A scalable and adaptive discretization for free surface incompressible flow simulation using overlapping Cartesian grids [electronic resource]

This dissertation presents a novel method for discretizing the incompressible Navier-Stokes equations. Within numerical simulation, the underlying spatial discretization plays a crucial role in defining the properties of a numerical method. In order to resolve solution features on multiple length scales, adaptive spatial discretizations using unstructured and block structured approaches have been developed. However, while many of these methods have resolved the mathematical issues, scaling these methods to large problems which require the use of large distributed memory systems has been problematic due to issues of domain decomposition and remeshing. Block structured approaches such as Adaptive Mesh Refinement (AMR) and Chimera grid methods have been particularly successful since they allow for accurate fluid-structure interaction and spatial adaptivity. However, their scalability has been limited when applied to more complicated problems due the large number of grids required in AMR methods and the complicated linear systems arising in Chimera methods. This dissertation introduces an overlapping grid method that uses moving and overlapping Cartesian grids with independently specified cell sizes to address adaptivity. Unlike AMR approaches, by allowing the Cartesian patches to be arbitrarily rotated and translated, far fewer patches are necessary to rasterize solid boundaries and flow features. This both improves cache coherency and greatly simplifies domain decomposition. Advection is handled with first and second order accurate semi-Lagrangian schemes in order to alleviate any time step restriction associated with small grid cell sizes. The primary focus of this dissertation, however, is the Poisson discretization used in order to solve the elliptic equation for the Navier-Stokes pressure or that resulting from the temporal discretization of parabolic terms such as viscosity. A novel discretization is introduced where the coupling terms are defined implicitly through the use of a Voronoi diagram computed for a subset of the Cartesian grid cell centers. The resulting Poisson discretization is second order accurate and requires the solution of a symmetric positive definite linear system. Each aspect of the approach is demonstrated independently on test problems in order to show efficacy and convergence before finally addressing a number of common test cases for incompressible flow with stationary and moving solid bodies. The method is then extended to handle two-way coupled fluid structure interaction using a monolithic immersed boundary approach. In order to represent free surfaces the particle level set method is adapted to overlapping grids. The method includes a new treatment for particles near grid boundaries with disparate cell sizes, and strategies to deal with issues of locality in the implementation of the level set and fast marching algorithms resulting from the different bandwidths of valid values near the interface on different grids. The resulting method is capable of simulating problems with solution features on disparate scales while efficiently exploiting distributed computational resources.

Advisor: Fedkiw Ronald P.
Henshaw William
Levis Philip advisor. Thesis advisor