有限元方法基礎(chǔ)理論

前言

　　it is thirty-eight years since the The Finite Element Method in Structural and ContinuumMechanics was first published. This book， which was the first dealing with the finiteelement method， provided the basis from which many further developments occurred.

內(nèi)容概要

This book is dedicated to our wives Helen, Mary Lou and Song and our families for their support and patience during the preparation of this book, and also to all of our students and colleagues who over the years have contributed to our knowledge of the finite element method. In particular we would like to mention Professor Eugenio Oniate and his group at CIMNE for their help, encouragement and support during the preparation process.

書(shū)籍目錄

Preface1  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  Problems2  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  Problems3  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 Problems4  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 Problems5  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  Problems6  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  Problems7  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  Problems8  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  Problems9  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  Problems10  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  Problems11  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  Problems12  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  Problems13  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  Problems14  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  Problems15  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  Problems16  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  Problems17  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  Problems18  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 remarks19  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 modulesAppendix A: Matrix algebraAppendix B: Tensor-indicial notation in the approximation of elasticity problemsAppendix C: Solution of simultaneous linear algebraic equationsAppendix D: Some integration formulae for a triangleAppendix E: Some integration formulae for a tetrahedronAppendix F: Some vector algebraAppendix G: Integration by parts in two or three dimensions (Green's theorem)Appendix H: Solutions exact at nodesAppendix I: Matrix diagonalization or lumpingAuthor indexSubject index

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