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1 Frequently Asked Questions about the Finite Element Method 1. What is the finite element method (FEM)? The FEM is a novel numerical method used to solve ordinary and partial differential equations. The method is based on the integration of the terms in the equation to be solved, in lieu of point discretization schemes like the finite difference method. The FEM utilizes the method of weighted residuals and integration by parts (Green-Gauss Theorem) to reduce second order derivatives to first order terms. The FEM has been used to solve a wide range of problems, and permits physical domains to be modeled directly using unstructured meshes typically based upon triangles or quadrilaterals in 2-D and tetrahedrons or hexahedrals in 3-D. The solution domain is discretized into individual elements these elements are operated upon individually and then solved globally using matrix solution techniques. 2. What is the history of the FEM? Early work on numerical solution of boundary-valued problems can be traced to the use of finite difference schemes; Southwell used such methods in his book published in the mid 1940 s. The beginnings of the finite element method actually stem from these early numerical methods and the frustration associated with attempting to use finite difference methods on more difficult, geometrically irregular problems. Beginning in the mid 1950s, efforts to solve continuum problems in elasticity using small, discrete "elements" to describe the overall behavior of simple elastic bars began to appear, and such techniques were initially applied to the aircraft industry. Actual coining of the term "finite element" appeared in a paper by Clough in The early use of finite elements lay in the application to structural-related problems. However, others soon recognized the versatility of the method and its underlying rich mathematical basis for application in non-structural areas. Since these early works, rapid growth in usage of the method has continued since the mid 1970s. Numerous articles and texts have been published, and new applications appear routinely in the literature. 3. What is the Method of Weighted Residuals, i.e., Galerkin s Method? The underlying mathematical basis of the finite element method first lies with the classical Rayleigh-Ritz and variational calculus procedures. These theories provided the reasons why the finite element method worked well for the class of problems in which variational statements could be obtained (e.g., linear diffusion type problems). However, as interest expanded in applying the finite element method to more types of problems, the use of classical theory to describe such problems became limited and could not be applied, e.g., fluid-related problems. Extension of the mathematical basis to non-linear and non-structural problems was achieved through the method of weighted residuals (MWR), originally conceived by Galerkin in the early 20th century. The MWR was found to provide the ideal theoretical basis for a much wider basis of problems as opposed to the Rayleigh-Ritz method. Basically, the method requires the governing

4 best commercial packages now on the market is COMSOL, which also allows users to write their own solver packages and PDEs. 10. Are there any recommended commercial FEM packages that are versatile in handling a wide range of problems? Any of the well known and widely versatile FEM codes now on the market are good it just depends on how comfortable the user is with the I/O part of the program. COMSOL, as mentioned before, is quite easy and very versatile handling a wide range of problem classes including fluid flow (with turbulence), heat transfer, structural analysis, electrodynamics, and general PDEs including species transport, chemical reactions, and groundwater/porous media flows. 11. Any suggested web sites for FEM? There are several recommended web sites: a. b. c. d How long does it take for me to be able to use a FEM program? Some programs allow you to solve problems fairly quickly. It is always highly recommended that work out the example problems generally provided by most commercial software. COMSOL, ANSYS, ALGOR, and NASTRAN all run on PCs. 13. Why would I want to use a FEM program? The versatility, ease of data input, and solution accuracy make the FEM one of the best numerical methods for solving engineering problems. FEM programs are the backbone of structural analyses, and are becoming more widely accepted for problems in which geometries are complex. 14. Is this a method that will soon become obsolete? The recent introduction of BEM and meshless methods would appear to indicate the eventual obsolescence of the FEM. However, these newer methods are still years away from being developed to the point of wide spread applicability found in FEM. The FEM will be around for many years to come. Recent advances with the inclusion of spectral schemes and adaptivity make it especially attractive now. 15. How expensive is a FEM code?

5 FEM codes range from those that can be found for free on the web to others costing many thousands of dollars. Those that run on PCs are generally inexpensive, yet provide powerful tools for solving a number of large scale problems. 16. What kind of hardware do I need to run a FEM code? A PC with a sufficiently fast processor, at least 256MB RAM, and at least 20GB of hard disk will permit many problems to be solved that once could only be run on mainframe computers. A suggested PC level for major FEM calculations is one with 1 GB RAM, 60 GB hard disk, and running with Pentium 4/3.2 GHz or better processors would provide more that adequate capabilities. The state-of-the-art in PC hardware is improving constantly; in a few years, even these suggested requirements will seem obsolete.

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