本文選題：Richards方程　切入點(diǎn)：半解析解　出處：《蘭州大學(xué)》2016年博士論文　論文類型：學(xué)位論文

【摘要】：Richards方程是描述土壤滲流過程的基本控制方程.土壤介質(zhì)的非均質(zhì)性、土壤導(dǎo)水率的高度非線性性以及區(qū)域和初、邊值條件的復(fù)雜性等諸多因素,為Richards方程數(shù)值求解帶來很多困難.傳統(tǒng)的有限差分方法或者有限元方法會導(dǎo)致計(jì)算結(jié)果出現(xiàn)非物理的數(shù)值震蕩或者彌散.尋求有效的Richards方程數(shù)值解是土壤水動力學(xué)研究的重點(diǎn)內(nèi)容.本文主要在以下幾個方面進(jìn)行了相關(guān)工作:在分層土壤滲流問題中,分層界面處土壤本構(gòu)關(guān)系發(fā)生變化,導(dǎo)水率及含水量出現(xiàn)跳躍,對Richards方程數(shù)值計(jì)算的穩(wěn)定性帶來挑戰(zhàn).我們針對分層土壤下一類Gardener-Basha型Richards方程,分析分層界面處本構(gòu)關(guān)系,得到形式上的解析解,然后離散時(shí)間層,通過迭代的方式,得到方程半解析解,避免了分層界面處的數(shù)值震蕩.進(jìn)一步,本文對上述方法設(shè)計(jì)并行計(jì)算格式.數(shù)值實(shí)驗(yàn)顯示,并行加速比實(shí)驗(yàn)值達(dá)到4.392,并行方式為完全可擴(kuò)展的.非飽和滲流變量(如:壓力水頭)在較小的空間和較短的時(shí)間范圍內(nèi)快速變化,是造成Richards方程數(shù)值求解困難的原因之一.提高數(shù)值格式精度、提高解在整個區(qū)域上的光滑性以及自適應(yīng)網(wǎng)格剖分是我們在解決這一問題時(shí)的基本策略.本文對空間1維采用3次B樣條基有限元,對空間2維采用5次Hermit型插值有限元,保證了解u在?上的整體光滑性,緩解非物理震蕩現(xiàn)象.進(jìn)一步,結(jié)合多重網(wǎng)格技術(shù),給出自適應(yīng)計(jì)算格式.數(shù)值實(shí)驗(yàn)顯示,基于上述方法得到的結(jié)果,對數(shù)值震蕩的控制優(yōu)于線性基有限元方法得到的結(jié)果.Richards方程刻畫的問題多為長時(shí)間問題.時(shí)間層采用具有保結(jié)構(gòu)性質(zhì)的數(shù)值方法可以有效提高數(shù)值穩(wěn)定性,保證解具備長時(shí)間良好數(shù)值性態(tài).本文對(1+1)維、(1+2)維Richards方程時(shí)間層用s級2s階全隱辛Runge-Kutta方法進(jìn)行數(shù)值離散,以便在積分過程中長時(shí)間保持系統(tǒng)的固有特性.數(shù)值實(shí)驗(yàn)顯示,相對于一般Runge-Kutta方法,該方法對Richards方程刻畫長時(shí)間問題表現(xiàn)出更高的數(shù)值穩(wěn)定性.考慮與Richards方程相關(guān)的一類凍土耦合模型 水熱耦合模型.該模型是一個科研項(xiàng)目中關(guān)注的模型.目前對這類耦合模型的數(shù)值求解多采用有限差分方法.本文對耦合模型的中的兩個基本方程,分析并整理本構(gòu)關(guān)系,給出基于差分方法的數(shù)值計(jì)算格式,在此基礎(chǔ)上,抽象模型方程,給出基于有限元方法的計(jì)算格式.Richards方程的實(shí)際應(yīng)用往往對應(yīng)海量計(jì)算,這使我們必須考慮如何提高并行計(jì)算效率.本文對使用廣泛的Gauss消去法,設(shè)計(jì)了一種列行調(diào)整雙向流水線并行算法,從通信時(shí)間、并行度、可擴(kuò)展性方面分析該算法的性能,并進(jìn)行了數(shù)值實(shí)驗(yàn).數(shù)值實(shí)驗(yàn)顯示,由于在通信階段數(shù)據(jù)執(zhí)行的并發(fā)度提高,其并行加速比可達(dá)到3.461,優(yōu)于傳統(tǒng)的行列劃分并行方式.
[Abstract]:The Richards equation is the basic governing equation for describing the soil seepage process, the heterogeneity of soil media, the high nonlinearity of soil water conductivity, and the complexity of regional and initial boundary conditions, and so on. The traditional finite difference method or finite element method will lead to non-physical numerical oscillation or dispersion of the results. Seeking effective numerical solution of Richards equation is soil hydrodynamic. The main contents of this paper are as follows: in the problem of layered soil seepage, The change of soil constitutive relation at the stratified interface, the jump of water conductivity and moisture content, brings a challenge to the stability of numerical calculation of Richards equation. We analyze the constitutive relation at the stratified interface for a class of Gardener-Basha type Richards equation under stratified soil. The formal analytical solution is obtained, and then the discrete time layer is discretized, and the semi-analytical solution of the equation is obtained by iterative method, which avoids the numerical oscillation at the layered interface. Furthermore, the parallel computing scheme is designed for the above methods. The numerical experiments show that, The parallel speedup ratio is 4.392, and the parallel mode is completely extensible. The unsaturated seepage variables (such as pressure head) change rapidly in smaller space and shorter time range. It is one of the reasons that the Richards equation is difficult to solve numerically. To improve the smoothness of the solution in the whole region and the adaptive mesh generation are the basic strategies for solving this problem. In this paper, we adopt the 3-order B-spline finite element method for the first dimension and the Hermit interpolation finite element method for the second dimension. Make sure you know you're here? Furthermore, combining with the technique of multi-grid, an adaptive computing scheme is given. Numerical experiments show that, based on the results obtained by the above method, The control of numerical oscillation is better than that of the results obtained by linear basis finite element method. The problems described by Richards equation are mostly long time problems. The numerical method with conserved structure property in time layer can effectively improve the numerical stability. It is guaranteed that the solution has good numerical behavior for a long time. In this paper, the time layer of the Richards equation is discretized by the Runge-Kutta method of order 2 s in order to preserve the inherent characteristics of the system for a long time during the integral process. Compared to the general Runge-Kutta method, This method has higher numerical stability for Richards equation to depict long time problems. A kind of coupled model of permafrost related to Richards equation is considered in this paper. The model is a model concerned in scientific research projects. The finite difference method is used to solve this kind of coupling model. In this paper, two basic equations in the coupled model are discussed. The constitutive relation is analyzed and arranged, and the numerical calculation scheme based on the difference method is given. On this basis, the model equation is abstracted, and the practical application of the Richards equation based on the finite element method is given. Therefore, we must consider how to improve the efficiency of parallel computing. In this paper, we design a column and row adjusted bidirectional pipeline parallel algorithm to analyze the performance of the algorithm in terms of communication time, parallelism and extensibility. The numerical experiments show that the parallel speedup ratio can reach 3.461 because of the increase of the concurrency of the data execution in the communication stage, which is superior to the traditional parallel method of column and column partition.
【學(xué)位授予單位】：蘭州大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O241.8

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