Few Necessary Considerations for the Use of the Finite Element Methods by Non-Specialists

摘要

The grounds of the finite element methods are presented with emphasis on their use. It is suggested that the variational method can be interpreted as a way of determining an interpolatory approximation for the solution of the problem. Conversely, a weighted residuals method may be considered as a tool to perform a curve fitting of the approximation to the actual solution. The presentation follows an approach focusing the numerical quality, of the expected results, in equations involving differences (e.g., differential equations). Formulation of the differential problem in integral terms, based on the calculus of variations, or on optimization principles, is presented. The problem reduces to determining the coefficients. Variational methods are based on the minimization of an integral. The Rayleigh-Ritz method is based on the assumption that relationship can be established between the differential equation of the problem and a Euler-Lagrange differential equation. The method of weighted residuals is presented. The linear and nonlinear cases are looked at. Other applications of the finite element method are given.