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Improvement in the fault boundary conditions for a staggered grid finite-difference method

机译：交错网格有限差分法对故障边界条件的改进

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摘要

The staggered grid finite-difference method is a powerful tool in seismology and is commonly used to study earthquake source dynamics. In the staggered grid finite-difference method stress and particle velocity components are calculated at different grid points, and a faulting problem is a mixed boundary problem, therefore different implementations of fault boundary conditions have been proposed. Viriuex and Madariaga (1982) chose the shear stress grid as the fault surface, however, this method has several problems: (1) Fault slip leakage outside the fault, and (2) the stress bump beyond the crack tip caused by S waves is not well resolved. Madariaga et al. (1998) solved the latter problem via thick fault implementation, but the former problem remains and causes a new issue; displacement discontinuity across the slip is not well modeled because of the artificial thickness of the fault. In the present study we improve the implementation of the fault boundary conditions in the staggered grid finite-difference method by using a fictitious surface to satisfy the fault boundary conditions. In our implementation, velocity (or displacement) grids are set on the fault plane, stress grids are shifted half grid spacing from the fault and stress on the fictitious surface in the rupture zone is given such that the interpolated stress on the fault is equal to the frictional stress. Within the area which does not rupture, stress on the fictitious surface is given a condition of no discontinuity of the velocity (or displacement). Fault normal displacement (or velocity) is given such that the normal stress on the fault is continuous across the fault. Artificial viscous damping is introduced on the fault to avoid vibration caused by onset of the slip. Our implementation has five advantages over previous versions: (1) No leakage of the slip prior to rupture and (2) a zero thickness fault, (3) stress on the fault is reliably calculated, (4) our implementation is suitable for the study of fault constitutive laws, as slip is defined as the difference between displacement on the plane of z = + 0 and that of z = - 0, and (5) cessation of slip is achieved correctly.
机译：交错网格有限差分法是地震学中的强大工具，通常用于研究震源动力学。在交错网格有限差分方法中，在不同网格点计算应力和粒子速度分量，并且故障问题是混合边界问题，因此提出了故障边界条件的不同实现方式。 Viriuex和Madariaga（1982）选择了剪应力网格作为断层面，但是这种方法存在几个问题：（1）断层外漏失滑动，（2）由S波引起的超出裂纹尖端的应力凸点是不好解决。 Madariaga等。 （１９９８）通过大量的故障实现解决了后一个问题，但是前一个问题仍然存在并引起了新的问题。由于断层的人为厚度，不能很好地模拟滑移上的位移不连续性。在本研究中，我们通过使用虚拟表面来满足故障边界条件，从而改进了交错网格有限差分方法中故障边界条件的实现。在我们的实现中，在断层平面上设置速度（或位移）网格，将应力网格从断层移开一半网格距离，并给出破裂区域虚拟表面上的应力，使得断层上的插值应力等于摩擦应力。在不破裂的区域内，虚拟表面上的应力被赋予速度（或位移）不间断的条件。给出断层法向位移（或速度），以使断层上的法向应力在整个断层上是连续的。在故障处引入了人工粘性阻尼，以避免由于打滑而引起的振动。与以前的版本相比，我们的实现具有五个优点：（1）破裂前没有泄漏泄漏;（2）厚度为零的断层;（3）可靠地计算了断层上的应力;（4）我们的实现适合于研究断层本构定律的定义是：滑动定义为z = + 0平面上的位移与z =-0平面上的位移之间的差，并且（5）正确实现了滑动的停止。

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