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

On the development of a wavenumber error reducing finite element model for solving the incompressible Navier-Stokes equations in parallel on GPU

高仕超(Neo Shih-Chao Kao)

指導教授 ： 許文翰

國立臺灣大學/工學院/工程科學及海洋工程學研究所/博士(2015年)

https://doi.org/10.6342/NTU.2015.02783

摘要

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。

關鍵字

波數 ； 有限元素 ； 流線上風 ； CUDA ； 圖形處理器

並列摘要

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.

並列關鍵字

wavenumber ； finite element ； streamline upwind ； CUDA ； GPU

參考文獻

[81] C. C. Fang, Development of a finite element method for inviscid Euler equations, Ph.D thesis, Department of Naval Architecture and Ocean Engineering, Taipei, Taiwan, 2001.........173

[1] R. Aris, Vector, Tensor, and the basic equations of fluid mechanics, Prentice-Hall Cliffs, Nj, 1962..........165

[2] G. K. Batchelor, An introduction to fluid dynamics, Cambridge University Press, 1967..........165

[4] C. A. J. Fletcher, Computational Techniques for Fluid Dynamics Volume I, Springer-Verlag press, New York, 1990..........165

[5] R. H. Pletcher, J. C. Tannehill, D. Anderson, Computational fluid mechanics and Heat transfer, CRC press, 2012..........165