Mesh-Free MLS-Based Error-Recovery Technique for Finite Element Incompressible Elastic Computations

Abstract

The finite element error and adaptive analysis are implemented in finite element procedures to increase the reliability of numerical analyses. In this paper, the mesh-free error-recovery technique based on moving least squares (MLS) interpolation is applied to recover the errors in the stresses and displacements of incompressible elastic finite element solutions and errors are estimated in energy norms. The effects of element types (triangular and quadrilateral elements) and the formation of patches (mesh-free patch, mesh-dependent element-based patch, and mesh-dependent node-based patch) for error recovery in MLS and conventional least-square interpolation-error quantification are also assessed in this study. Numerical examples of incompressible elasticity, including a problem with singularity, are studied to display the effectiveness and applicability of the mesh-free MLS interpolation-error recovery technique. The mixed formulation (displacement and pressure) is adopted for a finite element analysis of the incompressible elastic problem. The rate of convergence, the effectivity of the error estimation, and modified meshes for desired accuracy are used to assess the effectiveness of the error estimators. The error-convergence rates are computed in the original FEM solution, in the post-processed solution using mesh-free MLS-based displacement, stress recovery, mesh-dependent patch-based least-square-based displacement, and stress recovery (ZZ) as (0.9777, 2.2501, 2.0012, 1.6710 and 1.5436), and (0.9736, 2.0869, 1.6931, 1.8806 and 1.4973), respectively, for four-node quadrilateral, and six-node triangular meshes. It is concluded that displacement-based recovery was more effective in the finite element incompressible elastic analysis than stress-based recovery using mesh-free and mesh-dependent patches.