A methodology for adaptive finite element analysis: Towards an integrated computational environment

Abstract

This work introduces a methodology for selfadaptive numerical procedures, which relies on the various components of an integrated, object-oriented, computational environment involving pre-, analysis, and post-processing modules. A basic platform for numerical experiments and further development is provided, which allows implementation of new elements/error estimators and sensitivity analysis. A general implementation of the Superconvergent Patch Recovery (SPR) and the recently proposed Recovery by Equilibrium in Patches (REP) is presented. Both SPR and REP are compared and used for error estimation and for guiding the adaptive remeshing process. Moreover, the SPR is extended for calculating sensitivity quantities of ®rst and higher orders. The mesh (re-)generation process is accomplished by means of modern methods combining quadtree and Delaunay triangulation techniques. Surface mesh generation in arbitrary domains is performed automatically (i.e. with no user intervention) during the self-adaptive analysis using either quadrilateral or triangular elements. These ideas are implemented in the Finite Element System Technology in Adaptivity (FESTA) software. The effectiveness and versatility of FESTA are demonstrated by representative numerical examples illustrating the interconnections among ®nite element analysis, recovery procedures, error estimation/adaptivity and automatic mesh generation.Key words ®nite element analysis, error estimation, adaptivity, h-re®nement, sensitivity, superconvergent patch recovery (SPR), recovery by equilibrium in patches (REP), object oriented programming (OOP), interactive computer graphics. IntroductionThis work presents an integrated (object-oriented) computational environment for self-adaptive analyses of generic two-dimensional (2D) problems. This environment includes analysis procedures to insure a given level of accuracy according to certain criteria, and also the procedures to generate and modify the ®nite element discretization. This computational system, called FESTA (Finite Element System Technology in Adaptivity), involves ®ve main components (see shaded boxes in Figure 1):A graphical preprocessor, for de®ning the geometry of the problem, the initial ®nite element mesh (together with boundary conditions), and the main parameters used in a self-adaptive analysis. Here the geometrical model is dissociated from the ®nite element model. A ®nite element module for solving the current boundary value problem. The code has been developed so that it is highly modular, expandable, and user-friendly. Thus, it can be properly maintained and continued. Moreover, other users/developers should be able to modify the basic programming system to ®t their speci®c applications. An error estimation and sensitivity module. Discretization errors are estimated according to available recovery procedures, e.g. Zienkiewicz and Zhu (ZZ), superconvergent patch recovery (SPR) and recovery by equilibrium in patches (REP). Sensitivities of various orders (1st., 2nd. or higher) are calculated by means of...

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