本文選題：邊界元方法 + 滲流��； 參考：《哈爾濱工業(yè)大學(xué)》2014年碩士論文

【摘要】：大部分的工程問(wèn)題都可歸結(jié)為偏微分方程邊值問(wèn)題，但由于邊界條件復(fù)雜、物體性質(zhì)不均勻、幾何形狀不規(guī)則等原因，一般都求不出問(wèn)題的解析解，因此人們開(kāi)始關(guān)注求解問(wèn)題的近似解法——數(shù)值分析方法。邊界元方法作為數(shù)值分析方法的一種，雖然它的出現(xiàn)晚于有限元法，但以其獨(dú)特的優(yōu)勢(shì)在近五十年內(nèi)得到了快速的發(fā)展。如今，，邊界元引用的領(lǐng)域也越來(lái)越廣泛，如位勢(shì)問(wèn)題、彈性問(wèn)題、彈塑性問(wèn)題、動(dòng)力學(xué)問(wèn)題等。本論文采用直接邊界元法對(duì)四種滲流問(wèn)題進(jìn)行了研究。根據(jù)滲流介質(zhì)性質(zhì)的不同，即滲透系數(shù)的不同，四種滲流問(wèn)題分別為：均質(zhì)各向同性滲流問(wèn)題（k x ky，且都為常數(shù)）、正交各向異性滲流問(wèn)題（k x ky，但都為常數(shù)）、非飽和土滲流問(wèn)題（k k h se）、分層土的滲流問(wèn)題。本文除了完成四種滲流問(wèn)題的解決過(guò)程外，每類問(wèn)題也都編寫了matlab程序，并用算例進(jìn)行驗(yàn)證。具體的研究?jī)?nèi)容如下： （1）各類滲流問(wèn)題的邊界積分方程的推導(dǎo)。在各類滲流問(wèn)題的控制方程的基礎(chǔ)上，采用加權(quán)余量法及格林公式將偏微分方程轉(zhuǎn)化為邊界積分方程。其中非飽和土滲流問(wèn)題比較特殊，因?yàn)樗目刂品匠淌欠蔷�性的，所以積分方程中還有域內(nèi)積分項(xiàng)。 （2）邊界離散方法的研究。在前兩個(gè)滲流問(wèn)題中，都采用了常量元與線性元對(duì)邊界進(jìn)行離散，并對(duì)兩種離散方法最終的計(jì)算結(jié)果精度進(jìn)行對(duì)比，判斷優(yōu)劣。 （3）角點(diǎn)處法向流量不連續(xù)問(wèn)題的研究。當(dāng)采用線性元離散邊界，并用單元的兩端點(diǎn)作結(jié)點(diǎn)時(shí)會(huì)產(chǎn)生角點(diǎn)問(wèn)題，本文采用重結(jié)點(diǎn)法及重結(jié)點(diǎn)單未知量法解決，并針對(duì)不同滲流問(wèn)題得出兩種方法的優(yōu)缺點(diǎn)及適用情況。 （4）奇異積分的處理。對(duì)于邊界積分式中存在的奇異性一般為一階奇異性，本文采用解析算法處理。對(duì)于域內(nèi)積分式存在的奇異性，本文選擇先將積分進(jìn)行坐標(biāo)變換，即由直角坐標(biāo)積分轉(zhuǎn)化為為極坐標(biāo)積分，再采用半解析半數(shù)值法處理。
[Abstract]:Most engineering problems can be attributed to the boundary value problems of partial differential equations. However, due to the complex boundary conditions, uneven properties of objects and irregular geometry, the analytical solution of the problem can not be obtained. Therefore, people began to pay attention to the approximate solution-numerical analysis method. Boundary element method (BEM) is one of the numerical analysis methods. Although it appears later than the finite element method, it has been developed rapidly in the past 50 years with its unique advantages. Nowadays, boundary element references are more and more widely used, such as potential problem, elastic problem, elastoplastic problem, dynamic problem and so on. In this paper, the direct boundary element method is used to study four seepage problems. According to the different properties of percolation medium, that is, the permeability coefficient is different, The four seepage problems are: homogeneous isotropic seepage problem (k x ky,), orthotropic seepage problem (k x ky,), and unsaturated soil seepage problem (k k h se), layered soil seepage problem). In addition to the solution of four kinds of seepage problems, matlab programs are written for each kind of problems and verified by numerical examples. The main contents are as follows: (1) the derivation of boundary integral equations for various seepage problems. On the basis of governing equations of various seepage problems, partial differential equations are transformed into boundary integral equations by using weighted residual method and Green's formula. The problem of unsaturated soil seepage is quite special, because its governing equation is nonlinear, so there are integral terms in the integral equation. (2) the study of boundary discretization method. In the first two seepage problems, the constant element and the linear element are used to discretize the boundary, and the accuracy of the final calculation results of the two discrete methods are compared to judge the merits and demerits. (3) the study of the normal flow discontinuity problem at the corner point. When the boundary is discretized by linear element and the two ends of the element are used as the nodes, the corner problem will arise. In this paper, the double node method and the double node single unknown method are used to solve the problem. The advantages and disadvantages of the two methods and their application are obtained for different seepage problems. (4) the treatment of singular integrals. For the singularity in the boundary integral is generally the first order singularity, the analytic algorithm is used in this paper. For the singularity of the integral expression in the domain, this paper chooses to transform the integral into polar coordinate by means of coordinate transformation, and then use the semi-analytic semi-numerical method to deal with it.
【學(xué)位授予單位】：哈爾濱工業(yè)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2014
【分類號(hào)】：TU43;TV139.1

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