Modeling, Solving and Application for Topology Optimization of Continuum Structures: ICM Method Based on Step Function

By Yunkang Sui and Xirong Peng

Modelling, Solving and Applications for Topology Optimization of Continuum Structures: ICM Method Based on Step Function provides an introduction to the history of structural optimization, along with a summary of the existing state-of-the-art research on topology optimization of continuum structures. It systematically introduces basic concepts and principles of ICM method, also including modeling and solutions to complex engineering problems with different constraints and boundary conditions. The book features many numerical examples that are solved by the ICM method, helping researchers and engineers solve their own problems on topology optimization.

This valuable reference is ideal for researchers in structural optimization design, teachers and students in colleges and universities working, and majoring in, related engineering fields, and structural engineers.

• Offers a comprehensive discussion that includes both the mathematical basis and establishment of optimization models
• Centers on the application of ICM method in various situations with the introduction of easily coded software
• Provides illustrations of a large number of examples to facilitate the applications of ICM method across a variety of disciplines

• Language: English
• Publisher: Elsevier Science
• Release date: Aug 29, 2017
• ISBN: 9780128126561

Yunkang Sui

Professor, College of mechanical engineering and applied electronics technology in the Beijing University of Technology, Beijing, China. His research fields are structural-multidisciplinary optimization, computational mechanics and applied mathematical programming. One of his main contributions is the proposition of ICM (Independent Continuous and Mapping) Method for Topology Optimization of Continuum Structures e is member of ISSMO (International Society for Structural and Multidisciplinary Optimization), the vice chairman of Beijing society of mechanics and the deputy editor in chief of Journal Engineering Mechanics. He has presided over many projects supported by Natural Science Foundation of China and industrial fields. He has published more than 400 papers, 6 academic monographs and obtained more than 40 software copyrights. He won 4 science awards including the second-class national award in natural sciences of China and the third-class national science and technology progress award.

Book preview

Modeling, Solving and Application for Topology Optimization of Continuum Structures

ICM Method Based on Step Function

Yunkang Sui

Beijing University of Technology, Beijing, China

Xirong Peng

Hunan City University, Yiyang, China

Table of Contents

Cover image

Title page

Copyright

Dedication

Preface

Acknowledgment

Chapter 1. Exordium

Abstract

1.1 Research History on Structural Optimization Design

1.2 Research Progress in Topology Optimization of Continuum Structures

1.3 Concepts and Algorithms on Mathematical Programming

Chapter 2. Foundation of the ICM (independent, continuous and mapping) method

Abstract

2.1 Difficulties in Conventional Topology Optimization and Solution

2.2 Step Function and Hurdle Function—Bridge of Constructing Relationship Between Discrete Topology Variables and Element Performances

2.3 Fundamental Breakthrough—Polish Function Approaching to Step Function and Filter Function Approaching to Hurdle Function

2.4 ICM Method and its Application

2.5 Exploration of Performance of Polish Function and Filter Function

2.6 Exploration of Filter Function With High Precision

2.7 Breakthrough on Basic Conceptions in ICM Method

Chapter 3. Stress-constrained topology optimization for continuum structures

Abstract

3.1 ICM Method With Zero-Order Approximation Stresses and Solution of Model

3.2 Global Stress Constraints to Replace Stress Constraints

3.3 Topology Optimization of Continuum Structures With Strain Energy Constraints

3.4 Topology Optimization of Continuum Structures With Constraints of Distortional Strain Energy Density

3.5 Ill-Conditioned Loads and Their Solutions

3.6 Discussion on Stress Singularity

3.7 Examples

3.8 Summary

Chapter 4. Displacement-constrained topology optimization for continuum structures

Abstract

4.1 Explicit Approximation of Displacement Constraints

4.2 Establishment and Solution of Optimization Model With Displacement Constraints for Multiple Load Cases

4.3 ICM Method With Requirement of Discrete Topology Variables

4.4 Solutions for Checkerboard Patterns and Mesh-Dependent Problems

4.5 Examples

4.6 Summary

Chapter 5. Topology optimization for continuum structures with stress and displacement constraints

Abstract

5.1 Dimensionless Stress Constraints and Displacement Constraints

5.2 Establishment and Solution of Optimization Model With Stress Constraints and Displacement Constraints Under Multiple Load Cases

5.3 Examples

5.4 Summary

Chapter 6. Topology optimization for continuum structures with frequency constraints

Abstract

6.1 Explicit Approximation of Frequency Constraints

6.2 Establishment and Solution of Optimization Model With Frequency Constraints

6.3 Solutions for Checkerboard Patterns and Mesh Dependence Problems

6.4 Solutions for Localized Modes and Mode Switching Problems

6.5 Examples

6.6 Summary

Chapter 7. Topology optimization with displacement and frequency constraints for continuum structures

Abstract

7.1 Dimensionless Displacement and Frequency Constraints

7.2 Establishment and Solution of Optimization Model With Displacement and Frequency Constraints

7.3 Solutions for Numerical Unstable Problems

7.4 Examples

7.5 Summary

Chapter 8. Topology optimization for continuum structures under forced harmonic oscillation

Abstract

8.1 Sensitivity Analysis of Displacement Amplitude for Forced Harmonic Oscillation

8.2 Explicit Approximation of Displacement Amplitude Constraints

8.3 Establishment and Solution of Optimization Model With Displacement Amplitude Constraints for Forced Harmonic Oscillation

8.4 Examples

8.5 Summary

Chapter 9. Topology optimization with buckling constraints for continuum structures

Abstract

9.1 Basic Concepts for Buckling Analysis

9.2 Explicit Approximation of Buckling Constraints

9.3 Establishment and Solution of Topology Optimization Model of Continuum Structures With Buckling Constraints

9.4 Criterion of Selecting Upper Limit of Critical Buckling Force

9.5 Examples

9.6 Summary

Chapter 10. Other correlative methods

Abstract

10.1 Solid–Void Combined Element Method and Its Applications in Topology Optimization of Continuum Structures

10.2 Topology Optimization of Continuum Structures With Integration Constraints

10.3 Structural Topology Optimization With Parabolic Aggregation Function

10.4 Structural Topology Optimization With High-Quality Approximation of Step Function

10.5 Summary

References

Afterword

Index

Copyright

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Dedication

To our teacher

Academician Lingxi Qian’s soul

Preface

The structural optimization in this monograph refers to the optimal design problem of the artificial macrostructure bearing force loads. The topology is a more abstract mathematical concept than the geometry. Associated with the structural optimization, it represents a higher level of optimization than the structural cross-sectional or size optimization, shape or geometry optimization. Although the structural topology optimization undergoes the development from the traditional to the modern, it is essentially the decision problem of the solid or void. But the cross-sectional or size, shape, or geometry optimizations are the decision problems of the more or less.

The traditional structural topology optimization was put forward by Maxwell as early as the end of the 19th century, and was further researched by Michell in the early 20th century. The modern structural topology optimization was started in 1988. Bendsøe and Kikuchi were enlightened from the minimizing compliance optimization of the elastic thin plates carried out by Gengdong Cheng and Olhoff, and put forward the homogenization method. Therefore, it has the concept and solution method of the topology optimization of continuum structures. It is exciting that the optimal result of the topology optimization of continuum structures is the skeleton structures (truss and frame structures).

As the optimization problem of the solid or void, the traditional structural topology optimization studies skeleton structures to determine the presence or absence of nodes and nodal connections, which expresses the existence of the component. However, the modern structural topology optimization studies the solid or void of subregions in the space of continuum structures. Because the solid or void can be represented with 1 or 0, respectively, the structural topology optimization belongs to the mathematical programming based on the 0–1 discrete variables. It can be seen that it is the high-level structural optimization problem and is difficult to solve.

The structural optimization can be divided into three levels from low to high, according to the development levels. They can be described from engineering and mathematics, respectively:

The discipline on the structural optimization was developed in the early 1960s. Researches in the direction of the structural topology optimization were temporarily shelved and gave way to lower levels of structural cross-sectional optimization and shape optimization. In 1988, the direction of the structural topology optimization of continuum structures occurred; it was mentioned again and presented a more and hotter research trend.

Why has the structural topology optimization become a research hot topic? Many people think that the research direction has potentially immense economic benefits in the practice. Once structural topology optimization can be put into application, more structural materials can be saved than the structural cross-sectional and shape optimization. Despite this idea making sense, it should be considered deeply from the internal impetus of the mechanics development. The structural topology considered in the mathematics is the structural layout considered in the engineering. What is it corresponding to in the mechanics? We can answer this question from the viewpoint of the structure-bearing loads. The optimal topology structure or the optimal layout structure actually gives the corresponding optimal path of transferring loads. Or to put it more broadly, it gives the optimal path of the structure transferring loads or bearing mechanics responses; hereinafter, it is referred to as the optimal path of transferring loads and bearing responses.

Inevitably, for practical engineering components or structures, from the initial design to the final construction drawing detail design, the dominant path of transferring loads and bearing responses needs to be determined. A design is reasonable or not, which relates to a series of optimal indexes, such as whether it is economic, is safe, is high quality, and so on. And the basic method to solve the problem is the structural topology optimization method. This is why that scholars and engineers attach great importance to the research on structural topology optimization.

To sum up, the high-level problem, corresponding to optimal topology in mathematics and optimal layout in engineering, studies the optimal path of transferring loads and bearing responses in the mechanics. Accordingly, the low-level problem represents the size of the local path of transferring loads and bearing responses in the mechanics. The middle level problem increases the path’s bending degree or curvature change on the basis of the determined path of transferring loads and bearing responses. The high-level problem is the structural topology optimization problem, which adds challenging consideration in the mechanics: to design the path of transferring loads and bearing responses, to answer layout problem of the paths, namely, to solve the topology configuration in the mathematics.

The reason why the structural topology optimization can strongly attract the interest of researchers also lies in the explored space of the path of transferring loads and bearing responses that are expanded when the topology optimization extends from skeleton structures to continuum structures. Genius intuition of engineers in the configuration selection of skeleton structures gives place to the rational calculation of the solid or void of subregions of continuum structures for the first time.

Since 1988, more than 28 years have passed, the topology optimization of a continuum structures appears in a lot of methods, and encouraging gratifying progresses have been achieved. One of the important reasons is that people are not constrained by the algorithm of mathematical programming with discrete variables but turn to solve an approximated mathematical programming problem with continuous variables while they face structural topological optimization. Thus the differentiable algorithm with high efficiency can be applied appropriately. The detail strategy is to put topology variables to attach to or affiliate at low levels of variables of the structural optimization. The topology optimization of a skeleton and continuum is converted into the generalized cross-sectional or performance and generalized shape optimization problem.

Although these strategies are very well thought out and achieve fruitful results, there are also many aspects that need to be improved and developed:

1. Because structural topology optimization is converted into the generalized cross-sectional and generalized shape optimization problems, topology variables are not independent and the potential of its own independent level of optimization algorithm cannot be explored. Therefore, the optimization calculation efficiency needs to be improved further.

2. The existing research mainly focuses on the problem with structural compliance as the objective function under global constraints, for example, volume constraint, structural natural frequency constraint, etc. But for engineering structures, the stress, displacement, and vibration constraints all are very important, the topology optimization design regardless of those constraints is difficult to put into use in engineering practice.

3. The topology optimization of skeleton structures and that of continuum structures adopt different objective functions. The former mainly takes the minimizing structural weight as the objective function. The latter mainly takes the minimizing structural compliance as the objective function. The optimization models of both are not unified. Therefore, it is difficult to extend the effective model and algorithm of the topology optimization of skeleton structures to the topology optimization of continuum structures.

4. The structural compliance must correspond to a certain load case. If a problem with multiple load cases is dealt with, artificial conditions need to be inserted. It reduces the strict solution of the optimization problem to the trade-off solution involving many assumptions.

5. When a topology optimization model is established by the homogenization method, the variable density method or other methods, it is hard to find the approximated explicit functional relationship between the structural performance such as displacement and topological design variables. Even if the relationship is established, the optimization model is difficult to be solved by the conventional mathematical programming method because of too many topological design variables of the model.

To make progress at the above various aspects, the first author in the monograph put forward the concept of the independent continuous topological variable and the independent, continuous and mapping (ICM) method in 1996. Topology variables are abstracted from attached variables of the size or shape levels of optimization. Variables being independent of the specific element quantities are adopted to represent the solid or void of elements. It brings convenience for the establishment of the model. At the same time, the concept of the polish function and filter function is introduced. The 0–1 topology variable is approached by using the polish function. The discrete 0–1 independent topology variable is mapped to the continuous variable in the interval [0, 1]. The smooth mathematical model of the topology optimization problem is established. The solution efficiency is improved. At last, the continuous topology optimization model is solved by the appropriate differentiable optimization algorithm. The obtained optimal design variables in the interval [0, 1] are mapped back to the optimal discrete design variables.

The ICM method is suitable for any objective function; it can also solve the structural topology optimization problem with weight objective and under multiple load cases. Topology optimization models of skeleton structures and continuum structures are unified. The difficulty of dealing with problems under multiple load cases while structural compliance is taken as the objective function is overcome. The number of constraints is reduced, the scale of the solution model is decreased, and the computational efficiency is improved. In general, the continuous refers to the topology variable being continuous. The mapping contains three meanings:

1. The mapping between the discrete topology variable and continuous topology variable is established by the filter function to harmonize the contradiction between the independent and the continuous.

2. The mapping between the original model and the dual model is used while the optimization model is solved.

3. The reverse mapping from the continuum model to the discrete model is carried on after the optimization model is solved, and it is called as the inversion.

The ICM method has properties of simplicity and rationality. At the same time, it also has a mathematical explanation.

The ICM method can take the minimizing weight of structures as the objective function, which is standardized uniformly with the size, shape, and topology optimization. The ICM method not only can solve effectively the topology optimization problem of continuum structures with constraints such as the stress, displacement, stability, and vibration; it is also more suitable for practical engineering applications. It additionally unifies topology optimization models of skeleton structures and continuum structures. In particular, for the optimization problem with multiple load cases, because constraints under multiple load cases are included in the same optimization model, the trouble of combining paths of transferring loads and bearing responses under each single load case does not occur, and the optimal paths of transferring loads and bearing responses is sought rationally. The number of design variables is reduced by the introduction of the dual programming method. The optimization efficiency is improved. The number of iterations is reduced. In addition, the stress globalization method greatly reduces the calculation amount of the sensitivity analysis.

The ICM method has been studied since 1996. It is 8 years since Bendsøe and Kikuchi engaged in the study of the topology optimization of continuum structures. But our explorations follow the research direction created by the academician Lingxi Qian and we absorb academic nutrition of domestic and overseas counterparts. Twenty years is like a day, many researchers are trained in this research direction: two postdoctoral researchers, Jun Tie and Zhen Shang; nine PhD students, Deqing Yang, Xin Yu, Hongling Ye, Jiazheng Du, Xirong Peng, Xuesheng Zhang, Binchuang Bian, Donghai Xuan, Guilian Yi; and nine graduate students, Xuchun Ren, Zhichao Jia, Jianxin Liu, Shi Chen, Run Zhu, Hai Qiu, Xiaodi Liu, Jingxian Shen, and Junjie Li. Two monographs and a large number of papers have been published, and 48 counts of software copyright approved. Researches for many years are integrated in the monograph.

Chapter 1, Exordium, introduces the basic concept of the structural optimization, the progress and the basic theory of the topology optimization of continuum structures, and related mathematical programming contents used in later chapters. Chapter 2, Foundation of the ICM (Independent, Continuous and Mapping) Method, describes the theoretical basis of the ICM method. Chapter 3, Stress-constrained Topology Optimization for Continuum Structures, describes the topology optimization method of continuum structures with stress constraints. Chapter 4, Displacement-constrained Topology Optimization for Continuum Structures, describes the topology optimization method of continuum structures with displacement constraints. Chapter 5, Topology Optimization for Continuum Structures With Stress and Displacement Constraints, describes the topology optimization method of continuum structures with stress and displacement constraints. Chapter 6, Topology Optimization for Continuum Structures With Frequency Constraints, describes the topology optimization method of continuum structures with frequency constraints. Chapter 7, Topology Optimization With Displacement and Frequency Constraints for Continuum Structures, describes the topology optimization method of continuum structures with displacement and frequency constraints. Chapter 8, Topology Optimization for Continuum Structures Under Forced Harmonic Oscillation, describes the topology optimization method of continuum structures with forced harmonic oscillation constraints. Chapter 9, Topology Optimization With Buckling Constraints for Continuum Structures, describes the topology optimization method of continuum structures with buckling constraints. Chapter 10, Other Correlative Methods, introduces the solid–void combined element method and corresponding modeling method, the optimization method with the integration of constraints, the structural topology optimization method with the parabolic aggregation function, and the structural topology optimization method with the high-quality approximation of the step function.

Meditating after the book was finished, there are a few thoughts:

1. To achieve the tremendous development of the structure topology optimization, the development history needs to be reviewed and natures of all kinds of influential methods need to be insight into. It is more important that advantages need to be absorbed for further application and improvement and disadvantages must be avoided.

2. It should be begun from the root of the concept, break through the regular thinking tendency based on the basis, and investigate the root of the problem. To explore new ideas, it must be based on the basic concept to break through traditional ideas.

3. The structural topology optimization is different from the size and shape optimization. It is difficult to be solved because the decision of the more or less in the size and shape optimization is replaced with the decision of the solid or void in the structural topology optimization. It is bound to lead to the solution’s difficulty.

When the monograph is available, we sincerely thank the following organizations, enterprises and individuals.

1. National Science Foundation Committee of China. Our projects supported by it had: Curve approximated optimization theory of structural optimization design and solution method of ordinary differential equations (19172012). Modeling based on cumulative iteration information and optimal structure synthesizes theory and method (19472014). Unified mapping modeling and optimization of topology design for skeleton structures and continuum structures (10072005). Optimization of structure acquired bearing capacity (10472003). Damage minimization of car clash (10872012). Study on dynamic topology optimization of continuum structures based on independent topology variables (11072009). Structural topology optimization method with high precision approaching step function (11172013). Improvement and expansion of variable density method by merging ICM method on topological optimization of continuum structures (11672103).

2. The Ministry of Education of China. It supported a project by the specialized research fund for the doctoral program, which is Real-time implementation of structural optimal state under variable loads (63001015200701).

3. The Natural Science Foundation of Beijing. Our projects supported by it had: Practical modeling and solution technology for large-scale engineering structural optimization (3002002). Topology optimization method of rational path transferring forces for engineering structure (3042002). Structural optimization with fatigue constraints for pulsation vacuum sterilizer (3093019).

4. The Educational Committee of Beijing. Our projects supported by it had: Optimization design of complicated practical structures (05001015199904). Optimization and intelligent control implementation of structure acquired state (KM20 0410005019).

5. Relevant industry sectors of China. They have cooperated successfully with us.

6. American MSC Company, one of the international famous computational mechanics software companies. It provided cooperation and supports for the secondary development of the structural optimization algorithm on its software development platform.

7. The Beijing University of Technology. It provided hardware and software supports for the college of mechanical and electrical engineering, which offer jobs for first authors of the monograph. It made the numerical simulation of the structural and multidisciplinary optimization to be one of the main directions, which has become one of the research characteristic of engineering mechanics subjects. It funds the publication of the monograph.

8. Two postdoctoral researchers, nine PhD students, and nine postgraduate students mentioned above. They participated in the related researches of the ICM method and pay their hard work.

9. We would like to express special thanks to Dr. Xuchun Ren, Assistant Professor in Mechanical Engineering, Department of Georgia Southern University, [e-mail]. His research focuses on structure and multidisciplinary optimization including robust design and reliability-based design optimization. As an editor of this book, he made tremendous efforts to proofread this book and polish the sentences.

If there is no support from government projects and industry sector projects, no support from the international famous company on the computational mechanics software, no young students to participate in the research, our research is difficult to continue today. Therefore, as a gift filled with authors’ gratitude, we dedicate the monograph to the above organizations, enterprises, and individuals before it is offered to readers.

Yunkang Sui¹ and Xirong Peng², ¹Beijing University of Technology, Beijing, China, ²Hunan City University, Yiyang, Hunan, China

Written by author in September 10, 2016.

Acknowledgment

Thanks to Dr. Xuchun Ren for his tremendous efforts to proofread this book and polish the sentences.

Chapter 1

Exordium

Abstract

This chapter discusses the concept of structural optimization, its general definition, developments and methodologies, and relevant mathematical theories involved within the current research.

The text offers information on current research and ways in which reliable and efficient methods of design improvement of structure have gradually become an important branch of mechanics. Included in the chapter is an explanation of the structural optimization model and its classification into models of continuous variables, discrete variables, and continuous and discrete mixed variables. The author defines the three levels of structural optimization: size, shape, and topology.

The history of structural optimization is traced from the 1890s through present day development. Progress in topology optimization of continuum structures is detailed; other methods and solution strategies are also mentioned, including the evolutionary structural optimization (ESO) method, and the bi-directional evolutionary structural optimization (BESO) method.

Keywords

Structural optimization; optimization algorithms; evolutionary structural optimization; optimal criteria method; Lagrange multiplier; continuum structures

Chapter Outline

1.1 Research History on Structural Optimization Design 3

1.1.1 Classification and Hierarchy for Structural Optimization Design 3

1.1.2 Development of Structural Optimization 5

1.2 Research Progress in Topology Optimization of Continuum Structures 13

1.2.1 Numerical Methods Solving Problems of Topology Optimization of Continuum Structures 13

1.2.2 Solution Algorithms for Topology Optimization of Continuum Structures 21

1.3 Concepts and Algorithms on Mathematical Programming 22

1.3.1 Three Essential Factors of Structural Optimization Design 22

1.3.2 Models for Mathematical Programming 24

1.3.3 Linear Programming 26

1.3.4 Quadratic Programming 28

1.3.5 Kuhn–Tucker Conditions and Duality Theory 29

1.3.6 K-S Function Method 32

1.3.7 Theory of Generalized Geometric Programming 33

1.3.8 Higher Order Expansion Under Function Transformations and Monomial Higher Order Condensation Formula 35

With the development of science, technology and social productivity, human activities expand continuously into the space and ocean, and the research scope of the structural optimization extensively expands. Structural optimization design is becoming more and more important due to limited resources, intense engineering technological competitions, and environmental protection problems. Higher operating requirements are demanded for components of various high-precision and advanced devices. Designing structures and components to satisfy various constraints, therefore, provides both new opportunities and new challenges to structural engineers and mechanics researchers. On the other hand, the real-world simulations coupled in several physical fields are inevitably involved in structural and multidisciplinary optimization, greatly expanding the scope of structural optimization design.

Structural optimization aims at producing a safe and economic structural design subject to various load cases and structural materials. To obtain optimal design, not only mechanical properties such as strength, stiffness, stability, dynamic, and fatigue should be taken into account but also requirements of the application and operation such as manufacturing processes, construction conditions, and limits in the specifications of construction, manufacturing, and design should be satisfied. All requirements, conditions, and limits are expressed as constraints, whereas the economic index or a mechanical property is taken as the objective function. Design parameters, including design details such as the structure type and sizes, are taken as design variables. Henceforth, the optimization expression of a structural design is formed, and the mathematical model of the optimization can be further established. Finally, the optimization model is solved by optimization algorithms, and the optimal structure to satisfy the objective pursued by the user can be achieved automatically.

Structural optimization design is a synthetic subject involving computational mechanics, mathematical programming, computer science, and other engineering disciplines. It is highly comprehensive in theory and highly practical in method and technology; thus it is one of the important developments of the modern design method. Currently, applications of structural optimization design involve many fields, including aviation, aerospace, machinery, civil engineering, water conservancy, bridge, automobile, railway transportation, ships, warships, light industry, textile, energy, and military industry, to name just some. Engineering design problems should be solved properly, simultaneously pursuing better cost indicator of structure, the improvement of structure performances and enhancement on safety. Nonetheless, structural optimization design should meet the needs of the industrial production based on the accumulation of design experiences. Again, belonging to one of the synthesized and decision-making subjects, structural optimization design is founded on mathematical theory, method, and computer programming technology as well as its modeling technique.

In the 1960s, Schmit put forward the comprehensive design for structures by the mathematical programming. This marks the beginning of the structural optimization as an independent discipline. Hereafter, theory, method, and software of structural optimization design grew steadily. Over 50 years, the structural optimization has developed from the size optimization (or the so-called cross-section optimization in the initial stage), to the shape (or node) optimization, further to the topology optimization of skeletal structures, to the shape optimization and topology optimization of continuum structures. With a relative completion theoretical system formed and a great number of practical problems solved, huge economic and social benefits are created. However, the topology optimization design of continuum structures is still one of the hot spots due to emerging challenges from the lasting development and requirements of modern industry.

The authors believe that in the research of structural optimization design, engineering intuition and mechanical concepts should be closely combined with mathematical deduction; the analytic expression should be contrasted with geometrical intuition, which should be converted to an idea; and the conclusion of the low-dimensional space is sublimated to the high-dimensional space for rigorous developments of the theory in the structural optimization. Comprehensive, systematic researches on theory and numerical aspects should be carried out for the topology optimization of continuum structures. It is very important to grasp the key point, to hold the characteristic of the problem, and to analyze the essence through the phenomena during the researches.

In this chapter, the development history, the basic conception, and the classification of structural optimization are firstly summarized. Developments and methods of the topology optimization of continuum structures are then introduced. Finally, relevant mathematical theories involved with the research progresses in this monograph are presented.

1.1 Research History on Structural Optimization Design

1.1.1 Classification and Hierarchy for Structural Optimization Design

Structural optimization optimizes the structural design. Since the 1960s, with the rapid development of computer technology and the finite element method, researches on how to provide a reliable and efficient method to improve the design of the structures for engineers have gradually become an important branch of mechanics. According to the feature of design variables, the structural optimization model can be classified into the model with continuous variables, the model with discrete variables, and the model with continuous and discrete mixed variables. According to the scope of the structural design variables, structural optimization design in general is divided into three levels (Fig. 1.1): size optimization, shape optimization, and topology optimization. These correspond to the detail design, basic design, and conceptual design phases of the product design, respectively.

Figure 1.1 Levels of structural optimization.

Size optimization optimizes the sizes of components on the basis of specifying the structure type, topology, and shape. Its design variables can be the cross-sectional area of a rod, the thickness of a membrane or plate, a set of design parameters of a beam cross-section (such as the sizes of cross-section