發(fā)布時(shí)間：2018-02-24 20:18

  本文關(guān)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(jìn)收斂因子 波形松弛　出處：《昆明理工大學(xué)》2016年博士論文　論文類(lèi)型：學(xué)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(duì)離散微分方程后所得的代數(shù)方程組進(jìn)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(wú)關(guān),且計(jì)算成本與問(wèn)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(jì)算流體力學(xué)中一種高效的數(shù)值方法而受到廣泛關(guān)注和研究。本文依托高等學(xué)校博士學(xué)科點(diǎn)專(zhuān)項(xiàng)科研基金(優(yōu)先發(fā)展領(lǐng)域)(20135314130002)項(xiàng)目、國(guó)家自然科學(xué)基金面上項(xiàng)目(51279071),研究多重網(wǎng)格法在水力機(jī)械內(nèi)部流數(shù)值模擬方面的理論和應(yīng)用,重點(diǎn)是多重網(wǎng)格光滑理論中的局部Fourier分析方法,對(duì)數(shù)值求解不可壓縮流體控制方程的多重網(wǎng)格方法進(jìn)行收斂性分析。主要研究?jī)?nèi)容和創(chuàng)新如下：(1)結(jié)合水力機(jī)械流道湍流的流動(dòng)特點(diǎn),提出了多重網(wǎng)格算法及其誤差迭代的格式。基于局部Fourier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(xì)網(wǎng)格層Fourier組分之間的關(guān)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(jìn)收斂因子的理論計(jì)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(yīng)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(guò)程具有一般的迭代格式,所得結(jié)果具有代表性和應(yīng)用前景。(2)基于交錯(cuò)網(wǎng)格和非交錯(cuò)網(wǎng)格提出了求解Stokes流的離散格式,并對(duì)該離散系統(tǒng)實(shí)施兩種不同多重網(wǎng)格的松弛算法進(jìn)行了光滑分析：即聚松弛和分布松弛光滑分析。在交錯(cuò)網(wǎng)格的離散系統(tǒng)中實(shí)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(yīng)的光滑因子。其次,在非交錯(cuò)網(wǎng)格離散系統(tǒng)中,分別實(shí)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(guān)于附加人工壓力項(xiàng)參數(shù)的表達(dá)式。結(jié)果表明：松弛方法的收斂性與網(wǎng)格尺度無(wú)關(guān),而依賴(lài)于附加人工壓力項(xiàng)參數(shù)。(3)基于最優(yōu)紅黑Jacobi逐點(diǎn)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(duì)流擴(kuò)散方程的一階上迎風(fēng)離散格式,分析了對(duì)流占優(yōu)參數(shù)和擴(kuò)散參數(shù)對(duì)該離散格式的橢圓率影響,探索了對(duì)流擴(kuò)散方程各參數(shù)對(duì)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(chǔ)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(fēng)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(guò)局部Fourier分析方法,對(duì)獲得的離散方程的聚對(duì)稱(chēng)交替線(xiàn)Gauss-Seidel松弛的光滑性質(zhì)進(jìn)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(duì)稱(chēng)交替線(xiàn)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(duì)基于非定常不可壓縮流體的NS方程進(jìn)行基于交錯(cuò)網(wǎng)格離散系統(tǒng)實(shí)施多重網(wǎng)格分布松弛。通過(guò)局部Fourier分析,發(fā)現(xiàn)該離散系統(tǒng)的光滑性質(zhì)由時(shí)間依賴(lài)的對(duì)流擴(kuò)散算子決定,并對(duì)兩種處理時(shí)間依賴(lài)問(wèn)題的多重網(wǎng)格松弛,時(shí)空松弛和波形松弛進(jìn)行了系統(tǒng)研究。在交錯(cuò)網(wǎng)格上,提出了非定常不可壓縮流體NS方程僅對(duì)空間變量進(jìn)行離散的半離散格式,并對(duì)該離散系統(tǒng)實(shí)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(shí)間依賴(lài)的對(duì)流擴(kuò)散算子。通過(guò)局部Fourier分析,對(duì)時(shí)間依賴(lài)的對(duì)流擴(kuò)散問(wèn)題所使用的時(shí)空多重網(wǎng)格方法和波形多重網(wǎng)格方法進(jìn)行了光滑性分析。另一方面,在時(shí)空多重網(wǎng)格方法的光滑分析中,采用了時(shí)空離散格式,其中時(shí)間離散采用一階Euler向后格式,而空間離散采用一階上迎風(fēng)格式。提出了多重網(wǎng)格的粗化僅對(duì)空間粗化的半粗化方法以及與時(shí)空多重網(wǎng)格對(duì)應(yīng)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(shí)間依賴(lài)問(wèn)題轉(zhuǎn)化為帶有復(fù)參數(shù)的定常問(wèn)題,然后對(duì)應(yīng)用于波形多重網(wǎng)格方法的各種松弛進(jìn)行局部Fourier光滑分析。通過(guò)提出的兩種多重網(wǎng)格方法的光滑分析,研究了對(duì)流占優(yōu)參數(shù)和雷諾數(shù)對(duì)各種松弛算子光滑性的影響,給出了相應(yīng)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(dǎo)師負(fù)責(zé)的國(guó)家基金面上項(xiàng)目“水輪機(jī)旋轉(zhuǎn)湍流全歐拉并行多層網(wǎng)格模擬研究”等項(xiàng)目的算法設(shè)計(jì)和代碼開(kāi)發(fā)應(yīng)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(xué)位授予單位】：昆明理工大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類(lèi)號(hào)】：O241.82
，

本文編號(hào)：1531573