论文标题
由3D标量Helmholtz方程产生的BEM矩阵的频率提取
Frequency extraction for BEM-matrices arising from the 3D scalar Helmholtz equation
论文作者
论文摘要
标量Helmholtz方程的边界积分方程的离散化导致较大的致密线性系统。有效的边界元素方法（BEM），例如基于快速的多极方法（FMM）和$ \ hmat $的方法，重点关注这些系统中子块的结构性低级别近似值。众所周知，这些子块的等级随波数字的线性增加。我们基于提取绿色函数的已知阶段的bem-matrices探索BEM-Matrices的数据 - 比较表示。从代数上讲，这导致了$ \ hmat $的频率矩阵的Hadamard产品。我们表明，即使对于几何复杂的三维散射障碍，也可以使用少量频率样本确定此$ \ hmat $的频率依赖性。我们通过将自适应交叉近似与在连续频率维度中的适应性有理近似相结合，描述了表示形式的有效构造。我们表明，我们的数据 - 帕克斯表示允许以任何给定的频率有效地对完整的bem-matrix进行采样，因此，它可能是有效扫描例程的一部分。
The discretisation of boundary integral equations for the scalar Helmholtz equation leads to large dense linear systems. Efficient boundary element methods (BEM), such as the fast multipole method (FMM) and $\Hmat$ based methods, focus on structured low-rank approximations of subblocks in these systems. It is known that the ranks of these subblocks increase linearly with the wavenumber. We explore a data-sparse representation of BEM-matrices valid for a range of frequencies, based on extracting the known phase of the Green's function. Algebraically, this leads to a Hadamard product of a frequency matrix with an $\Hmat$. We show that the frequency dependency of this $\Hmat$ can be determined using a small number of frequency samples, even for geometrically complex three-dimensional scattering obstacles. We describe an efficient construction of the representation by combining adaptive cross approximation with adaptive rational approximation in the continuous frequency dimension. We show that our data-sparse representation allows to efficiently sample the full BEM-matrix at any given frequency, and as such it may be useful as part of an efficient sweeping routine.
