Isogeometric simulation of acoustic radiation

Abstract

In this paper we discuss the numerical solution of the Helmholtz equation with mixed boundary conditions on a 2D physical domain Ω. The so called radiation problem depends on the constant wavenumber k, that in some medical applications can be of order of thousands. For these values of k the classical Finite Element Method (FEM) faces up several numerical difficulties. To mitigate these limitations we apply the Isogeometric Analysis (IgA) to compute the approximated solution uh. Main steps of IgA are discussed and specific proposals for their fulfillment are addressed, with focus on some aspects not covered in available publications. In particular, we introduce a low distortion quadratic NURBS parametrization of Ω that represents exactly its boundary and contributes to the accuracy of uh. Our approach is non-isoparametric since uh is a bicubic tensor product polynomial B-spline function on Ω. This allows to improve the numerical solution refining the approximation space and keeping the coarser parametrization of the domain. Moreover, we discuss the role of the number of degrees of freedom in the directions perpendicular and longitudinal to wave front and its relationship with the noise and the shift in amplitude and phase of uh. The linear system derived from IgA discretization of the radiation problem is solved using GMRES and we show through experiments that the incomplete factorization of the Complex Shifted Laplacian provides a very good preconditioner. To solve the radiation problem, we have implemented IgA approach using the open source package GeoPDEs. A comparison with FEM is included, to provide evidence that IgA approach is superior since it is able to reduce significantly the pollution error, especially for high values of k, producing additionally smoother solutions which depend on less degrees of freedom.