A Hybrid Scheme Based on Finite Element/Volume Methods for Two Immiscible Fluid Flows

We have successfully extended our implicit hybrid finite element/volume (FE/FV) solver to flows involving two immiscible fluids. The solver is based on the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix-free implicit cell-centered FV method. The pressure Poisson equation is solved by the node-based Galerkin FE method for an auxiliary variable. The auxiliary variable is used to update the velocity field and the pressure field. The pressure field is carefully updated by taking into account the velocity divergence field. This updating strategy can be rigorously proven to be able to eliminate the unphysical pressure boundary layer and is crucial for the correct temporal convergence rate. Our current staggered-mesh scheme is distinct from other conventional ones in that we store the velocity components at cell centers and the auxiliary variable at vertices. The fluid interface is captured by solving an advection equation for the volume fraction of one of the fluids. The same matrix-free FV method, as the one used for momentum equations, is used to solve the advection equation. We will focus on the interface sharpening strategy to minimize the smearing of the interface over time. We have developed and implemented a global mass conservation algorithm that enforces the conservation of the mass for each fluid.

Finite-Element Simulations of Earthquakes

This thesis discusses simulations of earthquake ground motions using prescribed ruptures and dynamic failure. Introducing sliding degrees of freedom led to an innovative technique for numerical modeling of earthquake sources. This technique allows efficient implementation of both prescribed ruptures and dynamic failure on an arbitrarily oriented fault surface. Off the fault surface the solution of the three-dimensional, dynamic elasticity equation uses well known finite-element techniques. We employ parallel processing to efficiently compute the ground motions in domains containing millions of degrees of freedom.

Using prescribed ruptures we study the sensitivity of long-period near-source ground motions to five earthquake source parameters for hypothetical events on a strike-slip fault (M_w 7.0 to 7.1) and a thrust fault (M_w 6.6 to 7.0). The directivity of the ruptures creates large displacement and velocity pulses in the ground motions in the forward direction. We found a good match between the severity of the shaking and the shape of the near-source factor from the 1997 Uniform Building Code for strike-slip faults and thrust faults with surface rupture. However, for blind thrust faults the peak displacement and velocities occur up-dip from the region with the peak near-source factor. We assert that a simple modification to the formulation of the near-source factor improves the match between the severity of the ground motion and the shape of the near-source factor.

For simulations with dynamic failure on a strike-slip fault or a thrust fault, we examine what constraints must be imposed on the coefficient of friction to produce realistic ruptures under the application of reasonable shear and normal stress distributions with depth. We found that variation of the coefficient of friction with the shear modulus and the depth produces realistic rupture behavior in both homogeneous and layered half-spaces. Furthermore, we observed a dependence of the rupture speed on the direction of propagation and fluctuations in the rupture speed and slip rate as the rupture encountered changes in the stress field. Including such behavior in prescribed ruptures would yield more realistic ground motions.