弹性力学中微分方程的小波解法
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摘要
随着计算机技术的发展和完善以及实际应用的需要,科学和工程中的所有计算问题与计算机已经密不可分.弹性力学中的微分方程,由于其阶数较高,而且很多方程还无法找到其精确解,所以关于此类方程的数值解在工程中就显得尤为重要.小波分析作为数值分析的一种强有力的工具是二十世纪七十年代发展起来的一门新兴的数学学科,现在已广泛地应用到求解微分方程和积分方程中.本文对弹性力学中的微分方程的小波方法进行研究,得到了一系列结果.全文共分三章,每章的主要内容如下:
第一章简述本文所用小波分析的一些基础知识以及弹性力学的基本理论,对目前小波分析在微分方程中的应用现状进行探讨,提出应用正交小波基对微分方程进行求解的可行性.
第二章研究求解弹性地基梁的小波方法,主要分为四个部分进行说明:Winkler地基上梁的微分控制方程;线性Legendre多小波的定义及其性质和积分矩阵;用线性Legendre多小波求解Winkler地基上梁的方程;数值举例.
第三章研究求解弹性地基板的小波解法,主要分为五个部分:双参数弹性地基上板的微分控制方程:一维Sine-Cosine小波的定义及其微分矩阵;利用张量积构造二维的Sine-Cosine小波基并推导出其相应的两个微分矩阵;用Sine-Cosine小波求解双参数弹性地基上四边自由矩形板的挠度问题;数值举例.
With the development of computer technology and the requirements of practical application, all the problems arising in science and engineering field are closed related with computers. Due to the high rank of the differential equations in elastic mechanics and the difficulties of finding accurate solutions for these equations, the numerical solutions to these equations are especially important in engineering. As a powerful tool of numerical analysis, wavelet analysis was developed from 1970s and has been widely used to solve differential equations and integral equations. This thesis is focused on the research of wavelet methods to solve differential equations in elastic mechanics and a series of results are obtained. It is divided into 3 chapters and the main ideas of each chapter are as follows:
In Chapter 1, some general knowledge of wavelet analysis and basic theories of elastic mechanics are briefly discussed, the current applications of wavelet analysis in differential equations are investigated, and the feasibility of using orthogonal wavelets to solve differential equations is also proposed.
In Chapter 2, the wavelet method for solving problems of beams on elastic groundsill is investigated, and four aspects are illustrated: differential control equation of a beam on Winkler groundsill; the definition and property and integral operational matrix of the linear Legendre multi-wavelet; the numerical solutions to the differential control equation of a beam on Winkler groundsill solved by linear Legendre multi-wavelet; numerical examples.
In Chapter 3. the wavelet solutions to problems of laminas on elastic groundsill is researched and is divided into five sections: differential control equation of a plate on the two-parameter elastic groundsill; the definition and differential operational matrix of the Sine-Cosine wavelet: the construction of the two-dimensional Sine-Cosine wavelet using tensor product and two differential operational matrices; the numerical solutions to the differential equation of a rectangular plate on the two-parameter elastic groundsill solved by two-dimensional Sine-Cosine wavelet; numerical example.
第一章简述本文所用小波分析的一些基础知识以及弹性力学的基本理论,对目前小波分析在微分方程中的应用现状进行探讨,提出应用正交小波基对微分方程进行求解的可行性.
第二章研究求解弹性地基梁的小波方法,主要分为四个部分进行说明:Winkler地基上梁的微分控制方程;线性Legendre多小波的定义及其性质和积分矩阵;用线性Legendre多小波求解Winkler地基上梁的方程;数值举例.
第三章研究求解弹性地基板的小波解法,主要分为五个部分:双参数弹性地基上板的微分控制方程:一维Sine-Cosine小波的定义及其微分矩阵;利用张量积构造二维的Sine-Cosine小波基并推导出其相应的两个微分矩阵;用Sine-Cosine小波求解双参数弹性地基上四边自由矩形板的挠度问题;数值举例.
With the development of computer technology and the requirements of practical application, all the problems arising in science and engineering field are closed related with computers. Due to the high rank of the differential equations in elastic mechanics and the difficulties of finding accurate solutions for these equations, the numerical solutions to these equations are especially important in engineering. As a powerful tool of numerical analysis, wavelet analysis was developed from 1970s and has been widely used to solve differential equations and integral equations. This thesis is focused on the research of wavelet methods to solve differential equations in elastic mechanics and a series of results are obtained. It is divided into 3 chapters and the main ideas of each chapter are as follows:
In Chapter 1, some general knowledge of wavelet analysis and basic theories of elastic mechanics are briefly discussed, the current applications of wavelet analysis in differential equations are investigated, and the feasibility of using orthogonal wavelets to solve differential equations is also proposed.
In Chapter 2, the wavelet method for solving problems of beams on elastic groundsill is investigated, and four aspects are illustrated: differential control equation of a beam on Winkler groundsill; the definition and property and integral operational matrix of the linear Legendre multi-wavelet; the numerical solutions to the differential control equation of a beam on Winkler groundsill solved by linear Legendre multi-wavelet; numerical examples.
In Chapter 3. the wavelet solutions to problems of laminas on elastic groundsill is researched and is divided into five sections: differential control equation of a plate on the two-parameter elastic groundsill; the definition and differential operational matrix of the Sine-Cosine wavelet: the construction of the two-dimensional Sine-Cosine wavelet using tensor product and two differential operational matrices; the numerical solutions to the differential equation of a rectangular plate on the two-parameter elastic groundsill solved by two-dimensional Sine-Cosine wavelet; numerical example.
引文
[1]Polikar R.The wavelet tutorial,http://users.rowan.edu/polikar/WAVELETS /WTtutorial.html.
[2]I.Daubechies.小波十讲.李建平.杨万年译[M].北京:国防工业出版社.2004.
[3]Mallat S.G.A theory for multiresolution signal decomposition:the wavelet representation.IEEE Transactions on Pattern Analysis and Machine Intelligence.1989,11(7):674-693.
[4]孙延奎.小波分析及其应用[M].北京:机械工业出版社.2005.
[5]飞思科技产品研发中心.小波分析理论与MATLAB 7实现[M].北京:电子工业出版社.2005.
[6]石智.小波理论[M].西安:西安建筑科技大学.2004.12.
[7]戴道清.小波分析讲义.http://bbs.bossh.net/.2003.
[8]C.西得尼·伯罗斯,拉米什 A.戈皮那思,郭海涛.小波与小波变换导论[M].北京:机械工业出版社.2005.
[9]徐芝纶.弹性力学[M].北京:高等教育出版社.1990.
[10]寿楠椿.弹性薄板弯曲[M].北京:高等教育出版社.1986.
[11]朱加铭,欧贵宝,费纪生.弹性地基上矩形板弯曲的SS级数解[J].哈尔滨工程大学学报,(1995),16(2),66-73.
[12]朱加铭.弹性地基上矩形板弯曲的CC型级数解[J].应用数学与力学,(1995),16(6),553-561.
[13]钟阳,张永山.弹性地基上矩形薄板问题的Hamilton正则方程及解析解[J].固体力学学报,(2005),26(3),325-327.
[14]何芳社,黄义,郭雅云.双参数弹性地基板的动力问题[J].西安建筑科技大学学报,(2006),38(1),130-134.
[15]钟阳,王国新,孙爱民.弹性地基上四边自由矩形薄板振动分析有限积分变换法[J].大连理工大学学报,(2007),47(1),73-77.
[16]王春玲,黄义,贾继红.弹性半空间地基上四边自由矩形板稳态振动的解析解[J].Applied Mathematics and Mechanics(English Edition)(2006)27,111-119.
[17]黄义.弹性力学基础及有限单元法[M].北京:冶金工业出版社.1983.
[18]张雷顺,王俊林等,弹性力学及有限单元法[M].郑州:黄河水利出版社.2005.
[19]王元汉,邱先敏,张佑启.弹性地基板的等参有限元法计算[J].岩土工程学报,(1998),20(4),7-11.
[20]Teresa Reginska.Sideways heat equations and wavelets[J].Computational and Applied Mathematics.(1995)63,209-214.
[21]Jose Roberto Linhares de Mattos,Ernesto Prado Lopes.A wavelet Galerkin method applied to partial differential equations with variable coefficients[J].Electronic Journal of Differential Equations.(2003)10,211-225.
[22]U.Lepik.Numerical solution of differential equations using Haar wavelets [J].Mathematics and Computers in simulation.(2005)68,127-143.
[23]Bernd Fischer,Jurgen Prestin.Wavelets based on orthogonal polynomials [J].Mathematics of computation(1997)66,1593-1618
[24]F.Khellat,S.A.Yousefi.The linear Legendre mother wavelets operational matrix of integration and its application[J].Journal of the Franklin Institute.(2006)343,181-190.
[25]Xufeng Shang,Danfu Han.Numerical solution of Fredholm integral equations of the first kind by using linear Legendre multi-wavelets[J].Applied Mathematics and Computation.(2007)191:440-444.
[26]Sohrab Ali Yousefi.Legendre wavelets method for solving differential equations of Lane-Emden type[J].Applied Mathematics and Computation.(2006)181,1417-1422.
[27]David E.Newland.Harmonic wavelet analysis[J].Proc.R.Soc.Lond.A.(1993)443,203-225.
[28]C.Cattani.Harmonic wavelets towards the solution of nonlinear PDE [J].Computers and Mathematics with Applications.(2005)50,1191-1210.
[29]傅晓玲.插值小波自适应求解双曲型偏微分方程[J].北方工业大学学报.(2000)12,31-35.
[30]N.M.Bujurke,C.S.Salimath,R.B.Kudenatti,S.C.Shiralashetti.A waveletmultigrid method to solve elliptic partial differential equations[J].Applied Mathematics and Computation.(2007)185,667-680.
[31]Mohamed EI-Gamel.Comparison of the solutions obtained by Adomian decomposition and wavelet-Galerkin methods bo boundary -value problems[J].Applied Mathematics and Computation.(2007)186,652-664.
[32]M.Thamban nair.Wavelet-Galerkin method[M].QIP-short Term Course.2004.
[33]Wen-Sheng Chen,Wei Lin.Galerkin trigonnmetric wavelet methods for the natural boundary integral equations[J].Applied Mathematics and Computation.(2001)121,75-92.
[34]S.Lazaar,Pj.Ponenti,J.Liandrat,Ph.Tchamitchian.Wavelet algorithms for numerical resolution of partial differential equations[J].Computer mothods in applied mechanics and engineering.(1994)116,309-314.
[35]G.Chiavassa,M.Guichaoua,J.Liandrat.Two adaptive wavelet algorithms for non-linear parabolic partial differential equations[J].Computers and Fluids.(2002)31,467-480.
[36]Ryo Ikehata.Local energy decay for linear wave equations with variable coefficients [J].Journal of Mathematical Analysis and Applications.(2005)306,330-348.
[37]吴勃英,邓中兴.基于双正交小波基的热传导方程数值解法[J].哈尔滨理工大学学报.(1999)4,7-12.
[38]M.Tavassoli Kajani,M.Ghasemi,E.Babolian.Numerical solution of linear integro-differential equation by using Sine-Cosine wavelet[J].Applied Mathemaics and Computation.(2006)180,569-574.
[39]M.Razzaghi,S.Yousefi.Sine-Cosine wavelets operational matrix of integration and its applications in the calculus of variations[J].International Journal of Systems Science.(2002)33,805-810.
[40]M.Razzaghi,S.Yousefi.Legendre wavelets direct method for variational problem [J].Mathematics and Computers in Simulation.(2000)53,185-192.
[41]S.Yousefi,A.Banifatemi.Numerical solution of Fredholm integral equations by using CAS wavelets[J].Applied Mathematics and Computation.(2006)183,458-463.
[42]S.A.Yousefi.Numerical solution of Abel's integral equation by using Legendre wavelets[J].Applied Mathematics and Computation.(2006)175,574-580.
[43]Xufeng Shang,Danfu Han.Numerical solution of Fredholm integral equations of the first kind by using linear Legendre multi-wavelets[J].Applied Mathematics and Computation(2007)191,440-444.
[44]E.Douka,S.Loutridis,A.Trochidis.Crack identification in beams using wavelet analysis[J].International Journal of Solids and Structures.(2003)40,3557-3579.
[45]Quan Wang,Xiaomin Deng.Damage detection with spatial wavelets [J].International Journal of Solids and Structures(1999)36,3443-3468.
[46]Angelo Gentile,Arcangelo Messina.On the continuous wavelet transforms applied to discrete vibrational data for detecting open cracks in damaged beams [J].International Journal of Solids and Structures.(2003)40,295-315.
[47]马军星,王进.弹性地基梁小波有限元分析[J].系统仿真学报(2007)19,2183-2185.
[48]Byeong Hwa Kim,Heedai Kim,Taehyo Park.Nondestructive damage evaluation of plate using the multi-resolution analysis of two-dimensional Haar wavelet [J].Journal of Sound and Vibration.(2006)292,82-104.
[49]塞尔瓦杜雷 APS.土与结构共同作用的弹性分析[M].范文田译,北京:铁道出版社.1984.
[50]韩建刚,石智,黄义,暴瑛,有限长梁的B-样条小波解[J],西安科技学院学报,(2003)23(4),481-484.
[51]韩建国,石智,黄义,暴瑛.M-尺度函数及对弹性地基梁的求解[J].甘肃工业大学学报,(2003),29(4),107-109.
[52]王克林,黄义.弹性地基上的自由矩形板[J].计算结构力学与应用,(1985),2(2),47-58.
[53]季求知,曲庆璋.双参数弹性地基上四边自由矩形板问题[J].青岛建筑工程学院学报,(1996),17(3),85-93.
[54]生跃,黄义.双参数弹性地基上自由边矩形板[J].应用数学和力学,(1987),8(4),317-329.
[55]张威.MATLAB基础与编程入门[M].西安:西安电子科技大学出版社.2004.
[56]王沫然.MATLAB与科学计算(第2版)[M].北京:电子工业出版社.2006.
[2]I.Daubechies.小波十讲.李建平.杨万年译[M].北京:国防工业出版社.2004.
[3]Mallat S.G.A theory for multiresolution signal decomposition:the wavelet representation.IEEE Transactions on Pattern Analysis and Machine Intelligence.1989,11(7):674-693.
[4]孙延奎.小波分析及其应用[M].北京:机械工业出版社.2005.
[5]飞思科技产品研发中心.小波分析理论与MATLAB 7实现[M].北京:电子工业出版社.2005.
[6]石智.小波理论[M].西安:西安建筑科技大学.2004.12.
[7]戴道清.小波分析讲义.http://bbs.bossh.net/.2003.
[8]C.西得尼·伯罗斯,拉米什 A.戈皮那思,郭海涛.小波与小波变换导论[M].北京:机械工业出版社.2005.
[9]徐芝纶.弹性力学[M].北京:高等教育出版社.1990.
[10]寿楠椿.弹性薄板弯曲[M].北京:高等教育出版社.1986.
[11]朱加铭,欧贵宝,费纪生.弹性地基上矩形板弯曲的SS级数解[J].哈尔滨工程大学学报,(1995),16(2),66-73.
[12]朱加铭.弹性地基上矩形板弯曲的CC型级数解[J].应用数学与力学,(1995),16(6),553-561.
[13]钟阳,张永山.弹性地基上矩形薄板问题的Hamilton正则方程及解析解[J].固体力学学报,(2005),26(3),325-327.
[14]何芳社,黄义,郭雅云.双参数弹性地基板的动力问题[J].西安建筑科技大学学报,(2006),38(1),130-134.
[15]钟阳,王国新,孙爱民.弹性地基上四边自由矩形薄板振动分析有限积分变换法[J].大连理工大学学报,(2007),47(1),73-77.
[16]王春玲,黄义,贾继红.弹性半空间地基上四边自由矩形板稳态振动的解析解[J].Applied Mathematics and Mechanics(English Edition)(2006)27,111-119.
[17]黄义.弹性力学基础及有限单元法[M].北京:冶金工业出版社.1983.
[18]张雷顺,王俊林等,弹性力学及有限单元法[M].郑州:黄河水利出版社.2005.
[19]王元汉,邱先敏,张佑启.弹性地基板的等参有限元法计算[J].岩土工程学报,(1998),20(4),7-11.
[20]Teresa Reginska.Sideways heat equations and wavelets[J].Computational and Applied Mathematics.(1995)63,209-214.
[21]Jose Roberto Linhares de Mattos,Ernesto Prado Lopes.A wavelet Galerkin method applied to partial differential equations with variable coefficients[J].Electronic Journal of Differential Equations.(2003)10,211-225.
[22]U.Lepik.Numerical solution of differential equations using Haar wavelets [J].Mathematics and Computers in simulation.(2005)68,127-143.
[23]Bernd Fischer,Jurgen Prestin.Wavelets based on orthogonal polynomials [J].Mathematics of computation(1997)66,1593-1618
[24]F.Khellat,S.A.Yousefi.The linear Legendre mother wavelets operational matrix of integration and its application[J].Journal of the Franklin Institute.(2006)343,181-190.
[25]Xufeng Shang,Danfu Han.Numerical solution of Fredholm integral equations of the first kind by using linear Legendre multi-wavelets[J].Applied Mathematics and Computation.(2007)191:440-444.
[26]Sohrab Ali Yousefi.Legendre wavelets method for solving differential equations of Lane-Emden type[J].Applied Mathematics and Computation.(2006)181,1417-1422.
[27]David E.Newland.Harmonic wavelet analysis[J].Proc.R.Soc.Lond.A.(1993)443,203-225.
[28]C.Cattani.Harmonic wavelets towards the solution of nonlinear PDE [J].Computers and Mathematics with Applications.(2005)50,1191-1210.
[29]傅晓玲.插值小波自适应求解双曲型偏微分方程[J].北方工业大学学报.(2000)12,31-35.
[30]N.M.Bujurke,C.S.Salimath,R.B.Kudenatti,S.C.Shiralashetti.A waveletmultigrid method to solve elliptic partial differential equations[J].Applied Mathematics and Computation.(2007)185,667-680.
[31]Mohamed EI-Gamel.Comparison of the solutions obtained by Adomian decomposition and wavelet-Galerkin methods bo boundary -value problems[J].Applied Mathematics and Computation.(2007)186,652-664.
[32]M.Thamban nair.Wavelet-Galerkin method[M].QIP-short Term Course.2004.
[33]Wen-Sheng Chen,Wei Lin.Galerkin trigonnmetric wavelet methods for the natural boundary integral equations[J].Applied Mathematics and Computation.(2001)121,75-92.
[34]S.Lazaar,Pj.Ponenti,J.Liandrat,Ph.Tchamitchian.Wavelet algorithms for numerical resolution of partial differential equations[J].Computer mothods in applied mechanics and engineering.(1994)116,309-314.
[35]G.Chiavassa,M.Guichaoua,J.Liandrat.Two adaptive wavelet algorithms for non-linear parabolic partial differential equations[J].Computers and Fluids.(2002)31,467-480.
[36]Ryo Ikehata.Local energy decay for linear wave equations with variable coefficients [J].Journal of Mathematical Analysis and Applications.(2005)306,330-348.
[37]吴勃英,邓中兴.基于双正交小波基的热传导方程数值解法[J].哈尔滨理工大学学报.(1999)4,7-12.
[38]M.Tavassoli Kajani,M.Ghasemi,E.Babolian.Numerical solution of linear integro-differential equation by using Sine-Cosine wavelet[J].Applied Mathemaics and Computation.(2006)180,569-574.
[39]M.Razzaghi,S.Yousefi.Sine-Cosine wavelets operational matrix of integration and its applications in the calculus of variations[J].International Journal of Systems Science.(2002)33,805-810.
[40]M.Razzaghi,S.Yousefi.Legendre wavelets direct method for variational problem [J].Mathematics and Computers in Simulation.(2000)53,185-192.
[41]S.Yousefi,A.Banifatemi.Numerical solution of Fredholm integral equations by using CAS wavelets[J].Applied Mathematics and Computation.(2006)183,458-463.
[42]S.A.Yousefi.Numerical solution of Abel's integral equation by using Legendre wavelets[J].Applied Mathematics and Computation.(2006)175,574-580.
[43]Xufeng Shang,Danfu Han.Numerical solution of Fredholm integral equations of the first kind by using linear Legendre multi-wavelets[J].Applied Mathematics and Computation(2007)191,440-444.
[44]E.Douka,S.Loutridis,A.Trochidis.Crack identification in beams using wavelet analysis[J].International Journal of Solids and Structures.(2003)40,3557-3579.
[45]Quan Wang,Xiaomin Deng.Damage detection with spatial wavelets [J].International Journal of Solids and Structures(1999)36,3443-3468.
[46]Angelo Gentile,Arcangelo Messina.On the continuous wavelet transforms applied to discrete vibrational data for detecting open cracks in damaged beams [J].International Journal of Solids and Structures.(2003)40,295-315.
[47]马军星,王进.弹性地基梁小波有限元分析[J].系统仿真学报(2007)19,2183-2185.
[48]Byeong Hwa Kim,Heedai Kim,Taehyo Park.Nondestructive damage evaluation of plate using the multi-resolution analysis of two-dimensional Haar wavelet [J].Journal of Sound and Vibration.(2006)292,82-104.
[49]塞尔瓦杜雷 APS.土与结构共同作用的弹性分析[M].范文田译,北京:铁道出版社.1984.
[50]韩建刚,石智,黄义,暴瑛,有限长梁的B-样条小波解[J],西安科技学院学报,(2003)23(4),481-484.
[51]韩建国,石智,黄义,暴瑛.M-尺度函数及对弹性地基梁的求解[J].甘肃工业大学学报,(2003),29(4),107-109.
[52]王克林,黄义.弹性地基上的自由矩形板[J].计算结构力学与应用,(1985),2(2),47-58.
[53]季求知,曲庆璋.双参数弹性地基上四边自由矩形板问题[J].青岛建筑工程学院学报,(1996),17(3),85-93.
[54]生跃,黄义.双参数弹性地基上自由边矩形板[J].应用数学和力学,(1987),8(4),317-329.
[55]张威.MATLAB基础与编程入门[M].西安:西安电子科技大学出版社.2004.
[56]王沫然.MATLAB与科学计算(第2版)[M].北京:电子工业出版社.2006.