Multi-physics Optimization
Description
This book illustrates, in detail, the state of the art in the multidisciplinary science of multi-physics optimization. In a context of the perpetual search for improved industrial competitiveness, the evolution of product design and optimization methods and tools appears to be a strategic necessity in view of the imperative to reduce costs. In the aeronautics sector, resources are mainly focused on forecasting and controlling the costs incurred by failures that occur at commissioning, during the warranty period, and during aircraft operation. However, in the future, new contracts for the sale of aeronautical equipment will become increasingly oriented toward sales by the hour of operation.
The aim of this book is to propose new methods for reliability-based optimization, enabling an analysis of a system's life cycle. The V-cycle allows development phases to be viewed in terms of development time and levels of integration complexity.
Multi-physics Optimization is dedicated to optimization methods for multi-physics problems. Each chapter clearly sets out the techniques used and developed and accompanies them with illustrative examples. The book is aimed at students but is also a valuable resource for practicing engineers and research lecturers.
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Persons
Abdelkhalak El Hami is University Professor at INSA Rouen Normandie, France, and is the author (or co-author) of over sixty books. He is responsible for several European educational and/or research projects as a specialist in the fields of optimization, reliability and AI of multiphysics systems.
Mohamed Eid is University Professor at the Settat Faculty of Science and Technology, Hassan Premier University, Morocco. He is the author of several books, specializing in numerical optimization methods and system reliability.
Content
Preface ix
Chapter 1 Introduction to Optimization in Mechanics 1
1.1 Introduction 1
1.2 Problems of general dynamics 2
1.2.1 Finite element methods 4
1.2.2 Modal assumption method 5
1.2.3 Direct integration 7
1.3 Structural optimization 9
1.3.1 Design optimization 9
1.3.2 Shape optimization 10
1.3.3 Topological optimization 10
1.3.4 Definitions and formulation of an optimization problem 13
1.4 Structures with uncertain parameters 15
1.4.1 Monte Carlo simulation 16
1.4.2 Analytical method 16
1.4.3 Stochastic finite element method 16
1.4.4 Fuzzy logic method 17
1.4.5 Bibliography study of reliability methods 18
1.4.6 Bibliography study of reliability optimization methods 24
1.5 Conclusion 26
Chapter 2 Modal Synthesis and Reliability Optimization Methods 27
2.1 Introduction 27
2.2 State of the art of modal synthesis methods 27
2.2.1 Introduction 27
2.2.2 Model reduction method for elastodynamic systems 28
2.2.3 Fixed-interface method (Craig Bampton) 29
2.2.4 Reduction of junction degrees of freedom 31
2.3 Coupling of modal synthesis and RBDO methods 33
2.4 Numerical application 34
2.4.1 L-Plate embedded at the ends 34
2.4.2 Analysis of results 37
2.4.3 A vibrating connecting rod 38
2.4.4 Application: transient model reduction 42
2.5 Conclusion 44
Chapter 3 Modal Synthesis Methods and Stochastic Finite Element Methods 45
3.1 Introduction 45
3.2 Linear dynamical problems 46
3.2.1 Equations of motion 46
3.2.2 Solution in transient state 47
3.2.3 Solution in the harmonic domain 48
3.2.4 Direct integration 49
3.3 Modal synthesis methods 52
3.3.1 Introduction 52
3.3.2 Substructure assembly technique 54
3.3.3 Fixed-interface method 55
3.3.4 MacNeal's open-interface method 59
3.3.5 Open interface method 61
3.3.6 Hybrid method 64
3.3.7 Reduction of junction d.o.f 64
3.4 Stochastic finite element methods 67
3.4.1 Introduction 67
3.4.2 Discretization of random fields 67
3.4.3 Methods for calculating moments 69
3.5 Conclusion 78
Chapter 4 Fatigue and Reliability Optimization for Structures Subjected to Random Vibrations 79
4.1 Introduction 79
4.2 Fatigue damage analysis 80
4.2.1 Formulations and developments 80
4.2.2 Strategy for the fatigue damage analysis 83
4.3 Reliability optimization of structures subjected to random vibrations 84
4.3.1 Deterministic optimization 84
4.3.2 Reliability-based design optimization (RBDO) 85
4.3.3 Reliability optimization of structures subjected to random vibrations 92
4.4 Applications 94
4.4.1 Problem description 94
4.4.2 Results and discussion 97
4.5. Conclusion 101
Chapter 5 Optimization and Shaping 103
5.1 Introduction 103
5.2 Different approaches to process optimization 103
5.3 Characterization of forming processes by objective functions 107
5.4 Deterministic and probabilistic optimization of a T-shaped tube 107
5.4.1 Selection of objective function and definition of constraints 109
5.4.2 Deterministic formulation of the optimization problem 113
5.4.3 Probabilistic formulation of the optimization problem 116
5.4.4 Sensitivity of optimums to uncertainties 123
5.5 Deterministic and reliable optimization of a tube with two expansion areas 125
5.5.1 Deterministic formulation of the optimization problem 129
5.5.2 Reliability formulation of the optimization problem 130
5.5.3 Numerical results 130
Chapter 6 Reliability and FSI Optimization 133
6.1 Introduction 133
6.2 Reliability optimization in mechanics 134
6.2.1 Deterministic optimization 134
6.2.2 Different approaches to RBDO 135
6.2.3 Traditional approach 137
6.2.4 Hybrid approach 138
6.2.5 Hybrid frequency approach 140
6.3 Safest point (SP) method 142
6.4 Numerical simulation 146
6.4.1 Reliability calculation for an aircraft wing 146
6.4.2 RBDO application to the aircraft wing 148
Chapter 7 Reliability-based Optimization for Dental Implants 163
7.1 Introduction 163
7.2 Stochastic approach 164
7.2.1 Monte Carlo (MC) method 164
7.2.2 Generalized polynomial chaos (GPC) method 165
7.3 Deterministic design optimization 166
7.4 Reliability-based design optimization (RBDO) 166
7.4.1 Traditional method 167
7.4.2 Optimal safety factor (OSF) using GPC 168
7.5 Numerical result 170
7.5.1 2D dental implant 171
7.6 Conclusion 176
Appendices 177
Appendix 1 Binomial Distribution 179
Appendix 2 Geometric Distribution 181
Appendix 3 Poisson Distribution 183
Appendix 4 Exponential Distribution 185
Appendix 5 Normal Distribution 187
Appendix 6 Log-Normal Distribution 191
Appendix 7 Weibull Distribution 195
Appendix 8 Pareto Distribution 199
Appendix 9 Extreme Value Distribution 201
Appendix 10 Asymptotic Distributions 203
Appendix 11 Introduction to Optimization 209
Appendix 12 Notion of Statistics 221
References 235
Index 249
1
Introduction to Optimization in Mechanics
1.1. Introduction
The purpose of this chapter is to present a nonexhaustive state of the art of all of the fields that will be directly or indirectly covered in this book.
The first section of this chapter is dedicated to the general study of structural dynamics. This study will make it possible to define the notations essential to the calculation of dynamic responses, to the calculation of frequencies, eigenmodes and response functions. All of these aspects will be addressed in detail and through practical applications.
One of two conventional strategies can be used to solve the system of dynamical equilibrium equations of a structure [IMB 95]. The most common solution strategy in dynamic range is modal superposition, which is suitable for linear structures in which only the first modes are excited. On the other hand, direct solving methods use the integration of equations of motion, in order to deal with nonlinear structures. The latter can also be applied when the frequency content of the excitation covers a large number of modes of the mechanical structure under study.
The second section of this chapter is a presentation of a bibliographic study of structural optimization. The goal is to obtain the appropriate shapes for a part by minimizing a given criterion. In every area of structural mechanics, the impact of the proper design of a part is very important for its strength, service life and application use. This challenge is a daily one in high-tech sectors. The development of the art of engineering requires considerable effort to constantly improve structural design techniques. Optimization is essential in increasing performance and reducing the weight of aerospace and automotive vehicles, thus leading to substantial energy savings.
The last section of this chapter is dedicated to the description of the different tools for structural analysis with uncertain parameters. The taking into account of parameter uncertainty is a particularly critical problem in vibrational mechanics. However, the consideration of this effect can respond to different types of needs, where two main categories can be distinguished: analysis and design. In general, the modeled parts or structures must satisfy given specifications, which require, for example, that safety, reliability or comfort standards be observed.
During deterministic design, engineers try to find the best possible design among all of the potential solutions they have to study. This choice is based on cost as well as on improving product quality. In this case, the designer's objectives to get an optimal design are devised without worrying about the accuracy of the mechanical characteristics of the materials, the geometry and loading (uncertainty effects). The resulting optimal design may therefore represent an inadequate level of reliability. The objective of the approach integrating reliability analysis into the optimization problem and called RBDO (for Reliability Based Design Optimization) is to design structures while establishing the best compromise between cost and the proper functioning of the device
1.2. Problems of general dynamics
Figure 1.1. System O subject to forces.
The formulation of a problem of dynamics in small perturbations on a domain O of boundary (see Figure 1.1) and in a time interval [0, T] is:.
[1.1] [1.2]Initial conditions:
[1.3] [1.4]Boundary conditions:
[1.5] [1.6]u is the displacement vector, s and e are respectively the stress and strain tensor. ? is the density. Vectors g, f and u respectively represent the volume force, the external force and the imposed displacement. is the normal vector to the surface.
In the case of an isotropic elastic domain, the law of behavior is written as:.
[1.7a]l and m are the functions of Young's modulus and Poisson's ratio n:
[1.7b] [1.8]The dynamic problem presented above for this elastic example can be represented by the Navier equation as follows:
[1.9]where ?2 denotes the Laplacian operator: and "?" is the notation of the divergence operator:
1.2.1. Finite element methods
For structures with complex geometries, numerical methods such as the finite element method are used. In an elastodynamic problem, the displacements are generally expressed by a combination of vectors [GMÜ 97]:
[1.10]where [B(x)] is the matrix of shape functions, and {q (t )} is the vector of discrete real displacements, whose components are the discrete unknowns of the approximation.
In general, a structural mechanics problem, after discretization, is described by a system of second-order equations:.
[1.11]where N is the number of degrees of freedom of the system; M ( N × N ) is the positive definite symmetric mass matrix, C ( N × N ) and K ( N × N ) are the viscous damping and stiffness non-negative definite symmetric matrices respectively. F is the vector of the applied forces.
Mathematically, equation [1.11] represents a system of second-order differential equations that can be solved either by a direct integration method or by the mode superposition method
1.2.2. Modal assumption method
We apply the following modal transformation to system [1.11]:
[1.12]{p} is the vector of generalized coordinates; [F] is the modal matrix verifying the orthogonality properties: [F]T [M ][F] = I and [F]T [K][F] = [w2] with: , where wi is the natural angular frequency, equation [1.12] reverts to:.
[1.13]where {P} = [F]T {F} is the modal force vector.
It can be assumed that the damping matrix is proportional to the mass and stiffness matrix. This assumption, known as the Rayleigh assumption, is quite commonly used in structural analysis. It therefore postulates that:
[1.14] [1.15]which amounts to writing that:
[1.16]The decoupled system becomes:
[1.17] [1.18]The factor ?i is called the reduced damping coefficient for the i-th mode. a and ß are the initially unknown values and are calculated from the reduced damping coefficient ?i.
Figure 1.2. Damping coefficient graph.
In Figure 1.2, we graphically present the modal damping coefficient ?; note that the sum of the two functions is almost constant at damping over the frequency range of interest. Thereby, given the modal damping (?) and a frequency interval (f1 and f2), the two equations can be simultaneously solved to determine a and ß
[1.19]and
[1.20]1.2.3. Direct integration
There are many methods to integrate differential equations. The general procedure consists of discretizing time and formulating what happens at time "t+?t" as a function of what happens at time "t" based on Taylor expansions; in this part, we present the Newmark method and the Wilson method.
1.2.3.1. Newmark method
Newmark proposed a method where the velocity and displacements at t+?t are estimated with respect to and to accelerations . In addition, the displacement and velocity are developed in the Taylor series using the two independent parameters ß and ? as well as the time step [KLE 92]:
[1.21] [1.22]where and are the approximations of and respectively, and tn+1 = tn + ? t, with ?t being the time steps, the two independent parameters ß and ? ensure the accuracy and stability of the solution. When ? = 1/ 2. ß = (? + 0.5) / 4.
By carrying these equations forward into the equation of motion, the following matrix relation is obtained:
[1.23]with and
[1.24]The acceleration at time t = 0 is provided by the equilibrium conditions and the initial conditions on {q} and . The solution of equation [1.23] requires that a linear system be solved at each time step.
1.2.3.2. The Wilson-q method
This is an extension of a method in which the acceleration is assumed to linearly vary over the interval [ n ? t , ( n + 1) ? t ]; Wilson [KLE 92] assumes that this linear variation occurs over the interval [ n ? t , ( n + 1)...
