【作者】 罗鑫；

【导师】 黄晋；

【作者基本信息】 电子科技大学 ， 应用数学， 2013， 博士

【摘要】 积分方程在科学和工程技术问题中有着广泛应用。本文针对稳态热传导问题,利用间接边界元方法系统地研究了其边界积分方程的数值解。对各向异性热传导问题,采用机械求积法讨论了光滑域和凹角域情形,对各向同性热传导问题,利用校正求积法研究了三维轴对称域情形。在以下几方面展开研究工作，取得创新成果。1.研究了光滑域上带Dirichlet边界条件的达西方程数值解法。通过单层位势理论,利用达西方程的基本解,把达西方程转化为带对数奇性核的第一类弱奇异边界积分方程,然后借助Lyness与Sidi的弱奇异求积公式,结合中矩形公式,构造了求解含弱奇性核的第一类边界积分方程的机械求积法,直接估计离散方程本征值的上下界,得到离散矩阵的条件数为O(h1),从而表明机械求积法解第一类积分方程具有优秀的数值稳定性。基于Anselone的聚紧收敛理论证明了数值解的存在性和收敛性。此外,证明了近似解的误差有奇次幂的单参数渐近展开,得到机械求积法的收敛阶为O(h3),通过使用h3Richardson外推法,数值精度提高到O(h5)。利用导出的渐近后验误差估计构建了自适应算法。2.研究了多角域上带Dirichlet边界条件的达西方程数值解法。对于凹角域情形,解在凹点的奇异性将严重地影响近似解的精度,如何提高它的精度,成为数学家们长期关注的热点。Garlerkin有限元的精度一般是O(h1+r)(r <1),而配置法的精度更低。为消除对数奇异和角点奇异,使用了一个三角正弦变换,然后利用机械求积法得到边界积分方程的离散近似方程,并推出了关于误差的含奇数阶的多参数渐近展开式,其表明数值解的精度是O(h3),继而借助分裂外推法消去误差展式中的低阶项得到误差的高阶项,因此提高了机械求积法的收敛阶。3.研究了稳态完全各向异性热传导方程的数值算法。利用机械求积法,我们分别讨论了光滑域和多角域的情形,借助聚紧理论证明了机械求积法的收敛性。数值算例验证了适度增加边界节点数目，可进一步提高数值解的精度和稳定性。4.研究了校正求积法解三维轴对称稳态各向同性热传导问题。利用间接边界元,把三维各向同性热传导问题,也即轴对称Laplace问题,转化为一维边界积分方程,然后运用周期变换消除了解在角点处的奇性,进一步提高了校正求积法的精度,该求积方法简单易于实施。数值试验表明该算法的收敛阶为O(h3)。更多还原

【Abstract】 Integral equations have been used in many science and engineering problems. Inthis dissertation, indirect boundary element methods are systematically applied to studythe numerical solutions of the boundary integral equations of steady state heat conductionproblems with Dirichlet conditions. For the steady state anisotropic heat conduction prob-lems, we discussed the smooth domains and concave polygons by mechanical quadraturemethods, respectively. For the isotropic case, we study the three dimensional axisymmet-ric Laplace problems by modified quadrature method. These work are stated in detail asfollows.High accuracy mechanical quadrature methods are applied to solve boundary integralequations of anisotropic Darcy’s equations with Dirichlet conditions on smooth domains.By using single potential theory and fundamental solution of Darcy’s equations, Darcy’sequations can be converted into the first kind integral equation with a logarithmic and sin-gular kernel. Combining with middle point rule，the mechanical quadrature methods areconstructed for solving the weakly singular boundary integral equation system of the firstkind. By estimating the sup-bound and inf-bound of the eigenvalue expression of discretematrix, the condition number of discrete matrix is obtained only O(h1), which showthe mechanical quadrature methods possess the excellent stability. The convergence andstability are proved based on Anselone’s collective compact and asymptotical compacttheory. Furthermore, an asymptotical expansion with odd powers of errors for mechani-cal quadrature methods is presented, which possesses high accuracy order O(h3). Usingh3Richardson extrapolation algorithms, the accuracy order of the approximation can begreatly improved to O(h5), and an a posteriori error estimate can be achieved for con-structing self-adaptive algorithms.Numerical solutions for boundary integral equations of anisotropic Darcy’s equationswith Dirichlet conditions on polygonal boundaries are studied by mechanical quadraturemethods. In concave polygons, the solutions at concave points have singularities. Itsgreatly dampens the approximate accuracy. Hence, it has been critical point for math-ematicians to overcome the difficulty for a long time. The accuracy of Galerkin finitemethod is only O(h1+r)(r <1) and the accuracy of collocation methods are even lower. The logarithmic singularities at the corner points of the boundary can be removed bysin transformation. Then the discretion matrix by mechanical quadrature methods andthe asymptotic expansions of errors are obtained, which show the convergence order isO(h3). The convergence rate O(h5) can be achieved after using the splitting extrapola-tion methods.The fully anisotropic heat conduction problems with Dirichlet boundary conditionsby mechanical quadrature methods are investigated on smooth domains and polygonsrespectively. The convergence of mechanical quadrature methods are proved by Usingcollectively compact theory. Increasing the boundary nodes, the effect and stability willbe well.Modified quadrature method for three dimensional steady state isotropic heat con-duction problems are studied. By indirect boundary element method, the axisymmetricLaplace problems can be converted into the single boundary integral equations of the firstkind. Then, the periodic transformation can be used to remove the singularities of the cor-ners or endpoints. This technique improves the accuracy of modified quadrature method.The modified quadrature formula is simple to work for the singular integrals. Numericalexamples show the convergence rate of the modified quadrature method is O(h3).更多还原