A Post-Processor for Adaptive Meshing for Problems with Steep Gradient Areas

Abstract

Accuracy estimation and adaptive mesh refinement are among the main practical problems in finite element methods. Several methods to compute the accuracy of a finite element solution have been developed by many authors. Two important types of approaches can be discerned: one leads to the building of error indicators [1–6] and the other is based on the concept of error in constitutive relation [7–10]. Some methods for adaptive mesh refinement have also been developed but they are not very efficient for problems with steep gradients areas (singularities, for instance). In this paper we present a new method for adaptive mesh refinement which accounts for steep gradient areas automatically. This method is based on the concept of error in constitutive relation coupled with an h-version remeshing procedure. From a preliminary finite element analysis, we compute the accuracy and define an optimal mesh (accuracy equal to the error prescribed by the user and minimum element number) in a single step. We use local errors to detect the steep gradient areas of the problem, and the energy density of the finite element solution is used to take into account these areas automatically. This method has been implemented in our ESTEREF software, a post-processor which can be interfaced with any finite element code. As an illustration we show various examples of structures with steep gradient areas where the prescribed error is correctly achieved with a minimal computation cost.