Laplace方程数值解的小波边界元法研究
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摘要
小波边界元法是近几年发展起来的一种新型的数值计算方法。目前对它的研究还比较少,但这种方法从一开始就展现了它的独特优点和强劲生命力。本文主要是研究小波基函数在自然边界元方法中的应用。目的是解决自然边界元方法中存在的奇异积分的困难,减小计算量,提高计算精度。其基本思想是将微分方程经过自然边界归化得到与之等价的变分问题,再利用Galerkin-wavelet方法或小波插值法将其离散,得到具有独特优点的刚度矩阵,这样就能大大减小计算量。同时对数值解的误差估计作了进一步的分析。给出的算例表明了各种小波边界元方法的有效性。
论文共分五章。第一章介绍了边界元方法、自然边界归化理论以及小波分析理论,并给出了Laplace方程在典型域上通过自然边界归化所得的自然边界积分方程和Possion积分公式。
第二章针对上半平面Laplace方程的Neumann边值问题在求解过程中遇到的强奇异积分的困难,论文充分利用Shannon小波在频域上有限带宽的优良性质,采用Shannon小波边界元方法求解,使问题简化且得到的数值解比较精确。
第三章论文充分利用拟小波基在时域中光滑性高、快速衰减且具有插值性的特点,使其与自然边界元方法相结合,有效解决了解Laplace方程时存在的奇异积分困难。
第四章论文采用Hermite三次样条多小波边界元方法对上半平面Laplace方程的Neumann边值问题做了进一步的探讨,并通过算例验证了该方法的有效性。
第五章论文将Quak三角小波的尺度函数作为基函数离散自然边界积分方程,得到了仅由一项或两项表出的刚度矩阵系数的计算公式,从而使问题简化,给出的算例也表明了该方法的可行性。
In recent years, wavelet boundary element method has developed a new numerical method. Although there is few works about this method, it still shows unique advantages and strong vitality at the beginning. To overcome the shortcoming of singular integral difficulty existed in the natural boundary element method, in this paper, we mainly study the application of the wavelet base function in the natural boundary element method, it not only simplify the calculation process, but also improve the computation accuracy. The basic idea is transforming the differential equations into its equivalent variational problem by the natural boundary naturalization, and then discreting it by the Galerkin-wavelet method or wavelet interpolation to get the corresponding stiffness matrix with unique advantage, it substantially reducing computation. Meanwhile we make the further analysis to the error estimate of the numerical solution. The given example indicates the validity of each kind of wavelet boundary element method.
This paper includes five chapters. Chapter 1 introduces the boundary element method, the natural boundary naturalization theory and the wavelet analysis theory. The natural boundary integral equation of Laplace equation and the Possion integral formula by the natural boundary naturalization on the typical domain are also given.
In chapter 2, since there is difficulty in solving the Neumann boundary value problem of the Laplace equation on half-plane encounters strongly singular integral. In order to simplify the problem and get more precise numerical solution, we take full advantages of the Shannon wavelet limited bandwidth in frequency domain and use Shannon wavelet boundary element method to solve the problem.
In chapter 3, combining the properties of smooth, fast weaken and the interpolation of the wavelet base with the natural boundary element method, it makes the computation reduce and improves accuracy. And it effectively resolves the trouble of the singular integral when solving Laplace equation.
In chapter 4, we apply Hermite Cubic Spline Multi-wavelet Natural Boundary Element Method to the Neumann boundary value problem of the Laplace equation on the upper half-plane and make further research. Moreover, the given example proves that the method is effective and flexible.
In chapter 5, we applies the criterion of Quak Triangular Wavelet as the base function to discrete Natural Boundary integral equation, and we get the stiffness matrix coefficient formula expressed by one item or two items, which can simply the problem and get more precise solution.
论文共分五章。第一章介绍了边界元方法、自然边界归化理论以及小波分析理论,并给出了Laplace方程在典型域上通过自然边界归化所得的自然边界积分方程和Possion积分公式。
第二章针对上半平面Laplace方程的Neumann边值问题在求解过程中遇到的强奇异积分的困难,论文充分利用Shannon小波在频域上有限带宽的优良性质,采用Shannon小波边界元方法求解,使问题简化且得到的数值解比较精确。
第三章论文充分利用拟小波基在时域中光滑性高、快速衰减且具有插值性的特点,使其与自然边界元方法相结合,有效解决了解Laplace方程时存在的奇异积分困难。
第四章论文采用Hermite三次样条多小波边界元方法对上半平面Laplace方程的Neumann边值问题做了进一步的探讨,并通过算例验证了该方法的有效性。
第五章论文将Quak三角小波的尺度函数作为基函数离散自然边界积分方程,得到了仅由一项或两项表出的刚度矩阵系数的计算公式,从而使问题简化,给出的算例也表明了该方法的可行性。
In recent years, wavelet boundary element method has developed a new numerical method. Although there is few works about this method, it still shows unique advantages and strong vitality at the beginning. To overcome the shortcoming of singular integral difficulty existed in the natural boundary element method, in this paper, we mainly study the application of the wavelet base function in the natural boundary element method, it not only simplify the calculation process, but also improve the computation accuracy. The basic idea is transforming the differential equations into its equivalent variational problem by the natural boundary naturalization, and then discreting it by the Galerkin-wavelet method or wavelet interpolation to get the corresponding stiffness matrix with unique advantage, it substantially reducing computation. Meanwhile we make the further analysis to the error estimate of the numerical solution. The given example indicates the validity of each kind of wavelet boundary element method.
This paper includes five chapters. Chapter 1 introduces the boundary element method, the natural boundary naturalization theory and the wavelet analysis theory. The natural boundary integral equation of Laplace equation and the Possion integral formula by the natural boundary naturalization on the typical domain are also given.
In chapter 2, since there is difficulty in solving the Neumann boundary value problem of the Laplace equation on half-plane encounters strongly singular integral. In order to simplify the problem and get more precise numerical solution, we take full advantages of the Shannon wavelet limited bandwidth in frequency domain and use Shannon wavelet boundary element method to solve the problem.
In chapter 3, combining the properties of smooth, fast weaken and the interpolation of the wavelet base with the natural boundary element method, it makes the computation reduce and improves accuracy. And it effectively resolves the trouble of the singular integral when solving Laplace equation.
In chapter 4, we apply Hermite Cubic Spline Multi-wavelet Natural Boundary Element Method to the Neumann boundary value problem of the Laplace equation on the upper half-plane and make further research. Moreover, the given example proves that the method is effective and flexible.
In chapter 5, we applies the criterion of Quak Triangular Wavelet as the base function to discrete Natural Boundary integral equation, and we get the stiffness matrix coefficient formula expressed by one item or two items, which can simply the problem and get more precise solution.
引文
1李景湧.有限元法.北京:邮电大学出版社, 2004:51-87
2王勖成,邵敏.有限单元法基本原理和数值方法(第2版).北京:清华大学出版社, 1997:1-2
3谢贻权,何福保.弹性和塑性力学中的有限单元法.北京:机械工业出版社, 1981:1-5
4曹国金,姜弘道.无单元法研究和应用现状及动态.力学进展, 2002,32(4):526-53
5 Brebbia, C. A. ed. Recent Advances in Boundary Element Methods. Pentech Press, London, 1978
6 Brebbia, C. A. ed. New Development in Boundary Element Methods. CML Publications, Southampton, 1980
7 Brebbia, C. A. ed. Boundary Element Methods. Springer-Velag, Berlin, 1981
8嵇醒,臧跃龙,程玉民.边界元进展及通用程序.上海:同济大学出版社, 1997
9杜庆华.边界积分方程方法—边界元法.北京:高等教育出版社, 1989
10陈传璋.积分方程论及其应用.北京:科学技术文献出版社, 1987
11路见可,钟寿国.积分方程论.北京:高等教育出版社, 1990
12冯康.论微分与积分方程及有限与无限元.计算数学, 1980,2(1):100-105
13徐长发.科技应用中的微分变分模型.武汉:华中科技大学出版社, 2004
14徐长发,李红.偏微分方程数值解法.武汉:华中科技大学出版社, 2000
15冯康,余德浩.自然边界归化与区域分解.冯康文集,国防工业出版社, 1994:367- 371
16 K. Feng, D. H. Yu. Canonical Intergral Equation of Elliptic Boundary Value Problem and Their Numerical Solutions. Proc.of China-France Symp.on FEM(1982,Beijing), Science Press, Beijing, 1983,211-252
17 I. M. Gelfand, G. E. Shilov. Generalized Functions. Academic Press, New York, 1964
18余德浩.自然边界元的数学理论.北京:科学出版社, 1993:86-91
19 D. Yu. Natural Boundary Integral Method and Its Application. Beijing and Dordrecht:Science Press and Kluwer Academic Publishers, 2002
20沈有建.奇异积分方程的小波方法及其应用.中山大学博士学位论文, 2003
21祝家麟.椭圆边值问题的边界元分析.北京:科学出版社, 1991
22陈亚浙,吴兰成.数学物理方程讲义.北京:高等教育出版社, 2000
23崔锦泰.小波分析导论.西安:西安交通大学出版社, 1997
24徐长发,李国宽.实用小波方法.武汉:华中科技大学出版社, 2001
25 G. W. Wei. Quasi Wavelets and Quasi Interpolating Wavelets. Chem. Phys. Lett., 1998,296(3-4):215-222
26 G. W. Wei. Discrete Singular Convolution for the Solution of the Fokker-Planck Equations. J. Chem. Phy., 1999,110:8930-8942
27 G. W. Wei. A unified Approach for the Solution of the Fokker-Planck Equations. J. Phys. A., 2000,33:4935-4953
28 G. W. Wei, D. S. Zhang, S. C. Althorpe. Wavelet-distributed Approximating Method for Solving the Navier-Stokes Equation. Computer Physics Communications, 1998,115:18-24
29 D. C. Wan, G. W. Wei. The Study of Quasi Wavelets Based Numerical Method Applied to Burgers’equations. Appl. Math. Mech. Engl., 2000,21:1099-1110
30成礼智,王红霞,罗勇.小波的理论与应用.科学出版社, 2004
31 S.Mallat. Multiresolution Approximation and Wavelets. Trans of American Math Soc 1989,315:69-88
32 Y.Meyer. Wavelets:Algorithms and Application. SIAM, 1993
33 S.Mallat.信号处理的小波导引.杨力华等译.北京:机械工业出版社, 2002
34 J. Hadamard. Lectures on Cauchy’s Problems in Linear Partial Differential Equtions. Yale Uni.Press. New Haven, 1923
35陈文胜,张永虎.调和方程自然边界元的Shannon小波方法.高等学校计算数学学报, 2001,2:125-134
36徐耀群,孙明. Shannon小波混沌神经网络及其TSP问题的求解.控制理论与应用, 2008,25(3):574-577
37陈文胜.非平稳小波基与自然积分方程数值解.中山大学博士学位论文, 1998
38 R. Tao, B. Z. Li,Y. Wang. Spectral Analysis and Reconstruction for Periodic Nonuniformly Sampled Signals in Frac-tional Fourier Domain. IEEE Trans Signal Process, 2007, 55 (7) :3541-3547
39 G. W. Wei, Y. B. Zhao, Y. Xiang. Discrete Singular Convolution and Application to the Analysis of Plates with Internal Support. Part 1:Theory and algorithm. Int.J. Numer.Methods.Engrg., 2002,55:913-946
40万德成,韦国伟.用拟小波方法数值求解Burgers方程.应用数学和力学, 2000,21: 991-1001
41宋玲玲,缪守莉.测不准原理的分析.安徽农业技术师范学院学报, 1999,3:91-92
42 L. W. Qian, G. W. Wei. A Note on Regularized Shannon’s Sampling Formulae. Arxiv: math. NA/0005003, 2000,1:1-9
43 E. Quak. Trigonometric wavelets for Hermite interpolation. Math. Comput, 1996,65: 683-722
44 W. Dahmen, B. Han, R. Q. Jia, A. Kunoth. Biorthogonal Multiwavelets on the Interval: Cubic Hermite Splines. Constr. Approx., 2000,16:221-259
45 V. strela, G. Strang. Finte element multiwavelets. In Wavelet Applications in Signal and Image. ProcessingⅡ,SPIE Proceedings Volume 2303(A.F.Laine and M.A.Unser. eds). Society of Photo-Optical Industrial Engineers.Bellingham.WA.1994.202-213.
46徐长发.对数型奇异核小波展开的递推算法.华中理工大学学报, 1998,5:110-112
47 J. Wang. Spline Wavelets in Numerical Resolution of Partial Differenteal Equations. International Conference on Wavelet Analysis and its Application.AMS/IP Studies In Advanced Mathematics, 2002,25
48 J. C. Xu, W. C. Shann. Galerkin-wavelet Methods for Two-point Boundary Value Problems. Numer. Math. Anal, 1993,24:246-262
49吴勃英,何耀东. H2(1)空间中的离散小波变换.计算数学, 1999,21(2):209-214.
50 K. Feng. Asymptotic Radiation Conditions for Reduced Wave Eqution. J. Com. Math., 1984,2(2):130-138
51 W. Dahmen. Wavelet Methods for PDFs-some Recent Developments. J.Compur.Appl. Math., 2001,128:133-185
52余德浩.椭圆边界上的自然积分算子及各向异性外问题的耦合算法.计算数学, 2002, 24(3):375-384.
53 K. Amaratunga, J. R. Williams. Wavelet-Galerkin solution for one-dimensional partial differential equations. International Journal for Numerical Method in Engineering, 1994,37(16):2703-2716
54 R. P. Gilbert, W. Lin. Wavelet Solutions for Time Harmonic Acoustic Waves in a Finite Ocean. Journal of Computational Acoustic,1993,1(1):31-60
55 Z. Chen, C. A. Micchelli, Y. Xu. Fast Collocation Methods for Second Kind Integral Equations. Siam J. Numer. Anal, 2002,40(1):344-375
56邱建霞.增次广义Vandermonde矩阵.大学数学, 2005,3
57韩建民.广义Vandermonde行列式的计算.宝鸡文理学院学报, 2005,1
58 C. K. Chui. Wavelets Theory Algorithms and Applica-tions. California:Academic Press Inc, 1994
59郑小洋.小波系数与尺度的关系.重庆工学院学报, 2008,2
60祝强.奇异性检测中小波基的选取.山西科技大学学报, 2007,3
2王勖成,邵敏.有限单元法基本原理和数值方法(第2版).北京:清华大学出版社, 1997:1-2
3谢贻权,何福保.弹性和塑性力学中的有限单元法.北京:机械工业出版社, 1981:1-5
4曹国金,姜弘道.无单元法研究和应用现状及动态.力学进展, 2002,32(4):526-53
5 Brebbia, C. A. ed. Recent Advances in Boundary Element Methods. Pentech Press, London, 1978
6 Brebbia, C. A. ed. New Development in Boundary Element Methods. CML Publications, Southampton, 1980
7 Brebbia, C. A. ed. Boundary Element Methods. Springer-Velag, Berlin, 1981
8嵇醒,臧跃龙,程玉民.边界元进展及通用程序.上海:同济大学出版社, 1997
9杜庆华.边界积分方程方法—边界元法.北京:高等教育出版社, 1989
10陈传璋.积分方程论及其应用.北京:科学技术文献出版社, 1987
11路见可,钟寿国.积分方程论.北京:高等教育出版社, 1990
12冯康.论微分与积分方程及有限与无限元.计算数学, 1980,2(1):100-105
13徐长发.科技应用中的微分变分模型.武汉:华中科技大学出版社, 2004
14徐长发,李红.偏微分方程数值解法.武汉:华中科技大学出版社, 2000
15冯康,余德浩.自然边界归化与区域分解.冯康文集,国防工业出版社, 1994:367- 371
16 K. Feng, D. H. Yu. Canonical Intergral Equation of Elliptic Boundary Value Problem and Their Numerical Solutions. Proc.of China-France Symp.on FEM(1982,Beijing), Science Press, Beijing, 1983,211-252
17 I. M. Gelfand, G. E. Shilov. Generalized Functions. Academic Press, New York, 1964
18余德浩.自然边界元的数学理论.北京:科学出版社, 1993:86-91
19 D. Yu. Natural Boundary Integral Method and Its Application. Beijing and Dordrecht:Science Press and Kluwer Academic Publishers, 2002
20沈有建.奇异积分方程的小波方法及其应用.中山大学博士学位论文, 2003
21祝家麟.椭圆边值问题的边界元分析.北京:科学出版社, 1991
22陈亚浙,吴兰成.数学物理方程讲义.北京:高等教育出版社, 2000
23崔锦泰.小波分析导论.西安:西安交通大学出版社, 1997
24徐长发,李国宽.实用小波方法.武汉:华中科技大学出版社, 2001
25 G. W. Wei. Quasi Wavelets and Quasi Interpolating Wavelets. Chem. Phys. Lett., 1998,296(3-4):215-222
26 G. W. Wei. Discrete Singular Convolution for the Solution of the Fokker-Planck Equations. J. Chem. Phy., 1999,110:8930-8942
27 G. W. Wei. A unified Approach for the Solution of the Fokker-Planck Equations. J. Phys. A., 2000,33:4935-4953
28 G. W. Wei, D. S. Zhang, S. C. Althorpe. Wavelet-distributed Approximating Method for Solving the Navier-Stokes Equation. Computer Physics Communications, 1998,115:18-24
29 D. C. Wan, G. W. Wei. The Study of Quasi Wavelets Based Numerical Method Applied to Burgers’equations. Appl. Math. Mech. Engl., 2000,21:1099-1110
30成礼智,王红霞,罗勇.小波的理论与应用.科学出版社, 2004
31 S.Mallat. Multiresolution Approximation and Wavelets. Trans of American Math Soc 1989,315:69-88
32 Y.Meyer. Wavelets:Algorithms and Application. SIAM, 1993
33 S.Mallat.信号处理的小波导引.杨力华等译.北京:机械工业出版社, 2002
34 J. Hadamard. Lectures on Cauchy’s Problems in Linear Partial Differential Equtions. Yale Uni.Press. New Haven, 1923
35陈文胜,张永虎.调和方程自然边界元的Shannon小波方法.高等学校计算数学学报, 2001,2:125-134
36徐耀群,孙明. Shannon小波混沌神经网络及其TSP问题的求解.控制理论与应用, 2008,25(3):574-577
37陈文胜.非平稳小波基与自然积分方程数值解.中山大学博士学位论文, 1998
38 R. Tao, B. Z. Li,Y. Wang. Spectral Analysis and Reconstruction for Periodic Nonuniformly Sampled Signals in Frac-tional Fourier Domain. IEEE Trans Signal Process, 2007, 55 (7) :3541-3547
39 G. W. Wei, Y. B. Zhao, Y. Xiang. Discrete Singular Convolution and Application to the Analysis of Plates with Internal Support. Part 1:Theory and algorithm. Int.J. Numer.Methods.Engrg., 2002,55:913-946
40万德成,韦国伟.用拟小波方法数值求解Burgers方程.应用数学和力学, 2000,21: 991-1001
41宋玲玲,缪守莉.测不准原理的分析.安徽农业技术师范学院学报, 1999,3:91-92
42 L. W. Qian, G. W. Wei. A Note on Regularized Shannon’s Sampling Formulae. Arxiv: math. NA/0005003, 2000,1:1-9
43 E. Quak. Trigonometric wavelets for Hermite interpolation. Math. Comput, 1996,65: 683-722
44 W. Dahmen, B. Han, R. Q. Jia, A. Kunoth. Biorthogonal Multiwavelets on the Interval: Cubic Hermite Splines. Constr. Approx., 2000,16:221-259
45 V. strela, G. Strang. Finte element multiwavelets. In Wavelet Applications in Signal and Image. ProcessingⅡ,SPIE Proceedings Volume 2303(A.F.Laine and M.A.Unser. eds). Society of Photo-Optical Industrial Engineers.Bellingham.WA.1994.202-213.
46徐长发.对数型奇异核小波展开的递推算法.华中理工大学学报, 1998,5:110-112
47 J. Wang. Spline Wavelets in Numerical Resolution of Partial Differenteal Equations. International Conference on Wavelet Analysis and its Application.AMS/IP Studies In Advanced Mathematics, 2002,25
48 J. C. Xu, W. C. Shann. Galerkin-wavelet Methods for Two-point Boundary Value Problems. Numer. Math. Anal, 1993,24:246-262
49吴勃英,何耀东. H2(1)空间中的离散小波变换.计算数学, 1999,21(2):209-214.
50 K. Feng. Asymptotic Radiation Conditions for Reduced Wave Eqution. J. Com. Math., 1984,2(2):130-138
51 W. Dahmen. Wavelet Methods for PDFs-some Recent Developments. J.Compur.Appl. Math., 2001,128:133-185
52余德浩.椭圆边界上的自然积分算子及各向异性外问题的耦合算法.计算数学, 2002, 24(3):375-384.
53 K. Amaratunga, J. R. Williams. Wavelet-Galerkin solution for one-dimensional partial differential equations. International Journal for Numerical Method in Engineering, 1994,37(16):2703-2716
54 R. P. Gilbert, W. Lin. Wavelet Solutions for Time Harmonic Acoustic Waves in a Finite Ocean. Journal of Computational Acoustic,1993,1(1):31-60
55 Z. Chen, C. A. Micchelli, Y. Xu. Fast Collocation Methods for Second Kind Integral Equations. Siam J. Numer. Anal, 2002,40(1):344-375
56邱建霞.增次广义Vandermonde矩阵.大学数学, 2005,3
57韩建民.广义Vandermonde行列式的计算.宝鸡文理学院学报, 2005,1
58 C. K. Chui. Wavelets Theory Algorithms and Applica-tions. California:Academic Press Inc, 1994
59郑小洋.小波系数与尺度的关系.重庆工学院学报, 2008,2
60祝强.奇异性检测中小波基的选取.山西科技大学学报, 2007,3
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