小波有限元理论及其在结构工程中的应用
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摘要
小波理论是近些年形成和发展迅速的一种数学工具。它在科学技术界引起了越来越多的关注和重视，在工程应用领域，特别是在信号处理、图像处理、模式识别、语音识别、量子物理、地震勘测、流体力学、电磁场、CT成像、机械故障诊断与监控、分形、数值计算等领域，它被认为是近年来在工具及方法上的重大突破。基于小波变换的小波分析技术是正在发展的新的数学分支。虽有学者将小波应用于结构工程中的微分方程求解当中，但是由于Daubechies小波的尺度函数和小波函数均没有明确的表达式，在计算关联系数时则存在一定的困难，利用数值法计算关联系数有不稳定等一些缺点，还限制着其应用的广度。本论文目的就是在对小波理论进行比较系统的研究之后，寻求利用小波求解微分方程的新的方法，发现并选择合理的小波函数与传统的有限元法相结合，创造性地提出用于结构工程的小波有限元法。同时，为了分析由于结构的物理、几何参数和约束条件等不确定性而引发的工程结构性能的不确定性，提出了随机小波有限元法，从而使小波在结构工程中的应用更加全面。
本论文首先推导了Daubechies尺度函数导数或高阶导数的正确计算结果，给出了它的连续性的判定方式。由于Daubechies小波本身导数的连续性随着支集的增加而增大，解高阶微分方程时，就必须增加支集的长度，这会使计算复杂化。在保证导数的连续性和不增加支集长度的前提下，采用Daubechies尺度函数与B—样条尺度函数的卷积对原方法进行了改进。构造出M—尺度关系，并且证明通常所采用的小波求解微分方程的两尺度关系为其特例，最后利用三尺度样条小波，提出了采用小波伽辽金方法求解问题的方法。
经常采用的小波Galerkin方法对微分方程边界条件的处理，均是将边界条件作为附加方程补充到整体方程中，从而求解超越方程组得到原方程的解，这导致了求解过程出现的方程组个数与未知量个数不一致。鉴于以上的原因，本文构造了满足
西安建筑科技大学博士学位论文
区间上边界条件的HermiteB一样条尺度函数基，提出Galerkin法求解格式，并应用
于弹性地基上有限长梁和板问题，给出了数值结果。给出了B一样条小波函数及其基
本性质，并提出了B一样条小波与Galerkin方法相结合的求解列式。
首次提出了一种基于二类、三类变量广义变分原理的全域多变量小波有限元方
法。首先构造了便于边界条件处理的插值小波基，应用乘积型二元插值小波基来构
造梁、板、壳的广义变量场函数，通过二类、三类变量广义变分原理建立了多变量
小波有限元模型。在计算各种变量时，不需要利用其物理关系，也不必求导，可直
接计算其结果，因而各种变量均有足够的精度。但是这种全域的小波有限元方法仅
在具有规则形状的区域内求解时才能显示出它的优越性;同时，通过它对结构中的
梁、板、壳的求解过程可以看出，都是要通过各自的广义变分原理刁.能求得其解，
通用性较差。
为此，本文利用小波函数构造通常有限元法中的位移函数，得到了利用小波函
数表示的形函数，并用小波形函数首次创造性地推导出了分域的小波有限元列式。
这种分域的小波有限元方法在当前的文献中还未见到。此方法与全域的小波有限元
方法共同构成了小波有限元法的两种求解思路，可相互弥补各自方法中的不足之处。
将小波有限元法与Monte Carlo法相结合，首次提出了Monte Carlo随机小波有
限元法，它兼顾了小波有限元和Monte Carfo模拟的优点，使之可以解决较为复杂的
随机分析问题和检验其它的随机分析方法。并用它求解了具有不确定性的几何尺寸、
材料常数及荷载条件的薄板问题。以确定性变量的变分原理为基础，根据随机变量
的特性，利用摄动二次技术，结合小波形函数，提出了随机变量的变分原理及随机
小波有限元法。此方法可以有效地将材料、几何形状、外力和位移等边界条件的随
机变化特性直接引入到有限元公式中。
最后，本文提出了系统地分析了基岩震动作用下的场地土、基础和结构共同作
用的随机反应的方法。首先假定土层的剪切模量随深度线性变化，引入结构微分算
子，利用连续线性系统的随机振动理论，研究了非均匀场地土层地震的动力反应问
题。然后，利用场地土的随机反应的统计值得到考虑场地土、基础和结构共同作用
系统的随机动力反应。
Wavelet theory is a mathematic tool that is took shape and developed in recent. It has raised more attentions in engineer fields, specially in signal analysis, images handles, pattern recognition, phonetic recognition, quantum physics, earthquake reconnaissance, fluid mechanics, electromagnetic field, CT imagery, diagnosis and monitoring of machinery defect, fractal, numerical evaluation et al. It is considered as important breakthrough in methods and tools. Wavelet analysis based on wavelet transform is the most perfect combination of functional analysis, Fourier transform, spline analysis, harmonic analysis and numerical analysis at present, it is a new developing branch in mathematic. Although some scholars solved differential equations of structure engineering using wavelet, there have some defects. Such as the scaling function and wavelet function have not definit expression, some troubles existed in calculating connection coefficients, its values is instability in some times and so on. The object of th
is thesis is to find some new methods solving differential equation. Thesis combines usually finite element method (FEM) and wavelet analysis, wavelet FEM is put forth creativitily for structure engineering. At the same time, in order to analysis some indeterminacy of structure for example physics and geometric parameter, constraint conditions, stochastic wavelet FEM is proponed. Those methods make the applications of wavelet in structure engineering become more extensive.
At first, this thesis gives right calculation results of derivative of Daubechies scaling function, the determine fashion of continuity is rendered. In solving differential equations, we must increase supported assemble length since the continuity of Daubechies wavelet derivative increases as supported assemble increase, that results in calculation complexly. In guarding the continuity of derivative and not increasing
supported assemble, this paper uses convolution of Daubechies and B-spline scale functions to improve original method. M-scale function is constructed, two-scale function, which often is used to solve differential equations, is its special conditions. Three-scaling spline wavelets and wavelet Galerkin method are used to solve some problems.
Adding boundary conditions in whole equations in the wavelet Galerkin method frequently does with its of differential equations. The result can derive by solving transcendental equation, which make unknow quantity number and the number of system of equation become nonuniformity. For those, this thesis constructions the Hermite B-spline bases scale functions with boundary conditions on the interval. Combining with Garlerkin method, the steps are given to solve differential equation, as an example, it is used to deal with finite-length beam and plate problems. This thesis gives the B-spline wavelet and essential quality of it, proponing the solving array expression of combining B-spline wavelet and Galerkin method.
This thesis first presents a multivariable wavelet FEM that is based on Hellingger-Reissner and Hu-Washizu variational principle. The interpolating wavelet bases are constructed in order to deal with boundary conditions conveniently at first. Then the wavelet bases of duality in product form are used to construct the generalized field functions of beam, plate and shell. The model of multivariable wavelet FEM is built by Hellingger-Reissner and Hu-Washizu generalized variational principle. In calculating variables, the stress-strain relations and the differential calculation are not implemented, so all kinds of variable have enough precision. But this universe wavelet FEM displays its superiority only in regulation figure fields. Through the processing of solving beam, plate and shell in structure must transit generalized variational principle respective, so the commonality is difference.
The displacement functions in usually FEM are constructed with wavelet functions, then the form functions expression with wavelet functions. For the first time wavelet FEM of dividing
本论文首先推导了Daubechies尺度函数导数或高阶导数的正确计算结果，给出了它的连续性的判定方式。由于Daubechies小波本身导数的连续性随着支集的增加而增大，解高阶微分方程时，就必须增加支集的长度，这会使计算复杂化。在保证导数的连续性和不增加支集长度的前提下，采用Daubechies尺度函数与B—样条尺度函数的卷积对原方法进行了改进。构造出M—尺度关系，并且证明通常所采用的小波求解微分方程的两尺度关系为其特例，最后利用三尺度样条小波，提出了采用小波伽辽金方法求解问题的方法。
经常采用的小波Galerkin方法对微分方程边界条件的处理，均是将边界条件作为附加方程补充到整体方程中，从而求解超越方程组得到原方程的解，这导致了求解过程出现的方程组个数与未知量个数不一致。鉴于以上的原因，本文构造了满足
西安建筑科技大学博士学位论文
区间上边界条件的HermiteB一样条尺度函数基，提出Galerkin法求解格式，并应用
于弹性地基上有限长梁和板问题，给出了数值结果。给出了B一样条小波函数及其基
本性质，并提出了B一样条小波与Galerkin方法相结合的求解列式。
首次提出了一种基于二类、三类变量广义变分原理的全域多变量小波有限元方
法。首先构造了便于边界条件处理的插值小波基，应用乘积型二元插值小波基来构
造梁、板、壳的广义变量场函数，通过二类、三类变量广义变分原理建立了多变量
小波有限元模型。在计算各种变量时，不需要利用其物理关系，也不必求导，可直
接计算其结果，因而各种变量均有足够的精度。但是这种全域的小波有限元方法仅
在具有规则形状的区域内求解时才能显示出它的优越性;同时，通过它对结构中的
梁、板、壳的求解过程可以看出，都是要通过各自的广义变分原理刁.能求得其解，
通用性较差。
为此，本文利用小波函数构造通常有限元法中的位移函数，得到了利用小波函
数表示的形函数，并用小波形函数首次创造性地推导出了分域的小波有限元列式。
这种分域的小波有限元方法在当前的文献中还未见到。此方法与全域的小波有限元
方法共同构成了小波有限元法的两种求解思路，可相互弥补各自方法中的不足之处。
将小波有限元法与Monte Carlo法相结合，首次提出了Monte Carlo随机小波有
限元法，它兼顾了小波有限元和Monte Carfo模拟的优点，使之可以解决较为复杂的
随机分析问题和检验其它的随机分析方法。并用它求解了具有不确定性的几何尺寸、
材料常数及荷载条件的薄板问题。以确定性变量的变分原理为基础，根据随机变量
的特性，利用摄动二次技术，结合小波形函数，提出了随机变量的变分原理及随机
小波有限元法。此方法可以有效地将材料、几何形状、外力和位移等边界条件的随
机变化特性直接引入到有限元公式中。
最后，本文提出了系统地分析了基岩震动作用下的场地土、基础和结构共同作
用的随机反应的方法。首先假定土层的剪切模量随深度线性变化，引入结构微分算
子，利用连续线性系统的随机振动理论，研究了非均匀场地土层地震的动力反应问
题。然后，利用场地土的随机反应的统计值得到考虑场地土、基础和结构共同作用
系统的随机动力反应。
Wavelet theory is a mathematic tool that is took shape and developed in recent. It has raised more attentions in engineer fields, specially in signal analysis, images handles, pattern recognition, phonetic recognition, quantum physics, earthquake reconnaissance, fluid mechanics, electromagnetic field, CT imagery, diagnosis and monitoring of machinery defect, fractal, numerical evaluation et al. It is considered as important breakthrough in methods and tools. Wavelet analysis based on wavelet transform is the most perfect combination of functional analysis, Fourier transform, spline analysis, harmonic analysis and numerical analysis at present, it is a new developing branch in mathematic. Although some scholars solved differential equations of structure engineering using wavelet, there have some defects. Such as the scaling function and wavelet function have not definit expression, some troubles existed in calculating connection coefficients, its values is instability in some times and so on. The object of th
is thesis is to find some new methods solving differential equation. Thesis combines usually finite element method (FEM) and wavelet analysis, wavelet FEM is put forth creativitily for structure engineering. At the same time, in order to analysis some indeterminacy of structure for example physics and geometric parameter, constraint conditions, stochastic wavelet FEM is proponed. Those methods make the applications of wavelet in structure engineering become more extensive.
At first, this thesis gives right calculation results of derivative of Daubechies scaling function, the determine fashion of continuity is rendered. In solving differential equations, we must increase supported assemble length since the continuity of Daubechies wavelet derivative increases as supported assemble increase, that results in calculation complexly. In guarding the continuity of derivative and not increasing
supported assemble, this paper uses convolution of Daubechies and B-spline scale functions to improve original method. M-scale function is constructed, two-scale function, which often is used to solve differential equations, is its special conditions. Three-scaling spline wavelets and wavelet Galerkin method are used to solve some problems.
Adding boundary conditions in whole equations in the wavelet Galerkin method frequently does with its of differential equations. The result can derive by solving transcendental equation, which make unknow quantity number and the number of system of equation become nonuniformity. For those, this thesis constructions the Hermite B-spline bases scale functions with boundary conditions on the interval. Combining with Garlerkin method, the steps are given to solve differential equation, as an example, it is used to deal with finite-length beam and plate problems. This thesis gives the B-spline wavelet and essential quality of it, proponing the solving array expression of combining B-spline wavelet and Galerkin method.
This thesis first presents a multivariable wavelet FEM that is based on Hellingger-Reissner and Hu-Washizu variational principle. The interpolating wavelet bases are constructed in order to deal with boundary conditions conveniently at first. Then the wavelet bases of duality in product form are used to construct the generalized field functions of beam, plate and shell. The model of multivariable wavelet FEM is built by Hellingger-Reissner and Hu-Washizu generalized variational principle. In calculating variables, the stress-strain relations and the differential calculation are not implemented, so all kinds of variable have enough precision. But this universe wavelet FEM displays its superiority only in regulation figure fields. Through the processing of solving beam, plate and shell in structure must transit generalized variational principle respective, so the commonality is difference.
The displacement functions in usually FEM are constructed with wavelet functions, then the form functions expression with wavelet functions. For the first time wavelet FEM of dividing
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