Finite Element Analysis: Method, Verification and Validation

1 Introduction to FEM 3 1.1 An introductory problem 6 1.2 Generalized formulation 9 1.2.1 The exact solution 9 1.2.2 The principle of minimum potential energy 14 1.3 Approximate solutions 16 1.3.1 The standard polynomial space 17 1.3.2 Finite element spaces in one dimension 20 1.3.3 Computation of the coefficient matrices 22 1.3.4 Computation of the right hand side vector 26 1.3.5 Assembly 27 1.3.6 Condensation 30 1.3.7 Enforcement of Dirichlet boundary conditions 30 1.4 Post-solution operations 33 1.4.1 Computation of the quantities of interest 33 1.5 Estimation of error in energy norm 37 1.5.1 Regularity 38 1.5.2 A priori estimation of the rate of convergence 38 1.5.3 A posteriori estimation of error 40 1.5.4 Error in the extracted QoI 46 1.6 The choice of discretization in 1D 47 1.6.1 The exact solution lies in Hk(I), k . 1 > p 47 1.6.2 The exact solution lies in Hk(I), k . 1 <= p 49 1.7 Eigenvalue problems 52 1.8 Other finite element methods 57 1.8.1 The mixed method 59 1.8.2 Nitsche's method 60 2 Boundary value problems 63 2.1 Notation 63 2.2 The scalar elliptic boundary value problem 65 2.2.1 Generalized formulation 66 2.2.2 Continuity 68 2.3 Heat conduction 68 2.3.1 The differential equation 70 2.3.2 Boundary and initial conditions 71 2.3.3 Boundary conditions of convenience 73 2.3.4 Dimensional reduction 75 2.4 Linear elasticity - strong form 82 2.4.1 The Navier equations 86 2.4.2 Boundary and initial conditions 86 2.4.3 Symmetry, antisymmetry and periodicity 88 2.4.4 Dimensional reduction in linear elasticity 89 2.4.5 Incompressible elastic materials 93 2.5 Stokes flow 95 2.6 Elasticity - generalized formulation 96 2.6.1 The principle of minimum potential energy 98 2.6.2 The RMS measure of stress 100 2.6.3 The principle of virtual work 101 2.6.4 Uniqueness 102 2.7 Residual stresses 106 2.8 Chapter summary 108 3 Implementation 111 3.1 Standard elements in two dimensions 111 3.2 Standard polynomial spaces 111 3.2.1 Trunk spaces 111 3.2.2 Product spaces 112 3.3 Shape functions 112 3.3.1 Lagrange shape functions 113 3.3.2 Hierarchic shape functions 115 3.4 Mapping functions in two dimensions 118 3.4.1 Isoparametric mapping 118 3.4.2 Mapping by the blending function method 121 3.4.3 Mapping algorithms for high order elements 123 3.5 Finite element spaces in two dimensions 125 3.6 Essential boundary conditions 125 3.7 Elements in three dimensions 126 3.7.1 Mapping functions in three-dimensions 127 3.8 Integration and differentiation 129 3.8.1 Volume and area integrals 129 3.8.2 Surface and contour integrals 131 3.8.3 Differentiation 131 3.9 Stiffness matrices and load vectors 132 3.9.1 Stiffness matrices 133 3.9.2 Load vectors 134 3.10 Post-solution operations 135 3.11 Computation of the solution and its first derivatives 135 3.12 Nodal forces 137 3.12.1 Nodal forces in the h-version 137 3.12.2 Nodal forces in the p-version 140 3.12.3 Nodal forces and stress resultants 141 3.13 Chapter summary 142 4 Verification 143 4.1 Regularity in two and three dimensions 143 4.2 The Laplace equation in two dimensions 144 4.2.1 2D model problem, uEX element of Hk(), k . 1 > p 146 4.2.2 2D model problem, uEX element of Hk(), k . 1 <= p 148 4.2.3 Computation of the flux vector in a given point 151 4.2.4 Computation of the flux intensity factors 153 4.2.5 Material interfaces 158 4.3 The Laplace equation in three dimensions 160 4.4 Planar elasticity 164 4.4.1 Problems of elasticity on an L-shaped domain 165 4.4.2 Crack tip singularities in 2D 165 4.4.3 Forcing functions acting on boundaries 170 4.5 Robustness 172 4.6 Solution verification 177 5 Simulation 185 5.1 Development of a mathematical model 186 5.1.1 The Bernoulli-Euler beam model 187 5.1.2 Historical notes 188 5.2 FE modeling vs simulation 190 5.2.1 Numerical simulation 190 5.2.2 Finite element modeling 192 5.2.3 Calibration versus tuning 195 5.2.4 Simulation governance 196 5.2.5 Milestones in numerical simulation 197 5.2.6 Example: The Girkmann problem 199 5.2.7 Example: Fastened structural connection 203 5.2.8 Finite element model 210 5.2.9 Example: Coil spring with displacement boundary conditions 215 5.2.10 Example: Coil spring segment 220 6 Calibration, Validation and Ranking 225 6.1 Fatigue data 226 6.1.1 Equivalent stress 227 6.1.2 Statistical models 227 6.1.3 The effect of notches 228 6.1.4 Formulation of predictors of fatigue life 229 6.2 The predictors of Peterson and Neuber 230 6.2.1 The effect of notches - calibration 232 6.2.2 The effect of notches - validation 235 6.2.3 Updated calibration 237 6.2.4 The fatigue limit 240 6.2.5 Discussion 242 6.3 The predictor Galpha 243 6.3.1 Calibration of ß(V, alpha) 244 6.3.2 Ranking 246 6.3.3 Comparison of Galpha with Peterson's revised predictor 246 6.4 Biaxial test data 247 6.4.1 Axial, torsional and combined in-phase loading 248 6.4.2 The domain of calibration 249 6.4.3 Out-of-phase biaxial loading 252 6.4.4 Validation 255 6.4.5 Selection of the prior 256 6.4.6 Discussion 259 7 Beams, plates and shells 261 7.1 Beams 261 7.1.1 The Timoshenko beam 263 7.1.2 The Bernoulli-Euler beam 268 7.2 Plates 273 7.2.1 The Reissner-Mindlin plate 276 7.2.2 The Kirchhoff plate 281 7.2.3 The transverse variation of displacements 283 7.3 Shells 287 7.3.1 Hierarchic thin solid models 291 7.4 Chapter summary 295 8 Aspects of multiscale models 297 8.1 Unidirectional fiber-reinforced laminae 297 8.1.1 Determination of material constants 300 8.1.2 The coefficients of thermal expansion 300 8.1.3 Examples 301 8.1.4 Localization 304 8.1.5 Prediction of failure in composite materials 305 8.1.6 Uncertainties 307 8.2 Discussion 307 9 Non-linear models 309 9.1 Heat conduction 309 9.1.1 Radiation 309 9.1.2 Nonlinear material properties 310 9.2 Solid mechanics 310 9.2.1 Large strain and rotation 311 9.2.2 Structural stability and stress stiffening 314 9.2.3 Plasticity 321 9.2.4 Mechanical contact 327 9.3 Chapter summary 335 A Definitions 337 A.1 Normed linear spaces, linear functionals and bilinear forms 338 A.1.1 Normed linear spaces 338 A.1.2 Linear forms 339 A.1.3 Bilinear forms 339 A.2 Convergence in the space X 339 A.2.1 The space of continuous functions 339 A.2.2 The space Lp() 340 A.2.3 Sobolev space of order 1 340 A.2.4 Sobolev spaces of fractional index 341 A.3 The Schwarz inequality for integrals 342 B Proof of convergence 343 C Convergence in 3D 345 D Legendre polynomials 349 D.1 Shape functions based on Legendre polynomials 350 E Numerical quadrature 353 E.1 Gaussian quadrature 353 E.2 Gauss-Lobatto quadrature 355 F Polynomial mapping functions 357 F.1 Interpolation on surfaces 359 F.1.1 Interpolation on the standard quadrilateral element 359 F.1.2 Interpolation on the standard triangle 359 G Corner singularities 361 G.1 The Airy stress function 361 G.2 Stress-free edges 363 G.2.1 Symmetric eigenfunctions 364 G.2.2 Antisymmetric eigenfunctions 365 G.2.3 The L-shaped domain 366 G.2.4 Corner points 367 H Stress intensity factors 369 H.1 Singularities at crack tips 369 H.2 The contour integral method 370 H.3 The energy release rate 372 H.3.1 Symmetric (Mode I) loading 372 H.3.2 Antisymmetric (Mode II) loading 373 H.3.3 Combined (Mode I and Mode II) loading 373 H.3.4 Computation by the stiffness derivative method 374 I Fundamentals of data analysis 375 I.1 Statistical foundations 375 I.2 Test data 377 I.3 Statistical models 378 I.4 Ranking 387 I.5 Confidence intervals 387 J Fastener forces 389 K Useful algorithms 393 K.1 The traction vector 393 K.2 Transformation of vectors 394 K.3 Transformation of stresses 396 K.4 Principal stresses 396 K.5 The von Mises stress 397 K.6 Statically equivalent forces and moments 398 K.6.1 Technical formulas for stress 400