Biomechanics: Optimization, Uncertainties and Reliability

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.

Figure 1.2. Cantilever beam subject to free vibration.

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.

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.

Figure 1.9. Three different shapes for the same topology of a 17-bar truss.

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.

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.

Figure 1.15. a) Initial domain of the beam, b), c) and d) different resulting topologies.

Figure 1.16. Dimensions of the studied beam.

Figure 1.17. Boundary conditions.

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.

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.

Figure 2.12. Two types of hip prostheses a) without a shoulder (Model 1) and b) with a shoulder (Model 2).

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.

Figure 2.19. Dimensions of the disk under study.

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.

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.

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.

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.

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].

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 .

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].

Figure 5.12. Algorithm of the HCA method with deterministic- and reliability-based topology optimization of a cantilever beam [MOZ 06].

Figure 5.13. a) Initial configuration of a Michell-type structure and b) initial configuration of a 3-bar truss [MOZ 06].

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.

Figure 6.2. Sequence of deterministic design process integrating topology optimization.

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.

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.

Figure 6.17. a) Initial configuration and b) optimal configuration. For a color version of this figure, see www.iste.co.uk/kharmanda2/biomechanics.

Figure 6.18. Orthopantomogram of a 28-year-old male patient [KHA 16b].

Figure 6.19. Types of mini-plates.

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.

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