本文關鍵詞： 多重網(wǎng)格方法 局部Fourier分析 光滑性質(zhì) 聚松弛 分布松弛 漸進收斂因子 波形松弛　出處：《昆明理工大學》2016年博士論文　論文類型：學位論文

【摘要】：多重網(wǎng)格算法是偏微分方程數(shù)值求解的一種快速算法。主要針對離散微分方程后所得的代數(shù)方程組進行數(shù)值求解,在橢圓型偏微分方程的數(shù)值解中已被證明是最優(yōu)的數(shù)值算法,其收斂性與網(wǎng)格尺度的大小無關,且計算成本與問題的規(guī)模成正比。由于多重網(wǎng)格算法的優(yōu)越性,使得它成為計算流體力學中一種高效的數(shù)值方法而受到廣泛關注和研究。本文依托高等學校博士學科點專項科研基金(優(yōu)先發(fā)展領域)(20135314130002)項目、國家自然科學基金面上項目(51279071),研究多重網(wǎng)格法在水力機械內(nèi)部流數(shù)值模擬方面的理論和應用,重點是多重網(wǎng)格光滑理論中的局部Fourier分析方法,對數(shù)值求解不可壓縮流體控制方程的多重網(wǎng)格方法進行收斂性分析。主要研究內(nèi)容和創(chuàng)新如下：(1)結(jié)合水力機械流道湍流的流動特點,提出了多重網(wǎng)格算法及其誤差迭代的格式�；诰植縁ourier分析理論,分別定義了離散算子和松弛迭代算子的橢圓率和光滑因子,并利用不同粗、細網(wǎng)格層Fourier組分之間的關系,定義了新的不變子空間,分析了不同粗化方式下網(wǎng)格轉(zhuǎn)化算子的Fourier表述方式,研究了多重網(wǎng)格算法漸進收斂因子的理論計算方法,創(chuàng)新了兩色松弛在兩種不同的Fourier模態(tài)函數(shù)不變子空間中的光滑分析方法,得到了基于多色松弛矩陣的Fourier分析的理論表示,并以泊松方程為例給出了相應的分析結(jié)果。研究表明,基于多色松弛的多重網(wǎng)格光滑分析過程具有一般的迭代格式,所得結(jié)果具有代表性和應用前景。(2)基于交錯網(wǎng)格和非交錯網(wǎng)格提出了求解Stokes流的離散格式,并對該離散系統(tǒng)實施兩種不同多重網(wǎng)格的松弛算法進行了光滑分析：即聚松弛和分布松弛光滑分析。在交錯網(wǎng)格的離散系統(tǒng)中實現(xiàn)了多重網(wǎng)格分布松弛,發(fā)現(xiàn)該離散系統(tǒng)的光滑性取決于Laplace算子,并得到了相應的光滑因子。其次,在非交錯網(wǎng)格離散系統(tǒng)中,分別實施了多重網(wǎng)格分布松弛和聚松弛,在兩色松弛的Fourier諧波空間中,討論了這兩種松弛的光滑性質(zhì),得出光滑因子關于附加人工壓力項參數(shù)的表達式。結(jié)果表明：松弛方法的收斂性與網(wǎng)格尺度無關,而依賴于附加人工壓力項參數(shù)。(3)基于最優(yōu)紅黑Jacobi逐點松弛方法,從理論上分析了Possion方程兩層網(wǎng)格算法的收斂性。給出了對流擴散方程的一階上迎風離散格式,分析了對流占優(yōu)參數(shù)和擴散參數(shù)對該離散格式的橢圓率影響,探索了對流擴散方程各參數(shù)對多重網(wǎng)格光滑性和兩層網(wǎng)格收斂性的影響。在提出的理論方法基礎上,利用Riemann解的通量差分分裂法-Godunov方法處理Oseen流控制方程的離散,得到了基于一階上迎風格式的離散方程,并分析了使用多重網(wǎng)格方法求解該離散方程的V-循環(huán)算法和W-循環(huán)算法的收斂性,并通過局部Fourier分析方法,對獲得的離散方程的聚對稱交替線Gauss-Seidel松弛的光滑性質(zhì)進行了系統(tǒng)研究。結(jié)果表明：使用多重網(wǎng)格的兩層網(wǎng)格及三層網(wǎng)格算法求解具有不同Reynolds數(shù)的Oseen流,即便是在較高Reynolds數(shù)情況下,聚對稱交替線Gauss-Seidel松弛仍然具有很好的光滑性質(zhì),且W-循環(huán)算法收斂性比V-循環(huán)算法好。(4)首次對基于非定常不可壓縮流體的NS方程進行基于交錯網(wǎng)格離散系統(tǒng)實施多重網(wǎng)格分布松弛。通過局部Fourier分析,發(fā)現(xiàn)該離散系統(tǒng)的光滑性質(zhì)由時間依賴的對流擴散算子決定,并對兩種處理時間依賴問題的多重網(wǎng)格松弛,時空松弛和波形松弛進行了系統(tǒng)研究。在交錯網(wǎng)格上,提出了非定常不可壓縮流體NS方程僅對空間變量進行離散的半離散格式,并對該離散系統(tǒng)實施分布松弛,使得離散系統(tǒng)多重網(wǎng)格松弛的光滑性質(zhì)僅取決于時間依賴的對流擴散算子。通過局部Fourier分析,對時間依賴的對流擴散問題所使用的時空多重網(wǎng)格方法和波形多重網(wǎng)格方法進行了光滑性分析。另一方面,在時空多重網(wǎng)格方法的光滑分析中,采用了時空離散格式,其中時間離散采用一階Euler向后格式,而空間離散采用一階上迎風格式。提出了多重網(wǎng)格的粗化僅對空間粗化的半粗化方法以及與時空多重網(wǎng)格對應的各種松弛的局部Fourier分析方法。而在波形多重網(wǎng)格方法中,首先利用Laplace變換將時間依賴問題轉(zhuǎn)化為帶有復參數(shù)的定常問題,然后對應用于波形多重網(wǎng)格方法的各種松弛進行局部Fourier光滑分析。通過提出的兩種多重網(wǎng)格方法的光滑分析,研究了對流占優(yōu)參數(shù)和雷諾數(shù)對各種松弛算子光滑性的影響,給出了相應的最優(yōu)光滑因子和最佳松弛參數(shù)的選取方法。提出的理論和方法部分用于了由導師負責的國家基金面上項目“水輪機旋轉(zhuǎn)湍流全歐拉并行多層網(wǎng)格模擬研究”等項目的算法設計和代碼開發(fā)應用中,并獲得成功。
[Abstract]:The multigrid algorithm is a fast algorithm for solving the partial differential equations for discrete differential equation. The algebraic equations are solved numerically, in the numerical solution of elliptic partial differential equations has been proved to be the optimal numerical algorithm, its convergence and grid scale size, and the computing scale is proportional to the the cost and problems. Due to the superiority of the multigrid method, making it an efficient numerical method in computational fluid dynamics has received widespread attention and research. On the basis of Higher Education Research Fund for the doctoral program (priority areas) (20135314130002) project, the National Natural Science Foundation of China (51279071). Study on the theory and Application of multigrid method in numerical simulation of the internal flow in hydraulic machinery, especially the local Fourier multigrid smooth theory analysis method in the numerical Solution of multigrid method for incompressible fluid control equations of convergence analysis. The main research content and innovation are as follows: (1) according to the flow characteristics of turbulent flow in hydraulic machinery, proposed an iterative multigrid algorithm and its error format. Local Fourier analysis based on the theory of discrete elliptic operator and operator relaxation rate and smooth factor the definition of the use of different coarse and fine mesh, the relationship between Fourier components, the new definition of invariant subspace, analyzed the expression of Fourier grid transformation operator different coarsening method, calculation method of multigrid algorithm convergence factor theory, innovation and relaxation in two different the Fourier mode function invariant subspace smooth analysis method, obtained the relaxation matrix of Fourier color analysis based on the theory of representation, and with the Poisson equation is given. The corresponding analysis results. The research results show that the iterative multigrid relaxation process based on polychromatic smooth analysis with general, the results are representative and applications. (2) staggered and non staggered grid is proposed for discrete format based on Stokes stream, and the implementation of two different multigrid relaxation algorithm for the discrete the system of smooth Analysis: Poly relaxation and smooth distribution of relaxation analysis. In the discrete staggered grid system is implemented in the distributed relaxation multigrid, the discrete system depends on the smoothness of the Laplace operator, and the smooth factor accordingly. Secondly, on a non staggered grid discrete system, multi grid distribution respectively. Relaxation and relaxation in Fourier poly implementation, and relaxation in harmonic space, discusses the two kinds of relaxation of the smooth nature, the smooth factor on additional artificial pressure parameters The expression. The results show that the convergence and grid scale relaxation method to rely on additional artificial pressure parameters. (3) the optimal Jacobi point relaxation method based on theoretical analysis of the convergence of the two grid algorithm Possion equation. First order upwind discretization scheme is presented for convection diffusion equation and analyzed the influence of convection and diffusion parameters of the elliptic discrete format rate, exploring the various parameters of the convection diffusion equation of multi grid smoothness and two grid convergence effect. Based on the proposed method, the use of Riemann solution of the discrete flux difference splitting method -Godunov method Oseen flow control the obtained equation, discrete equations of first order upwind scheme based on the analysis, and the convergence of the use of multigrid method for solving the discrete equations of the V- cycle and W- cycle algorithm algorithm, and through the Bureau Fourier analysis method, symmetric alternating poly line Gauss-Seidel relaxation of smooth properties of discrete equations obtained were studied. The results show that using multi grid two grid and the three grid algorithm with different Reynolds number Oseen flow, even at high Reynolds number, poly symmetric alternating line Gauss-Seidel relaxation smooth still has good properties, and W- cycle convergence than V- cycle algorithm. (4) for the first time on the NS equation based on the unsteady incompressible fluid of staggered grid discrete system using multiple grid distribution. Based on relaxation through local Fourier analysis, found that the smooth nature of the convection of discrete systems by time the dependence of the diffusion operator, multigrid relaxation and dependence on two kinds of treatment time, temporal relaxation and waveform relaxation were studied. On staggered grid is proposed. The unsteady incompressible NS equations only the spatial variables for semi discrete scheme, implementation and distribution of relaxation of the discrete system, the smooth nature of the discrete system multigrid relaxation only depends on the time dependent convection diffusion operator. Through the local Fourier analysis, space-time multigrid method and multigrid method of diffusion wave the problem of time-dependent convection using the smoothness analysis. On the other hand, in the smooth temporal and spatial analysis of multigrid method, the temporal discrete format, in which time the first order discrete backward Euler format, and the space is discretized using first order upwind scheme is proposed. The semi coarsening method of multiple coarsening only the space grid coarsening and local Fourier relaxation and corresponding space-time multigrid method. The waveform in the multigrid method, we use Laplace Transform time dependent problem into a constant problem with complex parameters, and then the corresponding waveform relaxation for a variety of multigrid methods for local Fourier smooth analysis. Through the analysis of two kinds of smooth multigrid method proposed, studied the convection parameters and Reynolds number on various relaxation effects of smoothness operator selection method is given, the corresponding optimal smoothing factor and the optimum relaxation parameter. Some theories and methods for state funds by the tutor on the project of "rotating turbulent turbine Euler parallel multi grid simulation research" project, algorithm design and code design application, and achieved success.

【學位授予單位】：昆明理工大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O241.82
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本文編號：1531573