Numerical Solution of Helmholtz Equation by Boundary Elements Method

Boundary elements method is well-suited for computational acoustics. The main profit it brings is reduction in number of unknowns compared to FEM. However, application of BEM to sound scattering problems has shown, that this advantage is lost in case of high frequency waves. High frequencies require larger number of elements which contradicts the initial purpose of using BEM in solvers and may even result in situations, where solution does not exist for some exterior problems. A special choice of basis functions of exponential type can help to reduce computational cost for high wave numbers. The standard boundary integral formulation can further be improved using Burton-Miller scheme to avoid problems with existence of solution. 1 Helmholtz Boundary Equation Space-dependent part of time-harmonic acoustic wave is described by Helmholtz equation ∆u+ ku = 0 k = ω v (ω being frequency and v speed of sound in given media). Domain of solution is either interior or exterior of some closed curve (boundary) representing submerged obstacle in 2D. In case of the exterior problem an extra condition must be added to the condition on the body boundary, and it is so called Sommerfeld radiation condition, which ensures the uniqueness of solution to the scattering problem lim r→∞ r ( ∂u ∂r − iku ) = 0, r = |x|. Applying the second Green’s theorem to the problem as stated above we come to the boundary integral form of the Helmholtz equation valid in the interior and the exterior respectively