Table of Contents

Chapter 1　Introduction of Finite Element Method　1
1.1　Development Process of Finite Element Method　1
1.2　Computation Procedure of Finite Element Method　3
1.3　Main Contents of the Book　4
1.4　Exercises　5
Chapter 2　Fundamentals of Elasticity Mechanics　6
2.1　Displacements　7
2.2　Strains　8
2.3　Stresses　8
2.4　Geometric Equations　9
2.5　Constitutive Equations　10
2.6　Equilibrium Equations　12
2.6.1　Three-Dimensional Problems　12
2.6.2　Two-Dimensional Plane Stress and Strain Problems　13
2.6.3　Two-Dimensional Axisymmetric Problems　13
2.7　Boundary Conditions　13
2.8　Exercises　14
Chapter 3　Weak Form of Equivalent Integration　15
3.1　Weak Form of Equivalent Integration for Differential Equations　15
3.2　Weak Form of One-Dimensional Elasticity Problems　15
3.3　Finite Element Computation Based on Weak Form　17
3.3.1　Galerkin Method　17
3.3.2　Finite Element Computation　20
3.4　Global Assembly from One-Dimensional Elements　23
3.5　Treatments on Boundary Conditions　24
3.6　Exercises　28
Chapter 4　Elements and Shape Functions　29
4.1　One-Dimensional Lagrange Element　29
4.1.1　Linear Element with Two Nodes　29
4.1.2　Higher-Order Lagrange Element　29
4.1.3　Quadratic Lagrange Element　31
4.2　Two-Dimensional Triangle Element　32
4.2.1　Triangle with Three Nodes　32
4.2.2　Higher-Order Triangle Element　35
4.2.3　Quadratic Triangle Element　37
4.2.4　Cubic Triangle Element　37
4.3　Two-Dimensional Rectangle Element　37
4.3.1　Linear Rectangle Element with Four Nodes　37
4.3.2　Higher-Order Rectangle Element　39
4.3.3　Quadratic Rectangle Element　40
4.4　Three-Dimensional Tetrahedron Element　41
4.4.1　Linear Tetrahedron Element with Four Nodes　41
4.4.2　Higher-Order Tetrahedron Element　44
4.4.3　Quadratic Tetrahedron Element　44
4.4.4　Cubic Tetrahedron Element　44
4.5　Three-Dimensional Hexahedron Element　45
4.5.1　Hexahedron with Eight Nodes　45
4.5.2　Higher-Order Hexahedron Element　46
4.5.3　Quadratic Hexahedron Element　47
4.6　Exercises　48
Chapter 5　Isoparametric Element and Numerical Integration　50
5.1　Isoparametric Element　50
5.1.1　One-Dimensional Isoparametric Lagrange Element　50
5.1.2　Two-Dimensional Isoparametric Triangle Element　51
5.1.3　Two-Dimensional Isoparametric Rectangle Element　53
5.1.4　Three-Dimensional Isoparametric Tetrahedron Element　54
5.1.5　Three-Dimensional Isoparametric Hexahedron Element　56
5.1.6　Requirements of Isoparametric Element　58
5.2　Numerical Integration　59
5.2.1　One-Dimensional Integration for Lagrange Element　59
5.2.2　Two-Dimensional Integration for Triangle Element　62
5.2.3　Two-Dimensional Integration for Rectangle Element　63
5.2.4　Three-Dimensional Integration for Tetrahedron Element　64
5.2.5　Three-Dimensional Integration for Hexahedron Element　65
5.2.6　Required Order of Numerical Integration　66
5.3　Exercises　67
Chapter 6　Finite Element Computation Scheme of Elasticity Problems　68
6.1　Weak Form for General Elasticity Problems　68
6.2　Finite Element Method for Solving Elasticity Problems　71
6.3　Global Assembly from High-Dimensional Elements　72
6.4　Treatments on Boundary Conditions　78
6.5　Exercises　82
Chapter 7　Solutions of Linear Algebraic Equations　83
7.1　LU Decomposition Method　83
7.2　Exercises　87
Chapter 8　Error Estimation and Adaptive Analysis　88
8.1　Error Estimation of Finite Element Solutions　88
8.1.1　Error of Finite Element Solutions　88
8.1.2　Superconvergent Patch Recovery Method　88
8.2　Adaptive Finite Element Method　90
8.2.1　Categories of Adaptive Finite Element Method　90
8.2.2　h-Version Adaptive Finite Element Method　91
8.2.3　hp-Version Adaptive Finite Element Method　92
8.3　Exercises　94
Chapter 9　Programs of Finite Element Method　95
9.1　One-Dimensional Program of Beam Deformation　95
9.1.1　Main Program　96
9.1.2　Numerical Example　98
9.1.3　Interactive Interface　98
9.2　Two-Dimensional Program of Plane Strain Problem　108
9.2.1　Main Program　108
9.2.2　Numerical Example　112
9.2.3　Interactive Interface　114
9.3　Three-Dimensional Program of Solid Compression　124
9.3.1　Main Program　124
9.3.2　Numerical Example　128
9.3.3　Interactive Interface　131
9.4　Exercises　141
Appendix A. Keyword Index　143
Appendix B. Matrix Calculation　145
B.1　Definition　145
B.2　Matrix Addition or Subtraction　146
B.3　Transpose　146
B.4　Transpose of a Product　147
B.5　Inverse　147
B.6　Symmetric Matrices　148
B.7　Partitioning　148
Appendix C. Summary of Elements and Shape Functions　150
C.1　One-Dimensional Lagrange Element　150
C.2　Two-Dimensional Triangle Element　150
C.3　Two-Dimensional Rectangle Element　151
C.4　Three-Dimensional Tetrahedron Element　151
C.5　Three-Dimensional Hexahedron Element　152
Appendix D. Gaussian Integration Points and Weights　154
Appendix E. Exercise Solutions　158
References　174

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Chapter 1 Introduction of Finite Element Method
1.1 Development Process of Finite Element Method
Understanding the behaviour and mechanisms of nature is fundamental in scientific research. Owing to the limited capacity of the human mind, we fail to grasp the complexity of the world in its entirety. Thus, engineers, scientists, or even economists subdivide a system under investigation into individual components, or “elements”, the behaviour of which is easy to understand and reconstruct the original system from these components so that its behaviour can be naturally studied.
In many cases, we can use a limited number of well-defined components to get an effective model, and such problems are called “discrete”. In other cases, the subdivision continues indefinitely, and the problem can only be defined by the fictitious concept of infinitesimals (which are not rigorously defined in standard mathematical analysis). This leads to differential equations or equivalent statements, which contain an infinite number of elements. Such systems are correspondingly called “continuous”. With the development of digital computers, discrete problems can usually be solved easily, even with a large number of elements. Since the capacity of any computer is limited, continuous problems can only be solved accurately by mathematical manipulations. However, the mathematical techniques available for exact solutions are usually limited to oversimplified situations. In order to solve realistic continuous problems, various discretisation methods have been put forward. All these methods involve an approximation that, ideally, as the number of discrete variables increases, it approaches the limit of the true continuum solution.
Because most classical mathematical approximation methods and various direct approximation methods used in engineering belong to this category, it is difficult to determine the origin of the finite element method and the exact time of its invention. These approximate numerical methods definitely lay a crucial foundation for the generation of the finite element method, and promote the development of relevant numerical algorithms and numerical models in computational mechanics. The historical development of approximate numerical methods involved in the finite element method can also be obtained in some review papers. Figure 1.1 shows the evolution process, which leads to the concept of finite element analysis:
(1) Mathematicians and engineers have different views on the discretization of continuous problems. Since 1870, some mathematicians have developed general techniques that can be directly applied to differential equations, such as the trial functions (Ritz, 1909; Rayleigh, 1870), variational methods (Ritz, 1909; Rayleigh, 1870), weighted residuals (Biezeno and Koch, 1923; Galerkin, 1915; Gauss, 1795). Furthermore, using the trial function in each partition domain, the piecewise continuous trial functions were developed (Zienkiewicz and Cheung, 1964; Argyris and Kelsey, 1960; Prager and Synge, 1947; Courant, 1943).
(2) The differential equations can be approximately expressed by difference equations. In 1910, the classical difference method was established (Southwell, 1946; Liebman, 1918; Richardson, 1910). Then, based on the variational principle, the variational difference method was developed (Feng, 1965; Wilkins, 1969; Varga, 1962). It should be emphasised that, in China, K. Feng of the Chinese Academy of Sciences also proposed a discretisation numerical method based on the variational principle for solving elliptic partial differential equations in 1965. He is regarded as one of the pioneers in the basic theory of the finite element method: “Independently of parallel developments in the West, he (Feng) created a theory of the finite element method. He was instrumental in both the implementation of the method and the creation of its theoretical foundation using estimates in Sobolev spaces” (Lax, 1993).
(3) Some engineers and technicians directly used the method of analogy structure analysis to discretize the continuous domain, and the structural analog substitution was developed (Newmark, 1949; McHenry, 1943; Hrenikoff, 1941). Subsequently, the method of direct continuum elements was developed, and the representative research works are from Turner et al. (1956) and Clough (1960). Turner et al. proved that a more direct but equally intuitive replacement of properties could be made significantly and more effectively by considering that small portions or “elements” in a continuum behave in a simplified way. Clough used the term “finite element”, which was born based on the view of “direct analogy” in engineering, implying the direct use of a standard methodology applicable to discrete systems.