Efficient Simulation Of Two Kinds Of Nonlinear Dynamic Three Dimensional Problems Based On Improved FPM Meshless Algorithm

Nonlinear dynamic equations can be used to describe various phenomena in natural sciences or engineering sciences,such as the nonlinear Schrodinger equation(NLSE)that describes wave or quantum vortex phenomena,and the Cahn-Hilliard(C-H)equation that represents multiple phase separation phenomena.However,in most cases,it is difficult to obtain theoretical solutions of these nonlinear dynamic equations by analytical means,especially for high-dimensional and high-order problems.In this case,the numerical simulation method has become a powerful tool to solve these nonlinear dynamic equations.The current simulation methods mainly include grid-based methods and grid-free methods.The grid-based method has advantages when simulating problems in a regular area.However,there will be many difficulties when it comes to irregular areas or large deformation problems.Therefore,the pure mesh-free method has attracted the attention of many scholars because it does not depend on the grid.To solve the above problems,this thesis adopts the pure mesh-free finite point-set method(FPM)which does not depend on the grid and GPU parallel computing to carry out the numerical simulation and prediction research on the two types of important nonlinear dynamic equations(NLSE and C-H equations).At present,there are few reports on the finite point set method for solving high-dimensional and high-order nonlinear dynamic equations,and the traditional FPM method has low accuracy and low computational efficiency.Therefore,based on the improvement of the traditional FPM method,this thesis combines the CUDA parallel computing and the improved FPM method to study two types of nonlinear dynamic equations.The main research contents of this thesis are as follows:(1)For the solution of 2D/3D nonlinear Schrodinger equation,this thesis couples the time splitting format with FPM,and uses GPU parallel computing technology to propose an efficient locally refined FPM format(SS-FPM-GPU).First,introduce the second-order time splitting scheme to decompose the nonlinear Schrodinger equation into two parts:linear differential equation and nonlinear term;secondly,use local refinement method to improve the accuracy of numerical solution;finally,in order to reduce the calculation,use CUDA-C GPU parallel computing.In the numerical simulation,the numerical accuracy and convergence order of the proposed method are analyzed through the 2D/3D NLSE equation with analytical solution,and the GPU parallel computing efficiency is also discussed.Then the solitary wave collision process is predicted by solving the one-dimensional coupled NLSE problem which is compared with result by finite difference method(FDM).At last,the 2D/3D NLSE problem with no analytical solution is simulated and predicted.The numerical results show that the SS-FPM-GPU method proposed in this thesis can accurately and efficiently predict high-dimensional nonlinear wave phenomena(2)In order to accurately,stably and efficiently study the phase separation phenomenon in the multi-component system described by the multi-component C-H equation,this thesis presents an accelerated step-by-step FPM method based on GPU parallel computing.First,the fourth-order derivative is decomposed into two second-order derivatives,and the FPM format is used twice for discretization;secondly,the GPU parallel algorithm based on CUDA-C is applied to obtain the CH-FPM-GPU that can efficiently and stably solve the C-H equation.Subsequently,the accuracy of CH-FPM-GPU is verified by simulating the two-dimensional problem of radial symmetry and the three-dimensional problem of spherical symmetry.Finally,the two-phase separation phenomenon of three-element C-H system in complex irregular domain and the three-phase separation phenomenon of quaternary C-H system with practical significance are predicted by using CH-FPM-GPU method.The numerical results show that the CH-FPM-GPU method proposed in this thesis can accurately and efficiently predict the multiphase separation phenomenon in various irregular regions when it comes to two-dimensional and three-dimensional situations.