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An accelerated symmetric time-domain boundary element formulation for elasticity

机译：弹性加速对称时域边界元公式

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摘要

Wave propagation phenomena occur often in semi-infinite regions. It is well known that such problems can be handled well with the boundary element method (BEM). However, it is also known that the BEM, with its dense matrices, becomes prohibitive with respect to storage and computing time. Focusing on wave propagation problems, where a formulation in time domain is preferable, the mentioned limit of the method becomes evident. Several approaches, amongst them the adaptive cross approximation (ACA), have been developed in order to overcome these drawbacks mainly for elliptic problems.rnThe present work focuses on time dependent elastic problems, which are indeed not elliptic. The application of the presented fast boundary element formulation on such problems is enabled by introducing the well known Convolution Quadrature Method (CQM) as time stepping scheme. Thus, the solution of the time dependent problem ends up in the solution of a system of decoupled Laplace domain problems. This detour is worth since the resulting problems are again elliptic and, therefore, the ACA can be used in its standard fashion.rnThe main advantage of this approach of accelerating a time dependent BEM is that it can be easily applied to other fundamental solutions as, e.g., visco- or poroelasticity.
机译：波传播现象经常发生在半无限区域。众所周知，使用边界元方法（BEM）可以很好地解决此类问题。但是，众所周知，具有密集矩阵的BEM在存储和计算时间方面变得令人望而却步。针对波传播问题，在时域上的公式化是优选的，该方法的上述限制变得明显。为了克服主要针对椭圆问题的这些缺点，已经开发了几种方法，其中包括自适应交叉逼近（ACA）。目前的工作集中在时间相关的弹性问题上，这些问题确实不是椭圆的。通过引入众所周知的卷积正交方法（CQM）作为时间步进方案，可以将提出的快速边界元公式应用于此类问题。因此，与时间有关的问题的解决最终以解耦的拉普拉斯域问题的系统解决。这种绕行是值得的，因为由此产生的问题又是椭圆形的，因此，ACA可以以其标准方式使用。加速基于时间的BEM的这种方法的主要优点在于，它可以轻松地应用于其他基本解决方案，例如，例如，粘弹性或多孔弹性。

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