Enriched Meshfree Method for an Accurate Numerical Solution of the Motz Problem

Won-Tak Hong

doi:10.1155/2016/6324754

Abstract

We present an enriched meshfree solution of the Motz problem. The Motz problem has been known as a benchmark problem to verify the efficiency of numerical methods in the presence of a jump boundary data singularity at a point, where an abrupt change occurs for the boundary condition. We propose a singular basis function enrichment technique in the context of partition of unity based meshfree method. We take the leading terms of the local series expansion at the point singularity and use them as enrichment functions for the local approximation space. As a result, we obtain highly accurate leading coefficients of the Motz problem that are comparable to the most accurate numerical solution. The proposed singular enrichment technique is highly effective in the case of the local series expansion of the solution being known. The enrichment technique that is used in this study can be applied to monotone singularities (of typerαwithα<1) as well as oscillating singularities (of typerαsin⁡(ϵlog⁡r)). It is the first attempt to apply singular meshfree enrichment technique to the Motz problem.

Highlights

We present an enriched meshfree solution of the Motz problem
The Motz problem was first introduced by Motz [1]
The Motz problem has served as a benchmark problem to verify the efficiency of numerical methods in the presence of a singularity

Summary

Introduction

The Motz problem was first introduced by Motz [1]. Since it was introduced, the Motz problem has served as a benchmark problem to verify the efficiency of numerical methods in the presence of a singularity. There is partition of unity methods that use finite element mesh explicitly. In essence it is not purely meshfree. On top of that one can create highly regular basis functions that have compact support This enables us to solve higher order partial differential equations such as biharmonic or polyharmonic problems [27,28,29] without using the Hermite finite elements, which are difficult to implement in higher dimension. A 3D extension of the partition of unity based meshfree method, which uses compactly supported highly regular basis functions that have the Kronecker delta property, is available [30]. The enrichment technique that is introduced in this study can be used to solve problems that have a known local series expansion or even an asymptotic series expansion

An Enriched Meshfree Method

Numerical Results

Concluding Remark