Reliability in Biomechanics E-Book

Ghias Kharmanda

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 219

Veröffentlichungsjahr: 2016

Beschreibung

In this book, the authors present in detail several recent methodologies and algorithms that we have developed during the last fifteen years. The deterministic methods account for uncertainties through empirical safety factors, which implies that the actual uncertainties in materials, geometry and loading are not truly considered. This problem becomes much more complicated when considering biomechanical applications where a number of uncertainties are encountered in the design of prosthesis systems. This book implements improved numerical strategies and algorithms that can be applied only in biomechanical studies.

Leseprobe

Table of Contents

Cover

Title

Copyright

Preface

Introduction

Chapter 1: Basic Tools for Reliability Analysis

1.1. Introduction

1.2. Advantages of numerical simulation and optimization

1.3. Numerical simulation by finite elements

1.4. Optimization process

1.5. Sensitivity analysis

1.6. Conclusion

Chapter 2: Reliability Concept

2.1. Introduction

2.2. Basic functions and concepts for reliability analysis

2.3. System reliability

2.4. Statistical measures

2.5. Probability distributions

2.6. Reliability analysis

2.7. Conclusion

Chapter 3: Integration of Reliability Concept into Biomechanics

3.1. Introduction

3.2. Origin and categories of uncertainties

3.3. Uncertainties in biomechanics

3.4. Bone-related uncertainty

3.5. Bone developments and formulations

3.6. Characterization by experimentation of the bone’s mechanical properties

3.7. Conclusion

Chapter 4: Reliability Analysis of Orthopedic Prostheses

4.1. Introduction to orthopedic prostheses

4.2. Reliability analysis of the intervertebral disk

4.3. Reliability analysis of the hip prosthesis

4.4. Conclusion

Chapter 5: Reliability Analysis of Orthodontic Prostheses

5.1. Introduction to orthodontic prostheses

5.2. Anatomy of the temporomandibular joint

5.3. Numerical simulation of a non-fractured mandible

5.4. Reliability analysis of the fixation system of the fractured mandible

5.5. Conclusion

Appendices

Appendix 1: Matrix Calculation

Appendix 2: ANSYS Code for the Disk Implant

Appendix 3: ANSYS Code for the Stem Implant

Appendix 4: Probability of Failure/Reliability Index

Bibliography

Index

End User License Agreement

List of Tables

Chapter 1: Basic Tools for Reliability Analysis

Table 1.1. Objective function evaluation for different variable values

Table 1.2. Iterative values until convergence

Table 1.3. Number of iterations for the different methods

Chapter 2: Reliability Concept

Table 2.1. A set of 71 samples

Table 2.2. Different intervals and their centers

Table 2.3. A set of 21 values

Table 2.4. Different intervals and their frequency

Table 2.5. A set of 46 samples

Table 2.6. Different intervals and their centers

Table 2.7. A set of 62 samples

Table 2.8. The different intervals and their centers

Table 2.9. A set of 10 samples

Chapter 3: Integration of Reliability Concept into Biomechanics

Table 3.1. Experimental results for cortical bone and model validation

Table 3.2. Experimental results for trabecular bone and model validation

Chapter 4: Reliability Analysis of Orthopedic Prostheses

Table 4.1. Mechanical results of the studied intervertebral disk

Table 4.2. Probabilistic results of the studied intervertebral disk

Table 4.3. Mechanical properties of the used materials

Table 4.4. Output parameters for a direct simulation

Table 4.5. Results of the two parameters case

Table 4.6. Results of the six parameters case

Chapter 5: Reliability Analysis of Orthodontic Prostheses

Table 5.1. Muscle force components

Table 5.2. The material properties of the isotropic study

Table 5.3. The mean values and the MPP

Table 5.4. Output parameter values

Table 5.5. The material properties in the orthotropic study

Table 5.6. The mean value and the MPP

Table 5.7. The output parameters of the reliability analysis

List of Illustrations

Chapter 1: Basic Tools for Reliability Analysis

Figure 1.1. General process for developing a finite element model

Figure 1.2. Local optimal solutions and a global optimal solution

Figure 1.3. Simplified algorithm of the optimization process

Figure 1.4. Example of an unconstrained optimization problem

Figure 1.5. Example of a constrained optimization problem

Figure 1.6. Cylindrical reservoir (or tank)

Figure 1.7. Example of a linear problem

Figure 1.8. Example of a nonlinear problem

Figure 1.9. Model of the first constraint

Figure 1.10. Model of the second constraint

Figure 1.11. Model of the intersection of two constraints

Figure 1.12. Model of the intersection of the set of constraints

Figure 1.13. Model of the iso-values of the objective function with the set of constraints

Figure 1.14. Model of the optimal point

Figure 1.15. Objective function interpretation and the optimum value

Figure 1.16. Optimization strategies

Figure 1.17. Principle of the simplex method

Figure 1.18. Interpolation and the curve fitting approximation

Figure 1.19. Comparison of the descent gradient and conjugated gradient methods

Figure 1.20. Iteration (1) using DGM

Figure 1.21. Iteration (2) using DGM

Figure 1.22. Iteration (3) using DGM

Figure 1.23. Iteration (4) using DGM

Figure 1.24. Iteration (5) using DGM

Figure 1.25. Iteration (6) using DGM

Figure 1.26. Iterations 1 and 2 using CGM

Figure 1.27. Iteration (1) using NM and MNM

Figure 1.28. Different iterations for the different methods

Figure 1.29. Geometrical modeling of the optimal point

Figure 1.30. Geometrical modeling of the studied optimization problem

Figure 1.31. Geometric model of the optimization problem

Figure 1.32. Simple model of a cantilever beam

Figure 1.33. Implicit model of a cantilever beam

Figure 1.34. Numerical derivative as a function of the increment

Figure 1.35. Precision curve using FFD technique

Figure 1.36. Precision curve using BFD technique

Figure 1.37. Precision curve using CFD technique

Figure 1.38. Algorithm of precision, ε, with finite difference methods

Chapter 2: Reliability Concept

Figure 2.1. PDF as a function of time

Figure 2.2. CDF as a function of time

Figure 2.3. Reliability and failure probability functions

Figure 2.4. Series configuration

Figure 2.5. Parallel conjunction

Figure 2.6. Mixed configuration

Figure 2.7. Mixed system

Figure 2.8. Delta and star conjunctions

Figure 2.9. Mixed conjunction system

Figure 2.10. New form of the mixed conjunction system

Figure 2.11. Series conjunctions (DBEFG) and (DCG) replaced by (Ru) and (Rv), respectively

Figure 2.12. Transformation of the parallel conjunction (RuRv) to Rw

Figure 2.13. Distribution density

Figure 2.14. Probability density function for the uniform distribution

Figure 2.15. Cumulative distribution function for the uniform probability distribution

Figure 2.16. Reliability function for the uniform probability distribution

Figure 2.17. Cumulative density function for the given data

Figure 2.18. Probability density function (PDF) for the normal distribution

Figure 2.19. Cumulative distribution function (CDF) for the normal distribution

Figure 2.20. Reliability function for the normal distribution

Figure 2.21. Normal distribution of values

Figure 2.22. Probability density function for a lognormal distribution

Figure 2.23. Cumulative distribution function for a lognormal distribution

Figure 2.24. Reliability function for a lognormal distribution

Figure 2.25. Lognormal distribution of values

Figure 2.26. Statistical study of probability density functions

Figure 2.27. Summary of the margin of safety

Figure 2.28. Probabilities of failure

Figure 2.29. Shaft subject to a normal force

Figure 2.30. Geometrical modeling of the design point

Figure 2.31. Transformation between the physical space and the normalized space [KHA 08]

Figure 2.32. Algorithm for evaluating the reliability index

Figure 2.33. Monte-Carlo simulation in standard space

Figure 2.34. Distribution of samples

Chapter 3: Integration of Reliability Concept into Biomechanics

Figure 3.1. Design under uncertainty

Figure 3.2. Description of transversal isotropic behavior by a shell

Figure 3.3. Description of isotropic behavior by a cube

Figure 3.4. Three successive images of DIC compression technique

Figure 3.5. Bending test at three points

Figure 3.6. Sample of the bovine cortical bone

Figure 3.7. Compression test

Figure 3.8. Relation stress/rate

Chapter 4: Reliability Analysis of Orthopedic Prostheses

Figure 4.1. Sagittal curvatures of the spine

Figure 4.2. Disk prosthesis in situ

Figure 4.3. Intervertebral disk: a) Sagittal section of a healthy disk; b) illustration of the disk

Figure 4.4. Ligaments of the spine

Figure 4.5. Dorsal view of the posterior longitudinal ligament (PLL) after section of the pedicles

Figure 4.6. Artificial disk in situ

Figure 4.7. Disk prostheses approved by the FDA: a) Charité III and b) ProDisc

Figure 4.8. Position of the studied disk between lumbars L4 and L5

Figure 4.9. 2D model of the studied disk

Figure 4.10. Disk simulation strategy

Figure 4.11. Transformation of the geometry model into areas (surfaces)

Figure 4.12. Meshing model

Figure 4.13. Boundary conditions

Figure 4.14. Von-Mises stresses for the starting point (initial)

Figure 4.15. Geometric configuration for the optimal solution

Figure 4.16. Geometric configuration for the design point using the direct method

Figure 4.17. Geometric configuration for the design point using the classical method

Figure 4.18. a) Coxofemoral joint, frontal view, and b) Acetabulum in the coxal bone, lateral view

Figure 4.19. Proximal extremity: a) Frontal view, and b) Lateral view

Figure 4.20. Acetabular anatomy

Figure 4.21. Ilio-femoral and pubofemoral capsule and ligaments

Figure 4.22. Several hip muscles: a) anterior view, and b) posterior view, and c) Gluteus maximus and Gluteus medius, posterior view

Figure 4.23. Main parts of a total hip prosthesis

Figure 4.24. Metallic stem in 3D

Figure 4.25. Femoral part of the hip prosthesis with the different bone layers

Figure 4.26. Meshing model. For a color version of the figure, see www.iste.co.uk/kharmanda/biomechanics.zip

Figure 4.27. Boundary conditions: L1, L2 and L3 [KHA 16a]

Figure 4.28. Results of the initial simulation. For a color version of the figure, see www.iste.co.uk/kharmanda/biomechanics.zip

Figure 4.29. Probabilistic sensitivity analysis: a) L1; b) L2 and c) L3. For a color version of the figure, see www.iste.co.uk/kharmanda/biomechanics.zip

Figure 4.30. Reliability algorithm

Figure 4.31. Modeling of the problem in the physical space. For a color version of the figure, see www.iste.co.uk/kharmanda/biomechanics.zip

Figure 4.32. Problem modeling in the normalized space. For a color version of the figure, see www.iste.co.uk/kharmanda/biomechanics.zip

Chapter 5: Reliability Analysis of Orthodontic Prostheses

Figure 5.1. Temporomandibular joint

Figure 5.2. Articular surfaces

Figure 5.3. Temporal and mandibular articular bone regions

Figure 5.4. Meniscus, articular and synovial capsules

Figure 5.5. Secondary ligaments and the external lateral ligament

Figure 5.6. Elevator muscles: masseter, pterygoid and temporal

Figure 5.7. Depressor muscles, hyoid bone and mastoid process

Figure 5.8. Centric occlusion, rest position or malocclusion

Figure 5.9. Mandible subject to a bite force and fixed at its extremities

Figure 5.10. Mandible subject to bite force and muscle forces and fixed at its extremities

Figure 5.11. Boundary conditions for the muscle force exclusion case

Figure 5.12. Distribution of the von-Mises stresses in the case of excluding muscle forces. For a color version of the figure, see www.iste.co.uk/kharmanda/biomechanics.zip

Figure 5.13. Boundary conditions for the case of muscle force inclusion. For a color version of the figure, see www.iste.co.uk/kharmanda/biomechanics.zip

Figure 5.14. Distribution of von-Mises stresses in the case of including the muscle forces. For a color version of the figure, see www.iste.co.uk/kharmanda/biomechanics.zip

Figure 5.15. Orthopantomogram photo of a male patient, age 28 [KHA 16b]

Figure 5.16. Types of mini-plates

Figure 5.17. I-Mini-plate

Figure 5.18. Geometric model of the fractured mandible with mini-plates and screws

Figure 5.19. Meshing model of the fractured mandible with mini-plates and screws. For a color version of the figure, see www.iste.co.uk/kharmanda/biomechanics.zip

Figure 5.20. Boundary conditions for the studied homogeneous structure. For a color version of the figure, see www.iste.co.uk/kharmanda/biomechanics.zip

Figure 5.21. Transformation between the physical space and the normalized space

Figure 5.22. Reliability algorithm

Figure 5.23. Distribution of the von-Mises stresses. For a color version of the figure, see www.iste.co.uk/kharmanda/biomechanics.zip

Figure 5.24. Modeling of relative displacement. For a color version of the figure, see www.iste.co.uk/kharmanda/biomechanics.zip

Figure 5.25. Geometric model of the fractured mandible with mini-plates and screws. For a color version of the figure, see www.iste.co.uk/kharmanda/biomechanics.zip

Figure 5.26. Meshing model of the studied fractured mandible with mini-plates and screws. For a color version of the figure, see www.iste.co.uk/kharmanda/biomechanics.zip

Figure 5.27. Boundary conditions of the studied fractured mandible with composite bone tissues. For a color version of the figure, see www.iste.co.uk/kharmanda/biomechanics.zip

Figure 5.28. Transformation between the physical space and normalized space

Figure 5.29. Reliability algorithm

Figure 5.30. Distribution of the von-Mises stresses. For a color version of the figure, see www.iste.co.uk/kharmanda/biomechanics

Guide

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Reliability of Multiphysical Systems Set

coordinated byAbdelkhalak El Hami

Volume 3

Reliability in Biomechanics

Ghias Kharmanda

Abdelkhalak El Hami

First published 2016 in Great Britain and the United States by ISTE Ltd and John Wiley & 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 Ltd27-37 St George’s RoadLondon SW19 4EUUK

www.iste.co.uk

John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA

www.wiley.com

© ISTE Ltd 2016

The rights of Ghias Kharmanda and Abdelkhalak El Hami to be identified as the author of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2016952173

British Library Cataloguing-in-Publication Data

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

ISBN 978-1-78630-024-9

Preface

In the deterministic method, all parameters that have an uncertain nature are represented by unfavorable characteristic values, associated with the safety factor. The deterministic method uses a pessimistic margin determined by the consequences of a probable failure. Most of the time, this method leads to unjustified specifications, in particular for sensitive structures. Reliability analysis has become an invaluable tool for certain sectors such as the nuclear, space, aeronautical and automobile sectors. Failure phenomena must be addressed in detail in these sectors. We can distinguish between three types of studies:

– the reliability of structures;

– the reliability of systems;

– the reliability of results.

Faced with the deterministic method’s inability to take into account the diversity of physical phenomena that apply to structures, engineers have developed another method, better-suited to the uncertain nature of physical phenomena. In this method, the failure of a structure occurs if the probability of failure is greater than a preset threshold. This method is called the “probabilistic method”.

The probabilistic method is increasingly used in engineering as demonstrated by its different applications in industry. It is used to check that structures of known geometry have an acceptable level of probability, or to optimize the dimensions of a structure in order to respect certain fixed objectives, such as desired cost or the desired level of probability.

Beyond this, reliability analysis is an important tool for making decisions to establish a maintenance and inspection plan. It can also be used for the validation of standards and regulations. To perform reliability analysis, several methods can be used to give the probability of failure in an efficient and simple manner.

This work focuses on the tools necessary for integrating the concept of reliability into biomechanical applications, in particular the design of orthopedic and orthodontic prostheses. We are interested in the reliability of structures to integrate this into biomechanical applications, considering the uncertainty in loading, geometry and materials. Next, the reliability of systems is addressed considering several failure scenarios. The materials of living tissue are very difficult to deal with. For this, formulations with a high level of precision are presented. Finally, several recent methods are addressed to perform reliability analysis for biological applications, in particular orthopedic and orthodontic prostheses.

Acknowledgments

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 the massive moral support whilst this book was being written.

Ghias KHARMANDAAbdelkhalak EL HAMIAugust 2016

Introduction

Reliability analysis is a vast domain that contributes to understanding, modeling and forecasting the degradation and aging mechanisms liable to lead a component to failure and a system to breakdown. Understanding relationships between physical limitations, intrinsic faults, technological imperfections and environmental and internal constraint is the subject of this vast and complex activity. While it is not yet a subject we fully understand, this is something we must work toward. Reliability analysis can still lead to efficient solutions: adapting constraints to physical limitations, protecting the component from internal and external damage or conversely prompting the development of components to make them more robust regarding real constraints.

Research in biomechanics uses approaches in geometric and mechanical modeling, and experimental analysis similar to structural and material mechanics. However, there are numerous obstacles at all stages of characterization and personalization of the geometry, the materials whose behavior varies according to still poorly understood remodeling laws (such as growth, aging, etc.) as well as “in-service” mechanical loading. These key points are also relevant for the validation of models that are as mechanical as they are physiological. Such models require the development of quantitative methods of investigation of living things beyond the traditional domain of mechanics and anatomy, taking into account very strong clinical and ethical constraints. In this book, integrating the concepts of reliability is performed with the aim of dealing with the uncertainty of several aspects, including the loading and the properties of bone materials. The description of the behavior of biological tissue, and bone in particular, is very complex and remains controversial; even more so accurately, changing geometrical parameters during the design of prostheses. In the context of loading, understanding damage mechanisms can contribute to the design of prostheses, as well as improving them. Additionally, the mechanical properties of bone can change after surgery. First, bone is a living “material” whose characteristics vary over time, the type of bone considered, the sex of the individual, their age and health, etc. Furthermore, the modification of mechanical actions, induced, for example, by the installation of an implant, can provoke a radical change in its quality. Theories of bone remodeling and adaptation try to describe and predict its variation over time. Reliability analysis is applied to orthopedic prostheses (hip, knee, etc.) in order to evaluate their level of reliability and stabilization.

The effect of modeling choices on subsequent results has considerable consequences. Taking into account the anisotropy of the bone seems to be crucial in maxillary studies. The bone can be locally considered as an orthotropic material with two orthogonal planes of material symmetry. In a coordinate system linked to these two planes, its elastic behavior is thus defined with the help of nine constants or elastic moduli. However, this coordinate system can change from one point to another in an anatomical part such as the mandible, by orienting itself in a manner to guarantee itself maximum mechanical performance with respect to the usual loading to which it is subjected. A change in this loading due to, for example, the implantation of a prosthesis can lead to a reorientation of this orthotropic coordinate system. Reliability analysis is next applied to evaluate the stability of fixation systems used for mandible fractures.

This work is made up of five chapters:

The first chapter presents several basic tools for reliability analysis. Reliability is calculated by a specific optimization procedure. The optimization depends on the sensitivity analysis that determines the influence of all the input parameters on the output parameters. Furthermore, nothing in biomechanics can be described by analytic geometry. For this, the intervention of numerical simulation is an essential stage of solving complex problems. In order to estimate, with high precision, all these mechanical values and determine the reliability of a given solution, the finite element method appears to be a preferred numerical simulation tool. Thus, in this chapter, numerical simulation by finite elements is first presented. Then, a detailed presentation is given for optimization technology (classification and methods). Finally, the sensitivity analysis is presented as an indispensable tool for optimization and can also be used for the reliability in the following chapters.

The second chapter is dedicated to basic concepts of reliability. First, the history of reliability concept is presented to include the development of this concept over time. Basic statistical functions such as the probability density function and the cumulative distribution function are then examined along with the necessary statistical measures like the mean and standard deviation. Then, the reliability of systems is presented to include the case of multiple failure scenarios. Several applications of distribution laws are examined, in particular the uniform, normal and log-normal distributions. The reliability analysis approach is finally presented including optimization technology.

The third chapter is dedicated to the integration of reliability analysis in biomechanics at several levels: loading, geometry and materials. Then, the uncertainty of bones is examined along with the development of formulations relating the mechanical properties of bone for isotropic and orthotropic behavior. Different works are performed at two levels: macrostructure and microstructure. Experimental works were performed at the solids mechanics laboratory at the University of Lund to validate the developed formulation and extend it to dynamic cases.

The fourth chapter is based upon integrating reliability analysis into orthopedic prostheses to evaluate stabilization after surgical operation. An introduction to orthopedic prostheses is given. Then, two types of prostheses are examined to perform the reliability analysis. The first prosthesis studied is the intervertebral disk. An anatomical presentation of the lumbar region of the spine is given followed by a simple numerical treatment of the disk to show the application of reliability in a very simple manner by considering a problem of two geometrical variables. Next, a detailed study of the hip prosthesis is performed starting with an anatomical presentation of the hip. Finally, a strategy is presented for integrating reliability analysis into the hip prosthesis considering the uncertainty of the mechanical properties of the materials used. In this chapter, the numerical treatment is performed in two-dimensional in order to complete the reliability analysis in a simple and pedagogical manner.

The fifth chapter