有限元方法基础论第6版
目 录内容简介
Preface
1 The standard discrete system and origins of the finite element method
1.1 Introduction
1.2 The struraal 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 Virtual work as the weak form of equilibrium equations for analysis of solids or fluids
69
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， differential 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
88
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 elements - 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 elements
4.12 Other simple three-dimensional elements
4.13 Hierarchic polynomials in one dimension
4.14 Two- and three-dimensional， hierarchical elements of the rectangleor 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 coordinates
5.6 Evaluation of element matrices. Transformation in area and volume coordinates
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
332
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 elements withstandard 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-field 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 elementrobustness
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- 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 us7
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 element 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 （Greens theorem）
Appendix H： Solutions exact at nodes
Appendix I： Matrix diagonalization or lumping
Author index
Subject index
1 The standard discrete system and origins of the finite element method
1.1 Introduction
1.2 The struraal 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 Virtual work as the weak form of equilibrium equations for analysis of solids or fluids
69
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， differential 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
88
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 elements - 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 elements
4.12 Other simple three-dimensional elements
4.13 Hierarchic polynomials in one dimension
4.14 Two- and three-dimensional， hierarchical elements of the rectangleor 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 coordinates
5.6 Evaluation of element matrices. Transformation in area and volume coordinates
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
332
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 elements withstandard 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-field 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 elementrobustness
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- 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 us7
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 element 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 （Greens theorem）
Appendix H： Solutions exact at nodes
Appendix I： Matrix diagonalization or lumping
Author index
Subject index
目 录内容简介
《有限元方法基础论第6版》是一套在国际上颇具权威性的经典著作（共三卷），由有限元法的创始人Zienkiewicz教授和美国加州大学Taylor教授合作撰写，初版于1967年，多次修订再版，深受力学界和工程界科技人员的欢迎。《有限元方法基础论第6版》的特点是理论可靠，内容全面，既有基础理论，又有其具体应用。 第1卷目次：标准的离散系统和有限元方法的起源；弹性力学问题的直接方法；有限元概念的推广，Galerkin加权残数和变分法；‘标准的’和‘晋级的’单元形函数：Co连续性单元族；映射单元和数值积分—无限元和奇异元；线性弹性问题；场问题—热传导、电磁势、流体流动；自动网格生成；拼法试验，简缩积分和非协调元；混合公式和约束—完全场方法；不可压缩材料，混合方法和其它解法；多区域混合逼近-区域分解和“框架”方法；误差、恢复过程和误差估计；自适应有限元细分；以点为基础和单元分割的近似，扩展的有限元方法；时间维-场的半离散化、动力学问题以及分析解方法；时间维—时间的离散化近似；耦合系统；有限元分析和计算机处理。
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