有限元方法基础理论

Preface 1 The standard discrete system and origins of the finite element method 1.1 Introduction 1.2 The structural element and the structural system 1.3 Assembly and analysis of a structure 1.4 The boundary conditions 1.5 Electrical and fluid networks 1.6 The general pattern 1.7 The standard discrete system 1.8 Transformation of coordinates 1.9 Problems 2 A direct physical approach to problems in elasticity: plane stress 2.1 Introduction 2.2 Direct formulation of finite element characteristics 2.3 Generalization to the whole region - internal nodal force concept abandoned 2.4 Displacement approach as a minimization of total potential energy 2.5 Convergence criteria 2.6 Discretization error and convergence rate 2.7 Displacement functions with discontinuity between elements -non-conforming elements and the patch test 2.8 Finite element solution process 2.9 Numerical examples 2.10 Concluding remarks 2.11 Problems 3 Generalization of the finite element concepts. Galerkin-weighted residual and variational approaches 3.1 Introduction 3.2 Integral or 'weak' statements equivalent to the differential equations 3.3 Approximation to integral formulations: the weighted residual-Galerkin method 3.4 Vitual work as the 'weak form' of equilibrium equations for analysis of solids or fluids 3.5 Partial discretization 3.6 Convergence 3.7 What are 'variational principles' ? 3.8 'Natural' variational principles and their relation to governing differential equations 3.9 Establishment of natural variational principles for linear, self-adjoint, differentaal equations 3.10 Maximum, minimum, or a saddle point? 3.11 Constrained variational principles. Lagrange multipliers 3.12 Constrained variational principles. Penalty function and perturbed lagrangian methods 3.13 Least squares approximations 3.14 Concluding remarks - finite difference and boundary methods 3.15 Problems 4 Standard' and 'hierarchical' element shape functions: some general families of Co continuity 4.1 Introduction 4.2 Standard and hierarchical concepts 4.3 Rectangular elements - some preliminary considerations 4.4 Completeness of polynomials 4.5 Rectangular elements - Lagrange family 4.6 Rectangular dements - 'serendipity' family 4.7 Triangular element family 4.8 Line elements 4.9 Rectangular prisms - Lagrange family 4.10 Rectangular prisms - 'serendipity' family 4.11 Tetrahedral dements 4.12 Other simple three-dimensional elements 4.13 Hierarchic polynomials in one dimension 4.14 Two- and three-dimensional, hierarchical elements of the 'rectangle' or 'brick' type 4.15 Triangle and tetrahedron family 4.16 Improvement of conditioning with hierarchical forms 4.17 Global and local finite element approximation 4.18 Elimination of internal parameters before assembly - substructures 4.19 Concluding remarks 4.20 Problems 5 Mapped elements and numerical integration - 'infinite' and 'singularity elements' 5.1 Introduction 5.2 Use of 'shape functions' in the establishment of coordinate transformations 5.3 Geometrical conformity of elements 5.4 Variation of the unknown function within distorted, curvilinear elements. Continuity requirements 5.5 Evaluation of element matrices. Transformation in ε, η, ζ coordinates 5.6 Evaluation of element matrices. Transformation in area and volumecoordinates 5.7 Order of convergence for mapped elements 5.8 Shape functions by degeneration 5.9 Numerical integration - one dimensional 5.10 Numerical integration - rectangular (2D) or brick regions (3D) 5.11 Numerical integration - triangular or tetrahedral regions 5.12 Required order of numerical integration 5.13 Generation of finite element meshes by mapping. Blending functions 5.14 Infinite domains and infinite elements 5.15 Singular elements by mapping - use in fracture mechanics, etc. 5.16 Computational advantage of numerically integrated finite elements 5.17 Problems 6 Problems in linear elasticity 6.1 Introduction 6.2 Governing equations 6.3 Finite element approximation 6.4 Reporting of results: displacements, strains and stresses 6.5 Numerical examples 6.6 Problems 7 Field problems - heat conduction, electric and magnetic potential and fluid flow 7.1 Introduction 7.2 General quasi-harmonic equation 7.3 Finite element solution process 7.4 Partial discretization - transient problems 7.5 Numerical examples - an assessment of accuracy 7.6 Concluding remarks 7.7 Problems 8 Automatic mesh generation 8.1 Introduction 8.2 Two-dimensional mesh generation - advancing front method 8.3 Surface mesh generation 8.4 Three-dimensional mesh generation - Delaunay triangulation 8.5 Concluding remarks 8.6 Problems 9 The patch test, reduced integration, and non-conforming elements 9.1 Introduction 9.2 Convergence requirements 9.3 The simple patch test (tests A and B) - a necessary condition for convergence 9.4 Generalized patch test (test C) and the single-element test 9.5 The generality of a numerical patch test 9.6 Higher order patch tests 9.7 Application of the patch test to plane elasticity dements with 'standard' and 'reduced' quadrature 9.8 Application of the patch test to an incompatible element 9.9 Higher order patch test - assessment of robustness 9.10 Concluding remarks 9.11 Problems 10 Mixed formulation and constraints - complete field methods 10.1 Introduction 10.2 Discretization of mixed forms - some general remarks 10.3 Stability of mixed approximation. The patch test 10.4 Two-fidd mixed formulation in elasticity 10.5 Three-field mixed formulations in elasticity 10.6 Complementary forms with direct constraint 10.7 Concluding remarks - mixed formulation or a test of element 'robustness' 10.8 Problems 11 Incompressible problems, mixed methods and other procedures of solution 11.1 Introduction 11.2 Deviatoric stress and strain, pressure and volume change 11.3 Two-field incompressible elasticity (up form) 11.4 Three-field nearly incompressible elasticity (u-p-~o form) 11.5 Reduced and selective integration and its equivalence to penalized mixed problems 11.6 A simple iterative solution process for mixed problems: Uzawa method 11.7 Stabilized methods for some mixed elements failing the incompressibility patch test 11.8 Concluding remarks 11.9 Problems 12 Multidomain mixed approximations - domain decomposition and 'frame' methods 12.1 Introduction 12.2 Linking of two or more subdomains by Lagrange multipliers 12.3 Linking of two or more subdomains by perturbed lagrangian and penalty methods 12.4 Interface displacement 'frame' 12.5 Linking of boundary (or Trefftz)-type solution by the 'frame' of specified displacements 12.6 Subdomains with 'standard' elements and global functions 12.7 Concluding remarks 12.8 Problems 13 Errors, recovery processes and error estimates 13.1 Definition of errors 13.2 Superconvergence and optimal sampling points 13.3 Recovery of gradients and stresses 13.4 Superconvergent patch recovery -, SPR 13.5 Recovery by equilibration of patches - REP 13.6 Error estimates by recovery 13.7 Residual-based methods 13.8 Asymptotic behaviour and robustness of error estimators - the Babuska patch test 13.9 Bounds on quantities of interest 13.10 Which errors should concern us? 13.11 Problems 14 Adaptive finite element refinement 14.1 Introduction 14.2 Adaptive h-refinement 14.3 p-refinement and hp-refinement 14.4 Concluding remarks 14.5 Problems 15 Point-based and partition of unity approximations. Extended finite element methods 15.1 Introduction 15.2 Function approximation 15.3 Moving least squares approximations - restoration of continuity of approximation 15.4 Hierarchical enhancement of moving least squares expansions 15.5 Point collocation - finite point methods 15.6 Galerkin weighting and finite volume methods 15.7 Use of hierarchic and special functions based on standard finite elements satisfying the partition of unity requirement 15.8 Concluding remarks 15.9 Problems 16 The time dimension - semi-discretization of field and dynamic problems and analytical solution procedures 16.1 Introduction 16.2 Direct formulation of time-dependent problems with spatial finite element subdivision 16.3 General classification 16.4 Free response - eigenvalues for second-order problems and dynamic vibration 16.5 Free response - eigenvalues for first-order problems and heat conduction, etc. 16.6 Free response - damped dynamic eigenvalues 16.7 Forced periodic response 16.8 Transient response by analytical procedures 16.9 Symmetry and repeatability 16.10 Problems 17 The time dimension - discrete approximation in time 17.1 Introduction 17.2 Simple time-step algorithms for the first-order equation 17.3 General single-step algorithms for first- and second-order equations 17.4 Stability of general algorithms 17.5 Multistep recurrence algorithms 17.6 Some remarks on general performance of numerical algorithms 17.7 Time discontinuous Galerkin approximation 17.8 Concluding remarks 17.9 Problems 18 Coupled systems 18.1 Coupled problems - definition and classification 18.2 Fluid-structure interaction (Class I problems) 18.3 Soil-pore fluid interaction (Class II problems) 18.4 Partitioned single-phase systems - implicit--explicit partitions(Class I problems) 18.5 Staggered solution processes 18.6 Concluding remarks 19 Computer procedures for finite dement analysis 19.1 Introduction 19.2 Pre-processing module: mesh creation 19.3 Solution module 19.4 Post-processor module 19.5 User modules Appendix A: Matrix algebra Appendix B: Tensor-indicial notation in the approximation of elasticity problems Appendix C: Solution of simultaneous linear algebraic equations Appendix D: Some integration formulae for a triangle Appendix E: Some integration formulae for a tetrahedron Appendix F: Some vector algebra Appendix G: Integration by parts in two or three dimensions (Green's theorem) Appendix H: Solutions exact at nodes Appendix I: Matrix diagonalization or lumping Author index Subject index

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