Ltd and John Wiley amp Sons Inc
2 Biomechanics
historical terms each of them has been addressed in order of increasing difficulty and generality
With sizing optimization we are only able to modify the dimensions of an object whose shape and topology are fixed There can be no modification of the geometric model We speak of a homeomorphic transformation
Shape optimization involves making changes of shape which are compatible with a predetermined topology Typical shape optimization modifies the parametric representation of the boundaries of the domain By moving the boundaries of the domains we can seek the best solution out of all the structures obtained by homeomorphic transformation of the original object In this case it is clear that we can make a change to the transverse dimensions as well as a modification to the objectrsquos configuration but it is certainly not acceptable to modify its connectivity or its nature ndash in particular the number of components that it has The optimal object exhibits the same topology as the original object
With topology optimization we can fundamentally change the nature of the object The ldquotopologyrdquo refers to the number and position of the components of the domains Here the objectrsquos geometry is presented with no prerequisites as to the connectivity of the domains or the components present in the solution We take no initial information about the topology of the optimal shape
12 History of structural optimization
It was in the early 1960s that Schmit [SCH 60] and Fox [FOX 65] laid the foundations for a modern theory of structural optimization based on the concepts of mathematical programming and sensitivity analysis Paradoxically at the time ldquofully stressed designrdquo was the only widely used technique in practice although it lacked any theoretical justification other than empiricism and the engineersrsquo intuition It was Prager and Taylor [PRA 68] who set out variational methods and Lagrangian optimality conditions to justify the criteria of fully stressed design for a class of structural optimization problems The optimality conditions of the optimization problem were then used directly to construct an iterative resolution algorithm known as the ldquooptimality criteria methodrdquo
Originally structural optimization was mainly limited to the sizing optimization of trusses or gantries Thus sizing optimization of structures was the first field of application for optimality criteria When dealing with the problem of sizing we look at the transverse sections of the structural elements though their length and the
Introduction to Structural Optimization 3
location of their joints remain fixed During the late 1960s and early 1970s optimality criteria were soon adapted to large structures modeled using the finite-element method (FEM) [VEN 73] The optimality criteria method produced a few interesting results and a number of extensions have been presented since the 1970s including Venkayyarsquos generalized criteria method [VEN 73] or Rozvany and Zhoursquos [ROZ 91] discretized optimality criteria
Although since the 1970s attention has mainly been focused on sizing the problem of truss topology has also been studied by Prager [PRA 74] on a very restricted class of structures based on the concept of a truss put forward by Michell in 1904 [MIC 04] The problem lies in finding the best possible configuration so that the truss or gantry can convey forces to the foundations whilst minimizing a given performance and satisfying the design constraints Michellrsquos theory [MIC 04] is related to the topology of trusses made of bars of minimal mass The optimal solution from Michellrsquos point of view is composed solely of perpendicular bars which form a structure whose configuration is optimal for the maximum tensile- and compressive stresses All the configuration problems studied by Prager later on were solved analytically so the practical application of topology was very limited To remedy this shortcoming Rozvany [ROZ 76] invested a great deal of effort in developing new approaches to solve these configuration problems as automatically as possible
After that optimal truss topology was studied in greater depth by Kirsch [KIR 90] If we impose a very small minimum value for the cross-section then the ldquolayoutrdquo optimization of trusses and gantries can be approached as a conventional sizing problem on a very large scale The solution is then obtained by application of generalized optimality criteria for a variety of objective functions compliance movements tensions and eigenvalues [ZHO 91]
Finally the crux of the problem of truss topology optimization seems to have been identified by Bendsoslashe et al [BEN 91] The problem of truss topology is examined with an integral approach combining analysis and design simultaneously The problem of minimum compliance is transformed into a problem of non-differentiable optimization and then in the case of a truss of bars into a linear problem In this form it can be solved on very large structures using non-differentiable optimization methods [BEN 93a] an interior point method [BEN 93b] a dual method [BEC 94] or a penaltybarrier multiplier method [BEN 97]
The three main groups of structural optimization cannot be considered recent concepts but their integration into biomechanics is ndash especially topology optimization [FRA 10]
4 Biomechanics
13 Sizing optimization
131 Definition
With sizing optimization we can only modify the cross-section or transverse thickness of the components of structure whose shape and topology are fixed There can be no modification of the geometric model and its features Figure 11 shows a truss made of 17 bars with different cross-sections (circular rectangular and I-shaped)
Figure 11 Changing the dimensions whilst preserving the same topology of the section
Sizing optimization can be performed by considering the same topology to produce various dimensions For example when the cross-section is circular we merely need to vary the diameters to minimize an objective or several objectives under certain constraints
132 First works in sizing optimization
The problem of sizing optimization has benefited the most from this research so optimization of the transverse dimensions is today a reliable tool This problem was also extended to that of flexural elements [FLE 83] to improving performance when subjected to vibration and to the stability of balance Besides the transverse dimensions of the structural elements it is possible to vary their shape To the best of our knowledge few works have been devoted to the study of the optimal shape for a truss In this type of problem only the location of the structural joints is altered whilst the topology remains unchanged Svanberg [SVA 81] was one of the only people to carry out truss shape optimization on the basis of FEM analyses and solving using mathematical programming techniques
Introduction to Structural Optimization 5
133 Numerical application
1331 Description and modeling of the studied problem
Figure 12a shows a cantilever beam and its I-shaped cross-section in Figure 12b This beam is embedded at one end and subjected to free vibration The material from which the beam is made is structural steel which has a Youngrsquos modulus 200000E MPa= and Poissonrsquos ratio of 03ν = The density of the material is 6 37854 10 Kg mmρ minus= times The material exhibits linear elastic isotropic behavior The length of that beam is 300L mm= and the dimensions of the cross-section are 60 B mm= 100H mm= and 20T mm= (Figure 12b)
a) b)
Figure 12 Cantilever beam subject to free vibration
The objective of this study is to calculate the first four modes of resonance and then perform sizing optimization on the dimensions of the cross-section The problem of optimization therefore is to minimize the structural volume under the constraint of the first resonance frequency This problem can be formulated as follows
1
min ( ) ( ) 0
40 100 80 16010 30
w
Volume B H Ts t f B H T f
BHT
minus lele lele lele le
[11]
where 20wf Hz= is the maximum value of the first resonance frequency
1332 Numerical results
To perform sizing optimization on the ANSYS software for instance we consider the dimensions of the cross-section as optimization variables so as to obtain a parameterized model Then a direct simulation is performed as the heart of the optimization loop
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
vi Biomechanics
221 Structural optimization of the hip prosthesis 24 222 Sizing optimization of a 3D intervertebral disk prosthesis 42
23 Integration of structural optimization into orthodontic prosthesis design 47
231 Sizing optimization of a dental implant 47 232 Shape optimization of a mini-plate 49
24 Advanced integration of structural optimization into drilling surgery 52
241 Case of treatment of a crack with a single hole 53 242 Case of treatment of a crack with two holes 54
25 Conclusion 56
Chapter 3 Integration of Reliability into Structural Optimization 57
31 Introduction 57 32 Literature review of reliability-based optimization 58 33 Comparison between deterministic and reliability-based optimization 60
331 Deterministic optimization 61 332 Reliability-based optimization 63
34 Numerical application 64 341 Description and modeling of the studied problem 64 342 Numerical results 65
35 Approaches and strategies for reliability-based optimization 68
351 Mono-level approaches 68 352 Double-level approaches 68 353 Sequential decoupled approaches 68
36 Two points of view for developments of reliability-based optimization 69
361 Point of view of ldquoReliabilityrdquo 69 362 Point of view of ldquoOptimizationrdquo 70 363 Method efficiency 70
37 Philosophy of integration of the concept of reliability into structural optimization groups 72 38 Conclusion 73
Chapter 4 Reliability-based Design Optimization Model 75
41 Introduction 75 42 Classic method 76
Contents vii
421 Formulations 76 422 Optimality conditions 77 423 Algorithm 77 424 Advantages and disadvantages 79
43 Hybrid method 79 431 Formulation 79 432 Optimality conditions 82 433 Algorithm 84 434 Advantages and disadvantages 85
44 Improved hybrid method 86 441 Formulations 86 442 Optimality conditions 86 443 Algorithm 89 444 Advantages and disadvantages 90
45 Optimum safety factor method 91 451 Safety factor concept 91 452 Developments and optimality conditions 92 453 Algorithm 97 454 Advantages and disadvantages 98
46 Safest point method 98 461 Formulations 98 462 Algorithm 102 463 Advantages and disadvantages 104
47 Numerical applications 105 471 RBDO of a hook CM and HM 105 472 RBDO of a triangular plate HM amp IHM 107 473 RBDO of a console beam (sandwich beam) HM and OSF 110 474 RBDO of an aircraft wing HM amp SP 113
48 Classification of the methods developed 115 481 Numerical methods 115 482 Semi-numerical methods 116 483 Comparison between the numerical- and semi-numerical methods 118
49 Conclusion 119
Chapter 5 Reliability-based Topology Optimization Model 121
51 Introduction 121 52 Formulation and algorithm for the RBTO model 122
521 Formulation 122
viii Biomechanics
522 Algorithm 123 523 Validation of the RBTO code developed 125
53 Validation of the RBTO model 126 531 Analytical validation 126 532 Numerical validation 128
54 Variability of the reliability index 134 541 Example 1 MBB beam 136 542 Example 2 Cantilever beam 136 543 Example 3 Cantilever beam with double loads 136 544 Example 4 Cantilever beam with a transversal hole 136
55 Numerical applications for the RBTO model 137 551 Static analysis 138 552 Modal analysis 139 553 Fatigue analysis 141
56 Two points of view for integration of reliability into topology optimization 142
561 Point of view of ldquotopologyrdquo 144 562 Point of view of ldquoreliabilityrdquo 144 563 Numerical applications for the two points of view 146
57 Conclusion 152
Chapter 6 Integration of Reliability and Structural Optimization into Prosthesis Design 153
61 Introduction 153 62 Prosthesis design 154 63 Integration of topology optimization into prosthesis design 154
631 Importance of topology optimization in prosthesis design 155 632 Place of topology optimization in the prosthesis design chain 156
64 Integration of reliability and structural optimization into hip prosthesis design 157
641 Numerical application of the deterministic approach 158 642 Numerical application of the reliability-based approach 167
65 Integration of reliability and structural optimization into the design of mini-plate systems used to treat fractured mandibles 174
651 Numerical application of the deterministic approach 174 652 Numerical application of the reliability-based approach 181
Contents ix
66 Integration of reliability and structural optimization into dental implant design 184
661 Description and modeling of the problem 184 662 Numerical results 186
67 Conclusion 188
Appendices 189
Appendix 1 ANSYS Code for Stem Geometry 191
Appendix 2 ANSYS Code for Mini-Plate Geometry 197
Appendix 3 ANSYS Code for Dental Implant Geometry 201
Appendix 4 ANSYS Code for Geometry of Dental Implant with Bone 207
Bibliography 213
Index 229
Preface
The integration of structural optimization into biomechanics is a truly vast domain In this book we first focus on the integration of structural optimization into the design of orthopedic and orthodontic prostheses and also into drilling surgery Next we present the integration of reliability and structural optimization into the design of these prostheses which may be considered as a novel aspect introduced in this book The applications are made in 2D and in 3D considering the three major families of structural optimization sizing- shape- and topology optimization
In all domains of structural mechanics good design of a part is very important for its strength its lifetime and its use in service This is a challenge faced daily in sectors such as space research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve the techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile engines providing substantial energy savings The development of computer-aided design (CAD) techniques and optimization strategies is part of this context
Applying structural optimization is still somewhat complicated in certain domains Furthermore in deterministic structural optimization all parameters which are uncertain in nature are described by unfavorable characteristic values associated with safety coefficients The deterministic approach uses a pessimistic margin determined as a function of the consequences of a probable failure This approach often leads to unnecessary oversizing ndash particularly for sensitive structures
On the other hand researchers have developed a different approach which is better suited to uncertain physical phenomena In this approach the structure is deemed to have failed if the probability of failure is greater than a fixed threshold This is known as the ldquoprobabilistic approachrdquo The probabilistic approach is
xii Biomechanics
increasingly widely used in engineering as evidenced by the different applications in industry It is applied to check that the probability is sufficient when the structurersquos geometry is known or to optimize the sizing of the structure so as to respect certain fixed objectives such as a target cost or a required level of probability
Furthermore reliability analysis is an important tool in decision-making for establishing a maintenance- and inspection plan In addition it can be used in the validation of standards and regulations To perform reliability analysis various methods can be used to effectively and simply find the probability of failure Reliability analysis is a strategy used to evaluate the level of reliability without being able to control the design for a required reliability level For this reliability has become an important tool to be integrated into the process of structural optimization
This book also focuses on the necessary tools for the integration of reliability and structural optimization into biomechanics fields First the deterministic strategies of structural optimization are presented so we can implement them in structural design These deterministic strategies are applied in various domains in biomechanics including the design of orthopedic and orthodontic prostheses and drilling surgery Next reliability-based approaches pertaining to the integration of reliability into structural optimization are presented in detail with mechanical applications These reliability-based strategies are also applied in the design of orthopedic and orthodontic prostheses taking account of uncertainty in terms of geometry materials and load Finally system reliability strategies are also taken into account considering several failure scenarios
The book will provide invaluable support to teaching staff and researchers It is also intended for engineering students practising engineers and Masters students
Acknowledgements
We would like to thank all of those people who have in some way great or small contributed to the writing of this book ndash in particular Sophie Le Cann a researcher at the Biomedical Centre (BMC) at Lund University for her contribution in terms of biological language Heartfelt thanks go to our families to our students and to our colleagues for their massive moral support during the writing of this book
Ghias KHARMANDA Abdelkhalak EL-HAMI
October 2016
Introduction
This book begins with an introductory look at the fundamental principles of structural optimization before applying them to the field of biomechanics Then we present the different strategies for integrating reliability into structural optimization followed by the application of those strategies in biomechanics ndash particularly in terms of the design of orthopedic and orthodontic prostheses
In terms of structural optimization to illustrate the different techniques we can classify structural optimization into three main families
1) Sizing Optimization this model aims to improve a structural model whilst respecting the available resources (known as constraints or limitations) Sizing optimization is the particular case where we can only modify the cross section or transverse thickness of the components of a structure whose shape and topology are fixed There can be no modification of the geometric features andor models
2) Shape Optimization with this model it is possible to make changes to the shape provided they are compatible with a predefined topology This type of optimization modifies the parametric representation of the boundaries of the domain By moving those boundaries we try to find the best possible solution out of the set of all the configurations obtained by homeomorphic transformation of the original structure
3) Topology Optimization this model enables us to make more profound modifications to the shape of the structure Here the geometry of the part is examined with no prerequisites as to the connections of the domains or the structural elements present in the solution In order to optimize the topology we determine the structurersquos shape or transverse dimensions so some authors call topology optimization ldquogeneralized shape optimizationrdquo
xiv Biomechanics
In terms of reliability this concept can be integrated into all three families of structural optimization so we obtain a design that should be both optimal and reliable Sizing optimization shape optimization and topology optimization are generally classed as geometry optimization However the nature of the topology is non-quantitative in relation to shape and size For this reason this integration is divided into two models
1) RBDO Reliability-Based Design Optimization this model couples reliability analysis with sizing optimization (Reliability-Based Sizing Optimization) and also with shape optimization (Reliability-Based Shape Optimization) This coupling is a complex task requiring a lengthy computation time which seriously limits its applicability In addition during the process of reliability-based shape optimization the geometry of the structure is forced to change This coupling integrates different disciplines such as geometric modeling numerical simulation reliability analysis and optimization Thus the optimization problem becomes more complex The major difficulty lies in evaluating the reliability of the structure which in itself requires a specific optimization procedure The typical integration of reliability analysis into optimization methods is carried out in two spaces the normalized space of random variables and the physical space of design variables which requires a very significant computation time To solve this problem as we shall see there are a number of efficient methods
2) RBTO Reliability-Based Topology Optimization in the deterministic case we obtain a single optimal topology while the new RBTO model is able to generate several topologies as a function of a required reliability level However the coupling between reliability (which is quantitative) and topology (which is non-quantitative) requires the use of different methodologies relative to sizing optimization and shape optimization
In terms of biomechanics modeling is generally complicated on several different levels geometric description material properties and boundary conditions When performing the optimization process such modeling is needed with a new set of parameters being entered at each iteration In addition the problem becomes more complex when integrating reliability analysis which is itself performed by way of a particular optimization procedure In this case we couple several different software packages in order to integrate optimization and reliability into biomechanical applications Various strategies are presented in this book to simplify the coupling problem
This book is made up of six chapters presenting the different aspects of integration of reliability and structural optimization into biomechanics
Introduction xv
The first chapter offers an introduction to structural optimization and its main groups sizing optimization shape optimization and topology optimization The principles underpinning the three main types of structural optimization are presented with numerical applications for each type In structural design we have two different phases the conceptual phase and the detailed phase In the conceptual phase we use topology optimization to obtain an idea of the silhouette of the structure The detailed phase consists of shape optimization to obtain smooth geometry followed by sizing optimization to find the areas and thicknesses of the studied structure
The second chapter is devoted to the integration of structural optimization into biomechanics The integration is performed in this chapter for prosthesis design and drilling surgery Different 2D and 3D applications for orthopedic and orthodontic prostheses are presented in order to demonstrate the new strategies presented in this book and also to prove the advantages of those strategies Finally the integration of structural optimization into drilling surgery can be considered a novel aspect in relation to classic prosthesis design
The third chapter is given over to the integration of reliability into structural optimization Such integration can be viewed as a difficult task The difficulties arise in terms of coupling convergence and computation time In this chapter we present the basic principles of that integration and a literature review based on early attempts in this area A numerical application is presented to show the difference between deterministic- and reliability-based optimization Next two viewpoints are presented with the corresponding developed methods considering those two points of view to illustrate the advantages of the methods
The fourth chapter is devoted to the reliability-based design optimization (RBDO) model The developed methods are classified into two categories numerical and semi-numerical methods Numerical methods are generally applicable but require a high computation time Semi-numerical methods on the other hand are efficient in terms of computation time but can only be used in certain cases
The fifth chapter details reliability-based topology optimization (RBTO) This model enables designers and manufacturers to select the solution which is at once economical and reliable Two points of view are presented in the discussion of this model A number of applications in mechanics are presented in order to demonstrate the advantages of the RBTO model
The sixth chapter discusses the integration of reliability and structural optimization into orthopedic and orthodontic prostheses design considering two
xvi Biomechanics
approaches the deterministic approach and the reliability-based approach In orthopedic prostheses the models are presented using results drawn from the existing body of literature The aspect of materials characterization is also integrated into the design process In addition in orthodontics models are presented to test the stability of the prostheses in light of the constraints of dentistry The integration of reliability and structural optimization into the design of prostheses is performed very simply in order to aid understanding and implementation of the different strategies presented in this book
List of Abbreviations
AMA Approximate Moments Approach
AMV Advanced Mean Value
APDL ANSYS Parametric Design Language
AT Anterior Temporal (or Temporalis)
CAD Computer-Aided Design
CM Classical Method
CMV Conjugate Mean Value
CT Computed Tomography
DDO Deterministic Design Optimization
DLA Double-Level Approach
DM Deep Masseter
DO Deterministic Optimization
DTO Deterministic Topology Optimization
FE Finite Elements
FEM Finite Element Method
FORM First Order Reliability Method
HCA Hybrid Cellular Automata
HDS Hybrid Design Space
HM Hybrid Method
xviii Biomechanics
HMV Hybrid Mean Value
IAM Improved Austin-Moore
IHM Improved Hybrid Method
KKT Karush-Kuhn-Tucker
MBB Messerschmitt-Boumllkow-Blohm
MLA Mono-Level Approach
MP Medial Pterygoid
MPP Most Probable failure Point
MT Medial Temporal (or Temporalis)
OC Optimality Criteria
OSF Optimum Safety Factor
PDF Probability Density Function
PMA Performance Measure Approach
PT Posterior Temporal (or Temporalis)
RBDO Reliability-Based Design Optimization
RBO Reliability-Based Optimization
RBSO Reliability-Based Structural Optimization
RBTO Reliability-Based Topology Optimization
RIA Reliability Index Approach
SAP Sequential Approximate Programming
SDA Sequential Decoupled Approach
SED Strain Energy Density
SIMP Solid Isotropic Microstructure with Penalty
SM Superficial Masseter
SORA Sequential Optimization and Reliability Assessment
SORM Second Order Reliability Method
SP Safest Point
1
Introduction to Structural Optimization
11 Introduction
Structural optimization is a topic which affects many different physical domains ndash particularly solid mechanics ndash and which it is tricky to characterize as its formulations may have a number of different aspects Firstly a distinction regarding the way in which geometries are parameterized is presented and secondly a distribution pertaining to the intrinsic nature of optimization algorithms is established
Determining the appropriate shape for structural components is a crucially important problem for engineers In all areas of structural mechanics the impact of proper design of a part is very significant in terms of its strength its lifetime and its usage This is a challenge faced on a daily basis in the sectors of spatial research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile machinery providing substantial energy savings The different development of computer-aided design (CAD) techniques and optimization strategies is part of this context There has been keen interest in structural optimization for over thirty years Whilst it is still too infrequently applied in the conventional techniques used by research centers it is becoming more widely used as its reliability improves Having begun with the simplest of problems the field of application of structural optimization today extends to new and ever more interesting challenges
To illustrate the evolution of structural optimization techniques we can arbitrarily split structural optimization into three major groups (or families) In
Biomechanics Optimization Uncertainties and Reliability First Edition Ghias Kharmanda and Abdelkhalak El Hami copy ISTE Ltd 2017 Published by ISTE Ltd and John Wiley amp Sons Inc
2 Biomechanics
historical terms each of them has been addressed in order of increasing difficulty and generality
With sizing optimization we are only able to modify the dimensions of an object whose shape and topology are fixed There can be no modification of the geometric model We speak of a homeomorphic transformation
Shape optimization involves making changes of shape which are compatible with a predetermined topology Typical shape optimization modifies the parametric representation of the boundaries of the domain By moving the boundaries of the domains we can seek the best solution out of all the structures obtained by homeomorphic transformation of the original object In this case it is clear that we can make a change to the transverse dimensions as well as a modification to the objectrsquos configuration but it is certainly not acceptable to modify its connectivity or its nature ndash in particular the number of components that it has The optimal object exhibits the same topology as the original object
With topology optimization we can fundamentally change the nature of the object The ldquotopologyrdquo refers to the number and position of the components of the domains Here the objectrsquos geometry is presented with no prerequisites as to the connectivity of the domains or the components present in the solution We take no initial information about the topology of the optimal shape
12 History of structural optimization
It was in the early 1960s that Schmit [SCH 60] and Fox [FOX 65] laid the foundations for a modern theory of structural optimization based on the concepts of mathematical programming and sensitivity analysis Paradoxically at the time ldquofully stressed designrdquo was the only widely used technique in practice although it lacked any theoretical justification other than empiricism and the engineersrsquo intuition It was Prager and Taylor [PRA 68] who set out variational methods and Lagrangian optimality conditions to justify the criteria of fully stressed design for a class of structural optimization problems The optimality conditions of the optimization problem were then used directly to construct an iterative resolution algorithm known as the ldquooptimality criteria methodrdquo
Originally structural optimization was mainly limited to the sizing optimization of trusses or gantries Thus sizing optimization of structures was the first field of application for optimality criteria When dealing with the problem of sizing we look at the transverse sections of the structural elements though their length and the
Introduction to Structural Optimization 3
location of their joints remain fixed During the late 1960s and early 1970s optimality criteria were soon adapted to large structures modeled using the finite-element method (FEM) [VEN 73] The optimality criteria method produced a few interesting results and a number of extensions have been presented since the 1970s including Venkayyarsquos generalized criteria method [VEN 73] or Rozvany and Zhoursquos [ROZ 91] discretized optimality criteria
Although since the 1970s attention has mainly been focused on sizing the problem of truss topology has also been studied by Prager [PRA 74] on a very restricted class of structures based on the concept of a truss put forward by Michell in 1904 [MIC 04] The problem lies in finding the best possible configuration so that the truss or gantry can convey forces to the foundations whilst minimizing a given performance and satisfying the design constraints Michellrsquos theory [MIC 04] is related to the topology of trusses made of bars of minimal mass The optimal solution from Michellrsquos point of view is composed solely of perpendicular bars which form a structure whose configuration is optimal for the maximum tensile- and compressive stresses All the configuration problems studied by Prager later on were solved analytically so the practical application of topology was very limited To remedy this shortcoming Rozvany [ROZ 76] invested a great deal of effort in developing new approaches to solve these configuration problems as automatically as possible
After that optimal truss topology was studied in greater depth by Kirsch [KIR 90] If we impose a very small minimum value for the cross-section then the ldquolayoutrdquo optimization of trusses and gantries can be approached as a conventional sizing problem on a very large scale The solution is then obtained by application of generalized optimality criteria for a variety of objective functions compliance movements tensions and eigenvalues [ZHO 91]
Finally the crux of the problem of truss topology optimization seems to have been identified by Bendsoslashe et al [BEN 91] The problem of truss topology is examined with an integral approach combining analysis and design simultaneously The problem of minimum compliance is transformed into a problem of non-differentiable optimization and then in the case of a truss of bars into a linear problem In this form it can be solved on very large structures using non-differentiable optimization methods [BEN 93a] an interior point method [BEN 93b] a dual method [BEC 94] or a penaltybarrier multiplier method [BEN 97]
The three main groups of structural optimization cannot be considered recent concepts but their integration into biomechanics is ndash especially topology optimization [FRA 10]
4 Biomechanics
13 Sizing optimization
131 Definition
With sizing optimization we can only modify the cross-section or transverse thickness of the components of structure whose shape and topology are fixed There can be no modification of the geometric model and its features Figure 11 shows a truss made of 17 bars with different cross-sections (circular rectangular and I-shaped)
Figure 11 Changing the dimensions whilst preserving the same topology of the section
Sizing optimization can be performed by considering the same topology to produce various dimensions For example when the cross-section is circular we merely need to vary the diameters to minimize an objective or several objectives under certain constraints
132 First works in sizing optimization
The problem of sizing optimization has benefited the most from this research so optimization of the transverse dimensions is today a reliable tool This problem was also extended to that of flexural elements [FLE 83] to improving performance when subjected to vibration and to the stability of balance Besides the transverse dimensions of the structural elements it is possible to vary their shape To the best of our knowledge few works have been devoted to the study of the optimal shape for a truss In this type of problem only the location of the structural joints is altered whilst the topology remains unchanged Svanberg [SVA 81] was one of the only people to carry out truss shape optimization on the basis of FEM analyses and solving using mathematical programming techniques
Introduction to Structural Optimization 5
133 Numerical application
1331 Description and modeling of the studied problem
Figure 12a shows a cantilever beam and its I-shaped cross-section in Figure 12b This beam is embedded at one end and subjected to free vibration The material from which the beam is made is structural steel which has a Youngrsquos modulus 200000E MPa= and Poissonrsquos ratio of 03ν = The density of the material is 6 37854 10 Kg mmρ minus= times The material exhibits linear elastic isotropic behavior The length of that beam is 300L mm= and the dimensions of the cross-section are 60 B mm= 100H mm= and 20T mm= (Figure 12b)
a) b)
Figure 12 Cantilever beam subject to free vibration
The objective of this study is to calculate the first four modes of resonance and then perform sizing optimization on the dimensions of the cross-section The problem of optimization therefore is to minimize the structural volume under the constraint of the first resonance frequency This problem can be formulated as follows
1
min ( ) ( ) 0
40 100 80 16010 30
w
Volume B H Ts t f B H T f
BHT
minus lele lele lele le
[11]
where 20wf Hz= is the maximum value of the first resonance frequency
1332 Numerical results
To perform sizing optimization on the ANSYS software for instance we consider the dimensions of the cross-section as optimization variables so as to obtain a parameterized model Then a direct simulation is performed as the heart of the optimization loop
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
Contents vii
421 Formulations 76 422 Optimality conditions 77 423 Algorithm 77 424 Advantages and disadvantages 79
43 Hybrid method 79 431 Formulation 79 432 Optimality conditions 82 433 Algorithm 84 434 Advantages and disadvantages 85
44 Improved hybrid method 86 441 Formulations 86 442 Optimality conditions 86 443 Algorithm 89 444 Advantages and disadvantages 90
45 Optimum safety factor method 91 451 Safety factor concept 91 452 Developments and optimality conditions 92 453 Algorithm 97 454 Advantages and disadvantages 98
46 Safest point method 98 461 Formulations 98 462 Algorithm 102 463 Advantages and disadvantages 104
47 Numerical applications 105 471 RBDO of a hook CM and HM 105 472 RBDO of a triangular plate HM amp IHM 107 473 RBDO of a console beam (sandwich beam) HM and OSF 110 474 RBDO of an aircraft wing HM amp SP 113
48 Classification of the methods developed 115 481 Numerical methods 115 482 Semi-numerical methods 116 483 Comparison between the numerical- and semi-numerical methods 118
49 Conclusion 119
Chapter 5 Reliability-based Topology Optimization Model 121
51 Introduction 121 52 Formulation and algorithm for the RBTO model 122
521 Formulation 122
viii Biomechanics
522 Algorithm 123 523 Validation of the RBTO code developed 125
53 Validation of the RBTO model 126 531 Analytical validation 126 532 Numerical validation 128
54 Variability of the reliability index 134 541 Example 1 MBB beam 136 542 Example 2 Cantilever beam 136 543 Example 3 Cantilever beam with double loads 136 544 Example 4 Cantilever beam with a transversal hole 136
55 Numerical applications for the RBTO model 137 551 Static analysis 138 552 Modal analysis 139 553 Fatigue analysis 141
56 Two points of view for integration of reliability into topology optimization 142
561 Point of view of ldquotopologyrdquo 144 562 Point of view of ldquoreliabilityrdquo 144 563 Numerical applications for the two points of view 146
57 Conclusion 152
Chapter 6 Integration of Reliability and Structural Optimization into Prosthesis Design 153
61 Introduction 153 62 Prosthesis design 154 63 Integration of topology optimization into prosthesis design 154
631 Importance of topology optimization in prosthesis design 155 632 Place of topology optimization in the prosthesis design chain 156
64 Integration of reliability and structural optimization into hip prosthesis design 157
641 Numerical application of the deterministic approach 158 642 Numerical application of the reliability-based approach 167
65 Integration of reliability and structural optimization into the design of mini-plate systems used to treat fractured mandibles 174
651 Numerical application of the deterministic approach 174 652 Numerical application of the reliability-based approach 181
Contents ix
66 Integration of reliability and structural optimization into dental implant design 184
661 Description and modeling of the problem 184 662 Numerical results 186
67 Conclusion 188
Appendices 189
Appendix 1 ANSYS Code for Stem Geometry 191
Appendix 2 ANSYS Code for Mini-Plate Geometry 197
Appendix 3 ANSYS Code for Dental Implant Geometry 201
Appendix 4 ANSYS Code for Geometry of Dental Implant with Bone 207
Bibliography 213
Index 229
Preface
The integration of structural optimization into biomechanics is a truly vast domain In this book we first focus on the integration of structural optimization into the design of orthopedic and orthodontic prostheses and also into drilling surgery Next we present the integration of reliability and structural optimization into the design of these prostheses which may be considered as a novel aspect introduced in this book The applications are made in 2D and in 3D considering the three major families of structural optimization sizing- shape- and topology optimization
In all domains of structural mechanics good design of a part is very important for its strength its lifetime and its use in service This is a challenge faced daily in sectors such as space research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve the techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile engines providing substantial energy savings The development of computer-aided design (CAD) techniques and optimization strategies is part of this context
Applying structural optimization is still somewhat complicated in certain domains Furthermore in deterministic structural optimization all parameters which are uncertain in nature are described by unfavorable characteristic values associated with safety coefficients The deterministic approach uses a pessimistic margin determined as a function of the consequences of a probable failure This approach often leads to unnecessary oversizing ndash particularly for sensitive structures
On the other hand researchers have developed a different approach which is better suited to uncertain physical phenomena In this approach the structure is deemed to have failed if the probability of failure is greater than a fixed threshold This is known as the ldquoprobabilistic approachrdquo The probabilistic approach is
xii Biomechanics
increasingly widely used in engineering as evidenced by the different applications in industry It is applied to check that the probability is sufficient when the structurersquos geometry is known or to optimize the sizing of the structure so as to respect certain fixed objectives such as a target cost or a required level of probability
Furthermore reliability analysis is an important tool in decision-making for establishing a maintenance- and inspection plan In addition it can be used in the validation of standards and regulations To perform reliability analysis various methods can be used to effectively and simply find the probability of failure Reliability analysis is a strategy used to evaluate the level of reliability without being able to control the design for a required reliability level For this reliability has become an important tool to be integrated into the process of structural optimization
This book also focuses on the necessary tools for the integration of reliability and structural optimization into biomechanics fields First the deterministic strategies of structural optimization are presented so we can implement them in structural design These deterministic strategies are applied in various domains in biomechanics including the design of orthopedic and orthodontic prostheses and drilling surgery Next reliability-based approaches pertaining to the integration of reliability into structural optimization are presented in detail with mechanical applications These reliability-based strategies are also applied in the design of orthopedic and orthodontic prostheses taking account of uncertainty in terms of geometry materials and load Finally system reliability strategies are also taken into account considering several failure scenarios
The book will provide invaluable support to teaching staff and researchers It is also intended for engineering students practising engineers and Masters students
Acknowledgements
We would like to thank all of those people who have in some way great or small contributed to the writing of this book ndash in particular Sophie Le Cann a researcher at the Biomedical Centre (BMC) at Lund University for her contribution in terms of biological language Heartfelt thanks go to our families to our students and to our colleagues for their massive moral support during the writing of this book
Ghias KHARMANDA Abdelkhalak EL-HAMI
October 2016
Introduction
This book begins with an introductory look at the fundamental principles of structural optimization before applying them to the field of biomechanics Then we present the different strategies for integrating reliability into structural optimization followed by the application of those strategies in biomechanics ndash particularly in terms of the design of orthopedic and orthodontic prostheses
In terms of structural optimization to illustrate the different techniques we can classify structural optimization into three main families
1) Sizing Optimization this model aims to improve a structural model whilst respecting the available resources (known as constraints or limitations) Sizing optimization is the particular case where we can only modify the cross section or transverse thickness of the components of a structure whose shape and topology are fixed There can be no modification of the geometric features andor models
2) Shape Optimization with this model it is possible to make changes to the shape provided they are compatible with a predefined topology This type of optimization modifies the parametric representation of the boundaries of the domain By moving those boundaries we try to find the best possible solution out of the set of all the configurations obtained by homeomorphic transformation of the original structure
3) Topology Optimization this model enables us to make more profound modifications to the shape of the structure Here the geometry of the part is examined with no prerequisites as to the connections of the domains or the structural elements present in the solution In order to optimize the topology we determine the structurersquos shape or transverse dimensions so some authors call topology optimization ldquogeneralized shape optimizationrdquo
xiv Biomechanics
In terms of reliability this concept can be integrated into all three families of structural optimization so we obtain a design that should be both optimal and reliable Sizing optimization shape optimization and topology optimization are generally classed as geometry optimization However the nature of the topology is non-quantitative in relation to shape and size For this reason this integration is divided into two models
1) RBDO Reliability-Based Design Optimization this model couples reliability analysis with sizing optimization (Reliability-Based Sizing Optimization) and also with shape optimization (Reliability-Based Shape Optimization) This coupling is a complex task requiring a lengthy computation time which seriously limits its applicability In addition during the process of reliability-based shape optimization the geometry of the structure is forced to change This coupling integrates different disciplines such as geometric modeling numerical simulation reliability analysis and optimization Thus the optimization problem becomes more complex The major difficulty lies in evaluating the reliability of the structure which in itself requires a specific optimization procedure The typical integration of reliability analysis into optimization methods is carried out in two spaces the normalized space of random variables and the physical space of design variables which requires a very significant computation time To solve this problem as we shall see there are a number of efficient methods
2) RBTO Reliability-Based Topology Optimization in the deterministic case we obtain a single optimal topology while the new RBTO model is able to generate several topologies as a function of a required reliability level However the coupling between reliability (which is quantitative) and topology (which is non-quantitative) requires the use of different methodologies relative to sizing optimization and shape optimization
In terms of biomechanics modeling is generally complicated on several different levels geometric description material properties and boundary conditions When performing the optimization process such modeling is needed with a new set of parameters being entered at each iteration In addition the problem becomes more complex when integrating reliability analysis which is itself performed by way of a particular optimization procedure In this case we couple several different software packages in order to integrate optimization and reliability into biomechanical applications Various strategies are presented in this book to simplify the coupling problem
This book is made up of six chapters presenting the different aspects of integration of reliability and structural optimization into biomechanics
Introduction xv
The first chapter offers an introduction to structural optimization and its main groups sizing optimization shape optimization and topology optimization The principles underpinning the three main types of structural optimization are presented with numerical applications for each type In structural design we have two different phases the conceptual phase and the detailed phase In the conceptual phase we use topology optimization to obtain an idea of the silhouette of the structure The detailed phase consists of shape optimization to obtain smooth geometry followed by sizing optimization to find the areas and thicknesses of the studied structure
The second chapter is devoted to the integration of structural optimization into biomechanics The integration is performed in this chapter for prosthesis design and drilling surgery Different 2D and 3D applications for orthopedic and orthodontic prostheses are presented in order to demonstrate the new strategies presented in this book and also to prove the advantages of those strategies Finally the integration of structural optimization into drilling surgery can be considered a novel aspect in relation to classic prosthesis design
The third chapter is given over to the integration of reliability into structural optimization Such integration can be viewed as a difficult task The difficulties arise in terms of coupling convergence and computation time In this chapter we present the basic principles of that integration and a literature review based on early attempts in this area A numerical application is presented to show the difference between deterministic- and reliability-based optimization Next two viewpoints are presented with the corresponding developed methods considering those two points of view to illustrate the advantages of the methods
The fourth chapter is devoted to the reliability-based design optimization (RBDO) model The developed methods are classified into two categories numerical and semi-numerical methods Numerical methods are generally applicable but require a high computation time Semi-numerical methods on the other hand are efficient in terms of computation time but can only be used in certain cases
The fifth chapter details reliability-based topology optimization (RBTO) This model enables designers and manufacturers to select the solution which is at once economical and reliable Two points of view are presented in the discussion of this model A number of applications in mechanics are presented in order to demonstrate the advantages of the RBTO model
The sixth chapter discusses the integration of reliability and structural optimization into orthopedic and orthodontic prostheses design considering two
xvi Biomechanics
approaches the deterministic approach and the reliability-based approach In orthopedic prostheses the models are presented using results drawn from the existing body of literature The aspect of materials characterization is also integrated into the design process In addition in orthodontics models are presented to test the stability of the prostheses in light of the constraints of dentistry The integration of reliability and structural optimization into the design of prostheses is performed very simply in order to aid understanding and implementation of the different strategies presented in this book
List of Abbreviations
AMA Approximate Moments Approach
AMV Advanced Mean Value
APDL ANSYS Parametric Design Language
AT Anterior Temporal (or Temporalis)
CAD Computer-Aided Design
CM Classical Method
CMV Conjugate Mean Value
CT Computed Tomography
DDO Deterministic Design Optimization
DLA Double-Level Approach
DM Deep Masseter
DO Deterministic Optimization
DTO Deterministic Topology Optimization
FE Finite Elements
FEM Finite Element Method
FORM First Order Reliability Method
HCA Hybrid Cellular Automata
HDS Hybrid Design Space
HM Hybrid Method
xviii Biomechanics
HMV Hybrid Mean Value
IAM Improved Austin-Moore
IHM Improved Hybrid Method
KKT Karush-Kuhn-Tucker
MBB Messerschmitt-Boumllkow-Blohm
MLA Mono-Level Approach
MP Medial Pterygoid
MPP Most Probable failure Point
MT Medial Temporal (or Temporalis)
OC Optimality Criteria
OSF Optimum Safety Factor
PDF Probability Density Function
PMA Performance Measure Approach
PT Posterior Temporal (or Temporalis)
RBDO Reliability-Based Design Optimization
RBO Reliability-Based Optimization
RBSO Reliability-Based Structural Optimization
RBTO Reliability-Based Topology Optimization
RIA Reliability Index Approach
SAP Sequential Approximate Programming
SDA Sequential Decoupled Approach
SED Strain Energy Density
SIMP Solid Isotropic Microstructure with Penalty
SM Superficial Masseter
SORA Sequential Optimization and Reliability Assessment
SORM Second Order Reliability Method
SP Safest Point
1
Introduction to Structural Optimization
11 Introduction
Structural optimization is a topic which affects many different physical domains ndash particularly solid mechanics ndash and which it is tricky to characterize as its formulations may have a number of different aspects Firstly a distinction regarding the way in which geometries are parameterized is presented and secondly a distribution pertaining to the intrinsic nature of optimization algorithms is established
Determining the appropriate shape for structural components is a crucially important problem for engineers In all areas of structural mechanics the impact of proper design of a part is very significant in terms of its strength its lifetime and its usage This is a challenge faced on a daily basis in the sectors of spatial research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile machinery providing substantial energy savings The different development of computer-aided design (CAD) techniques and optimization strategies is part of this context There has been keen interest in structural optimization for over thirty years Whilst it is still too infrequently applied in the conventional techniques used by research centers it is becoming more widely used as its reliability improves Having begun with the simplest of problems the field of application of structural optimization today extends to new and ever more interesting challenges
To illustrate the evolution of structural optimization techniques we can arbitrarily split structural optimization into three major groups (or families) In
Biomechanics Optimization Uncertainties and Reliability First Edition Ghias Kharmanda and Abdelkhalak El Hami copy ISTE Ltd 2017 Published by ISTE Ltd and John Wiley amp Sons Inc
2 Biomechanics
historical terms each of them has been addressed in order of increasing difficulty and generality
With sizing optimization we are only able to modify the dimensions of an object whose shape and topology are fixed There can be no modification of the geometric model We speak of a homeomorphic transformation
Shape optimization involves making changes of shape which are compatible with a predetermined topology Typical shape optimization modifies the parametric representation of the boundaries of the domain By moving the boundaries of the domains we can seek the best solution out of all the structures obtained by homeomorphic transformation of the original object In this case it is clear that we can make a change to the transverse dimensions as well as a modification to the objectrsquos configuration but it is certainly not acceptable to modify its connectivity or its nature ndash in particular the number of components that it has The optimal object exhibits the same topology as the original object
With topology optimization we can fundamentally change the nature of the object The ldquotopologyrdquo refers to the number and position of the components of the domains Here the objectrsquos geometry is presented with no prerequisites as to the connectivity of the domains or the components present in the solution We take no initial information about the topology of the optimal shape
12 History of structural optimization
It was in the early 1960s that Schmit [SCH 60] and Fox [FOX 65] laid the foundations for a modern theory of structural optimization based on the concepts of mathematical programming and sensitivity analysis Paradoxically at the time ldquofully stressed designrdquo was the only widely used technique in practice although it lacked any theoretical justification other than empiricism and the engineersrsquo intuition It was Prager and Taylor [PRA 68] who set out variational methods and Lagrangian optimality conditions to justify the criteria of fully stressed design for a class of structural optimization problems The optimality conditions of the optimization problem were then used directly to construct an iterative resolution algorithm known as the ldquooptimality criteria methodrdquo
Originally structural optimization was mainly limited to the sizing optimization of trusses or gantries Thus sizing optimization of structures was the first field of application for optimality criteria When dealing with the problem of sizing we look at the transverse sections of the structural elements though their length and the
Introduction to Structural Optimization 3
location of their joints remain fixed During the late 1960s and early 1970s optimality criteria were soon adapted to large structures modeled using the finite-element method (FEM) [VEN 73] The optimality criteria method produced a few interesting results and a number of extensions have been presented since the 1970s including Venkayyarsquos generalized criteria method [VEN 73] or Rozvany and Zhoursquos [ROZ 91] discretized optimality criteria
Although since the 1970s attention has mainly been focused on sizing the problem of truss topology has also been studied by Prager [PRA 74] on a very restricted class of structures based on the concept of a truss put forward by Michell in 1904 [MIC 04] The problem lies in finding the best possible configuration so that the truss or gantry can convey forces to the foundations whilst minimizing a given performance and satisfying the design constraints Michellrsquos theory [MIC 04] is related to the topology of trusses made of bars of minimal mass The optimal solution from Michellrsquos point of view is composed solely of perpendicular bars which form a structure whose configuration is optimal for the maximum tensile- and compressive stresses All the configuration problems studied by Prager later on were solved analytically so the practical application of topology was very limited To remedy this shortcoming Rozvany [ROZ 76] invested a great deal of effort in developing new approaches to solve these configuration problems as automatically as possible
After that optimal truss topology was studied in greater depth by Kirsch [KIR 90] If we impose a very small minimum value for the cross-section then the ldquolayoutrdquo optimization of trusses and gantries can be approached as a conventional sizing problem on a very large scale The solution is then obtained by application of generalized optimality criteria for a variety of objective functions compliance movements tensions and eigenvalues [ZHO 91]
Finally the crux of the problem of truss topology optimization seems to have been identified by Bendsoslashe et al [BEN 91] The problem of truss topology is examined with an integral approach combining analysis and design simultaneously The problem of minimum compliance is transformed into a problem of non-differentiable optimization and then in the case of a truss of bars into a linear problem In this form it can be solved on very large structures using non-differentiable optimization methods [BEN 93a] an interior point method [BEN 93b] a dual method [BEC 94] or a penaltybarrier multiplier method [BEN 97]
The three main groups of structural optimization cannot be considered recent concepts but their integration into biomechanics is ndash especially topology optimization [FRA 10]
4 Biomechanics
13 Sizing optimization
131 Definition
With sizing optimization we can only modify the cross-section or transverse thickness of the components of structure whose shape and topology are fixed There can be no modification of the geometric model and its features Figure 11 shows a truss made of 17 bars with different cross-sections (circular rectangular and I-shaped)
Figure 11 Changing the dimensions whilst preserving the same topology of the section
Sizing optimization can be performed by considering the same topology to produce various dimensions For example when the cross-section is circular we merely need to vary the diameters to minimize an objective or several objectives under certain constraints
132 First works in sizing optimization
The problem of sizing optimization has benefited the most from this research so optimization of the transverse dimensions is today a reliable tool This problem was also extended to that of flexural elements [FLE 83] to improving performance when subjected to vibration and to the stability of balance Besides the transverse dimensions of the structural elements it is possible to vary their shape To the best of our knowledge few works have been devoted to the study of the optimal shape for a truss In this type of problem only the location of the structural joints is altered whilst the topology remains unchanged Svanberg [SVA 81] was one of the only people to carry out truss shape optimization on the basis of FEM analyses and solving using mathematical programming techniques
Introduction to Structural Optimization 5
133 Numerical application
1331 Description and modeling of the studied problem
Figure 12a shows a cantilever beam and its I-shaped cross-section in Figure 12b This beam is embedded at one end and subjected to free vibration The material from which the beam is made is structural steel which has a Youngrsquos modulus 200000E MPa= and Poissonrsquos ratio of 03ν = The density of the material is 6 37854 10 Kg mmρ minus= times The material exhibits linear elastic isotropic behavior The length of that beam is 300L mm= and the dimensions of the cross-section are 60 B mm= 100H mm= and 20T mm= (Figure 12b)
a) b)
Figure 12 Cantilever beam subject to free vibration
The objective of this study is to calculate the first four modes of resonance and then perform sizing optimization on the dimensions of the cross-section The problem of optimization therefore is to minimize the structural volume under the constraint of the first resonance frequency This problem can be formulated as follows
1
min ( ) ( ) 0
40 100 80 16010 30
w
Volume B H Ts t f B H T f
BHT
minus lele lele lele le
[11]
where 20wf Hz= is the maximum value of the first resonance frequency
1332 Numerical results
To perform sizing optimization on the ANSYS software for instance we consider the dimensions of the cross-section as optimization variables so as to obtain a parameterized model Then a direct simulation is performed as the heart of the optimization loop
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
viii Biomechanics
522 Algorithm 123 523 Validation of the RBTO code developed 125
53 Validation of the RBTO model 126 531 Analytical validation 126 532 Numerical validation 128
54 Variability of the reliability index 134 541 Example 1 MBB beam 136 542 Example 2 Cantilever beam 136 543 Example 3 Cantilever beam with double loads 136 544 Example 4 Cantilever beam with a transversal hole 136
55 Numerical applications for the RBTO model 137 551 Static analysis 138 552 Modal analysis 139 553 Fatigue analysis 141
56 Two points of view for integration of reliability into topology optimization 142
561 Point of view of ldquotopologyrdquo 144 562 Point of view of ldquoreliabilityrdquo 144 563 Numerical applications for the two points of view 146
57 Conclusion 152
Chapter 6 Integration of Reliability and Structural Optimization into Prosthesis Design 153
61 Introduction 153 62 Prosthesis design 154 63 Integration of topology optimization into prosthesis design 154
631 Importance of topology optimization in prosthesis design 155 632 Place of topology optimization in the prosthesis design chain 156
64 Integration of reliability and structural optimization into hip prosthesis design 157
641 Numerical application of the deterministic approach 158 642 Numerical application of the reliability-based approach 167
65 Integration of reliability and structural optimization into the design of mini-plate systems used to treat fractured mandibles 174
651 Numerical application of the deterministic approach 174 652 Numerical application of the reliability-based approach 181
Contents ix
66 Integration of reliability and structural optimization into dental implant design 184
661 Description and modeling of the problem 184 662 Numerical results 186
67 Conclusion 188
Appendices 189
Appendix 1 ANSYS Code for Stem Geometry 191
Appendix 2 ANSYS Code for Mini-Plate Geometry 197
Appendix 3 ANSYS Code for Dental Implant Geometry 201
Appendix 4 ANSYS Code for Geometry of Dental Implant with Bone 207
Bibliography 213
Index 229
Preface
The integration of structural optimization into biomechanics is a truly vast domain In this book we first focus on the integration of structural optimization into the design of orthopedic and orthodontic prostheses and also into drilling surgery Next we present the integration of reliability and structural optimization into the design of these prostheses which may be considered as a novel aspect introduced in this book The applications are made in 2D and in 3D considering the three major families of structural optimization sizing- shape- and topology optimization
In all domains of structural mechanics good design of a part is very important for its strength its lifetime and its use in service This is a challenge faced daily in sectors such as space research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve the techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile engines providing substantial energy savings The development of computer-aided design (CAD) techniques and optimization strategies is part of this context
Applying structural optimization is still somewhat complicated in certain domains Furthermore in deterministic structural optimization all parameters which are uncertain in nature are described by unfavorable characteristic values associated with safety coefficients The deterministic approach uses a pessimistic margin determined as a function of the consequences of a probable failure This approach often leads to unnecessary oversizing ndash particularly for sensitive structures
On the other hand researchers have developed a different approach which is better suited to uncertain physical phenomena In this approach the structure is deemed to have failed if the probability of failure is greater than a fixed threshold This is known as the ldquoprobabilistic approachrdquo The probabilistic approach is
xii Biomechanics
increasingly widely used in engineering as evidenced by the different applications in industry It is applied to check that the probability is sufficient when the structurersquos geometry is known or to optimize the sizing of the structure so as to respect certain fixed objectives such as a target cost or a required level of probability
Furthermore reliability analysis is an important tool in decision-making for establishing a maintenance- and inspection plan In addition it can be used in the validation of standards and regulations To perform reliability analysis various methods can be used to effectively and simply find the probability of failure Reliability analysis is a strategy used to evaluate the level of reliability without being able to control the design for a required reliability level For this reliability has become an important tool to be integrated into the process of structural optimization
This book also focuses on the necessary tools for the integration of reliability and structural optimization into biomechanics fields First the deterministic strategies of structural optimization are presented so we can implement them in structural design These deterministic strategies are applied in various domains in biomechanics including the design of orthopedic and orthodontic prostheses and drilling surgery Next reliability-based approaches pertaining to the integration of reliability into structural optimization are presented in detail with mechanical applications These reliability-based strategies are also applied in the design of orthopedic and orthodontic prostheses taking account of uncertainty in terms of geometry materials and load Finally system reliability strategies are also taken into account considering several failure scenarios
The book will provide invaluable support to teaching staff and researchers It is also intended for engineering students practising engineers and Masters students
Acknowledgements
We would like to thank all of those people who have in some way great or small contributed to the writing of this book ndash in particular Sophie Le Cann a researcher at the Biomedical Centre (BMC) at Lund University for her contribution in terms of biological language Heartfelt thanks go to our families to our students and to our colleagues for their massive moral support during the writing of this book
Ghias KHARMANDA Abdelkhalak EL-HAMI
October 2016
Introduction
This book begins with an introductory look at the fundamental principles of structural optimization before applying them to the field of biomechanics Then we present the different strategies for integrating reliability into structural optimization followed by the application of those strategies in biomechanics ndash particularly in terms of the design of orthopedic and orthodontic prostheses
In terms of structural optimization to illustrate the different techniques we can classify structural optimization into three main families
1) Sizing Optimization this model aims to improve a structural model whilst respecting the available resources (known as constraints or limitations) Sizing optimization is the particular case where we can only modify the cross section or transverse thickness of the components of a structure whose shape and topology are fixed There can be no modification of the geometric features andor models
2) Shape Optimization with this model it is possible to make changes to the shape provided they are compatible with a predefined topology This type of optimization modifies the parametric representation of the boundaries of the domain By moving those boundaries we try to find the best possible solution out of the set of all the configurations obtained by homeomorphic transformation of the original structure
3) Topology Optimization this model enables us to make more profound modifications to the shape of the structure Here the geometry of the part is examined with no prerequisites as to the connections of the domains or the structural elements present in the solution In order to optimize the topology we determine the structurersquos shape or transverse dimensions so some authors call topology optimization ldquogeneralized shape optimizationrdquo
xiv Biomechanics
In terms of reliability this concept can be integrated into all three families of structural optimization so we obtain a design that should be both optimal and reliable Sizing optimization shape optimization and topology optimization are generally classed as geometry optimization However the nature of the topology is non-quantitative in relation to shape and size For this reason this integration is divided into two models
1) RBDO Reliability-Based Design Optimization this model couples reliability analysis with sizing optimization (Reliability-Based Sizing Optimization) and also with shape optimization (Reliability-Based Shape Optimization) This coupling is a complex task requiring a lengthy computation time which seriously limits its applicability In addition during the process of reliability-based shape optimization the geometry of the structure is forced to change This coupling integrates different disciplines such as geometric modeling numerical simulation reliability analysis and optimization Thus the optimization problem becomes more complex The major difficulty lies in evaluating the reliability of the structure which in itself requires a specific optimization procedure The typical integration of reliability analysis into optimization methods is carried out in two spaces the normalized space of random variables and the physical space of design variables which requires a very significant computation time To solve this problem as we shall see there are a number of efficient methods
2) RBTO Reliability-Based Topology Optimization in the deterministic case we obtain a single optimal topology while the new RBTO model is able to generate several topologies as a function of a required reliability level However the coupling between reliability (which is quantitative) and topology (which is non-quantitative) requires the use of different methodologies relative to sizing optimization and shape optimization
In terms of biomechanics modeling is generally complicated on several different levels geometric description material properties and boundary conditions When performing the optimization process such modeling is needed with a new set of parameters being entered at each iteration In addition the problem becomes more complex when integrating reliability analysis which is itself performed by way of a particular optimization procedure In this case we couple several different software packages in order to integrate optimization and reliability into biomechanical applications Various strategies are presented in this book to simplify the coupling problem
This book is made up of six chapters presenting the different aspects of integration of reliability and structural optimization into biomechanics
Introduction xv
The first chapter offers an introduction to structural optimization and its main groups sizing optimization shape optimization and topology optimization The principles underpinning the three main types of structural optimization are presented with numerical applications for each type In structural design we have two different phases the conceptual phase and the detailed phase In the conceptual phase we use topology optimization to obtain an idea of the silhouette of the structure The detailed phase consists of shape optimization to obtain smooth geometry followed by sizing optimization to find the areas and thicknesses of the studied structure
The second chapter is devoted to the integration of structural optimization into biomechanics The integration is performed in this chapter for prosthesis design and drilling surgery Different 2D and 3D applications for orthopedic and orthodontic prostheses are presented in order to demonstrate the new strategies presented in this book and also to prove the advantages of those strategies Finally the integration of structural optimization into drilling surgery can be considered a novel aspect in relation to classic prosthesis design
The third chapter is given over to the integration of reliability into structural optimization Such integration can be viewed as a difficult task The difficulties arise in terms of coupling convergence and computation time In this chapter we present the basic principles of that integration and a literature review based on early attempts in this area A numerical application is presented to show the difference between deterministic- and reliability-based optimization Next two viewpoints are presented with the corresponding developed methods considering those two points of view to illustrate the advantages of the methods
The fourth chapter is devoted to the reliability-based design optimization (RBDO) model The developed methods are classified into two categories numerical and semi-numerical methods Numerical methods are generally applicable but require a high computation time Semi-numerical methods on the other hand are efficient in terms of computation time but can only be used in certain cases
The fifth chapter details reliability-based topology optimization (RBTO) This model enables designers and manufacturers to select the solution which is at once economical and reliable Two points of view are presented in the discussion of this model A number of applications in mechanics are presented in order to demonstrate the advantages of the RBTO model
The sixth chapter discusses the integration of reliability and structural optimization into orthopedic and orthodontic prostheses design considering two
xvi Biomechanics
approaches the deterministic approach and the reliability-based approach In orthopedic prostheses the models are presented using results drawn from the existing body of literature The aspect of materials characterization is also integrated into the design process In addition in orthodontics models are presented to test the stability of the prostheses in light of the constraints of dentistry The integration of reliability and structural optimization into the design of prostheses is performed very simply in order to aid understanding and implementation of the different strategies presented in this book
List of Abbreviations
AMA Approximate Moments Approach
AMV Advanced Mean Value
APDL ANSYS Parametric Design Language
AT Anterior Temporal (or Temporalis)
CAD Computer-Aided Design
CM Classical Method
CMV Conjugate Mean Value
CT Computed Tomography
DDO Deterministic Design Optimization
DLA Double-Level Approach
DM Deep Masseter
DO Deterministic Optimization
DTO Deterministic Topology Optimization
FE Finite Elements
FEM Finite Element Method
FORM First Order Reliability Method
HCA Hybrid Cellular Automata
HDS Hybrid Design Space
HM Hybrid Method
xviii Biomechanics
HMV Hybrid Mean Value
IAM Improved Austin-Moore
IHM Improved Hybrid Method
KKT Karush-Kuhn-Tucker
MBB Messerschmitt-Boumllkow-Blohm
MLA Mono-Level Approach
MP Medial Pterygoid
MPP Most Probable failure Point
MT Medial Temporal (or Temporalis)
OC Optimality Criteria
OSF Optimum Safety Factor
PDF Probability Density Function
PMA Performance Measure Approach
PT Posterior Temporal (or Temporalis)
RBDO Reliability-Based Design Optimization
RBO Reliability-Based Optimization
RBSO Reliability-Based Structural Optimization
RBTO Reliability-Based Topology Optimization
RIA Reliability Index Approach
SAP Sequential Approximate Programming
SDA Sequential Decoupled Approach
SED Strain Energy Density
SIMP Solid Isotropic Microstructure with Penalty
SM Superficial Masseter
SORA Sequential Optimization and Reliability Assessment
SORM Second Order Reliability Method
SP Safest Point
1
Introduction to Structural Optimization
11 Introduction
Structural optimization is a topic which affects many different physical domains ndash particularly solid mechanics ndash and which it is tricky to characterize as its formulations may have a number of different aspects Firstly a distinction regarding the way in which geometries are parameterized is presented and secondly a distribution pertaining to the intrinsic nature of optimization algorithms is established
Determining the appropriate shape for structural components is a crucially important problem for engineers In all areas of structural mechanics the impact of proper design of a part is very significant in terms of its strength its lifetime and its usage This is a challenge faced on a daily basis in the sectors of spatial research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile machinery providing substantial energy savings The different development of computer-aided design (CAD) techniques and optimization strategies is part of this context There has been keen interest in structural optimization for over thirty years Whilst it is still too infrequently applied in the conventional techniques used by research centers it is becoming more widely used as its reliability improves Having begun with the simplest of problems the field of application of structural optimization today extends to new and ever more interesting challenges
To illustrate the evolution of structural optimization techniques we can arbitrarily split structural optimization into three major groups (or families) In
Biomechanics Optimization Uncertainties and Reliability First Edition Ghias Kharmanda and Abdelkhalak El Hami copy ISTE Ltd 2017 Published by ISTE Ltd and John Wiley amp Sons Inc
2 Biomechanics
historical terms each of them has been addressed in order of increasing difficulty and generality
With sizing optimization we are only able to modify the dimensions of an object whose shape and topology are fixed There can be no modification of the geometric model We speak of a homeomorphic transformation
Shape optimization involves making changes of shape which are compatible with a predetermined topology Typical shape optimization modifies the parametric representation of the boundaries of the domain By moving the boundaries of the domains we can seek the best solution out of all the structures obtained by homeomorphic transformation of the original object In this case it is clear that we can make a change to the transverse dimensions as well as a modification to the objectrsquos configuration but it is certainly not acceptable to modify its connectivity or its nature ndash in particular the number of components that it has The optimal object exhibits the same topology as the original object
With topology optimization we can fundamentally change the nature of the object The ldquotopologyrdquo refers to the number and position of the components of the domains Here the objectrsquos geometry is presented with no prerequisites as to the connectivity of the domains or the components present in the solution We take no initial information about the topology of the optimal shape
12 History of structural optimization
It was in the early 1960s that Schmit [SCH 60] and Fox [FOX 65] laid the foundations for a modern theory of structural optimization based on the concepts of mathematical programming and sensitivity analysis Paradoxically at the time ldquofully stressed designrdquo was the only widely used technique in practice although it lacked any theoretical justification other than empiricism and the engineersrsquo intuition It was Prager and Taylor [PRA 68] who set out variational methods and Lagrangian optimality conditions to justify the criteria of fully stressed design for a class of structural optimization problems The optimality conditions of the optimization problem were then used directly to construct an iterative resolution algorithm known as the ldquooptimality criteria methodrdquo
Originally structural optimization was mainly limited to the sizing optimization of trusses or gantries Thus sizing optimization of structures was the first field of application for optimality criteria When dealing with the problem of sizing we look at the transverse sections of the structural elements though their length and the
Introduction to Structural Optimization 3
location of their joints remain fixed During the late 1960s and early 1970s optimality criteria were soon adapted to large structures modeled using the finite-element method (FEM) [VEN 73] The optimality criteria method produced a few interesting results and a number of extensions have been presented since the 1970s including Venkayyarsquos generalized criteria method [VEN 73] or Rozvany and Zhoursquos [ROZ 91] discretized optimality criteria
Although since the 1970s attention has mainly been focused on sizing the problem of truss topology has also been studied by Prager [PRA 74] on a very restricted class of structures based on the concept of a truss put forward by Michell in 1904 [MIC 04] The problem lies in finding the best possible configuration so that the truss or gantry can convey forces to the foundations whilst minimizing a given performance and satisfying the design constraints Michellrsquos theory [MIC 04] is related to the topology of trusses made of bars of minimal mass The optimal solution from Michellrsquos point of view is composed solely of perpendicular bars which form a structure whose configuration is optimal for the maximum tensile- and compressive stresses All the configuration problems studied by Prager later on were solved analytically so the practical application of topology was very limited To remedy this shortcoming Rozvany [ROZ 76] invested a great deal of effort in developing new approaches to solve these configuration problems as automatically as possible
After that optimal truss topology was studied in greater depth by Kirsch [KIR 90] If we impose a very small minimum value for the cross-section then the ldquolayoutrdquo optimization of trusses and gantries can be approached as a conventional sizing problem on a very large scale The solution is then obtained by application of generalized optimality criteria for a variety of objective functions compliance movements tensions and eigenvalues [ZHO 91]
Finally the crux of the problem of truss topology optimization seems to have been identified by Bendsoslashe et al [BEN 91] The problem of truss topology is examined with an integral approach combining analysis and design simultaneously The problem of minimum compliance is transformed into a problem of non-differentiable optimization and then in the case of a truss of bars into a linear problem In this form it can be solved on very large structures using non-differentiable optimization methods [BEN 93a] an interior point method [BEN 93b] a dual method [BEC 94] or a penaltybarrier multiplier method [BEN 97]
The three main groups of structural optimization cannot be considered recent concepts but their integration into biomechanics is ndash especially topology optimization [FRA 10]
4 Biomechanics
13 Sizing optimization
131 Definition
With sizing optimization we can only modify the cross-section or transverse thickness of the components of structure whose shape and topology are fixed There can be no modification of the geometric model and its features Figure 11 shows a truss made of 17 bars with different cross-sections (circular rectangular and I-shaped)
Figure 11 Changing the dimensions whilst preserving the same topology of the section
Sizing optimization can be performed by considering the same topology to produce various dimensions For example when the cross-section is circular we merely need to vary the diameters to minimize an objective or several objectives under certain constraints
132 First works in sizing optimization
The problem of sizing optimization has benefited the most from this research so optimization of the transverse dimensions is today a reliable tool This problem was also extended to that of flexural elements [FLE 83] to improving performance when subjected to vibration and to the stability of balance Besides the transverse dimensions of the structural elements it is possible to vary their shape To the best of our knowledge few works have been devoted to the study of the optimal shape for a truss In this type of problem only the location of the structural joints is altered whilst the topology remains unchanged Svanberg [SVA 81] was one of the only people to carry out truss shape optimization on the basis of FEM analyses and solving using mathematical programming techniques
Introduction to Structural Optimization 5
133 Numerical application
1331 Description and modeling of the studied problem
Figure 12a shows a cantilever beam and its I-shaped cross-section in Figure 12b This beam is embedded at one end and subjected to free vibration The material from which the beam is made is structural steel which has a Youngrsquos modulus 200000E MPa= and Poissonrsquos ratio of 03ν = The density of the material is 6 37854 10 Kg mmρ minus= times The material exhibits linear elastic isotropic behavior The length of that beam is 300L mm= and the dimensions of the cross-section are 60 B mm= 100H mm= and 20T mm= (Figure 12b)
a) b)
Figure 12 Cantilever beam subject to free vibration
The objective of this study is to calculate the first four modes of resonance and then perform sizing optimization on the dimensions of the cross-section The problem of optimization therefore is to minimize the structural volume under the constraint of the first resonance frequency This problem can be formulated as follows
1
min ( ) ( ) 0
40 100 80 16010 30
w
Volume B H Ts t f B H T f
BHT
minus lele lele lele le
[11]
where 20wf Hz= is the maximum value of the first resonance frequency
1332 Numerical results
To perform sizing optimization on the ANSYS software for instance we consider the dimensions of the cross-section as optimization variables so as to obtain a parameterized model Then a direct simulation is performed as the heart of the optimization loop
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
Contents ix
66 Integration of reliability and structural optimization into dental implant design 184
661 Description and modeling of the problem 184 662 Numerical results 186
67 Conclusion 188
Appendices 189
Appendix 1 ANSYS Code for Stem Geometry 191
Appendix 2 ANSYS Code for Mini-Plate Geometry 197
Appendix 3 ANSYS Code for Dental Implant Geometry 201
Appendix 4 ANSYS Code for Geometry of Dental Implant with Bone 207
Bibliography 213
Index 229
Preface
The integration of structural optimization into biomechanics is a truly vast domain In this book we first focus on the integration of structural optimization into the design of orthopedic and orthodontic prostheses and also into drilling surgery Next we present the integration of reliability and structural optimization into the design of these prostheses which may be considered as a novel aspect introduced in this book The applications are made in 2D and in 3D considering the three major families of structural optimization sizing- shape- and topology optimization
In all domains of structural mechanics good design of a part is very important for its strength its lifetime and its use in service This is a challenge faced daily in sectors such as space research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve the techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile engines providing substantial energy savings The development of computer-aided design (CAD) techniques and optimization strategies is part of this context
Applying structural optimization is still somewhat complicated in certain domains Furthermore in deterministic structural optimization all parameters which are uncertain in nature are described by unfavorable characteristic values associated with safety coefficients The deterministic approach uses a pessimistic margin determined as a function of the consequences of a probable failure This approach often leads to unnecessary oversizing ndash particularly for sensitive structures
On the other hand researchers have developed a different approach which is better suited to uncertain physical phenomena In this approach the structure is deemed to have failed if the probability of failure is greater than a fixed threshold This is known as the ldquoprobabilistic approachrdquo The probabilistic approach is
xii Biomechanics
increasingly widely used in engineering as evidenced by the different applications in industry It is applied to check that the probability is sufficient when the structurersquos geometry is known or to optimize the sizing of the structure so as to respect certain fixed objectives such as a target cost or a required level of probability
Furthermore reliability analysis is an important tool in decision-making for establishing a maintenance- and inspection plan In addition it can be used in the validation of standards and regulations To perform reliability analysis various methods can be used to effectively and simply find the probability of failure Reliability analysis is a strategy used to evaluate the level of reliability without being able to control the design for a required reliability level For this reliability has become an important tool to be integrated into the process of structural optimization
This book also focuses on the necessary tools for the integration of reliability and structural optimization into biomechanics fields First the deterministic strategies of structural optimization are presented so we can implement them in structural design These deterministic strategies are applied in various domains in biomechanics including the design of orthopedic and orthodontic prostheses and drilling surgery Next reliability-based approaches pertaining to the integration of reliability into structural optimization are presented in detail with mechanical applications These reliability-based strategies are also applied in the design of orthopedic and orthodontic prostheses taking account of uncertainty in terms of geometry materials and load Finally system reliability strategies are also taken into account considering several failure scenarios
The book will provide invaluable support to teaching staff and researchers It is also intended for engineering students practising engineers and Masters students
Acknowledgements
We would like to thank all of those people who have in some way great or small contributed to the writing of this book ndash in particular Sophie Le Cann a researcher at the Biomedical Centre (BMC) at Lund University for her contribution in terms of biological language Heartfelt thanks go to our families to our students and to our colleagues for their massive moral support during the writing of this book
Ghias KHARMANDA Abdelkhalak EL-HAMI
October 2016
Introduction
This book begins with an introductory look at the fundamental principles of structural optimization before applying them to the field of biomechanics Then we present the different strategies for integrating reliability into structural optimization followed by the application of those strategies in biomechanics ndash particularly in terms of the design of orthopedic and orthodontic prostheses
In terms of structural optimization to illustrate the different techniques we can classify structural optimization into three main families
1) Sizing Optimization this model aims to improve a structural model whilst respecting the available resources (known as constraints or limitations) Sizing optimization is the particular case where we can only modify the cross section or transverse thickness of the components of a structure whose shape and topology are fixed There can be no modification of the geometric features andor models
2) Shape Optimization with this model it is possible to make changes to the shape provided they are compatible with a predefined topology This type of optimization modifies the parametric representation of the boundaries of the domain By moving those boundaries we try to find the best possible solution out of the set of all the configurations obtained by homeomorphic transformation of the original structure
3) Topology Optimization this model enables us to make more profound modifications to the shape of the structure Here the geometry of the part is examined with no prerequisites as to the connections of the domains or the structural elements present in the solution In order to optimize the topology we determine the structurersquos shape or transverse dimensions so some authors call topology optimization ldquogeneralized shape optimizationrdquo
xiv Biomechanics
In terms of reliability this concept can be integrated into all three families of structural optimization so we obtain a design that should be both optimal and reliable Sizing optimization shape optimization and topology optimization are generally classed as geometry optimization However the nature of the topology is non-quantitative in relation to shape and size For this reason this integration is divided into two models
1) RBDO Reliability-Based Design Optimization this model couples reliability analysis with sizing optimization (Reliability-Based Sizing Optimization) and also with shape optimization (Reliability-Based Shape Optimization) This coupling is a complex task requiring a lengthy computation time which seriously limits its applicability In addition during the process of reliability-based shape optimization the geometry of the structure is forced to change This coupling integrates different disciplines such as geometric modeling numerical simulation reliability analysis and optimization Thus the optimization problem becomes more complex The major difficulty lies in evaluating the reliability of the structure which in itself requires a specific optimization procedure The typical integration of reliability analysis into optimization methods is carried out in two spaces the normalized space of random variables and the physical space of design variables which requires a very significant computation time To solve this problem as we shall see there are a number of efficient methods
2) RBTO Reliability-Based Topology Optimization in the deterministic case we obtain a single optimal topology while the new RBTO model is able to generate several topologies as a function of a required reliability level However the coupling between reliability (which is quantitative) and topology (which is non-quantitative) requires the use of different methodologies relative to sizing optimization and shape optimization
In terms of biomechanics modeling is generally complicated on several different levels geometric description material properties and boundary conditions When performing the optimization process such modeling is needed with a new set of parameters being entered at each iteration In addition the problem becomes more complex when integrating reliability analysis which is itself performed by way of a particular optimization procedure In this case we couple several different software packages in order to integrate optimization and reliability into biomechanical applications Various strategies are presented in this book to simplify the coupling problem
This book is made up of six chapters presenting the different aspects of integration of reliability and structural optimization into biomechanics
Introduction xv
The first chapter offers an introduction to structural optimization and its main groups sizing optimization shape optimization and topology optimization The principles underpinning the three main types of structural optimization are presented with numerical applications for each type In structural design we have two different phases the conceptual phase and the detailed phase In the conceptual phase we use topology optimization to obtain an idea of the silhouette of the structure The detailed phase consists of shape optimization to obtain smooth geometry followed by sizing optimization to find the areas and thicknesses of the studied structure
The second chapter is devoted to the integration of structural optimization into biomechanics The integration is performed in this chapter for prosthesis design and drilling surgery Different 2D and 3D applications for orthopedic and orthodontic prostheses are presented in order to demonstrate the new strategies presented in this book and also to prove the advantages of those strategies Finally the integration of structural optimization into drilling surgery can be considered a novel aspect in relation to classic prosthesis design
The third chapter is given over to the integration of reliability into structural optimization Such integration can be viewed as a difficult task The difficulties arise in terms of coupling convergence and computation time In this chapter we present the basic principles of that integration and a literature review based on early attempts in this area A numerical application is presented to show the difference between deterministic- and reliability-based optimization Next two viewpoints are presented with the corresponding developed methods considering those two points of view to illustrate the advantages of the methods
The fourth chapter is devoted to the reliability-based design optimization (RBDO) model The developed methods are classified into two categories numerical and semi-numerical methods Numerical methods are generally applicable but require a high computation time Semi-numerical methods on the other hand are efficient in terms of computation time but can only be used in certain cases
The fifth chapter details reliability-based topology optimization (RBTO) This model enables designers and manufacturers to select the solution which is at once economical and reliable Two points of view are presented in the discussion of this model A number of applications in mechanics are presented in order to demonstrate the advantages of the RBTO model
The sixth chapter discusses the integration of reliability and structural optimization into orthopedic and orthodontic prostheses design considering two
xvi Biomechanics
approaches the deterministic approach and the reliability-based approach In orthopedic prostheses the models are presented using results drawn from the existing body of literature The aspect of materials characterization is also integrated into the design process In addition in orthodontics models are presented to test the stability of the prostheses in light of the constraints of dentistry The integration of reliability and structural optimization into the design of prostheses is performed very simply in order to aid understanding and implementation of the different strategies presented in this book
List of Abbreviations
AMA Approximate Moments Approach
AMV Advanced Mean Value
APDL ANSYS Parametric Design Language
AT Anterior Temporal (or Temporalis)
CAD Computer-Aided Design
CM Classical Method
CMV Conjugate Mean Value
CT Computed Tomography
DDO Deterministic Design Optimization
DLA Double-Level Approach
DM Deep Masseter
DO Deterministic Optimization
DTO Deterministic Topology Optimization
FE Finite Elements
FEM Finite Element Method
FORM First Order Reliability Method
HCA Hybrid Cellular Automata
HDS Hybrid Design Space
HM Hybrid Method
xviii Biomechanics
HMV Hybrid Mean Value
IAM Improved Austin-Moore
IHM Improved Hybrid Method
KKT Karush-Kuhn-Tucker
MBB Messerschmitt-Boumllkow-Blohm
MLA Mono-Level Approach
MP Medial Pterygoid
MPP Most Probable failure Point
MT Medial Temporal (or Temporalis)
OC Optimality Criteria
OSF Optimum Safety Factor
PDF Probability Density Function
PMA Performance Measure Approach
PT Posterior Temporal (or Temporalis)
RBDO Reliability-Based Design Optimization
RBO Reliability-Based Optimization
RBSO Reliability-Based Structural Optimization
RBTO Reliability-Based Topology Optimization
RIA Reliability Index Approach
SAP Sequential Approximate Programming
SDA Sequential Decoupled Approach
SED Strain Energy Density
SIMP Solid Isotropic Microstructure with Penalty
SM Superficial Masseter
SORA Sequential Optimization and Reliability Assessment
SORM Second Order Reliability Method
SP Safest Point
1
Introduction to Structural Optimization
11 Introduction
Structural optimization is a topic which affects many different physical domains ndash particularly solid mechanics ndash and which it is tricky to characterize as its formulations may have a number of different aspects Firstly a distinction regarding the way in which geometries are parameterized is presented and secondly a distribution pertaining to the intrinsic nature of optimization algorithms is established
Determining the appropriate shape for structural components is a crucially important problem for engineers In all areas of structural mechanics the impact of proper design of a part is very significant in terms of its strength its lifetime and its usage This is a challenge faced on a daily basis in the sectors of spatial research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile machinery providing substantial energy savings The different development of computer-aided design (CAD) techniques and optimization strategies is part of this context There has been keen interest in structural optimization for over thirty years Whilst it is still too infrequently applied in the conventional techniques used by research centers it is becoming more widely used as its reliability improves Having begun with the simplest of problems the field of application of structural optimization today extends to new and ever more interesting challenges
To illustrate the evolution of structural optimization techniques we can arbitrarily split structural optimization into three major groups (or families) In
Biomechanics Optimization Uncertainties and Reliability First Edition Ghias Kharmanda and Abdelkhalak El Hami copy ISTE Ltd 2017 Published by ISTE Ltd and John Wiley amp Sons Inc
2 Biomechanics
historical terms each of them has been addressed in order of increasing difficulty and generality
With sizing optimization we are only able to modify the dimensions of an object whose shape and topology are fixed There can be no modification of the geometric model We speak of a homeomorphic transformation
Shape optimization involves making changes of shape which are compatible with a predetermined topology Typical shape optimization modifies the parametric representation of the boundaries of the domain By moving the boundaries of the domains we can seek the best solution out of all the structures obtained by homeomorphic transformation of the original object In this case it is clear that we can make a change to the transverse dimensions as well as a modification to the objectrsquos configuration but it is certainly not acceptable to modify its connectivity or its nature ndash in particular the number of components that it has The optimal object exhibits the same topology as the original object
With topology optimization we can fundamentally change the nature of the object The ldquotopologyrdquo refers to the number and position of the components of the domains Here the objectrsquos geometry is presented with no prerequisites as to the connectivity of the domains or the components present in the solution We take no initial information about the topology of the optimal shape
12 History of structural optimization
It was in the early 1960s that Schmit [SCH 60] and Fox [FOX 65] laid the foundations for a modern theory of structural optimization based on the concepts of mathematical programming and sensitivity analysis Paradoxically at the time ldquofully stressed designrdquo was the only widely used technique in practice although it lacked any theoretical justification other than empiricism and the engineersrsquo intuition It was Prager and Taylor [PRA 68] who set out variational methods and Lagrangian optimality conditions to justify the criteria of fully stressed design for a class of structural optimization problems The optimality conditions of the optimization problem were then used directly to construct an iterative resolution algorithm known as the ldquooptimality criteria methodrdquo
Originally structural optimization was mainly limited to the sizing optimization of trusses or gantries Thus sizing optimization of structures was the first field of application for optimality criteria When dealing with the problem of sizing we look at the transverse sections of the structural elements though their length and the
Introduction to Structural Optimization 3
location of their joints remain fixed During the late 1960s and early 1970s optimality criteria were soon adapted to large structures modeled using the finite-element method (FEM) [VEN 73] The optimality criteria method produced a few interesting results and a number of extensions have been presented since the 1970s including Venkayyarsquos generalized criteria method [VEN 73] or Rozvany and Zhoursquos [ROZ 91] discretized optimality criteria
Although since the 1970s attention has mainly been focused on sizing the problem of truss topology has also been studied by Prager [PRA 74] on a very restricted class of structures based on the concept of a truss put forward by Michell in 1904 [MIC 04] The problem lies in finding the best possible configuration so that the truss or gantry can convey forces to the foundations whilst minimizing a given performance and satisfying the design constraints Michellrsquos theory [MIC 04] is related to the topology of trusses made of bars of minimal mass The optimal solution from Michellrsquos point of view is composed solely of perpendicular bars which form a structure whose configuration is optimal for the maximum tensile- and compressive stresses All the configuration problems studied by Prager later on were solved analytically so the practical application of topology was very limited To remedy this shortcoming Rozvany [ROZ 76] invested a great deal of effort in developing new approaches to solve these configuration problems as automatically as possible
After that optimal truss topology was studied in greater depth by Kirsch [KIR 90] If we impose a very small minimum value for the cross-section then the ldquolayoutrdquo optimization of trusses and gantries can be approached as a conventional sizing problem on a very large scale The solution is then obtained by application of generalized optimality criteria for a variety of objective functions compliance movements tensions and eigenvalues [ZHO 91]
Finally the crux of the problem of truss topology optimization seems to have been identified by Bendsoslashe et al [BEN 91] The problem of truss topology is examined with an integral approach combining analysis and design simultaneously The problem of minimum compliance is transformed into a problem of non-differentiable optimization and then in the case of a truss of bars into a linear problem In this form it can be solved on very large structures using non-differentiable optimization methods [BEN 93a] an interior point method [BEN 93b] a dual method [BEC 94] or a penaltybarrier multiplier method [BEN 97]
The three main groups of structural optimization cannot be considered recent concepts but their integration into biomechanics is ndash especially topology optimization [FRA 10]
4 Biomechanics
13 Sizing optimization
131 Definition
With sizing optimization we can only modify the cross-section or transverse thickness of the components of structure whose shape and topology are fixed There can be no modification of the geometric model and its features Figure 11 shows a truss made of 17 bars with different cross-sections (circular rectangular and I-shaped)
Figure 11 Changing the dimensions whilst preserving the same topology of the section
Sizing optimization can be performed by considering the same topology to produce various dimensions For example when the cross-section is circular we merely need to vary the diameters to minimize an objective or several objectives under certain constraints
132 First works in sizing optimization
The problem of sizing optimization has benefited the most from this research so optimization of the transverse dimensions is today a reliable tool This problem was also extended to that of flexural elements [FLE 83] to improving performance when subjected to vibration and to the stability of balance Besides the transverse dimensions of the structural elements it is possible to vary their shape To the best of our knowledge few works have been devoted to the study of the optimal shape for a truss In this type of problem only the location of the structural joints is altered whilst the topology remains unchanged Svanberg [SVA 81] was one of the only people to carry out truss shape optimization on the basis of FEM analyses and solving using mathematical programming techniques
Introduction to Structural Optimization 5
133 Numerical application
1331 Description and modeling of the studied problem
Figure 12a shows a cantilever beam and its I-shaped cross-section in Figure 12b This beam is embedded at one end and subjected to free vibration The material from which the beam is made is structural steel which has a Youngrsquos modulus 200000E MPa= and Poissonrsquos ratio of 03ν = The density of the material is 6 37854 10 Kg mmρ minus= times The material exhibits linear elastic isotropic behavior The length of that beam is 300L mm= and the dimensions of the cross-section are 60 B mm= 100H mm= and 20T mm= (Figure 12b)
a) b)
Figure 12 Cantilever beam subject to free vibration
The objective of this study is to calculate the first four modes of resonance and then perform sizing optimization on the dimensions of the cross-section The problem of optimization therefore is to minimize the structural volume under the constraint of the first resonance frequency This problem can be formulated as follows
1
min ( ) ( ) 0
40 100 80 16010 30
w
Volume B H Ts t f B H T f
BHT
minus lele lele lele le
[11]
where 20wf Hz= is the maximum value of the first resonance frequency
1332 Numerical results
To perform sizing optimization on the ANSYS software for instance we consider the dimensions of the cross-section as optimization variables so as to obtain a parameterized model Then a direct simulation is performed as the heart of the optimization loop
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
Preface
The integration of structural optimization into biomechanics is a truly vast domain In this book we first focus on the integration of structural optimization into the design of orthopedic and orthodontic prostheses and also into drilling surgery Next we present the integration of reliability and structural optimization into the design of these prostheses which may be considered as a novel aspect introduced in this book The applications are made in 2D and in 3D considering the three major families of structural optimization sizing- shape- and topology optimization
In all domains of structural mechanics good design of a part is very important for its strength its lifetime and its use in service This is a challenge faced daily in sectors such as space research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve the techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile engines providing substantial energy savings The development of computer-aided design (CAD) techniques and optimization strategies is part of this context
Applying structural optimization is still somewhat complicated in certain domains Furthermore in deterministic structural optimization all parameters which are uncertain in nature are described by unfavorable characteristic values associated with safety coefficients The deterministic approach uses a pessimistic margin determined as a function of the consequences of a probable failure This approach often leads to unnecessary oversizing ndash particularly for sensitive structures
On the other hand researchers have developed a different approach which is better suited to uncertain physical phenomena In this approach the structure is deemed to have failed if the probability of failure is greater than a fixed threshold This is known as the ldquoprobabilistic approachrdquo The probabilistic approach is
xii Biomechanics
increasingly widely used in engineering as evidenced by the different applications in industry It is applied to check that the probability is sufficient when the structurersquos geometry is known or to optimize the sizing of the structure so as to respect certain fixed objectives such as a target cost or a required level of probability
Furthermore reliability analysis is an important tool in decision-making for establishing a maintenance- and inspection plan In addition it can be used in the validation of standards and regulations To perform reliability analysis various methods can be used to effectively and simply find the probability of failure Reliability analysis is a strategy used to evaluate the level of reliability without being able to control the design for a required reliability level For this reliability has become an important tool to be integrated into the process of structural optimization
This book also focuses on the necessary tools for the integration of reliability and structural optimization into biomechanics fields First the deterministic strategies of structural optimization are presented so we can implement them in structural design These deterministic strategies are applied in various domains in biomechanics including the design of orthopedic and orthodontic prostheses and drilling surgery Next reliability-based approaches pertaining to the integration of reliability into structural optimization are presented in detail with mechanical applications These reliability-based strategies are also applied in the design of orthopedic and orthodontic prostheses taking account of uncertainty in terms of geometry materials and load Finally system reliability strategies are also taken into account considering several failure scenarios
The book will provide invaluable support to teaching staff and researchers It is also intended for engineering students practising engineers and Masters students
Acknowledgements
We would like to thank all of those people who have in some way great or small contributed to the writing of this book ndash in particular Sophie Le Cann a researcher at the Biomedical Centre (BMC) at Lund University for her contribution in terms of biological language Heartfelt thanks go to our families to our students and to our colleagues for their massive moral support during the writing of this book
Ghias KHARMANDA Abdelkhalak EL-HAMI
October 2016
Introduction
This book begins with an introductory look at the fundamental principles of structural optimization before applying them to the field of biomechanics Then we present the different strategies for integrating reliability into structural optimization followed by the application of those strategies in biomechanics ndash particularly in terms of the design of orthopedic and orthodontic prostheses
In terms of structural optimization to illustrate the different techniques we can classify structural optimization into three main families
1) Sizing Optimization this model aims to improve a structural model whilst respecting the available resources (known as constraints or limitations) Sizing optimization is the particular case where we can only modify the cross section or transverse thickness of the components of a structure whose shape and topology are fixed There can be no modification of the geometric features andor models
2) Shape Optimization with this model it is possible to make changes to the shape provided they are compatible with a predefined topology This type of optimization modifies the parametric representation of the boundaries of the domain By moving those boundaries we try to find the best possible solution out of the set of all the configurations obtained by homeomorphic transformation of the original structure
3) Topology Optimization this model enables us to make more profound modifications to the shape of the structure Here the geometry of the part is examined with no prerequisites as to the connections of the domains or the structural elements present in the solution In order to optimize the topology we determine the structurersquos shape or transverse dimensions so some authors call topology optimization ldquogeneralized shape optimizationrdquo
xiv Biomechanics
In terms of reliability this concept can be integrated into all three families of structural optimization so we obtain a design that should be both optimal and reliable Sizing optimization shape optimization and topology optimization are generally classed as geometry optimization However the nature of the topology is non-quantitative in relation to shape and size For this reason this integration is divided into two models
1) RBDO Reliability-Based Design Optimization this model couples reliability analysis with sizing optimization (Reliability-Based Sizing Optimization) and also with shape optimization (Reliability-Based Shape Optimization) This coupling is a complex task requiring a lengthy computation time which seriously limits its applicability In addition during the process of reliability-based shape optimization the geometry of the structure is forced to change This coupling integrates different disciplines such as geometric modeling numerical simulation reliability analysis and optimization Thus the optimization problem becomes more complex The major difficulty lies in evaluating the reliability of the structure which in itself requires a specific optimization procedure The typical integration of reliability analysis into optimization methods is carried out in two spaces the normalized space of random variables and the physical space of design variables which requires a very significant computation time To solve this problem as we shall see there are a number of efficient methods
2) RBTO Reliability-Based Topology Optimization in the deterministic case we obtain a single optimal topology while the new RBTO model is able to generate several topologies as a function of a required reliability level However the coupling between reliability (which is quantitative) and topology (which is non-quantitative) requires the use of different methodologies relative to sizing optimization and shape optimization
In terms of biomechanics modeling is generally complicated on several different levels geometric description material properties and boundary conditions When performing the optimization process such modeling is needed with a new set of parameters being entered at each iteration In addition the problem becomes more complex when integrating reliability analysis which is itself performed by way of a particular optimization procedure In this case we couple several different software packages in order to integrate optimization and reliability into biomechanical applications Various strategies are presented in this book to simplify the coupling problem
This book is made up of six chapters presenting the different aspects of integration of reliability and structural optimization into biomechanics
Introduction xv
The first chapter offers an introduction to structural optimization and its main groups sizing optimization shape optimization and topology optimization The principles underpinning the three main types of structural optimization are presented with numerical applications for each type In structural design we have two different phases the conceptual phase and the detailed phase In the conceptual phase we use topology optimization to obtain an idea of the silhouette of the structure The detailed phase consists of shape optimization to obtain smooth geometry followed by sizing optimization to find the areas and thicknesses of the studied structure
The second chapter is devoted to the integration of structural optimization into biomechanics The integration is performed in this chapter for prosthesis design and drilling surgery Different 2D and 3D applications for orthopedic and orthodontic prostheses are presented in order to demonstrate the new strategies presented in this book and also to prove the advantages of those strategies Finally the integration of structural optimization into drilling surgery can be considered a novel aspect in relation to classic prosthesis design
The third chapter is given over to the integration of reliability into structural optimization Such integration can be viewed as a difficult task The difficulties arise in terms of coupling convergence and computation time In this chapter we present the basic principles of that integration and a literature review based on early attempts in this area A numerical application is presented to show the difference between deterministic- and reliability-based optimization Next two viewpoints are presented with the corresponding developed methods considering those two points of view to illustrate the advantages of the methods
The fourth chapter is devoted to the reliability-based design optimization (RBDO) model The developed methods are classified into two categories numerical and semi-numerical methods Numerical methods are generally applicable but require a high computation time Semi-numerical methods on the other hand are efficient in terms of computation time but can only be used in certain cases
The fifth chapter details reliability-based topology optimization (RBTO) This model enables designers and manufacturers to select the solution which is at once economical and reliable Two points of view are presented in the discussion of this model A number of applications in mechanics are presented in order to demonstrate the advantages of the RBTO model
The sixth chapter discusses the integration of reliability and structural optimization into orthopedic and orthodontic prostheses design considering two
xvi Biomechanics
approaches the deterministic approach and the reliability-based approach In orthopedic prostheses the models are presented using results drawn from the existing body of literature The aspect of materials characterization is also integrated into the design process In addition in orthodontics models are presented to test the stability of the prostheses in light of the constraints of dentistry The integration of reliability and structural optimization into the design of prostheses is performed very simply in order to aid understanding and implementation of the different strategies presented in this book
List of Abbreviations
AMA Approximate Moments Approach
AMV Advanced Mean Value
APDL ANSYS Parametric Design Language
AT Anterior Temporal (or Temporalis)
CAD Computer-Aided Design
CM Classical Method
CMV Conjugate Mean Value
CT Computed Tomography
DDO Deterministic Design Optimization
DLA Double-Level Approach
DM Deep Masseter
DO Deterministic Optimization
DTO Deterministic Topology Optimization
FE Finite Elements
FEM Finite Element Method
FORM First Order Reliability Method
HCA Hybrid Cellular Automata
HDS Hybrid Design Space
HM Hybrid Method
xviii Biomechanics
HMV Hybrid Mean Value
IAM Improved Austin-Moore
IHM Improved Hybrid Method
KKT Karush-Kuhn-Tucker
MBB Messerschmitt-Boumllkow-Blohm
MLA Mono-Level Approach
MP Medial Pterygoid
MPP Most Probable failure Point
MT Medial Temporal (or Temporalis)
OC Optimality Criteria
OSF Optimum Safety Factor
PDF Probability Density Function
PMA Performance Measure Approach
PT Posterior Temporal (or Temporalis)
RBDO Reliability-Based Design Optimization
RBO Reliability-Based Optimization
RBSO Reliability-Based Structural Optimization
RBTO Reliability-Based Topology Optimization
RIA Reliability Index Approach
SAP Sequential Approximate Programming
SDA Sequential Decoupled Approach
SED Strain Energy Density
SIMP Solid Isotropic Microstructure with Penalty
SM Superficial Masseter
SORA Sequential Optimization and Reliability Assessment
SORM Second Order Reliability Method
SP Safest Point
1
Introduction to Structural Optimization
11 Introduction
Structural optimization is a topic which affects many different physical domains ndash particularly solid mechanics ndash and which it is tricky to characterize as its formulations may have a number of different aspects Firstly a distinction regarding the way in which geometries are parameterized is presented and secondly a distribution pertaining to the intrinsic nature of optimization algorithms is established
Determining the appropriate shape for structural components is a crucially important problem for engineers In all areas of structural mechanics the impact of proper design of a part is very significant in terms of its strength its lifetime and its usage This is a challenge faced on a daily basis in the sectors of spatial research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile machinery providing substantial energy savings The different development of computer-aided design (CAD) techniques and optimization strategies is part of this context There has been keen interest in structural optimization for over thirty years Whilst it is still too infrequently applied in the conventional techniques used by research centers it is becoming more widely used as its reliability improves Having begun with the simplest of problems the field of application of structural optimization today extends to new and ever more interesting challenges
To illustrate the evolution of structural optimization techniques we can arbitrarily split structural optimization into three major groups (or families) In
Biomechanics Optimization Uncertainties and Reliability First Edition Ghias Kharmanda and Abdelkhalak El Hami copy ISTE Ltd 2017 Published by ISTE Ltd and John Wiley amp Sons Inc
2 Biomechanics
historical terms each of them has been addressed in order of increasing difficulty and generality
With sizing optimization we are only able to modify the dimensions of an object whose shape and topology are fixed There can be no modification of the geometric model We speak of a homeomorphic transformation
Shape optimization involves making changes of shape which are compatible with a predetermined topology Typical shape optimization modifies the parametric representation of the boundaries of the domain By moving the boundaries of the domains we can seek the best solution out of all the structures obtained by homeomorphic transformation of the original object In this case it is clear that we can make a change to the transverse dimensions as well as a modification to the objectrsquos configuration but it is certainly not acceptable to modify its connectivity or its nature ndash in particular the number of components that it has The optimal object exhibits the same topology as the original object
With topology optimization we can fundamentally change the nature of the object The ldquotopologyrdquo refers to the number and position of the components of the domains Here the objectrsquos geometry is presented with no prerequisites as to the connectivity of the domains or the components present in the solution We take no initial information about the topology of the optimal shape
12 History of structural optimization
It was in the early 1960s that Schmit [SCH 60] and Fox [FOX 65] laid the foundations for a modern theory of structural optimization based on the concepts of mathematical programming and sensitivity analysis Paradoxically at the time ldquofully stressed designrdquo was the only widely used technique in practice although it lacked any theoretical justification other than empiricism and the engineersrsquo intuition It was Prager and Taylor [PRA 68] who set out variational methods and Lagrangian optimality conditions to justify the criteria of fully stressed design for a class of structural optimization problems The optimality conditions of the optimization problem were then used directly to construct an iterative resolution algorithm known as the ldquooptimality criteria methodrdquo
Originally structural optimization was mainly limited to the sizing optimization of trusses or gantries Thus sizing optimization of structures was the first field of application for optimality criteria When dealing with the problem of sizing we look at the transverse sections of the structural elements though their length and the
Introduction to Structural Optimization 3
location of their joints remain fixed During the late 1960s and early 1970s optimality criteria were soon adapted to large structures modeled using the finite-element method (FEM) [VEN 73] The optimality criteria method produced a few interesting results and a number of extensions have been presented since the 1970s including Venkayyarsquos generalized criteria method [VEN 73] or Rozvany and Zhoursquos [ROZ 91] discretized optimality criteria
Although since the 1970s attention has mainly been focused on sizing the problem of truss topology has also been studied by Prager [PRA 74] on a very restricted class of structures based on the concept of a truss put forward by Michell in 1904 [MIC 04] The problem lies in finding the best possible configuration so that the truss or gantry can convey forces to the foundations whilst minimizing a given performance and satisfying the design constraints Michellrsquos theory [MIC 04] is related to the topology of trusses made of bars of minimal mass The optimal solution from Michellrsquos point of view is composed solely of perpendicular bars which form a structure whose configuration is optimal for the maximum tensile- and compressive stresses All the configuration problems studied by Prager later on were solved analytically so the practical application of topology was very limited To remedy this shortcoming Rozvany [ROZ 76] invested a great deal of effort in developing new approaches to solve these configuration problems as automatically as possible
After that optimal truss topology was studied in greater depth by Kirsch [KIR 90] If we impose a very small minimum value for the cross-section then the ldquolayoutrdquo optimization of trusses and gantries can be approached as a conventional sizing problem on a very large scale The solution is then obtained by application of generalized optimality criteria for a variety of objective functions compliance movements tensions and eigenvalues [ZHO 91]
Finally the crux of the problem of truss topology optimization seems to have been identified by Bendsoslashe et al [BEN 91] The problem of truss topology is examined with an integral approach combining analysis and design simultaneously The problem of minimum compliance is transformed into a problem of non-differentiable optimization and then in the case of a truss of bars into a linear problem In this form it can be solved on very large structures using non-differentiable optimization methods [BEN 93a] an interior point method [BEN 93b] a dual method [BEC 94] or a penaltybarrier multiplier method [BEN 97]
The three main groups of structural optimization cannot be considered recent concepts but their integration into biomechanics is ndash especially topology optimization [FRA 10]
4 Biomechanics
13 Sizing optimization
131 Definition
With sizing optimization we can only modify the cross-section or transverse thickness of the components of structure whose shape and topology are fixed There can be no modification of the geometric model and its features Figure 11 shows a truss made of 17 bars with different cross-sections (circular rectangular and I-shaped)
Figure 11 Changing the dimensions whilst preserving the same topology of the section
Sizing optimization can be performed by considering the same topology to produce various dimensions For example when the cross-section is circular we merely need to vary the diameters to minimize an objective or several objectives under certain constraints
132 First works in sizing optimization
The problem of sizing optimization has benefited the most from this research so optimization of the transverse dimensions is today a reliable tool This problem was also extended to that of flexural elements [FLE 83] to improving performance when subjected to vibration and to the stability of balance Besides the transverse dimensions of the structural elements it is possible to vary their shape To the best of our knowledge few works have been devoted to the study of the optimal shape for a truss In this type of problem only the location of the structural joints is altered whilst the topology remains unchanged Svanberg [SVA 81] was one of the only people to carry out truss shape optimization on the basis of FEM analyses and solving using mathematical programming techniques
Introduction to Structural Optimization 5
133 Numerical application
1331 Description and modeling of the studied problem
Figure 12a shows a cantilever beam and its I-shaped cross-section in Figure 12b This beam is embedded at one end and subjected to free vibration The material from which the beam is made is structural steel which has a Youngrsquos modulus 200000E MPa= and Poissonrsquos ratio of 03ν = The density of the material is 6 37854 10 Kg mmρ minus= times The material exhibits linear elastic isotropic behavior The length of that beam is 300L mm= and the dimensions of the cross-section are 60 B mm= 100H mm= and 20T mm= (Figure 12b)
a) b)
Figure 12 Cantilever beam subject to free vibration
The objective of this study is to calculate the first four modes of resonance and then perform sizing optimization on the dimensions of the cross-section The problem of optimization therefore is to minimize the structural volume under the constraint of the first resonance frequency This problem can be formulated as follows
1
min ( ) ( ) 0
40 100 80 16010 30
w
Volume B H Ts t f B H T f
BHT
minus lele lele lele le
[11]
where 20wf Hz= is the maximum value of the first resonance frequency
1332 Numerical results
To perform sizing optimization on the ANSYS software for instance we consider the dimensions of the cross-section as optimization variables so as to obtain a parameterized model Then a direct simulation is performed as the heart of the optimization loop
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
xii Biomechanics
increasingly widely used in engineering as evidenced by the different applications in industry It is applied to check that the probability is sufficient when the structurersquos geometry is known or to optimize the sizing of the structure so as to respect certain fixed objectives such as a target cost or a required level of probability
Furthermore reliability analysis is an important tool in decision-making for establishing a maintenance- and inspection plan In addition it can be used in the validation of standards and regulations To perform reliability analysis various methods can be used to effectively and simply find the probability of failure Reliability analysis is a strategy used to evaluate the level of reliability without being able to control the design for a required reliability level For this reliability has become an important tool to be integrated into the process of structural optimization
This book also focuses on the necessary tools for the integration of reliability and structural optimization into biomechanics fields First the deterministic strategies of structural optimization are presented so we can implement them in structural design These deterministic strategies are applied in various domains in biomechanics including the design of orthopedic and orthodontic prostheses and drilling surgery Next reliability-based approaches pertaining to the integration of reliability into structural optimization are presented in detail with mechanical applications These reliability-based strategies are also applied in the design of orthopedic and orthodontic prostheses taking account of uncertainty in terms of geometry materials and load Finally system reliability strategies are also taken into account considering several failure scenarios
The book will provide invaluable support to teaching staff and researchers It is also intended for engineering students practising engineers and Masters students
Acknowledgements
We would like to thank all of those people who have in some way great or small contributed to the writing of this book ndash in particular Sophie Le Cann a researcher at the Biomedical Centre (BMC) at Lund University for her contribution in terms of biological language Heartfelt thanks go to our families to our students and to our colleagues for their massive moral support during the writing of this book
Ghias KHARMANDA Abdelkhalak EL-HAMI
October 2016
Introduction
This book begins with an introductory look at the fundamental principles of structural optimization before applying them to the field of biomechanics Then we present the different strategies for integrating reliability into structural optimization followed by the application of those strategies in biomechanics ndash particularly in terms of the design of orthopedic and orthodontic prostheses
In terms of structural optimization to illustrate the different techniques we can classify structural optimization into three main families
1) Sizing Optimization this model aims to improve a structural model whilst respecting the available resources (known as constraints or limitations) Sizing optimization is the particular case where we can only modify the cross section or transverse thickness of the components of a structure whose shape and topology are fixed There can be no modification of the geometric features andor models
2) Shape Optimization with this model it is possible to make changes to the shape provided they are compatible with a predefined topology This type of optimization modifies the parametric representation of the boundaries of the domain By moving those boundaries we try to find the best possible solution out of the set of all the configurations obtained by homeomorphic transformation of the original structure
3) Topology Optimization this model enables us to make more profound modifications to the shape of the structure Here the geometry of the part is examined with no prerequisites as to the connections of the domains or the structural elements present in the solution In order to optimize the topology we determine the structurersquos shape or transverse dimensions so some authors call topology optimization ldquogeneralized shape optimizationrdquo
xiv Biomechanics
In terms of reliability this concept can be integrated into all three families of structural optimization so we obtain a design that should be both optimal and reliable Sizing optimization shape optimization and topology optimization are generally classed as geometry optimization However the nature of the topology is non-quantitative in relation to shape and size For this reason this integration is divided into two models
1) RBDO Reliability-Based Design Optimization this model couples reliability analysis with sizing optimization (Reliability-Based Sizing Optimization) and also with shape optimization (Reliability-Based Shape Optimization) This coupling is a complex task requiring a lengthy computation time which seriously limits its applicability In addition during the process of reliability-based shape optimization the geometry of the structure is forced to change This coupling integrates different disciplines such as geometric modeling numerical simulation reliability analysis and optimization Thus the optimization problem becomes more complex The major difficulty lies in evaluating the reliability of the structure which in itself requires a specific optimization procedure The typical integration of reliability analysis into optimization methods is carried out in two spaces the normalized space of random variables and the physical space of design variables which requires a very significant computation time To solve this problem as we shall see there are a number of efficient methods
2) RBTO Reliability-Based Topology Optimization in the deterministic case we obtain a single optimal topology while the new RBTO model is able to generate several topologies as a function of a required reliability level However the coupling between reliability (which is quantitative) and topology (which is non-quantitative) requires the use of different methodologies relative to sizing optimization and shape optimization
In terms of biomechanics modeling is generally complicated on several different levels geometric description material properties and boundary conditions When performing the optimization process such modeling is needed with a new set of parameters being entered at each iteration In addition the problem becomes more complex when integrating reliability analysis which is itself performed by way of a particular optimization procedure In this case we couple several different software packages in order to integrate optimization and reliability into biomechanical applications Various strategies are presented in this book to simplify the coupling problem
This book is made up of six chapters presenting the different aspects of integration of reliability and structural optimization into biomechanics
Introduction xv
The first chapter offers an introduction to structural optimization and its main groups sizing optimization shape optimization and topology optimization The principles underpinning the three main types of structural optimization are presented with numerical applications for each type In structural design we have two different phases the conceptual phase and the detailed phase In the conceptual phase we use topology optimization to obtain an idea of the silhouette of the structure The detailed phase consists of shape optimization to obtain smooth geometry followed by sizing optimization to find the areas and thicknesses of the studied structure
The second chapter is devoted to the integration of structural optimization into biomechanics The integration is performed in this chapter for prosthesis design and drilling surgery Different 2D and 3D applications for orthopedic and orthodontic prostheses are presented in order to demonstrate the new strategies presented in this book and also to prove the advantages of those strategies Finally the integration of structural optimization into drilling surgery can be considered a novel aspect in relation to classic prosthesis design
The third chapter is given over to the integration of reliability into structural optimization Such integration can be viewed as a difficult task The difficulties arise in terms of coupling convergence and computation time In this chapter we present the basic principles of that integration and a literature review based on early attempts in this area A numerical application is presented to show the difference between deterministic- and reliability-based optimization Next two viewpoints are presented with the corresponding developed methods considering those two points of view to illustrate the advantages of the methods
The fourth chapter is devoted to the reliability-based design optimization (RBDO) model The developed methods are classified into two categories numerical and semi-numerical methods Numerical methods are generally applicable but require a high computation time Semi-numerical methods on the other hand are efficient in terms of computation time but can only be used in certain cases
The fifth chapter details reliability-based topology optimization (RBTO) This model enables designers and manufacturers to select the solution which is at once economical and reliable Two points of view are presented in the discussion of this model A number of applications in mechanics are presented in order to demonstrate the advantages of the RBTO model
The sixth chapter discusses the integration of reliability and structural optimization into orthopedic and orthodontic prostheses design considering two
xvi Biomechanics
approaches the deterministic approach and the reliability-based approach In orthopedic prostheses the models are presented using results drawn from the existing body of literature The aspect of materials characterization is also integrated into the design process In addition in orthodontics models are presented to test the stability of the prostheses in light of the constraints of dentistry The integration of reliability and structural optimization into the design of prostheses is performed very simply in order to aid understanding and implementation of the different strategies presented in this book
List of Abbreviations
AMA Approximate Moments Approach
AMV Advanced Mean Value
APDL ANSYS Parametric Design Language
AT Anterior Temporal (or Temporalis)
CAD Computer-Aided Design
CM Classical Method
CMV Conjugate Mean Value
CT Computed Tomography
DDO Deterministic Design Optimization
DLA Double-Level Approach
DM Deep Masseter
DO Deterministic Optimization
DTO Deterministic Topology Optimization
FE Finite Elements
FEM Finite Element Method
FORM First Order Reliability Method
HCA Hybrid Cellular Automata
HDS Hybrid Design Space
HM Hybrid Method
xviii Biomechanics
HMV Hybrid Mean Value
IAM Improved Austin-Moore
IHM Improved Hybrid Method
KKT Karush-Kuhn-Tucker
MBB Messerschmitt-Boumllkow-Blohm
MLA Mono-Level Approach
MP Medial Pterygoid
MPP Most Probable failure Point
MT Medial Temporal (or Temporalis)
OC Optimality Criteria
OSF Optimum Safety Factor
PDF Probability Density Function
PMA Performance Measure Approach
PT Posterior Temporal (or Temporalis)
RBDO Reliability-Based Design Optimization
RBO Reliability-Based Optimization
RBSO Reliability-Based Structural Optimization
RBTO Reliability-Based Topology Optimization
RIA Reliability Index Approach
SAP Sequential Approximate Programming
SDA Sequential Decoupled Approach
SED Strain Energy Density
SIMP Solid Isotropic Microstructure with Penalty
SM Superficial Masseter
SORA Sequential Optimization and Reliability Assessment
SORM Second Order Reliability Method
SP Safest Point
1
Introduction to Structural Optimization
11 Introduction
Structural optimization is a topic which affects many different physical domains ndash particularly solid mechanics ndash and which it is tricky to characterize as its formulations may have a number of different aspects Firstly a distinction regarding the way in which geometries are parameterized is presented and secondly a distribution pertaining to the intrinsic nature of optimization algorithms is established
Determining the appropriate shape for structural components is a crucially important problem for engineers In all areas of structural mechanics the impact of proper design of a part is very significant in terms of its strength its lifetime and its usage This is a challenge faced on a daily basis in the sectors of spatial research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile machinery providing substantial energy savings The different development of computer-aided design (CAD) techniques and optimization strategies is part of this context There has been keen interest in structural optimization for over thirty years Whilst it is still too infrequently applied in the conventional techniques used by research centers it is becoming more widely used as its reliability improves Having begun with the simplest of problems the field of application of structural optimization today extends to new and ever more interesting challenges
To illustrate the evolution of structural optimization techniques we can arbitrarily split structural optimization into three major groups (or families) In
Biomechanics Optimization Uncertainties and Reliability First Edition Ghias Kharmanda and Abdelkhalak El Hami copy ISTE Ltd 2017 Published by ISTE Ltd and John Wiley amp Sons Inc
2 Biomechanics
historical terms each of them has been addressed in order of increasing difficulty and generality
With sizing optimization we are only able to modify the dimensions of an object whose shape and topology are fixed There can be no modification of the geometric model We speak of a homeomorphic transformation
Shape optimization involves making changes of shape which are compatible with a predetermined topology Typical shape optimization modifies the parametric representation of the boundaries of the domain By moving the boundaries of the domains we can seek the best solution out of all the structures obtained by homeomorphic transformation of the original object In this case it is clear that we can make a change to the transverse dimensions as well as a modification to the objectrsquos configuration but it is certainly not acceptable to modify its connectivity or its nature ndash in particular the number of components that it has The optimal object exhibits the same topology as the original object
With topology optimization we can fundamentally change the nature of the object The ldquotopologyrdquo refers to the number and position of the components of the domains Here the objectrsquos geometry is presented with no prerequisites as to the connectivity of the domains or the components present in the solution We take no initial information about the topology of the optimal shape
12 History of structural optimization
It was in the early 1960s that Schmit [SCH 60] and Fox [FOX 65] laid the foundations for a modern theory of structural optimization based on the concepts of mathematical programming and sensitivity analysis Paradoxically at the time ldquofully stressed designrdquo was the only widely used technique in practice although it lacked any theoretical justification other than empiricism and the engineersrsquo intuition It was Prager and Taylor [PRA 68] who set out variational methods and Lagrangian optimality conditions to justify the criteria of fully stressed design for a class of structural optimization problems The optimality conditions of the optimization problem were then used directly to construct an iterative resolution algorithm known as the ldquooptimality criteria methodrdquo
Originally structural optimization was mainly limited to the sizing optimization of trusses or gantries Thus sizing optimization of structures was the first field of application for optimality criteria When dealing with the problem of sizing we look at the transverse sections of the structural elements though their length and the
Introduction to Structural Optimization 3
location of their joints remain fixed During the late 1960s and early 1970s optimality criteria were soon adapted to large structures modeled using the finite-element method (FEM) [VEN 73] The optimality criteria method produced a few interesting results and a number of extensions have been presented since the 1970s including Venkayyarsquos generalized criteria method [VEN 73] or Rozvany and Zhoursquos [ROZ 91] discretized optimality criteria
Although since the 1970s attention has mainly been focused on sizing the problem of truss topology has also been studied by Prager [PRA 74] on a very restricted class of structures based on the concept of a truss put forward by Michell in 1904 [MIC 04] The problem lies in finding the best possible configuration so that the truss or gantry can convey forces to the foundations whilst minimizing a given performance and satisfying the design constraints Michellrsquos theory [MIC 04] is related to the topology of trusses made of bars of minimal mass The optimal solution from Michellrsquos point of view is composed solely of perpendicular bars which form a structure whose configuration is optimal for the maximum tensile- and compressive stresses All the configuration problems studied by Prager later on were solved analytically so the practical application of topology was very limited To remedy this shortcoming Rozvany [ROZ 76] invested a great deal of effort in developing new approaches to solve these configuration problems as automatically as possible
After that optimal truss topology was studied in greater depth by Kirsch [KIR 90] If we impose a very small minimum value for the cross-section then the ldquolayoutrdquo optimization of trusses and gantries can be approached as a conventional sizing problem on a very large scale The solution is then obtained by application of generalized optimality criteria for a variety of objective functions compliance movements tensions and eigenvalues [ZHO 91]
Finally the crux of the problem of truss topology optimization seems to have been identified by Bendsoslashe et al [BEN 91] The problem of truss topology is examined with an integral approach combining analysis and design simultaneously The problem of minimum compliance is transformed into a problem of non-differentiable optimization and then in the case of a truss of bars into a linear problem In this form it can be solved on very large structures using non-differentiable optimization methods [BEN 93a] an interior point method [BEN 93b] a dual method [BEC 94] or a penaltybarrier multiplier method [BEN 97]
The three main groups of structural optimization cannot be considered recent concepts but their integration into biomechanics is ndash especially topology optimization [FRA 10]
4 Biomechanics
13 Sizing optimization
131 Definition
With sizing optimization we can only modify the cross-section or transverse thickness of the components of structure whose shape and topology are fixed There can be no modification of the geometric model and its features Figure 11 shows a truss made of 17 bars with different cross-sections (circular rectangular and I-shaped)
Figure 11 Changing the dimensions whilst preserving the same topology of the section
Sizing optimization can be performed by considering the same topology to produce various dimensions For example when the cross-section is circular we merely need to vary the diameters to minimize an objective or several objectives under certain constraints
132 First works in sizing optimization
The problem of sizing optimization has benefited the most from this research so optimization of the transverse dimensions is today a reliable tool This problem was also extended to that of flexural elements [FLE 83] to improving performance when subjected to vibration and to the stability of balance Besides the transverse dimensions of the structural elements it is possible to vary their shape To the best of our knowledge few works have been devoted to the study of the optimal shape for a truss In this type of problem only the location of the structural joints is altered whilst the topology remains unchanged Svanberg [SVA 81] was one of the only people to carry out truss shape optimization on the basis of FEM analyses and solving using mathematical programming techniques
Introduction to Structural Optimization 5
133 Numerical application
1331 Description and modeling of the studied problem
Figure 12a shows a cantilever beam and its I-shaped cross-section in Figure 12b This beam is embedded at one end and subjected to free vibration The material from which the beam is made is structural steel which has a Youngrsquos modulus 200000E MPa= and Poissonrsquos ratio of 03ν = The density of the material is 6 37854 10 Kg mmρ minus= times The material exhibits linear elastic isotropic behavior The length of that beam is 300L mm= and the dimensions of the cross-section are 60 B mm= 100H mm= and 20T mm= (Figure 12b)
a) b)
Figure 12 Cantilever beam subject to free vibration
The objective of this study is to calculate the first four modes of resonance and then perform sizing optimization on the dimensions of the cross-section The problem of optimization therefore is to minimize the structural volume under the constraint of the first resonance frequency This problem can be formulated as follows
1
min ( ) ( ) 0
40 100 80 16010 30
w
Volume B H Ts t f B H T f
BHT
minus lele lele lele le
[11]
where 20wf Hz= is the maximum value of the first resonance frequency
1332 Numerical results
To perform sizing optimization on the ANSYS software for instance we consider the dimensions of the cross-section as optimization variables so as to obtain a parameterized model Then a direct simulation is performed as the heart of the optimization loop
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
Introduction
This book begins with an introductory look at the fundamental principles of structural optimization before applying them to the field of biomechanics Then we present the different strategies for integrating reliability into structural optimization followed by the application of those strategies in biomechanics ndash particularly in terms of the design of orthopedic and orthodontic prostheses
In terms of structural optimization to illustrate the different techniques we can classify structural optimization into three main families
1) Sizing Optimization this model aims to improve a structural model whilst respecting the available resources (known as constraints or limitations) Sizing optimization is the particular case where we can only modify the cross section or transverse thickness of the components of a structure whose shape and topology are fixed There can be no modification of the geometric features andor models
2) Shape Optimization with this model it is possible to make changes to the shape provided they are compatible with a predefined topology This type of optimization modifies the parametric representation of the boundaries of the domain By moving those boundaries we try to find the best possible solution out of the set of all the configurations obtained by homeomorphic transformation of the original structure
3) Topology Optimization this model enables us to make more profound modifications to the shape of the structure Here the geometry of the part is examined with no prerequisites as to the connections of the domains or the structural elements present in the solution In order to optimize the topology we determine the structurersquos shape or transverse dimensions so some authors call topology optimization ldquogeneralized shape optimizationrdquo
xiv Biomechanics
In terms of reliability this concept can be integrated into all three families of structural optimization so we obtain a design that should be both optimal and reliable Sizing optimization shape optimization and topology optimization are generally classed as geometry optimization However the nature of the topology is non-quantitative in relation to shape and size For this reason this integration is divided into two models
1) RBDO Reliability-Based Design Optimization this model couples reliability analysis with sizing optimization (Reliability-Based Sizing Optimization) and also with shape optimization (Reliability-Based Shape Optimization) This coupling is a complex task requiring a lengthy computation time which seriously limits its applicability In addition during the process of reliability-based shape optimization the geometry of the structure is forced to change This coupling integrates different disciplines such as geometric modeling numerical simulation reliability analysis and optimization Thus the optimization problem becomes more complex The major difficulty lies in evaluating the reliability of the structure which in itself requires a specific optimization procedure The typical integration of reliability analysis into optimization methods is carried out in two spaces the normalized space of random variables and the physical space of design variables which requires a very significant computation time To solve this problem as we shall see there are a number of efficient methods
2) RBTO Reliability-Based Topology Optimization in the deterministic case we obtain a single optimal topology while the new RBTO model is able to generate several topologies as a function of a required reliability level However the coupling between reliability (which is quantitative) and topology (which is non-quantitative) requires the use of different methodologies relative to sizing optimization and shape optimization
In terms of biomechanics modeling is generally complicated on several different levels geometric description material properties and boundary conditions When performing the optimization process such modeling is needed with a new set of parameters being entered at each iteration In addition the problem becomes more complex when integrating reliability analysis which is itself performed by way of a particular optimization procedure In this case we couple several different software packages in order to integrate optimization and reliability into biomechanical applications Various strategies are presented in this book to simplify the coupling problem
This book is made up of six chapters presenting the different aspects of integration of reliability and structural optimization into biomechanics
Introduction xv
The first chapter offers an introduction to structural optimization and its main groups sizing optimization shape optimization and topology optimization The principles underpinning the three main types of structural optimization are presented with numerical applications for each type In structural design we have two different phases the conceptual phase and the detailed phase In the conceptual phase we use topology optimization to obtain an idea of the silhouette of the structure The detailed phase consists of shape optimization to obtain smooth geometry followed by sizing optimization to find the areas and thicknesses of the studied structure
The second chapter is devoted to the integration of structural optimization into biomechanics The integration is performed in this chapter for prosthesis design and drilling surgery Different 2D and 3D applications for orthopedic and orthodontic prostheses are presented in order to demonstrate the new strategies presented in this book and also to prove the advantages of those strategies Finally the integration of structural optimization into drilling surgery can be considered a novel aspect in relation to classic prosthesis design
The third chapter is given over to the integration of reliability into structural optimization Such integration can be viewed as a difficult task The difficulties arise in terms of coupling convergence and computation time In this chapter we present the basic principles of that integration and a literature review based on early attempts in this area A numerical application is presented to show the difference between deterministic- and reliability-based optimization Next two viewpoints are presented with the corresponding developed methods considering those two points of view to illustrate the advantages of the methods
The fourth chapter is devoted to the reliability-based design optimization (RBDO) model The developed methods are classified into two categories numerical and semi-numerical methods Numerical methods are generally applicable but require a high computation time Semi-numerical methods on the other hand are efficient in terms of computation time but can only be used in certain cases
The fifth chapter details reliability-based topology optimization (RBTO) This model enables designers and manufacturers to select the solution which is at once economical and reliable Two points of view are presented in the discussion of this model A number of applications in mechanics are presented in order to demonstrate the advantages of the RBTO model
The sixth chapter discusses the integration of reliability and structural optimization into orthopedic and orthodontic prostheses design considering two
xvi Biomechanics
approaches the deterministic approach and the reliability-based approach In orthopedic prostheses the models are presented using results drawn from the existing body of literature The aspect of materials characterization is also integrated into the design process In addition in orthodontics models are presented to test the stability of the prostheses in light of the constraints of dentistry The integration of reliability and structural optimization into the design of prostheses is performed very simply in order to aid understanding and implementation of the different strategies presented in this book
List of Abbreviations
AMA Approximate Moments Approach
AMV Advanced Mean Value
APDL ANSYS Parametric Design Language
AT Anterior Temporal (or Temporalis)
CAD Computer-Aided Design
CM Classical Method
CMV Conjugate Mean Value
CT Computed Tomography
DDO Deterministic Design Optimization
DLA Double-Level Approach
DM Deep Masseter
DO Deterministic Optimization
DTO Deterministic Topology Optimization
FE Finite Elements
FEM Finite Element Method
FORM First Order Reliability Method
HCA Hybrid Cellular Automata
HDS Hybrid Design Space
HM Hybrid Method
xviii Biomechanics
HMV Hybrid Mean Value
IAM Improved Austin-Moore
IHM Improved Hybrid Method
KKT Karush-Kuhn-Tucker
MBB Messerschmitt-Boumllkow-Blohm
MLA Mono-Level Approach
MP Medial Pterygoid
MPP Most Probable failure Point
MT Medial Temporal (or Temporalis)
OC Optimality Criteria
OSF Optimum Safety Factor
PDF Probability Density Function
PMA Performance Measure Approach
PT Posterior Temporal (or Temporalis)
RBDO Reliability-Based Design Optimization
RBO Reliability-Based Optimization
RBSO Reliability-Based Structural Optimization
RBTO Reliability-Based Topology Optimization
RIA Reliability Index Approach
SAP Sequential Approximate Programming
SDA Sequential Decoupled Approach
SED Strain Energy Density
SIMP Solid Isotropic Microstructure with Penalty
SM Superficial Masseter
SORA Sequential Optimization and Reliability Assessment
SORM Second Order Reliability Method
SP Safest Point
1
Introduction to Structural Optimization
11 Introduction
Structural optimization is a topic which affects many different physical domains ndash particularly solid mechanics ndash and which it is tricky to characterize as its formulations may have a number of different aspects Firstly a distinction regarding the way in which geometries are parameterized is presented and secondly a distribution pertaining to the intrinsic nature of optimization algorithms is established
Determining the appropriate shape for structural components is a crucially important problem for engineers In all areas of structural mechanics the impact of proper design of a part is very significant in terms of its strength its lifetime and its usage This is a challenge faced on a daily basis in the sectors of spatial research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile machinery providing substantial energy savings The different development of computer-aided design (CAD) techniques and optimization strategies is part of this context There has been keen interest in structural optimization for over thirty years Whilst it is still too infrequently applied in the conventional techniques used by research centers it is becoming more widely used as its reliability improves Having begun with the simplest of problems the field of application of structural optimization today extends to new and ever more interesting challenges
To illustrate the evolution of structural optimization techniques we can arbitrarily split structural optimization into three major groups (or families) In
Biomechanics Optimization Uncertainties and Reliability First Edition Ghias Kharmanda and Abdelkhalak El Hami copy ISTE Ltd 2017 Published by ISTE Ltd and John Wiley amp Sons Inc
2 Biomechanics
historical terms each of them has been addressed in order of increasing difficulty and generality
With sizing optimization we are only able to modify the dimensions of an object whose shape and topology are fixed There can be no modification of the geometric model We speak of a homeomorphic transformation
Shape optimization involves making changes of shape which are compatible with a predetermined topology Typical shape optimization modifies the parametric representation of the boundaries of the domain By moving the boundaries of the domains we can seek the best solution out of all the structures obtained by homeomorphic transformation of the original object In this case it is clear that we can make a change to the transverse dimensions as well as a modification to the objectrsquos configuration but it is certainly not acceptable to modify its connectivity or its nature ndash in particular the number of components that it has The optimal object exhibits the same topology as the original object
With topology optimization we can fundamentally change the nature of the object The ldquotopologyrdquo refers to the number and position of the components of the domains Here the objectrsquos geometry is presented with no prerequisites as to the connectivity of the domains or the components present in the solution We take no initial information about the topology of the optimal shape
12 History of structural optimization
It was in the early 1960s that Schmit [SCH 60] and Fox [FOX 65] laid the foundations for a modern theory of structural optimization based on the concepts of mathematical programming and sensitivity analysis Paradoxically at the time ldquofully stressed designrdquo was the only widely used technique in practice although it lacked any theoretical justification other than empiricism and the engineersrsquo intuition It was Prager and Taylor [PRA 68] who set out variational methods and Lagrangian optimality conditions to justify the criteria of fully stressed design for a class of structural optimization problems The optimality conditions of the optimization problem were then used directly to construct an iterative resolution algorithm known as the ldquooptimality criteria methodrdquo
Originally structural optimization was mainly limited to the sizing optimization of trusses or gantries Thus sizing optimization of structures was the first field of application for optimality criteria When dealing with the problem of sizing we look at the transverse sections of the structural elements though their length and the
Introduction to Structural Optimization 3
location of their joints remain fixed During the late 1960s and early 1970s optimality criteria were soon adapted to large structures modeled using the finite-element method (FEM) [VEN 73] The optimality criteria method produced a few interesting results and a number of extensions have been presented since the 1970s including Venkayyarsquos generalized criteria method [VEN 73] or Rozvany and Zhoursquos [ROZ 91] discretized optimality criteria
Although since the 1970s attention has mainly been focused on sizing the problem of truss topology has also been studied by Prager [PRA 74] on a very restricted class of structures based on the concept of a truss put forward by Michell in 1904 [MIC 04] The problem lies in finding the best possible configuration so that the truss or gantry can convey forces to the foundations whilst minimizing a given performance and satisfying the design constraints Michellrsquos theory [MIC 04] is related to the topology of trusses made of bars of minimal mass The optimal solution from Michellrsquos point of view is composed solely of perpendicular bars which form a structure whose configuration is optimal for the maximum tensile- and compressive stresses All the configuration problems studied by Prager later on were solved analytically so the practical application of topology was very limited To remedy this shortcoming Rozvany [ROZ 76] invested a great deal of effort in developing new approaches to solve these configuration problems as automatically as possible
After that optimal truss topology was studied in greater depth by Kirsch [KIR 90] If we impose a very small minimum value for the cross-section then the ldquolayoutrdquo optimization of trusses and gantries can be approached as a conventional sizing problem on a very large scale The solution is then obtained by application of generalized optimality criteria for a variety of objective functions compliance movements tensions and eigenvalues [ZHO 91]
Finally the crux of the problem of truss topology optimization seems to have been identified by Bendsoslashe et al [BEN 91] The problem of truss topology is examined with an integral approach combining analysis and design simultaneously The problem of minimum compliance is transformed into a problem of non-differentiable optimization and then in the case of a truss of bars into a linear problem In this form it can be solved on very large structures using non-differentiable optimization methods [BEN 93a] an interior point method [BEN 93b] a dual method [BEC 94] or a penaltybarrier multiplier method [BEN 97]
The three main groups of structural optimization cannot be considered recent concepts but their integration into biomechanics is ndash especially topology optimization [FRA 10]
4 Biomechanics
13 Sizing optimization
131 Definition
With sizing optimization we can only modify the cross-section or transverse thickness of the components of structure whose shape and topology are fixed There can be no modification of the geometric model and its features Figure 11 shows a truss made of 17 bars with different cross-sections (circular rectangular and I-shaped)
Figure 11 Changing the dimensions whilst preserving the same topology of the section
Sizing optimization can be performed by considering the same topology to produce various dimensions For example when the cross-section is circular we merely need to vary the diameters to minimize an objective or several objectives under certain constraints
132 First works in sizing optimization
The problem of sizing optimization has benefited the most from this research so optimization of the transverse dimensions is today a reliable tool This problem was also extended to that of flexural elements [FLE 83] to improving performance when subjected to vibration and to the stability of balance Besides the transverse dimensions of the structural elements it is possible to vary their shape To the best of our knowledge few works have been devoted to the study of the optimal shape for a truss In this type of problem only the location of the structural joints is altered whilst the topology remains unchanged Svanberg [SVA 81] was one of the only people to carry out truss shape optimization on the basis of FEM analyses and solving using mathematical programming techniques
Introduction to Structural Optimization 5
133 Numerical application
1331 Description and modeling of the studied problem
Figure 12a shows a cantilever beam and its I-shaped cross-section in Figure 12b This beam is embedded at one end and subjected to free vibration The material from which the beam is made is structural steel which has a Youngrsquos modulus 200000E MPa= and Poissonrsquos ratio of 03ν = The density of the material is 6 37854 10 Kg mmρ minus= times The material exhibits linear elastic isotropic behavior The length of that beam is 300L mm= and the dimensions of the cross-section are 60 B mm= 100H mm= and 20T mm= (Figure 12b)
a) b)
Figure 12 Cantilever beam subject to free vibration
The objective of this study is to calculate the first four modes of resonance and then perform sizing optimization on the dimensions of the cross-section The problem of optimization therefore is to minimize the structural volume under the constraint of the first resonance frequency This problem can be formulated as follows
1
min ( ) ( ) 0
40 100 80 16010 30
w
Volume B H Ts t f B H T f
BHT
minus lele lele lele le
[11]
where 20wf Hz= is the maximum value of the first resonance frequency
1332 Numerical results
To perform sizing optimization on the ANSYS software for instance we consider the dimensions of the cross-section as optimization variables so as to obtain a parameterized model Then a direct simulation is performed as the heart of the optimization loop
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
xiv Biomechanics
In terms of reliability this concept can be integrated into all three families of structural optimization so we obtain a design that should be both optimal and reliable Sizing optimization shape optimization and topology optimization are generally classed as geometry optimization However the nature of the topology is non-quantitative in relation to shape and size For this reason this integration is divided into two models
1) RBDO Reliability-Based Design Optimization this model couples reliability analysis with sizing optimization (Reliability-Based Sizing Optimization) and also with shape optimization (Reliability-Based Shape Optimization) This coupling is a complex task requiring a lengthy computation time which seriously limits its applicability In addition during the process of reliability-based shape optimization the geometry of the structure is forced to change This coupling integrates different disciplines such as geometric modeling numerical simulation reliability analysis and optimization Thus the optimization problem becomes more complex The major difficulty lies in evaluating the reliability of the structure which in itself requires a specific optimization procedure The typical integration of reliability analysis into optimization methods is carried out in two spaces the normalized space of random variables and the physical space of design variables which requires a very significant computation time To solve this problem as we shall see there are a number of efficient methods
2) RBTO Reliability-Based Topology Optimization in the deterministic case we obtain a single optimal topology while the new RBTO model is able to generate several topologies as a function of a required reliability level However the coupling between reliability (which is quantitative) and topology (which is non-quantitative) requires the use of different methodologies relative to sizing optimization and shape optimization
In terms of biomechanics modeling is generally complicated on several different levels geometric description material properties and boundary conditions When performing the optimization process such modeling is needed with a new set of parameters being entered at each iteration In addition the problem becomes more complex when integrating reliability analysis which is itself performed by way of a particular optimization procedure In this case we couple several different software packages in order to integrate optimization and reliability into biomechanical applications Various strategies are presented in this book to simplify the coupling problem
This book is made up of six chapters presenting the different aspects of integration of reliability and structural optimization into biomechanics
Introduction xv
The first chapter offers an introduction to structural optimization and its main groups sizing optimization shape optimization and topology optimization The principles underpinning the three main types of structural optimization are presented with numerical applications for each type In structural design we have two different phases the conceptual phase and the detailed phase In the conceptual phase we use topology optimization to obtain an idea of the silhouette of the structure The detailed phase consists of shape optimization to obtain smooth geometry followed by sizing optimization to find the areas and thicknesses of the studied structure
The second chapter is devoted to the integration of structural optimization into biomechanics The integration is performed in this chapter for prosthesis design and drilling surgery Different 2D and 3D applications for orthopedic and orthodontic prostheses are presented in order to demonstrate the new strategies presented in this book and also to prove the advantages of those strategies Finally the integration of structural optimization into drilling surgery can be considered a novel aspect in relation to classic prosthesis design
The third chapter is given over to the integration of reliability into structural optimization Such integration can be viewed as a difficult task The difficulties arise in terms of coupling convergence and computation time In this chapter we present the basic principles of that integration and a literature review based on early attempts in this area A numerical application is presented to show the difference between deterministic- and reliability-based optimization Next two viewpoints are presented with the corresponding developed methods considering those two points of view to illustrate the advantages of the methods
The fourth chapter is devoted to the reliability-based design optimization (RBDO) model The developed methods are classified into two categories numerical and semi-numerical methods Numerical methods are generally applicable but require a high computation time Semi-numerical methods on the other hand are efficient in terms of computation time but can only be used in certain cases
The fifth chapter details reliability-based topology optimization (RBTO) This model enables designers and manufacturers to select the solution which is at once economical and reliable Two points of view are presented in the discussion of this model A number of applications in mechanics are presented in order to demonstrate the advantages of the RBTO model
The sixth chapter discusses the integration of reliability and structural optimization into orthopedic and orthodontic prostheses design considering two
xvi Biomechanics
approaches the deterministic approach and the reliability-based approach In orthopedic prostheses the models are presented using results drawn from the existing body of literature The aspect of materials characterization is also integrated into the design process In addition in orthodontics models are presented to test the stability of the prostheses in light of the constraints of dentistry The integration of reliability and structural optimization into the design of prostheses is performed very simply in order to aid understanding and implementation of the different strategies presented in this book
List of Abbreviations
AMA Approximate Moments Approach
AMV Advanced Mean Value
APDL ANSYS Parametric Design Language
AT Anterior Temporal (or Temporalis)
CAD Computer-Aided Design
CM Classical Method
CMV Conjugate Mean Value
CT Computed Tomography
DDO Deterministic Design Optimization
DLA Double-Level Approach
DM Deep Masseter
DO Deterministic Optimization
DTO Deterministic Topology Optimization
FE Finite Elements
FEM Finite Element Method
FORM First Order Reliability Method
HCA Hybrid Cellular Automata
HDS Hybrid Design Space
HM Hybrid Method
xviii Biomechanics
HMV Hybrid Mean Value
IAM Improved Austin-Moore
IHM Improved Hybrid Method
KKT Karush-Kuhn-Tucker
MBB Messerschmitt-Boumllkow-Blohm
MLA Mono-Level Approach
MP Medial Pterygoid
MPP Most Probable failure Point
MT Medial Temporal (or Temporalis)
OC Optimality Criteria
OSF Optimum Safety Factor
PDF Probability Density Function
PMA Performance Measure Approach
PT Posterior Temporal (or Temporalis)
RBDO Reliability-Based Design Optimization
RBO Reliability-Based Optimization
RBSO Reliability-Based Structural Optimization
RBTO Reliability-Based Topology Optimization
RIA Reliability Index Approach
SAP Sequential Approximate Programming
SDA Sequential Decoupled Approach
SED Strain Energy Density
SIMP Solid Isotropic Microstructure with Penalty
SM Superficial Masseter
SORA Sequential Optimization and Reliability Assessment
SORM Second Order Reliability Method
SP Safest Point
1
Introduction to Structural Optimization
11 Introduction
Structural optimization is a topic which affects many different physical domains ndash particularly solid mechanics ndash and which it is tricky to characterize as its formulations may have a number of different aspects Firstly a distinction regarding the way in which geometries are parameterized is presented and secondly a distribution pertaining to the intrinsic nature of optimization algorithms is established
Determining the appropriate shape for structural components is a crucially important problem for engineers In all areas of structural mechanics the impact of proper design of a part is very significant in terms of its strength its lifetime and its usage This is a challenge faced on a daily basis in the sectors of spatial research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile machinery providing substantial energy savings The different development of computer-aided design (CAD) techniques and optimization strategies is part of this context There has been keen interest in structural optimization for over thirty years Whilst it is still too infrequently applied in the conventional techniques used by research centers it is becoming more widely used as its reliability improves Having begun with the simplest of problems the field of application of structural optimization today extends to new and ever more interesting challenges
To illustrate the evolution of structural optimization techniques we can arbitrarily split structural optimization into three major groups (or families) In
Biomechanics Optimization Uncertainties and Reliability First Edition Ghias Kharmanda and Abdelkhalak El Hami copy ISTE Ltd 2017 Published by ISTE Ltd and John Wiley amp Sons Inc
2 Biomechanics
historical terms each of them has been addressed in order of increasing difficulty and generality
With sizing optimization we are only able to modify the dimensions of an object whose shape and topology are fixed There can be no modification of the geometric model We speak of a homeomorphic transformation
Shape optimization involves making changes of shape which are compatible with a predetermined topology Typical shape optimization modifies the parametric representation of the boundaries of the domain By moving the boundaries of the domains we can seek the best solution out of all the structures obtained by homeomorphic transformation of the original object In this case it is clear that we can make a change to the transverse dimensions as well as a modification to the objectrsquos configuration but it is certainly not acceptable to modify its connectivity or its nature ndash in particular the number of components that it has The optimal object exhibits the same topology as the original object
With topology optimization we can fundamentally change the nature of the object The ldquotopologyrdquo refers to the number and position of the components of the domains Here the objectrsquos geometry is presented with no prerequisites as to the connectivity of the domains or the components present in the solution We take no initial information about the topology of the optimal shape
12 History of structural optimization
It was in the early 1960s that Schmit [SCH 60] and Fox [FOX 65] laid the foundations for a modern theory of structural optimization based on the concepts of mathematical programming and sensitivity analysis Paradoxically at the time ldquofully stressed designrdquo was the only widely used technique in practice although it lacked any theoretical justification other than empiricism and the engineersrsquo intuition It was Prager and Taylor [PRA 68] who set out variational methods and Lagrangian optimality conditions to justify the criteria of fully stressed design for a class of structural optimization problems The optimality conditions of the optimization problem were then used directly to construct an iterative resolution algorithm known as the ldquooptimality criteria methodrdquo
Originally structural optimization was mainly limited to the sizing optimization of trusses or gantries Thus sizing optimization of structures was the first field of application for optimality criteria When dealing with the problem of sizing we look at the transverse sections of the structural elements though their length and the
Introduction to Structural Optimization 3
location of their joints remain fixed During the late 1960s and early 1970s optimality criteria were soon adapted to large structures modeled using the finite-element method (FEM) [VEN 73] The optimality criteria method produced a few interesting results and a number of extensions have been presented since the 1970s including Venkayyarsquos generalized criteria method [VEN 73] or Rozvany and Zhoursquos [ROZ 91] discretized optimality criteria
Although since the 1970s attention has mainly been focused on sizing the problem of truss topology has also been studied by Prager [PRA 74] on a very restricted class of structures based on the concept of a truss put forward by Michell in 1904 [MIC 04] The problem lies in finding the best possible configuration so that the truss or gantry can convey forces to the foundations whilst minimizing a given performance and satisfying the design constraints Michellrsquos theory [MIC 04] is related to the topology of trusses made of bars of minimal mass The optimal solution from Michellrsquos point of view is composed solely of perpendicular bars which form a structure whose configuration is optimal for the maximum tensile- and compressive stresses All the configuration problems studied by Prager later on were solved analytically so the practical application of topology was very limited To remedy this shortcoming Rozvany [ROZ 76] invested a great deal of effort in developing new approaches to solve these configuration problems as automatically as possible
After that optimal truss topology was studied in greater depth by Kirsch [KIR 90] If we impose a very small minimum value for the cross-section then the ldquolayoutrdquo optimization of trusses and gantries can be approached as a conventional sizing problem on a very large scale The solution is then obtained by application of generalized optimality criteria for a variety of objective functions compliance movements tensions and eigenvalues [ZHO 91]
Finally the crux of the problem of truss topology optimization seems to have been identified by Bendsoslashe et al [BEN 91] The problem of truss topology is examined with an integral approach combining analysis and design simultaneously The problem of minimum compliance is transformed into a problem of non-differentiable optimization and then in the case of a truss of bars into a linear problem In this form it can be solved on very large structures using non-differentiable optimization methods [BEN 93a] an interior point method [BEN 93b] a dual method [BEC 94] or a penaltybarrier multiplier method [BEN 97]
The three main groups of structural optimization cannot be considered recent concepts but their integration into biomechanics is ndash especially topology optimization [FRA 10]
4 Biomechanics
13 Sizing optimization
131 Definition
With sizing optimization we can only modify the cross-section or transverse thickness of the components of structure whose shape and topology are fixed There can be no modification of the geometric model and its features Figure 11 shows a truss made of 17 bars with different cross-sections (circular rectangular and I-shaped)
Figure 11 Changing the dimensions whilst preserving the same topology of the section
Sizing optimization can be performed by considering the same topology to produce various dimensions For example when the cross-section is circular we merely need to vary the diameters to minimize an objective or several objectives under certain constraints
132 First works in sizing optimization
The problem of sizing optimization has benefited the most from this research so optimization of the transverse dimensions is today a reliable tool This problem was also extended to that of flexural elements [FLE 83] to improving performance when subjected to vibration and to the stability of balance Besides the transverse dimensions of the structural elements it is possible to vary their shape To the best of our knowledge few works have been devoted to the study of the optimal shape for a truss In this type of problem only the location of the structural joints is altered whilst the topology remains unchanged Svanberg [SVA 81] was one of the only people to carry out truss shape optimization on the basis of FEM analyses and solving using mathematical programming techniques
Introduction to Structural Optimization 5
133 Numerical application
1331 Description and modeling of the studied problem
Figure 12a shows a cantilever beam and its I-shaped cross-section in Figure 12b This beam is embedded at one end and subjected to free vibration The material from which the beam is made is structural steel which has a Youngrsquos modulus 200000E MPa= and Poissonrsquos ratio of 03ν = The density of the material is 6 37854 10 Kg mmρ minus= times The material exhibits linear elastic isotropic behavior The length of that beam is 300L mm= and the dimensions of the cross-section are 60 B mm= 100H mm= and 20T mm= (Figure 12b)
a) b)
Figure 12 Cantilever beam subject to free vibration
The objective of this study is to calculate the first four modes of resonance and then perform sizing optimization on the dimensions of the cross-section The problem of optimization therefore is to minimize the structural volume under the constraint of the first resonance frequency This problem can be formulated as follows
1
min ( ) ( ) 0
40 100 80 16010 30
w
Volume B H Ts t f B H T f
BHT
minus lele lele lele le
[11]
where 20wf Hz= is the maximum value of the first resonance frequency
1332 Numerical results
To perform sizing optimization on the ANSYS software for instance we consider the dimensions of the cross-section as optimization variables so as to obtain a parameterized model Then a direct simulation is performed as the heart of the optimization loop
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
Introduction xv
The first chapter offers an introduction to structural optimization and its main groups sizing optimization shape optimization and topology optimization The principles underpinning the three main types of structural optimization are presented with numerical applications for each type In structural design we have two different phases the conceptual phase and the detailed phase In the conceptual phase we use topology optimization to obtain an idea of the silhouette of the structure The detailed phase consists of shape optimization to obtain smooth geometry followed by sizing optimization to find the areas and thicknesses of the studied structure
The second chapter is devoted to the integration of structural optimization into biomechanics The integration is performed in this chapter for prosthesis design and drilling surgery Different 2D and 3D applications for orthopedic and orthodontic prostheses are presented in order to demonstrate the new strategies presented in this book and also to prove the advantages of those strategies Finally the integration of structural optimization into drilling surgery can be considered a novel aspect in relation to classic prosthesis design
The third chapter is given over to the integration of reliability into structural optimization Such integration can be viewed as a difficult task The difficulties arise in terms of coupling convergence and computation time In this chapter we present the basic principles of that integration and a literature review based on early attempts in this area A numerical application is presented to show the difference between deterministic- and reliability-based optimization Next two viewpoints are presented with the corresponding developed methods considering those two points of view to illustrate the advantages of the methods
The fourth chapter is devoted to the reliability-based design optimization (RBDO) model The developed methods are classified into two categories numerical and semi-numerical methods Numerical methods are generally applicable but require a high computation time Semi-numerical methods on the other hand are efficient in terms of computation time but can only be used in certain cases
The fifth chapter details reliability-based topology optimization (RBTO) This model enables designers and manufacturers to select the solution which is at once economical and reliable Two points of view are presented in the discussion of this model A number of applications in mechanics are presented in order to demonstrate the advantages of the RBTO model
The sixth chapter discusses the integration of reliability and structural optimization into orthopedic and orthodontic prostheses design considering two
xvi Biomechanics
approaches the deterministic approach and the reliability-based approach In orthopedic prostheses the models are presented using results drawn from the existing body of literature The aspect of materials characterization is also integrated into the design process In addition in orthodontics models are presented to test the stability of the prostheses in light of the constraints of dentistry The integration of reliability and structural optimization into the design of prostheses is performed very simply in order to aid understanding and implementation of the different strategies presented in this book
List of Abbreviations
AMA Approximate Moments Approach
AMV Advanced Mean Value
APDL ANSYS Parametric Design Language
AT Anterior Temporal (or Temporalis)
CAD Computer-Aided Design
CM Classical Method
CMV Conjugate Mean Value
CT Computed Tomography
DDO Deterministic Design Optimization
DLA Double-Level Approach
DM Deep Masseter
DO Deterministic Optimization
DTO Deterministic Topology Optimization
FE Finite Elements
FEM Finite Element Method
FORM First Order Reliability Method
HCA Hybrid Cellular Automata
HDS Hybrid Design Space
HM Hybrid Method
xviii Biomechanics
HMV Hybrid Mean Value
IAM Improved Austin-Moore
IHM Improved Hybrid Method
KKT Karush-Kuhn-Tucker
MBB Messerschmitt-Boumllkow-Blohm
MLA Mono-Level Approach
MP Medial Pterygoid
MPP Most Probable failure Point
MT Medial Temporal (or Temporalis)
OC Optimality Criteria
OSF Optimum Safety Factor
PDF Probability Density Function
PMA Performance Measure Approach
PT Posterior Temporal (or Temporalis)
RBDO Reliability-Based Design Optimization
RBO Reliability-Based Optimization
RBSO Reliability-Based Structural Optimization
RBTO Reliability-Based Topology Optimization
RIA Reliability Index Approach
SAP Sequential Approximate Programming
SDA Sequential Decoupled Approach
SED Strain Energy Density
SIMP Solid Isotropic Microstructure with Penalty
SM Superficial Masseter
SORA Sequential Optimization and Reliability Assessment
SORM Second Order Reliability Method
SP Safest Point
1
Introduction to Structural Optimization
11 Introduction
Structural optimization is a topic which affects many different physical domains ndash particularly solid mechanics ndash and which it is tricky to characterize as its formulations may have a number of different aspects Firstly a distinction regarding the way in which geometries are parameterized is presented and secondly a distribution pertaining to the intrinsic nature of optimization algorithms is established
Determining the appropriate shape for structural components is a crucially important problem for engineers In all areas of structural mechanics the impact of proper design of a part is very significant in terms of its strength its lifetime and its usage This is a challenge faced on a daily basis in the sectors of spatial research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile machinery providing substantial energy savings The different development of computer-aided design (CAD) techniques and optimization strategies is part of this context There has been keen interest in structural optimization for over thirty years Whilst it is still too infrequently applied in the conventional techniques used by research centers it is becoming more widely used as its reliability improves Having begun with the simplest of problems the field of application of structural optimization today extends to new and ever more interesting challenges
To illustrate the evolution of structural optimization techniques we can arbitrarily split structural optimization into three major groups (or families) In
Biomechanics Optimization Uncertainties and Reliability First Edition Ghias Kharmanda and Abdelkhalak El Hami copy ISTE Ltd 2017 Published by ISTE Ltd and John Wiley amp Sons Inc
2 Biomechanics
historical terms each of them has been addressed in order of increasing difficulty and generality
With sizing optimization we are only able to modify the dimensions of an object whose shape and topology are fixed There can be no modification of the geometric model We speak of a homeomorphic transformation
Shape optimization involves making changes of shape which are compatible with a predetermined topology Typical shape optimization modifies the parametric representation of the boundaries of the domain By moving the boundaries of the domains we can seek the best solution out of all the structures obtained by homeomorphic transformation of the original object In this case it is clear that we can make a change to the transverse dimensions as well as a modification to the objectrsquos configuration but it is certainly not acceptable to modify its connectivity or its nature ndash in particular the number of components that it has The optimal object exhibits the same topology as the original object
With topology optimization we can fundamentally change the nature of the object The ldquotopologyrdquo refers to the number and position of the components of the domains Here the objectrsquos geometry is presented with no prerequisites as to the connectivity of the domains or the components present in the solution We take no initial information about the topology of the optimal shape
12 History of structural optimization
It was in the early 1960s that Schmit [SCH 60] and Fox [FOX 65] laid the foundations for a modern theory of structural optimization based on the concepts of mathematical programming and sensitivity analysis Paradoxically at the time ldquofully stressed designrdquo was the only widely used technique in practice although it lacked any theoretical justification other than empiricism and the engineersrsquo intuition It was Prager and Taylor [PRA 68] who set out variational methods and Lagrangian optimality conditions to justify the criteria of fully stressed design for a class of structural optimization problems The optimality conditions of the optimization problem were then used directly to construct an iterative resolution algorithm known as the ldquooptimality criteria methodrdquo
Originally structural optimization was mainly limited to the sizing optimization of trusses or gantries Thus sizing optimization of structures was the first field of application for optimality criteria When dealing with the problem of sizing we look at the transverse sections of the structural elements though their length and the
Introduction to Structural Optimization 3
location of their joints remain fixed During the late 1960s and early 1970s optimality criteria were soon adapted to large structures modeled using the finite-element method (FEM) [VEN 73] The optimality criteria method produced a few interesting results and a number of extensions have been presented since the 1970s including Venkayyarsquos generalized criteria method [VEN 73] or Rozvany and Zhoursquos [ROZ 91] discretized optimality criteria
Although since the 1970s attention has mainly been focused on sizing the problem of truss topology has also been studied by Prager [PRA 74] on a very restricted class of structures based on the concept of a truss put forward by Michell in 1904 [MIC 04] The problem lies in finding the best possible configuration so that the truss or gantry can convey forces to the foundations whilst minimizing a given performance and satisfying the design constraints Michellrsquos theory [MIC 04] is related to the topology of trusses made of bars of minimal mass The optimal solution from Michellrsquos point of view is composed solely of perpendicular bars which form a structure whose configuration is optimal for the maximum tensile- and compressive stresses All the configuration problems studied by Prager later on were solved analytically so the practical application of topology was very limited To remedy this shortcoming Rozvany [ROZ 76] invested a great deal of effort in developing new approaches to solve these configuration problems as automatically as possible
After that optimal truss topology was studied in greater depth by Kirsch [KIR 90] If we impose a very small minimum value for the cross-section then the ldquolayoutrdquo optimization of trusses and gantries can be approached as a conventional sizing problem on a very large scale The solution is then obtained by application of generalized optimality criteria for a variety of objective functions compliance movements tensions and eigenvalues [ZHO 91]
Finally the crux of the problem of truss topology optimization seems to have been identified by Bendsoslashe et al [BEN 91] The problem of truss topology is examined with an integral approach combining analysis and design simultaneously The problem of minimum compliance is transformed into a problem of non-differentiable optimization and then in the case of a truss of bars into a linear problem In this form it can be solved on very large structures using non-differentiable optimization methods [BEN 93a] an interior point method [BEN 93b] a dual method [BEC 94] or a penaltybarrier multiplier method [BEN 97]
The three main groups of structural optimization cannot be considered recent concepts but their integration into biomechanics is ndash especially topology optimization [FRA 10]
4 Biomechanics
13 Sizing optimization
131 Definition
With sizing optimization we can only modify the cross-section or transverse thickness of the components of structure whose shape and topology are fixed There can be no modification of the geometric model and its features Figure 11 shows a truss made of 17 bars with different cross-sections (circular rectangular and I-shaped)
Figure 11 Changing the dimensions whilst preserving the same topology of the section
Sizing optimization can be performed by considering the same topology to produce various dimensions For example when the cross-section is circular we merely need to vary the diameters to minimize an objective or several objectives under certain constraints
132 First works in sizing optimization
The problem of sizing optimization has benefited the most from this research so optimization of the transverse dimensions is today a reliable tool This problem was also extended to that of flexural elements [FLE 83] to improving performance when subjected to vibration and to the stability of balance Besides the transverse dimensions of the structural elements it is possible to vary their shape To the best of our knowledge few works have been devoted to the study of the optimal shape for a truss In this type of problem only the location of the structural joints is altered whilst the topology remains unchanged Svanberg [SVA 81] was one of the only people to carry out truss shape optimization on the basis of FEM analyses and solving using mathematical programming techniques
Introduction to Structural Optimization 5
133 Numerical application
1331 Description and modeling of the studied problem
Figure 12a shows a cantilever beam and its I-shaped cross-section in Figure 12b This beam is embedded at one end and subjected to free vibration The material from which the beam is made is structural steel which has a Youngrsquos modulus 200000E MPa= and Poissonrsquos ratio of 03ν = The density of the material is 6 37854 10 Kg mmρ minus= times The material exhibits linear elastic isotropic behavior The length of that beam is 300L mm= and the dimensions of the cross-section are 60 B mm= 100H mm= and 20T mm= (Figure 12b)
a) b)
Figure 12 Cantilever beam subject to free vibration
The objective of this study is to calculate the first four modes of resonance and then perform sizing optimization on the dimensions of the cross-section The problem of optimization therefore is to minimize the structural volume under the constraint of the first resonance frequency This problem can be formulated as follows
1
min ( ) ( ) 0
40 100 80 16010 30
w
Volume B H Ts t f B H T f
BHT
minus lele lele lele le
[11]
where 20wf Hz= is the maximum value of the first resonance frequency
1332 Numerical results
To perform sizing optimization on the ANSYS software for instance we consider the dimensions of the cross-section as optimization variables so as to obtain a parameterized model Then a direct simulation is performed as the heart of the optimization loop
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
xvi Biomechanics
approaches the deterministic approach and the reliability-based approach In orthopedic prostheses the models are presented using results drawn from the existing body of literature The aspect of materials characterization is also integrated into the design process In addition in orthodontics models are presented to test the stability of the prostheses in light of the constraints of dentistry The integration of reliability and structural optimization into the design of prostheses is performed very simply in order to aid understanding and implementation of the different strategies presented in this book
List of Abbreviations
AMA Approximate Moments Approach
AMV Advanced Mean Value
APDL ANSYS Parametric Design Language
AT Anterior Temporal (or Temporalis)
CAD Computer-Aided Design
CM Classical Method
CMV Conjugate Mean Value
CT Computed Tomography
DDO Deterministic Design Optimization
DLA Double-Level Approach
DM Deep Masseter
DO Deterministic Optimization
DTO Deterministic Topology Optimization
FE Finite Elements
FEM Finite Element Method
FORM First Order Reliability Method
HCA Hybrid Cellular Automata
HDS Hybrid Design Space
HM Hybrid Method
xviii Biomechanics
HMV Hybrid Mean Value
IAM Improved Austin-Moore
IHM Improved Hybrid Method
KKT Karush-Kuhn-Tucker
MBB Messerschmitt-Boumllkow-Blohm
MLA Mono-Level Approach
MP Medial Pterygoid
MPP Most Probable failure Point
MT Medial Temporal (or Temporalis)
OC Optimality Criteria
OSF Optimum Safety Factor
PDF Probability Density Function
PMA Performance Measure Approach
PT Posterior Temporal (or Temporalis)
RBDO Reliability-Based Design Optimization
RBO Reliability-Based Optimization
RBSO Reliability-Based Structural Optimization
RBTO Reliability-Based Topology Optimization
RIA Reliability Index Approach
SAP Sequential Approximate Programming
SDA Sequential Decoupled Approach
SED Strain Energy Density
SIMP Solid Isotropic Microstructure with Penalty
SM Superficial Masseter
SORA Sequential Optimization and Reliability Assessment
SORM Second Order Reliability Method
SP Safest Point
1
Introduction to Structural Optimization
11 Introduction
Structural optimization is a topic which affects many different physical domains ndash particularly solid mechanics ndash and which it is tricky to characterize as its formulations may have a number of different aspects Firstly a distinction regarding the way in which geometries are parameterized is presented and secondly a distribution pertaining to the intrinsic nature of optimization algorithms is established
Determining the appropriate shape for structural components is a crucially important problem for engineers In all areas of structural mechanics the impact of proper design of a part is very significant in terms of its strength its lifetime and its usage This is a challenge faced on a daily basis in the sectors of spatial research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile machinery providing substantial energy savings The different development of computer-aided design (CAD) techniques and optimization strategies is part of this context There has been keen interest in structural optimization for over thirty years Whilst it is still too infrequently applied in the conventional techniques used by research centers it is becoming more widely used as its reliability improves Having begun with the simplest of problems the field of application of structural optimization today extends to new and ever more interesting challenges
To illustrate the evolution of structural optimization techniques we can arbitrarily split structural optimization into three major groups (or families) In
Biomechanics Optimization Uncertainties and Reliability First Edition Ghias Kharmanda and Abdelkhalak El Hami copy ISTE Ltd 2017 Published by ISTE Ltd and John Wiley amp Sons Inc
2 Biomechanics
historical terms each of them has been addressed in order of increasing difficulty and generality
With sizing optimization we are only able to modify the dimensions of an object whose shape and topology are fixed There can be no modification of the geometric model We speak of a homeomorphic transformation
Shape optimization involves making changes of shape which are compatible with a predetermined topology Typical shape optimization modifies the parametric representation of the boundaries of the domain By moving the boundaries of the domains we can seek the best solution out of all the structures obtained by homeomorphic transformation of the original object In this case it is clear that we can make a change to the transverse dimensions as well as a modification to the objectrsquos configuration but it is certainly not acceptable to modify its connectivity or its nature ndash in particular the number of components that it has The optimal object exhibits the same topology as the original object
With topology optimization we can fundamentally change the nature of the object The ldquotopologyrdquo refers to the number and position of the components of the domains Here the objectrsquos geometry is presented with no prerequisites as to the connectivity of the domains or the components present in the solution We take no initial information about the topology of the optimal shape
12 History of structural optimization
It was in the early 1960s that Schmit [SCH 60] and Fox [FOX 65] laid the foundations for a modern theory of structural optimization based on the concepts of mathematical programming and sensitivity analysis Paradoxically at the time ldquofully stressed designrdquo was the only widely used technique in practice although it lacked any theoretical justification other than empiricism and the engineersrsquo intuition It was Prager and Taylor [PRA 68] who set out variational methods and Lagrangian optimality conditions to justify the criteria of fully stressed design for a class of structural optimization problems The optimality conditions of the optimization problem were then used directly to construct an iterative resolution algorithm known as the ldquooptimality criteria methodrdquo
Originally structural optimization was mainly limited to the sizing optimization of trusses or gantries Thus sizing optimization of structures was the first field of application for optimality criteria When dealing with the problem of sizing we look at the transverse sections of the structural elements though their length and the
Introduction to Structural Optimization 3
location of their joints remain fixed During the late 1960s and early 1970s optimality criteria were soon adapted to large structures modeled using the finite-element method (FEM) [VEN 73] The optimality criteria method produced a few interesting results and a number of extensions have been presented since the 1970s including Venkayyarsquos generalized criteria method [VEN 73] or Rozvany and Zhoursquos [ROZ 91] discretized optimality criteria
Although since the 1970s attention has mainly been focused on sizing the problem of truss topology has also been studied by Prager [PRA 74] on a very restricted class of structures based on the concept of a truss put forward by Michell in 1904 [MIC 04] The problem lies in finding the best possible configuration so that the truss or gantry can convey forces to the foundations whilst minimizing a given performance and satisfying the design constraints Michellrsquos theory [MIC 04] is related to the topology of trusses made of bars of minimal mass The optimal solution from Michellrsquos point of view is composed solely of perpendicular bars which form a structure whose configuration is optimal for the maximum tensile- and compressive stresses All the configuration problems studied by Prager later on were solved analytically so the practical application of topology was very limited To remedy this shortcoming Rozvany [ROZ 76] invested a great deal of effort in developing new approaches to solve these configuration problems as automatically as possible
After that optimal truss topology was studied in greater depth by Kirsch [KIR 90] If we impose a very small minimum value for the cross-section then the ldquolayoutrdquo optimization of trusses and gantries can be approached as a conventional sizing problem on a very large scale The solution is then obtained by application of generalized optimality criteria for a variety of objective functions compliance movements tensions and eigenvalues [ZHO 91]
Finally the crux of the problem of truss topology optimization seems to have been identified by Bendsoslashe et al [BEN 91] The problem of truss topology is examined with an integral approach combining analysis and design simultaneously The problem of minimum compliance is transformed into a problem of non-differentiable optimization and then in the case of a truss of bars into a linear problem In this form it can be solved on very large structures using non-differentiable optimization methods [BEN 93a] an interior point method [BEN 93b] a dual method [BEC 94] or a penaltybarrier multiplier method [BEN 97]
The three main groups of structural optimization cannot be considered recent concepts but their integration into biomechanics is ndash especially topology optimization [FRA 10]
4 Biomechanics
13 Sizing optimization
131 Definition
With sizing optimization we can only modify the cross-section or transverse thickness of the components of structure whose shape and topology are fixed There can be no modification of the geometric model and its features Figure 11 shows a truss made of 17 bars with different cross-sections (circular rectangular and I-shaped)
Figure 11 Changing the dimensions whilst preserving the same topology of the section
Sizing optimization can be performed by considering the same topology to produce various dimensions For example when the cross-section is circular we merely need to vary the diameters to minimize an objective or several objectives under certain constraints
132 First works in sizing optimization
The problem of sizing optimization has benefited the most from this research so optimization of the transverse dimensions is today a reliable tool This problem was also extended to that of flexural elements [FLE 83] to improving performance when subjected to vibration and to the stability of balance Besides the transverse dimensions of the structural elements it is possible to vary their shape To the best of our knowledge few works have been devoted to the study of the optimal shape for a truss In this type of problem only the location of the structural joints is altered whilst the topology remains unchanged Svanberg [SVA 81] was one of the only people to carry out truss shape optimization on the basis of FEM analyses and solving using mathematical programming techniques
Introduction to Structural Optimization 5
133 Numerical application
1331 Description and modeling of the studied problem
Figure 12a shows a cantilever beam and its I-shaped cross-section in Figure 12b This beam is embedded at one end and subjected to free vibration The material from which the beam is made is structural steel which has a Youngrsquos modulus 200000E MPa= and Poissonrsquos ratio of 03ν = The density of the material is 6 37854 10 Kg mmρ minus= times The material exhibits linear elastic isotropic behavior The length of that beam is 300L mm= and the dimensions of the cross-section are 60 B mm= 100H mm= and 20T mm= (Figure 12b)
a) b)
Figure 12 Cantilever beam subject to free vibration
The objective of this study is to calculate the first four modes of resonance and then perform sizing optimization on the dimensions of the cross-section The problem of optimization therefore is to minimize the structural volume under the constraint of the first resonance frequency This problem can be formulated as follows
1
min ( ) ( ) 0
40 100 80 16010 30
w
Volume B H Ts t f B H T f
BHT
minus lele lele lele le
[11]
where 20wf Hz= is the maximum value of the first resonance frequency
1332 Numerical results
To perform sizing optimization on the ANSYS software for instance we consider the dimensions of the cross-section as optimization variables so as to obtain a parameterized model Then a direct simulation is performed as the heart of the optimization loop
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
List of Abbreviations
AMA Approximate Moments Approach
AMV Advanced Mean Value
APDL ANSYS Parametric Design Language
AT Anterior Temporal (or Temporalis)
CAD Computer-Aided Design
CM Classical Method
CMV Conjugate Mean Value
CT Computed Tomography
DDO Deterministic Design Optimization
DLA Double-Level Approach
DM Deep Masseter
DO Deterministic Optimization
DTO Deterministic Topology Optimization
FE Finite Elements
FEM Finite Element Method
FORM First Order Reliability Method
HCA Hybrid Cellular Automata
HDS Hybrid Design Space
HM Hybrid Method
xviii Biomechanics
HMV Hybrid Mean Value
IAM Improved Austin-Moore
IHM Improved Hybrid Method
KKT Karush-Kuhn-Tucker
MBB Messerschmitt-Boumllkow-Blohm
MLA Mono-Level Approach
MP Medial Pterygoid
MPP Most Probable failure Point
MT Medial Temporal (or Temporalis)
OC Optimality Criteria
OSF Optimum Safety Factor
PDF Probability Density Function
PMA Performance Measure Approach
PT Posterior Temporal (or Temporalis)
RBDO Reliability-Based Design Optimization
RBO Reliability-Based Optimization
RBSO Reliability-Based Structural Optimization
RBTO Reliability-Based Topology Optimization
RIA Reliability Index Approach
SAP Sequential Approximate Programming
SDA Sequential Decoupled Approach
SED Strain Energy Density
SIMP Solid Isotropic Microstructure with Penalty
SM Superficial Masseter
SORA Sequential Optimization and Reliability Assessment
SORM Second Order Reliability Method
SP Safest Point
1
Introduction to Structural Optimization
11 Introduction
Structural optimization is a topic which affects many different physical domains ndash particularly solid mechanics ndash and which it is tricky to characterize as its formulations may have a number of different aspects Firstly a distinction regarding the way in which geometries are parameterized is presented and secondly a distribution pertaining to the intrinsic nature of optimization algorithms is established
Determining the appropriate shape for structural components is a crucially important problem for engineers In all areas of structural mechanics the impact of proper design of a part is very significant in terms of its strength its lifetime and its usage This is a challenge faced on a daily basis in the sectors of spatial research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile machinery providing substantial energy savings The different development of computer-aided design (CAD) techniques and optimization strategies is part of this context There has been keen interest in structural optimization for over thirty years Whilst it is still too infrequently applied in the conventional techniques used by research centers it is becoming more widely used as its reliability improves Having begun with the simplest of problems the field of application of structural optimization today extends to new and ever more interesting challenges
To illustrate the evolution of structural optimization techniques we can arbitrarily split structural optimization into three major groups (or families) In
Biomechanics Optimization Uncertainties and Reliability First Edition Ghias Kharmanda and Abdelkhalak El Hami copy ISTE Ltd 2017 Published by ISTE Ltd and John Wiley amp Sons Inc
2 Biomechanics
historical terms each of them has been addressed in order of increasing difficulty and generality
With sizing optimization we are only able to modify the dimensions of an object whose shape and topology are fixed There can be no modification of the geometric model We speak of a homeomorphic transformation
Shape optimization involves making changes of shape which are compatible with a predetermined topology Typical shape optimization modifies the parametric representation of the boundaries of the domain By moving the boundaries of the domains we can seek the best solution out of all the structures obtained by homeomorphic transformation of the original object In this case it is clear that we can make a change to the transverse dimensions as well as a modification to the objectrsquos configuration but it is certainly not acceptable to modify its connectivity or its nature ndash in particular the number of components that it has The optimal object exhibits the same topology as the original object
With topology optimization we can fundamentally change the nature of the object The ldquotopologyrdquo refers to the number and position of the components of the domains Here the objectrsquos geometry is presented with no prerequisites as to the connectivity of the domains or the components present in the solution We take no initial information about the topology of the optimal shape
12 History of structural optimization
It was in the early 1960s that Schmit [SCH 60] and Fox [FOX 65] laid the foundations for a modern theory of structural optimization based on the concepts of mathematical programming and sensitivity analysis Paradoxically at the time ldquofully stressed designrdquo was the only widely used technique in practice although it lacked any theoretical justification other than empiricism and the engineersrsquo intuition It was Prager and Taylor [PRA 68] who set out variational methods and Lagrangian optimality conditions to justify the criteria of fully stressed design for a class of structural optimization problems The optimality conditions of the optimization problem were then used directly to construct an iterative resolution algorithm known as the ldquooptimality criteria methodrdquo
Originally structural optimization was mainly limited to the sizing optimization of trusses or gantries Thus sizing optimization of structures was the first field of application for optimality criteria When dealing with the problem of sizing we look at the transverse sections of the structural elements though their length and the
Introduction to Structural Optimization 3
location of their joints remain fixed During the late 1960s and early 1970s optimality criteria were soon adapted to large structures modeled using the finite-element method (FEM) [VEN 73] The optimality criteria method produced a few interesting results and a number of extensions have been presented since the 1970s including Venkayyarsquos generalized criteria method [VEN 73] or Rozvany and Zhoursquos [ROZ 91] discretized optimality criteria
Although since the 1970s attention has mainly been focused on sizing the problem of truss topology has also been studied by Prager [PRA 74] on a very restricted class of structures based on the concept of a truss put forward by Michell in 1904 [MIC 04] The problem lies in finding the best possible configuration so that the truss or gantry can convey forces to the foundations whilst minimizing a given performance and satisfying the design constraints Michellrsquos theory [MIC 04] is related to the topology of trusses made of bars of minimal mass The optimal solution from Michellrsquos point of view is composed solely of perpendicular bars which form a structure whose configuration is optimal for the maximum tensile- and compressive stresses All the configuration problems studied by Prager later on were solved analytically so the practical application of topology was very limited To remedy this shortcoming Rozvany [ROZ 76] invested a great deal of effort in developing new approaches to solve these configuration problems as automatically as possible
After that optimal truss topology was studied in greater depth by Kirsch [KIR 90] If we impose a very small minimum value for the cross-section then the ldquolayoutrdquo optimization of trusses and gantries can be approached as a conventional sizing problem on a very large scale The solution is then obtained by application of generalized optimality criteria for a variety of objective functions compliance movements tensions and eigenvalues [ZHO 91]
Finally the crux of the problem of truss topology optimization seems to have been identified by Bendsoslashe et al [BEN 91] The problem of truss topology is examined with an integral approach combining analysis and design simultaneously The problem of minimum compliance is transformed into a problem of non-differentiable optimization and then in the case of a truss of bars into a linear problem In this form it can be solved on very large structures using non-differentiable optimization methods [BEN 93a] an interior point method [BEN 93b] a dual method [BEC 94] or a penaltybarrier multiplier method [BEN 97]
The three main groups of structural optimization cannot be considered recent concepts but their integration into biomechanics is ndash especially topology optimization [FRA 10]
4 Biomechanics
13 Sizing optimization
131 Definition
With sizing optimization we can only modify the cross-section or transverse thickness of the components of structure whose shape and topology are fixed There can be no modification of the geometric model and its features Figure 11 shows a truss made of 17 bars with different cross-sections (circular rectangular and I-shaped)
Figure 11 Changing the dimensions whilst preserving the same topology of the section
Sizing optimization can be performed by considering the same topology to produce various dimensions For example when the cross-section is circular we merely need to vary the diameters to minimize an objective or several objectives under certain constraints
132 First works in sizing optimization
The problem of sizing optimization has benefited the most from this research so optimization of the transverse dimensions is today a reliable tool This problem was also extended to that of flexural elements [FLE 83] to improving performance when subjected to vibration and to the stability of balance Besides the transverse dimensions of the structural elements it is possible to vary their shape To the best of our knowledge few works have been devoted to the study of the optimal shape for a truss In this type of problem only the location of the structural joints is altered whilst the topology remains unchanged Svanberg [SVA 81] was one of the only people to carry out truss shape optimization on the basis of FEM analyses and solving using mathematical programming techniques
Introduction to Structural Optimization 5
133 Numerical application
1331 Description and modeling of the studied problem
Figure 12a shows a cantilever beam and its I-shaped cross-section in Figure 12b This beam is embedded at one end and subjected to free vibration The material from which the beam is made is structural steel which has a Youngrsquos modulus 200000E MPa= and Poissonrsquos ratio of 03ν = The density of the material is 6 37854 10 Kg mmρ minus= times The material exhibits linear elastic isotropic behavior The length of that beam is 300L mm= and the dimensions of the cross-section are 60 B mm= 100H mm= and 20T mm= (Figure 12b)
a) b)
Figure 12 Cantilever beam subject to free vibration
The objective of this study is to calculate the first four modes of resonance and then perform sizing optimization on the dimensions of the cross-section The problem of optimization therefore is to minimize the structural volume under the constraint of the first resonance frequency This problem can be formulated as follows
1
min ( ) ( ) 0
40 100 80 16010 30
w
Volume B H Ts t f B H T f
BHT
minus lele lele lele le
[11]
where 20wf Hz= is the maximum value of the first resonance frequency
1332 Numerical results
To perform sizing optimization on the ANSYS software for instance we consider the dimensions of the cross-section as optimization variables so as to obtain a parameterized model Then a direct simulation is performed as the heart of the optimization loop
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
xviii Biomechanics
HMV Hybrid Mean Value
IAM Improved Austin-Moore
IHM Improved Hybrid Method
KKT Karush-Kuhn-Tucker
MBB Messerschmitt-Boumllkow-Blohm
MLA Mono-Level Approach
MP Medial Pterygoid
MPP Most Probable failure Point
MT Medial Temporal (or Temporalis)
OC Optimality Criteria
OSF Optimum Safety Factor
PDF Probability Density Function
PMA Performance Measure Approach
PT Posterior Temporal (or Temporalis)
RBDO Reliability-Based Design Optimization
RBO Reliability-Based Optimization
RBSO Reliability-Based Structural Optimization
RBTO Reliability-Based Topology Optimization
RIA Reliability Index Approach
SAP Sequential Approximate Programming
SDA Sequential Decoupled Approach
SED Strain Energy Density
SIMP Solid Isotropic Microstructure with Penalty
SM Superficial Masseter
SORA Sequential Optimization and Reliability Assessment
SORM Second Order Reliability Method
SP Safest Point
1
Introduction to Structural Optimization
11 Introduction
Structural optimization is a topic which affects many different physical domains ndash particularly solid mechanics ndash and which it is tricky to characterize as its formulations may have a number of different aspects Firstly a distinction regarding the way in which geometries are parameterized is presented and secondly a distribution pertaining to the intrinsic nature of optimization algorithms is established
Determining the appropriate shape for structural components is a crucially important problem for engineers In all areas of structural mechanics the impact of proper design of a part is very significant in terms of its strength its lifetime and its usage This is a challenge faced on a daily basis in the sectors of spatial research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile machinery providing substantial energy savings The different development of computer-aided design (CAD) techniques and optimization strategies is part of this context There has been keen interest in structural optimization for over thirty years Whilst it is still too infrequently applied in the conventional techniques used by research centers it is becoming more widely used as its reliability improves Having begun with the simplest of problems the field of application of structural optimization today extends to new and ever more interesting challenges
To illustrate the evolution of structural optimization techniques we can arbitrarily split structural optimization into three major groups (or families) In
Biomechanics Optimization Uncertainties and Reliability First Edition Ghias Kharmanda and Abdelkhalak El Hami copy ISTE Ltd 2017 Published by ISTE Ltd and John Wiley amp Sons Inc
2 Biomechanics
historical terms each of them has been addressed in order of increasing difficulty and generality
With sizing optimization we are only able to modify the dimensions of an object whose shape and topology are fixed There can be no modification of the geometric model We speak of a homeomorphic transformation
Shape optimization involves making changes of shape which are compatible with a predetermined topology Typical shape optimization modifies the parametric representation of the boundaries of the domain By moving the boundaries of the domains we can seek the best solution out of all the structures obtained by homeomorphic transformation of the original object In this case it is clear that we can make a change to the transverse dimensions as well as a modification to the objectrsquos configuration but it is certainly not acceptable to modify its connectivity or its nature ndash in particular the number of components that it has The optimal object exhibits the same topology as the original object
With topology optimization we can fundamentally change the nature of the object The ldquotopologyrdquo refers to the number and position of the components of the domains Here the objectrsquos geometry is presented with no prerequisites as to the connectivity of the domains or the components present in the solution We take no initial information about the topology of the optimal shape
12 History of structural optimization
It was in the early 1960s that Schmit [SCH 60] and Fox [FOX 65] laid the foundations for a modern theory of structural optimization based on the concepts of mathematical programming and sensitivity analysis Paradoxically at the time ldquofully stressed designrdquo was the only widely used technique in practice although it lacked any theoretical justification other than empiricism and the engineersrsquo intuition It was Prager and Taylor [PRA 68] who set out variational methods and Lagrangian optimality conditions to justify the criteria of fully stressed design for a class of structural optimization problems The optimality conditions of the optimization problem were then used directly to construct an iterative resolution algorithm known as the ldquooptimality criteria methodrdquo
Originally structural optimization was mainly limited to the sizing optimization of trusses or gantries Thus sizing optimization of structures was the first field of application for optimality criteria When dealing with the problem of sizing we look at the transverse sections of the structural elements though their length and the
Introduction to Structural Optimization 3
location of their joints remain fixed During the late 1960s and early 1970s optimality criteria were soon adapted to large structures modeled using the finite-element method (FEM) [VEN 73] The optimality criteria method produced a few interesting results and a number of extensions have been presented since the 1970s including Venkayyarsquos generalized criteria method [VEN 73] or Rozvany and Zhoursquos [ROZ 91] discretized optimality criteria
Although since the 1970s attention has mainly been focused on sizing the problem of truss topology has also been studied by Prager [PRA 74] on a very restricted class of structures based on the concept of a truss put forward by Michell in 1904 [MIC 04] The problem lies in finding the best possible configuration so that the truss or gantry can convey forces to the foundations whilst minimizing a given performance and satisfying the design constraints Michellrsquos theory [MIC 04] is related to the topology of trusses made of bars of minimal mass The optimal solution from Michellrsquos point of view is composed solely of perpendicular bars which form a structure whose configuration is optimal for the maximum tensile- and compressive stresses All the configuration problems studied by Prager later on were solved analytically so the practical application of topology was very limited To remedy this shortcoming Rozvany [ROZ 76] invested a great deal of effort in developing new approaches to solve these configuration problems as automatically as possible
After that optimal truss topology was studied in greater depth by Kirsch [KIR 90] If we impose a very small minimum value for the cross-section then the ldquolayoutrdquo optimization of trusses and gantries can be approached as a conventional sizing problem on a very large scale The solution is then obtained by application of generalized optimality criteria for a variety of objective functions compliance movements tensions and eigenvalues [ZHO 91]
Finally the crux of the problem of truss topology optimization seems to have been identified by Bendsoslashe et al [BEN 91] The problem of truss topology is examined with an integral approach combining analysis and design simultaneously The problem of minimum compliance is transformed into a problem of non-differentiable optimization and then in the case of a truss of bars into a linear problem In this form it can be solved on very large structures using non-differentiable optimization methods [BEN 93a] an interior point method [BEN 93b] a dual method [BEC 94] or a penaltybarrier multiplier method [BEN 97]
The three main groups of structural optimization cannot be considered recent concepts but their integration into biomechanics is ndash especially topology optimization [FRA 10]
4 Biomechanics
13 Sizing optimization
131 Definition
With sizing optimization we can only modify the cross-section or transverse thickness of the components of structure whose shape and topology are fixed There can be no modification of the geometric model and its features Figure 11 shows a truss made of 17 bars with different cross-sections (circular rectangular and I-shaped)
Figure 11 Changing the dimensions whilst preserving the same topology of the section
Sizing optimization can be performed by considering the same topology to produce various dimensions For example when the cross-section is circular we merely need to vary the diameters to minimize an objective or several objectives under certain constraints
132 First works in sizing optimization
The problem of sizing optimization has benefited the most from this research so optimization of the transverse dimensions is today a reliable tool This problem was also extended to that of flexural elements [FLE 83] to improving performance when subjected to vibration and to the stability of balance Besides the transverse dimensions of the structural elements it is possible to vary their shape To the best of our knowledge few works have been devoted to the study of the optimal shape for a truss In this type of problem only the location of the structural joints is altered whilst the topology remains unchanged Svanberg [SVA 81] was one of the only people to carry out truss shape optimization on the basis of FEM analyses and solving using mathematical programming techniques
Introduction to Structural Optimization 5
133 Numerical application
1331 Description and modeling of the studied problem
Figure 12a shows a cantilever beam and its I-shaped cross-section in Figure 12b This beam is embedded at one end and subjected to free vibration The material from which the beam is made is structural steel which has a Youngrsquos modulus 200000E MPa= and Poissonrsquos ratio of 03ν = The density of the material is 6 37854 10 Kg mmρ minus= times The material exhibits linear elastic isotropic behavior The length of that beam is 300L mm= and the dimensions of the cross-section are 60 B mm= 100H mm= and 20T mm= (Figure 12b)
a) b)
Figure 12 Cantilever beam subject to free vibration
The objective of this study is to calculate the first four modes of resonance and then perform sizing optimization on the dimensions of the cross-section The problem of optimization therefore is to minimize the structural volume under the constraint of the first resonance frequency This problem can be formulated as follows
1
min ( ) ( ) 0
40 100 80 16010 30
w
Volume B H Ts t f B H T f
BHT
minus lele lele lele le
[11]
where 20wf Hz= is the maximum value of the first resonance frequency
1332 Numerical results
To perform sizing optimization on the ANSYS software for instance we consider the dimensions of the cross-section as optimization variables so as to obtain a parameterized model Then a direct simulation is performed as the heart of the optimization loop
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
1
Introduction to Structural Optimization
11 Introduction
Structural optimization is a topic which affects many different physical domains ndash particularly solid mechanics ndash and which it is tricky to characterize as its formulations may have a number of different aspects Firstly a distinction regarding the way in which geometries are parameterized is presented and secondly a distribution pertaining to the intrinsic nature of optimization algorithms is established
Determining the appropriate shape for structural components is a crucially important problem for engineers In all areas of structural mechanics the impact of proper design of a part is very significant in terms of its strength its lifetime and its usage This is a challenge faced on a daily basis in the sectors of spatial research aeronautics the automobile industry naval competition fine mechanics precision mechanics or artwork in civil engineering and so on To develop the art of the engineer requires enormous effort to continuously improve techniques for designing structures Optimization is of primary importance in improving the performance and reducing the weight of aerospace- and automobile machinery providing substantial energy savings The different development of computer-aided design (CAD) techniques and optimization strategies is part of this context There has been keen interest in structural optimization for over thirty years Whilst it is still too infrequently applied in the conventional techniques used by research centers it is becoming more widely used as its reliability improves Having begun with the simplest of problems the field of application of structural optimization today extends to new and ever more interesting challenges
To illustrate the evolution of structural optimization techniques we can arbitrarily split structural optimization into three major groups (or families) In
Biomechanics Optimization Uncertainties and Reliability First Edition Ghias Kharmanda and Abdelkhalak El Hami copy ISTE Ltd 2017 Published by ISTE Ltd and John Wiley amp Sons Inc
2 Biomechanics
historical terms each of them has been addressed in order of increasing difficulty and generality
With sizing optimization we are only able to modify the dimensions of an object whose shape and topology are fixed There can be no modification of the geometric model We speak of a homeomorphic transformation
Shape optimization involves making changes of shape which are compatible with a predetermined topology Typical shape optimization modifies the parametric representation of the boundaries of the domain By moving the boundaries of the domains we can seek the best solution out of all the structures obtained by homeomorphic transformation of the original object In this case it is clear that we can make a change to the transverse dimensions as well as a modification to the objectrsquos configuration but it is certainly not acceptable to modify its connectivity or its nature ndash in particular the number of components that it has The optimal object exhibits the same topology as the original object
With topology optimization we can fundamentally change the nature of the object The ldquotopologyrdquo refers to the number and position of the components of the domains Here the objectrsquos geometry is presented with no prerequisites as to the connectivity of the domains or the components present in the solution We take no initial information about the topology of the optimal shape
12 History of structural optimization
It was in the early 1960s that Schmit [SCH 60] and Fox [FOX 65] laid the foundations for a modern theory of structural optimization based on the concepts of mathematical programming and sensitivity analysis Paradoxically at the time ldquofully stressed designrdquo was the only widely used technique in practice although it lacked any theoretical justification other than empiricism and the engineersrsquo intuition It was Prager and Taylor [PRA 68] who set out variational methods and Lagrangian optimality conditions to justify the criteria of fully stressed design for a class of structural optimization problems The optimality conditions of the optimization problem were then used directly to construct an iterative resolution algorithm known as the ldquooptimality criteria methodrdquo
Originally structural optimization was mainly limited to the sizing optimization of trusses or gantries Thus sizing optimization of structures was the first field of application for optimality criteria When dealing with the problem of sizing we look at the transverse sections of the structural elements though their length and the
Introduction to Structural Optimization 3
location of their joints remain fixed During the late 1960s and early 1970s optimality criteria were soon adapted to large structures modeled using the finite-element method (FEM) [VEN 73] The optimality criteria method produced a few interesting results and a number of extensions have been presented since the 1970s including Venkayyarsquos generalized criteria method [VEN 73] or Rozvany and Zhoursquos [ROZ 91] discretized optimality criteria
Although since the 1970s attention has mainly been focused on sizing the problem of truss topology has also been studied by Prager [PRA 74] on a very restricted class of structures based on the concept of a truss put forward by Michell in 1904 [MIC 04] The problem lies in finding the best possible configuration so that the truss or gantry can convey forces to the foundations whilst minimizing a given performance and satisfying the design constraints Michellrsquos theory [MIC 04] is related to the topology of trusses made of bars of minimal mass The optimal solution from Michellrsquos point of view is composed solely of perpendicular bars which form a structure whose configuration is optimal for the maximum tensile- and compressive stresses All the configuration problems studied by Prager later on were solved analytically so the practical application of topology was very limited To remedy this shortcoming Rozvany [ROZ 76] invested a great deal of effort in developing new approaches to solve these configuration problems as automatically as possible
After that optimal truss topology was studied in greater depth by Kirsch [KIR 90] If we impose a very small minimum value for the cross-section then the ldquolayoutrdquo optimization of trusses and gantries can be approached as a conventional sizing problem on a very large scale The solution is then obtained by application of generalized optimality criteria for a variety of objective functions compliance movements tensions and eigenvalues [ZHO 91]
Finally the crux of the problem of truss topology optimization seems to have been identified by Bendsoslashe et al [BEN 91] The problem of truss topology is examined with an integral approach combining analysis and design simultaneously The problem of minimum compliance is transformed into a problem of non-differentiable optimization and then in the case of a truss of bars into a linear problem In this form it can be solved on very large structures using non-differentiable optimization methods [BEN 93a] an interior point method [BEN 93b] a dual method [BEC 94] or a penaltybarrier multiplier method [BEN 97]
The three main groups of structural optimization cannot be considered recent concepts but their integration into biomechanics is ndash especially topology optimization [FRA 10]
4 Biomechanics
13 Sizing optimization
131 Definition
With sizing optimization we can only modify the cross-section or transverse thickness of the components of structure whose shape and topology are fixed There can be no modification of the geometric model and its features Figure 11 shows a truss made of 17 bars with different cross-sections (circular rectangular and I-shaped)
Figure 11 Changing the dimensions whilst preserving the same topology of the section
Sizing optimization can be performed by considering the same topology to produce various dimensions For example when the cross-section is circular we merely need to vary the diameters to minimize an objective or several objectives under certain constraints
132 First works in sizing optimization
The problem of sizing optimization has benefited the most from this research so optimization of the transverse dimensions is today a reliable tool This problem was also extended to that of flexural elements [FLE 83] to improving performance when subjected to vibration and to the stability of balance Besides the transverse dimensions of the structural elements it is possible to vary their shape To the best of our knowledge few works have been devoted to the study of the optimal shape for a truss In this type of problem only the location of the structural joints is altered whilst the topology remains unchanged Svanberg [SVA 81] was one of the only people to carry out truss shape optimization on the basis of FEM analyses and solving using mathematical programming techniques
Introduction to Structural Optimization 5
133 Numerical application
1331 Description and modeling of the studied problem
Figure 12a shows a cantilever beam and its I-shaped cross-section in Figure 12b This beam is embedded at one end and subjected to free vibration The material from which the beam is made is structural steel which has a Youngrsquos modulus 200000E MPa= and Poissonrsquos ratio of 03ν = The density of the material is 6 37854 10 Kg mmρ minus= times The material exhibits linear elastic isotropic behavior The length of that beam is 300L mm= and the dimensions of the cross-section are 60 B mm= 100H mm= and 20T mm= (Figure 12b)
a) b)
Figure 12 Cantilever beam subject to free vibration
The objective of this study is to calculate the first four modes of resonance and then perform sizing optimization on the dimensions of the cross-section The problem of optimization therefore is to minimize the structural volume under the constraint of the first resonance frequency This problem can be formulated as follows
1
min ( ) ( ) 0
40 100 80 16010 30
w
Volume B H Ts t f B H T f
BHT
minus lele lele lele le
[11]
where 20wf Hz= is the maximum value of the first resonance frequency
1332 Numerical results
To perform sizing optimization on the ANSYS software for instance we consider the dimensions of the cross-section as optimization variables so as to obtain a parameterized model Then a direct simulation is performed as the heart of the optimization loop
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
2 Biomechanics
historical terms each of them has been addressed in order of increasing difficulty and generality
With sizing optimization we are only able to modify the dimensions of an object whose shape and topology are fixed There can be no modification of the geometric model We speak of a homeomorphic transformation
Shape optimization involves making changes of shape which are compatible with a predetermined topology Typical shape optimization modifies the parametric representation of the boundaries of the domain By moving the boundaries of the domains we can seek the best solution out of all the structures obtained by homeomorphic transformation of the original object In this case it is clear that we can make a change to the transverse dimensions as well as a modification to the objectrsquos configuration but it is certainly not acceptable to modify its connectivity or its nature ndash in particular the number of components that it has The optimal object exhibits the same topology as the original object
With topology optimization we can fundamentally change the nature of the object The ldquotopologyrdquo refers to the number and position of the components of the domains Here the objectrsquos geometry is presented with no prerequisites as to the connectivity of the domains or the components present in the solution We take no initial information about the topology of the optimal shape
12 History of structural optimization
It was in the early 1960s that Schmit [SCH 60] and Fox [FOX 65] laid the foundations for a modern theory of structural optimization based on the concepts of mathematical programming and sensitivity analysis Paradoxically at the time ldquofully stressed designrdquo was the only widely used technique in practice although it lacked any theoretical justification other than empiricism and the engineersrsquo intuition It was Prager and Taylor [PRA 68] who set out variational methods and Lagrangian optimality conditions to justify the criteria of fully stressed design for a class of structural optimization problems The optimality conditions of the optimization problem were then used directly to construct an iterative resolution algorithm known as the ldquooptimality criteria methodrdquo
Originally structural optimization was mainly limited to the sizing optimization of trusses or gantries Thus sizing optimization of structures was the first field of application for optimality criteria When dealing with the problem of sizing we look at the transverse sections of the structural elements though their length and the
Introduction to Structural Optimization 3
location of their joints remain fixed During the late 1960s and early 1970s optimality criteria were soon adapted to large structures modeled using the finite-element method (FEM) [VEN 73] The optimality criteria method produced a few interesting results and a number of extensions have been presented since the 1970s including Venkayyarsquos generalized criteria method [VEN 73] or Rozvany and Zhoursquos [ROZ 91] discretized optimality criteria
Although since the 1970s attention has mainly been focused on sizing the problem of truss topology has also been studied by Prager [PRA 74] on a very restricted class of structures based on the concept of a truss put forward by Michell in 1904 [MIC 04] The problem lies in finding the best possible configuration so that the truss or gantry can convey forces to the foundations whilst minimizing a given performance and satisfying the design constraints Michellrsquos theory [MIC 04] is related to the topology of trusses made of bars of minimal mass The optimal solution from Michellrsquos point of view is composed solely of perpendicular bars which form a structure whose configuration is optimal for the maximum tensile- and compressive stresses All the configuration problems studied by Prager later on were solved analytically so the practical application of topology was very limited To remedy this shortcoming Rozvany [ROZ 76] invested a great deal of effort in developing new approaches to solve these configuration problems as automatically as possible
After that optimal truss topology was studied in greater depth by Kirsch [KIR 90] If we impose a very small minimum value for the cross-section then the ldquolayoutrdquo optimization of trusses and gantries can be approached as a conventional sizing problem on a very large scale The solution is then obtained by application of generalized optimality criteria for a variety of objective functions compliance movements tensions and eigenvalues [ZHO 91]
Finally the crux of the problem of truss topology optimization seems to have been identified by Bendsoslashe et al [BEN 91] The problem of truss topology is examined with an integral approach combining analysis and design simultaneously The problem of minimum compliance is transformed into a problem of non-differentiable optimization and then in the case of a truss of bars into a linear problem In this form it can be solved on very large structures using non-differentiable optimization methods [BEN 93a] an interior point method [BEN 93b] a dual method [BEC 94] or a penaltybarrier multiplier method [BEN 97]
The three main groups of structural optimization cannot be considered recent concepts but their integration into biomechanics is ndash especially topology optimization [FRA 10]
4 Biomechanics
13 Sizing optimization
131 Definition
With sizing optimization we can only modify the cross-section or transverse thickness of the components of structure whose shape and topology are fixed There can be no modification of the geometric model and its features Figure 11 shows a truss made of 17 bars with different cross-sections (circular rectangular and I-shaped)
Figure 11 Changing the dimensions whilst preserving the same topology of the section
Sizing optimization can be performed by considering the same topology to produce various dimensions For example when the cross-section is circular we merely need to vary the diameters to minimize an objective or several objectives under certain constraints
132 First works in sizing optimization
The problem of sizing optimization has benefited the most from this research so optimization of the transverse dimensions is today a reliable tool This problem was also extended to that of flexural elements [FLE 83] to improving performance when subjected to vibration and to the stability of balance Besides the transverse dimensions of the structural elements it is possible to vary their shape To the best of our knowledge few works have been devoted to the study of the optimal shape for a truss In this type of problem only the location of the structural joints is altered whilst the topology remains unchanged Svanberg [SVA 81] was one of the only people to carry out truss shape optimization on the basis of FEM analyses and solving using mathematical programming techniques
Introduction to Structural Optimization 5
133 Numerical application
1331 Description and modeling of the studied problem
Figure 12a shows a cantilever beam and its I-shaped cross-section in Figure 12b This beam is embedded at one end and subjected to free vibration The material from which the beam is made is structural steel which has a Youngrsquos modulus 200000E MPa= and Poissonrsquos ratio of 03ν = The density of the material is 6 37854 10 Kg mmρ minus= times The material exhibits linear elastic isotropic behavior The length of that beam is 300L mm= and the dimensions of the cross-section are 60 B mm= 100H mm= and 20T mm= (Figure 12b)
a) b)
Figure 12 Cantilever beam subject to free vibration
The objective of this study is to calculate the first four modes of resonance and then perform sizing optimization on the dimensions of the cross-section The problem of optimization therefore is to minimize the structural volume under the constraint of the first resonance frequency This problem can be formulated as follows
1
min ( ) ( ) 0
40 100 80 16010 30
w
Volume B H Ts t f B H T f
BHT
minus lele lele lele le
[11]
where 20wf Hz= is the maximum value of the first resonance frequency
1332 Numerical results
To perform sizing optimization on the ANSYS software for instance we consider the dimensions of the cross-section as optimization variables so as to obtain a parameterized model Then a direct simulation is performed as the heart of the optimization loop
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
Introduction to Structural Optimization 3
location of their joints remain fixed During the late 1960s and early 1970s optimality criteria were soon adapted to large structures modeled using the finite-element method (FEM) [VEN 73] The optimality criteria method produced a few interesting results and a number of extensions have been presented since the 1970s including Venkayyarsquos generalized criteria method [VEN 73] or Rozvany and Zhoursquos [ROZ 91] discretized optimality criteria
Although since the 1970s attention has mainly been focused on sizing the problem of truss topology has also been studied by Prager [PRA 74] on a very restricted class of structures based on the concept of a truss put forward by Michell in 1904 [MIC 04] The problem lies in finding the best possible configuration so that the truss or gantry can convey forces to the foundations whilst minimizing a given performance and satisfying the design constraints Michellrsquos theory [MIC 04] is related to the topology of trusses made of bars of minimal mass The optimal solution from Michellrsquos point of view is composed solely of perpendicular bars which form a structure whose configuration is optimal for the maximum tensile- and compressive stresses All the configuration problems studied by Prager later on were solved analytically so the practical application of topology was very limited To remedy this shortcoming Rozvany [ROZ 76] invested a great deal of effort in developing new approaches to solve these configuration problems as automatically as possible
After that optimal truss topology was studied in greater depth by Kirsch [KIR 90] If we impose a very small minimum value for the cross-section then the ldquolayoutrdquo optimization of trusses and gantries can be approached as a conventional sizing problem on a very large scale The solution is then obtained by application of generalized optimality criteria for a variety of objective functions compliance movements tensions and eigenvalues [ZHO 91]
Finally the crux of the problem of truss topology optimization seems to have been identified by Bendsoslashe et al [BEN 91] The problem of truss topology is examined with an integral approach combining analysis and design simultaneously The problem of minimum compliance is transformed into a problem of non-differentiable optimization and then in the case of a truss of bars into a linear problem In this form it can be solved on very large structures using non-differentiable optimization methods [BEN 93a] an interior point method [BEN 93b] a dual method [BEC 94] or a penaltybarrier multiplier method [BEN 97]
The three main groups of structural optimization cannot be considered recent concepts but their integration into biomechanics is ndash especially topology optimization [FRA 10]
4 Biomechanics
13 Sizing optimization
131 Definition
With sizing optimization we can only modify the cross-section or transverse thickness of the components of structure whose shape and topology are fixed There can be no modification of the geometric model and its features Figure 11 shows a truss made of 17 bars with different cross-sections (circular rectangular and I-shaped)
Figure 11 Changing the dimensions whilst preserving the same topology of the section
Sizing optimization can be performed by considering the same topology to produce various dimensions For example when the cross-section is circular we merely need to vary the diameters to minimize an objective or several objectives under certain constraints
132 First works in sizing optimization
The problem of sizing optimization has benefited the most from this research so optimization of the transverse dimensions is today a reliable tool This problem was also extended to that of flexural elements [FLE 83] to improving performance when subjected to vibration and to the stability of balance Besides the transverse dimensions of the structural elements it is possible to vary their shape To the best of our knowledge few works have been devoted to the study of the optimal shape for a truss In this type of problem only the location of the structural joints is altered whilst the topology remains unchanged Svanberg [SVA 81] was one of the only people to carry out truss shape optimization on the basis of FEM analyses and solving using mathematical programming techniques
Introduction to Structural Optimization 5
133 Numerical application
1331 Description and modeling of the studied problem
Figure 12a shows a cantilever beam and its I-shaped cross-section in Figure 12b This beam is embedded at one end and subjected to free vibration The material from which the beam is made is structural steel which has a Youngrsquos modulus 200000E MPa= and Poissonrsquos ratio of 03ν = The density of the material is 6 37854 10 Kg mmρ minus= times The material exhibits linear elastic isotropic behavior The length of that beam is 300L mm= and the dimensions of the cross-section are 60 B mm= 100H mm= and 20T mm= (Figure 12b)
a) b)
Figure 12 Cantilever beam subject to free vibration
The objective of this study is to calculate the first four modes of resonance and then perform sizing optimization on the dimensions of the cross-section The problem of optimization therefore is to minimize the structural volume under the constraint of the first resonance frequency This problem can be formulated as follows
1
min ( ) ( ) 0
40 100 80 16010 30
w
Volume B H Ts t f B H T f
BHT
minus lele lele lele le
[11]
where 20wf Hz= is the maximum value of the first resonance frequency
1332 Numerical results
To perform sizing optimization on the ANSYS software for instance we consider the dimensions of the cross-section as optimization variables so as to obtain a parameterized model Then a direct simulation is performed as the heart of the optimization loop
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
4 Biomechanics
13 Sizing optimization
131 Definition
With sizing optimization we can only modify the cross-section or transverse thickness of the components of structure whose shape and topology are fixed There can be no modification of the geometric model and its features Figure 11 shows a truss made of 17 bars with different cross-sections (circular rectangular and I-shaped)
Figure 11 Changing the dimensions whilst preserving the same topology of the section
Sizing optimization can be performed by considering the same topology to produce various dimensions For example when the cross-section is circular we merely need to vary the diameters to minimize an objective or several objectives under certain constraints
132 First works in sizing optimization
The problem of sizing optimization has benefited the most from this research so optimization of the transverse dimensions is today a reliable tool This problem was also extended to that of flexural elements [FLE 83] to improving performance when subjected to vibration and to the stability of balance Besides the transverse dimensions of the structural elements it is possible to vary their shape To the best of our knowledge few works have been devoted to the study of the optimal shape for a truss In this type of problem only the location of the structural joints is altered whilst the topology remains unchanged Svanberg [SVA 81] was one of the only people to carry out truss shape optimization on the basis of FEM analyses and solving using mathematical programming techniques
Introduction to Structural Optimization 5
133 Numerical application
1331 Description and modeling of the studied problem
Figure 12a shows a cantilever beam and its I-shaped cross-section in Figure 12b This beam is embedded at one end and subjected to free vibration The material from which the beam is made is structural steel which has a Youngrsquos modulus 200000E MPa= and Poissonrsquos ratio of 03ν = The density of the material is 6 37854 10 Kg mmρ minus= times The material exhibits linear elastic isotropic behavior The length of that beam is 300L mm= and the dimensions of the cross-section are 60 B mm= 100H mm= and 20T mm= (Figure 12b)
a) b)
Figure 12 Cantilever beam subject to free vibration
The objective of this study is to calculate the first four modes of resonance and then perform sizing optimization on the dimensions of the cross-section The problem of optimization therefore is to minimize the structural volume under the constraint of the first resonance frequency This problem can be formulated as follows
1
min ( ) ( ) 0
40 100 80 16010 30
w
Volume B H Ts t f B H T f
BHT
minus lele lele lele le
[11]
where 20wf Hz= is the maximum value of the first resonance frequency
1332 Numerical results
To perform sizing optimization on the ANSYS software for instance we consider the dimensions of the cross-section as optimization variables so as to obtain a parameterized model Then a direct simulation is performed as the heart of the optimization loop
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
Introduction to Structural Optimization 5
133 Numerical application
1331 Description and modeling of the studied problem
Figure 12a shows a cantilever beam and its I-shaped cross-section in Figure 12b This beam is embedded at one end and subjected to free vibration The material from which the beam is made is structural steel which has a Youngrsquos modulus 200000E MPa= and Poissonrsquos ratio of 03ν = The density of the material is 6 37854 10 Kg mmρ minus= times The material exhibits linear elastic isotropic behavior The length of that beam is 300L mm= and the dimensions of the cross-section are 60 B mm= 100H mm= and 20T mm= (Figure 12b)
a) b)
Figure 12 Cantilever beam subject to free vibration
The objective of this study is to calculate the first four modes of resonance and then perform sizing optimization on the dimensions of the cross-section The problem of optimization therefore is to minimize the structural volume under the constraint of the first resonance frequency This problem can be formulated as follows
1
min ( ) ( ) 0
40 100 80 16010 30
w
Volume B H Ts t f B H T f
BHT
minus lele lele lele le
[11]
where 20wf Hz= is the maximum value of the first resonance frequency
1332 Numerical results
To perform sizing optimization on the ANSYS software for instance we consider the dimensions of the cross-section as optimization variables so as to obtain a parameterized model Then a direct simulation is performed as the heart of the optimization loop
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
6 Biomechanics
13321 Direct simulation
Figures 13a and b show the geometric model and meshing model of the cross-section of the beam under examination At the start the mesh is created in 2D using the linear element (PLANE42 - 4-node)
a) b)
Figure 13 a) Geometric model and b) meshing model of the cross-section For a color version of this figure see
wwwistecoukkharmanda2biomechanicszip
Next a 3D model is constructed and meshed using a linear element SOLID45 - (8-node) Figure 14 shows the boundary conditions where one of its ends is fixed
Figure 14 Boundary conditions For a color version of this figure see wwwistecoukkharmanda2biomechanicszip
