## 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

ISBN:9780750663205

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**

Preface

Overview

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

References

**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

References

**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

References

**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

References

**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

Transformations

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

References

**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

References

**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

References

**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

References

**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

References

**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

References

**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

References

**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

References

**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

References

**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

References

**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

References

**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

References

**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

References

**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

References

**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

References

**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

Partitioning

The Eigenvalue Problem

**Appendix B – Tensor-Indicial Notation in the Approximation of Elasticity Problems**

Introduction

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

References

**Appendix C – Solution of Simultaneous Linear Algebraic Equations**

Direct Solution

Iterative Solution

References

**Appendix D – Some Integration Formulae for a Triangle**

** Appendix E – Some Integration Formulae for a Tetrahedron**

** Appendix F – Some Vector Algebra**

Overview

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**

Overview

References

**Appendix I – Matrix Diagonalization or Lumping**

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