Numerical test for the Neumann problem of interior BVP of plane elasticity

Abstract

Following the concept of the patch test in FEM (finite element method), several numerical tests are suggested for the Neumann problem of interior BVP (boundary value problem) of plane elasticity by using different techniques. We choose the elliptic plate as an object in the test. Along the boundary of the plate, two groups of boundary traction are assumed. The first group is derived from the constant stress solution (or constant strain solution), and the second group is derived from the linear stress distribution. From displacement solution, we can evaluate the normal peripheral stress along the boundary, which is compared with the result from an exact solution. In addition, several error estimation factors are defined to evaluate the achieved accuracy. Extensive numerical examples and computed results are provided.

Introduction

A degenerate scale problem was studied [1]. The degenerate scales have significance influence on the conventional BIE (boundary integral equation) solutions. In addition, a necessary and sufficient BIE was suggested.

It was pointed out that in the Neumann boundary value problem for interior region in the BIE of plane elasticity, the solution for displacement is not unique [2]. Several techniques were suggested for removal the rigid body motion in the solution.

A study of errors appearing in traction boundary value problems on simply connected domains solved by the symmetric Galerkin boundary element method (SGBEM) was presented [3]. Two methods for the removal of rigid body motions from the null space of the discretized SGBEM system matrix were analyzed. The degenerate scale problem arising from the Dirichlet BVP of plane elasticity was studied in detail [4].

A general formulation for the static stiffness was analytically derived using the dual integral formulations [5]. It is found that the same stiffness matrix is derived by using the integral equation no matter what the rigid body mode and the complementary solutions are superimposed in the fundamental solution. According to the Fichera's idea, authors enriched the conventional BIE by adding constants and corresponding constraints [6]. The range deficiency of the integral operator for the solution space in the degenerate-scale problem for the 2D elasticity in the BIE was analytically studied. The studied problem was devoted to a Dirichlet BVP for an interior region.

The error estimation schemes are classified into the residual type, the interpolation type, the integral equation type, the node sensitivity type, and the solution difference type [7]. Those error estimation schemes were analyzed in detail.

It is well known that the patch test plays an important role in the FEM (finite element method) [8]. Typically, the prescribed exact solution in FEM consists of displacements that vary as linear functions in space (called a constant strain solution). The elements pass the patch test if the finite element solution is the same as the exact solution.

Following the concept of the patch test in FEM [8], several numerical tests are suggested for the Neumann problem of interior BVP of plane elasticity by using different techniques. We choose the elliptic plate as an object in the test. Along the boundary of the plate, two groups of boundary traction are assumed. Among them, the first group is derived from the constant stress solution (or constant strain solution), and the second group is derived from the linear stress distribution. The boundary tractions from the mentioned stress fields are the input data in the BIE. After making discretization to the relevant BIE, we can evaluate displacements along the boundary. It is well known the displacement solution along the boundary is not unique. Three different techniques are assumed for evaluating a unique displacement solution. From the displacement solution, we can evaluate the normal peripheral stress along the boundary which is denoted by σ_T,num. The stress component σ_T,num is compared with stress component σ_T,ext which is derived from the exact solution. In addition, several error estimation factors are assumed to evaluate the achieved accuracy. Extensive numerical examples and computed results are provided in the paper.