Introduction to Finite Element Analysis : Formulation, Verification and Validation

About the Authors xiiiPreface xv 1 Introduction 11.1 Numerical simulation 21.2 Why is numerical accuracy important? 111.3 Chapter summary 14 2 An outline of the finite element method 172.1 Mathematical models in one dimension 172.2 Approximate solution 292.3 Generalized formulation in one dimension 332.4 Finite element approximations 382.5 FEM in one dimension 442.6 Properties of the generalized formulation 672.7 Error estimation based on extrapolation 732.8 Extraction methods 752.9 Laboratory exercises 772.10 Chapter summary 77 3 Formulation of mathematical models 793.1 Notation 793.2 Heat conduction 813.3 The scalar elliptic boundary value problem 923.4 Linear elasticity 933.5 Incompressible elastic materials 1033.6 Stokes' flow 1053.7 The hierarchic view of mathematical models 1063.8 Chapter summary 106 4 Generalized formulations 1094.1 The scalar elliptic problem 1094.2 The principle of virtual work 1154.3 Elastostatic problems 1174.4 Elastodynamic models 1334.5 Incompressible materials 1404.6 Chapter summary 143 5 Finite element spaces 1455.1 Standard elements in two dimensions 1455.2 Standard polynomial spaces 1465.3 Shape functions 1475.4 Mapping functions in two dimensions 1525.5 Elements in three dimensions 1575.6 Integration and differentiation 1585.7 Stiffness matrices and load vectors 1625.8 Chapter summary 164 6 Regularity and rates of convergence 1676.1 Regularity 1676.2 Classification 1706.3 The neighborhood of singular points 1736.4 Rates of convergence 1936.5 Chapter summary 212 7 Computation and verification of data 2157.1 Computation of the solution and its first derivatives 2157.2 Nodal forces 2177.3 Verification of computed data 2227.4 Flux and stress intensity factors 2287.5 Chapter summary 235 8 What should be computed and why? 2378.1 Basic assumptions 2388.2 Conceptualization: drivers of damage accumulation 2388.3 Classical models of metal fatigue 2408.4 Linear elastic fracture mechanics 2508.5 On the existence of a critical distance 2528.6 Driving forces for damage accumulation 2538.7 Cycle counting 2548.8 Validation 2558.9 Chapter summary 257 9 Beams, plates and shells 2619.1 Beams 2619.2 Plates 2749.3 Shells 2839.4 The Oak Ridge experiments 2889.5 Chapter summary 296 10 Nonlinear models 29710.1 Heat conduction 29710.2 Solid mechanics 29810.3 Chapter summary 312 A Definitions 313A.1 Norms and seminorms 313A.2 Normed linear spaces 314A.3 Linear functionals 314A.4 Bilinear forms 314A.5 Convergence 315A.6 Legendre polynomials 315A.7 Analytic functions 316A.8 The Schwarz inequality for integrals 317 B Numerical quadrature 319B.1 Gaussian quadrature 320B.2 Gauss-Lobatto quadrature 321 C Properties of the stress tensor 323C.1 The traction vector 323C.2 Principal stresses 324C.3 Transformation of vectors 325C.4 Transformation of stresses 326 D Computation of stress intensity factors 329D.1 The contour integral method 329D.2 The energy release rate 330 E Saint-Venant's principle 335E.1 Green's function for the Laplace equation 335E.2 Model problem 336 F Solutions for selected exercises 343 Bibliography 363 Index 369