Equilibrium method for postprocessing and error estimation in the finite element method

Abstract

Modeling of elastic thin-walled bcams, plates and shells as 1D and 2D boundary value problems is valid in undisturbed subdomains. Disturbances near supports and free edges, in the vicinity of concentrated loads and at t hickness jumps cannot be described by 1D and 2D BVP's. In these disturbed subdomai ns dimensional (d)-adaptivity and possibly model (m)-adaptivity have to be performed and coupled with mixed h- and/or p-adaptivity by hicrarchically expandcd test spaces iu ordcr to guarantee a reliablc and efficient overall solution. Using residual error estimators coupled with anisotropic crror estimation and mesh refinemcnt, a.ll efficient adaptive calculation is possible. This residual estimator is based on stress jumps along the internal boundaries and residua of the field equation in L₂norms. In this paper, we introduce an equilibrium method for calculation of the internal tractions on local patches using orthogonality conditions. These t ractions are equilibrated with respect to the global equilibrium condition of forces and bending momcnts. We derive a new error estimation based on jumps between t be lIew t ractions and the tractions calculated with the stresses of the current finite elemcnt solution solution. This posteriori equilibrium method (PEM) is based of the local calculation of improved stress tractions along the internal boundaries of clement patches with continuity coudition in normal directions. The in troduction of new tractions is a method which can be regarded as a stepwise hybrid displacement method or as Trefftz method for a Neumann problem of clement patches. An additional and important advantage is the local numerical solution and the model error estimation based on the equilibrated tractions.