高精度小波数值方法及其在结构非线性分析中的应用

发布时间：2018-06-25 01:19

  本文选题：微分方程 + 积分方程　； 参考：《兰州大学》2016年博士论文

【摘要】：小波数值方法是近二十多年来发展起来的一类新兴数值方法。随着其自身的发展,小波数值方法的应用范围越来越广泛。而发展统一求解弱非线性和强非线性问题的小波方法这一重要课题也越来越受到重视。立足于小波封闭解法的基础之上,本文拓展了小波方法在具有非线性、奇异性及微分积分算子共存的复杂力学问题中的应用。另外,通过改进小波逼近方式和提出新的求解思路,本文针对一般非线性初值问题和边值问题分别提出了新的高精度小波算法。本文首先介绍了紧支正交的Coiflet小波函数基及其具有拟插值特性的小波逼近公式,它们是小波封闭解法的理论基础。接着介绍了构造有限区间上平方可积函数Coiflet小波逼近公式的边界延拓技术,它是小波数值方法的应用基础。数值研究表明消失矩数目为6的Coiflet是现有小波方法较好的基函数选择。在这些基础之上,本文通过将非线性项中的导数定义为新函数,拓展了现有小波方法在一维和二维拟线性微分方程中的应用,以及结合分部积分和函数变换等技术和小波伽辽金法,还提出了非线性奇异积分方程的几类高精度小波方法。而通过十余个具体数值算例和与其他方法的对比均显示了这些小波方法在计算精度和收敛性方面的优势。非线弹性梁杆的大挠度弯曲屈曲问题和矩形薄板的大变形问题均是现代工程中的典型结构非线性问题,细胞特异性粘附问题是具有弹性-随机耦合特性的非线性生物力学问题。本文发展的小波方法提供了定量求解这些问题的技术。在分析屈曲问题时,小波方法得到的离散代数方程组形式简单,便于结合扩展系统法来直接求解屈曲问题中的临界荷载。在分析大变形问题时,小波方法相对于传统的有限元方法具有更高的计算效率且不出现剪力锁死现象。在分析粘附问题时,小波方法提供了稳定状态下细胞间归一化的力与界面位移非线性关系的定量描述。同时可以注意到在具体的求解过程中,本文的小波方法均能处理任意形式的非线性项以及具有对问题非线性强弱特征不敏感的特性。最后通过推导基于Coiflet的数值微分公式,提高了有限区间上平方可积函数小波逼近公式的逼近精度。在此基础之上,本文构造了一般非线性初值问题的小波时间积分法,并结合空间离散的小波伽辽金法提出了非线性初边值问题的小波时空统一求解法。理论分析表明,该小波时间积分法具有N阶精度和良好的稳定性。数值算例则表明,该小波方法适用于追踪激波或者孤立波等剧烈变化的时空演化问题。另外,本文还提出了求解一般边值问题的新的高精度小波积分配点法。理论分析和数值算例均表明,该小波积分配点法的收敛速度大约为O(2~(-nN)),n为小波分解尺度,N为Coiflet小波消失矩阶数。与之前的小波伽辽金法相比,小波积分配点法不仅提高了方程的求解精度而且其收敛阶数与方程的阶数无关。
[Abstract]:The wavelet numerical method is a new kind of new numerical method developed in the last more than 20 years. With its own development, the application range of the wavelet numerical method is becoming more and more extensive. And the important topic of the wavelet method to develop the unified solution to the weak nonlinear and strong nonlinear problems is becoming more and more important. Based on the wavelet closed method On the basis of this, this paper extends the application of wavelet method to the complex mechanics problem with nonlinear, singular and differential integral operators. In addition, a new high precision wavelet algorithm is proposed for the general nonlinear initial value problem and the boundary value problem by improving the method of wavelet approximation and the new solution. This paper first introduces the Coiflet wavelet function base of tight branch orthogonal and the wavelet approximation formula with quasi interpolation properties. They are the theoretical basis of the wavelet closed method. Then, the boundary extension technique for constructing the Coiflet wavelet approximation formula of the square integrable function on the finite interval is introduced. It is the application basis of the small wave numerical method. Coiflet with the number of vanishing moments is 6 is a better basis function choice for the existing wavelet method. On these basis, by defining the derivative in the nonlinear term as a new function, the application of the existing wavelet method in the one and two dimensional quasilinear differential equations is extended, as well as the combination of partial integral and function transformation and small wave gamma. Several high precision wavelet methods for nonlinear singular integral equations are also proposed by the Liao and Jin method. Through more than ten specific numerical examples and the comparison with other methods, the advantages of these methods in calculating precision and convergence are shown. The large deflection flexural buckling of non linear elastic beams and the large deformation of rectangular thin plates It is a typical structural nonlinear problem in modern engineering. The problem of cell specific adhesion is a nonlinear biomechanical problem with elastic random coupling characteristics. The wavelet method developed in this paper provides a quantitative solution to these problems. In the analysis of the problem of buckling, the discrete algebraic equations obtained by the wavelet method are simple. In the analysis of the large deformation problem, the wavelet method has higher computational efficiency and no shear locking phenomenon when analyzing the large deformation problem. When analyzing the adhesion problem, the wavelet method provides the force and the interface position of the cell normalization under the stable state. At the same time, it can be noted that in the specific solving process, the wavelet method in this paper can both deal with any form of nonlinear term and is insensitive to the nonlinear strong and weak characteristic of the problem. Finally, the square integrable function on the finite interval is improved by deriving the Coiflet based numerical differential common formula. On this basis, the wavelet time integration method for the general nonlinear initial value problem is constructed, and the wavelet space-time unified solution method for nonlinear initial boundary value problem is proposed with the space discrete wavelet Galerkin method. The theoretical analysis shows that the wavelet time integration method has the N order accuracy and good quality. The numerical example shows that the wavelet method is suitable for the spatio-temporal evolution of intense changes in the shock wave or the solitary wave. In addition, a new high precision wavelet integral point method for solving general boundary value problems is also proposed. Both theoretical analysis and numerical examples show that the convergence rate of the wavelet integral point method is about O (2~). -nN)), n is a wavelet decomposition scale, and N is the order of vanishing moment of Coiflet wavelets. Compared with the previous small Galerkin method, the wavelet integral point method not only improves the solution accuracy of the equation, but also has nothing to do with the order of the equation.
【学位授予单位】：兰州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O344.1;O241
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本文编号：2063856