梁板结构等非线性问题的小波封闭解法
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摘要
自然科学和工程技术中的许多非线性问题都可以用非线性微分方程这一基本的数学模型来表征,因而非线性微分方程的求解技术是研究非线性科学过程中不可回避的一个环节。虽然自非线性科学诞生伊始,各种的求解方法,包括解析方法和数值方法就被源源不断的开发出来,但现有方法在处理非线性微分方程,尤其是定量求解强非线性问题时仍然存在着诸多的不足。一个重要的原因就是这些方法无法将方程中非线性项的低阶与高阶信息解耦,从而导致舍去的解的高阶项对低阶近似解的求解产生了很大的影响,即低阶近似解依赖于舍去的高阶项。因而随着非线性效应的增强,解的精度将会显著的下降甚至出现解不收敛等问题。因此如何获得强非线性系统的高精度近似解已成为非线性科学研究中的一个至关重要的课题。
小波分析是数学的一个新的分支,在时域与频域空间均具有强大的局部识别能力,目前在许多领域如图像处理,故障诊断及方程的数值求解中已展示出强大的优越性与生命力。具体到微分方程的数值求解领域,基于多分辨分析的小波级数具有层层嵌套逐步逼近L2空间的能力,此外,小波函数还具有正交性、紧支集与光滑可导等数值特性,因而以小波函数或尺度函数为基函数的小波逼近格式可以稳定而快速的逼近任一平方可积函数。至关重要的是广义小波高斯积分法实现了在逼近非线性项时低阶与高阶信息的解耦,为获得非线性微分方程的封闭解奠定了基础。本文以此为基础针对梁板结构等具体的非线性问题研究相应的封闭小波算法。
首先通过在边界处采用基于泰勒级数展开的延拓处理,构造出了一种可与任意边界条件相协调的改进小波尺度基函数。基于此基函数的小波逼近格式,可以有效的逼近定义在有限区间上的函数,克服了经典小波级数在边界处出现跳跃或抖动等缺陷。在此基础上,结合广义小波高斯积分法,在逼近定义在有限区间上的任意非线性函数时,可以将非线性算子等价的作用在展开系数上,从而获得了非线性项的小波逼近格式中的系数的显示表征形式。
随后以该修正后的区间小波展开格式为试函数,运用伽辽金方法研究了圆薄板的大挠度弯曲问题,讨论了板壳理论与薄膜理论的过渡问题。在具体运用中,通过将边界条件嵌入到小波级数中,获得了满足本质边界条件的小波展开格式,避免了普通小波伽辽金方法需要处理边界条件的麻烦；同时非线性项的展开系数可以用待求的系数显式的表征出来,实现了其低阶与高阶信息的解耦,为获得方程的封闭解奠定了基础。最后通过与精确解的对比,表明该修正小波伽辽金方法能够获得强非线性微分方程的高精度近似解,并且该近似解的精度对非线性的强弱特征并不敏感,换句话说该方法在求解强非线性问题时仍然能够保证极高的数值精度。
在小波伽辽金方法中需要计算尺度函数与其导数的乘积的积分,即连接系数,而在这一过程中不可避免的要引入数值误差。为克服这一缺陷,通过将微分方程转换为积分方程,然后采用基于本文构造的区间小波数值积分法,获得了方程的高精度近似解。数值算例表明该方法具有比现有小波伽辽金方法更高的精度,并且大大的减少了计算量。
针对具有奇异项的非线性微分方程,利用拉普拉斯正、逆两次变换可将其转化为含有卷积核函数的非奇异方程。在该转化过程中,虽然无法获得非线性项在拉普拉斯像空间的显示表征格式,但通过随后的拉普拉斯逆变换,非线性项的像函数可以消掉,即在最后的计算中并不需要用到该像函数。基于本文构造的区间小波逼近格式,通过在空间上采用配点法离散,在时间上直接采用数值积分法,最后可将原非线性微分方程转化为以展开系数为未知量的非线性代数方程组。同时针对力学领域的绝大部分问题,如振动问题,扩散问题,由于其卷积核函数具有在零点处连续且函数值为零的特性,当前求解步的结果可以由前述求解步的结果显示表征出来,故在求解该非线性代数方程组时只需进行简单的迭代计算,而无需求解方程组,避免了大量的矩阵运算,极大的提高了计算效率。
基于上述小波直接迭代法,研究了非线性分数阶振动、扩散与波动问题。并将当前的数值结果与精确解及采用Adomain分解法、差分法的数值结果进行了对比。此外通过定量求解包含幂次非线性与非幂次非线性项的两类分数阶初／边值问题,分析讨论了不同非线性情形下小波直接迭代法的有效性。数值结果证明了该方法的可靠性,并且展示了该数值解具有极高的数值精度且对非线性项的形式及强弱不敏感。
最后基于改进的Coiflets小波尺度函数变换建立了表面粘贴感应压电片、致动压电片及中间嵌入粘弹性材料薄膜层的矩形层合板结构的主动控制模式。在这一方法中,位移可由感应压电片的电流信号显示表征出来。同时由于小波尺度函数具有低通滤波特性,能自动滤除噪声,有效的避免了观测溢出与控制溢出等传统控制模式中存在的问题,极大的提高了控制系统的稳定性。仿真结果表明主动控制回路可以有效的控制系统的低频振动,而对于高频振动的控制效果显著减弱,而粘弹性薄膜层则可有效的抑制系统的高频振动,故而这一混合控制系统具有较高的鲁棒性和稳定性。
Nonlinear differential equations are mathematical models of many nonlinear problems in natural sciences and engineering. Method of solving nonlinear differential equations cann't be avoided in the studying of nonlinear science. Since the beginning of nonlinear science, many methods including analytical and numerical methods have been developed, but the existing numerical methods in solving nonlinear differential equations, especially in the quantitative solution of strongly nonlinear problems, there are still many deficiencies. One of the most important reason is that the existing numerical methods can't make high-level information and low-level information decoupling in the nonlinear equation, leading to that the discarding of high-order terms have influence in the low-order approximate solution, therefore, when the nonlinear effection increases to a certain extent, the accuracy of existing numerical methods for solution is not enough and even the solution doesn't converge. So, how to obtain a high-precision approximate solution for strongly nonlinear systems is an important topic in nonlinear science.
Wavelet analysis is a new branch of mathematics, and has excellent recognition capabilities in the time domain and frequency domain. It has been used in many fields such as image processing, fault diagnosis and the numerical solution of equations, which have demonstrated the superiority and vitality. Specific to the field of numerical solution of differential equations, based on multi-resolution analysis, wavelet series has special approximation characteristics of the layers of nested space, in addition, the wavelet function has characteristics like compactly supported, orthogonal, smooth conductivity and so on. Therefore, the wavelet approximation of an arbitrary square-integrable function based on wavelet function or scaling function has stable and fast approach advantages. Further, the generalized-Gaussian-quadrature method in wavelet analysis solve the problem that the high-level information and the low-level information cann't be decoupled in the existing numerical methods, which makes a theoretical foundation in order to obtain the closed solution of equations. This dissertation presents closed solution of nonlinear equations of the beams/plates based on wavelet.
First, construct a coordinated improved wavelet scaling function associated with the arbitrary boundary conditions through the basis of the Taylor series expansion on boundarys. Wavelet approximation based on the modified scaling function, can be an effective approximation of a function defined on a finite interval, avoid the undesired jump or wiggle phenomenon near the boundary points when the wavelet-based method is employed to solve a boundary-value problem. After that, according to the generalized-Gaussian-quadrature method in wavelet analysis, it has the properties of making the symbols of nonlinear function on the expanded coefficients when approximate arbitrary nonlinear functions. Then, a wavelet approximation of the nonlinear terms with explicit expansion of an arbitrary function defined on a bounded interval is obtained.
Apply the modified wavelet approximation with explicit expansion to the wavelet Galerkin method, which solved the problem about bending of circular thin plate with nonlinear characterization and the transitional problem of film theory successfully. With the approximation, boundary values and boundary derivatives of the unknown function to be approximated can be explicitly embedded in the resulting scaling function expansions, which avoid the trouble of handling the boundary conditions in traditional Galerkin method. And the expansion coefficients in the approximation of arbitrary nonlinear term can be explicitly expressed by the unknown coefficients, laid the foundation to obtain the closed solution of equations. Finally, by comparison with the exact solution, show that the modified wavelet Galerkin method has good numerical accuracy in solving strongly nonlinear equations, and the accauracy of the approximate solution isn't sensitive to the nonlinear characteristics. It means that the modified wavelet Galerkin method still has high numerical accuracy even the nonlinearity is strong.
Ones need to calculate the multiplied of scaling function and its derivative in the classical wavelet Galerkin method, which is inevitable to introduce numerical errors. To overcome this shortcoming, by means of that differential equations can be transformed into integral equations and the interval wavelet numerical integration method proposed, we can obtain a high-precision approximate solution of equations. Numerical example shows that this method greatly reduces the amount of computation and improves the numerical accuracy comparing to the classical wavelet Galerkin method.
Consider that there are singular integral in differential equations with fractional order, we use the Laplace transform and inverse transform to convert them into the second type Voltera integral equations with non-singular kernels. During the Laplace transform process, it is difficult to obtain the explicit expression of the nonlinear terms in the Laplace image space, but by the inverse transform it can be eliminate, which means the nonlinear terms are expressed by the symbols of their Laplace transform. By the interval wavelet numerical integration method proposed, and by using the direct numerical integration method in time, while the collocation method is using for discrete space, the original nonlinear differential equation can be changed into nonlinear algebraic equations. At the same time, most of the problems in mechanics, such as the vibration problem, diffusion problem, due to its conveolution kernel function is continuous at zero and the corresponding function value is zero, so the results of current solution step can be explicitly expressed by the results of the aforementioned solution, the proposed method does not involve any matrix inversions and can be implemented as simple as the linear multi-step methods for solving the nonlinear fractional equations, which improved the efficiency of the computation.
The wavelet direct iterative method is applied for numerical solution of nonlinear fractional vibration, diffusion and wave equations, and compared the numerical results with the exact solution, results obtained by the Adomain decomposition method and the difference method. The quantitative numerical results obtained in solving fractional differential equations with either integer order or non-integer order nonlinear terms or both, and the effectiveness of the proposed wavelet-based method is analyzed and discussed with different nonlinear characterization, indicate that this method has high accuracy no matter the nonlinearity is weak or strong.
Based on the modified scaling function transform of the Coiflets wavelet theory, an one-to-one explicit relation between the electric signals measured in piezoelectric sensors and deflection of the laminated plate is established in this paper, and proposes a hybrid active-passive control strategy for suppressing vibrations of laminated rectangular plates bonded with distributed piezoelectric sensors and actuators via thin viscoelastic bonding layers. Due to that the reconstruction function of the scaling function has the ability of automatical filtering high order frequency signals of vibration or disturbance, the phenomenon of instability which caused from the spilling over of measurement and controller, doesn't occur in the control strategy proposed. Numerical simulation results shows that the hybrid active-passive control strategy can effectively control the low frequency vibration in the control loop of the system. Moreover, the existence of thin viscoelastic bonding layers can further improve robustness and reliability of the system through dissipating the energy of any other possible noise partially induced by numerical errors during the control process.
小波分析是数学的一个新的分支,在时域与频域空间均具有强大的局部识别能力,目前在许多领域如图像处理,故障诊断及方程的数值求解中已展示出强大的优越性与生命力。具体到微分方程的数值求解领域,基于多分辨分析的小波级数具有层层嵌套逐步逼近L2空间的能力,此外,小波函数还具有正交性、紧支集与光滑可导等数值特性,因而以小波函数或尺度函数为基函数的小波逼近格式可以稳定而快速的逼近任一平方可积函数。至关重要的是广义小波高斯积分法实现了在逼近非线性项时低阶与高阶信息的解耦,为获得非线性微分方程的封闭解奠定了基础。本文以此为基础针对梁板结构等具体的非线性问题研究相应的封闭小波算法。
首先通过在边界处采用基于泰勒级数展开的延拓处理,构造出了一种可与任意边界条件相协调的改进小波尺度基函数。基于此基函数的小波逼近格式,可以有效的逼近定义在有限区间上的函数,克服了经典小波级数在边界处出现跳跃或抖动等缺陷。在此基础上,结合广义小波高斯积分法,在逼近定义在有限区间上的任意非线性函数时,可以将非线性算子等价的作用在展开系数上,从而获得了非线性项的小波逼近格式中的系数的显示表征形式。
随后以该修正后的区间小波展开格式为试函数,运用伽辽金方法研究了圆薄板的大挠度弯曲问题,讨论了板壳理论与薄膜理论的过渡问题。在具体运用中,通过将边界条件嵌入到小波级数中,获得了满足本质边界条件的小波展开格式,避免了普通小波伽辽金方法需要处理边界条件的麻烦；同时非线性项的展开系数可以用待求的系数显式的表征出来,实现了其低阶与高阶信息的解耦,为获得方程的封闭解奠定了基础。最后通过与精确解的对比,表明该修正小波伽辽金方法能够获得强非线性微分方程的高精度近似解,并且该近似解的精度对非线性的强弱特征并不敏感,换句话说该方法在求解强非线性问题时仍然能够保证极高的数值精度。
在小波伽辽金方法中需要计算尺度函数与其导数的乘积的积分,即连接系数,而在这一过程中不可避免的要引入数值误差。为克服这一缺陷,通过将微分方程转换为积分方程,然后采用基于本文构造的区间小波数值积分法,获得了方程的高精度近似解。数值算例表明该方法具有比现有小波伽辽金方法更高的精度,并且大大的减少了计算量。
针对具有奇异项的非线性微分方程,利用拉普拉斯正、逆两次变换可将其转化为含有卷积核函数的非奇异方程。在该转化过程中,虽然无法获得非线性项在拉普拉斯像空间的显示表征格式,但通过随后的拉普拉斯逆变换,非线性项的像函数可以消掉,即在最后的计算中并不需要用到该像函数。基于本文构造的区间小波逼近格式,通过在空间上采用配点法离散,在时间上直接采用数值积分法,最后可将原非线性微分方程转化为以展开系数为未知量的非线性代数方程组。同时针对力学领域的绝大部分问题,如振动问题,扩散问题,由于其卷积核函数具有在零点处连续且函数值为零的特性,当前求解步的结果可以由前述求解步的结果显示表征出来,故在求解该非线性代数方程组时只需进行简单的迭代计算,而无需求解方程组,避免了大量的矩阵运算,极大的提高了计算效率。
基于上述小波直接迭代法,研究了非线性分数阶振动、扩散与波动问题。并将当前的数值结果与精确解及采用Adomain分解法、差分法的数值结果进行了对比。此外通过定量求解包含幂次非线性与非幂次非线性项的两类分数阶初／边值问题,分析讨论了不同非线性情形下小波直接迭代法的有效性。数值结果证明了该方法的可靠性,并且展示了该数值解具有极高的数值精度且对非线性项的形式及强弱不敏感。
最后基于改进的Coiflets小波尺度函数变换建立了表面粘贴感应压电片、致动压电片及中间嵌入粘弹性材料薄膜层的矩形层合板结构的主动控制模式。在这一方法中,位移可由感应压电片的电流信号显示表征出来。同时由于小波尺度函数具有低通滤波特性,能自动滤除噪声,有效的避免了观测溢出与控制溢出等传统控制模式中存在的问题,极大的提高了控制系统的稳定性。仿真结果表明主动控制回路可以有效的控制系统的低频振动,而对于高频振动的控制效果显著减弱,而粘弹性薄膜层则可有效的抑制系统的高频振动,故而这一混合控制系统具有较高的鲁棒性和稳定性。
Nonlinear differential equations are mathematical models of many nonlinear problems in natural sciences and engineering. Method of solving nonlinear differential equations cann't be avoided in the studying of nonlinear science. Since the beginning of nonlinear science, many methods including analytical and numerical methods have been developed, but the existing numerical methods in solving nonlinear differential equations, especially in the quantitative solution of strongly nonlinear problems, there are still many deficiencies. One of the most important reason is that the existing numerical methods can't make high-level information and low-level information decoupling in the nonlinear equation, leading to that the discarding of high-order terms have influence in the low-order approximate solution, therefore, when the nonlinear effection increases to a certain extent, the accuracy of existing numerical methods for solution is not enough and even the solution doesn't converge. So, how to obtain a high-precision approximate solution for strongly nonlinear systems is an important topic in nonlinear science.
Wavelet analysis is a new branch of mathematics, and has excellent recognition capabilities in the time domain and frequency domain. It has been used in many fields such as image processing, fault diagnosis and the numerical solution of equations, which have demonstrated the superiority and vitality. Specific to the field of numerical solution of differential equations, based on multi-resolution analysis, wavelet series has special approximation characteristics of the layers of nested space, in addition, the wavelet function has characteristics like compactly supported, orthogonal, smooth conductivity and so on. Therefore, the wavelet approximation of an arbitrary square-integrable function based on wavelet function or scaling function has stable and fast approach advantages. Further, the generalized-Gaussian-quadrature method in wavelet analysis solve the problem that the high-level information and the low-level information cann't be decoupled in the existing numerical methods, which makes a theoretical foundation in order to obtain the closed solution of equations. This dissertation presents closed solution of nonlinear equations of the beams/plates based on wavelet.
First, construct a coordinated improved wavelet scaling function associated with the arbitrary boundary conditions through the basis of the Taylor series expansion on boundarys. Wavelet approximation based on the modified scaling function, can be an effective approximation of a function defined on a finite interval, avoid the undesired jump or wiggle phenomenon near the boundary points when the wavelet-based method is employed to solve a boundary-value problem. After that, according to the generalized-Gaussian-quadrature method in wavelet analysis, it has the properties of making the symbols of nonlinear function on the expanded coefficients when approximate arbitrary nonlinear functions. Then, a wavelet approximation of the nonlinear terms with explicit expansion of an arbitrary function defined on a bounded interval is obtained.
Apply the modified wavelet approximation with explicit expansion to the wavelet Galerkin method, which solved the problem about bending of circular thin plate with nonlinear characterization and the transitional problem of film theory successfully. With the approximation, boundary values and boundary derivatives of the unknown function to be approximated can be explicitly embedded in the resulting scaling function expansions, which avoid the trouble of handling the boundary conditions in traditional Galerkin method. And the expansion coefficients in the approximation of arbitrary nonlinear term can be explicitly expressed by the unknown coefficients, laid the foundation to obtain the closed solution of equations. Finally, by comparison with the exact solution, show that the modified wavelet Galerkin method has good numerical accuracy in solving strongly nonlinear equations, and the accauracy of the approximate solution isn't sensitive to the nonlinear characteristics. It means that the modified wavelet Galerkin method still has high numerical accuracy even the nonlinearity is strong.
Ones need to calculate the multiplied of scaling function and its derivative in the classical wavelet Galerkin method, which is inevitable to introduce numerical errors. To overcome this shortcoming, by means of that differential equations can be transformed into integral equations and the interval wavelet numerical integration method proposed, we can obtain a high-precision approximate solution of equations. Numerical example shows that this method greatly reduces the amount of computation and improves the numerical accuracy comparing to the classical wavelet Galerkin method.
Consider that there are singular integral in differential equations with fractional order, we use the Laplace transform and inverse transform to convert them into the second type Voltera integral equations with non-singular kernels. During the Laplace transform process, it is difficult to obtain the explicit expression of the nonlinear terms in the Laplace image space, but by the inverse transform it can be eliminate, which means the nonlinear terms are expressed by the symbols of their Laplace transform. By the interval wavelet numerical integration method proposed, and by using the direct numerical integration method in time, while the collocation method is using for discrete space, the original nonlinear differential equation can be changed into nonlinear algebraic equations. At the same time, most of the problems in mechanics, such as the vibration problem, diffusion problem, due to its conveolution kernel function is continuous at zero and the corresponding function value is zero, so the results of current solution step can be explicitly expressed by the results of the aforementioned solution, the proposed method does not involve any matrix inversions and can be implemented as simple as the linear multi-step methods for solving the nonlinear fractional equations, which improved the efficiency of the computation.
The wavelet direct iterative method is applied for numerical solution of nonlinear fractional vibration, diffusion and wave equations, and compared the numerical results with the exact solution, results obtained by the Adomain decomposition method and the difference method. The quantitative numerical results obtained in solving fractional differential equations with either integer order or non-integer order nonlinear terms or both, and the effectiveness of the proposed wavelet-based method is analyzed and discussed with different nonlinear characterization, indicate that this method has high accuracy no matter the nonlinearity is weak or strong.
Based on the modified scaling function transform of the Coiflets wavelet theory, an one-to-one explicit relation between the electric signals measured in piezoelectric sensors and deflection of the laminated plate is established in this paper, and proposes a hybrid active-passive control strategy for suppressing vibrations of laminated rectangular plates bonded with distributed piezoelectric sensors and actuators via thin viscoelastic bonding layers. Due to that the reconstruction function of the scaling function has the ability of automatical filtering high order frequency signals of vibration or disturbance, the phenomenon of instability which caused from the spilling over of measurement and controller, doesn't occur in the control strategy proposed. Numerical simulation results shows that the hybrid active-passive control strategy can effectively control the low frequency vibration in the control loop of the system. Moreover, the existence of thin viscoelastic bonding layers can further improve robustness and reliability of the system through dissipating the energy of any other possible noise partially induced by numerical errors during the control process.
引文
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