Recovery of equilibrium on star patches using a partition of unity technique

Abstract

SUMMARYA technique for recovering equilibrated element stresses is developed for finite element models of structural mechanics problems. The data for the method consist of the prescribed loading and the stress and displacement fields resulting from a conventional compatible finite element model. Local problems are defined for each star of elements, via the introduction of fictitious body forces and strains. These problems can be solved independently for equilibrium in a process that can be easily parallelized. The quality of the solutions is assessed, for two-dimensional linear elastic problems, by using them to compute bounds of the error of the finite element solutions, in terms of both global and local quantities of interest.

A recovery-explicit error estimator in energy norm for linear elasticity

“…Zienkiewicz and Zhu [18] introduced the so-called Zienkiewicz and Zhu (ZZ) error estimator (14) which consists of substituting the unknown field σ by the recovered stress field σ * , evaluated with a recovery procedure, in expression (13).…”

“…The second branch of error estimators, related with the concept of dual analysis, makes use of two solutions, one compatible and one equilibrated. Some of these error estimators solve two global problems in parallel [10] whereas other post-process the FE solution [11,12,13]. Under this group we can also include the error estimators based on the Constitutive Relation Error (CRE) introduced by Ladevèze and Leguillon [11] and followed by several contributions for many applications, see for example [14,15,16,17].…”

Cartesian grid FEM (cgFEM): High performance h-adaptive FE analysis with efficient error control. Application to structural shape optimization

“…In this case, the system solves both formulations together, the displacement-based and stress-based formulations. Some authors [86,89,88] maintain that despite the higher computational cost, these mixed formulations, which directly provide kinematically and statically admissible solutions, have important advantages in quality of the solution and in the possibility to directly obtain very accurate error bounds in energy norm.…”

“…Therefore, it will affect to the body forces b and also it will bring up a new term, first introduced in [88]: the fictitious body forces, L T N k i σ h . Their role is to ensure that the forces applied to each patch satisfy global equilibrium for the isolated patch.…”

“…This type of error estimators can also be used to directly obtain upper error bounds. Some of these error estimators solve two global problems in parallel [86] whereas others post-process the FE solution [81,87,88]. The main characteristic of these error bounding techniques is that the error is evaluated by comparing the two solutions, which are complementary in nature, and whose errors are orthogonal, see [89].…”