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

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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.