A comparison of techniques for overcoming non‐uniqueness of boundary integral equations for the collocation partition of unity method in two‐dimensional acoustic scattering

Abstract

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The Wave-Matching Boundary Integral Equation — An energy approach to Galerkin BEM for acoustic wave propagation problems

“…The CHIEF and Burton-Miller approaches were compared for POU-BEM in [40]. Here the authors concluded that CHIEF outperformed Burton-Miller, though implementation details such as regularising coordinate transforms appear to have had a significant effect.…”

“…Here the authors concluded that CHIEF outperformed Burton-Miller, though implementation details such as regularising coordinate transforms appear to have had a significant effect. It is worth pointing out here that regularisation of the hypersingular operator, which occurs in the Burton-Miller formulation, is rather challenging for collocation schemes, as was used in [40]. In contrast, Galerkin schemes allow the hypersingular operator to be rewritten in a form that only contains a weak singularity [41,42].…”

“…We would now regard them as entries from the discrete Dirichletto-Neumann map, since they apply to individual basis functions within the approximation space rather than the total solution. They can also be used to simplify (40), again using in place of , to find the reflection coefficient that would occur if the plane had admittance :…”

“…Using in the exponent of (80) resolves this issue and was demonstrated to be accurate, but produces an approximation space that is so badly conditioned that it is impossible to see any difference in the condition numbers due to the different boundary operators. Very high condition numbers are the norm in POU-BEM and specialist matrix solvers are required to overcome them [40], but that does not help with this analysis. Using , as is instead done here, gives better conditioning since all the basis functions with a particular value of are orthogonal when multiplied by a single envelope function.…”

Extended isogeometric boundary element method (XIBEM) for three-dimensional medium-wave acoustic scattering problems

“…Diwan et al [34] have recently suggested that CHIEF is a suitable solution to the non-uniqueness problem and is simpler to implement than dealing with the hyper-singular integrals of the Burton-Miller formulation [4]. In addition to the complex integrals of the Burton-Miller formulation, there is another problem that must be addressed: some NURBS meshes have locations at which control points are coincident (e.g.…”