桥梁结构分析的广义变分原理—Daubechies条件小波法研究
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摘要
小波理论是20世纪80年代出现的一个新兴数学分支，是近年来在工具及方法上的重大突破，它已被广泛地应用在科学技术和工程计算等各个领域。其中，以Daubechies小波使用最广，影响最为深远，在解决诸如应力大梯度等奇异问题中，较其它小波函数有明显的优势。基于Daubechies小波的小波Ritz法、小波Galerkin法以及小波有限元法近年来一直受到国内外学者的高度重视。但直到目前为止，小波理论在结构工程中的应用还很不完善，尤其是Daubechies小波在诸如联系系数的计算精度不高、位移转换矩阵奇异、高阶消失矩基函数无法使用以及高精度小尺度函数空间难以应用等方面遇到很大困难。因此，如何应用小波理论，特别是Daubechies小波进行结构工程计算，提高计算精度，克服上述缺陷，发挥其独特的优势，具有重要的理论意义和显著的实用价值。
本文在系统研究小波数值计算方法及已有小波有限元的基础上，以Daubechies小波为切入点，以桥梁结构工程计算为主要应用方向，以传统Ritz法和Galerkin法为主要手段，将小波分析的多分辨思想与条件变分原理相结合，成功构造出可直接用于工程结构分析的全求解域条件小波Ritz法和条件小波Galerkin法，并进一步构造出基于条件变分和二类变量广义变分的单元刚度矩阵的条件小波有限元法。
本论文首先简要介绍了小波理论的发展现状及其在数值计算领域的应用情况，并系统介绍了小波分析的基础理论及Daubechies小波的数学特性，推导了Daubechies小波尺度函数、小波函数及其相关导数、积分、内积和现有联系系数的计算过程，阐明了现有联系系数计算方法中存在的问题，提出了提高联系系数计算精度的有效方法。
现有的Daubechies小波有限元法中，为方便边界条件的引入，均在小波待定系数与单元内部节点位置之间设置了位移转换矩阵，从而将小波有限元问题转化为常规有限元问题，方便了小波单元的使用。但也正是由于位移转换矩阵的存在，使得Daubechies小波单元难以实现高精度计算，在结构工程计算方面的应用受到限制。本文在分析传统Daubechies小波有限元法所存在问题的基础上，结合传统Ritz法、Galerkin法和广义变分原理，首次提出了条件小波Ritz法和条件小波Galerkin法，并构造出基于条件变分和二类变量广义变分的单元刚度矩阵的条件小波有限元法和条件小波混合有限元法，构建出条件小波单元求解矩阵，给出条件小波总体刚度求解矩阵的组装方法。从而避免了由于转换矩阵奇异而造成精度下降且计算结果不易收敛的问题，提升了小波Ritz法和小波Galerkin法的求解精度，使小波分析的“显微”特性得以充分发挥，并为应力大梯度问题和工程奇异问题的有效求解提供了强有力的计算手段。同时，编制典型算例，从各个方面对条件小波分析方法在计算精度、稳定性、求解速度以及在处理应力大梯度等奇异问题上的有效性进行全面测试。
桥梁桩基础是桥梁工程中典型构件，其内力计算的准确与否将直接关系到整个桥梁结构的安全。本文针对桥梁桩基础计算模型的特点，首次提出并推导了一类可用于桩基础计算的联系系数，同时首次将二类变量的混合能量原理引入Daubechies小波小波有限元法中，以进一步提高结构内力的求解精度。最后，利用上述结果，对桥梁桩基础的典型模型进行了计算。
本文还编制了大量的数值计算子程序和计算例程，几乎囊括了Daubechies小波有关结构工程数值计算的所有方面，这些程序的编制，不仅验证了本文的相关结论，同时，也为后续进一步拓展Daubechies小波在结构工程数值计算领域的应用空间打下坚实的基础。
Wavelet theory is a new developing branch in mathematics appeared in the 1980's. In recent years, it is considered as an important breakthrough in methods and tools. And it has been employed in various fields of science and technology and engineering calculation. Especially, Daubechies wavelet has been used the most widely and has the most far-reaching influence because it has obvious superiority in solving some odd state problems, for example, high stress gradient. Wavelet Ritz, wavelet Galerkin and wavelet Finite Element Method (FEM) which are based on Daubechies wavelet have been attached high importance by scholars both at home and abroad in recent yeas. But until now, the application of wavelet theory in structure engineering is far from perfect. Quite a lot difficulties exist in the employing of Daubechies wavelet, such as low computation accuracy of connection coefficients, odd state of displacement transformation matrix, unable to use primary function of higher order vanishing moments, and hard to employ small scaling function space of high accuracy. So how to use wavelet theory, especially Daubechies wavelet to conduct structural computation, improve computation accuracy, overcome the above shortcomings, exploit its special advantages, has significant theoretical importance and obvious practical value.
The thesis first briefly introduces the present developing situation of wavelet theory and its application process in numerical calculation; and also systematically introduces the basic theory of wavelet analysis and the mathematic properties of Daubechies wavelet; and then infers the scaling function, wavelet function and correlation derivative, integral, and inner product and the calculating process of existing connection coefficient. Then it clarifies the problem in the existing calculatoin methods of connection coefficients, and also puts forward the effective methods to improve the computation accuracy of connection coefficients.
In the present Daubechies wavelet FEM, the displacement transformation matrix has been set between undetermined wavelet coefficients and element internal nodes for the convenience of leading boundary condition. It will transfer the wavelet finite element problem to normal finite element problem, so make the using of wavelet elements convenient. But just because of the existing of displacement transformation matrix, high accuracy computation of Daubechies wavelet elements is hard to achieve, so its application in structural computation has been greatly limited.
Based on analyzing the problems existing in traditional Daubechies wavelet FEM, combining traditional Ritz method , Galerkin method with generalized variational principle, this thesis for the first time puts forward conditional wavelet Ritz method and conditional wavelet Galerkin method, and constructs conditional wavelet FEM of element stiffness matrix based on conditional variational principle and Hu-Washizu generalized variational principle. Furthermore, the author also constructs stiffness matrix of conditional wavelet and provide the assembling methods of massive stiffness matrix. So it can be avoided that the computation accuracy is lowered and the calculation result is hard to be convergent for the odd state of transformation matrix.
The computation accuracy of wavelet Ritz method and wavelet Galerkin method has been improved so the special 'microscopic' property could be fully expressed. An effective method is offered also to solve the problem of high stress gradient and some other odd state problems in engineering. At the same time, the author presents the typical examples to exam the accuracy, stability, calculating speed and its effectiveness for solving the odd state problems like high stress gradient from all respects.
Pile is a typical member in bridge structure. The precision of calculating its internal force will affect the whole bridge structure's safety. This thesis for the first time presents a new kind of connection coefficient which can be used in computation of pile foundation; and it also leads the Hu-Washizu principle into Daubechies wavelet FEM to increase the computation accuracy of structural internal force. At last it calculates the typical model of bridge pile foundation using the above methods.
The author also writes a great amount of numeric computational sub-programs and programs too, almost include all respects of structural numeric computation using Daubechies wavelet. These programs not only test and verify the correlation results in this thesis, but also lay a strong foundation for expanding the application space of Daubechies wavelet in structural numeric computation.
本文在系统研究小波数值计算方法及已有小波有限元的基础上，以Daubechies小波为切入点，以桥梁结构工程计算为主要应用方向，以传统Ritz法和Galerkin法为主要手段，将小波分析的多分辨思想与条件变分原理相结合，成功构造出可直接用于工程结构分析的全求解域条件小波Ritz法和条件小波Galerkin法，并进一步构造出基于条件变分和二类变量广义变分的单元刚度矩阵的条件小波有限元法。
本论文首先简要介绍了小波理论的发展现状及其在数值计算领域的应用情况，并系统介绍了小波分析的基础理论及Daubechies小波的数学特性，推导了Daubechies小波尺度函数、小波函数及其相关导数、积分、内积和现有联系系数的计算过程，阐明了现有联系系数计算方法中存在的问题，提出了提高联系系数计算精度的有效方法。
现有的Daubechies小波有限元法中，为方便边界条件的引入，均在小波待定系数与单元内部节点位置之间设置了位移转换矩阵，从而将小波有限元问题转化为常规有限元问题，方便了小波单元的使用。但也正是由于位移转换矩阵的存在，使得Daubechies小波单元难以实现高精度计算，在结构工程计算方面的应用受到限制。本文在分析传统Daubechies小波有限元法所存在问题的基础上，结合传统Ritz法、Galerkin法和广义变分原理，首次提出了条件小波Ritz法和条件小波Galerkin法，并构造出基于条件变分和二类变量广义变分的单元刚度矩阵的条件小波有限元法和条件小波混合有限元法，构建出条件小波单元求解矩阵，给出条件小波总体刚度求解矩阵的组装方法。从而避免了由于转换矩阵奇异而造成精度下降且计算结果不易收敛的问题，提升了小波Ritz法和小波Galerkin法的求解精度，使小波分析的“显微”特性得以充分发挥，并为应力大梯度问题和工程奇异问题的有效求解提供了强有力的计算手段。同时，编制典型算例，从各个方面对条件小波分析方法在计算精度、稳定性、求解速度以及在处理应力大梯度等奇异问题上的有效性进行全面测试。
桥梁桩基础是桥梁工程中典型构件，其内力计算的准确与否将直接关系到整个桥梁结构的安全。本文针对桥梁桩基础计算模型的特点，首次提出并推导了一类可用于桩基础计算的联系系数，同时首次将二类变量的混合能量原理引入Daubechies小波小波有限元法中，以进一步提高结构内力的求解精度。最后，利用上述结果，对桥梁桩基础的典型模型进行了计算。
本文还编制了大量的数值计算子程序和计算例程，几乎囊括了Daubechies小波有关结构工程数值计算的所有方面，这些程序的编制，不仅验证了本文的相关结论，同时，也为后续进一步拓展Daubechies小波在结构工程数值计算领域的应用空间打下坚实的基础。
Wavelet theory is a new developing branch in mathematics appeared in the 1980's. In recent years, it is considered as an important breakthrough in methods and tools. And it has been employed in various fields of science and technology and engineering calculation. Especially, Daubechies wavelet has been used the most widely and has the most far-reaching influence because it has obvious superiority in solving some odd state problems, for example, high stress gradient. Wavelet Ritz, wavelet Galerkin and wavelet Finite Element Method (FEM) which are based on Daubechies wavelet have been attached high importance by scholars both at home and abroad in recent yeas. But until now, the application of wavelet theory in structure engineering is far from perfect. Quite a lot difficulties exist in the employing of Daubechies wavelet, such as low computation accuracy of connection coefficients, odd state of displacement transformation matrix, unable to use primary function of higher order vanishing moments, and hard to employ small scaling function space of high accuracy. So how to use wavelet theory, especially Daubechies wavelet to conduct structural computation, improve computation accuracy, overcome the above shortcomings, exploit its special advantages, has significant theoretical importance and obvious practical value.
The thesis first briefly introduces the present developing situation of wavelet theory and its application process in numerical calculation; and also systematically introduces the basic theory of wavelet analysis and the mathematic properties of Daubechies wavelet; and then infers the scaling function, wavelet function and correlation derivative, integral, and inner product and the calculating process of existing connection coefficient. Then it clarifies the problem in the existing calculatoin methods of connection coefficients, and also puts forward the effective methods to improve the computation accuracy of connection coefficients.
In the present Daubechies wavelet FEM, the displacement transformation matrix has been set between undetermined wavelet coefficients and element internal nodes for the convenience of leading boundary condition. It will transfer the wavelet finite element problem to normal finite element problem, so make the using of wavelet elements convenient. But just because of the existing of displacement transformation matrix, high accuracy computation of Daubechies wavelet elements is hard to achieve, so its application in structural computation has been greatly limited.
Based on analyzing the problems existing in traditional Daubechies wavelet FEM, combining traditional Ritz method , Galerkin method with generalized variational principle, this thesis for the first time puts forward conditional wavelet Ritz method and conditional wavelet Galerkin method, and constructs conditional wavelet FEM of element stiffness matrix based on conditional variational principle and Hu-Washizu generalized variational principle. Furthermore, the author also constructs stiffness matrix of conditional wavelet and provide the assembling methods of massive stiffness matrix. So it can be avoided that the computation accuracy is lowered and the calculation result is hard to be convergent for the odd state of transformation matrix.
The computation accuracy of wavelet Ritz method and wavelet Galerkin method has been improved so the special 'microscopic' property could be fully expressed. An effective method is offered also to solve the problem of high stress gradient and some other odd state problems in engineering. At the same time, the author presents the typical examples to exam the accuracy, stability, calculating speed and its effectiveness for solving the odd state problems like high stress gradient from all respects.
Pile is a typical member in bridge structure. The precision of calculating its internal force will affect the whole bridge structure's safety. This thesis for the first time presents a new kind of connection coefficient which can be used in computation of pile foundation; and it also leads the Hu-Washizu principle into Daubechies wavelet FEM to increase the computation accuracy of structural internal force. At last it calculates the typical model of bridge pile foundation using the above methods.
The author also writes a great amount of numeric computational sub-programs and programs too, almost include all respects of structural numeric computation using Daubechies wavelet. These programs not only test and verify the correlation results in this thesis, but also lay a strong foundation for expanding the application space of Daubechies wavelet in structural numeric computation.
引文
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