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完整後設資料紀錄
DC 欄位 | 值 | 語言 |
---|---|---|
dc.contributor.advisor | 許文翰 | |
dc.contributor.author | Neo Shih-Chao Kao | en |
dc.contributor.author | 高仕超 | zh_TW |
dc.date.accessioned | 2021-05-14T17:45:16Z | - |
dc.date.available | 2015-12-01 | |
dc.date.available | 2021-05-14T17:45:16Z | - |
dc.date.copyright | 2015-12-01 | |
dc.date.issued | 2015 | |
dc.date.submitted | 2015-10-30 | |
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dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/4686 | - |
dc.description.abstract | 本篇論文中,我們首先提出一能得到對流項的最小波數誤差,且能增進對流
項之穩定性之流線上風有限元素模型。為了驗證論文中所提出的有限元素模 型,本論文測試了許多具實解及典型的純量傳輸方程及高雷諾數及高瑞利數 的黏性不可壓縮流 Navier-Stokes 方程的測試問題。由結果可知,本文所提出 之模型,在所有的測試問題中均能有相當好的精確度及收斂斜率。 在使用迭代法求解經由有限元素方法離散三維空間架構下不可壓縮流 Navier-Stokes 方程後所得到的非對稱非正定矩陣方程時,為了避免 Lanczos 及選主元(pivoting)過程而導致求解發散或者求解不收斂問題,我們提出了避 免求解原始矩陣方程,改以求解經由在原始矩陣方程兩側乘上其轉置矩陣所 得到的正規矩陣方程的策略。正規矩陣方程為一對稱且正定的型式,因此可 利用具求解效率高及無條件收斂特性的共軛梯度(CG)迭代法來求解以得到無 條件穩定收斂的答案。本文亦提出了兩種基於多項式型式的預條件子來降低 因正規化過程中而大幅提高的條件數以加快收斂速度。由數值測試結果,顯 示本文所提出的求解策略效率比起傳統使用 BICGSTAB 及 GMRES 迭代法求 解原始矩陣方程還更來得穩定且更有效率。 為 了 加 速 計 算 成 本 繁 重 的 有 限 元 素 方 法 用 以 求 解 不 可 壓 縮 流 Navier-Stokes 方程,本文遂將所發展的有限元素計算程式執行在比起中央處 理器(CPU)有更高的浮點數運算效能及更大的記憶體帶寬的圖形處理器(GPU) 上。此外,本文提出了一些最佳化策略以最佳化其計算效能。對於測試的典 型拉穴流問題,本論文所提出的 CPU/GPU 異構平行計算方法的計算加速比及 效能比均具相當良好的結果。 最後,本論文將所發展的三維有限元素不可壓縮流體求解器用以求解九 十度彎管流及後向階梯流體問題,其計算結果與前人所模擬與實驗之結果均 相當的吻合。顯示本文所發展的基於 GPU 的有限元素流體求解器為一精準且 可信賴的計算工具。 | zh_TW |
dc.description.abstract | In this dissertation, a new streamline upwind finite element model which
accommodates a minimum wavenumber error for convection terms shown in the transport equation, is presented to enhance convective stability. The validity of the proposed finite element model is justified by solving several problems amenable to analytical and benchmark solutions at high Reynolds and Rayleigh numbers. The results with good accuracy and spatial rate of convergence are demonstrated for all the investigated problems. To avoid Lanczos or pivoting breakdown while solving the resulting large-scaled un-symmetric and indefinite matrix equations using the mixed finite element formulation, the matrix equations have been modified by pre-multiplying matrix with its transpose counterpart. The resulting normalized matrix system becomes symmetric and positive-definite. A computationally efficient conjugate gradient Krylov iterative solver can be therefore applied to get the unconditionally convergent solution. To improve the slow convergence behavior arising from the increased condition number, the two polynomial-based pre-conditioners are adopted. The numerical results show that the performance of the pre-conditioned conjugate-gradient solver for the normalized system is better than the two common used BICGSTAB and GMRES solvers for the original matrix equations. In order to accelerate the time-consuming finite element calculations for the incompressible Navier-Stokes equations, the developed finite element program has been implemented on a hybrid CPU/GPU platform endowed with its high floating-points arithmetic operation performance and large memory bandwidth compared to that implemented in CPU. Moreover, some optimization strategies are introduced in order to optimize the speedup performance. The resulting speedup and efficiency are good for the simulation of benchmark lid-driven cavity flow problem. Finally, the proposed GPU-based finite element fluid solver was used to investigate the three-dimensional 90 bend curved flow and three-dimensional backward-facing step flow problems. The results simulated from the proposed GPU-based finite element flow solver agree well with other numerical and experimental results. It shows that the proposed GPU-based finite element solver is accurate and reliable for use. | en |
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dc.description.tableofcontents | 1 Introduction 1
1.1 Finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Parallel computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Outlines of this study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Incompressible Navier-Stokes equations 9 2.1 Velocity-pressure variables formulation . . . . . . . . . . . . . . . . . . 9 2.2 Streamfunction-vorticity formulation . . . . . . . . . . . . . . . . . . . . 11 2.3 Velocity-vorticity formulation . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Solution algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5 Computation challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3 Finite element model for the transport equation 18 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Governing equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 Weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.4 Finite element model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.4.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.4.2 Streamline Upwind Petrov-Galerkin model . . . . . . . . . . . . 23 3.4.3 Modified wavenumber error optimizing model . . . . . . . . . . 24 3.5 Numerical studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.5.1 Verification study . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.5.2 Skew problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.5.3 Smith & Hutton problem . . . . . . . . . . . . . . . . . . . . . . 28 3.5.4 Rotation shaped cone problem . . . . . . . . . . . . . . . . . . . 28 3.5.5 Mixing of warm and cold fluids problem . . . . . . . . . . . . . 29 4 Finite element model for the incompressible Navier-Stokes equations 41 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 i4.3 Incompressibility constraint condition . . . . . . . . . . . . . . . . . . . 44 4.4 Interpolation function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.5 Finite element formulation . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.6 Verification study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.6.1 Steady-state analytical verification problem . . . . . . . . . . . . 49 4.6.2 Transient analytical verification problem . . . . . . . . . . . . . 49 4.7 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.7.1 Lid-driven cavity problem . . . . . . . . . . . . . . . . . . . . . 50 4.7.2 Backward facing step flow problem . . . . . . . . . . . . . . . . 50 4.7.3 Natural convection problem . . . . . . . . . . . . . . . . . . . . 51 5 Iterative solver for three-dimensional Navier-Stokes finite element equations 73 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.2 Interpolation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.3 The finite element formulation . . . . . . . . . . . . . . . . . . . . . . . 75 5.4 Newton linearization of the nonlinear convection term . . . . . . . . . . . 76 5.5 The normalization procedure . . . . . . . . . . . . . . . . . . . . . . . . 78 5.6 The element-by-element technique . . . . . . . . . . . . . . . . . . . . . 79 5.7 Implementation of Dirichlet boundary condition on elementary matrix . . 81 5.8 Implementation of polynomial-based pre-conditioner . . . . . . . . . . . 84 5.9 Iterative matrix solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.9.1 The pre-conditioned CG solver . . . . . . . . . . . . . . . . . . . 88 5.9.2 The pre-conditioned BICGSTAB solver . . . . . . . . . . . . . . 90 5.9.3 The pre-conditioned GMRES solver . . . . . . . . . . . . . . . . 92 5.10 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.10.1 Verification study . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.10.2 Lid-driven cavity flow in a cube . . . . . . . . . . . . . . . . . . 95 5.11 Conclusion remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6 Parallel computing on GPU 109 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.2 GPU architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.2.1 Memory hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.3 CUDA programming model . . . . . . . . . . . . . . . . . . . . . . . . 115 6.4 CUDA Fortran programming . . . . . . . . . . . . . . . . . . . . . . . . 116 6.5 Implementation of iterative solvers on GPU . . . . . . . . . . . . . . . . 119 6.5.1 Inner product operation on GPU . . . . . . . . . . . . . . . . . . 120 6.5.2 Matrix-vector product operation on GPU . . . . . . . . . . . . . 121 6.6 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.6.1 Global memory coalescing . . . . . . . . . . . . . . . . . . . . . 123 ii6.6.2 Shared memory . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.7 Description of GPU code . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.8 Numerical study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.8.1 2D verification study . . . . . . . . . . . . . . . . . . . . . . . . 126 6.8.2 3D verification study . . . . . . . . . . . . . . . . . . . . . . . . 126 6.8.3 Lid-driven cavity flow problem . . . . . . . . . . . . . . . . . . . 127 6.9 Performance analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7 Applications 146 7.1 Three-dimensional 90 bend curved duct flow problem . . . . . . . . . . . 146 7.1.1 Problem description and validation . . . . . . . . . . . . . . . . 147 7.1.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7.2 Three-dimensional backward facing step flow problem . . . . . . . . . . 149 7.2.1 Problem description and numerical results . . . . . . . . . . . . . 150 8 Concluding Remarks 163 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .165 | |
dc.language.iso | en | |
dc.title | 發展一波數誤差最佳化有限元素 GPU 平行計算模型
以求解不可壓縮 Navier-Stokes 方程式 | zh_TW |
dc.title | On the development of a wavenumber error reducing
finite element model for solving the incompressible Navier-Stokes equations in parallel on GPU | en |
dc.type | Thesis | |
dc.date.schoolyear | 104-1 | |
dc.description.degree | 博士 | |
dc.contributor.oralexamcommittee | 陳宜良,林文偉,王偉仲,李汶樺,黃楓南 | |
dc.subject.keyword | 波數,有限元素,流線上風,CUDA,圖形處理器, | zh_TW |
dc.subject.keyword | wavenumber,finite element,streamline upwind,CUDA,GPU, | en |
dc.relation.page | 178 | |
dc.rights.note | 同意授權(全球公開) | |
dc.date.accepted | 2015-10-30 | |
dc.contributor.author-college | 工學院 | zh_TW |
dc.contributor.author-dept | 工程科學及海洋工程學研究所 | zh_TW |
顯示於系所單位: | 工程科學及海洋工程學系 |
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