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Convolution quadrature boundary element method for quasi-static visco- and poroelastic continua

机译:准静态粘弹性和多孔弹性连续体的卷积正交边界元方法

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The main difference of convolution quadrature method (CQM)-based boundary element formulations to usual time-stepping BE formulations is the way to solve the convolution integral appearing in most time-dependent integral equations. In the CQM formulation, the convolution integrals are approximated by a quadrature rule whose weights are determined by the Laplace transformed fundamental solutions and a multi-step method. So, there is no need of a time domain fundamental solution. For quasi-static problems in visco- or poroelasticity time-dependent fundamental solutions are available, but these fundamental solutions are highly complicated yielding to very sensitive algorithms. Especially in viscoelasticity, for every rheological model a separate fundamental solution must be deduced. Here, firstly, viscoelastic as well as poroelastic constitutive equations are recalled and, then, the respective integral equations are presented. Applying the usual spatial discretization and using the CQM for the temporal discretization yields the final time-stepping algorithm. The proposed methodology is tested by two simple examples considering creep behavior in viscoelasticity and consolidation processes in poroelasticity. The algorithm shows no stability problems and behaves well over a broad range of time step sizes.
机译:基于卷积正交方法(CQM)的边界元素公式与常规时间步长BE公式的主要区别是解决大多数时间相关积分方程中出现的卷积积分的方法。在CQM公式中,卷积积分由正交规则近似,其权重由拉普拉斯变换的基本解和多步法确定。因此,不需要时域基本解决方案。对于粘弹性或多孔弹性中的准静态问题,可以使用时间相关的基本解,但是这些基本解非常复杂,导致生成非常敏感的算法。特别是在粘弹性方面,对于每种流变模型,必须推导出单独的基本解决方案。在这里,首先,回顾粘弹性以及多孔弹性的本构方程,然后给出各个积分方程。应用通常的空间离散化并将CQM用于时间离散化会产生最终的时间步长算法。所提出的方法通过两个简单的示例进行了测试,这些示例考虑了粘弹性的蠕变行为和多孔弹性的固结过程。该算法没有显示稳定性问题,并且在较大的时间步长范围内表现良好。

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