L2-Norm Based a Posteriori Error Estimates of Compressible and Nearly-Incompressible Elastic Finite Element Solutions

Abstract

The displacement and stress-based error estimates in a posteriori error recovery of compressible and nearly-incompressible elastic finite element solutions is investigated in the present study. The errors in the finite element solutions, i.e., in displacement and stress, at local and global levels are computed in L2-norm of quantity of interest, namely, displacements and gradients. The error estimation techniques are based on the least square fitting of higher order polynomials to stress and displacement in a patch comprising of node/elements surrounding and including the node/elements under consideration. The benchmark examples of compressible and incompressible elastic bodies, with known solutions employing triangular discretization schemes, are implemented to measure the finite element errors in displacements and gradients. The mixed formulation involving displacement and pressure is used for incompressible elastic analysis. The performance of error estimation is measured in terms of convergence properties, effectivity and mesh required for predefined precision. The error convergence rate, in FEM original solution, recovered solution using displacement recovery-based and stress-based error recovery technique for stresses, are obtained as (1.9714, 2.8999, and 2.5018) and (0.9818, 1.7805, and 1.4952) respectively for compressible and incompressible self-loaded elastic plate benchmark example using higher order triangular elements. It is concluded from the study that displacement fitting technique for extracting higher order derivatives shows a very effective technique for recovery of compressible and nearly-incompressible finite element analysis errors.