| Summary: | 碩士 === 國立成功大學 === 機械工程學系 === 104 === The application of Computational Fluid Dynamics (CFD) is an important part of any engineering analysis, with the Finite Volume Method (FVM) a key tool in modern CFD. Modern FVM simulations can be categorized into explicit simulations – where fluxes for the next time step are computed based on older values – or implicit simulations, where the new state and new fluxes across cell interfaces are computed simultaneously. This paper focuses on the application of the direction decoupled Quiet Direct Simulation (QDS) approach to implicit calculation, allowing the use of larger CFL numbers without resulting in instabilities. The governing equations are non-linear – conventional attempts at implementing an implicit approach norming involve the linearization of the governing equations. The QDS approach is fundamentally non-linear, hence this research centers on the solution of the non-linear equations. Hence, the Newton Raphson approach is employed to solve the non-linear equations for both the Equilibrium Flux Method (EFM) and the Quiet Direct Simulation (QDS). The Newton Raphson approach requires the inverse of the Jacobian of the residual functions. The Jacobian matrix is computed using a simple Finite Difference approach, while the Bi-Conjugate Stabilised (BiCGStab) method is employed for finding the Jacobian inverse. The Jacobian matrix has a considerable amount of non-zero elements, which not only influences computation time but also wastes a large amount of memory. To overcome this obstacle we employ the Compressed Sparse Matrix (CSR) storage technique.
In order to validate the implicit implementation of the QDS solver, we discuss the one-dimensional shock tube problem, a two-dimensional four contact interaction, the two dimensional four shock interaction and two dimensional blast wave problem, in addition to a hypersonic flow over a forward facing step. After comparison of the results against the EFM and existing results, we describe the parallel performance of the OpenMP implementation. The overall conclusions for the study can by summarized by (1) The allowable CFL number is greater for the implicit implementation than its equivalent explicit implementation – however, using a larger CFL number demonstrates the lower accuracy due to the use of a first order time discretization, (2) The speedup obtained using OpenMP with 16 cores is approximately 13.6x that of a single core, and demonstrating a high degree of parallel efficiency, (3) the application of CSR is critical to the efficient application and parallelization of the implicit solution.
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