一种使用快速多极子方法（FMM）和$\mathcal{H}$矩阵技术的3D-ACA加速的弹性动力学时域边界元方法
A 3D-ACA accelerated time domain boundary element method for elastodynamics using FMM and $\mathcal{H}$-matrix techniques
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中文总结 AI 辅助
研究线性弹性动力学时域BEM，用gCQ方法，在拉普拉斯域评估BEM矩阵生成三维张量。采用3D-ACA计算低秩近似，在频率切片内用$\mathcal{H}$矩阵方法或FMM进一步压缩，通过实例比较研究并分析感应电机结构振动。
中文摘要 AI 辅助
本文考虑了初始条件为零的线性弹性动力学的时域边界元方法（BEM）。空间离散采用标准低阶边界元，时间离散采用广义卷积求积（gCQ）方法。gCQ框架需要在选定轮廓上的几个复频率处评估拉普拉斯域中的BEM矩阵，生成一个三维张量。为降低存储和计算成本，使用3D-自适应交叉近似（3D-ACA）计算张量的低秩近似。在每个频率切片内，使用基于$\mathcal{H}$矩阵方法的经典ACA算法或基于切比雪夫插值的快速多极子方法（FMM）进一步压缩BEM矩阵。通过两个学术例子对所有提出的方法进行了比较研究，并分析了感应电机的结构振动。
英文摘要
The time-domain Boundary Element Method (BEM) for linear elastodynamics with vanishing initial conditions is considered. Spatial discretization uses standard low-order boundary elements, while temporal discretization employs the generalized Convolution Quadrature (gCQ) method. The gCQ framework requires evaluating BEM matrices in the Laplace domain at several complex frequencies along a chosen contour, producing a three-dimensional tensor with one spatial matrix slice per frequency. To reduce storage and computational cost, a low rank approximation of the tensor is computed using 3D-Adaptive Cross Approximation (3D-ACA), extending the classical ACA to handle both the additional frequency dimension and the tensorial structure of elastodynamics. Within each frequency slice, the BEM matrices are further compressed using either the classical ACA algorithm using the $\mathcal{H}$-matrix approach or a Chebyshev interpolation based Fast Multipole Method (FMM). A comparative study of all proposed methods is carried out using two academic examples, and the structural vibration of an induction machine is analyzed.
