Abstract

In this part of the paper we shall use the formulation given in the first part to assess the quality of recovery based error estimators using two recovery methods, i.e. Superconvergent Patch Recovery (SPR) and Recovery by Equilibrium in Patches (REP). The recovery methods have been shown to be asymptotically robust and superconvergent when applied to two dimensional problems. In this study we shall examine the behavior of the recovery methods on several three dimensional mesh patterns for patches located either inside or at boundaries. This is performed by first finding an asymptotic finite element solution, irrespective of boundary conditions at far ends of the domain, and then applying the recovery methods. The test procedure near kinked boundaries is explained in a step by step manner. The results are given in a series of tables and figures for various cases of three dimensional mesh patterns. It has been experienced that the full superconvergent property is generally lost due to presence of boundary layer solution and the definition of the recoveries near boundaries though the results of the robustness test is still within an acceptable range.

Abstract

In this report dynamic solution of unbounded domains using pure finite element method is presented. The problems of concern are those with governing differential equations of constant coefficient. When the grid is of repeatable pattern the solution of an unbounded domain is reducible to a solution over a smaller domain with a grid consisting of few numbers of repeatable patterns. It is shown that by use of the proportionality property, having roots in governing equations, both conditions required for a unique solution, i.e. the decay and radiation of energy, are met through a spectral formulation. As a key point, a consistent transformation approach is proposed in order to employ such a spectral formulation for the solution. The transformation technique is analogous to those conventionally used for solution of partial differential equations but of course in a matrix form. To demonstrate the applicability of method, Green’s functions for two dimensional scalar and elastic wave equations are obtained numerically in frequency domain. The method is also capable of giving Green’s functions for dynamic solution of domains with repeatable material properties. Comprehensive discussions are given for accuracy and convergence of the solution.