Biomechanics ebook

Table of Contents

Cover

Title

Copyright

Preface

Introduction

List of Abbreviations

1 Introduction to Structural Optimization

1.1. Introduction

1.2. History of structural optimization

1.3. Sizing optimization

1.4. Shape optimization

1.5. Topology optimization

1.6. Conclusion

2 Integration of Structural Optimization into Biomechanics

2.1. Introduction

2.2. Integration of structural optimization into orthopedic prosthesis design

2.3. Integration of structural optimization into orthodontic prosthesis design

2.4. Advanced integration of structural optimization into drilling surgery

2.5. Conclusion

3 Integration of Reliability into Structural Optimization

3.1. Introduction

3.2. Literature review of reliability-based optimization

3.3. Comparison between deterministic and reliability-based optimization

3.4. Numerical application

3.5. Approaches and strategies for reliability-based optimization

3.6. Two points of view for developments of reliability-based optimization

3.7. Philosophy of integration of the concept of reliability into structural optimization groups

3.8. Conclusion

4 Reliability-based Design Optimization Model

4.1. Introduction

4.2. Classic method

4.3. Hybrid method

4.4. Improved hybrid method

4.5. Optimum safety factor method

4.6. Safest point method

4.7. Numerical applications

4.8. Classification of the methods developed

4.9. Conclusion

5 Reliability-based Topology Optimization Model

5.1. Introduction

5.2. Formulation and algorithm for the RBTO model

5.3. Validation of the RBTO model

5.4. Variability of the reliability index

5.5. Numerical applications for the RBTO model

5.6. Two points of view for integration of reliability into topology optimization

5.7. Conclusion

6 Integration of Reliability and Structural Optimization into Prosthesis Design

6.1. Introduction

6.2. Prosthesis design

6.3. Integration of topology optimization into prosthesis design

6.4. Integration of reliability and structural optimization into hip prosthesis design

6.5. Integration of reliability and structural optimization into the design of mini-plate systems used to treat fractured mandibles

6.6. Integration of reliability and structural optimization into dental implant design

6.7. Conclusion

Appendices

Appendix 1: ANSYS Code for Stem Geometry

Appendix 2: ANSYS Code for Mini-Plate Geometry

Appendix 3: ANSYS Code for Dental Implant Geometry

Appendix 4: ANSYS Code for Geometry of Dental Implant with Bone

Bibliography

Index

End User License Agreement

List of Illustrations

1 Introduction to Structural Optimization

Figure 1.1.

Changing the dimensions whilst preserving the same topology of the section

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Figure 1.2.

Cantilever beam subject to free vibration

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Figure 1.3.

a) Geometric model and b) meshing model of the cross-section. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 1.4.

Boundary conditions. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 1.5.

First four modes of resonance. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 1.6.

General approach to FEM simulation in dynamics

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Figure 1.7.

a) Initial configuration and b) optimal configuration. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 1.8.

Algorithm used for sizing optimization of the studied beam

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Figure 1.9.

Three different shapes for the same topology of a 17-bar truss

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Figure 1.10.

Dimensions of the plate being studied. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 1.11.

a) Geometric model and b) meshing model with the boundary conditions. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 1.12.

General approach of FEM simulation in statics

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Figure 1.13.

Stress distribution in a) the initial and b) the optimal configurations. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 1.14.

Algorithm used for shape optimization of the studied plate

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Figure 1.15.

a) Initial domain of the beam, b), c) and d) different resulting topologies

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Figure 1.16.

Dimensions of the studied beam

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Figure 1.17.

Boundary conditions

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Figure 1.18.

Resulting topology. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 1.19.

Algorithm used for topology optimization of the studied beam. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 1.20.

Layout configuration for shape optimization. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

2 Integration of Structural Optimization into Biomechanics

Figure 2.1.

Dimensions of the studied stem. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.2.

Boundary conditions of the stem studied in the tests

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Figure 2.3.

Boundary conditions of the studied stem. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.4.

Stress distribution in the a) initial and b) optimal configurations. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.5.

Boundary conditions using ANSYS software. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.6.

a) Geometric model and b) resulting topology considering the entire stem as an optimization domain. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.7.

a) Geometric model and b) resulting topology considering the lower part of the studied stem as an optimization domain. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.8.

a) Geometric model and b) resulting topology considering the inner surface of the lower part of the studied stem as an optimization domain. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.9.

Strategy of structural optimization in hip prosthesis design. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.10.

Sagittal cross-section of the proximal femur with the different areas of bone tissue. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.11.

Switch from a surface description to a volumetric description

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Figure 2.12.

Two types of hip prostheses a) without a shoulder (Model 1) and b) with a shoulder (Model 2)

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Figure 2.13.

a) Boundary conditions of the hip prosthesis without shoulders with the bone, b) the bone tissues and c) the meshing model [KHA 12]. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.14.

a) Boundary conditions of the hip prosthesis with a shoulder with the bone, b) the bone tissues and c) the meshing model [KHA 12]. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.15.

Maximum values of stresses for the different components when considering the first case of materials. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.16.

Maximum values of the stresses on the different components considering the model case of materials. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.17.

The maximum values of the stresses on the different components when considering the third case of materials. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.18.

a) The curvature of the spine (lumbar lordosis), b) the part studied between lumbar vertebrae L4 and L5 and c) the components of the studied disk

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Figure 2.19.

Dimensions of the disk under study

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Figure 2.20.

a) Meshing model, b) two contact surfaces (upper and lower) and c) boundary conditions. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.21.

Von Mises stress distribution of the a) initial and b) optimal configurations. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.22.

The studied part between lumbar vertebrae L4 and L5 with a) the initial and b) the optimal disk configurations [KHA 12]. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.23.

Dental implant in the bone

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Figure 2.24.

a) Geometric model, b) meshing model, c) boundary conditions and d) distribution of von Mises stresses. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.25.

Initial dimensions of the studied mini-plate

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Figure 2.26.

Modeling of the boundary conditions of the 2D mini-plate. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.27.

Optimal configuration of the mini-plate

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Figure 2.28.

a) CT image, b) 3D geometric modeling of the fractured femur, c) μCT model and d) simplified 2D model. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.29.

Distribution of the principal strain average in the granular zone with a single hole for the a) initial and b) optimal configurations. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.30.

Sensitivity diagram of compliance and principal strain average of the granular zone with a single hole. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.31.

Principal strain average distribution in the granular zone with two holes. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 2.32.

Sensitivity diagram of the compliance and the principal strain average of the granular zone with two holes. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

3 Integration of Reliability into Structural Optimization

Figure 3.1.

Physical and normed spaces

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Figure 3.2.

Geometric interpretation of deterministic and reliability-based optimization solutions. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 3.3.

a) Geometry of the structure and b) mesh adopted. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 3.4.

Stress distribution: a) before and b) after optimization. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 3.5.

RIA versus PMA

.

Figure 3.6.

Geometric interpretation of FORM. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 3.7.

Geometric interpretation of SORM. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 3.8.

Integration of the eliability concept into sizing-, shape- and topology optimization. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

4 Reliability-based Design Optimization Model

Figure 4.1.

Algorithm of the classic method

Figure 4.2.

HDS for a normal distribution law [KHA 02d]

.

Figure 4.3.

HDS for a log-normal distribution law [KHA 07d]

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Figure 4.4.

HDS for a uniform distribution law [KHA 07d]

.

Figure 4.5.

Algorithm for the HM [KHA 02a]

.

Figure 4.6.

Algorithm for the IHM [MOH 06b]

.

Figure 4.7.

Design point and optimal solution obtained by OSF

.

Figure 4.8.

OSF algorithm

.

Figure 4.9.

SP at the eigenfrequency (displacement/frequency relationship) for a) non-symmetric case and b) symmetric case

.

Figure 4.10.

SP algorithm for the non-symmetric case where

Figure 4.11.

SP algorithm for the symmetric case where

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Figure 4.12.

DDO and RBDO iterations for CM and HM (80% reduction)

Figure 4.13.

Dimensions of the triangular plate. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 4.14.

Mesh and boundary conditions of the studied triangular plate. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 4.15.

Sandwich beam subject to distributed loads. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 4.16.

Cross-section of the airplane wing

Figure 4.17.

Modes of the aircraft wing. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 4.18.

Numerical RBDO methods

.

Figure 4.19.

Semi-numerical RBDO methods

.

5 Reliability-based Topology Optimization Model

Figure 5.1.

RBTO algorithm

.

Figure 5.2.

a) Initial domain, b) resulting topology using ANSYS software, c) resulting topology using the code developed in MATLAB [KHA 11c]. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 5.3.

Analytical validation of the RBTO model [KHA 04e]

.

Figure 5.4.

a) Deterministic topology, b) reliability-based topology, c) truss modeling of the DTO model, d) truss modeling of the RBTO model [KHA 11c]

.

Figure 5.5.

a) Initial domain of an MBB beam, subject to a distributed load; b) deterministic topology c) reliability-based topology [KHA 11c]

.

Figure 5.6.

a) and b) Initial configurations in deterministic topology and reliability-based topology; c) and d) Optimal shapes when considering the DTO and RBTO models [KHA 11c]. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 5.7.

Approximation of variability (objective function/reliability index). For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 5.8.

DTO and RBDO results for static analysis [KHA 11b]

.

Figure 5.9.

DTO and RBDO results for modal analysis [KHA 11b]

.

Figure 5.10.

DTO and RBDO results for fatigue analysis [KHA 11b]

.

Figure 5.11.

Two points of view of for integration of reliability into topology optimization [KHA 08b]

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Figure 5.12.

Algorithm of the HCA method with deterministic- and reliability-based topology optimization of a cantilever beam [MOZ 06]

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Figure 5.13.

a) Initial configuration of a Michell-type structure and b) initial configuration of a 3-bar truss [MOZ 06]

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Figure 5.14.

Topologies and damage distributions: (a,c) DTO and (b,d) RBTO models. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 5.15.

Topologies and damage distributions: (a,c) DTO and (b,d) RBTO models. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 5.16.

Topologies and damage distributions with (a, c) DTO and (b, d) RBTO models. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 5.17.

Topologies and damage distributions: a,c) DTO and b,d) RBTO models. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

6 Integration of Reliability and Structural Optimization into Prosthesis Design

Figure 6.1.

Illustrative example of structural optimization groups

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Figure 6.2.

Sequence of deterministic design process integrating topology optimization

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Figure 6.3.

a) 3D geometric model of the studied stem, b) model with solid stem and c) model with holed stem. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.4.

Boundary conditions: L1, L2 and L3 [KHA 16a]. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.5.

a) Geometric model of the studied system; b), c) and d) resulting topologies for loading scenarios L1, L2 and L3, respectively. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.6.

Two forms of the classic Austin-Moore models

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Figure 6.7.

Geometric models for a) the solid stem and b) the IAM stem. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.8.

Distribution of von Mises stresses for the solid stem, considering the three loading scenarios a) L1, b) L2 and c) L3 and for the IAM stem, considering d) L3. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.9.

Distribution of the von Mises stresses, considering the first loading scenario in 3D for a) the solid stem, b) the implant–bone interface of the solid stem, c) the IAM stem and d) the implant–bone interface of the IAM stem. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.10.

Distribution of the von Mises stresses, considering the second loading scenario in 3D for a) the solid stem, b) the implant–bone interface of the solid stem, c) the IAM stem and d) the implant–bone interface of the IAM stem. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.11.

Distribution of the von Mises stresses, considering the third loading scenario in 3D for a) the solid stem, b) the implant–bone interface of the solid stem, c) the IAM stem, d) the implant–bone interface in the IAM stem. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.12.

Comparison of the values of the objective function for solid and IAM stems, considering the three loading scenarios. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.13.

a) 3D geometric model of the studied stem and b) implant–bone interface in 2D. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.14.

Distribution of von Mises stresses for loading scenarios a) L1, b) L2 and c) L3. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.15.

Reliability algorithm

.

Figure 6.16.

RBDO algorithm

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Figure 6.17.

a) Initial configuration and b) optimal configuration. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics

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Figure 6.18.

Orthopantomogram of a 28-year-old male patient [KHA 16b]

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Figure 6.19.

Types of mini-plates

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Figure 6.20.

Mandible subject to a bite force and to the forces of the muscles, embedded at its endpoints. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.21.

Initial domain of the mini-plate used for topology optimization, with the sums of the applied forces. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.22.

a) Resulting topology and b) input domain for shape optimization. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.23.

2D modeling of the boundary conditions for the mini-plate. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.24.

a) Initial dimensions of the studied mini-plate and b) its optimal configuration. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.25.

I-Mini-plate. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.26.

a) Initial and b) optimal configurations for the studied mini-plate

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Figure 6.27.

Boundary conditions for the case of inclusion of the muscular forces. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.28.

Distribution of the von Mises stresses for the optimal configuration. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.29.

Structural optimization strategy. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.30.

Modeling of the distance between the two surfaces of the fractured mandible. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.31.

Sensitivity of the strain energy with respect to the different forces. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.32.

a) DTO and b) RBTO layouts. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.33.

a) 2D geometric model with optimization variables and b) 3D geometric model considering two different materials. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.34.

Boundary conditions. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Figure 6.35.

Distribution of the von Mises stress for a) the implant–bone ensemble and for b) the dental implant. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

List of Tables

2 Integration of Structural Optimization into Biomechanics

Table 2.1.

Mechanical properties of different zones of the proximal part of the femur [SEN 05]

Table 2.2.

Mechanical properties of the alloys used

Table 2.3.

Numerical results on the hip prosthesis for a stem without a shoulder. For a color version of this table, see www.iste.co.uk/kharmanda2/biomechanics.zip

Table 2.4.

Numerical results of the hip prosthesis for the stem with a shoulder. For a color version of this table, see www.iste.co.uk/kharmanda2/biomechanics.zip

Table 2.5.

Stresses on the different parts of the disk prosthesis before and after the optimization process [KHA 12]. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.zip

Table 2.6.

Different input and output parameters corresponding to the optimization process of the studied mini-plate

3 Integration of Reliability into Structural Optimization

Table 3.1.

Initial values for the studied hook

Table 3.2.

Design point and optimal solution for DO and RBO regarding the hook

Table 3.3.

DO and RBO results

4 Reliability-based Design Optimization Model

Table 4.1.

OSFs for the normal, log-normal and uniform distribution laws

.

Table 4.2.

The equations of the SP solutions for normal, log-normal and uniform distribution laws

Table 4.3.

Design point and optimal solution for RBDO of the hook

Table 4.4.

Results for RBDO of the hook

Table 4.5.

HM and IHM results

Table 4.6.

OSFs

Table 4.7.

Design point and optimal solution for the studied beam

Table 4.8.

Results of the HM and SP methods

Table 4.9.

Overview of the advantages and disadvantages of numerical and semi-numerical RBDO methods

5 Reliability-based Topology Optimization Model

Table 5.1.

DTO and RBTO results for the cantilever beam

Table 5.2.

Input parameters and compliance of the studied cantilever beam using the DTO and RBTO models

Table 5.3.

Initial and optimal values for the first case (7-bar structure)

Table 5.4.

Initial and optimal values in the second case (12-bar structure)

Table 5.5.

Input parameters and compliance of the MBB beam, studied using the DTO and RBTO models

Table 5.6.

Initial and optimal values when considering the DTO model

Table 5.7.

Initial values and optimal values when considering the RBTO model

Table 5.8.

Topology generation [KHA 04e]

Table 5.9.

Input and output parameters for DTO and RBTO for static analysis

Table 5.10.

Input and output parameters from DTO and RBTO for modal analysis

Table 5.11.

Input and output parameters in DTO and RBTO for fatigue analysis

Table 5.12.

Variability of reliability index on a Michell-type structure and a 3-bar truss using the HCA method [MOZ 06]

Table 5.13.

DTO and RBTO results for case 1

Table 5.14.

DTO and RBTO results for case 2

Table 5.15.

DTO and RBTO results for case 1

Table 5.16.

DTO and RBTO results for case 2

6 Integration of Reliability and Structural Optimization into Prosthesis Design

Table 6.1.

Results of shape optimization for the different models

Table 6.2.

Results of 3D simulation

Table 6.3.

Mechanical properties and maximal values of von Mises stresses

Table 6.4.

Results of reliability analysis on the stem using the two formulations

Table 6.5.

RBDO results of the studied stem

Table 6.6.

Components of muscle forces

Table 6.7.

Mean values and sensitivities

Landmarks

Cover

Table of Contents

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G1

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G5

Reliability of Multiphysical Systems Set

coordinated by

Abdelkhalak El Hami

Volume 5

Biomechanics

Optimization, Uncertainties and Reliability

First published 2017 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

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Library of Congress Control Number: 2016952066

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A CIP record for this book is available from the British Library

ISBN 978-1-78630-025-6

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 – 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 “probabilistic approach”. The probabilistic approach is 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 structure’s 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 – 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 KHARMANDAAbdelkhalak EL-HAMIOctober 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 – 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 and/or 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 structure’s shape or transverse dimensions, so some authors call topology optimization “generalized shape optimization”.

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.