Some applications of the Greens' function method in mechanics

Abstract

Almost each applied mechanical question leads to some boundary value problem of mathematical physics for the body of a complicated shape, whose boundary surface does not coincide with the coordinate surfaces of the selected coordinate system. Mathematical physics, however, has traditionally developed methods (separation of variables, variational, integral transformations, etc.) which are basically suitable for bodies bounded by simple coordinate surfaces.The finite differences technique applied to problems for bodies of complicated shape requires too large arrays for the storing of interior data so it cannot compete with the widely used method of finite elements. p]The basic idea of the Boundary-integral equation (BIE) method is to represent the unknown solution of the given problem in terms of the surface integral, the kernel of which is the fundamental solution of the governing operator. This method recently became rather popular in mechanics because it has some known [13] advantages in comparison with the finite element analysis.The main distinction between the potential approach which is described in our paper and the BIE method is that here instead of the fundamental solution for the kernel of the potential we are using the Green's matrix of some domain for which the given one is only some portion. Therefore corresponding potentials satisfy the boundary conditions on the part of the bounding surface automatically. Subsequently, we are left only with the necessity of treating the remaining part of the boundary. This approach allows us to solve the problems [2–6, 14–18] for bodies of very complicated geometry.The first section of the paper contains an explanation of the basic idea of the approach. The numerical examples which are presented in this section applied to the elastic torsion problem.Two- and three-dimensional steady-state heat conduction problems are presented in the second section. There one can find, for instance, the case of the layered strip with arbitrary holes.The last section contains a description of the algorithm for constructing of the Green's matrix for the sandwich type of elastic body. Some numerical examples for the homogeneous and layered strip with holes of various shapes are shown.Other possible applications of the potential method described are briefly reviewed in the conclusion of the paper.