This project will provide both analysis and functioning open-source software to evaluate the utility and applicability of the Extended Finite Element Method (XFEM) to the problem of dynamic Earthquake rupture on complex non-planar faults. While standard finite element methods are applicable to many earthquake physics problems, the requirement that the mesh be conformal to a complex, non-volume forming network of faults is a fundamental difficulty, particularly in 3-dimensions. The XFEM, provides a potentially powerful alternative by encoding discontinuous basis functions into the approximation space without requiring faults to coincide with mesh edges, an approach that naturally fits the earthquake rupture problem.
Preliminary work, however, demonstrates several complications that arise in this application. Standard techniques for frictional failure do not work with the XFEM, so new weak formulations of failure must be developed. In addition, discrete singularities, which are effectively removed in quasi-static engineering rupture problems, become crucial in dynamic repeated rupture. These discrete problems can fundamentally affect event statistics, and must be avoided.
The investigators will address these complications via analysis, computation, and physical intuition. Specifically, they propose two possibilities for weak failure criteria, and plan to test them with a series of benchmark problems. Additionally, they propose analysis of numerical accuracy of the weak frictional criteria, with the goal of better understanding error in rupture propagation in these discrete systems. In a second component, they plan to derive bounds for the existence of mesh-fault interaction artifacts. Using these bounds, they will either determine enrichment schemes that eliminate the artifacts or develop meshing schemes to avoid them. Finally, the above work will culminate in a proof of concept for the XFEM in repeated rupture problems, and they will work to better understand event complexity in complex fault systems. These problems provide an excellent opportunity for collaboration between computational and applied mathematicians and earthquake physicists.
Broader Significance: Understanding the dynamics and probability of Earthquakes on realistically complex fault networks is a fundamental science and engineering problem that has direct consequences for improved estimates of Earthquake hazards. Advanced computational models, combined with observations, provide an important tool for exploring and understanding these systems. A critical component of such models, however is the geometric description and accurate modeling of failure on non-planar faults which poses significant challenges for traditional finite-element methods. This project will investigate an alternative method that allows the description of the fault network to be only loosely coupled to the computational mesh. If this method is successful, it promises to significantly increase the ease of describing and composing these problems and is better suited to exploring the dynamics of fault networks, particularly under the uncertainty of fault location. The proposed research will also contribute to the computational infrastructure for modeling the dynamics of brittle systems and earthquake genesis. The resulting open source software will be distributed through the Computational Infrastructure for Geodynamics (CIG: www.geodynamics.org), and be accessible to a broad community of researchers with impact beyond the immediate realm of Earthquake physics.
