發(fā)布時(shí)間：2018-11-07 18:48

【摘要】：包氣帶是連接大氣降水、地表水與地下水的重要紐帶,并且對(duì)于保護(hù)干旱、半干旱地區(qū)的生態(tài)環(huán)境具有重大意義。由于描述非飽和帶水分運(yùn)移的動(dòng)力學(xué)參數(shù)具有高度的非線性,導(dǎo)致了非飽和帶水分運(yùn)移的Richard方程同樣也具有高度的非線性,因此求解該類方程極其困難。通常在相對(duì)簡單的定解條件下,可以獲得描述包氣帶水分運(yùn)移問題的解析解,然而在研究較為復(fù)雜條件下的非飽和帶水流問題時(shí),當(dāng)前最為有效的方法還是采用數(shù)值模擬方法。Richard方程在空間上是一個(gè)二階橢圓型方程,但是對(duì)于時(shí)間是一階的,具有拋物線型方程的性質(zhì)。所以在使用有限差分法及有限元法等傳統(tǒng)的數(shù)值模擬方法求解時(shí),容易造成數(shù)值彌散、數(shù)值振蕩及質(zhì)量不守恒等一系列問題。上世紀(jì)80年代陳景仁提出了一種新的數(shù)值計(jì)算方法——有限分析法。有限分析法的基本思想是:將控制方程的局部解析解組成整體數(shù)值解,這樣得到的解可以比較好的保持原有問題的物理特性,格式通過自動(dòng)調(diào)節(jié)有限分析系數(shù)來反映對(duì)流與擴(kuò)散效應(yīng),可以獲得單調(diào)無震蕩性解,數(shù)值穩(wěn)定性高。因此,本文應(yīng)用有限分析法求解Richard方程,探討包氣帶水分運(yùn)移問題。本文提出了四種形式包氣帶在均質(zhì)條件下水分運(yùn)移的有限分析計(jì)算格式及一種非均質(zhì)條件下的有限分析計(jì)算格式。通過定性定量化的研究,可以獲得以下幾個(gè)方面的成果及結(jié)論:1.在數(shù)學(xué)上嚴(yán)格證明了有限分析計(jì)算格式的穩(wěn)定性及收斂性,即在理論上說明了有限分析法是無條件穩(wěn)定及收斂的。2.均質(zhì)條件下四種形式的有限分析計(jì)算格式中,除了負(fù)壓型的有限分析計(jì)算格式外,其它三種形式的有限分析計(jì)算格式都可以獲得穩(wěn)定性高、收斂性好及精度高的數(shù)值解,并且能夠較好的控制質(zhì)量平衡誤差。通過與解析解進(jìn)行比較,混合型Richard方程的有限分析計(jì)算格式可以獲得計(jì)算精度最高的數(shù)值解并且能夠最好的控制質(zhì)量平衡誤差;含水量型Richard方程及Kirchhoff變換的Richard方程的有限分析計(jì)算格式次之,負(fù)壓型Richard方程的有限分析計(jì)算格式最差。因此不推薦使用負(fù)壓型Richard方程的有限分析計(jì)算格式。3.混合型Richard方程、含水量型Richard方程及Kirchhoff變換的Richard方程等三種有限分析計(jì)算格式在相對(duì)粗糙的空間步長條件下仍然可以獲得精度較高的數(shù)值解。4.首次應(yīng)用有限分析法推導(dǎo)了基于Kirchhoff變換后負(fù)壓型Richard方程的有限分析分析計(jì)算格式,用來求解包氣帶非均質(zhì)條件水分運(yùn)移問題。通過和解析解對(duì)比可知,本文推導(dǎo)的算法可以很好的解決Kirchhoff變換后,變量在非均質(zhì)界面不連續(xù)的問題,并且該算法在較為寬松步長條件下也能夠獲得較高精度的數(shù)值解。然而使用有限差分法(應(yīng)用幾何平均計(jì)算內(nèi)部節(jié)點(diǎn)的滲透系數(shù))求解包氣帶非均質(zhì)條件水流問題時(shí),對(duì)空間步長的要求較高,在非均質(zhì)界面上會(huì)引起巨大的局部單點(diǎn)誤差,并且其還會(huì)影響整個(gè)計(jì)算域的計(jì)算精度。5.應(yīng)用有限分析法模擬了一個(gè)室內(nèi)實(shí)驗(yàn)。有限分析法的模擬結(jié)果與實(shí)測(cè)值擬合度較高,說明了有限分析法能夠應(yīng)用于求解相對(duì)復(fù)雜的實(shí)際問題。
[Abstract]:The package gas zone is an important link between the atmospheric precipitation, the surface water and the ground water, and is of great significance to the protection of the ecological environment in the arid and semi-arid areas. Because the dynamic parameters describing the water migration of unsaturated zone have a high degree of non-linearity, the Richard's equation, which results in the water migration of unsaturated zone, also has a high degree of non-linearity, so it is extremely difficult to solve this kind of equation. In this paper, the analytical solution to the problem of water migration in the air zone of the bag can be obtained under relatively simple fixed conditions. However, in the case of the water flow in unsaturated zone under the complicated conditions, the most effective method is to adopt the numerical simulation method. Richard's equation is a second-order elliptic equation in space, but it has a parabolic equation for time. Therefore, a series of problems such as numerical dispersion, numerical oscillation and non-conservation of quality can be easily caused when using the traditional numerical simulation method such as the finite difference method and the finite element method. In the 1980s, Chen Jingren proposed a new numerical method of numerical calculation _ finite analytic method. The basic idea of the finite analytic method is to form the whole numerical solution by the local analytic solution of the control equation, so that the obtained solution can maintain the physical property of the original problem, and the format can reflect the convection and the diffusion effect by automatically adjusting the limited analysis coefficient, and the numerical stability is high. In this paper, the finite analytic method is applied to solve the problem of water migration in the package gas zone. In this paper, the finite analysis and calculation scheme of the water migration under the homogeneous condition and the finite analytic calculation method under the non-homogeneous condition are presented in this paper. Through the qualitative and quantitative study, the following results and conclusions can be obtained: 1. In this paper, the stability and the convergence of the finite analytic calculation form are strictly proved, that is, the finite analytic method is proved to be unconditionally stable and convergent. In addition to the limited analysis and calculation of the negative pressure type, the finite analysis and calculation form of the four forms of homogeneous condition can obtain the numerical solution with high stability, good convergence and high precision, and can control the quality balance error better. By comparison with the analytical solution, the finite analysis and calculation form of the mixed-type Richard equation can obtain the numerical solution with the highest calculation accuracy and the best control quality balance error; the water content type Richard equation and the Kirchhoff-transformed Richard equation are the second in the finite analysis and calculation format, The finite analysis of the negative pressure type Richard's equation is the worst. Therefore, the limited analysis and calculation format of the negative pressure type Richard equation is not recommended. A numerical solution with high accuracy can still be obtained under the condition of relatively rough space steps, such as the mixed-type Richard's equation, the water-content-type Richard equation and the Kirchhoff-transformed Richard's equation. The finite analysis and analysis of the negative pressure type Richard's equation based on Kirchhoff's transformation are derived for the first time. The method is used to solve the problem of moisture migration in the non-homogeneous condition of the package gas zone. Compared with the analytical solution, the proposed algorithm can solve the problem that the variable is not continuous in the non-homogeneous interface after the Kirchhoff transformation, and the algorithm can obtain high-precision numerical solution under the condition of looser step. However, using the finite difference method (using the geometric mean to calculate the permeability coefficient of the internal node), the requirement of the space step size is high, and a great local single-point error can be caused on the non-homogeneous interface. and it also affects the calculation accuracy of the whole computing domain. The application of finite analytic method to the simulation of an indoor experiment. The simulation results of the finite analytic method are higher than that of the measured values, and the finite analytic method can be used to solve the relative complex practical problems.
【學(xué)位授予單位】：長安大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：P641.2

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