Adaptive boundary element computation of acoustic radiation and scattering problems in two dimensions

Abstract

In this paper we carry out boundary element computations of the Helmholtz equation in two dimensions, in the context of time-harmonic exterior acoustics. The purpose is to demonstrate cost savings engendered through adaptivity for propagating solutions at moderate wave numbers. The computations are performed on meshes of constant boundary elements, and are adapted to the solution by locally changing element sizes (h-version). Burton and Miller approach is employed to solve the exterior problems for all wave numbers. Two error indicators obtained from the dual integral equations in conjuction with the exact error indicator are used for local error estimation, which are essential ingredients for all adaptive mesh schemes in BEM. Computational experiments are performed for the two-dimensional exterior acoustics problems. The three error tracking curves are in good agreement with their shapes. Two examples show that the adaptive mesh based on the error indicators converge faster than does uniform mesh discretization. 本文以邊界元素法求解二維外域聲場問題，配合Burton and Miller法解決了外域聲場數值共振的問題。並使用自適性網格切割的策略將誤差較大的問題邊界重新切割網格，提高求解的效率。所使用的自適性網格切割策略是屬於h-型。本文所採取的誤差指標為對偶積分方程式的第一式(UT)的殘餘量、第二式(LM)的殘餘量與解析差量等三種來做為局部誤差的估計，而成為自適性網格切割的策略的判斷依據。最後以兩個數值算例說明了使用自適性網格切割將提高解的收斂效率，驗證本法的可行性。