Boundary Element Method

This lecture enables the students to understand the basic approach using the boundary element method. They will know about the arising difficulties and the limits of this method. Furthermore they can distinguish the approach from other methods such as FEM or the SEA. With the help of exercises and a computer workshop the students will be able to apply this method on structures like membranes or plates.

The Boundary Element Method is a numerical method to solve engineering problems as wave propagation or sound radiation. However, it can also be applied to static and dynamic problems. The application of the BEM is advantageous in case of large or infinite domains. Typical applications arise in fractural mechanics, soil mechanics or acoustics. The Boundary Element method is, as well as the Finite Element Method, a numerical method for differential equations. Key point of the boundary element method is to describe the differential equation in their weak form by the principle of the weighted residuals. The resulting integral equation over the domain is then transferred to a boundary integral (integral along the boundary) by Green’s theorem. The dimension of the problem is reduced by one. Only the displacements and loads along the boundary are unknown. The system of equations can be interpreted as influence functions of the boundary forces and displacements (Betti). Furthermore, the fundamental solution of the homogenous differential equation is used as the weighting function for the weighted residuals. Finally the boundary is discretised and the set of boundary integrals can be solved numerically. The lecture covers the derivation of the boundary element method as well as their application to static and dynamic problems. Thematic outline: Differential equations Approximate solutions by weighted residual - Weighted residual - Galerkin - Collocation - Ritz Boundary Element Method - Derivation of the BEM: residual, Gauß integration - Bettis law - Fundamental solution (Greens function) - weighting functions - form functions Difficulties - How to determine Greens function? - Singularities of Greens function - singular integrals - integration along boundaries Solution of the system of equations - evaluation of the unknown deflections or boundary loads -alternative evaluation of the singular integrals Application of the Boundary-Element technique to dynamics - influence function - convolution - reciprocity in dynamics - Gauß function in dynamics - further volume and boundary integral - form functions in dynamics - evaluation of reciprocity - stationary loads