Green's Functions and Finite Elements $c

Cover......Page 1 Green’s Functions and Finite Elements......Page 3 Preface......Page 5 Contents......Page 8 6 Appendix......Page 13 1.1 What are Green's Functions?......Page 14 1.2 The Importance of Green's Functions for FE-Analysis......Page 15 1.3.2 Green's Function......Page 16 1.3.3 Finite Elements......Page 18 1.3.4 Finite Elements and Green's Function......Page 19 1.4 Entanglement......Page 21 1.4.1 Functionals......Page 24 1.4.2 Proof......Page 28 1.5 Goal Oriented Refinement......Page 30 1.6 Model Adaptivity......Page 31 1.6.1 Local & Global......Page 38 1.7 How to Calculate Influence Functions with Finite Elements......Page 39 References......Page 46 2.1 Elements of Functional Analysis......Page 48 2.1.1 Notation......Page 49 2.1.2 Vector Spaces and Scalar Product......Page 50 2.1.3 Linear Functionals......Page 51 2.1.4 Projection......Page 53 2.1.5 Variational Problems......Page 54 2.1.6 Equivalent Norms......Page 55 2.1.8 Sobolev Spaces......Page 58 2.2.2 The Laplace Operator......Page 61 2.2.3 Linear Self-Adjoint Operators......Page 65 2.3 Duality......Page 66 2.3.1 Linear Algebra......Page 67 2.3.2 Vectors and Linear Functionals......Page 69 2.4 Influence Functions......Page 70 2.4.1 Influence Function for u(x)......Page 71 2.4.2 Influence Function for u'(x)......Page 73 2.4.3 Weak Influence Function for u(x)......Page 75 2.4.4 A Sequence that Converges to G1......Page 79 2.4.5 Elevators and Escalators......Page 81 2.4.6 Influence Functions in Higher Dimensions......Page 83 2.4.7 Weak Influence Functions......Page 85 2.4.8 Non-Zero Boundary Values......Page 86 2.4.9 Average Values of Stresses......Page 87 2.5 Properties of Green's Functions......Page 91 2.5.2 Maxwell......Page 94 2.5.3 Modes of Decay......Page 95 2.5.4 Dipoles and Monopoles......Page 97 2.5.5 Multipole Expansion......Page 99 2.5.6 Infinite Energy......Page 103 2.5.7 Genealogy of Influence Functions......Page 104 2.6 Sobolev's Embedding Theorem......Page 105 2.7 Fundamental Solutions......Page 107 2.7.1 Influence Function......Page 108 2.8 Ill-Posed Problems......Page 109 2.9 Nonlinear Problems......Page 110 2.9.1 Lagrange Multiplier......Page 111 2.9.2 Lagrange Multiplier and Linear Algebra......Page 112 2.9.3 Nonlinear Functionals......Page 113 2.9.4 Nonlinear Problems......Page 115 2.10 Mixed Problems......Page 118 References......Page 119 3.1 Poisson Equation......Page 121 3.2 The FE-Load Case ph......Page 123 3.2.2 Notation......Page 125 3.3.1 Betti's Theorem: Extended......Page 129 3.3.2 Tottenham's Equation......Page 130 3.3.3 Maxwell's Theorem: Extended......Page 133 3.4 Proxies......Page 135 3.4.1 Gh = G on mathcal......Page 136 3.4.2 δh = δ on mathcal......Page 137 3.4.3 Jh(u) = J(u) on mathcal......Page 140 3.4.4 Summary......Page 144 3.5 Dirac Energy......Page 146 3.6 Generalized Green's Functions......Page 150 3.6.1 Arbitrary Deltas......Page 151 3.7 Influence Functions for Integral Values......Page 155 3.7.1 Nodal Forces......Page 156 3.8 Weak Influence Functions......Page 158 3.9 Weak Influence Functions Have More Choices......Page 162 3.10 Nodal Form of Influence Functions......Page 164 3.10.2 Sensitivity Plots......Page 166 3.11 Nodal Values of Green's Functions......Page 169 3.11.2 Influence Function for p(x)......Page 170 3.11.3 The Foot Print of ph......Page 173 3.12 The Inverse Stiffness Matrix......Page 174 3.12.1 Examples......Page 176 3.13 Condition of a Stiffness Matrix......Page 178 3.13.1 The Triple Product......Page 181 3.14 Interpolation......Page 182 3.14.1 The Nodal Vector uI......Page 184 3.15 Infinite Stresses......Page 189 3.15.1 Archimedes' Lever......Page 190 3.15.2 Continuous Beam......Page 193 3.15.3 Cantilever Plate......Page 194 3.16 Why Do Singularities Matter?......Page 196 3.17 Nature Makes No Jumps: Finite Elements Do......Page 197 3.18 Influence Functions for Support Reactions......Page 199 3.18.1 Global Equilibrium......Page 201 3.19 The Path the Load Takes......Page 203 3.20 The Path the Influence Function takes......Page 206 3.21 Mixed Problems......Page 208 3.21.1 Tottenham's Equation for Mixed Problems......Page 213 3.22 Condensation of a Stiffness Matrix......Page 214 3.23 p-Method......Page 216 References......Page 219 4.1 Asymptotic Error Analysis......Page 221 4.2 Goal-Oriented Refinement......Page 224 4.3 Comparison......Page 225 4.4 Primal and Dual Error......Page 227 4.5 An Analysis of the Goal-Oriented Error Estimator......Page 228 4.6 The Algebra of the Residuals......Page 230 4.7 Goal-Oriented Refinement for Nonlinear Problems......Page 231 4.7.1 Estimates......Page 232 4.7.3 Implementation......Page 235 4.8 Drift......Page 238 4.9 Combination of Modeling and Discretization Error......Page 240 4.10 Pollution......Page 241 4.11 Gauss Points and Green's Functions......Page 248 References......Page 250 5.1 Linear Algebra......Page 252 5.2.1 Determining uc......Page 254 5.2.2 Determining Effects......Page 255 5.3.1 One Entry on the Diagonal Changes, kii + Δk......Page 258 5.3.2 The Inverse of the Updated Stiffness Matrix Kc......Page 259 5.5 Force Method......Page 261 5.5.1 Notation......Page 263 5.5.2 Changes on the Diagonal......Page 264 5.5.3 The Inverse......Page 265 5.6 Example......Page 266 5.6.1 What It Means......Page 267 5.6.2 The Inverse of ΔKe......Page 269 5.6.3 Example......Page 272 5.6.4 Collapse......Page 275 5.7 Functionals......Page 276 5.7.1 The Gradient of a Functional......Page 277 5.8 Weak Formulations and the d-Term......Page 278 5.8.1 Linear Problems......Page 279 5.8.2 The Error in Functionals......Page 281 5.9.1 Continuous and Discrete Case......Page 288 5.9.2 Long & Strong and Short & Weak......Page 290 5.10 The Approximation approx......Page 291 5.11 Linearization......Page 293 5.12.1 Focus on a Point......Page 296 5.12.2 Focus on an Element......Page 305 5.12.3 Beams......Page 311 5.12.4 Poisson Problem......Page 313 5.12.5 Kirchhoff Plates......Page 315 5.12.6 Analysis......Page 316 5.13 Equations for the Unknown Stresses on Ωe......Page 317 5.13.1 Computational Aspects......Page 319 5.14 Adjoint Method of Sensitivity Analysis......Page 323 5.15 Linear Versus Nonlinear......Page 325 References......Page 327 6.1 Nonlinear Elasticity......Page 328 6.1.1 Linearization......Page 330 6.2 Software......Page 331 Reference......Page 332 Index......Page 333 This book elucidates how Finite Element methods look like from the perspective of Green’s functions, and shows new insights into the mathematical theory of Finite Elements. Practically, this new view on Finite Elements enables the reader to better assess solutions of standard programs and to find better model of a given problem. The book systematically introduces the basic concepts how Finite Elements fulfill the strategy of Green’s functions and how approximating of Green’s functions. It discusses in detail the discretization error and shows that are coherent with the strategy of “goal oriented refinement”. The book also gives much attention to the dependencies of FE solutions from the parameter set of the model. This book shows how Finite Elements fulfill the strategy of Green's functions, helping the reader to better assess solutions of standard programs and to find the optimal model of a given problem. Covers discretization error and "goal oriented refinement."