Adaptive Finite Element Method of Lines with Local Mesh Refinement in Maximum Norm Based on Element Energy Projection Method

Abstract

The reliable and efficient self-adaptive analysis is a modern goal of various numerical computations. Most adaptivity methods, however, adopt energy norm to measure errors, which may not be the most natural and convenient means, e.g., for problems with locally singular gradient of displacement. Based on the Element Energy Projection (EEP) super-convergent technique in the Finite Element Method of Lines (FEMOL) which is a general and powerful semi-discrete method, reliable error estimates of displacements in maximum norm can be obtained anywhere on the FEMOL mesh and hence adaptive FEMOL by maximum norm becomes feasible. However, to tackle singularity problems effectively and efficiently, an automatic and flexible local mesh refinement strategy is required to generate meshes of high quality for more efficient adaptive FEMOL analysis. Taking the two-dimensional Poisson equation as the model problem, the paper firstly introduces the FEMOL and EEP methods with interface sides resulting from local mesh refinement. Then a local mesh refinement strategy and corresponding adaptive algorithm are presented. The numerical results given show that the proposed adaptive FEMOL with local mesh refinement can produce displacement solutions satisfying the specified tolerances in maximum norm and the adaptively-generated meshes reasonably reflect the local difficulties inherent in the physical problems without much redundant accuracy.

An adaptive nonlinear finite element analysis of minimal surface problem based on element energy projection technique

“…If e h S =e h L < w, the element is to be subdivided into four elements [Figure 4(d)], otherwise, the element is to be subdivided only into two elements [Figure 4(c)] by inserting a dividing "line", the position of which is determined in j direction by using an error-averaging method (Yuan et al, 2019) to make the areas of the squared errors on the two sides roughly equal to each other.…”

“…Based on EEP method, a series of adaptive FE analyses have been developed and have successfully applied to various 1D FE (Yuan et al, 2014 and 2D linear FE adaptive analyses (Yuan et al, 2019;Dong et al, 2019). However, for 2D nonlinear problems, especially the MSP studied in this paper, the corresponding adaptive FE analysis has been absent.…”

A New Nodal Stress Recovery Technique in Finite Element Method Using Colliding Bodies Optimization Algorithm

“…For the low order 3D FEM, Sharma et al [14] improved stress recovery. Wang et al [15] employed superconvergence analysis for solving Maxwell's equations and Yuan et al [16] developed a new adaptive finite element method (AFEM) based on element energy projection method. This field advanced by efforts of Patton et al [17] on laminated composite plastic and endeavors of Chi et al [18].…”