Improved SPH Parallel Algorithm And Numerical Simulation Research For Two Types Of Nonlinear Dynamic Equations

The study of numerical algorithms for nonlinear dynamical equations is a hot topic in the field of computational mechanics.It is well-known that the exact solutions of nonlinear dynamic equations in high dimensional conditions are generally difficult to obtain analytically,for example,nonlinear transient heat conduction equations that describe complex temperature changes(thermodynamic problems described in the real domain)and complex physical phenomena.Nonlinear Schrodinger equation(quantum mechanical problems described in complex domain).However,when the numerical methods based on meshes are used to solve the nonlinear dynamic equations in high-dimensional complex regions,the numerical simulation is complicated.In recent years,the smooth particle dynamics(SPH)method has been widely used in various fields of computational mechanics as a pure meshless method with its advantages that it is completely independent of the mesh.However,the numerical study of the SPH method for high-dimensional complex nonlinear dynamic equations is still rare.The existing SPH methods have the disadvantages of low accuracy,poor stability,and low computational efficiency.In view of this,firstly,in order to improve the accuracy and improving the stability of the traditional SPH method when solving the higher partial derivative partial differential equations,a correction without high-order nuclear derivative calculations based on the Taylor series expansion and stabilization "windward" thought SPH method is established;Secondly,for solving the three-dimensional variable-coefficient transient heat conduction problem,the SPH method is expanded and applied.Based on the MPI parallel computing technology,a three-dimensional correction Parallel SPH(CPSPH-3D)method for solving three-dimensional complex problems is proposed;Third,when the above CPSPH method is extended to the solution of the nonlinear Schrodinger equation in the complex domain,combined with the second-order splitting scheme,a fast and accurate solution to the nonlinear Schrodinger equation is given.Splitting modified parallel SPH(SS-CPSPH)method;Finally,using the SS-CPSPH method described above,a two-dimensional and three-dimensional nonlinear Schrodinger equation was simulated.The main work of this paper is as follows:(1)In order to improve the precision of solving the partial differential equations by the traditional SPH method,the first order based on the Taylor expansion is introduced.In order to improve the stability of the SPH method,the idea of "wind up" stabilization is introduced,and a modified SPH method without high-order nuclear derivative calculation is given.The solution with the analytical solution heat conduction equation verifies that the proposed modified SPH method has second-order accuracy and better numerical stability.The numerical results show that compared with the traditional SPH method,the proposed revised SPH method is more accurate and faster,and it is also with better numerical stability.(2)Extending the above-mentioned modified SPH method to the solution of the three-dimensional variable-coefficient transient heat conduction problem to improve.The computational efficiency,combined with the easy parallel programming features of the SPH method and the parallel computing technology based on MPI,presents a three-dimensional modified parallel SPH(CPSPH-3D)method.Then,the transient heat transfer equation with different coefficients under different boundary conditions is solved,and the computational efficiency under different CPU numbers is compared.Numerical results show that the CPSPH-3D method is accurate and efficient when solveing the variable-coefficient heat transfer problem with Dirichlet or Neumann boundary;CPSPH-3D method has approximate second-order accuracy,better convergence and stability when solving three-dimensional equations;based on MPI parallel computing technology,it increases the CPU in the case of particle number encryption.The number can significantly improve the computational efficiency.(3)Using the proposed CPSPH-3D method for the heat transfer process of three-dimensional non-uniform gradient functional materials Simulations were performed and compared with the results of grid class methods.The results show that the proposed CPSPH-3D method is stable and reliable for the simulation of transient heat transfer problems with three-dimensional variable coefficients without mixed boundary of analytical solutions.Good convergence;The proposed method is used to accurately predict the evolution of temperature with time under the influence of different parameters in 3D functionally graded materials.(4)When using the modified parallel SPH method to directly solve the nonlinear Schrodinger equation in the complex domain,In the presence of both nonlinear terms and source terms,numerical instability tends to occur in long-term simulations.Therefore,by introducing the idea of splitting scheme and combining the above-mentioned modified SPH method with the splitting scheme,a modified parallel SPH(SS-CPSPH)method based on the splitting scheme suitable for solving the nonlinear Schrodinger equation efficiently is presented.Then,several nonlinear Schrodinger equations were studied numerically,and compared with the reliable values,and the computational efficiency under different CPU numbers was demonstrated.The numerical results show that the proposed SS-CPSPH method can accurately and efficiently solve the high-dimensional nonlinear Schrodinger equation,and has second-order accuracy and good convergence.