The Application Of Lattice Boltzmann Method In Several Typical Partial Differential Equations And High Velocity Compaction Process

Partial differential equation(PDE) has extensive application in science and engi-neering technology and many practical problems can be described by partial differen-tial equations. But the analytical solutions of these equations are difficult to find out, we always looking for their numerical solutions through all sorts of methods. Lattice Boltzmann method evolved out of ideas that has been intensely investigated since1980s which is used to solve the fluid mechanics problems and also can be used to solve some partial differential equations. By choosing appropriate collision or equilibrium distri-bution, the lattice Boltzmann model is able to recover the PDE of interest. Lattice Boltzmann method which use micro model to simulate the macro transport phenomena has made successful applications in many fields.Firstly, the development of lattice Boltzmann method and their applications in nu-merical solution of PDE and powder forming process is summarized. The basic prin-ciple and common models of lattice Boltzmann method are introduced. The lattice Boltzmann equation is deduced from Newton equation in detail.Secondly, though directly taking second order multi-scale expansion of the addi-tional terms such that the complex partial derivative terms can be omitted in the evolu-tion equation, the lattice Boltzmann method was used to solve nonlinear heat conduction type equations and nonlinear Klein-Gordon equation. The Fisher equation,Newell-Whitehead equation,FitzHugh-Nagumo equation,two and three-order nonlinear Klein-Gordon equation are calculated. The results show that the lattice Boltzmann method can effectively solve the above equations and can achieve second order precision. In solving three-order nonlinear Klein-Gordon equation, the calculating results are more effective than other numerical results in the large amplitude periodic boundary condition.Thirdly, the stability of the one-dimensional Goldstein-Taylor model's finite dif-ference lattice Boltzmann scheme is proofed in the meaning of L2norm under the pe-riodic boundary condition. According to the definition of macro quantity, the DIQ2lattice Boltzmann model of the one-dimensional Burgers equation can be rewritten as a three levels finite difference scheme and the stability conditions of the finite difference scheme is analyzed. We compare the results of our three levels finite difference scheme with D1Q2, D1Q3model and the Fourier series solutions and find that the results match with the results of D1Q2, D1Q3model, the global relative error is even smaller. When the viscosity coefficient is very small, the Fourier series solution is divergence because of the shock wave but the three levels finite difference scheme can solve the equation accurately.Finally, according to the technical characteristic and the relevant references, the multi-speeds compressible lattice Boltzmann method is used to simulate the high veloc-ity compaction process first time. The rebound and reflection mixed boundary condition is adopted to simulate the lubrication and friction role in the process. According to the variation of powder's density, the original constant relaxation time is changed to an re-laxation time function which is consisted of initial density and maximum ideal density. Two-dimensional numerical simulation of high velocity compaction process is realized. The density evolution process and stress wave transformation process are vividly pre-sented.