连续体结构拓扑优化的建模、求解和应用——基于阶跃函数的ICM方法

构优化学科发展于 20世纪60年代初。结构拓扑优化方向的研究暂时被束之高阁，让位于低层次的结构截面、形状优化研究，直到 1988年连续体结构拓扑优化方向的出现，它才被重新提及，并且呈现了越来越热的研究趋势。 

连续体结构拓扑优化的建模、求解和应用，能够使航空、航天、汽车、船舶、土木、水利、交通和轻工业等领域的研究者和工程师受益，去解决各自的结构拓扑优化问题，在初步设计阶段就能够找到经济与安全的平衡点。
本书以作者多年的研究工作为基础，系统地介绍连续体结构拓扑优化 ICM（独立、连续和映射）方法的基本概念和原理，为具有不同约束条件和边界条件的复杂工程问题提供了建立模型和求解的途径。本书还对结构拓扑优化*前沿的研究领域提供 ICM方法的应用，以及将其应用到计算固体力学的范围。
本书可供上述工程领域的优化研究、设计人员及高等院校师生参考。

前言

致谢

第1章
绪论1

1.1结构优化设计研究的历史发展 3

1.1.1结构优化设计的分类和层次 3

1.1.2结构优化的发展 5

1.2连续体结构拓扑优化的研究进展 13

1.2.1连续体结构拓扑优化的数值方法 13

1.2.2连续体结构拓扑优化的求解方法 21

1.3数学规划概念与算法 22

1.3.1结构优化设计的三要素 22

1.3.2数学规划模型 24

1.3.3 线性规划 26

1.3.4 二次规划 28

1.3.5 库恩 -塔克条件和对偶理论29

1.3.6 K-S函数方法 32

1.3.7广义几何规划理论 33

1.3.8函数变换下的函数高阶展式和单项高阶缩并公式
35

第2章 ICM（独立、连续、映射）方法的基础37

2.1   传统拓扑优化问题的困难
39

2.2   阶跃函数和跨栏函数——构造离散拓扑变量与单元性能关系的桥

 

梁 41

2.3   根本性突破的关键——磨光函数逼近阶跃函数和过滤函数逼近跨

 

栏函数 43

2.3.1 磨光函数 44

2.3.2 过滤函数 45

2.3.3过滤函数使拓扑优化的求解可以操作 46

2.3.4四个函数的相互关系 46

2.4 ICM方法及其应用 47

2.4.1映射识别与全程识别量 47

2.4.2几种典型的磨光函数和过滤函数 49

2.4.3各种性能识别的快慢和参数确定 52

2.4.4过滤函数中指数函数参数到对数函数参数的变换
60

2.4.5 基于 ICM方法建立结构拓扑优化模型
63

2.4.6映射的反演 64

2.5对磨光函数和过滤函数性能的深入探讨 66

2.5.1划分磨光函数（和过滤函数）的类型 66

2.5.2类型判别定理 67

2.5.3磨光函数与过滤函数的对应定理 67

2.6对高精度过滤函数的深入探讨 69

2.6.1高精度过滤函数的使用准则 69

2.6.2根据高精度左磨函数构造快滤函数的方法 70

2.6.3指数类快滤函数的参数选取 74

2.7 ICM方法在基本概念上的突破 76

第3章
连续体应力约束下的拓扑优化 79

3.1 基于 ICM方法的应力零阶近似处理和模型求解82

3.1.1应力约束零阶近似的连续体结构拓扑优化模型
82

3.1.2应力零阶近似策略下连续体结构拓扑优化求解
82

3.1.3求解算法中的其他策略 84

3.1.4 数值算例 87

3.2替代应力约束的全局化应力约束 89

3.2.1应力约束全局化策略 89

3.2.2应变能约束修正系数的引入 93

3.3.3利用昀小二乘法求修正系数 94

3.2.4数值模拟确定修正系数 95

3.2.5许用应力对连续体拓扑结构的影响 96

3.2.6多工况下应变能约束的修正系数 100

3.2.7结构许用应变能的求得 100

3.3结构应变能约束下的连续体结构拓扑优化
104

3.4结构畸变比能约束下的连续体结构拓扑优化
108

3.4.1         应力约束转化为结构畸变比能约束的全局化策略及其修正 108

3.4.2         修正结构全局畸变比能约束下的连续体结构拓扑优化 ICM模型 

3.5载荷病态问题及其解决 112

3.5.1三类载荷病态现象  113

3.5.2以结构应变能作为权系数处理载荷  114

3.5.3工况间有载荷病态但工况内无载荷病态  116

3.5.4仅在工况内有载荷病态  116

3.5.5工况间有载荷病态同时某工况内亦有载荷病态
118

3.6应力奇异性的探讨 119

3.7数值算例  119

3.7.1 算例 1  119

3.7.2 算例 2 121

3.7.3 算例 3 126

3.7.4 算例 4 128

3.7.5 算例 5 130

3.7.6 算例 6：工程应用——赵州桥的拓扑优化133

3.8本章小结 136

第4章
连续体位移约束拓扑优化 139

4.1位移约束的显式化 141

4.1.1位移敏度分析的直接法 141

4.1.2位移敏度分析的伴随法 142

4.1.3利用一阶泰勒近似将位移约束显式化 145

4.1.4利用莫尔定理将位移约束显式化 145

4.1.5两种显式化方式的一致性 146

4.2多工况位移约束下优化模型的建立与求解
147

4.3考虑离散性条件目标的 ICM方法 150 

4.4解决棋盘格式及网格依赖 151

4.4.1棋盘格现象及网格依赖性 151

4.4.2使用过滤方法消除棋盘格现象及解决网格依赖性
154

4.5数值算例 156

4.5.1 算例 1 156

4.5.2 算例 2 159

4.5.3 算例 3 159

4.5.4 算例 4 163

4.5.5 算例 5 165

4.6本章小结 169

第5章
连续体应力与位移约束拓扑优化 171

5.1应力约束和位移约束的无量纲化 172

5.2多工况连续体应力与位移约束拓扑优化模型的建立与求解 174

5.3数值算例 179

5.3.1 算例 1 179

5.3.2 算例 2 183

5.3.3 算例 3 187

5.3.4 算例 4 191

5.4本章小结 196

第6章
连续体频率约束拓扑优化 199

6.1频率约束的近似显式化 200

6.2频率约束下连续体拓扑优化模型的求解 203

6.3解决棋盘格式及网格依赖 204

6.4局部模态及模态交换问题 204

6.4.1局部模态问题 204

6.4.2解决局部模态问题 206

6.4.3模态交换问题 207

6.4.4解决模态交换问题 209

6.5数值算例 210

6.5.1 算例 1 210

6.5.2 算例 2 213

6.5.3 算例 3 215

目录

6.5.4 算例 4 216

6.6本章小结 221

第7章
连续体位移与频率约束拓扑优化 223

7.1频率与位移约束的无量纲化 224

7.2位移与频率约束连续体拓扑优化模型的建立与求解 226

7.3数值不稳定问题的求解策略 227

7.3.1解决棋盘格式及网格依赖 227

7.3.2局部模态及模态交换等问题的处理 227

7.4数值算例 228

7.4.1 算例 1 228

7.4.2 算例 2 229

7.4.3 算例 3 233

7.5本章小结 236

第8章
连续体强迫谐振动下拓扑优化 237

8.1强迫谐振动下位移幅值灵敏度分析 238

8.1.1强迫谐振动下位移幅值敏度分析常见方法
238

8.1.2无阻尼强迫谐振动下位移幅值敏度分析 239

8.1.3有阻尼强迫谐振动下位移幅值敏度分析 242

8.1.4矩阵导数的计算 245

8.1.5 数值算例 246

8.2位移幅值约束的近似显式化 251

8.3强迫谐振动下位移幅值约束拓扑优化的建模与求解 254

8.4数值算例 255

8.4.1 算例 1 255

8.4.2 算例 2 255

8.5本章小结 262

第9章
连续体屈曲约束拓扑优化 263

9.1屈曲分析的基本概念 265

9.2屈曲约束的显式化 267

9.3连续体屈曲约束拓扑优化建模与求解 270

9.4屈曲临界力上限的选取准则 270

9.4.1一阶临界力上限与拓扑结构重量的关系 271

9.4.2二阶临界力上限与拓扑结构重量的关系 275

9.4.3三阶临界力上限与拓扑结构重量的关系 277

9.5数值算例 281

9.5.1 算例 1 281

9.5.2 算例 2 286

9.5.3 算例 3 288

9.5.4 算例 4 290

9.6本章小结 294

第10章           其他相关的方法 295

10.1有无复合体方法及其对于连续体拓扑优化的应用
296

10.1.1       有无复合体平面膜单元
297

10.1.2       有无复合体的许用应力
298

10.1.3       有无复合体平面膜单元对位移的贡献 299

10.1.4       有无复合体平面膜结构在应力与位移约束下的拓扑优化300

10.1.5      
数值算例 301

10.2约束集成化的连续体结构拓扑优化 304

10.2.1       应力约束集成化建模与求解
304

10.2.2       位移约束集成化建模与求解
311

10.2.3       应力和位移约束集成化建模与求解 316

10.3抛物型凝聚函数的结构拓扑优化方法 319

10.3.1       抛物型凝聚函数 319

10.3.2       抛物型凝聚函数对于约束的集成化 322

10.4阶跃函数高精度逼近的结构拓扑优化方法
327

10.5本章小结 333

参考文献335

跋355

索引363

本专著涉及的结构优化（ Structural Optimization）是指承受力载荷的宏观人造结构的优化设计问题。拓扑（ Topology）作为比几何（ Geometry）更抽象的数学概念，同结构优化联系在一起，表征了高于结构截面或尺寸、形状或几何层次的优化，尽管结构拓扑优化经历了从传统到现代的发展，本质上都是有与无的决策问题，而截面或尺寸、几何或形状层次的优化却是多与少的决策问题。如果说传统的结构拓扑优化早在19世纪末就被 Maxwell提出，又在 20世纪初由 Michell做进一步研究，那么，现代的结构拓扑优化应该说是在 1988年才开始的， Bends.e和 Kikuchi从程耿东和 Olhoff对于昀小柔顺性实心弹性薄板优化的研究中得到启迪，提出了均匀化（ Homogenization）方法，于是开始有了连续体拓扑优化的概念和求解途径。令人振奋的是，连续体拓扑优化的昀终结果趋于骨架类结构（桁架、框架结构）! 作为有/无优化问题，传统的结构拓扑优化研究骨架类结构，确定节点和节点联结，从而表达了构件的存在与否。然而，现代的结构拓扑优化研究连续体结构所在空间的子域的有或无。由于有和无分别用 1和0表示，结构拓扑优化属于0-1型离散变量数学规划，可见它是难于求解的高层次结构优化问题。结构优化从低到高的发展可以分成三个层次，皆可从工程和数学上分别进行描述：视角 Viewpoint  低层次 Low level  中层次 Middle level  高层次 High level  工程观点 Engineering viewpoint  截面 Cross Section  形状 Shape  布局 Layout  数学观点 Mathematical viewpoint  尺寸 Size  几何 Geometry 拓扑 Topology  
结构优化学科发展于 20世纪60年代初。结构拓扑优化方向的研究暂时被束之高阁，让位于低层次的结构截面、形状优化研究，直到 1988年连续体结构拓扑优化方向的出现，它才被重新提及，并且呈现了越来越热的研究趋势。为什么结构拓扑优化研究方向能够成为热点？不少人认为该方向有着实用上潜在的极大经济效益，结构拓扑优化一旦能付诸应用，将比结构的截面和形状优化节省更多的结构用材。尽管这一想法很有道理，还是应当从力学学科内在的发展动力去进行深入的思考：数学上考虑的结构拓扑是工程上考虑的结构布局，在力学上对应着什么？我们可以从结构承受力载荷作用的视点来回答这一问题，昀优拓扑结构或昀优结构布局实际上给出了对应的昀佳传力路径，或者说得更宽泛一点：给出了结构传递力学载荷或承受力学响应的昀佳通路，以下简称为“传承载响的昀佳通路”。不可避免地，对于实际工程构件或结构，从提出昀初的设计方案到昀终的施工图设计，都要确定一个基本的传承载响的主导性通路，其设计是否合理，关系到是否经济、是否安全、是否优质等一系列昀优的指标，而解决问题的根本手段则是采用结构拓扑优化方法，这也就是为什么学者和工程师那样重视结构拓扑优化研究的原因。综上所述，对应于数学上的昀优拓扑和工程上的昀优布局这一高层次问题，力学上则是研究传承力学载响的昀佳通路。相应地，低层次问题，在力学上表示传承载响的各个局部通路的大小；中层次问题，对于已经确定的传承载响通路，增加了其弯曲程度或曲率变化大小的考虑；高层次问题即结构拓扑优化问题，在力学上又增加了具有挑战性的思考——要设计传承载响的通路，回答通路的布置问题，亦即解决数学拓扑的构成。结构拓扑优化能够具有强烈吸引研究者兴趣的魅力，还在于拓扑优化由骨架类结构向连续体结构的开拓，扩大了传承载响昀佳通路的探索空间，工程师在骨架类结构构型选取的天才直觉，次让位于连续体子域有无的理性计算！从1988年至今， 28年过去了，连续体结构拓扑优化的研究出现了精彩纷呈的大量方法，有了令人鼓舞的可喜进展。其中一个重要的原因在于，面对结构拓扑优化人们没有受限于离散变量数学规划的算法，而是转而求解一个近似的连续变量的数学规划问题，从而得心应手地运用求解效率高的可微性算法。具体策略是，把拓扑变量挂靠和依附在结构低层次变量上，骨架和连续体拓扑优化分别转化为广义截面或性能和广义形状优化问题。尽管这些都是非常灵巧的做法，并且确实取得了丰硕的成果，但是还有许多尚待改进和发展之处： 1．由于结构拓扑优化处理为广义截面与广义形状优化问题，拓扑变量不再独立，未能发掘出其本身独立优化层次算法的潜能，因此优化计算的效率有待进一步提高。 
2．己有主要研究在整体约束下例如体积约束、结构自振频率约束等，以结构柔顺性（ Compliance）作为目标函效，然而实际结构中应力、位移、振动约束是非常重要的，不考虑它们的设计是难以使拓扑优化付诸工程实际使用的。 
3．骨架结构与连续体结构的拓扑优化采用了不同的目标函数，前者多以结构重量极小化为目标，后者多以结构柔顺性昀小为目标，二者优化模型基本上不统一，因此很难将骨架结构拓扑优化有效的模型及解法推广到连续体结构拓扑优化中。 
4．柔顺性对应着某一工况，若用之处理多工况问题则要加进许多人为的条件，使优化问题的严格求解降低为含有许多假定的权衡解答。 
5．当利用均匀化方法、变密度法或其他方法建立拓扑优化模型时，很难找到位移约束与拓扑设计变量之间的近似显函数关系。即使建立了这一关系，也由于该模型的拓扑设计变量过多，利用常规的数学规划方法难以求解。

为了在以上各点取得进展，本书作者在 1996年提出了“独立连续”拓扑变量的概念和“独立连续映射”方法（ Independent Continuous and Mapping Method)，简称 ICM方法。将拓扑变量从依附截面优化层次或形状层次优化的变量中抽取出来，以独立于单元具体特征参数的变量来表征单元的“有”与“无”，为模型的建立带来方便；同时引入磨光函数和过滤函数的概念，利用磨光函数逼近实际的 0、1拓扑变量，将离散的 0-1独立拓扑变量映射为 [0,1]区间上的连续变量，建立了拓扑优化问题光滑的数学模型，提高了求解效率；之后利用合适的可微性优化算法求解此连续化的拓扑优化模型，将得到的在区间 [0,1]上的昀优设计变量映射反演回离散的昀优设计变量。ICM方法适合任何目标函数，也能够解决以重量为目标、多工况下的结构拓扑优化问题，使骨架结构与连续体结构拓扑优化模型获得了统一，克服了用柔顺性为目标函数难以处理多工况的困难，减少了约束数目，降低了求解规模，提高了计算效率。概言之，连续是指拓扑变量是连续的；映射包含三层含义，一是为了协调独立和连续之间的矛盾，借助过滤函数建立离散拓扑变量和连续拓扑变量之间的映射，二是指优化模型的求解用到了原模型和对偶模型之间的映射，三是求解之后由连续模型向离散模型的逆向映射，又称为反演（ Inversion）。ICM方法具有简洁性、合理性，同时也有数学上的解释。 ICM方法可以取结构重量为昀轻化的目标，从而将截面优化、形状优化和拓扑优化的目标统一规范化，ICM方法不仅有效地解决了应力、位移、稳定、振动等约束下的连续体结构拓扑优化问题，从而更有利于工程实际应用，也实现了骨架类结构和连续体结构拓扑优化模型的统一，尤其在处理多工况问题时，将多工况的约束放在同一个模型中，不再出现单个工况“传承载响的路径”组合的困扰，理性地寻找到了昀佳的“传承载响的路径”；引入对偶规划方法减少了设计变量的数目，提高了优化的效率，减少了迭代次数；另外，应力全局化方法大大地降低了灵敏度分析的工作量。虽然 ICM方法是从 1996年以来展开研究的，较 Bends.e和 Kikuchi晚了8年才从事连续体结构拓扑优化研究，但是我们探索追随恩师钱令希院士开创的研究方向，汲取国内外同行研究的学术营养，二十年如一日，在该方向上培养了铁军、尚珍 2位博士后研究人员，杨德庆、于新、叶红玲、杜家政、彭细荣、张学胜、边炳传、宣东海、易桂莲 9位博士研究生，任旭春、贾志超、刘建信、陈实、朱润、邱海、刘晓迪、沈静娴、李俊杰 9位硕士研究生，出版了 2本专著和发表了大量论文，申请获批了 48项软件著作权。本专著就是多年来研究成果的集成。第1章介绍了结构优化的基本概念，连续体结构拓扑优化的进展及其基本理论，以及以后章节中需要用到的相关的数学规划方面的内容；第 2章叙述了 ICM方法的理论基础；第 3章叙述了连续体应力约束拓扑优化方法；第4章叙述了连续体位移约束拓扑优化方法；第 5章叙述了连续体应力与位移约束拓扑优化方法；第 6章叙述了连续体频率约束拓扑优化方法；第 7章叙述了连续体位移与频率约束拓扑优化方法；第8章叙述了连续体强迫谐振动下拓扑优化方法；第 9章叙述了连续体屈曲约束拓扑优化方法；第 10章介绍了有无复合体和建模优化方法、约束集成化的建模优化方法、抛物型凝聚函数的结构拓扑优化方法以及阶跃函数高精度逼近的结构拓扑优化方法。掩卷静思，有几点感受： 1．结构拓扑优化欲取得跨跃性的进展，一方面要回顾发展的历程，洞悉各种具有影响力方法的本质，更要扬长避短、为我所用。 

2．应当从概念之根着手，从基础上突破思维定势，追究问题的根本。欲发掘出新的思路，必须从基本概念入手上予以观念的突破。 

3．结构拓扑优化不同于截面和形状优化，它求解的高难度就因为以“多/少决策”代替“有/无决策”，必然导致勉为其难。在本专著面世之时，我们衷心地感谢如下单位和个人： 
1．国家自然科学基金委。它资助我们的课题有：结构优化设计曲线寻优理论逼近论和常微分方程组解法（ 19172012）、累积迭代信息的模型化和昀优化的结构综合理论和方法（ 19472014）、骨架与连续体结构拓扑设计的统一映射模型化和昀优化（ 10072005）、结构后天承载能力的昀优化
（10472003）、汽车撞击时损伤的昀小化（ 10872012）、基于独立拓扑变量的连续体结构动力拓扑优化方法的研究 (11072009)、高精度逼近阶跃函数的结构拓扑优化方法(11172013)、融合连续体结构拓扑优化 ICM法提升与扩展变密度法(11672103)。 2．中国*。它资助了博士学科点专项科研基金课题：变荷载下结构昀优状态的实时实现（63001015200701）。 
3．北京自然科学基金委。它资助的课题有：大型工程结构优化实用的建模和求解技术（3002002）、工程结构传力路径合理化的拓扑优化方法（3042002）、脉动真空灭菌器疲劳约束下的结构优化（ 3093019）。 4．北京市*。它资助的课题有：复杂实用结构的优化设计（ 05001015199904）、结构的后天状态优化和智能控制实现（ KM20 0410005019）。 5．有关的中国产业部门。我们进行了成功的合作。 6．国际著名的计算力学软件美国 MSC公司。对在其软件平台上进行的结构优化二次开发予以了合作和支持。
7．北京工业大学。它对于作者所在的机电学院的软、硬件支持，使结构与多学科优化的数值模拟成为其主攻方向之一，也成为工程力学学科的研究特色之一。它对于本专著的出版予以了资助。 
8．前述 2位博士后研究人员、 9位博士研究生和 9位硕士研究生。他们参加了 ICM方法的相关研究工作，付出了辛勤的劳动。 

9、我们将特别强调对于美国 Georgia Southern大学机械工程系的助教授任旭春博士（ [email protected]）的感谢。他的研究集中在结构和多学科优化，包括健壮性设计和基于可靠性的设计优化。作为本书的一位编辑，在修正本书的英文表达和润色句子方面，付出了巨大的努力。如果没有来自政府项目和工业部门课题的资助，没有国际著名的计算力学软件公司对于作者所在学科的支持，没有青年学子们的参加研究，我们的研究是很难一直持续到今天的。因此，在将这本专著奉献给广大读者之前，我们把它作为满怀感恩之心的礼品，首先敬献给上述单位和个人。隋允康彭细荣 2018年4月
致谢 感谢任旭春博士在修正本书的英文表达和润色句子方面付出的巨大努力。

Exordium 1 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 oper-ating requirements are demanded for components of various high-precision and advanced devices. Designing structures and components to satisfy various con-straints, therefore, provides both new opportunities and new challenges to struc-tural 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 1 
application and operation such as manufacturing processes, construction condi-tions, and limits in the specifications of construction, manufacturing, and design should be satisfied. All requirements, conditions, and limits are expressed as con-straints, whereas the economic index or a mechanical property is taken as the objective function. Design parameters, including design details such as the struc-ture 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 pur-sued 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 enhance-ment 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 opti-mization as an independent discipline. Hereafter, theory, method, and software of structural optimization design grew steadily. Over 50 years, the structural optimi-zation 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 theo-retical system formed and a great number of practical problems solved, huge eco-nomic 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, engi-neering intuition and mechanical concepts should be closely combined with math-ematical 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 devel-opments of the theory in the structural optimization. Comprehensive, systematic researches on theory and numerical aspects should be carried out for the topology 1.1 Research History on Structural Optimization Design 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 classifi-cation of structural optimization are firstly summarized. Developments and meth-ods 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 optimi-zation 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 optimi-zation, shape optimization, and topology optimization. These correspond to the detail design, basic design, and conceptual design phases of the product design, respectively. 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 para-meters of a beam cross-section (such as the sizes of cross-section or quantities of a cross-section: area, bending moment of inertias in two directions, torsion moment of inertia, bending modulus, shear modulus, or torsion modulus), etc. [1]. Geometry optimization or shape optimization optimizes shapes of structural boundaries on the basis of specifying the structure type and topology. It belongs to the moving boundary problem. For continuum structures, structural boundaries are usually described by geometrical curves (such as line, arc, and spline) with a set of changeable parameters. The structural boundaries are adjusted when these parameters are changed. For truss structures, nodal coordinates are usually taken as design variables. The topology optimization changes structural topology in the design area to optimize a structural performance index and satisfy constraints on the stress, displacement, frequency, and so on under given loads and boundary conditions. For skeletal structures (including truss and frame), the Structural optimization levels  (1) Size optimization for skeletal structures  (2) Size optimization for continuum structures  (3) Geometry optimization for skeletal structures  Initial figure     Optimal figure     Structural optimization levels  (4) Shape optimization for continuum structures  (5) Topology optimization for skeletal structures  (6) Topology optimization for continuum structures  Initial figure     Optimal figure     
FIGURE 1.1 Levels of structural optimization. presence or absence of nodes and components are taken as design variables. For continuum structures, the solid or void of subregions in the design is taken as a design variable. Compared with the size optimization and geometrical optimization, the struc-tural topology optimization not only has more undetermined parameters but also its topology variables have more influence on the optimization objective. Thus, greater economic benefits can be obtained. It is more attractive to engineering designers and has become a researching hot spot in the field of current structural optimization design. Due to design variables not being specific sizes or nodal coordinates, but the solid or void of subregions on the independent level, the diffi-culty of topology optimization is significant and is recognized as one of the most challenging topics in the field of current structural optimization [1 4]. Kirsch [5 11], who long engaged in the study of structural optimization design, 1.1 Research History on Structural Optimization Design considers the topology design problem to be the most difficult task in structural optimization. Optimization methods are still in the development stage. Applications of optimization methods in design practice are relatively fewer. This field urgently needs further improvement and development. The development of generic algorithms is still a challenge. Similar statements are also widely visible in the recent references [12 15]. The authors think it is very important to understand the structural topology optimization from the view of engineers. That is, the optimal topology of the topology optimization of continuum structures is in fact the reasonable paths of transferring loads and bearing responses. In the earlier researches, we understand it as the reasonable paths of transferring loads. The earlier understanding is intui-tive for static topology optimization problems but is not precise enough for dynamic topology optimization problems. As a result, the understanding is revised in this monograph: the optimal topology of the topology optimization of contin-uum structures can be understood as the reasonable paths of transferring loads or the reasonable paths of bearing responses. By combining two aspects, it can be called succinctly the reasonable paths of transferring loads and bearing responses. 
1.1.2 DEVELOPMENT OF STRUCTURAL OPTIMIZATION The history of structural optimization can be traced back to Maxwell (1890)’s studies on the layout optimization of the truss. Thereafter, Michell [16] studied the layout optimization with stress constraints for the truss with coplanar forces applied to specified locations. The condition of the optimal truss with the lightest weight should be satisfied is obtained and later is called the Michell criterion. It is a milestone in the theory of structural optimization design. In essence, the Michell truss is a very advanced research in the field of structural topology opti-mization and still belongs to research directions at the highest levels. The size optimization is the lowest level of optimization. Although it is the lowest level of structural optimization, it not only has the value of engineering application but also provides precious basic experiences for deeply understanding the structural optimization problem and various optimization algorithms. It was in 1960, 56 years after the Michell truss had been put forward, that structural optimi-zation design became a subject. Schmit first established the mathematical model of the optimization design for elastic structures under multiple load cases [17] and put forward the solution method based on mathematical programming. Thereafter, a new phase of structural optimization design began. Why were there no followers at that time after Michell published his papers and the area became an advanced research? Why has Schmit published his papers, making a clarion call, caused numerous scholars to follow up immediately? The reason is the meth-odology and research tools. The idea of the criterion is the basis of Michell’s method. He put forward the idea of material economic optimum for truss structures (“frame structures” is used in his paper; it should be “truss structures”). At that time, there are no the finite element method and the mathematical programming. But they are the basis of two methodologies—the mechanics analy-sis establishing the optimization model and the mathematical optimization solving the optimization model, which makes structural optimization design science. If we say that Maxwell and Michell put forward an advantage direction at a high level when the foundation of structural optimization design had not yet been built, then Schmit’s contribution is that he captured keenly the two methodologies constructing the architecture of structural optimization design just as they came out. There are many scholars following him. The architecture of structural optimi-zation design is constructed layer by layer from size optimization to shape optimi-zation and to topology optimization. It has the effect that someone raises his arm and is followed up. Except for two soft tools—two methodologies, there is an indispensable hard tool—the development of the computer. Thereafter, for the structural optimization problem with stress, displacement, and frequency con-straints, various methods are adopted to solve it, including linear programming (LP), gradient projection, feasible direction, penalty function, and other methods. Because the mathematical programming theory is directly used to solve those pro-blems and there is no clear conception of establishing the optimization model, the amount of calculation is huge during the iteration process. When directly using the mathematical programming theory, there is no high-efficiency algorithm due to lack of considering the mechanical characteristics of optimization problems. Thus, more effective ways to solve optimization problems are continually sought. In 1968, Venkayya [18] and Gellatly [19] put forward the optimal criteria method. The iterative mode of design variables is selected according to the pre-scribed optimal criteria. The convergence is speeded up. Although this method is not rigorous in the theory aspect, its program is easy to implement, and the amount of calculation is small. Actually, the success of the criterion method is inevitable. When the mathematical programming method had not yet become an independent subject, mechanicians and engineers always put forward optimization criteria based on the mechanics conception or engineering intuition. Among vari-ous optimization criteria, there are the Michell truss criterion, the structure full stress criterion, and the component synchronous failure criterion as typical crite-ria. If we say that those early optimization criteria are perceptual criteria, then those optimization criteria appearing after 1969 can be called rational criteria. Accompanying the development of structural optimization criterion methods, structural optimization programming methods are developing continually. In 1976, Schmit [20] divided design variables into several groups by the link-ing of design variables. The number of independent design variables was reduced. Invalid constraints were removed after every structure analysis, and the computa-tional efficiency was improved. In 1979, Fleury first introduced the duality theory into the structural optimization problem [21]. By adopting the separable dual pro-gramming to solve the optimization problem, calculation results similar to the optimal criterion method were obtained. In 1980, Fleury and Schmit put forward the mixed optimality criteria method [22]. Some critical stress constraints were selected as effective stress constraints by using the virtual load method; other 1.1 Research History on Structural Optimization Design stress constraints were converted into upper and lower bounds. The number of iterations in the method has nothing to do with the number of design variables, thus the calculation efficiency is high. In every iterative process, the effective and noneffective constraints, the active and passive design variables are determined, the number design variables and the number of constraints are reduced, and there-fore the computational efficiency is improved further. The development of struc-tural optimization programming methods makes the number of iterations drop to the same level with structural optimization criterion methods. Why is there such a result? The key point is that the structural optimization approximation models were established, either consciously or unconsciously. By turning back to look at the criterion method, we find that there is an approximated explicit optimization model lurking behind every criterion. Therefore, in the late 1970s, the structural optimization programming method and criterion method met. There are also some scholars who have yet to reach the confluence stage of both methods. For example, Khan put forward the strict-est constraint method [23]. In every iteration process, based on structural stress analysis, the strictest constraint is picked out from all constraints. The design point is migrated to the strictest constraint plane by using the scaling step. Therefore, only an effective constraint needs to be considered in every iteration. The amount of calculation is reduced greatly. The strictest constraint method sometimes fails. For example, if the optimal point locates at two or more con-straints at the same time, iterative oscillation will occur and the solution process will not be convergent. Researchers in China have proposed many new methods. In 1973, in the sym-posium on the mechanics programming organized by the Chinese Academy of Sciences, Lingxi Qian presented the academic report “New developments on the optimization theory and method of the structural mechanics.” This attracted extensive attention and responses in the mechanics and engineering field in China [2]. Since the 1980s, for the minimizing weight optimization problem of compli-cated structures simulated by different types of finite elements, Lingxi Qian et al. have taken the reciprocal of the cross-sectional size as the design variable. The objective function is expressed by the second-order Taylor expansion. Constraints are expanded by linear approximations. The iterative mode of the design variable, including the Lagrange multiplier, is derived by using Kuhn Tucker conditions. The nonlinear programming method and the design criterion method are com-bined [2,24]. The stress constraints and displacement constraints are dealt with separately. The number of structural reanalysis is reduced further. Lingxi Qian led a team in the Dalian University of Technology to develop “structural optimi-zation design with multiple elements, multiple load cases, and multiple con-straints—DDDU system” [25,26]. Combining the mechanics conception and the mathematical programming method, some traditional difficulties are overcome. The sequential quadratic programming (SQP) algorithm for structural optimiza-tion is developed. In 1983, Guangyuan Wang and Da Huo put forward the structural two-phase optimization method [27,28]. In this method, structural optimization design is divided into two stages. In the first stage, the criterion conditions are fully satis-fied. In the second stage, the lightest design of the structure is solved. Two stages iterate alternatively. Renwei Xia and Ming Zhou studied the dual algorithm on the basis of the second-order approximation of functions [29], and put forward the generalized intermediate variables approximation method for the geometry optimi-zation of truss structures [30]. Yunkang Sui improved the Newton method and the dual method by using two-point rational approximation [31,32]. The most effective approximated analytical method and its approximation method are found by using curve optimization theory to replace linear optimization theory [33,34]. The sequence rational programming method of the nonlinear programming is studied by the equivalent LP problem and equivalent quadratic programming (QP) prob-lem, respectively [35]. A convenient and practical rational approximation method is put forward. By taking advantage of the information in the previous iteration, the waste of the repeated analysis and information is avoided; the efficiency of the optimization algorithm is improved. Huanchun Sun et al. discussed the discrete structural optimization problems [36,37]. Templeman and Yates constructed the multisegment element for the rod element of truss structures. Each segment of the multisegment element corresponds to an area in the discrete set. The discrete area design variable is thus converted skillfully to the corresponding continuous rod length design variable. The optimization model is solved by LP [38]. Considering that the number of design variables increases dramatically after the discrete vari-able is converted to the continuous variable in the method, Yunkang Sui and Kejian Peng [39] modified the method to construct the two-segment element near the continuous optimal solution. Under the condition of the total length of the rod element remaining constant, the number of continuous rod length design variables is the same as the number of the discrete cross-section; thereby, the method by Templeman and Yates is improved. In addition, the above method cannot be applied on the beam element as its internal forces are changed along the elemental length. Yunkang Sui and Yongming Lin constructed the infinite combination of the infinitesimal multiple segment element, thus improving the conversion method by Templeman and Yates to be applicable for the discrete variable optimization of frame structures [40,41]. Yunkang Sui also extended the method to the discrete size optimization of structures simulated by arbitrary types of finite elements [1]. Early works regarding shape optimization began from researches carried out by Zienkiewicz and Compell in 1973. They took nodal coordinates as design vari-ables to describe the shape of a dam structure. The isoparametric element is adopted in the structural analysis. The sequential linear programming (SLP) method is adopted to solve the shape optimization problem of the dam structure [42]. In the same year, Deslva adopted the same mathematical method and struc-ture analysis method to optimize the disk shape of the turbine [43]. However, for the method of taking nodal coordinates as design variables, the scale of the solved problem is limited and the solution precision is also affected because the number 1.1 Research History on Structural Optimization Design of design variables is usually large. Therefore, some scholars have put forward the method described structural shape by some fixed function. For example, in 1974, Vitiello took coefficients of the polynomial function as design variables, namely, a polynomial function was adopted to describe the distribution of thick-ness [44]. In this way, the scale of the problem was greatly reduced. Mature parameter optimization methods can also be adopted to solve the problem. Similar researches were also carried out by Ramarkishman [45]. The method of describing structural shape by a specified function adds an additional man-made constraint on the structures. It places the shape of the design structures in a fixed mode and can be selected at the optimum in small-scale changes. A more general method is to take functions as design variables. At first, the shape functions are defined. Then, the function to describe the initial shape of the structure is obtained from a linear combination of the shape functions with a set of undeter-mined parameters. Those undetermined parameters of the shape functions are taken as design variables. In 1979, Haug [46] proposed the variational principle of the shape optimization and solved the shape optimization problem of one-dimensional and two-dimensional plates. In 1985, Haug and Choi proposed the material derivative sensitivity analysis method of shape optimization. The sensitivity analysis formula in the form of boundary integral and surface integral is presented [47]. In 1986, Belegundu developed the shape optimization method based on the natural design variable and the shape function [48]. A series of hypothetical loads applied to the struc-ture are taken as natural design variables. The displacements produced by hypo-thetical loads are added to the initial shape to yield a new shape. The linear relationship between nodal displacements of meshes and the design variables pro-duced in the finite element analysis is established. The shape optimization prob-lem of elastic planes is solved. In 1987, Helder and Rodrganes studied the shape optimization of elastic body by the mixed variational formula. By adopting the mixed finite discretization method, a Euler Lagrange formula based on the vir-tual work principle was put forward [49]. Oueau and Trompette studied the shape optimization problem of axisymmetric structures under symmetric or asymmetric loads. The objective function is to make local stresses along borders that are uni-form and to reduce the stress concentration. The six-or eight-node isoparametric elements are used for the structural analysis. The algorithm cooperates with the automatic mesh generation program. Improvements on the derivative calculation of the stress and stiffness matrix were put forward. Parts of the shape of a heli-copter rotor are optimized by the algorithm [50]. Ming Zhou and Rozvany com-bined the COC theory and the finite element method, and proposed an iterative COC algorithm. The problem with simple constraints can be solved very well, and the calculation efficiency is very high so that the large-scale problem can be solved [51,52]. In engineering practice, the optimization problem with multiple load cases often needs to be solved. In 1982, Botkin studied the structural shape optimization with multiple load cases [53]. The basic idea is to calculate and normalize the difference of the expected value and the calculated value of the strain energy den-sity under various load cases separately in the same iteration process. The mesh updating area is determined according to those results. Meshes are automatically updated to prepare for the next iteration. For large-scale complex structures, only parts of the structural shape are often allowed to be changed. The whole structure is divided into many parts that are grouped into two categories: changeable parts and unchangeable parts. Only changeable parts need to be regenerated meshes in every iteration, which can reduce the amount of calculation of the finite element analysis and sensitivity analysis dramatically. In addition, the whole structure can be divided into several substructures (regions); internal degrees of freedom of the region are squeezed onto the boundary of the region, and then the relationship between the structure and the interface is established. In 1988, Huang and Huang put forward the substructure method of structural system optimization design [54]. In 1991, Botkin and Yang applied the substructure method of shape optimi-zation of the three-dimensional solid [55]. There are also many scholars devoted to the research of the theory and method of the shape optimization in China. Yunkang Sui, Xicheng Wang, and Bei Wang put forward the secondary control method to overcome the difficulty of design vari-ables controlling meshes in shape optimization software development. In the first stage, natural design variables determine coordinates of key points by the boundary shape function. In the second stage, key points determine coordinates of the mesh nodes by parameters coordinates of design variables of mesh nodes [25,26,56 58]. Yunkang Sui introduced some basic principles of mathematical programming into structural optimization and developed further from the view of methodology. According to the mapping inversion principle of the relationship, the dual algo-rithm of LP and geometric programming and the Lemke algorithm of QP were ana-lyzed. The sequential mapping method is put forward to construct the algorithm solving the generalized QP [1]. Gengdong Cheng pointed out several difficulties existing in shape optimization. The shape optimization of the profile of the railway wheel and that of the turbine disc of an aero-engine were carried out [59]. Gengdong Cheng and Olhoff studied the shape optimization of the uniformity of the microstructure material [60]. Yuanxian Gu studied the sensitivity analysis of the shape optimization and the structural forming, and applied it to shape optimiza-tion with thermal stress constraints [61]. Jie Xing and Ping Cai developed the inte-grated software system FSOPD by using variational sensitivity analysis technology and the virtual load method. A shape optimization method was also put forward. The method takes the essential derivative and the variational sensitivity analysis as its foundation and defines the local and global velocity field. The integral of the sensitivity analysis is carried out in local regions, and the amount of calculation is reduced [62]. Weihong Zhang put forward the parameterized structural shape opti-mization design method by selecting automatically independent design variables [63,64], and a mesh disturbance physical analysis method was established based on finite element analysis. The sensitivity analysis method for size variables was established based on a simple proportion calculation [65]. The multiple