文件名称：Structural Sensitivity Analysis and Optimization 1--Linear Systems
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Structural Sensitivity Analysis Optimization
Kyung K. Choi，Nam H. Kim， Structural Sensitivity Analysis and Optimization 1 Linear Systems， 2005 Springer Science+Business Media, Inc. Contents 1: Linear Systems Preface ..................................................................................................................... vii PART I Structural Design and Analysis 1 Introduction to Structural Design .................................................................. 3 1.1 Elements of Structural Design ................................................................................ 3 1.2 Structural Modeling and Design Parameterization ................................................. 9 1.2.1 Structural Modeling............................................................................................ 9 1.2.2 Design Parameterization .................................................................................... 9 1.2.3 Three-Bar Truss Example .................................................................................. 13 1.3 Structural Analysis .................................................................................................. 15 1.4 Finite Element Analysis .......................................................................................... 16 1.5 Structural Design Sensitivity Analysis.................................................................... 20 1.5.1 Methods of Structural Design Sensitivity Analysis ........................................... 21 1.5.2 Finite Difference Method ................................................................................... 22 1.5.3 Discrete Method................................................................................................. 24 1.5.4 Continuum Method ............................................................................................ 27 1.5.5 Summary of Design Sensitivity Analysis Approaches....................................... 29 1.6 Second-Order Design Sensitivity Analysis ............................................................. 30 1.7 Design Optimization ............................................................................................... 31 1.7.1 Linear Programming Method ............................................................................. 31 1.7.2 Unconstrained Optimization Problems .............................................................. 32 1.7.3 Constrained Optimization Problems .................................................................. 34 2 Variational Methods of Structural Systems................................................... 37 2.1. Introduction............................................................................................................. 37 2.2. Energy Method........................................................................................................ 39 2.3. Variational Formulation and the Principle of Virtual Work ................................... 45 2.4. Hamilton's Principle................................................................................................ 49 2.5. Eigenvalue Problem ................................................................................................ 51 2.6. Frequency Response Problem................................................................................. 54 2.6.1. Structural Response............................................................................................ 54 2.6.2. Acoustic Response ............................................................................................. 56 2.7. Thermoelastic Problem ........................................................................................... 60 2.7.1. Thermal Analysis ............................................................................................... 60 2.7.2. Elastic Analysis .................................................................................................. 61 3 Variational Equations and Finite Element Methods .................................... 63 3.1 Energy Bilinear and Load Linear Forms of Static Problems .................................. 64 3.1.1. Truss Component ............................................................................................... 64 3.1.2. Beam Component............................................................................................... 66 3.1.3. Plate Component ................................................................................................ 71 3.1.4. Elastic Solid ....................................................................................................... 75 3.1.5. Deflection of a Membrane.................................................................................. 80 3.1.6. Torsion of an Elastic Shaft ................................................................................. 81 3.1.7. General Form of Static Variational Equations ................................................... 81 3.2 Vibration and Buckling of Elastic Systems............................................................. 83 3.3 Finite Element Structural Equations ....................................................................... 88 3.3.1. Truss Element..................................................................................................... 88 3.3.2. Beam Element .................................................................................................... 90 3.3.3. Plate Element ..................................................................................................... 96 3.3.4. Three-Dimensional Elastic Solid ....................................................................... 101 3.4 Global Matrix Equations for the Finite Element Method ....................................... 103 3.4.1. Construction of Global Matrices ........................................................................ 103 3.4.2. Variational Principles for Discrete Structural Systems...................................... 108 3.4.3. Reduced Matrix Equation of Structural Mechanics ........................................... 110 3.4.4. Variational Equations for Discrete Structural Systems...................................... 111 3.4.5. Numerical Integration ........................................................................................ 114 PART II Design Sensitivity Analysis of Linear Structural Systems 4 Discrete Design Sensitivity Analysis ............................................................... 119 4.1 Static Response Design Sensitivity......................................................................... 121 4.1.1 Statement of the Problem ................................................................................... 121 4.1.2 Design Sensitivity Analysis with Reduced Stiffness Matrix.............................. 122 4.1.3 Design Sensitivity Analysis with Generalized Stiffness Matrix ........................ 124 4.1.4 Computational Considerations ........................................................................... 128 4.1.5 Second-Order Design Sensitivity Analysis ........................................................ 130 4.1.6 Examples ............................................................................................................ 135 4.2 Design Sensitivity of the Eigenvalue Problem........................................................ 142 4.2.1 Eigenvalue Design Sensitivity Analysis ............................................................ 143 4.2.2 Design Sensitivity Analysis of Eigenvectors ..................................................... 146 4.2.3 Second-Order Design Sensitivity of a Simple Eigenvalue................................. 149 4.2.4 Systematic Occurrence of Repeated Eigenvalues in Structural Optimization ... 149 4.2.5 Directional Derivatives of Repeated Eigenvalues.............................................. 154 4.2.6 Examples ............................................................................................................ 159 4.3 Transient Dynamic Response Design Sensitivity ................................................... 162 4.3.1 Design Sensitivity Analysis of Damped Elastic Structures................................ 162 4.3.2 Design Sensitivity Analysis of Undamped Structures ....................................... 165 4.3.3 Modal Reduction Method Using Ritz Vectors ................................................... 168 4.3.4 Functionals in a Structural Dynamic Design...................................................... 169 5 Continuum Sizing Design Sensitivity Analysis .............................................. 171 5.1 Design Sensitivity Analysis of Static Response...................................................... 172 5.1.1 Differentiability of Energy Bilinear Forms and Static Response....................... 172 5.1.2 Adjoint Variable Design Sensitivity Analysis.................................................... 174 5.1.3 Analytical Examples of Static Design Sensitivity.............................................. 177 5.1.4 Numerical Considerations .................................................................................. 189 5.1.5 Numerical Examples .......................................................................................... 194 5.2 Eigenvalue Design Sensitivity ................................................................................ 199 5.2.1 Differentiability of Energy Bilinear Forms and Eigenvalues............................. 199 5.2.2 Analytical Examples of Eigenvalue Design Sensitivity..................................... 202 5.2.3 Numerical Considerations .................................................................................. 208 5.2.4 Numerical Examples of Eigenvalue Design Sensitivity..................................... 208 5.3 Transient Dynamic Response Design Sensitivity ................................................... 209 5.3.1 Design Sensitivity of Structural Dynamics Performance................................... 209 5.3.2 Analytical Examples .......................................................................................... 212 5.4 Frequency Response Design Sensitivity ................................................................. 217 5.4.1 Design Sensitivity Analysis of Frequency Response ......................................... 217 5.4.2 Analytical Examples .......................................................................................... 220 5.4.3 Numerical Examples .......................................................................................... 225 5.5 Structural-Acoustic Design Sensitivity Analysis .................................................... 230 5.5.1 Design Sensitivity Analysis of Structural-Acoustic Response........................... 230 5.5.2 Analytical Example ............................................................................................ 232 5.5.3 Numerical Examples .......................................................................................... 234 6 Continuum Shape Design Sensitivity Analysis .............................................. 243 6.1 Material Derivatives for Shape Design Sensitivity Analysis .................................. 243 6.1.1 Material Derivative ............................................................................................ 244 6.1.2 Basic Material Derivative Formulas .................................................................. 248 6.2 Static Response Design Sensitivity Analysis .......................................................... 256 6.2.1 Differentiability of Bilinear Forms and Static Response ................................... 256 6.2.2 Adjoint Variable Design Sensitivity Analysis.................................................... 258 6.2.3 Boundary Method for Static Design Sensitivity................................................. 260 6.2.4 Shape Design Sensitivity Analysis of Local Performance Measures................. 282 6.2.5 Domain Shape Design Sensitivity Method ........................................................ 285 6.2.6 Parameterization of Design Boundary ............................................................... 290 6.2.7 Regularity of Design Velocity Field .................................................................. 291 6.2.8 Numerical Examples .......................................................................................... 396 6.3 Eigenvalue Shape Design Sensitivity Analysis....................................................... 301 6.3.1 Differentiability of Bilinear Forms and Eigenvalues ......................................... 301 6.3.2 Boundary and Domain Methods of Eigenvalue Design Sensitivity................... 304 6.4 Frequency Response Problem................................................................................. 309 6.4.1 Design Sensitivity of Frequency Response ........................................................ 309 6.4.2 Numerical Examples .......................................................................................... 311 6.5 Thermoelastic Problem ........................................................................................... 322 6.5.1 Design Sensitivity Analysis of Thermal Systems .............................................. 322 6.5.2 Design Sensitivity Analysis of Structural Systems ............................................ 324 6.5.3 Adjoint Variable Method in the Thermoelastic Problem ................................... 326 6.6 Second-Order Shape Design Sensitivity Analysis .................................................. 328 6.6.1 Second-Order Material Derivative Formulas ..................................................... 329 6.6.2 Direct Differentiation Method............................................................................ 333 6.6.3 Hybrid Method................................................................................................... 335 6.6.4 Numerical Examples .......................................................................................... 337 7 Configuration Design Sensitivity Analysis..................................................... 347 7.1. Material Derivatives for Configuration Design Sensitivity Analysis ..................... 348 7.1.1. Line Design Component..................................................................................... 349 7.1.2. Surface Design Component................................................................................ 353 7.1.3. Material Derivative of a General Functional...................................................... 354 7.2. Configuration Design Sensitivity Analysis ............................................................. 355 7.2.1. Variation of the Static Response ........................................................................ 355 7.2.2. Eigenvalue Problems.......................................................................................... 358 7.2.3. Analytical Examples .......................................................................................... 360 7.3. Numerical Methods in Configuration Design Sensitivity Analysis ........................ 369 7.3.1. Linear Approximation between Design Parameterization and Design Velocity Field..................................................................................................... 370 7.3.2. Regularity of Design Velocity Fields................................................................. 375 7.3.3. Numerical Examples .......................................................................................... 379 7.4. Structural-Acoustic Problem................................................................................... 389 7.4.1. Variations for the Configuration Design ............................................................ 389 7.4.2. Design Sensitivity Analysis ............................................................................... 393 7.4.3. Design Components ........................................................................................... 395 7.4.4. Analytical Example ............................................................................................ 399 7.4.5. Numerical Example............................................................................................ 400 7.5. Configuration Design Theory for Curved Structure ............................................... 402 7.5.1. Geometric Mapping ........................................................................................... 402 7.5.2. Degenerated Shell Formulation.......................................................................... 405 7.5.3. Material Derivative Formulas ............................................................................ 407 7.5.4. Direct Differentiation Method............................................................................ 408 7.5.5. Adjoint Variable Method ................................................................................... 409 7.5.6. Numerical Example............................................................................................ 410 Appendix..................................................................................................................... 415 A.1 Matrix Calculus Notation........................................................................................ 415 A.2 Basic Function Spaces............................................................................................. 417 A.2.1 Rk; k-Dimensional Euclidean Space ................................................................... 417 A.2.2 Cm(??); m-Times Continuously Differentiable Functions on ??.......................... 420 A.2.3 L2(??); The Space of Lebesgue Square Integrable Functions ............................. 422 A.2.4 L??(??); Space of Essentially Bounded, Lebesgue-Measurable Functions .......... 425 A.2.5 Hm(??); Sobolev Space of Order m..................................................................... 425 A.2.6 0 ( ) m H ?? ; Sobolev m-Space with Compact Support ............................................. 426 A.2.7 The Sobolev Imbedding Theorem...................................................................... 427 A.2.8 Trace Operator ................................................................................................... 428 A.2.9 Product Spaces ................................................................................................... 428 A.3 Differentials and Derivatives in Normed Space...................................................... 429 A.3.1 Mappings in Normed Spaces.............................................................................. 429 A.3.2 Variations and Directional Derivatives .............................................................. 430 A.3.3 Fréchet Differential and Derivative.................................................................... 431 A.3.4 Partial Derivatives and the Chain Rule of Differentiation ................................. 432 References................................................................................................................... 435 Index ..................................................................................................................... 441