了得網研究生_連續體結構拓撲優化的建模、求解和應用—

編輯推薦

構優化學科發展於 20世紀60年代初。結構拓撲優化方向的研究暫時被束之高閣，讓位於低層次的結構截面、形狀優化研究，直到 1988年連續體結構拓撲優化方向的出現，它纔被重新提及，並且呈現了越來越熱的研究趨勢。

內容簡介

連續體結構拓撲優化的建模、求解和應用，能夠使航空、航天、汽車、船舶、土木、水利、交通和輕工業等領域的研究者和工程師受益，去解決各自的結構拓撲優化問題，在初步設計階段就能夠找到經濟與安全的平衡點。
本書以作者多年的研究工作為基礎，繫統地介紹連續體結構拓撲優化 ICM（獨立、連續和映射）方法的基本概念和原理，為具有不同約束條件和邊界條件的復雜工程問題提供了建立模型和求解的途徑。本書還對結構拓撲優化*前沿的研究領域提供 ICM方法的應用，以及將其應用到計算固體力學的範圍。
本書可供上述工程領域的優化研究、設計人員及高等院校師生參考。

前言
致謝
第1章緒論1
1.1結構優化設計研究的歷史發展 3
1.1.1結構優化設計的分類和層次 3
1.1.2結構優化的發展 5
1.2連續體結構拓撲優化的研究進展 13
1.2.1連續體結構拓撲優化的數值方法 13
1.2.2連續體結構拓撲優化的求解方法 21
1.3數學規劃概念與算法 22
1.3.1結構優化設計的三要素 22
1.3.2數學規劃模型 24
1.3.3 線性規劃 26
1.3.4 二次規劃 28

前言

致謝

第1章
緒論1

1.1結構優化設計研究的歷史發展 3

1.1.1結構優化設計的分類和層次 3

1.1.2結構優化的發展 5

1.2連續體結構拓撲優化的研究進展 13

1.2.1連續體結構拓撲優化的數值方法 13

1.2.2連續體結構拓撲優化的求解方法 21

1.3數學規劃概念與算法 22

1.3.1結構優化設計的三要素 22

1.3.2數學規劃模型 24

1.3.3 線性規劃 26

1.3.4 二次規劃 28

1.3.5 庫恩 -塔克條件和對偶理論29

1.3.6 K-S函數方法 32

1.3.7廣義幾何規劃理論 33

1.3.8函數變換下的函數高階展式和單項高階縮並公式
35

第2章 ICM（獨立、連續、映射）方法的基礎37

2.1 傳統拓撲優化問題的困難
39

2.2 階躍函數和跨欄函數——構造離散拓撲變性能關繫的橋

梁 41

2.3 根本性突破的關鍵——磨光函數逼近階躍函數和過濾函數逼近跨

欄函數 43

2.3.1 磨光函數 44

2.3.2 過濾函數 45

2.3.3過濾函數使拓撲優化的求解可以操作 46

2.3.4四個函數的相互關繫 46

2.4 ICM方法及其應用 47

2.4.1映射識別與全程識別量 47

2.4.2幾種典型的磨光函數和過濾函數 49

2.4.3各種性能識別的快慢和參數確定 52

2.4.4過濾函數中指數函數參數到對數函數參數的變換
60

2.4.5 基於 ICM方法建立結構拓撲優化模型
63

2.4.6映射的反演 64

2.5對磨光函數和過濾函數性能的深入探討 66

2.5.1劃分磨光函數（和過濾函數）的類型 66

2.5.2類型判別定理 67

2.5.3磨光函數與過濾函數的對應定理 67

2.6對高精度過濾函數的深入探討 69

2.6.1高精度過濾函數的使用準則 69

2.6.2根據高精度左磨函數構造快濾函數的方法 70

2.6.3指數類快濾函數的參數選取 74

2.7 ICM方法在基本概念上的突破 76

第3章
連續體應力約束下的拓撲優化 79

3.1 基於 ICM方法的應力零階近似處理和模型求解82

3.1.1應力約束零階近似的連續體結構拓撲優化模型
82

3.1.2應力零階近似策略下連續體結構拓撲優化求解
82

3.1.3求解算法中的其他策略 84

3.1.4 數值算例 87

3.2替代應力約束的全局化應力約束 89

3.2.1應力約束全局化策略 89

3.2.2應變能約束修正繫數的引入 93

3.3.3利用昀小二乘法求修正繫數 94

3.2.4數值模擬確定修正繫數 95

3.2.5許用應力對連續體拓撲結構的影響 96

3.2.6多工況下應變能約束的修正繫數 100

3.2.7結構許用應變能的求得 100

3.3結構應變能約束下的連續體結構拓撲優化
104

3.4結構畸變比能約束下的連續體結構拓撲優化
108

3.4.1 應力約束轉化為結構畸變比能約束的全局化策略及其修正 108

3.4.2 修正結構全局畸變比能約束下的連續體結構拓撲優化 ICM模型

3.5載荷病態問題及其解決 112

3.5.1三類載荷病態現像 113

3.5.2以結構應變能作為權繫數處理載荷 114

3.5.3工況間有載荷病態但工況內無載荷病態 116

3.5.4僅在工況內有載荷病態 116

3.5.5工況間有載荷病態同時某工況內亦有載荷病態
118

3.6應力奇異性的探討 119

3.7數值算例 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：工程應用——趙州橋的拓撲優化133

3.8本章小結 136

第4章
連續體位移約束拓撲優化 139

4.1位移約束的顯式化 141

4.1.1位移敏度分析的直接法 141

4.1.2位移敏度分析的伴隨法 142

4.1.3利用一階泰勒近似將位移約束顯式化 145

4.1.4利用莫爾定理將位移約束顯式化 145

4.1.5兩種顯式化方式的一致性 146

4.2多工況位移約束下優化模型的建立與求解
147

4.3考慮離散性條件目標的 ICM方法 150

4.4解決棋盤格式及網格依賴 151

4.4.1棋盤格現像及網格依賴性 151

4.4.2使用過濾方法消除棋盤格現像及解決網格依賴性
154

4.5數值算例 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本章小結 169

第5章
連續體應力與位移約束拓撲優化 171

5.1應力約束和位移約束的無量綱化 172

5.2多工況連續體應力與位移約束拓撲優化模型的建立與求解 174

5.3數值算例 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本章小結 196

第6章
連續體頻率約束拓撲優化 199

6.1頻率約束的近似顯式化 200

6.2頻率約束下連續體拓撲優化模型的求解 203

6.3解決棋盤格式及網格依賴 204

6.4局部模態及模態交換問題 204

6.4.1局部模態問題 204

6.4.2解決局部模態問題 206

6.4.3模態交換問題 207

6.4.4解決模態交換問題 209

6.5數值算例 210

6.5.1 算例 1 210

6.5.2 算例 2 213

6.5.3 算例 3 215

6.5.4 算例 4 216

6.6本章小結 221

第7章
連續體位移與頻率約束拓撲優化 223

7.1頻率與位移約束的無量綱化 224

7.2位移與頻率約束連續體拓撲優化模型的建立與求解 226

7.3數值不穩定問題的求解策略 227

7.3.1解決棋盤格式及網格依賴 227

7.3.2局部模態及模態交換等問題的處理 227

7.4數值算例 228

7.4.1 算例 1 228

7.4.2 算例 2 229

7.4.3 算例 3 233

7.5本章小結 236

第8章
連續體強迫諧振動下拓撲優化 237

8.1強迫諧振動下位移幅值靈敏度分析 238

8.1.1強迫諧振動下位移幅值敏度分析常見方法
238

8.1.2無阻尼強迫諧振動下位移幅值敏度分析 239

8.1.3有阻尼強迫諧振動下位移幅值敏度分析 242

8.1.4矩陣導數的計算 245

8.1.5 數值算例 246

8.2位移幅值約束的近似顯式化 251

8.3強迫諧振動下位移幅值約束拓撲優化的建模與求解 254

8.4數值算例 255

8.4.1 算例 1 255

8.4.2 算例 2 255

8.5本章小結 262

第9章
連續體屈曲約束拓撲優化 263

9.1屈曲分析的基本概念 265

9.2屈曲約束的顯式化 267

9.3連續體屈曲約束拓撲優化建模與求解 270

9.4屈曲臨界力上限的選取準則 270

9.4.1一階臨界力上限與拓撲結構重量的關繫 271

9.4.2二階臨界力上限與拓撲結構重量的關繫 275

9.4.3三階臨界力上限與拓撲結構重量的關繫 277

9.5數值算例 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本章小結 294

第10章其他相關的方法 295

10.1有無復合體方法及其對於連續體拓撲優化的應用
296

10.1.1 有無復合體平
297

10.1.2 有無復合體的許用應力
298

10.1.3 有無復合體平對位移的貢獻 299

10.1.4 有無復合體平面膜結構在應力與位移約束下的拓撲優化300

10.1.5
數值算例 301

10.2約束集成化的連續體結構拓撲優化 304

10.2.1 應力約束集成化建模與求解
304

10.2.2 位移約束集成化建模與求解
311

10.2.3 應力和位移約束集成化建模與求解 316

10.3拋物型凝聚函數的結構拓撲優化方法 319

10.3.1 拋物型凝聚函數 319

10.3.2 拋物型凝聚函數對於約束的集成化 322

10.4階躍函數高精度逼近的結構拓撲優化方法
327

10.5本章小結 333

參考文獻335

跋355

索引363

前言

本專著涉及的結構優化（ Structural Optimization）是指承受力載荷的宏觀人造結構的優化設計問題。拓撲（ Topology）作為比幾何（ Geometry）更抽像的數學概念，同結構優化聯繫在一起，表征了高於結構截面或尺寸、形狀或幾何層次的優化，盡管結構拓撲優化經歷了從傳統到現代的發展，本質上都是有與無的決策問題，而截面或尺寸、幾何或形狀層次的優化卻是多與少的決策問題。如果說傳統的結構拓撲優化早在19世紀末就被 Maxwell提出，又在 20世紀初由 Michell做進一步研究，那麼，現代的結構拓撲優化應該說是在 1988年纔開始的， Bends.e和 Kikuchi從程耿東和 Olhoff對於昀小柔順性實心彈性薄板優化的研究中得到啟迪，提出了均勻化（ Homogenization）方法，於是開始有了連續體拓撲優化的概念和求解途徑。令人振奮的是，連續體拓撲優化的昀終結果趨於骨架類結構（桁架、框架結構）! 作為有/無優化問題，傳統的結構拓撲優化研究骨架類結構，確定節點和節點聯結，從而表達了構件的存在與否。然而，現代的結構拓撲優化研究連續體結構所在空間的子域的有或無。由於有和無分別用 1和0表示，結構拓撲優化屬於0-1型離散變量數學規劃，可見它是難於求解的高層次結構優化問題。結構優化從低到高的發展可以分成三個層次，皆可從工程和數學上分別進行描述：視角 Viewpoint 低層次 Low level 中層次 Middle level 高層次 High level 工程觀點 Engineering viewpoint 截面 Cross Section 形狀 Shape 布局 Layout 數學觀點 Mathematical viewpoint 尺寸 Size 幾何 Geometry 拓撲 Topology本專著涉及的結構優化（ Structural Optimization）是指承受力載荷的宏觀人造結構的優化設計問題。拓撲（ Topology）作為比幾何（ Geometry）更抽像的數學概念，同結構優化聯繫在一起，表征了高於結構截面或尺寸、形狀或幾何層次的優化，盡管結構拓撲優化經歷了從傳統到現代的發展，本質上都是有與無的決策問題，而截面或尺寸、幾何或形狀層次的優化卻是多與少的決策問題。如果說傳統的結構拓撲優化早在19世紀末就被 Maxwell提出，又在 20世紀初由 Michell做進一步研究，那麼，現代的結構拓撲優化應該說是在 1988年纔開始的， Bends.e和 Kikuchi從程耿東和 Olhoff對於昀小柔順性實心彈性薄板優化的研究中得到啟迪，提出了均勻化（ Homogenization）方法，於是開始有了連續體拓撲優化的概念和求解途徑。令人振奮的是，連續體拓撲優化的昀終結果趨於骨架類結構（桁架、框架結構）! 作為有/無優化問題，傳統的結構拓撲優化研究骨架類結構，確定節點和節點聯結，從而表達了構件的存在與否。然而，現代的結構拓撲優化研究連續體結構所在空間的子域的有或無。由於有和無分別用 1和0表示，結構拓撲優化屬於0-1型離散變量數學規劃，可見它是難於求解的高層次結構優化問題。結構優化從低到高的發展可以分成三個層次，皆可從工程和數學上分別進行描述：視角 Viewpoint 低層次 Low level 中層次 Middle level 高層次 High level 工程觀點 Engineering viewpoint 截面 Cross Section 形狀 Shape 布局 Layout 數學觀點 Mathematical viewpoint 尺寸 Size 幾何 Geometry 拓撲 Topology
結構優化學科發展於 20世紀60年代初。結構拓撲優化方向的研究暫時被束之高閣，讓位於低層次的結構截面、形狀優化研究，直到 1988年連續體結構拓撲優化方向的出現，它纔被重新提及，並且呈現了越來越熱的研究趨勢。為什麼結構拓撲優化研究方向能夠成為熱點？不少人認為該方向有著實用上潛在的極大經濟效益，結構拓撲優化一旦能付諸應用，將比結構的截面和形狀優化更多的結構用材。盡管這一想法很有道理，還是應當從力學學科內在的發展動力去進行深入的思考：數學上考慮的結構拓撲是工程上考慮的結構布局，在力學上對應著什麼？我們可以從結構承受力載荷作用的視點來回答這一問題，昀優拓撲結構或昀優結構布局實際上給出了對應的昀佳傳力路徑，或者說得更寬泛一點：給出了結構傳遞力學載荷或承受力學響應的昀佳通路，以下簡稱為“傳承載響的昀佳通路”。不可避免地，對於實際工程構件或結構，從提出昀初的設計方案到昀終的施工圖設計，都要確定一個基本的傳承載響的主導性通路，其設計是否合理，關繫到是否經濟、是否安全、是否優質等一繫列昀優的指標，而解決問題的根本手段則是采用結構拓撲優化方法，這也就是為什麼學者和工程師那樣重視結構拓撲優化研究的原因。綜上所述，對應於數學上的昀優拓撲和工程上的昀優布局這一高層次問題，力學上則是研究傳承力學載響的昀佳通路。相應地，低層次問題，在力學上表示傳承載響的各個局部通路的大小；中層次問題，對於已經確定的傳承載響通路，增加了其彎曲程度或曲率變化大小的考慮；高層次問題即結構拓撲優化問題，在力學上又增加了具有挑戰性的思考——要設計傳承載響的通路，回答通路的布置問題，亦即解決數學拓撲的構成。結構拓撲優化能夠具有強烈吸引研究者興趣的魅力，還在於拓撲優化由骨架類結構向連續體結構的開拓，擴大了傳承載響昀佳通路的探索空間，工程師在骨架類結構構型選取的天纔直覺，次讓位於連續體子域有無的理性計算！從1988年至今， 28年過去了，連續體結構拓撲優化的研究出現了精彩紛呈的大量方法，有了令人鼓舞的可喜進展。其中一個重要的原因在於，面對結構拓撲優化人們沒有受限於離散變量數學規劃的算法，而是轉而求解一個近似的連續變量的數學規劃問題，從而得心應手地運用求解效率高的可微性算法。具體策略是，把拓撲變量掛靠和依附在結構低層次變量上，骨架和連續體拓撲優化分別轉化為廣義截面或性能和廣義形狀優化問題。盡管這些都是非常靈巧的做法，並且確實取得了豐碩的成果，但是還有許多尚待改進和發展之處： 1．由於結構拓撲優化處理為廣義截面與廣義形狀優化問題，拓撲變量不再獨立，未能發掘出其本身獨立優化層次算法的潛能，因此優化計算的效率有待進一步提高。
2．己有主要研究在整體約束下例如體積約束、結構自振頻率約束等，以結構柔順性（ Compliance）作為目標函效，然而實際結構中應力、位移、振動約束是非常重要的，不考慮它們的設計是難以使拓撲優化付諸工程實際使用的。
3．骨架結構與連續體結構的拓撲優化采用了不同的目標函數，前者多以結構重量極小化為目標，後者多以結構柔順性昀小為目標，二者優化模型基本上不統一，因此很難將骨架結構拓撲優化有效的模型及解法推廣到連續體結構拓撲優化中。
4．柔順性對應著某一工況，若用之處理多工況問題則要加進許多人為的條件，使優化問題的嚴格求解降低為含有許多假定的權衡解答。
5．當利用均勻化方法、變密度法或其他方法建立拓撲優化模型時，很難找到位移約束與拓撲設計變量之間的近似顯函數關繫。即使建立了這一關繫，也由於該模型的拓撲設計變量過多，利用常規的數學規劃方法難以求解。

為了在以上各點取得進展，本書作者在 1996年提出了“獨立連續”拓撲變量的概念和“獨立連續映射”方法（ Independent Continuous and Mapping Method)，簡稱 ICM方法。將拓撲變量從依附截面優化層次或形狀層次優化的變量中抽取出來，以獨具體特征參數的變量來的“有”與“無”，為模型的建立帶來方便；同時引入磨光函數和過濾函數的概念，利用磨光函數逼近實際的 0、1拓撲變量，將離散的 0-1獨立拓撲變量映射為 [0,1]區間上的連續變量，建立了拓撲優化問題光滑的數學模型，提高了求解效率；之後利用合適的可微性優化算法求解此連續化的拓撲優化模型，將得到的在區間 [0,1]上的昀優設計變量映射反演回離散的昀優設計變量。ICM方法適合任何目標函數，也能夠解決以重量為目標、多工況下的結構拓撲優化問題，使骨架結構與連續體結構拓撲優化模型獲得了統一，克服了用柔順性為目標函數難以處理多工況的困難，減少了約束數目，降低了求解規模，提高了計算效率。概言之，連續是指拓撲變量是連續的；映射包含三層含義，一是為了協調獨立和連續之間的矛盾，借助過濾函數建立離散拓撲變量和連續拓撲變量之間的映射，二是指優化模型的求解用到了原模型和對偶模型之間的映射，三是求解之後由連續模型向離散模型的逆向映射，又稱為反演（ Inversion）。ICM方法具有簡潔性、合理性，同時也有數學上的解釋。 ICM方法可以取結構重量為昀輕化的目標，從而將截面優化、形狀優化和拓撲優化的目標統一規範化，ICM方法不僅有效地解決了應力、位移、穩定、振動等約束下的連續體結構拓撲優化問題，從而更有利於工程實際應用，也實現了骨架類結構和連續體結構拓撲優化模型的統一，尤其在處理多工況問題時，將多工況的約束放在同一個模型中，不再出現單個工況“傳承載響的路徑”組合的困擾，理性地尋找到了昀佳的“傳承載響的路徑”；引入對偶規劃方法減少了設計變量的數目，提高了優化的效率，減少了迭代次數；另外，應力全局化方法大大地降低了靈敏度分析的工作量。雖然 ICM方法是從 1996年以來展開研究的，較 Bends.e和 Kikuchi晚了8年纔從事連續體結構拓撲優化研究，但是我們探索追隨恩師錢令希院士開創的研究方向，汲取國內外同行研究的學術營養，二十年如一日，在該方向上培養了鐵軍、尚珍 2位博士後研究人員，楊德慶、於新、葉紅玲、杜家政、彭細榮、張學勝、邊炳傳、宣東海、易桂蓮 9位博士研究生，任旭春、賈志超、劉建信、陳實、朱潤、邱海、劉曉迪、瀋靜嫻、李俊傑 9位碩士研究生，出版了 2本專著和發表了大量論文，申請獲批了 48項軟件著作權。本專著就是多年來研究成果的集成。第1章介紹了結構優化的基本概念，連續體結構拓撲優化的進展及其基本理論，以及以後章節中需要用到的相關的數學規劃方面的內容；第 2章敘述了 ICM方法的理論基礎；第 3章敘述了連續體應力約束拓撲優化方法；第4章敘述了連續體位移約束拓撲優化方法；第 5章敘述了連續體應力與位移約束拓撲優化方法；第 6章敘述了連續體頻率約束拓撲優化方法；第 7章敘述了連續體位移與頻率約束拓撲優化方法；第8章敘述了連續體強迫諧振動下拓撲優化方法；第 9章敘述了連續體屈曲約束拓撲優化方法；第 10章介紹了有無復合體和建模優化方法、約束集成化的建模優化方法、拋物型凝聚函數的結構拓撲優化方法以及階躍函數高精度逼近的結構拓撲優化方法。掩卷靜思，有幾點感受： 1．結構拓撲優化欲取得跨躍性的進展，一方面要回顧發展的歷程，洞悉各種具有影響力方法的本質，更要揚長避短、為我所用。

2．應當從概念之根著手，從基礎上突破思維定勢，追究問題的根本。欲發掘出新的思路，必須從基本概念入手上予以觀念的突破。

3．結構拓撲優化不同於截面和形狀優化，它求解的高難度就因為以“多/少決策”代替“有/無決策”，必然導致勉為其難。在本專著面世之時，我們衷心地感謝如下單位和個人：
1．國家自然科學基金委。它資助我們的課題有：結構優化設計曲線尋優理論逼近論和常微分方程組解法（ 19172012）、累積迭代信息的模型化和昀優化的結構綜合理論和方法（ 19472014）、骨架與連續體結構拓撲設計的統一映射模型化和昀優化（ 10072005）、結構後天承載能力的昀優化
（10472003）、汽車撞擊時損傷的昀小化（ 10872012）、基於獨立拓撲變量的連續體結構動力拓撲優化方法的研究 (11072009)、高精度逼近階躍函數的結構拓撲優化方法(11172013)、融合連續體結構拓撲優化 ICM法提升與擴展變密度法(11672103)。 2．中國*。它資助了博士學科點專項科研基金課題：變荷載下結構昀優狀態的實時實現（63001015200701）。
3．北京自然科學基金委。它資助的課題有：大型工程結構優化實用的建模和求解技術（3002002）、工程結構傳力路徑合理化的拓撲優化方法（3042002）、脈動真空滅菌器疲勞約束下的結構優化（ 3093019）。 4．北京市*。它資助的課題有：復雜實用結構的優化設計（ 05001015199904）、結構的後天狀態優化和智能控制實現（ KM20 0410005019）。 5．有關的中國產業部門。我們進行了成功的合作。 6．國際著名的計算力學軟件美國 MSC公司。對在其軟件平臺上進行的結構優化二次開發予以了合作和支持。
7．北京工業大學。它對於作者所在的機電學院的軟、硬件支持，使結構與多學科優化的數值模擬成為其主攻方向之一，也成為工程力學學科的研究特色之一。它對於本專著的出版予以了資助。
8．前述 2位博士後研究人員、 9位博士研究生和 9位碩士研究生。他們參加了 ICM方法的相關研究工作，付出了辛勤的勞動。

9、我們將特別強調對於美國 Georgia Southern大學機械工程繫的助教授任旭春博士（ xren@georgiasouthern.edu）的感謝。他的研究集中在結構和多學科優化，包括健壯性設計和基於可靠性的設計優化。作為本書的一位編輯，在修正本書的英文表達和潤色句子方面，付出了巨大的努力。如果沒有來自政府項目和工業部門課題的資助，沒有國際著名的計算力學軟件公司對於作者所在學科的支持，沒有青年學子們的參加研究，我們的研究是很難一直持續到今天的。因此，在將這本專著奉獻給廣大讀者之前，我們把它作為滿懷感恩之心的禮品，首先敬獻給上述單位和個人。隋允康彭細榮 2018年4月
致謝感謝任旭春博士在修正本書的英文表達和潤色句子方面付出的巨大努力。

在線試讀

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
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 const

商品搜索

商品分类

【醫學】

【各大出版社】