Stable and efficient methods for the interaction of shock waves with solid bodies [electronic resource]

This thesis considers complex scenarios involving two-way coupled interactions between compressible fluids and solid bodies under extreme conditions where monolithic, as opposed to partitioned, schemes are preferred for maintaining stability. When considering such problems, spurious numerical cavitation can be quite common and have deleterious consequences on the flow field stability, accuracy, etc. Thus, it is desirable to devise numerical methods that maintain the positivity of important physical quantities such as density, internal energy and pressure. It is shown that for an arbitrary flux function, one can enforce conditions on the time step in order to preserve positivity by solving a linear equation for density fluxes and a quadratic equation for energy fluxes. The proposed formulation is independent of the underlying equation of state. After deriving the method for forward Euler time integration, it is further extended to higher order accurate Runge-Kutta methods. Although the scheme works well in general, there are some cases where no lower bound on the size of the allowable time step exists. Thus, to prevent the size of the time step from becoming arbitrarily small, a conservative flux clamping scheme which is also positivity preserving is introduced. Exploiting the generality of the formulation, a positivity preserving scheme for a semi-implicit approach to time integration that solves a symmetric positive definite linear system to determine the pressure associated with an equation of state is then designed. Finally, this modified semi-implicit approach is extended to monolithic two-way solid-fluid coupling problems for modeling fluid structure interactions such as those generated by blast waves impacting complex solid objects. Next, the dissertation presents a novel method for treating bubbles in incompressible flow that relies on the conservative advection of bubble mass and an associated equation of state in order to determine pressure boundary conditions inside each bubble. It is shown that executing this algorithm in a traditional manner leads to stability issues similar to those seen for partitioned methods for solid-fluid coupling. Therefore, the problem is reformulated monolithically. This method is then extended to model both large and small scale bubble dynamics. Small under-resolved bubbles are evolved using Lagrangian particles that are monolithically two-way coupled to the surrounding flow in a manner that closely approximates the analytic bubble oscillation frequency while converging to the analytic volume as predicted by the well-known Rayleigh-Plesset equation. Drawing motivation from modeling small scale bubble dynamics, the monolithic solid-fluid coupling presented above is further extended to handle sub-grid rigid bodies while preserving positivity. Finally, the dissertation presents two approaches for simulating reduced order models in the context of visual computing. Unlike the previous methods presented in this thesis, these approaches are targeted towards real-time applications, and while they provide visually plausible results, the results might not always be physically accurate. First, a novel method for the efficient denting and bending of rigid bodies without the need for expensive finite element simulations is presented. Denting is achieved by deforming the triangulated surface of the target body based on a dent map computed on-the-fly from the projectile body using a Z-buffer algorithm with varying degrees of smoothing. On the other hand, bending is addressed by augmenting a rigid body with an articulated skeleton which is used to drive skinning weights for the bending deformation. Second, a novel sharp-crease bending element for the folding and wrinkling of surfaces and volumes is presented. The key idea is to cut the object along a given curve using the virtual node algorithm creating new degrees of freedom, while subsequently reattaching the resulting pieces eliminating the translational degrees of freedom so that adjacent pieces may only rotate or bend about the cut.

Advisor: Fedkiw Ronald P.
Lele Sanjiva K. (Sanjiva Keshava)
Levis Philip advisor. Thesis advisor