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

Modeling of elastic thin-walled beams, 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 thickness jumps cannot be described by 1D and 2D BVP’s. In these disturbed subdomains dimensional (d)-adaptivity and possibly model (m)-adaptivity have to be performed and coupled with mixed hand/or p-adaptivity by hierarchically expanded test spaces in order to guarantee a reliable and efficient overall solution. Using residual error estimators coupled with anisotropic error estimation and mesh refinement, an efficient adaptive calculation is possible. This residual estimator is based on stress jumps along the internal boundaries and residua of the field equation in L2 norms. In this paper, we introduce an equilibrium method for calculation of the internal tractions on local patches using orthogonality conditions. These tractions are equilibrated with respect to the global equilibrium condition of forces and bending moments. We derive a new error estimation based on jumps between the new tractions and the tractions calculated with the stresses of the current finite element solution solution. This posterior equilibrium method (PEM) is based on the local calculation of improved stress tractions along the internal boundaries of element patches with continuity condition in normal directions. The introduction 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 element patches. An additional and important advantage is the local numerical solution and the model error estimation based on the equilibrated tractions.