title: Solving the hypersingular boundary integral equation for the Burton and Miller formulation.
abstract: This paper presents an easy numerical implementation of the Burton and Miller (BM) formulation, where the hypersingular Helmholtz integral is regularized by identities from the associated Laplace equation and thus needing only the evaluation of weakly singular integrals. The Helmholtz equation and its normal derivative are combined directly with combinations at edge or corner collocation nodes not used when the surface is not smooth. The hypersingular operators arising in this process are regularized and then evaluated by an indirect procedure based on discretized versions of the Calderón identities linking the integral operators for associated Laplace problems. The method is valid for acoustic radiation and scattering problems involving arbitrarily shaped three-dimensional bodies. Unlike other approaches using direct evaluation of hypersingular integrals, collocation points still coincide with mesh nodes, as is usual when using conforming elements. Using higher-order shape functions (with the boundary element method model size kept fixed) reduces the overall numerical integration effort while increasing the solution accuracy. To reduce the condition number of the resulting BM formulation at low frequencies, a regularized version α = ik/(k(2 )+ λ) of the classical BM coupling factor α = i/k is proposed. Comparisons with the combined Helmholtz integral equation Formulation method of Schenck are made for four example configurations, two of them featuring non-smooth surfaces.
authors.affiliation: Laboratoire de Mécanique des Structures et des Systèmes Couplés, Conservatoire National des Arts et Métiers, 292 Rue Saint Martin, 75141 Paris Cedex 03, France.
authors.forename: Christophe
authors.initials: C
authors.lastname: Langrenne
authors.affiliation: Laboratoire de Mécanique des Structures et des Systèmes Couplés, Conservatoire National des Arts et Métiers, 292 Rue Saint Martin, 75141 Paris Cedex 03, France.
authors.forename: Alexandre
authors.initials: A
authors.lastname: Garcia
authors.affiliation: POEMS (ENSTA ParisTech, CNRS, INRIA, Université Paris-Saclay), 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France.
authors.forename: Marc
authors.initials: M
authors.lastname: Bonnet