The Theory of Density Representation of Finite Element Meshes. Examples of Density Operators with Quadrilateral Elements in the Mapped Domain

A method to represent nite element meshes in any dimension by functions called mesh density functions is described. The method is based on a generalization of the notion of mesh grading. The idea of the mesh density function is to represent a mesh by the density of elements in all points of the domain. The density representation is part of a methodology for adaptive solution of linear and non linear parameter dependent problems. The general theory is given for any space dimension, and examples are presented for one and two dimensional problems with point singularities. 1. Introduction Many real life problems involve one or more physical parameters. The goal of the solution process can be either to nd some optimal values of the parameters or to study the behavior of the solution over a range of parameter values. The purpose can for example be to nd areas where the stability properties of the system under consideration changes, or to make sure that stresses in a mechanical system stay within acceptable limits for all required load cases. Parametrized problems have in common that many problems are being solved in each run, and that normally problems solved consecutively are very similar in structure and solution. Many methods have been developed to take advantage of this fact. In the mathematical literature they are known as continuation methods while in the engineering literature also the names incremental or incremental loading methods are used. Many articles and books on the subject exist. Here only two