用边界结点法求解非齐次双调和方程的Cauchy问题
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摘要
在科学和工程技术中,许多实际问题归结为求解偏微分方程的反问题。本文考虑对于椭圆型偏微分方程的定解问题而言,给出的边界条件不足的Cauchy型反问题。
用边界元法来求解椭圆方程型方程的边值问题,需要已知足够的边界条件,否则问题就是不适定的,会产生很大误差,若采用边界结点法则可以克服这种不适定性[46]。这里的边界结点法是指除了在所研究的区域的边界上分布结点外,还要通过在区域之外分布若干虚拟源点,在所研究的区域之内选取若干内点,以此来求解未给出边界条件的那部分边界上结点的未知函数值。这种方法要通过选取合适的径向基函数的线性组合来表示特解,再利用微分算子(Laplace算子、重调和算子等)的基本解形成满足已知边界条件的线性组合来表示问题的通解,这样的解适合整个边界以及区域内部。
本文主要针对二维的非齐次双调和方程的Cauchy反问题用边界结点法求解,利用部分已知的边界条件来推导解的线性组合的待定系数,从而得出适用于全部求解域的解的表达式。求待定系数时,由于所选取的虚拟源点和边界结点数目不匹配,因此我们将该问题转化为一个最小二乘问题;对于Cauchy问题的不适定性,本文使用常用的正则化方法即奇异值分解法,来求解该最小二乘问题所对应的病态线性方程组。
本文的数值试验考察了边界光滑和分片光滑的不同区域的情况,并对给出的准确数据以及有噪声的数据所得到得结果进行了分析,分析了几个影响结果准确度的参数。数值试验的结果表明了使用边界结点法求解非齐次双调和方程Cauchy问题的有效性,计算效率高、结果精确度高,还观察到计算结果的误差随着数据中的噪声的减小而收敛。
Many practical problems in engineering and science can be classified as inverse problems of partial differential equations. In this thesis, we consider the inverse problem of the Cauchy kind that to the problem for determining solution of elliptic partial differential equations, the boundary conditions are not sufficient.
When using boundary element method to solve the boundary value problem of elliptic equations, we need to know sufficient boundary conditions, otherwise, the problem is ill-posed, and the solution is not proper. This ill-posed problem can be avoided by using the Boundary Knot Method[46]. By the Boundary knot method, besides the boundary knots, we place some virtual source points outside the domain, and select some inter points inside the domain in order to help us get the unknown values on the boundary; we should choose the linear combination of some proper radial basis functions to express the particular solution, and using the fundamental solution of partial differential operators such as Laplace operator and biharmonic operator etc. to produce a linear combination to represent the general solution, which satisfies the known boundary condition.
In this thesis, we use the Boundary Knot Method to solve the 2-D Cauchy inverse problem of non-homogeneous biharmonic equation. We obtain the coefficients in the combination expression depending on the known boundary data to form the representation of the solution. If the number of the virtual source points cannot match the number of boundary knots, we need to use the least square method. Meanwhile, since the Cauchy problem is ill-posed, we choose the singular value decomposition method to solve the linear equations corresponding to the least square problem.
Numerical examples were performed to demonstrate the efficiency and efficacy of the boundary knot method. The examples with both smooth and piecewise smooth boundary and with both exact and noisy known data are tested. Several parameters were checked to find out the influence to the numerical results. Numerical tests show that the method is computationally efficient, accurate, stable and convergent with respect to the decrease of the noise in the data.
用边界元法来求解椭圆方程型方程的边值问题,需要已知足够的边界条件,否则问题就是不适定的,会产生很大误差,若采用边界结点法则可以克服这种不适定性[46]。这里的边界结点法是指除了在所研究的区域的边界上分布结点外,还要通过在区域之外分布若干虚拟源点,在所研究的区域之内选取若干内点,以此来求解未给出边界条件的那部分边界上结点的未知函数值。这种方法要通过选取合适的径向基函数的线性组合来表示特解,再利用微分算子(Laplace算子、重调和算子等)的基本解形成满足已知边界条件的线性组合来表示问题的通解,这样的解适合整个边界以及区域内部。
本文主要针对二维的非齐次双调和方程的Cauchy反问题用边界结点法求解,利用部分已知的边界条件来推导解的线性组合的待定系数,从而得出适用于全部求解域的解的表达式。求待定系数时,由于所选取的虚拟源点和边界结点数目不匹配,因此我们将该问题转化为一个最小二乘问题;对于Cauchy问题的不适定性,本文使用常用的正则化方法即奇异值分解法,来求解该最小二乘问题所对应的病态线性方程组。
本文的数值试验考察了边界光滑和分片光滑的不同区域的情况,并对给出的准确数据以及有噪声的数据所得到得结果进行了分析,分析了几个影响结果准确度的参数。数值试验的结果表明了使用边界结点法求解非齐次双调和方程Cauchy问题的有效性,计算效率高、结果精确度高,还观察到计算结果的误差随着数据中的噪声的减小而收敛。
Many practical problems in engineering and science can be classified as inverse problems of partial differential equations. In this thesis, we consider the inverse problem of the Cauchy kind that to the problem for determining solution of elliptic partial differential equations, the boundary conditions are not sufficient.
When using boundary element method to solve the boundary value problem of elliptic equations, we need to know sufficient boundary conditions, otherwise, the problem is ill-posed, and the solution is not proper. This ill-posed problem can be avoided by using the Boundary Knot Method[46]. By the Boundary knot method, besides the boundary knots, we place some virtual source points outside the domain, and select some inter points inside the domain in order to help us get the unknown values on the boundary; we should choose the linear combination of some proper radial basis functions to express the particular solution, and using the fundamental solution of partial differential operators such as Laplace operator and biharmonic operator etc. to produce a linear combination to represent the general solution, which satisfies the known boundary condition.
In this thesis, we use the Boundary Knot Method to solve the 2-D Cauchy inverse problem of non-homogeneous biharmonic equation. We obtain the coefficients in the combination expression depending on the known boundary data to form the representation of the solution. If the number of the virtual source points cannot match the number of boundary knots, we need to use the least square method. Meanwhile, since the Cauchy problem is ill-posed, we choose the singular value decomposition method to solve the linear equations corresponding to the least square problem.
Numerical examples were performed to demonstrate the efficiency and efficacy of the boundary knot method. The examples with both smooth and piecewise smooth boundary and with both exact and noisy known data are tested. Several parameters were checked to find out the influence to the numerical results. Numerical tests show that the method is computationally efficient, accurate, stable and convergent with respect to the decrease of the noise in the data.
引文
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[4] J. William Helton, A numerical method for computing the structured singular value[J], Systems & Control Letters,1988,10:21-26.
[5]夏志皋著.弹性力学及其数值方法[M].上海:同济大学出版社,1997.
[6] D. Holnapy, M. Sz?ts, A. Botár. A generalization of the method of finite differences[J], Information Processing Letters,1973,2:55-59.
[7] Avron Douglis. A finite-difference method for generalized radial transport equations[J], Journal of Differential Equations.1967,3:451-481.
[8] N. N. Kuznetsov. A finite difference method for solving the cauchy problem for a quasi-linear first-order equation[J], USSR Computational Mathematics and Mathematical Physics. 1977, 17:114-127.
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[10]张鸿庆,王鸣著.有限元的数学理论[M].北京:科学出版社,1991.
[11]屈钧利著.工程结构的有限元方法[M].西安:西北工业大学出版社,2004.
[12]陈传淼著.有限元超收敛构造理论[M].长沙:湖南科学技术出版社,2001.
[13] N.C. Knowles. Finite element analysis[J], Computer-Aided Design, 1985,16:134-160.
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[15]祝家麟,袁政强著.边界元分析[M].北京:科学出版社,2009.
[16]祝家麟著.椭圆边值问题的边界元分析[M].北京:科学出版社,1991.
[17]孙焕纯,张立洲,许强,张耀明著.无奇异边界元法[M].大连:大连理工大学出版社, 1999.
[18]申光宪著.边界元法[M].北京:机械工业出版社,1998
[19] Djuro M. Misljenovic. Boundary element method and wave equation[J], Applied Mathematical Modelling. 1982,6:205-208.
[20] Masataka Tanaka, Kouji Yamagiwa. A boundary element method for some inverse problems in elastodynamics[J], Applied Mathematical Modelling. 1989,13:307-312.
[21] G. Gospodinov, D. Ljutskanov. The boundary element method applied to plates[J], Applied Mathematical Modelling. 1982,6:237-244.
[22] Atluri SN, Shen S. The Meshless Local Petrov-Galerkin (MLPG) Method [M]. California: Tech.Science Press, 2002.
[23] Liu GR. Meshfree Methods: Moving Beyond The Finite Method [M]. Singapore: CRC Press, 2002.
[24] Liu GR, Liu MB. Smoothed Particle Hydrodynamics: A Meshfree Particle Method [M]. Singapore: World Scientific, 2003.
[25] Li SF, Liu WK. Meshfree Particle Methods [M]. Berlin: Springer, 2004.
[26] Atluri SN. The Meshless Method (MLPG) for Domain & BIE Discretizations [M]. California: Tech. Science Press, 2004.
[27]刘更,刘天祥,谢琴.无网格方法及其应用[M].西安:西北工业大学出版社, 2005.
[28] C.A.Brebbia.J.C.F.Telles.L.C.Wrobel,Boudary Element techniques–Theory and Applications in Engineering[M].Springer-Verlag,Berlin Heidelberg New York Tokyo,1984.
[29]杜庆华著.边界积分方程方法-边界元法[M].北京:高等教育出版社, 1989.
[30]冯康,石钟慈著.弹性结构的数学理论[M].北京:科学出版社, 1989.
[31]余德浩著.自然边界元方法的数学理论[M].北京:科学出版社,1993.
[32]翟景春,Laplace方程边值问题的一种数值解法[J],山东科学,1995(3):21-29.
[33]陈浩,二维Laplace方程解的性质[J],工程数学,1995(12)2:95-99.
[34]祝家麟,边界元方法中的奇异性[J].重庆建筑工程学院学报1985(13)2:90—102.
[35]谷超豪等,数学物理方程[M].北京:人民教育出版社,1979.
[36]胡建伟,汤怀民著.微分数值方法[M].北京:科学出版社.1999.1.
[37]吴宗敏.散乱数据拟合的模型、方法和理论[M].北京:科学出版社, 2006.
[38] Liu GR, Gu YT. A point interpolation method for two-dimensional solids [J]. Int. J. Numer. Meth. Eng., 2001, 50: 937-951.
[39] Liu WK, Jun S, Zhang YF. Reproducing kernel particle methods [J]. Int. J. Numer. Meth. Fluid., 1995, 20: 1081-1106.
[40] Michael A. Golberg. The method of fundamental solutions for Poisson’s equation[J]. Engineering Analysis with Boundary Elements. 1995,16:205-213.
[41] Babu?ka I, Melenk JM. The partition of unity methods [J]. Int. J. Numer. Meth. Eng., 1997, 40: 727-758.
[42]张雄,刘岩,无网格法[M].北京:清华大学出版社,2004.
[43] Kress. R. Linear Integral Equations[M]. Springer-Verlag: Berlin, Heidelberg, NewYork, 1989.
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