An Introduction to Linear and Nonlinear Finite Element Analysis: A Computational Approach

Abstract

9R1. An Introduction to Linear and Nonlinear Finite Element Analysis: A Computational Approach. - Edited by PK Kythe and Dongming Wei (Dept of Math, Univ of New Orleans, New Orleans LA 70148-0001). Birkhauser, Boston. 2004. 445 pp. ISBN 0-8176-4308-7. $79.95.Reviewed by M Okrouhlik (Dept of Solids, Inst of Thermomech, Acad of Sci, Dolejskova 5, 182 00 Prague 8, Czech Republic).The book is intended as a textbook for undergraduate and graduate students from engineering, geophysics, and applied mathematics. Very broad spectrum of engineering problems is covered—the examples are taken from structural, mechanical, electrical, and chemical fields. Generally, the book deals with linear and nonlinear problems in radiation, heat transfer, mechanics of elastic and plastic media, continuum mechanics, non-Newtonian fluid flows, and electromagnetics. The book is composed of 14 chapters, 6 appendices, a bibliography, and subject index. Chapter 1 deals with weak variational formulation of a boundary value problem, Galerkin and Rayleigh-Ritz weighted residual methods. Chapter 2 presents one-dimensional local and global interpolation functions. Chapter 3 explains the Galerkin method and applies it to a one-dimensional second-order equation using liner and quadratic elements. Chapter 4 treats one-dimensional fourth-order equation (beam). Chapter 5 introduces linear triangular and bilinear four-node rectangular elements. Chapter 6 is devoted to two-dimensional problems with a single scalar variable. Stiffness matrix and load vector are derived. In Chapter 7 the two-dimensional boundary value problems as heat exchange, torsion, and seepage are treated. Chapter 8 deals with axisymmetric linear and nonlinear heat transfer problems in solid and fluids. Chapter 9 is devoted to transient problems and numerical time integration. Chapter 10 treats nonlinear problems in one dimension—radiation heat transfer, stress in elastoplastic bars, non-Newtonian fluid flow in between parallel plates, and turbulent flows in tubes. For the numerical solution the Newton method, the method of the steepest descent and conjugate gradient methods are used. Chapter 11 presents the steady-state problems of plane elasticity. Linear triangular and bilinear rectangular elements are treated; stiffness matrices and load vectors are derived. Assembling is clearly described but implementation considerations are not considered. Chapter 12 introduces the penalty method. It is applied to the treatment of both Newtonian and power law non-Newtonian Stokes flow. Chapter 13 deals with vibration analysis—elastic rods, Euler beams, and in-plane vibration of an elastic plate are treated. The last chapter contains the computer codes in Mathematica, Matlab, and Fortran. They form a vivid complement to selected problems appearing in the book. Results of computer runs are presented in the tabular form. There are six appendices, labeled A to G, in the book. They contain useful complementary information and are subsequently devoted to overview of classical integration formulas, evaluation of stiffness matrices for triangular and rectangular elements for a chosen geometry, time-step marching algorithms (forward and backward difference schemes, Cranck-Nicolson formulas, Newmark method, etc). Also the concept of isoparametric elements is briefly explained, Green identities are shown and Gauss quadrature formulas are derived. In the last appendix the classical minimization methods (method of the steepest gradient and conjugate gradient method) are presented. There are 87 examples and 148 exercises and 152 figures in the book. Solving the problems both SI and imperial units are used. The finite element analysis is clearly presented with rigorous mathematical treatment of its background and accompanied by numerous examples. Finite element methodology is covered briefly being saliently combined with a proper computer implementation. Programs and subroutines written in Fortran, Matlab, and Mathematica, treating the examples presented in the book, form a consistent part of the book. The book is carefully edited and printed. The only misprint I found is on p. 403 where there is a missing zero term in a matrix. From the reviewer’s point of view the book has high educational value stemming from the fact that a very large spectrum of engineering topics is covered. As such the book might well be a good purchase both for university libraries and individuals.