Date of Award

Spring 1-1-2016

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Computer Science

First Advisor

Xiao-Chuan Cai

Second Advisor

Jed Brown

Third Advisor

Elizabeth R. Jessup

Fourth Advisor

Vitaly Kheyfets

Fifth Advisor

Congming Li

Abstract

High-resolution simulation of fluid-structure interaction (FSI) problems on supercomputers has many applications including our targeting application in hemodynamics, but most existing methods and software do not scale to a large number of processor cores (more than O(104)) when the computational domain is complex and highly unstructured meshes have to be employed. In this thesis, we develop a highly scalable parallel nonlinear solver for fluid-structure interaction problems on unstructured meshes with complex geometry in 3D. The proposed approach consists of several ingredients: a monolithic coupling technique for the fluid and the solid equations, a fully implicit nonlinear solver, and a highly scalable multilevel overlapping Schwarz preconditioner based on isogeometric coarse spaces.

In the monolithic coupling approach, the incompressible Navier-Stokes equations for the fluid is formulated in an arbitrary Lagrangian-Euler framework in which an additional partial differential equation is used to describe the motion of the fluid domain. For the solid, a linear or nonlinear elasticity equation is described in the Lagrangian configuration, and the coupling conditions between the fluid and the solid are implicitly enforced on the wet interface. The fluid system is discretized by a finite element method together with SUPG stabilization, and the fluid domain motion and the solid equations are discretized by a finite element method. The semi-discretized system is further discretized in time by implicit schemes such as the backward Euler and the second-order backward difference formula. The coupled FSI system of equations is highly nonlinear and consists of hyperbolic, elliptic, parabolic components in a single system. We develop an inexact Newton-Krylov-Schwarz (NKS) method for solving the coupled FSI system of equations. The focus is on the scalability and the robustness of the algorithms with respect to different physics parameters on 3D unstructured meshes with complex geometry and on supercomputers with a large number of processor cores.

The parallel scalability of the NKS method depends almost completely on how the Jacobian system is solved and how a preconditioner is constructed. The main contribution of the thesis is a scalable preconditioner based on new non-standard coarse spaces. The new approach is developed based on the observation that the geometry of the fine mesh, including the fluid mesh and the solid mesh, is vital for the fast convergence of the linear solver. Therefore, a geometry preserving coarsening algorithm is introduced to generate isogeometric coarse meshes that share the same geometry as the fine mesh. The basic idea of the new coarsening algorithm is to retain all the vertices on the boundaries and the wet interface for preserving the geometry of the fine mesh and to coarsen the interior of the fine mesh as much as possible for saving the compute time on coarse levels. The quality of isogeometric coarse meshes is bad in the traditional sense but it is acceptable to be part of the multilevel Schwarz preconditioner. The overall algorithm equipped with isogeometric coarse spaces is scalable. Note that the overall solution accuracy is not affected by the quality of the coarse meshes, instead, it is determined completely by the fine mesh. We show numerically that the proposed algorithm and implementation are highly scalable in terms of the numbers of linear and nonlinear iterations and the total compute time on a supercomputer with more than 10,000 processor cores for several problems with hundreds of millions of unknowns.

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