Improvement of the superconvergent patch recovery technique by the use of constraint equations: the SPR‐C technique

Abstract

SUMMARYThe superconvergent patch recovery (SPR) technique is widely used in the evaluation of a recovered stress field r * from the finite element solution r fe . Several modifications of the original SPR technique have been proposed. A new improvement of the SPR technique, called SPR-C technique (Constrained SPR), is presented in this paper. This new technique proposes the use of the appropriate constraint equations in order to obtain stress interpolation polynomials in the patch r * p that locally satisfy the equations that should be satisfied by the exact solution. As a result the evaluated expressions for r * p will satisfy the internal equilibrium and compatibility equations in the whole patch and the boundary equilibrium equation at least in vertex boundary nodes and, under certain circumstances, along the whole boundary of the patch coinciding with the boundary of the domain. The results show that the use of this technique considerably improves the accuracy of the recovered stress field r * and therefore the local effectivity of the ZZ error estimator.

A modified perturbed Lagrangian formulation for contact problems

“…The smooth stress field is based on the SPR (Superconvergent Patch Recovery) first proposed in [59] and improved in [43] to include constraint equations that must be fulfilled by the exact solution. Here we recall the main features of the smooth stress field calculation.…”

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

“…Other techniques looking for equilibrated recovered solutions for upper bounding purposes can be found in [46,47,48,49], but always presenting small lacks of equilibrium even at patch level, thus preventing the strict upper bound property. More recently, Ródenas and coworkers introduced the so-called SPR-C technique [28], where the "C" stands for "constraint", which was later applied in the XFEM framework by Ródenas et al [34] and finally adapted to geometry-mesh independent FE formulations [35], such as the cgFEM. We will here show the main features of the SPR-C technique.…”

“…For further details of the SPR-C recovery process see [28,30]. Note that the difference between this approach and the one proposed by Zienkiewicz and Zhu [18] is that in this last case, they only retain the recovered stress valueσ * i (x i ) at the node, but in our approach, based on the Conjoint Polynomial enhancement, we retain the full polynomial,σ * i (x), that defines the stress field at each patch.…”

“…Because of the use of the SPR-C technique, the lack of equilibrium along the boundary is either null or negligible, thus we can neglect the terms related to the lack of equilibrium along the boundaries r * σ L 2 (Γ N ) . Then, with minimal loss in accuracy: (28) where C = C Ω . Now, we rewrite expression (15) composed by the ZZ error estimator (14) and the correction terms (16) as follows:…”

“…The publication of the original SPR technique was followed by several works aimed to improve its quality, see for example [25,26,27]. Ródenas et al proposed to add constraints to impose local equilibrium and local compatibility to the recovered solution in the FEM framework [28] bringing up the SPR-C technique that was also adapted to the eXtended Finite Element Method (XFEM) framework [29,30].…”