Marc Duflot

Review: Meshless methods: A review and computer implementation aspects

The aim of this manuscript is to give a practical overview of meshless methods (for solid mechanics) based on global weak forms through a simple and well-structured MATLAB code, to illustrate our discourse. The source code is available for download on our website and should help students and researchers get started with some of the basic meshless methods; it includes intrinsic and extrinsic enrichment, point collocation methods, several boundary condition enforcement schemes and corresponding test cases. Several one and two-dimensional examples in elastostatics are given including weak and strong discontinuities and testing different ways of enforcing essential boundary conditions.

Meshless methods: review and key computer implementation aspects

The aim of this manuscript is to give a practical overview of meshless methods (for solid mechanics) based on global weak forms through a simple and well-structured MATLAB code, to illustrate our discourse. The source code is available for download on our website and should help students and researchers get started with some of the basic meshless methods; it includes intrinsic and extrinsic enrichment, point collocation methods, several boundary condition enforcement schemes and corresponding test cases. Several one and two-dimensional examples in elastostatics are given including weak and strong discontinuities and testing different ways of enforcing essential boundary conditions.

On the role of enrichment and statical admissibility of recovered fields in a posteriori error estimation for enriched finite element methods

Purpose – This paper aims at assessing the effect of (1) the statical admissibility of the recovered solution; (2) the ability of the nrecovered solution to represent the singular solution; on the accuracy, local and global effectivity of recovery-based error estimators nfor enriched finite element methods (e.g. the extended finite element method, XFEM). nDesign/methodology/approach – We study the performance of two recovery techniques. The first is a recently developed nsuperconvergent patch recovery procedure with equilibration and enrichment (SPR-CX). nThe second is known as the extended moving least squares recovery (XMLS), which enriches the recovered solutions but does not nenforce equilibrium constraints. Both are extended recovery techniques as the polynomial basis used in the recovery process is nenriched with singular terms for a better description of the singular nature of the solution. nFindings – Numerical results comparing the convergence and the effectivity index of both techniques with those obtained without the nenrichment enhancement clearly show the need for the use of extended recovery techniques in Zienkiewicz-Zhu type error nestimators for this class of problems. nThe results also reveal significant improvements in the effectivities yielded by statically admissible recovered solutions. nOriginality/value – This work shows that both extended recovery procedures and statical admissibility are key to an accurate nassessment of the quality of enriched finite element approximations.