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  本文選題：軸對稱邊界積分方程　切入點：機械求積法　出處：《電子科技大學》2017年博士論文　論文類型：學位論文

【摘要】：科學與工程問題中的大量數(shù)學模型都歸結(jié)于求解域是旋轉(zhuǎn)體的微分方程邊值問題。這類問題稱為軸對稱問題,是目前研究的熱點。本文旨在通過邊界元方法把這類問題轉(zhuǎn)化為軸對稱的邊界積分方程,利用機械求積法系統(tǒng)討論了軸對稱彈性靜力學邊界積分方程、軸對稱達西邊界積分方程、軸對稱非線性Laplace邊界積分方程和軸對稱泊松邊界積分方程的數(shù)值解法,取得的成果如下:1、研究了軸對稱彈性靜力學方程帶Dirichlet邊值條件的數(shù)值解法。通過單層位勢理論,利用軸對稱彈性靜力學方程的基本解,把彈性靜力學方程轉(zhuǎn)化為帶有對數(shù)弱奇異核的第一類邊界積分方程。由于軸對稱問題的邊界大部分是非光滑的,所以邊界積分方程的解在角點處具有奇異性,利用三角周期變換消除了解在角點處的奇性。利用Lyness和Sidi的弱奇異求積公式,結(jié)和中矩形數(shù)值積分公式,構造了求解具有弱奇異核的第一類邊界積分方程的機械求積法。利用Anselone的聚緊收斂理論證明了數(shù)值解的存在性和收斂性,還證明了數(shù)值解的誤差具有(?38)(6)的收斂階。2、研究了軸對稱達西方程帶Dirichlet邊值條件的數(shù)值解法。利用單層位勢理論及空間坐標變換,將軸對稱達西方程轉(zhuǎn)化為第一類的帶有對數(shù)弱奇異核的邊界積分方程。為了提高數(shù)值解的精度,利用三角周期變換消除邊界積分方程的解在角點處的奇性。利用機械求積法求解第一類的弱奇異的邊界積分方程,得到解的誤差具有奇數(shù)階的多參數(shù)漸近展開式,其給出了數(shù)值解的精度為(?38)(6)。利用分裂外推算法消去誤差展開式中的低階項得到高階項,提高數(shù)值解的收斂階。聚緊理論證明了機械求積法的收斂性。3、研究了軸對稱非線性Laplace方程的數(shù)值解法。利用直接邊界積分方程法和軸對稱Laplace方程的基本解,將具有非線性邊值條件的軸對稱Laplace方程轉(zhuǎn)化為軸對稱的非線性邊界積分方程,該積分方程具有弱奇異核。利用機械求積法和牛頓迭代法求解非線性的邊界積分方程,得到數(shù)值解的誤差具有奇數(shù)階的單參數(shù)漸近展開式,其給出了數(shù)值解的精度為(?3)。利用外推算法提高數(shù)值解的收斂精度階為(?5)。利用Stepleman定理證明了非線性近似方程解的存在性和穩(wěn)定性。4、研究了軸對稱泊松方程帶Dirichlet邊值條件的數(shù)值解法。利用軸對稱泊松方程的特解,軸對稱泊松方程可以導出軸對稱Laplace方程,利用單層位勢理論,將導出方程轉(zhuǎn)化為第一類的帶有對數(shù)弱奇異核的邊界積分方程。利用三角變換消除解在角點處的奇性,利用機械求積法離散邊界積分方程,得到數(shù)值解的誤差具有奇數(shù)階的多參數(shù)漸近展開式,其給出了數(shù)值解的精度為(?38)(6)。通過分裂外推算法消去展開式中的低階項得到高階項提高數(shù)值解的精度為(?58)(6)。多個數(shù)值算例驗證了我們的理論分析。
[Abstract]:A large number of mathematical models in scientific and engineering problems are attributed to boundary value problems of differential equations in which the domain is a rotating body. Such problems are called axisymmetric problems. The purpose of this paper is to transform this kind of problems into axisymmetric boundary integral equations by boundary element method, and to discuss the axisymmetric elastic boundary integral equations systematically by means of mechanical quadrature method. Numerical solution of axisymmetric Darcy boundary integral equation, axisymmetric nonlinear Laplace boundary integral equation and axisymmetric Poisson boundary integral equation, The results obtained are as follows: 1. The numerical solution of axisymmetric elastic statics equation with Dirichlet boundary condition is studied. The basic solution of axisymmetric elastic statics equation is obtained by using the theory of single layer potential and the basic solution of axisymmetric elastic statics equation. The elastic statics equation is transformed into the first kind boundary integral equation with logarithmic weakly singular kernel. Because the boundary of axisymmetric problem is mostly nonsmooth, the solution of the boundary integral equation is singular at the corner point. The singularity of the solution at the corner is eliminated by using the triangular periodic transformation. The weak singular quadrature formula of Lyness and Sidi, the numerical integral formula of the junction and the middle rectangle are used. A mechanical quadrature method for solving the first kind of boundary integral equation with weakly singular kernel is constructed. The existence and convergence of the numerical solution are proved by using Anselone's convergence theory, and the error of the numerical solution is proved. In this paper, the numerical solution of axisymmetric Darcy equation with Dirichlet boundary value condition is studied. The single layer potential theory and space coordinate transformation are used. The axisymmetric Darcy equation is transformed into the first class boundary integral equation with logarithmic weakly singular kernel. The singularity of the solution of the boundary integral equation at the corner point is eliminated by means of the triangular periodic transformation. By using the mechanical quadrature method, the weak singular boundary integral equation of the first kind is solved, and the error of the solution is obtained by the multi-parameter asymptotic expansion with odd order. The accuracy of the numerical solution is obtained. Using the split extrapolation method to eliminate the lower order term in the error expansion, the higher order term is obtained. The convergence of mechanical quadrature method is proved by convergence theory. The numerical solution of axisymmetric nonlinear Laplace equation is studied. The direct boundary integral equation method and the basic solution of axisymmetric Laplace equation are used. The axisymmetric Laplace equation with nonlinear boundary value condition is transformed into axisymmetric nonlinear boundary integral equation, which has a weak singular kernel. The mechanical quadrature method and Newton iterative method are used to solve the nonlinear boundary integral equation. A one-parameter asymptotic expansion with odd order error of the numerical solution is obtained, and the accuracy of the numerical solution is obtained. Using the extrapolation method to improve the convergence accuracy of the numerical solution is? 5. The existence and stability of solutions of nonlinear approximate equations are proved by using Stepleman theorem. The numerical solution of axisymmetric Poisson equation with Dirichlet boundary value condition is studied, and the special solution of axisymmetric Poisson equation is obtained. The axisymmetric Poisson equation can be derived from the axisymmetric Laplace equation. By using the single-layer potential theory, the derived equation is transformed into the first kind of boundary integral equation with logarithmic weakly singular kernels. The singularity of the solution at the corner point is eliminated by triangular transformation. By using the mechanical quadrature method to discretize the boundary integral equation, a multi-parameter asymptotic expansion with odd order error of the numerical solution is obtained. The accuracy of the numerical solution is obtained. By using the split extrapolation method to eliminate the lower order term in the expansion, the higher order term is obtained to improve the accuracy of the numerical solution. A number of numerical examples verify our theoretical analysis.
【學位授予單位】：電子科技大學
【學位級別】：博士
【學位授予年份】：2017
【分類號】：O241.83

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