Comparison of finite difference and boundary integral solutions to three-dimensional spontaneous rupture
Comparison of finite difference and boundary integral solutions to three-dimensional spontaneous rupture
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DOI：
10.1029/2005jb003813
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发表时间：
2005-12-23
影响因子：
3.9
通讯作者：
Liu, Y
中科院分区：
文献类型：
--
作者：
Day, SM;Dalguer, LA;Liu, Y
在摩擦界面上自发传播的剪切裂纹已被证明是一种有用的自然地震的理想情况。相应的边值问题是非线性的，通常需要计算密集的数值方法来求解。评估数值方法的收敛性和准确性是具有挑战性的，因为我们缺乏适当的分析解决方案进行比较。作为其他评估方法的补充，我们比较了两种独立的数值方法，有限差分方法和边界积分（BI）方法得到的解。有限差分实现，称为DFM，使用故障不连续的分割节点牵引公式。BI实现采用应力传递函数的谱表示。三维（3-D）测试问题涉及自发破裂在平面界面上的传播，该界面由线性滑移削弱摩擦控制，本质上定义了一个内聚规律。为了先验地理解这个问题和类似问题所需的空间分辨率，我们回顾并结合了一些简单的内聚区大小估计，这些估计与模拟中观察到的大小相当吻合。我们根据破裂时间、最终滑移和峰值滑移率的均方根差异评估了两种方法之间的一致性，并将这些差异与数值解中观察到的黏聚带分辨率的中值和最小值测量相关联。当它们的网格间距Delta x足够小时，BI和DFM方法对三维自发破裂测试问题给出了几乎无法区分的解决方案，从而使解决方案充分解决了内聚区，BI至少有三个点，DFM至少有五个节点点。此外，对于单独采取的两种方法中的每一种，结果中与网格相关的差异作为Delta x的幂律衰减，每种方法具有相同的收敛速率，计算显然收敛到一个共同的网格区间不变解。这一结果为两种方法的准确性提供了强有力的证据。此外，本文提出的具体解，由于在两种截然不同的数值方法之间具有明显的网格无关性和一致性，可能对测试用于自发破裂问题的新数值方法有用。
[1] The spontaneously propagating shear crack on a frictional interface has proven to be a useful idealization of a natural earthquake. The corresponding boundary value problems are nonlinear and usually require computationally intensive numerical methods for their solution. Assessing the convergence and accuracy of the numerical methods is challenging, as we lack appropriate analytical solutions for comparison. As a complement to other methods of assessment, we compare solutions obtained by two independent numerical methods, a finite difference method and a boundary integral (BI) method. The finite difference implementation, called DFM, uses a traction-at-split-node formulation of the fault discontinuity. The BI implementation employs spectral representation of the stress transfer functional. The three-dimensional (3-D) test problem involves spontaneous rupture spreading on a planar interface governed by linear slip-weakening friction that essentially defines a cohesive law. To get a priori understanding of the spatial resolution that would be required in this and similar problems, we review and combine some simple estimates of the cohesive zone sizes which correspond quite well to the sizes observed in simulations. We have assessed agreement between the methods in terms of the RMS differences in rupture time, final slip, and peak slip rate and related these to median and minimum measures of the cohesive zone resolution observed in the numerical solutions. The BI and DFM methods give virtually indistinguishable solutions to the 3-D spontaneous rupture test problem when their grid spacing Delta x is small enough so that the solutions adequately resolve the cohesive zone, with at least three points for BI and at least five node points for DFM. Furthermore, grid-dependent differences in the results, for each of the two methods taken separately, decay as a power law in Delta x, with the same convergence rate for each method, the calculations apparently converging to a common, grid interval invariant solution. This result provides strong evidence for the accuracy of both methods. In addition, the specific solution presented here, by virtue of being demonstrably grid-independent and consistent between two very different numerical methods, may prove useful for testing new numerical methods for spontaneous rupture problems.