Studies Of Lattice Boltzmann Method For Elastic Wave Equation

The Lattice Boltzmann method(LBM) has received extensive attention as a new numerical simulation method from the day of its birth. It is a mesoscopic numerical simulation between micro and macro, which provides a new thought for building the system models. Therefore, it has the following advantages: its algorithm is simple and easy, and the boundary conditions are easier to be treated simply and effectively. These advantages make it stand out from other simulation methods, and its application and development is amazing in many research fields.In this paper, a lattice Boltzmann model is constructed for elastic wave equation which is based on the lattice Boltzmann method from the dynamic lame equation. And the numerical simulations are given. The paper is organized as follows:In the first chapter, we reviewed the history, basic model and basic equation of the lattice Boltzmann method including its research status in various fields. In particular, we describe its application in the field of partial differential equations and solid mechanics.Then, in the second chapter, the basic mathematical theories of LBM are presented in detail. Here, we obtain a series of partial differential equations respectively from time multi-scale and multi-scale space. Furthermore, we have established a lattice Boltzmann model for elastic wave equation. And the different treatment methods and related error analysis are given.In the third chapter, we present the fundamental solutions for elastic wave equation, and propose a lattice Boltzmann model for P wave. Further, we simulated two-dimensional P wave problem conditions single-point wave source, the initial source, multi-source and in random media. By comparing the numerical results with the analytical solutions, we can see that the lattice Boltzmann method can be used to simulate P wave problems completely.In the fourth chapter, on the one hand, the classic questions of isotropic elastic wave propagate in the half space are simulated by lattice Boltzmann model. The numerical examples show that the numerical results and analytical results are consistent. On the other hand, we also simulate the propagation of elastic waves in bilayer isotropic medium. From the comparisons between the numerical results and the analysis solutions, we can see that the result is basically the same. The simulation results are acceptable.Finally, we give conclusions in the fifth chapter.