本文選題：保角變換 + 模擬電荷法��； 參考：《昆明理工大學(xué)》2017年碩士論文

【摘要】：保角變換是復(fù)變函數(shù)的一個(gè)基本問題,廣泛應(yīng)用于物理學(xué)與工學(xué).保角變換的主要求解方法有解析法和數(shù)值計(jì)算法.解析法的基礎(chǔ)是Riemann存在與唯一性定理,指出保角變換函數(shù)的存在性,但是并不能給出函數(shù)的具體形式.因此,對于許多復(fù)雜的區(qū)域必須通過數(shù)值計(jì)算方法來求變換函數(shù).目前的保角變換分為兩類:一類是單連通區(qū)域的保角變換,另一類是多連通區(qū)域的保角變換.本文研究基于模擬電荷法多連通區(qū)域的數(shù)值保角變換計(jì)算法,主要工作有:(1)對保角變換的相關(guān)概念和定理進(jìn)行介紹.(2)給出了以心臟線和星形線為外邊界的多連通區(qū)域的模擬電荷點(diǎn)和約束點(diǎn)的取法.(3)通過對于雙連通和有界多連通區(qū)域的約束方程進(jìn)行預(yù)處理,構(gòu)造了一個(gè)對稱正定線形方程組.進(jìn)而,分別利用Runge-Kutta法和(n,m)-Pade迭代法求解對稱正定線性方程組得到電量和變換半徑,構(gòu)造了近似高精度的保角變換函數(shù).(4)對基于模擬電荷法的無界多連通區(qū)域的數(shù)值保角變換計(jì)算法進(jìn)行了詳細(xì)闡述,將本文的方法運(yùn)用到了流體力學(xué),并模擬出了圓柱體周圍的流線.最后,對文中提出的數(shù)值保角變換改善法其進(jìn)行了數(shù)值實(shí)驗(yàn),數(shù)值結(jié)果表明了本文算法改善數(shù)值保角變換精度的有效性.
[Abstract]:Conformal transformation is a basic problem of complex function, which is widely used in physics and engineering. The main methods of conformal transformation are analytical method and numerical method. The analytical method is based on the existence and uniqueness theorem of Riemann, and points out the existence of conformal transformation function, but it can not give the concrete form of the function. Therefore, the transformation function must be solved by numerical method for many complex regions. At present, conformal transformation is divided into two categories: one is conformal transformation of simply connected region and the other is conformal transformation of multi-connected region. In this paper, the numerical conformal transformation method based on the simulated charge method for multi-connected regions is studied. The main work of this paper is to introduce the concepts and theorems of conformal transformation. (2) the method of selecting the simulated charge points and constraint points of multi-connected regions with the outer boundary of heart line and star line is given. The constraint equation of the connected region is pretreated, A system of symmetric positive definite linear equations is constructed. Furthermore, the electric quantity and the transformation radius are obtained by using the Runge-Kutta method and the Pade iteration method, respectively, for solving the symmetric positive definite linear equations. The conformal transformation function of approximate high precision is constructed. The numerical conformal transformation calculation method of unbounded multi-connected region based on simulated charge method is described in detail. The method in this paper is applied to hydrodynamics. The streamline around the cylinder is simulated. Finally, numerical experiments are carried out to improve the accuracy of the numerical conformal transform proposed in this paper. The numerical results show that the proposed algorithm is effective in improving the accuracy of the numerical conformal transformation.
【學(xué)位授予單位】：昆明理工大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O241

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本文編號(hào)：1786955