CMG Research: Analysis and Application of XFEM to dynamic rupture processes in Earthquake physics

该项目将提供分析和功能开源软件，以评估扩展有限元法（XFEM）在复杂非平面断层上的动态地震爆发问题中的实用性和适用性。虽然标准有限元方法适用于许多地震物理问题，但要求网格与复杂的、非体积形成的断层网络共形是一个基本的困难，特别是在三维空间中。XFEM提供了一种潜在的强大替代方法，它将不连续的基函数编码到近似空间中，而不需要断层与网格边缘重合，这种方法自然适合地震破裂问题。然而，初步的工作显示了该应用程序中出现的几个复杂问题。摩擦破坏的标准技术不适用于XFEM，因此必须开发新的弱破坏公式。此外，在准静态工程破裂问题中被有效去除的离散奇异点，在动态重复破裂中是至关重要的。这些离散的问题会从根本上影响事件统计，必须避免。研究人员将通过分析、计算和物理直觉来解决这些复杂问题。具体来说，他们提出了两种弱失效准则的可能性，并计划用一系列基准问题对它们进行测试。此外，他们提出了对弱摩擦准则的数值精度分析，目的是更好地理解这些离散系统中破裂传播的误差。在第二部分，他们计划推导网格-断层交互工件存在的边界。使用这些边界，他们将确定消除工件的充实方案或开发网格方案来避免它们。最后，上述工作将在重复破裂问题中的XFEM概念验证中达到高潮，并且他们将更好地理解复杂故障系统中的事件复杂性。这些问题为计算和应用数学家以及地震物理学家之间的合作提供了绝佳的机会。更广泛的意义：在实际复杂的断层网络上了解地震的动力学和概率是一个基本的科学和工程问题，它对改进地震危险性的估计有直接的影响。先进的计算模型与观测相结合，为探索和理解这些系统提供了重要的工具。然而，这种模型的一个关键组成部分是对非平面断层的几何描述和精确建模，这对传统的有限元方法提出了重大挑战。该项目将研究一种替代方法，该方法允许故障网络的描述仅与计算网格松散耦合。如果这种方法是成功的，它有望大大增加描述和组成这些问题的便利性，并且更适合于探索故障网络的动态，特别是在故障定位的不确定性下。提出的研究也将有助于建立脆性系统动力学和地震成因的计算基础设施。由此产生的开源软件将通过地球动力学计算基础设施（CIG: www.geodynamics.org）发布，并可供广泛的研究人员访问，其影响超出了地震物理学的直接领域。

This project will provide both analysis and functioning open-sourcesoftware to evaluate the utility and applicability of the ExtendedFinite Element Method (XFEM) to the problem of dynamic Earthquakerupture on complex non-planar faults. While standard finite elementmethods are applicable to many earthquake physics problems, therequirement that the mesh be conformal to a complex, non-volumeforming network of faults is a fundamental difficulty, particularly in3-dimensions. The XFEM, provides a potentially powerful alternativeby encoding discontinuous basis functions into the approximation spacewithout requiring faults to coincide with mesh edges, an approach thatnaturally fits the earthquake rupture problem.Preliminary work, however, demonstrates several complications thatarise in this application. Standard techniques for frictional failuredo not work with the XFEM, so new weak formulations of failure must bedeveloped. In addition, discrete singularities, which are effectivelyremoved in quasi-static engineering rupture problems, become crucialin dynamic repeated rupture. These discrete problems canfundamentally affect event statistics, and must be avoided.The investigators will address these complications via analysis,computation, and physical intuition. Specifically, they propose twopossibilities for weak failure criteria, and plan to test them with aseries of benchmark problems. Additionally, they propose analysis ofnumerical accuracy of the weak frictional criteria, with the goal ofbetter understanding error in rupture propagation in these discretesystems. In a second component, they plan to derive bounds for theexistence of mesh-fault interaction artifacts. Using these bounds, theywill either determine enrichment schemes that eliminate the artifactsor develop meshing schemes to avoid them. Finally, the above workwill culminate in a proof of concept for the XFEM in repeated ruptureproblems, and they will work to better understand event complexity incomplex fault systems. These problems provide an excellentopportunity for collaboration between computational and appliedmathematicians and earthquake physicists.Broader Significance: Understanding the dynamics and probability ofEarthquakes on realistically complex fault networks is a fundamentalscience and engineering problem that has direct consequences forimproved estimates of Earthquake hazards. Advanced computationalmodels, combined with observations, provide an important tool forexploring and understanding these systems. A critical component ofsuch models, however is the geometric description and accuratemodeling of failure on non-planar faults which poses significantchallenges for traditional finite-element methods. This project willinvestigate an alternative method that allows the description of thefault network to be only loosely coupled to the computational mesh.If this method is successful, it promises to significantly increasethe ease of describing and composing these problems and is bettersuited to exploring the dynamics of fault networks, particularly underthe uncertainty of fault location. The proposed research will alsocontribute to the computational infrastructure for modeling thedynamics of brittle systems and earthquake genesis. The resulting opensource software will be distributed through the ComputationalInfrastructure for Geodynamics (CIG: www.geodynamics.org), and beaccessible to a broad community of researchers with impact beyond theimmediate realm of Earthquake physics.