| Summary: | 碩士 === 國立中央大學 === 數學研究所 === 97 === Various numerical simulations of physical phenomena in some engineering applications often require fast, reliable, accurate numerical solutions of unsteady 3D incompressible Navier-Stokes equations defined on a complex geometry. To resolve the details of the solution in the boundary layer region, high resolution meshes are often required, which implies the need for large-scale parallel computing. Even though years of research have been spent on finding such a suitable method for solving Navier-Stokes equations on very fine meshes for a wide range of Reynolds number, it remains a difficult computing task. The goal of this thesis is to study some parallel scalable algorithms for solving large sparse nonlinear systems of equations arising from the discretization of unsteady incompressible Navier-Stokes equations, where a stabilized finite element method and a family of implicit ODE integrators are employed for the spatial and temporal discretizations, respectively. Our parallel algorithm is based on a Newton-Krylov-Schwarz algorithm, which consists of three key components: an inexact Newton method with backtracking as the nonlinear linear solver, a Krylov subspace method as the linear solver for the Jacobian systems, and a parallel overlapping Schwarz domain decomposition method as a preconditioner to accelerate the convergence rate of the linear solver. In addition, our parallel flow solver implemented by PETSc is integrated with other pre-processing and post-processing software packages. These packages include (1) A Cubit and C language based 3D unstructured finite element mesh generator; (2) a mesh partitioner, ParMETIS for the purpose of parallel processing; (3) A ParaView based scientific visualization for displaying numerical results and conducting data analysis. We report the parallel performance of our algorithms for solving three-dimensional start-up lid-driven rectangular cavity flows, which are tested on some parallel machines in Taiwan. We also present an application of the parallel flow solver to simulate numerically micromixing in a microfluidic system.
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