An Introduction to the Finite Element Method

Abstract

This is a unique book on finite element methods (FEM) and applications. In this book, FEM is presented to the reader as a variational-based technique of solving differential equations. Usually FEM is introduced to the reader via solid mechanics approach. In the present day, FEM has been successfully employed in fluid mechanics, heat transfer, and acoustics. FEM overcomes some of the problems associated with varia-tional methods (Ritz and weighted residual methods) and provides systematic procedures for the derviation method of the approximation method. First, a geometrically complex domain is represented as a collection of geometrical subdomains. Over each FE, the approximation functions are derived employing the basic concept that any continuous function can be represented by a linear combination of algebraic polynomials. These approximation functions are derived using the ideas from interpolation theory. Thus, they are denoted as interpolation functions. The values of the solutions at a finite number of prescribed points in the boundary or interior are called nodes. Interpolation theory is one of the important subjects of this book. This volume consists of five chapters and three computer programs. The initial chapter introduces the subject, presents some historical comments and explains some basic concepts. Chapter 2 introduces the reader to the variational formulation and approximation. Beginning with the differential equation and its variants, this continues with boundary and initial value problems, gradient and divergence theories. The next section presents the variation formulation of boundary value problems, including Poisson equation. This leads to the varia-tional method of approximation (Ritz and weighted residuals) and a concluding section on time-dependent problems. The author discusses the forward difference (Eulers, Crank-Nicholson Scheme, Galerkin method) and backward difference schemes. This is accompanied by the stability concept and a number of illustrative examples. The next chapter is a lengthy one. It introduces the reader to finite element analysis of one-dimensional second-order equations and their respective problems. Opening the next chapter is the one-dimensional second-order equation. The author delves into the derivation and assembly of element equations, imposition of boundary conditions and solutions of the foregoing. This leads to the one-dimensional fourth-order equation. We next read about the various errors introduced into FE solution of a given differential equation. They are: (a) boundary errors, (b) quadrature and finite arithmetic errors, and (c) approximation errors. These are further developed via illustrative examples. A short illuminating example explaining the time-dependent second-order problem is considered. The next important section develops the one-dimensional isoparametric element. The latter is explained in great clarity and the various numerical analysis schemes are explained. They are: (a) Newton-Cotes quadrature, and (b) Gauss-Legendre quadrature. The latter being the most productive. The concluding section attempts to answer the question of computer implementation. Beginning with the pre-processor, this goes forth to the processor (calculation of element matrices). The program consists of an assembly of a banded matrix form and imposition of the boundary conditions. The three computer programs (FEM1D, PLATE and FEM2D) accompanied by a lengthy discussion of their contents, use and aid in the solution of the illustrative examples. A most interesting chapter and well worth reading! Chapter 4 is the lengthiest and reports on finite element analysis of two-dimensional problems. The second-order equation involving a scaled-value function opens the chapter. Starting with the variational formulation, this continues with FE formulation, integration functions with application to three-node triangular element and a four-node rectangular element. After assembling the element matrices, the author plunges ahead with the makeup of the element matrice accompanied by clear-cut examples showing the power of the finite element tool. Comments on mesh generation are next in order. This involves discretization of a given domain, generation of FE data, and imposition of boundary conditions. Jumping ahead, we encounter the triangular and rectangular elements and the serendipity elements. This is a forerunner of the section on second-order multivariable equations. The resulting coupled equations are provided by the following examples. They are: (a) the plane elastic deformation of a linear elastic solid, (b) flow of an incompressible viscous fluid, and (c) bending of elastic plates with transverse shear strains. Continuing, we meet the time-dependent problems and are then introduced to heat transfer problem and incompressible viscous fluid flow. In the latter, mention is made of a fixed model plus the derivation and employment of the penalty model This section ends with temporal approximations. It is then compared to the finite-difference and finite elements solutions plus an exact solution of the heat conduction problem. The next section proceeds with isoparametric elements and methods of numerical integration. The chapter concludes with computer implementation as to element calculation. Extensive application of the three computer programs are provided. They acquaint the reader with the powerful FE method. This is an excellent chapter and should be read thoroughly. The author announces that the last chapter is a prelude to advanced topics. Starting with alternate formulation (least squares and mixed formulation), we move ahead into the solution of eigenvalue problems and nonlinear problems. Three-dimensional problems are touched upon very lightly. In summary, this is a good book. The reviewer believes that the three-dimensional section should be greatly expanded and extended. Mention should have been made of the Wilson 6 and Houbalt methods in the solution of transient problems. Additional topics of interest that are missing are the variable nodes 348