- 學位論文
拉格朗日-尤拉混合粒子法以解不可壓縮黏性流之改進
Improvement on the Mixed Lagrangian-Eulerian Particle Method for Solving Incompressible Viscous Flow Problem
摘要
移動粒子壓力網格法（MPPM）架構下所發展的拉格朗日-尤拉粒子法（MLEP）中，粒子-網格資料交換步驟在此研究中以所發展的最佳化形參數之徑向基函數（RBF）計算以改善精度。在本方法中，最佳形參數是以嚴格方式自欲插值函數之區域分佈得出，並代入多項式化的多元二次曲面（multiquadric）標量徑向基函数。而在將速度從網格轉移至粒子時，則使用無散度向量徑向基函数以維持質量守恆。為對應對流項中的非線性，本方法中使用SIMPLE方式的壓力-速度結合，以便能在計算中使用更大的時間步。壓力方程計算方式則使用RBF在有限差分式格式（RBF-FD）的方式離散。當中的RBF-FD最佳形參數則以能消除前置離散誤差為目的的方式計算得出，由此則可在不需要增加計算點下達到高階準確。在測試問題的計算結果顯示，當使用了用本方法推算出的最佳化形參數后，RBF及RBF-FD方式的準確度及收斂速度都有明顯的提高。一些二維流體基準問題在使用本MLEP方法以更大的庫朗數及相對粗糙的網格求解時，所得到的結果仍舊與參考文獻中的結果接近。
並列摘要
In this study, the mixed Lagrangian-Eulerian particle (MLEP) method has been improved on top of the previous moving particle with pressure mesh (MPPM) framework by focusing on the development of radial basis function (RBF) with optimum shape parameter to improve the accuracy of the data transfer between particle and grid. The optimum shape parameter of the polynomilized multiquadric scalar RBF is calculate rigorously by considering the local distribution of the function to be interpolated. To preserve the continuity constraint, the vector RBF function with divergence-free property is applied when transferring information from grid to particle. The SIMPLE type pressure-velocity coupling is implemented to take the non-linearity of the convection term into account, and allows the use of larger time step size. The solution of pressure equation is improved by using the RBF-FD, the finite-differencing type scheme derived from the polynomialized RBF. The shape parameter of the RBF-FD is calculated in such a way that the leading error term of the discretized equation is eliminated, and thus achieving higher-order accuracy. From the results of test cases, it is found that by applying the calculated shape parameter to the RBF and RBF-FD schemes, the solution accuracy and rate of convergence can be greatly improved. Several benchmark two-dimensional problems are solved by using present MLEP scheme, and the results are comparable with the one in reference solution despite the larger Courant number and relatively coarse grid used in present scheme.