标题： 时间域边界元方法中FMM和$\mathcal{H}$矩阵的三维ACA比较 显示英文标题

标题： Comparison of FMM and $\mathcal{H}$-matrix based 3D-ACA for a time domain boundary element method

摘要： 考虑具有零初始条件的齐次波动方程的时间域边界元方法（BEM）。 对于时间离散化，使用Lopez-Fernandez和Sauter开发的广义卷积求积方法（gCQ）。 空间离散化经典地使用低阶形状函数进行。 本质上，gCQ需要在几个复数频率下建立相应椭圆问题的拉普拉斯域边界元矩阵。 因此，得到一个系统矩阵数组。 这个系统矩阵数组可以被解释为一个三维数据数组，应通过数据稀疏表示进行近似，对此可以应用广义自适应交叉近似（3D-ACA）。 三维数据数组的秩被自适应地增加，直到达到预定的精度。 在纯代数层面上，决定三维数据数组的低秩近似是否足够接近原始矩阵。 在对应于每个频率的BEM计算的数据切片中，可以使用标准的$\mathcal{H}$矩阵方法结合ACA或快速多极子（FMM）方法。 数据数组的第三维代表复数频率。 因此，该算法不仅在两个空间维度上对数据数组进行稀疏化，还自适应地检测哪些矩阵块需要多少频率。 在切片中使用ACA或FMM的两种版本进行了简要讨论。 本文的主要贡献是两者在存储和计算时间节省方面的比较。 电机器声音散射的例子表明，这两种技术都允许在现实世界的问题中使用时域BEM。 显示英文摘要

摘要： The time domain Boundary Element Method (BEM) for the homogeneous wave equation with vanishing initial conditions is considered. For the temporal discretisation, the generalized convolution quadrature method (gCQ) developed by Lopez-Fernandez and Sauter is used. The spatial discretisation is done classically using low-order shape functions. Essentially, the gCQ requires to establish boundary element matrices of the corresponding elliptic problem in Laplace domain at several complex frequencies. Consequently, an array of system matrices is obtained. This array of system matrices can be interpreted as a three-dimensional array of data which should be approximated by a data-sparse representation, for which the generalised Adaptive Cross Approximation (3D-ACA) can be applied. The rank of the three-dimensional data array is increased adaptively until a prescribed accuracy is obtained. On a pure algebraic level, it is decided whether a low-rank approximation of the three-dimensional data array is close enough to the original matrix. Within the data slices corresponding to the BEM calculations for each frequency, either the standard $\mathcal{H}$-matrices approach with ACA or a fast multipole (FMM) approach can be used. The third dimension of the data array represents the complex frequencies. Hence, the algorithm does not only sparsify the data array in the two spatial dimensions but also adaptively detects how much frequencies are necessary for which matrix block. he two versions, either using ACA or FMM within the slices, are briefly discussed. The main contribution of this paper is a comparison of both with respect to savings in storage and computing time. The example of the sound scattering of an electric machine shows that both techniques allow to utilise a time-domain BEM in real-world problems.