弹性波方程的格子Boltzmann方法研究

发布时间：2018-05-02 15:46

本文选题：格子Boltzmann方法 + 弹性波　；参考：《吉林大学》2016年博士论文

【摘要】：格子Boltzmann方法(LBM)做为一种新兴的数值模拟方法,自诞生之日就受到了广泛的关注。它是一种介于微观和宏观之间的细观数值模拟方法,为系统模型的建立提供了一个全新的思路。其算法简便易行、处理复杂边界简单有效等优点,使得它在众多数值模拟方法中脱颖而出,在很多研究领域中都得到了令人惊喜的应用和发展。在本文中,我们从动态Lame方程出发,用格子Boltzmann方法为弹性波方程构造了相应的格子Boltzmann模型,并对一些例子进行了数值模拟。本文主要有如下几点内容:首先,在第一章中我们简单的叙述了格子Boltzmann方法的历史发展过程、基本方程和基本模型,并且描述了格子Boltzmann方法在各个领域的研究现状。尤其提到了在偏微分方程和固体力学领域当中的应用,这也是本文所主要关注的研究方向。然后,在第二章中我们详细的介绍了格子Boltzmann方法的基本理论,分别通过空间多尺度展开和时间多尺度展开得到了格子Boltzmann方法的系列偏微分方程,进而建立了弹性波方程的格子Boltzmann模型,并且给出了不同的处理方法以及相关的误差分析。在第三章中,我们给出了弹性波方程的基本解并建立了P波的格子Boltzmann模型。针对二维P波问题,包括单点波源、初始震源、多震源以及在随机介质中的P波传播进行了数值模拟。通过得到的数值结果与解析解进行比较,我们可以看出格子Boltzmann方法是完全可以模拟P波问题。在第四章中,一方面,我们对经典的半空间各向同性的弹性波的传播进行了数值模拟,其数值结果与经典例子结果一致;另一方面,我们也对双层各向同性介质中的弹性波的传播进行了数值模拟,把得到的数值结果与解析结果进行了比较,模拟的结果是可以接受的。最后,在第五章中我们给出了本文的结论。
[Abstract]:As a new numerical simulation method, lattice Boltzmann method has been paid more and more attention since its birth. It is a mesoscopic numerical simulation method between micro and macro, which provides a new idea for the establishment of system model. The algorithm is simple and efficient, which makes it stand out in many numerical simulation methods, and has been applied and developed in many research fields. In this paper, we construct the lattice Boltzmann model for the elastic wave equation by using the lattice Boltzmann method, starting from the dynamic Lame equation, and carry out numerical simulation for some examples. The main contents of this paper are as follows: in the first chapter, we briefly describe the historical development process, basic equations and basic models of the lattice Boltzmann method, and describe the current research situation of the lattice Boltzmann method in various fields. The application in the field of partial differential equation and solid mechanics is especially mentioned, which is also the main research direction of this paper. Then, in the second chapter, we introduce the basic theory of lattice Boltzmann method in detail, and obtain a series of partial differential equations of lattice Boltzmann method by space multi-scale expansion and time-multi-scale expansion, respectively. Furthermore, the lattice Boltzmann model of elastic wave equation is established, and different processing methods and related error analysis are given. In chapter 3, we give the basic solution of elastic wave equation and establish the lattice Boltzmann model of P-wave. Two dimensional P wave problems including single point source, initial source, multiple source and P wave propagation in random medium are numerically simulated. By comparing the numerical results with the analytical solutions, we can see that the lattice Boltzmann method is completely capable of simulating P-wave problems. In the fourth chapter, on the one hand, we simulate the propagation of classical isotropic elastic waves in half space, and the numerical results are consistent with those of classical examples, on the other hand, We also numerically simulate the propagation of elastic waves in double-layer isotropic media. The numerical results are compared with the analytical results. The simulation results are acceptable. Finally, in the fifth chapter, we give the conclusion of this paper.
【学位授予单位】：吉林大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.8

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