A Finite Element Primer
Many of today's finite element systems are intended to be robust and friendly; and may be used by engineers and scientists who are unfamiliar with the finite element method, its virtues and its vices. Indeed, some finite element systems may be embedded in a CAD/CAE package and the user may be a production engineer, designer, or systems analyst who not only does not understand the characteristics of finite element systems, but may very well not even wish to know.
The problems of the novice have been identified and addressed in this primer. The aim is not to write just another finite element text book. There are scores of these, and many are daunting for the new reader since, not unaturally, many go into algebraic detail of specific elements or they highlight areas of current research. The aim is also not to write a detailed instruction manual, since this cannot be done without referring to a specific system. Indeed, all commerical systems have their own introductory and users' manuals, together with more advanced programmers' manuals in some cases, and these are usually comprehensive. They do not, however, discuss the various vices or failings of the system, for obvious reasons.
This Primer will try to explain the basis of the Finite Element Method, stressing the essentially simple fundamental concepts without digressing into lengthy algebra or oversmart mathematics. There will be no functional analysis in Hilbert spaces, and Lagrange multipliers will just be mentioned, even though they are used sparingly. Some algebra is inevitable, otherwise the text becomes a descriptive routine rather like explaining the fitting together of pieces of a jigsaw, and this can lead to all sorts of confusion. However, the algebra can be kept concise using matrix notation which is obligatory in discussing these methods. The Primer will strip some of the mysteries from the method and in particular will explain precisely what the method does exactly, and what it does approximately, and as a consequence which errors are important and which are not. Some errors are very useful sources of guidance.
Following fundamental chapters into the nature of the method for static problems and various types of structures, the nature of proprietary finite element systems will be discussed, together with those features which the user has the right to expect. The latter parts of the Primer will extend the method to dynamic problems, non-linear and elasto-plastic and buckling problems, heat transfer, and in Chapter 12 examples will be given to illustrate some of the modelling difficulties encountered in making a finite element idealisation in the first place.
It will be assumed that the reader has a basic understanding of applied mechanics and that differentiation or integration are not completely foreign concepts. The validity of a material's constitutive laws will not be questioned. The opening section even runs through a quick introduction to matrices and their manipulation- we realise that nowadays most engineers are familiar with matrix algebra but it does also give us an opportunity to gently introduce the notation used in this text.
In most chapters, the fundamentals behind the finite element method are very briefly discussed. It is possible of course to omit this, and simply state the equations which are necessary to formulate the numerical models. However, it has been shown that a grasp of the fundamentals is useful in judging errors, choosing idealisations, and even understanding user-manuals! Such introductions can be omitted if familiar.
If the reader therefore finds eoms parts of the Primer to elementary, we apologise. If any find parts a little demanding, then perhaps the balance is about right.
2. Structural Analysis
3. General Continuum
5. Two-dimensional Membranes
6. Bricks, Plates and Shells
7. Mesh Specification
8. Assembly and Solution
9. Results Processing
11. Nonlinear Analysis
13. Other Field Problems