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Please use this identifier to cite or link to this item: http://ntour.ntou.edu.tw:8080/ir/handle/987654321/52424

Title: 結合廣義有限差分法與GPU平行計算模擬水動力問題
Using Generalized Finite Difference Method and GPU Parallel Computation for Hydrodynamics Problems
Authors: Wang, Shih-Hua
王士華
Contributors: NTOU:Department of Harbor and River Engineering
國立臺灣海洋大學:河海工程學系
Keywords: 無網格法;平行計算;圖形處理器;廣義有限差分法;沖激問題;奈維爾-史托克斯方程式;淺水波方程式
meshless method;parallel computing;graphics processing unit(GPU);generalized finite difference method;sloshing problem;Navier-Stokes equations;shallow water equations
Date: 2017
Issue Date: 2019-11-18T08:03:57Z
Abstract: 近年來圖形處理器(graphics processing unit, GPU)的計算能力隨著半導體製程的進步持續上升,圖形處理器在電腦模擬數值計算的應用與研究日漸增加。為了測試圖形處理器平行計算的可能發展與應用潛力,本論文延續前人開發完成的無網格法數值模擬模式,針對三個不同特性之水動力問題,利用MATLAB平行計算工具箱(parallel computing toolbox)提供的圖形處理器平行計算功能,開發高效率之平行計算數值模擬模式,並且分別依據本論文三個圖形處理器平行化程式之平行計算效率進行評估,來了解圖形處理器平行計算對無網格法模式應用於分析水動力問題時之可行性與效率提升值。 本論文中三個研究主題之空間離散方法使用的是廣義有限差分法(generalized finite difference method),依據不同水動力問題之控制方程式再配合其他數值方法進行分析求解。廣義有限差分法具有無須建置網格也無須數值積分的特性,可大幅增加數值模式之計算效率並簡化計算模式。廣義有限差分法在計算域中劃定子區域,並計算出子區域中之不同權重係數,任意點位上之空間微分項都能利用鄰近點的物理量以線性權重累加來表示;除此之外,廣義有限差分法在進行控制方程式離散之後,所形成之線性代數系統是一個稀疏矩陣,因此在大尺度水動力學問題之分析時,對於圖形處理器有限的記憶體空間而言具有極大的優勢。因此,本論文所採用之空間離散方法為無網格法中之廣義有限差分法。 本論文的第一個研究主題為二維沖激問題(sloshing problem),其控制方程式為拉普拉斯方程式(Laplace equation),除了廣義有限差分法之外,本研究使用半拉格朗日法(semi-Lagrangian method)計算移動點位,再搭配顯式尤拉法(explicit Euler method)離散時間微分項,以形成一個簡單且有效率之無網格法模擬模式;本論文的第二個研究主題為方形穴室中之黏性不可壓縮流體流場,其控制方程式為奈維爾-史托克斯方程式(Navier-Stokes equations),本研究採用投影法(projection method)進行時間微分項之離散,以有效分離速度與壓力之計算。而本論文的第三個研究主題為淺水波方程式(shallow water equations),使用分離係數矩陣法(split-coefficient matrix method)將控制方程式轉換成具有波傳方向的形式,接著利用二階龍格-庫塔法(Runge-Kutta method)離散時間微分項。本論文針對以上三個不同問題之控制方程式,以廣義有限差分法配合各自不同之數值技巧,以準確地求解相對應之水動力學問題。 在圖形處理器平行計算方面,由於圖形處理器設計架構本身具有高度可平行化之特性,各領域已陸續開始使用圖形處理器來克服以往數值計算耗費時間過於冗長的問題,而本論文之研究目的就是使用圖形處理器平行計算,來改善數值模式於水動力學問題模擬之計算效率,使平行化之數值模式能更加快速計算出模擬結果。在本論文中,將上述三種不同之水動力問題,依據前人研究發展之無網格法電腦模擬模式,並配合圖形處理器進行平行化程式撰寫與測試。針對三個不同特性之水動力問題,每一個問題皆與前人研究結果進行驗證來確認模式的準確性,再研究使用不同總點數進行模擬時,其圖形處理器之平行計算效率;最後,針對不同水動力問題之平行計算效率進行評比與討論。
With the rapid development of computing power of graphics processing unit (GPU) in recent years due to semiconductor process and technology development, the applications of GPU in computer simulation and academic research has been gradually increased. In this thesis, we used the GPU parallel computing function in MATLAB parallel computing toolbox to develop numerical model and evaluate parallel efficiency of the GPU parallel computation in the applications of meshless method for hydrodynamics problems. In this thesis, we adopted the generalized finite difference method (GFDM) for spatial discretization in three numerical simulation models. In order to analyze governing equations in these three considered hydrodynamics problems, different numerical schemes are adopted to combine with the GFDM. The GFDM does not need to generate mesh and implement numerical integration, which can greatly increase the computational efficiency of numerical simulation and simplify the computational model. In addition, the GFDM uses the concept of star in the computational domain. When the weighting coefficients in the star can be obtained, the spatial derivative terms at any point can be expressed as the linear combinations of weighted functional values within the star. In comparing to other meshless methods, a sparse matrix can be formed in the GFDM rather than a full matrix. Therefore, the GFDM is very suitable for the parallel computation by GPU with limited memory space to analyze large-scale problems. Based on advantages of the GFDM and the GPU parallel computation, we adopted the GFDM for spatial discretization in this thesis to form efficient GPU parallel computation codes. The first hydrodynamics problem in this thesis is the sloshing problem, and its governing equation is Laplace equation. In addition to the GFDM, the semi-Lagrangian method is used to calculate the movement of computational nodes, and the explicit Euler method is adopted for temporal discretization. The second hydrodynamics problem of this study is the flow field of viscous incompressible fluid in a square cavity, and its governing equations is the Navier-Stokes equations. Additionally, the third problem is the shallow water equations. We adopted the split-coefficient matrix method to convert the governing equations to the characteristic form, and the second-order Runge-Kutta method is used for temporal discretization. Due to the GPU computing capacity rapidly increasing, the GPU design architecture itself can be highly parallelized and the users in various backgrounds start to use GPU to analyze problems, which cost too much labor and time in conventional numerical simulation. The purpose of this thesis is to use the parallel computation of GPU to improve the efficiency of numerical models for three hydrodynamic problems, so the parallelized numerical models can quickly calculate the simulation results. Since the accuracy and stability of the GFDM-based meshless numerical models have been verified in previous researches, the numerical models with GPU parallel computation are developed and tested in this thesis. For each of the three different hydrodynamics problems, the numerical solutions of each example are compared with the results in some previous study to verify the accuracy of the model, and GPU parallel computing efficiency is studied by using different numbers of total points. Finally, we evaluated and discussed the efficiency of GPU parallel computation for these three hydrodynamics problems.
URI: http://ethesys.lib.ntou.edu.tw/cgi-bin/gs32/gsweb.cgi?o=dstdcdr&s=G0010452006.id
http://ntour.ntou.edu.tw:8080/ir/handle/987654321/52424
Appears in Collections:[河海工程學系] 博碩士論文

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