發展與驗證虛擬網格切割模組於平行化直接模擬蒙地卡羅法程式

Abstract

本研究主要目的是發展與驗證虛擬網格切割模組(VMR)於平行化直接模擬摩地卡羅法程式(PDSC)。在DSMC模擬中，網格被使用於分子的碰撞與巨觀性質的採樣，而網格的尺寸必須小於局部平均自由路徑。然而，在模擬之前，並無法知道局部平均自由路徑的分佈。在以前，我們研究室也曾經發展過數個網格切割法，使用h-refinement的方法於非結構性網格中。然而，在被精細化的非結構性網格中，分子的追蹤是很困難的，而且網格的品質通常很難保持。在本文中，我們將利用transient adaptive sub-cells (TAS)的想法，發展出一個新的非結構性網格切割法於PDSC中。這是一個二階虛擬網格切割法(two-level Virtual Mesh Refinement)。網格將在第一次DSMC模擬中被切割。被切割的網格稱做虛擬網格(virtual refined cells)，它們類似結構性網格。因此，分子的追蹤將變的有效率。這些虛擬網格被使用於分子的碰撞與採樣。另外，我們使用蒙地卡羅積分法去計算每一個虛擬網格的面積。5,000*Nvc的particles數，對於每一個虛擬網格面積的計算可達到0.1%的誤差，並且使用12個CPU計算300,000個虛擬網格面積所花的計算時間為12.5分鐘。僅有一個包含初始網格質心的虛擬網格將被輸出作為初始網格的結果。這樣的一個方式將保持著記憶體使用上的精簡，並且加上使用動態網格分解 (dynamic domain decomposition)去減少模擬所需要的龐大時間。 最後，模擬兩個二維流場題目，包括極超音速流(M-12)流過一矩形物(argon gas, velocity=1413 m/s, temperature=40 K and Kn=0.05, number density=1.29E21 m-3)，以及極超音速流(M-10)流過一圓柱物(cylinder)(D=0.3048 m, argon gas, velocity=2634.1 m/s, temperature=200 K and Kn=0.0091, number density=4.274E20 m-3)，並且單獨使用四邊形、三角形，以及四邊形與三角行的混合網格，進而去驗證程式的正確性。從這些模擬的結果顯示，使用VMR不僅可以獲得benchmark的結果，而且也減少了三到四倍的計算時間。

The objective of this thesis is to develop and verify a virtual mesh refinement module (VMR), based on a new concept, in a parallelized direct simulation Monte Carlo code (PDSC). Cells are used for particle collisions and sampling of macroscopic properties in a DSMC simulation, in which the sizes have to be much smaller than the local mean free path. Unfortunately, it is generally impossible to know the distribution of local mean free path before the simulation. Previously, in our group we have developed several mesh refinement techniques in DSMC, which were based on the concept of h-refinement to unstructured grids. However, particle tracing on the refined unstructured mesh becomes inefficient and mesh quality is generally difficult to maintain. In this thesis, we will utilize the concept of transient adaptive sub-cells (TAS) proposed by Tseng et al. and propose a new type of mesh refinement on unstructured grids for DSMC simulation. This method is a two-level virtual mesh refinement, in which the background mesh is refined based on an initial DSMC simulation. The virtual refined cells are arranged in a way similar to the structured grid, which makes the particle tracing on them very efficient, unlike on unstructured grids. These virtual cells are used for particle collision and sampling. In addition, area of each virtual refined cell is calculated using the Monte Carlo integration method. Approximately 5,000*Nvc particles are required to reach 0.1% error for area calculations of all the virtual refined cells, which takes about 12.5 minutes of computational time for ~300,000 virtual refined cells using 12 processors. Only a virtual refined cell, which includes centroid of the background cell, we output only this data in each background cell. In this way, the original grid data structure is retained and memory cost is comparably low and using dynamic domain decomposition (DDD) to reduce computational time. Finally, two two-dimensional test cases, which are Mach-12 hypersonic flow past a block (argon gas, velocity=1413 m/s, temperature=40 K and Kn=0.05, number density=1.29E21 m-3) and Mach-10 hypersonic flow past a circular cylinder (D=0.3048 m, argon gas, velocity=2634.1 m/s, temperature=200 K and Kn=0.0091, number density=4.274E20 m-3), including quadrilateral, triangular and mixed triangular-quadrilateral mesh have demonstrated in the thesis to show the robustness of this new mesh-refining algorithm. Results of cylinder simulation show that the case using VMR not only can faithfully reproduce the benchmark case, but also can reduce computational time from 15 hours (benchmark) to 3.5 hours (quadrilateral mesh), 4.5 hours (triangular mesh) and 5 hours (mixed quadrilateral-triangular mesh).