B-Spline FEM for Time-Harmonic Acoustic Scattering and Propagation

Abstract

We study the application of a B-splines Finite Element Method (FEM) to time-harmonic scattering acoustic problems. The infinite space is truncated by a fictitious boundary and second-order Absorbing Boundary Conditions (ABCs) are applied. The truncation error is included in the exact solution so that the reported error is an indicator of the performance of the numerical method, in particular of the size of the pollution error. Numerical results performed with high-order basis functions (third or fourth order) showed no visible pollution error even for very high frequencies. To prove the ability of the method to increase its accuracy in the high frequency regime, we show how to implement a high-order Padé-type ABC on the fictitious outer boundary. The above-mentioned properties combined with exact geometrical representation make B-Spline FEM a very promising platform to solve high-frequency acoustic problems.

Corner treatments for high-order local absorbing boundary conditions in high-frequency acoustic scattering

“…Nevertheless, in the complete plane-wave solution, the coefficients corresponding to the evanescent modes get smaller, and they do not contribute too much to the global error (see e.g. [55])…”

Highly accurate acoustic scattering: Isogeometric Analysis coupled with local high order Farfield Expansion ABC

“…The weak form (16) is obtained by applying the radial derivatives of Helmholtz operator (15) to Karp expansion, multiplying by a test functionv ∈ H 1 (S R ), integrating on S R and applying integration by parts with respect to the angular variable θ.…”

Non Uniform Rational B-Splines and Lagrange approximations for time-harmonic acoustic scattering: accuracy and absorbing boundary conditions

“…The pollution error of IGA is really lower than that of FEM for the same order p and same number of degrees of freedom, providing excellent accuracy in mid-to high-frequencies. The low pollution error of IGA was also observed in [38] for very high frequencies but was not compared to the pollution error in FEM before.…”

“…Since this problem is symmetric, we can fix the incidence direction to d¼ ð1; 0Þ T . The modified exact solution can be obtained as a summation of modal solutions [38]…”

“…Within this framework, NURBS are used as basis functions but are truncated. The performance of B-splines FEM and adaptive PHT-spline IGA for solving exterior time-harmonic acoustic problems were studied in [38,39]. In particular, it was shown that the pollution error is well controlled for a fixed discretization density and order of the basis functions p � 3.…”