三维轴对称边界积分方程的高精度算法

发布时间：2018-03-04 13:46

本文选题：轴对称边界积分方程　切入点：机械求积法　出处：《电子科技大学》2017年博士论文　论文类型：学位论文

【摘要】：科学与工程问题中的大量数学模型都归结于求解域是旋转体的微分方程边值问题。这类问题称为轴对称问题,是目前研究的热点。本文旨在通过边界元方法把这类问题转化为轴对称的边界积分方程,利用机械求积法系统讨论了轴对称弹性静力学边界积分方程、轴对称达西边界积分方程、轴对称非线性Laplace边界积分方程和轴对称泊松边界积分方程的数值解法,取得的成果如下:1、研究了轴对称弹性静力学方程带Dirichlet边值条件的数值解法。通过单层位势理论,利用轴对称弹性静力学方程的基本解,把弹性静力学方程转化为带有对数弱奇异核的第一类边界积分方程。由于轴对称问题的边界大部分是非光滑的,所以边界积分方程的解在角点处具有奇异性,利用三角周期变换消除了解在角点处的奇性。利用Lyness和Sidi的弱奇异求积公式,结和中矩形数值积分公式,构造了求解具有弱奇异核的第一类边界积分方程的机械求积法。利用Anselone的聚紧收敛理论证明了数值解的存在性和收敛性,还证明了数值解的误差具有(?38)(6)的收敛阶。2、研究了轴对称达西方程带Dirichlet边值条件的数值解法。利用单层位势理论及空间坐标变换,将轴对称达西方程转化为第一类的带有对数弱奇异核的边界积分方程。为了提高数值解的精度,利用三角周期变换消除边界积分方程的解在角点处的奇性。利用机械求积法求解第一类的弱奇异的边界积分方程,得到解的误差具有奇数阶的多参数渐近展开式,其给出了数值解的精度为(?38)(6)。利用分裂外推算法消去误差展开式中的低阶项得到高阶项,提高数值解的收敛阶。聚紧理论证明了机械求积法的收敛性。3、研究了轴对称非线性Laplace方程的数值解法。利用直接边界积分方程法和轴对称Laplace方程的基本解,将具有非线性边值条件的轴对称Laplace方程转化为轴对称的非线性边界积分方程,该积分方程具有弱奇异核。利用机械求积法和牛顿迭代法求解非线性的边界积分方程,得到数值解的误差具有奇数阶的单参数渐近展开式,其给出了数值解的精度为(?3)。利用外推算法提高数值解的收敛精度阶为(?5)。利用Stepleman定理证明了非线性近似方程解的存在性和稳定性。4、研究了轴对称泊松方程带Dirichlet边值条件的数值解法。利用轴对称泊松方程的特解,轴对称泊松方程可以导出轴对称Laplace方程,利用单层位势理论,将导出方程转化为第一类的带有对数弱奇异核的边界积分方程。利用三角变换消除解在角点处的奇性,利用机械求积法离散边界积分方程,得到数值解的误差具有奇数阶的多参数渐近展开式,其给出了数值解的精度为(?38)(6)。通过分裂外推算法消去展开式中的低阶项得到高阶项提高数值解的精度为(?58)(6)。多个数值算例验证了我们的理论分析。
[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.
【学位授予单位】：电子科技大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O241.83

【参考文献】