Superconvergent patch recovery with equilibrium and conjoint interpolant enhancements

Abstract

The superconvergent patch derivative recovery method of Zienkiewicz and Zhu is enhanced by adding the squares of the residuals of the equilibrium equation and natural boundary conditions. In addition, a new conjoint polynomial for interpolating the local patch stresses over the element which significantly improves the local projection scheme is presented. Results show that in the 4-node quadrilateral, the equilibrium and boundary condition residuals usually improve accuracy but not the rate of convergence, whereas in the 9-node quadrilateral, results are mixed. The conjoint polynomial always improves the accuracy of the derivative field within the element as compared to the standard nodal interpolation, particularly in 4-node quadrilaterals.

Interlaminar stress recovery for three-dimensional finite elements

“…Many investigators have modified this procedure to include satisfaction of boundary conditions [24,25,26]. For a more accurate prediction of the transverse stresses in laminated composites and shells, a Modified Super-convergent Patch Recovery (MSPR) technique has been derived to obtain accurate nodal in-plane stresses which, subsequently, are used in the integration along the thickness of the equilibrium equations for evaluating the transverse shear and normal stresses [27].…”

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

“…In general they couple the stress components in order to be able to add constraints that improve the quality of the recovered field. Wiberg and Abdulwahab [45,25] proposed to take into account the equilibrium of the recovered field by using a penalty method, Blacker and Belytschko [26] introduced the "Conjoint Polynomial Enhancement" to improve the recovered field along the boundaries. 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.…”

“…Once the local field, at each patch, is obtained, a continuous field for the whole domain is evaluated using a partition of unity procedure (defined in [26] as Conjoint Polynomial enhancement) that, at any point x, properly weights the stress polynomials σ * i obtained from patches corresponding to each one of the vertex nodes i of the element containing x. Thus, the field σ * σ is constructed as a linear weighting of the contributions of each patch using linear shape functions N i associated with the n v vertex nodes according to the following expression:…”

“…The recovery-based error estimators are robust, easy to implement and are used in commercial codes. 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].…”

A separated representation of an error indicator for the mesh refinement process under the proper generalized decomposition framework

“…Generally, it is well known that the main drawback of recovery type techniques is the lack of accuracy along the boundaries of the domain. Further improvements to the SPR technique have been introduced [24][25][26][27] in order to prevent the lack of accuracy of the smoothed field along the boundaries of the domain and also to increase the accuracy into the domain.…”