Introduction to modern Galerkin-type boundary element methods with an application from mechanical engineering

The Galerkin-type boundary element method (BEM) is an discretization proced ure for integral equations, represents itself however compared with classic al integral equation methods as an universal tool for the solution of pract ical engineering problems and can be coupled very easily with finite elemen t substructures. The BEM, whose main advantage lies in the fact that only a surface mesh must be generated, is superior to FEM in special applications , i.e. in elastostatics (notch problems) and fracture mechanics. In this pa per the individual steps to solving an elliptical boundary value problem of 3-D linear elasticity theory by way of an equivalent system of boundary in tegral equations will be explained. For the mathematical investigation of e lliptical differential equations and integral equations, the theory of Sobo lev spaces has proved to be especially suitable. Basic terms to Sobolev spa ces will be introduced so that the reader does not have to refer to textboo ks for new terms. The transformation of elliptical boundary value problems to systems of singular and hypersingular integral equations will be explain ed with help of a Calderon projector, which is defined by using fundamental solutions. The discretization of the obtained integral equations with the Galerkin-type BEM will be presented. Finally the approximation of non-linea r problems by using the Galerkin-type BEM will be shown. A numerical test f or a strength problem will be discussed shortly.