兰州大学机构库 >土木工程与力学学院
小波级数逼近方式的改进及其在时空动力学求解中的应用
Alternative TitleAn Improved Wavelet Approximation Method and Its Application to Spatio-Temporal Dynamics
李飞跃
Thesis Advisor周又和
2015-05-28
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name硕士
KeywordCoiflets小波 非线性 Klein-Gordon方程 动力控制 压电结构
Abstract科学与工程中许多非线性问题需要用非线性微分方程这样一种基本数学模型进行表征,因而发展非线性微分方程的简单高效的求解方法对研究非线性问题至关重要。本文介绍了一种逼近具有紧支区间的任意平方可积函数的Coiflets小波展开公式。该逼近格式可使方程中非线性项的展开系数能被显式地表达,并且截断误差不会影响所保留的近似部分的精度。同时,在边界处采用了一种新的基于泰勒级数展开的边界延拓处理。结合伽辽金方法,给出了非线性微分方程的统一求解格式。Coiflets小波逼近公式的精度。提出了一种新的边界处理方式,采用Coiflets小波逼近公式来计算边界处泰勒展开式中的边界导数,推导得到了边界导数关于函数区间内函数值的表达式。数值计算结果表明,这样一种新的边界处理方式不仅能够更加有效地抑制边界跳跃现象而且相对于原来的边界处理方式具有更高的精度。其次,利用小波伽辽金法求解了Klein-Gordon方程,选择具有驻波解、行波解的方程和经典的Sine-Gordon方程作为了算例。对于时间维度,本文用龙格库塔法求解。将数值解与方程的精确解进行对比,显示出该方法具有很高精度,可以满足求解问题的实际需求。最后,结合复合材料的层合板理论和压电控制理论,建立了完全基于Coiflets小波理论的压电智能梁式板动力控制过程。数值模拟结果显示,在相同条件下,本文中的控制方法能够在更短时间内使梁式板的振幅衰减到零。
Other AbstractNonlinear differential equations are mathematical models of many nonlinear problems in sciences and engineering. Method of solving nonlinear differential equations become extremely important in the studying of nonlinear science. We introduce an expansion formula of Coiflets approximations for square-integrable functions. The expansion coefficient of nonlinear term can be expressed explicitly, and the precision of the expansion is independent of truncation error. Consruct a new extansion method through the basis of the Taylor series expansion on boundarys. We obtained a wavelet tmethod for uniformly solving nonlinear differential equations according to the Galerkin method. Firstly, we verified the accuracy of Coiflets wavelet approximation formula and proposed a new boundary treatment method in this paper. By substituting Taylor expansion into scale function expansion and representing derivative values at the boundary by function values within the interval, we use the unknown values at the nodes expressed the boundary derivative values explicitly. The numerical results show that the new expression form able to inhibition the undesired jump or wiggle phenomenon near the boundary points more effectively. Finally, by comparison with the original approximation formula, the improved wavelet approximation has higher accuracy. Secondly, we solve the Klein-Gordon equations with wavelet-Galerkin method and choose equations with standing wave solutions, traveling wave solutions and the classical sine-Gordon equation as examples. The Runge-Kutta method is used in the time dimension. Finally, by comparison with the exact soltion , show that the wavelet-Galerkin method has good numerical accuracy, and can meet the practical requirement of solving problems. Finally, according to composite laminated plate theory and the piezoelectric control theory, we introduced the dynamic control process of the beam-plate based on wavelet theory. Piezoelectric patches are pasted on the both side of the beam, as sensors and actuators. The displacement and speed of the beam can be obtained according to the charge and current in piezoelectric patches, further the control voltage applied on the beam can be got through the negative feedback rate. A dynamic control model for the piezoelectric plates is formulated by the improved Coiflets wavelet approximation. Under the same conditions, the numerical results show that the control method in this paper can make the amplitude of the plate d...
URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/225853
Collection土木工程与力学学院
Recommended Citation
GB/T 7714
李飞跃. 小波级数逼近方式的改进及其在时空动力学求解中的应用[D]. 兰州. 兰州大学,2015.
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