A Trefftz function approximation in local boundary integral equations

Abstract

In the present paper the Trefftz function as a test function is used to derive the local boundary integral equations (LBIE) for linear elasticity. Since Trefftz functions are regular, much less requirements are put on numerical integration than in the conventional boundary integral method. The moving least square (MLS) approximation is applied to the displacement field. Then, the traction vectors on the local boundaries are obtained from the gradients of the approximated displacements by using Hooke's law. Nodal points are randomly spread on the domain of the analysed body. The present method is a truly meshless method, as it does not need a finite element mesh, either for purposes of interpolation of the solution variables, or for the integration of the energy. Two ways are presented to formulate the solution of boundary value problems. In the first one the local boundary integral equations are written in all nodes (interior and boundary nodes). In the second way the LBIE are written only at the interior nodes and at the nodes on the global boundary the prescribed values of displacements and/or tractions are identified with their MLS approximations. Numerical examples for a square patch test and a cantilever beam are presented to illustrate the implementation and performance of the present method.