Prologue
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The finite element method is a numerical method for solving problems of engineering and mathematical physics. Typical problem areas of interest in engineering and mathematical physics that are solvable by use of the finite element method include:
1. structural analysis
* heat transfer
* fluid flow
* mass transport
* electromagnetic potential.

For problems involving complicated geometries, loadings, and material properties, it is generally not possible to obtain analytical mathematical solutions. Analytical solutions are those given by a mathematical expression that yields the values of the desired uknown quantities at any location in a body (here total structure or physical
system of interest) and are thus valid for an infinite number of locations in the body. These analytical solutions generally require the solution of ordinary or partial differential equations, which, because of the complicated geometries, loadings, and material properties, are not usually obtainable. Hence we need to rely on numerical methods, such as the finite element method, for acceptable solutions. The finite element formulation of the problem results in a system of simultaneous algebraic equations for solution, rather than requiring the solution of differential equations. These numerical methods yield approximate values of the unknowns at discrete numbers of points in the continuum. Hence this process of modeling a body by dividing it into an equivalent system of smaller bodies or units (finite elements) interconnected at points common to two or more elements (nodal points or nodes) and/or boundary lines and/or surfaces is called discretisation. In the finite element method, instead of solving the
problem for the entire body in one operation, we formulate the equations for each finite element and combine them to obtain the solution of the whole body.

Briefly, the solution for structural problems typically refers to determining the displacement at each node and the stresses within each element making up the structure that is subjected to applied loads. In nonstructural problems, the nodal unknowns may, for instance, be temperatures or fluid pressures due to thermal or fluid fluxes.

Introduction

This chapter first presents a brief history of the development of the finite element method. You will see from this historical account that the method has become a practical one for solving engineering problems only in the past 50 years (paralleling the developments associated with the modern high-speed electronic digital computer). This historical account is followed by an introduction to matrix notation; then we describe the need for matrix methods (as made practical by the development of the modern digital computer) in formulating the equations for solution. This section discusses both the role of the digital computer in solving the large systems of simultaneous algebraic equations associated with complex problems and the development of numerous computer programs based on the finite element method. Next, a general description of the steps involved in obtaining a solution to a problem is provided.
This description includes discussion of the types of elements available for a finite element method solution. Various representative applications are then presented to illustrate the capacity of the method to solve problems, such as those involving complicated geometries, several different materials) and irregular loadings. Chapter 1
also lists some of the advantages of the finite element method in solving problems of engineering and mathematical physics. Finally, we present numerous features of computer programs based on the finite element method.

Brief History

This section presents a brief history of the finite element method as applied to both structural and nonstructural areas of engineering and to mathematical physics. References cited here are intended to augment this short introduction to the historical background.

The modem development of the finite element method began in the 1940s in the field of structural engineering with the work by Hrennikoff [1] in 1941 and McHenry [2] in 1943, who used a lattice of line (one-dimensional) elements (bars and beams) for the solution of stresses in continuous solids. In a paper published in 1943 but not widely recognised for many years, Courant [3] proposed setting up the solution of stresses in a variational form. Then he introduced piecewise interpolation (or shape) functions over triangular subregions making up the whole region as a method to obtain approximate numerical solutions. In 1947 Levy [4] developed the flexibility or force method, and in 1953 his work [5] suggested that another method (the stiffness or displacement method) could be a promising alternative for use in analysing statically redundant aircraft structures. However, his equations were cumbersome to
solve by hand, and thus the method became popular only with the advent of the high-speed digital computer.

In 1954 Argyris and Kelsey [6, 7] developed matrix structural analysis methods using energy principles. This development illustrated the important role that energy principles would play in the finite element method. The first treatment of two-dimensional elements was by Turner et al. [8] in 1956. They derived stiffness matrices for truss elements, beam elements, and two-dimensional triangular and rectangular elements in plane stress and outlined the procedure commonly known as the direct stiffness method for obtaining the total structure stiffness matrix. Along with the development of the high-speed digital computer in the early 1950s, the work of Turner et al. [8] prompted further development of finite element stiffness equations expressed in matrix notation. The phrase finite element was
introduced by Clough [9] in 1960 when both triangular and rectangular elements were used for plane stress analysis.
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A fiat, rectangular-plate bending-element stiffness matrix was develope<;l by
Melosh [10] in 1961. This was followed by deVelopment of the curved-shell bendingelement stiffness matrix for axisymmetric shells and pressure vessels by Grafton and
Strome [II] in 1963.
Extension of the finite element method to three-dimensional problems with the
development of a tetrahedral stiffness matrix was done by Martin [12] in 1961; by
Gallagher et al. [13} in 1962, and by Melosh [14] in 1963. Additional three-dimensional
elements were studied by Argyris (15) in 1964. The special case of axisymmetric solids
was considered by Clough and Rashid (16] and Wilson [17] in 1965.
Most of the finite element work up, to the early 1960s dealt with small strains
and small displacements) elastic material behavior, and static loadings. However,
Jarge deflection and thermal analysis were considered by Turner et a1. [18] in 1960
and material nonlinearities by Gallagher.et al. [13] in 1962, whereas buckling problems were initially treated by Gallagher and Padlog [19] in 1963. Zienkiewicz et aL
[20J extended the method to visco-e1a~ticity problems in 1968.
In 1965 Archer {2l J considered dynamic analysis in the development
the
consistent-mass matrix, which is applicable to analysis of distributed-mass systems
such as bars and beams in structural analysis.
With Melosh's [I4] realization in 1963 that the finite element method could be
set up in terms of a variational formulation, it began to be used to solve nonstructural
applications. Field problems, such as determination of the torsion of a shaft,
fluid How, and heat conduction, were solved by Zienkiewicz and Cheung [22] in
1965, Martin {23J in 1968, and Wilson and Nickel [24J in 1966.
Further extension of the method was made possible by the adaptation of weighted
residual methods, first by Szabo and Lee (25} in 1969 to derive the previously known
elasticity equations used in structural analysis and then by Zienkiewicz and Parekh [26J
in 1970 for transient field problems. It was then recognized that when direct fonnulations and variational formulations are difficult or not possible to use, the method of
weighted residuals may at times be appropriate. For example, in 1977 Lyness et al. [27]
applied the method of weighted residuals to the determination of magnetic field.
In 1976 Belytschko [28,29] considered problems associated ,with Iarge-displacement
nonlinear dynamic behavior, and improved numerical techniques. for solving the
resulting systems of equations. For more OD' these topics, consult the text, by
Belytschko, Liu, and Moran [58J.
A relatively new field ota.pplication of the finite element method is that ofbioen~
gineering {30, 31]. This field is still troubled by such difficulties as nonlinear materials,
geometric nonJinearities, and other complexities still being discovered.
From the early 19508 to the present, enormous advances have been made in the
application of the finite element method to solve complicated engineering problems.
Engineers, applied mathematicians, and other scientists win undoubtedly continue to

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1 Introduction

develop new applications. For an extensive bibliography on the finite element method.
consult the work of Kardestuncer [32]~ Clough [33}, or Noor 157].

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