The Finite Element Method: Its Basis and Fundamentals, Sixth Edition | by O. C. Zienkiewicz, R. L. Taylor and J.Z. Zhu | ISBN: 9780750663205. Physics Engineering Books. Problems in Linear Elasticity. Heat Conduction, Electric and Magnetic Potential and Fluid Flow

The Finite Element Method: Its Basis and Fundamentals, Sixth Edition
by O. C. Zienkiewicz, R. L. Taylor and J.Z. Zhu
Amply supplemented by exercises, worked solutions and computer algorithms, this formidable resource covers the theory and application of FEM, the basis of the method, its application to advanced solid and structural mechanics and to computational fluid dynamics.
The Finite Element Method—Its Basis and Fundamentals, Sixth Edition
Resources to Accompany This Book

Chapter 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

Chapter 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

Chapter 3 – Generalization of the Finite Element Concepts. Galerkin-Weighted Residual and Variational Approaches
3.1: Introduction
Weighted Residual Methods

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
3.5: Partial Discretization
3.6: Convergence
Variational Principles

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
3.13: Least Squares Approximations
3.14: Concluding Remarks — Finite Difference and Boundary Methods
3.15: Problems

Chapter 4 – ‘Standard’ and ‘Hierarchical’ Element Shape Functions: Some General Families of C0 Continuity
4.1: Introduction
4.2: Standard and Hierarchical Concepts
Part 1. ‘Standard’ Shape Functions—Two-Dimensional Elements

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
One-Dimensional Elements

4.8 Line Elements
Three-Dimensional Elements

4.9 Rectangular Prisms — Lagrange Family
4.10: Rectangular Prisms — ‘Serendipity’ Family
4.11: Tetrahedral Elements
4.12: Other Simple Three-Dimensional Elements
Part 2. Hierarchical Shape Functions

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

Chapter 5 – Mapped Elements and Numerical Integration — ‘Infinite’ and ‘Singularity Elements’
5.1: Introduction
Parametric Curvilinear Coordinates

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

Chapter 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

Chapter 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

Chapter 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

Chapter 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 Elements 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

Chapter 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 Element ‘Robustness’
10.8: Problems

Chapter 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 (u-p Form)
11.4: Three-Field Nearly Incompressible Elasticity (u-p-?v 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

Chapter 12 – Multidomain Mixed Approximations – Domain Decomposition and ‘Frame’ Methods
12.1: Introduction
Domain Decompostion Methods

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
Frame 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

Chapter 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 Babuška Patch Test
13.9: Bounds on Quantities of Interest
13.10: Which Errors Should Concern Us?
13.11: Problems

Chapter 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

Chapter 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

Chapter 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
Eigenvalues and Analytical Solution Procedures

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

Chapter 17 – The Time Dimension — Discrete Approximation in Time
17.1: Introduction
Single-Step Algorithms

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
Multistep Methods

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

Chapter 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

Chapter 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
Definition of a Matrix
Matrix Addition or Subtraction
Transpose of a Matrix
Inverse of a Matrix
A Sum of Products
Transpose of a Product
Symmetric Matrices
The Eigenvalue Problem

Appendix B – Tensor-Indicial Notation in the Approximation of Elasticity Problems
Indicial Notation: Summation Convention
Derivatives and Tensorial Relations
Coordinate Transformation
Equilibrium and Energy
Elastic Constitutive Equations
Finite Element Approximation
Relation Between Indicial and Matrix Notation

Appendix C – Solution of Simultaneous Linear Algebraic Equations
Direct Solution
Iterative Solution

Appendix D – Some Integration Formulae for a Triangle
Appendix E – Some Integration Formulae for a Tetrahedron
Appendix F – Some Vector Algebra
Addition and Subtraction
‘Scalar’ Products
Length of Vector
Direction Cosines
‘Vector’ or Cross Product
Elements of Area and Volume

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

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