Multidisciplinary Design Optimization Of Automotive Structures
Linköping Studies in Science and Technology Licentiate Thesis No. 1578 Multidisciplinary Design Optimization of Automotive Structures Rebecka Domeij Bäckryd LIU-TEK-LIC-2013:12 Division of Solid Mechanics Department of Management and Engineering Linköping University SE-581 83 Linköping, Sweden February 2013
Linköping Studies in Science and Technology, Licentiate Thesis No. 1578 LIU-TEK-LIC-2013:12 ISBN 978-91-7519-688-6 ISSN 0280-7971 Printed by: LiU-Tryck, Linköping, Sweden, 2013 Distributed by: Division of Solid Mechanics Linköping University SE-581 83 Linköping, Sweden Cover: The picture is reproduced by courtesy of Volvo Cars and illustrates a CAD model of the body of Volvo V40. Copyright 2013 Rebecka Domeij Bäckryd No part of this publication may be reproduced, stored in a retrieval system, or be transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the author.
Preface The work presented in this thesis has been carried out at the Division of Solid Mechanics at Linköping University. It has been part of the ProOpt project, which has been financed by the Swedish foundation for strategic research through the ProViking programme. I would like to thank my supervisor Professor Larsgunnar Nilsson for his encouraging words along the way and for believing in me. All my PhD student colleagues have also been a valuable support. This research has been performed in collaboration with PhD student Ann-Britt Ryberg, who first worked at Saab Automobile AB and thereafter at Combitech AB. I would like to thank Ann-Britt for all the time we have spent on the phone discussing different aspects of this research. My working days have been more productive and much less lonely thanks to our collaboration. I would like to thank my parents, sisters, and friends for all the encouragement and support they have given me. Most importantly, my husband Emmanuel is always there for me, loving and supporting me. Moreover, my wonderful, intense, and strong-minded children, Elna and Alvin, bring so much happiness into my life. This thesis is dedicated to the memory of Vilgot, whom I joyfully expected but painfully lost during the time this work was performed. Rebecka Domeij Bäckryd Linköping, February 2013 iii
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Abstract Multidisciplinary design optimization (MDO) can be used as an effective tool to improve the design of automotive structures. Large-scale MDO problems typically involve several groups who must work concurrently and autonomously for reasons of efficiency. When performing MDO, a large number of designs need to be rated. Detailed simulation models used to assess automotive design proposals are often computationally expensive to evaluate. A useful MDO process must distribute work to the groups involved and be computationally efficient. In this thesis, MDO methods are assessed in relation to the characteristics of automotive structural applications. Single-level optimization methods have a single optimizer, while multi-level optimization methods have a distributed optimization process. Collaborative optimization and analytical target cascading are possible choices of multi-level optimization methods for automotive structures. They distribute the design process, but are complex. One approach to handle the computationally demanding simulation models involves metamodel-based design optimization (MBDO), where metamodels are used as approximations of the detailed models during optimization studies. Metamodels can be created by individual groups prior to the optimization process, and therefore also offer a way of distributing work. A single-level optimization method in combination with metamodels is concluded to be the most straightforward way of implementing MDO into the development of automotive structures. Keywords: multidisciplinary design optimization (MDO); single-level optimization methods; multi-level optimization methods; metamodel-based design optimization (MBDO); automotive structures v
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List of Papers The following papers have been appended to this thesis: I. R. D. Bäckryd, A.-B. Ryberg, and L. Nilsson (2013). Multidisciplinary design optimization methods for automotive structures, Submitted. II. A.-B. Ryberg, R. D. Bäckryd, and L. Nilsson (2013). A metamodel-based multidisciplinary design optimization process for automotive structures, Submitted. Author’s contribution: The work resulting in the two appended papers has been a joint effort by Ann-Britt Ryberg and me. I have borne primary responsibility for the literature survey and the writing process of the first paper. Concerning the second paper, I have actively participated in the writing process and in the application example. vii
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Contents Preface iii Abstract v List of Papers vii Contents ix Part I – Theory and Background 1 Introduction 3 2 Optimization Concepts 7 2.1 Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Structural Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Multidisciplinary Design Optimization Problems 11 3.1 Problem Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1.1 Terminology of Decomposed Systems . . . . . . . . . . . . . . . . . . . 3.1.2 Aspect-Based and Object-Based Decomposition . . . . . . . . . . . . . . 3.1.3 Coupling Breadth and Coupling Strength. . . . . . . . . . . . . . . . . . 12 14 15 3.2 Automotive Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 Comparison between Automotive and Aerospace Problems . . . . . . . . . . 17 4 Metamodel-Based Design Optimization 19 4.1 Design of Experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 Variable Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.3 Metamodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.3.1 Artificial Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Metamodel Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 23 4.4 Stochastic Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . 24 ix
4.4.1 Evolutionary Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Simulated Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Multidisciplinary Design Optimization Methods 24 25 27 5.1 Single-Level Optimization Methods . . . . . . . . . . . . . . . . . . . . . . 27 5.1.1 Multidisciplinary Feasible . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Individual Discipline Feasible . . . . . . . . . . . . . . . . . . . . . . . 28 29 5.2 Multi-Level Optimization Methods . . . . . . . . . . . . . . . . . . . . . . . 30 5.2.1 5.2.2 5.2.3 5.2.4 Concurrent Subspace Optimization . . Bi-Level Integrated System Synthesis Collaborative Optimization . . . . . . Analytical Target Cascading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Metamodel-Based MDO of Automotive Structures 30 33 37 41 45 6.1 Requirements on MDO Methods . . . . . . . . . . . . . . . . . . . . . . . . 45 6.2 Suitable MDO Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6.3 A Metamodel-Based MDO Process . . . . . . . . . . . . . . . . . . . . . . . 48 7 Conclusions 51 8 Review of Appended Papers 53 Bibliography 55 Part II – Appended Papers Paper I Multidisciplinary design optimization methods for automotive structures . . . . . . 63 Paper II A metamodel-based multidisciplinary design optimization process for automotive structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 x
Part I Theory and Background
Introduction 1 Automotive companies are exposed to tough competition and continuously strive to improve their products in order to maintain their position on the market. The aim of multidisciplinary design optimization (MDO) is to find the best possible design taking into account several disciplines simultaneously. Introducing MDO can thus aid in the search for better products. During large-scale automotive product development, several design groups are responsible for different aspects or parts of the product. The aspects or parts cannot be considered to be isolated entities as they mutually influence one another. The groups must therefore interact during the development. The term “group” is used here to denote both the administrative unit and a team working with a specific task. Traditionally, the aim of the design process is to meet a certain number of requirements by repeated parallel development phases with intermediate data synchronizations between the groups. A traditional approach leads to a feasible design, but, in all probability, not to an optimal one. The goal of MDO is to find the optimal design taking into account two or more disciplines simultaneously using a formalized optimization methodology. A discipline is an aspect of the product, and different disciplines are typically handled by different groups. Consequently, performing MDO generally involves several groups. For MDO to be efficient, these groups must work concurrently and autonomously, placing restrictions on the choice of MDO method. Concurrency implies using human and computational resources in parallel, and autonomy means letting groups make design decisions and govern methods and tools within their areas of responsibility. Multidisciplinary design optimization evolved as a new engineering field in the area of aerospace structural optimization, where disciplines strongly interacting with the structural ones were included in the optimization process (Agte et al., 2010). Kroo and Manning (2000) describe the development of MDO in terms of three generations. Initially, all disciplines were integrated into a single optimization loop. As the MDO problem size grew, the second generation of MDO methods was developed. Analyses were distributed, which enabled computational resources to be used in parallel, but coordinated by an optimizer. Both the first and second generations of MDO methods are single-level optimization methods, which rely on a central optimizer making all the design decisions. When MDO was applied to even larger problems, the need for distributing the decision- 3
CHAPTER 1. INTRODUCTION making process became apparent. The third generation of MDO methods includes the multi-level optimization methods, where the optimization process as such is distributed. These different approaches are illustrated in Figure 1.1. Multidisciplinary design optimization has primarily been used for aerospace and automotive applications (Agte et al., 2010). The industrial usage is smaller than was first expected since MDO is found to be complex (Alexandrov, 2005). The difficulties associated with MDO depend on the nature of the problem in question. When using MDO for industrial applications, it might not be feasible to reach the global optimum. However, finding the global optimum is not necessarily the ultimate goal from an engineering point of view. Instead, the aim is often to gain an increased understanding of the system, which can be used to obtain a better design. a) b) c) System Optimizer Optimizer Optimizer and Analyzer Analyzer Analyzer Subspace Subspace Optimizer Optimizer and Analyzer and Analyzer Figure 1.1a) Single-level optimization method with integrated analyses (first generation of MDO methods). b) Single-level optimization method with distributed analyses (second generation of MDO methods). c) Multi-level optimization method (third generation of MDO methods). Detailed simulation models, typically finite element (FE) models, are used to assess automotive design proposals, and these are often computationally expensive to evaluate. A large number of evaluations are needed when performing optimization. Direct optimization, which implies evaluations of the detailed simulation models during the optimization process, may require more computational resources than are available. Metamodels can be used as approximations of the detailed models during optimization studies. They can be evaluated quickly and are developed based on a series of evaluations of the detailed models. Metamodel-based design optimization (MBDO) can therefore be an efficient alternative for automotive structural applications. Multidisciplinary design optimization studies of automotive structures with full vehicle models are still rather rare. However, one type of such a study found in the literature involves minimizing the mass of the vehicle body considering noise, vibration and harshness (NVH) and crashworthiness. Examples are described by Craig et al. (2002), Sobieszczanski-Sobieski et al. (2001), and Kodiyalam et al. (2004), which are all executed using metamodels and single-level optimization methods. There are automotive MDO studies that use direct optimization, e.g. Duddeck (2008), but metamodel-based design 4
CHAPTER 1. INTRODUCTION optimization is the most common approach when computationally expensive simulation models are involved. Multi-level optimization methods are rarely used for automotive applications. The aim of this research is to find suitable methods for large-scale MDO of automotive structures involving computationally expensive simulation models. The work has been performed within the ProOpt project, which has the goal to develop methods for optimization-driven design. The first part of the thesis contains theory and background, and it is organized as follows. First, some basic optimization concepts are covered. Next, the nature of MDO problems is discussed, specifically focusing on automotive structural applications. Motivated by the need for metamodels when performing MDO of automotive structures, MBDO is thereafter given special attention. A number of commonly used MDO methods are then accounted for. This part of the thesis is summed up in a discussion concerning suitable ways of performing metamodel-based MDO of automotive structures. Finally, conclusions are drawn and the two appended papers are reviewed. 5
CHAPTER 1. INTRODUCTION 6
Optimization Concepts 2 Some basic optimization concepts are introduced in this chapter. First, a general optimization problem is defined. Next, structural optimization is given special attention as the focus of this thesis is to find MDO methods suitable for automotive structural applications. Finally, a short discussion concerning optimization algorithms is provided. 2.1 Optimization Problem A general optimization problem can be formulated as min 𝐱 subject to 𝑓(𝐱) 𝐠(𝐱) 𝟎 𝐡(𝐱) 𝟎 𝐱 lower 𝐱 𝐱 upper . (2.1) The goal is to find the values of the design variables x that minimize the objective function f. In general, the optimization problem has a number of inequality and equality constraints that must be fulfilled, represented by the vectors g and h respectively. The objective and constraint functions depend on the design variables x. The design variables must be kept within the upper and lower limits, called x upper and x lower . The general formulation can be recast into the simpler form min 𝐱 subject to 𝑓(𝐱) 𝐠(𝐱) 𝟎. (2.2) In this latter formulation, the inequality constraints g contain all three types of constraints in the former formulation. The design variables can be continuous or discrete, meaning that they can take any value, or only certain discrete values, between the upper and lower limits. Design points that fulfil all constraints are feasible, while all other design points are unfeasible. An unconstrained optimization problem lacks constraints, as opposed to a constrained optimization problem. The problem is convex if the objective function is a convex 7
CHAPTER 2. OPTIMIZATION CONCEPTS function and the feasible region, defined by the constraints, is a convex set. The solution of an optimization problem is called the global optimum. In a non-convex optimization problem, a number of local optima different from the global optimum may exist, while the only optimum in a convex optimization problem is the global one. 2.2 Structural Optimization A structure is a body, or an assemblage of bodies, that can support loads. The following definition of structural optimization is provided by Gallagher (1973, p. 7): “Structural optimization seeks the selection of design variables to achieve, within the limits (constraints) placed on the structural behaviour, geometry, or other factors, its goal of optimality defined by the objective function for specified loading or environmental conditions.” Three types of structural optimization can be distinguished: size, shape, and topology optimization (Bendsøe and Sigmund, 2003). In size optimization, the design variables represent structural properties, e.g. sheet thicknesses or material parameters. In shape optimization on the other hand, the design variables represent the shape of material boundaries. Topology optimization is the most general form of structural optimization and is used to find where material should be placed to be most effective. A typical automotive structural optimization problem is a size optimization problem with sheet thicknesses and material parameters as design variables, the mass as objective function, and a number of performance measures as constraints. For feasible designs, the governing equations of the analyzed structure must be fulfilled. These can either be included as constraints in the optimization formulation, or be handled externally to the optimization procedure and then made sure to be fulfilled for every analyzed design point. In this thesis, the fulfilment of the governing equations is ensured by analyzers. An analyzer is for example an FE model or the corresponding metamodel. For a vector of design variables x, the analyzer fulfils the governing equations and returns a number of responses r, see Figure 2.1. These responses can be used to evaluate the objective and constraint functions for that specific vector of design variables, which is indicated in the reformulation of Equation (2.2) below. min 𝐱 subject to 8 𝑓(𝐫(𝐱)) 𝐠(𝐫(𝐱)) 𝟎 (2.3)
CHAPTER 2. OPTIMIZATION CONCEPTS x r Analyzer Figure 2.1 Illustration of an analyzer that fulfils the governing equations. 2.3 Optimization Algorithms Optimization problems can be solved using optimization algorithms consisting of iterative search processes. Various types of algorithms are available, and which type is suitable depends on the nature of the problem at hand. A way of classifying optimization algorithms refers to the kind of gradient information used. Zero-order algorithms only use objective and constraint function values, while first-order and second-order algorithms also use first-order and second-order gradients, respectively, and are collectively referred to as gradient-based algorithms. One type of zero-order optimization algorithms is stochastic, or heuristic, algorithms. These are based on a random generation of points used for local search procedures and are typically inspired by some phenomenon from nature. Stochastic algorithms have a good chance of finding the global optimum and are well suited for discrete optimization problems, but require evaluations of a large number of design points (Venter, 2010). Examples of stochastic algorithms are evolutionary algorithms, simulated annealing, and particle swarm optimization. More information concerning stochastic optimization algorithms is given in Section 4.4. Many gradient-based optimization algorithms use an iterative two-step procedure to reach a local optimum. Gradient information is first used to find a search direction, and a linesearch is then performed to determine the step size. For non-convex problems, there is no guarantee that a local optimum also is the global one. However, an improved estimate of the global optimum can be found if several search procedures with different starting points are performed. Gradient-based optimization algorithms generally require few design point evaluations to reach a local optimum, but both function and gradient evaluations are needed. Moreover, they can exhibit difficulties solving discrete optimization problems and also be susceptible to numerical noise (Venter, 2010). 9
CHAPTER 2. OPTIMIZATION CONCEPTS 10
Multidisciplinary Design Optimization Problems 3 Multidisciplinary design optimization is a formalized methodology used to perform optimization of a product considering several disciplines simultaneously. Giesing and Barthelemy (1998, p. 2) provide the following definition of MDO: “A methodology for the design of complex engineering systems and subsystems that coherently exploits the synergism of mutually interacting phenomena.” In general, a better design can be found when considering the interactions between different aspects of a product than when considering them as isolated entities, something which is taken advantage of when using MDO. An MDO problem can be expressed by the general optimization formulation in Equation (2.2). The problem becomes multidisciplinary if the design variables, objective function, and constraints affect different disciplines. Within one discipline, many different loadcases can be considered. A loadcase is a specific configuration that is evaluated using an analyzer, e.g. a simulation of a crash scenario using an FE model, and will here be denoted subspace. The MDO methodology can just as well be applied to different loadcases within one single discipline, and the problem is then not truly multidisciplinary. However, the idea of finding a better solution by taking advantage of the interactions between subspaces still remains. When solving large-scale MDO problems, some kind of problem decomposition is required. There are two main motivations for decomposing a problem according to Kodiyalam and Sobieszczanski-Sobieski (2001), namely concurrency and autonomy. Concurrency is achieved through distribution of the problem so that human and computational resources can work on the problem in parallel. Autonomy can be attained if individual groups responsible for certain parts of the problem are granted freedom to make their own design decisions and to govern methods and tools. In this chapter, different aspects related to problem decomposition are considered. Thereafter, the nature of automotive MDO problems is discussed with focus on structural applications. Several MDO methods were developed for aerospace applications. In order to assess whether or not these methods are suitable for automotive structural applications, this chapter also contains a comparison between automotive and aerospace MDO problems. 11
CHAPTER 3. MULTIDISCIPLINARY DESIGN OPTIMIZATION PROBLEMS 3.1 Problem Decomposition For single-level optimization methods, decomposition is achieved through distributing the analyses to subspace analyzers, enabling computational resources to be used in parallel. For multi-level optimization methods, the optimization process as such is distributed to subspace optimizers that communicate with a system optimizer, making it possible for individual groups to work on the problem concurrently and autonomously. 3.1.1 Terminology of Decomposed Systems The implications of problem decomposition and the associated terminology are presented in this section. Even if the decomposition is fundamentally different for single-level and multi-level optimization methods, a unified terminology can be used. Each subspace has a number of variables, indicated by the vector x j for subspace j. The union of the variables in all subspaces is the original set of design variables x. The variables in the different subspaces are in general not disjoint. Variables that are unique to a specific subspace are called local variables, denoted by the vector x lj for subspace j. The collection of local variables in all subspaces is termed x l . There are also a number of shared variables, and x sj indicates the vector of shared variables in subspace j, where each component is present in at least one other subspace. The union of shared variables in all subspaces is denoted by x s . An illustration of local and shared variables can be found in Figure 3.1. a) b) xs1 xl1 Subspace 1 x1 Subspace 3 x3 Subspace 2 x2 xl2 Subspace 1 xs3 Subspace 2 Subspace 3 xs2 xl3 Figure 3.1 Illustration of local and shared variables in three subspaces a) The variables x 1, x 2, and x 3 are not disjoint. b) The intersection of x s1 and x s2 are shared variables present in both subspace 1 and subspace 2, while the intersection of x s1, x s2, and x s3 are shared variables present in all three subspaces. 12
CHAPTER 3. MULTIDISCIPLINARY DESIGN OPTIMIZATION PROBLEMS When a problem is decomposed, it is necessary to handle the couplings between the resulting subspaces. Coupling variables are defined as output from one subspace needed as input to another subspace. The notion of coupled subspaces indicates the presence of coupling variables. The vector yij consists of output from subspace j that is input into subspace i. y*j denotes all coupling variables output from subspace j and yj* all coupling variables input to subspace j. The collection of all coupling variables is indicated by the vector y. The presence of coupling variables complicates the problem considerably. Consistency of coupling variables means that the input yij to subspace i is the same as the output yij from subspace j, and it is obtained using an iterative approach. This is referred to as multidisciplinary feasibility by Cramer et al. (1994), but since feasibility in an optimization context refers to the fulfilment of the constraints, the term multidisciplinary consistency is used here. Individual discipline consistency, also renamed from the definition by Cramer et al., refers to the situation when the governing equations of each subspace are fulfilled, but the coupling variables are not necessarily consistent. This term is used when defining the individual discipline feasible method in Section 5.1.2. Decomposition of a system into three subspaces is illustrated in Figure 3.2. The local, shared, and coupling variables are given as input to each subspace. The output from the subspaces is used by the optimizer for a single-level optimization method, or by the system optimizer for a multi-level optimization method, to solve and coordinate the MDO problem. xl1, xs1 xl3, xs3 xl2, xs2 y21 Subspace 1 y12 y32 Subspace 2 y23 Subspace 3 y31 y13 Figure 3.2 Decomposition of a system into three subspaces. The optimization problem in Equation (2.3) will now be reformulated using the terminology introduced. The objective function f is assumed to be computed from a combination of subspace data fj , where fj is a function of the responses rj from subspace j. Furthermore, the constraints are assumed to be separable, which means that each constraint belongs only to one subspace. The constraints g j are also functions of the 13
CHAPTER 3. MULTIDISCIPLINARY DESIGN OPTIMIZATION PROBLEMS responses rj from subspace j. These assumptions hold throughout this thesis. The resulting optimization formulation is then min 𝐱 𝑓(𝑓1 (𝐫1(𝐱 𝑙1 , 𝐱 𝑠1 , 𝐲1 )), 𝑓2 (𝐫2 (𝐱𝑙2 , 𝐱 𝑠2 , 𝐲2 )), , 𝑓𝑛 (𝐫𝑛 (𝐱 𝑙𝑛 , 𝐱 𝑠𝑛 , 𝐲𝑛 ))) subject to 𝐠 𝑗 𝐫𝑗 (𝐱 𝑙𝑗 , 𝐱 𝑠𝑗 , 𝐲𝑗 ) 𝟎, 𝑗 1,2, , 𝑛, (3.1) where n is the number of subspaces. 3.1.2 Aspect-Based and Object-Based Decomposition A system can be decomposed in different ways (Sobieszczanski-Sobieski and Haftka, 1987). Aspect-based decomposition refers to dividing the system into different disciplines, see Figure 3.3. The system will then naturally consist of two levels: one top level and one for all the disciplines. System Discipline 1 Discipline 3 Discipline 2 Figure 3.3 Illustration of aspect-based decomposition. Object-based decomposition means dividing the system into its constituent subsystems, which can, in turn, be divided into smaller subsystems or components, see Figure 3.4. A system decomposed by object can have an arbitrary number of levels. System Subsystem 1 Component 1.1 Component 1.2 Subsystem 2 Component 3.1 Figure 3.4 Illustration of object-based decomposition. 14 Subsystem 3 Component 3.2
CHAPTER 3. MULTIDISCIPLINARY DESIGN OPTIMIZATION PROBLEMS 3.1.3 Coupling Breadth and Coupling Strength Decomposing MDO problems can be more or less efficient. The terms coupling breadth and coupling strength can be employed to classify coupled MDO problems in order to gain an understanding of the effectiveness of decomposition (Agte et al., 2010). The coupling breadth is defined by the number of coupling variables and the coupling strength is a measure of how much a change in a coupling variable, output from one subspace, affects the subspace that it is input to. Coupling Strength For visualization purposes, the coupling breadth can be plotted against the coupling strength, see Figure 3.5. Agte et al. (2010) discuss how to look upon MDO problems in the four different corners of the graph. The discussion focuses specifically on the suitability of multi-level optimization methods. The existing methods are particularly suitable for problems in the upper left corner that have a strong but narrow coupling. Decomposition is least complicated in the lower left corner where the subspaces are weakly coupled. Problems in the lower right corner have many but weak couplings, some of which may be neglected in order to obtain an effective decomposition. In the upper right corner on the other hand, subspaces are so widely and strongly coupled that it may be preferable to merge them. Coupling Breadth Figure 3.5 Coupling breadth versus coupling strength. 3.2 Automotive Problems For automotive structural applications, a typical MDO problem de
multi-level optimization methods have a distributed optimization process. ollaborative C optimization and analytical target cascading are possible choices of multi-level optimization methods for automotive structures. They distribute the design process, but are complex. One approach to handle the computationally demanding simulation models
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2. Topology Optimization Method Based on Variable Density 2.1. Basic Theory There are three kinds of structure optimization, they are: size optimization, shape optimization and topology op-timization. Three optimization methods correspond to the three stages of the product design process, namely the
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Since the eld { also referred to as black-box optimization, gradient-free optimization, optimization without derivatives, simulation-based optimization and zeroth-order optimization { is now far too expansive for a single survey, we focus on methods for local optimization of continuous-valued, single-objective problems.
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