Finite Elements and Approximation

O. C. Zienkiewicz, K. Morgan

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486318011
• Topic: Tecnología e ingeniería
• Subtopic: Ingeniería general

A powerful tool for the approximate solution of differential equations, the finite element is extensively used in industry and research. This book offers students of engineering and physics a comprehensive view of the principles involved, with numerous illustrative examples and exercises.
Starting with continuum boundary value problems and the need for numerical discretization, the text examines finite difference methods, weighted residual methods in the context of continuous trial functions, and piecewise defined trial functions and the finite element method. Additional topics include higher order finite element approximation, mapping and numerical integration, variational methods, and partial discretization and time-dependent problems. A survey of generalized finite elements and error estimates concludes the text.

CHAPTER ONE

Continuum Boundary Value Problems and the Need for Numerical Discretization. Finite Difference Methods

1.1. INTRODUCTION

While searching for a quantitative description of physical phenomena, the engineer or the physicist establishes generally a system of ordinary or partial differential equations valid in a certain region (or domain) and imposes on this system suitable boundary and initial conditions. At this stage the mathematical model is complete, and for practical applications “merely” a solution for a particular set of numerical data is needed. Here, however, come the major difficulties, as only the very simplest forms of equations, within geometrically trivial boundaries, are capable of being solved exactly with available mathematical methods. Ordinary differential equations with constant coefficients are one of the few examples for which standard solution procedures are available—and even here, with a large number of dependent variables, considerable difficulties are encountered.

To overcome such difficulties and to enlist the aid of the most powerful tool developed in this century—the digital computer—it is necessary to recast the problem in a purely algebraic form, involving only the basic arithmetic operations. To achieve this, various forms of discretization of the continuum problem defined by the differential equations can be used. In such a discretization the infinite set of numbers representing the unknown function or functions is replaced by a finite number of unknown parameters, and this process, in general, requires some form of approximation.

Of the various forms of discretization which are possible, one of the simplest is the finite difference process. In this chapter we describe some of the essentials of this process to set the stage, but the remainder of this book is concerned with various trial function approximations falling under the general classification of finite element methods. The reader will find later that even the finite difference process can be included as a subclass of this more general category.

Before proceeding further we shall focus our attention on some particular problems which will serve as a basis for later examples. It is clearly impossible to deal in detail in...