Ch 1 - An Introduction to the Use of Finite Element Procedures

1.1 Introduction
1.2 Physical Problems, Mathematical Models, and the Finite Element Solution
1.3 Finite Element Analysis as an Integral Part of Computer-Aided Design
1.4 A Proposal on How to Study Finite Element Methods

1.1 Introduction

The procedures are employed extensively in the analysis of solids and structures and of heat transfer and fluids, and indeed, finite element methods are useful in virtually every field of engineering analysis.

The development of finite element methods for the solution of practical engineering problems began with the advent of the digital computer. That is, the essence of a finite element method solution of an engineering problem is that a set of governing algebraic equations is established and solved, and it was only through the use of the digital computer that this process could be rendered effective and given general applicability. These two properties - effectiveness and general applicability in engineering analysis - are inherent in the theory used and have been developed to a high degree for practical computations, so that finite element methods have found wide appeal in engineering practice.

As is often the case with original developments, it is rather difficult to quote an exact "date of invention," but the roots of the finite element method can be traced back to three separate research groups :

Applied Mathematicians : R. Courant
Physicist : J.L Synge
Engineers : J.H Argyris and S. Kelsey

The finite element method in engineering was initially developed on a physical basis for the analysis of problems in structural mechanics. However, it was soon recognized that the technique could be applied equally well to be the solution of many other classes of problems.

To introduce the topics of this book we consider three important items in the following sections of this chapter.

First, we discuss the important point that in any analysis we always select a mathematical model of a physical problem and then we solve that model. The finite element method is employed to solve very complex mathematical models, but it is important to realize that the finite element solution can never can give more information than that contained in the
mathematical model.

Then we discuss the importance of finite element analysis in the complete process of computer-aided design (CAD). This is where finite element analysis procedures have their greatest utility and where an engineer is most likely to  encounter the use of finite element methods.

Finally, we address the question of how to study finite element methods. 

1.2 Physical Problems, Mathematical Models, and the Finite Element Solution

The finite element method is used to solve physical problems in engineering analysis and design. Since the finite element method technique is numerical procedure, it is necessary to assess the solution accuracy. It is clear that the finite element solution will solve only the selected mathematical model and that all assumptions in this model will be reflected in the predicted response. We cannot expect any more information in the prediction of physical phenomena than the information contained in the mathematical model. Hence the choice of an appropriate mathematical model is crucial and completely determines the insight into the actual physical problem that we can obtain by the analysis.

The process of finite element analysis.

For example, a beam structure (using engineering terminology) may first be analyzed using Bernoulli beam theory, then Timoshenko beam theory, then two-dimensional plane stress theory, and finally using a fully three-dimensional continuum model, and in each case non-linear effects may be included.

1.3 Finite Element Analysis as an Integral Part of Computer-Aided Design

In the early use of finite element methods, only specific structures were analyzed, mainly in the aerospace and civil engineering industries. However, once the full potential of finite element methods was realized and the use of computers increased in engineering design environments, emphasis in research and development was placed upon making the use of finite element methods as integral part of the design process in mechanical, civil and aeronautical engineering.

We venture to comment on the future of finite element methods in computer aided design. Surely, many engineering designers do not have time to study finite element methods in depth or finite element precedures in general. Theri sole objective is to use these techniques to enhance the design product.

1.4 A Proposal on how to Study Finite Element Methods

1. To learn the proper use of finite element methods for the solution of complex problems.
2. To understand finite element methods in depth so as to be able to pursue research on finite element methods.

It is particular importance during the study of finite element procedures that good judgment be developed and exercised concerning the quality of any finite element solution and of any mathematical model considered.