Nitsche Extended Finite Element Methods for Earthquake Simulation

Ethan Coon, PhD Thesis

Thesis, PDF
Defense slides, PDF

Abstract:

Current research indicates that fault geometry plays an important role in the physics of both earthquake dynamics and lithostatic deformation. Accurate depiction of the fault geometry, including many, rough faults within a computational mesh has proven a difficult task for current mesh generation techniques. Generating meshes which incorporate fault surfaces while maintaining good computational properties limits fault system simulation.

The eXtended Finite Element Method (XFEM) provides an alternate way of including fault system geometry in computations. We demonstrate a new method for fault system science, using Nitsche's method to incorporate the mixed boundary conditions needed for rupture. The XFEM is then used to discretize these equations. In this method, faults are included independently of the background mesh through the introduction of discontinuous functions into the approximation space. Basis functions which enable stress singularities at fault tips are also added to the approximation space. An inversion process is used to calculate stress and update friction and the rupture front.

We demonstrate the method on a hierarchy of benchmarks, and see good agreement with established solutions when faults are on element edges. The solution is shown to be equally accurate when coordinate systems are rotated so that faults are no longer coincident with element edges. In this verification, we test and compare several choices in the method, including tip functions and methods for calculating frictional tractions. Finally, the method is demonstrated through two classes of problems with implications for fault mechanics. First we solve static problems on a fault model of southern California, and compare results to slip rate distribution data. Our model agrees with the basic slip-distribution on vertical faults in the region. Finally we solve problems of repeated rupture, modeling the earthquake cycle over the course of many earthquakes. This thesis acts as a proof of concept for extended finite element methods in crustal deformation models, and uses the method to begin to explore the role of both individual fault nonplanarity and fault system structure in brittle deformation. In doing so, we identify strengths and weaknesses of the method, and find it to demonstrate great potential for simulations on complex fault networks,