Research On Efficient Numerical Methods And Applications For Thermal Diffusion Of Multicomponent Alloys

Diffusion equation is a kind of important partial differential equation,which plays an important role in describing transport phenomenon and other problems.The multicomponent alloy with excellent performance is widely used in practical life.Its design,processing and use are closely related to diffusion.In this paper,aiming at the problems related to thermal diffusion of multicomponent alloys,the maximum principle preserving and the efficient numerical algorithm for preserving positive and physical properties of its nonlinear differential equation model are studied,and the algorithms are applied to the calculation of diffusion kinetics of multicomponent alloys.The main results are as follows:Firstly,for a class of semilinear diffusion equations,starting from the linear problem with non-smooth initial values,the causes of non-physical oscillation of numerical solutions are analyzed.Based on the finite difference method,the semi-discrete scheme of the stabilization equation is established.Using the time exponential difference method,the first-and second-order schemes of the unconditional maximum principle preserving for semilinear diffusion equations are constructed.The present work analyzes six kinds of common numerical algorithms for calculating matrix exponents,deduces sufficient conditions for the maxima preservation of the established exponential difference scheme,and gives a method for constructing the maximum principle preserving scheme of general problems.The maximum principle preserving of the proposed algorithm are verified by numerical examples,and the algorithm are applied to multi-grain growth simulation.Secondly,for a class of nonsmooth initial value problems of quasilinear diffusion equations,the Picard iterative scheme of a single nonlinear diffusion equation is established based on the construction principle of the positive preserving scheme.According to the banded sparse feature of the nonlinear algebraic equations of the iterative scheme,the scheme is accelerated by time high order extrapolation and space explicit Jacobi iteration,and a positive correction strategy is adopted to realize the efficient solution of a single quasilinear diffusion equation.A numerical example is given.Using the same method,the full implicit positive preserving efficient scheme of nonlinear diffusion equations is constructed and numerical experiments are carried out.Numerical examples demonstrate the accuracy,positivity and conservation of the proposed algorithm.Based on the proposed algorithm,the full-dimensional diffusion behaviors of binary,ternary and quaternary alloys are simulated efficiently.Thirdly,based on Fick’s law and mass conservation law,the nonlinear differential equations model of moving phase boundary of multicomponent alloy is given.Due to the difficulty of tracking interface in the fixed grid method for solving the moving phase boundary problem and the large amount of computation in the moving grid method,the binary system moving interface problem was transformed into the fixed boundary problem based on Landau transformation.The discrete scheme of the completely nonlinear convection-diffusion equation is established by using the finite volume method,and an efficient iterative scheme based on Picard method is constructed.The conservation of mass of the scheme is proved and verified by numerical examples.Using the same method,the iterative scheme is extended to the multivariate case,and some numerical experiments are carried out.Numerical examples demonstrate the reliability of the proposed algorithm.The correlation algorithm is extended to solve the linear compound growth simulation problem,and the numerical simulation of the linear compound growth process is realized.Fourthly,in order to solve the problems of low precision and insufficient use of data information in traditional calculation methods of diffusion coefficients of binary and multicomponent systems,the least squares model of inverse problem of diffusion coefficients of multicomponent alloys based on physical information is established by Tikhonov regularization method from the perspectives of multicomponent and multidimensional systems.For the nonlinear diffusion equations of the forward problem,a fast numerical algorithm is established based on finite element in space and rational approximation scheme for time,which ensures high efficiency of inversion calculation.For the regularized optimization problem,the method of combining gradient descent and line search is adopted to ensure the accuracy of parameter optimization.Based on the essential parallelism of forward problem solving and descending direction calculation,the basic module and algorithm flow of high-throughput diffusion coefficient calculation algorithm for multicomponent alloys are designed.The reliability and efficiency of the algorithm are verified by numerical examples and application cases.The algorithm has been widely used in the efficient calculation of the diffusion coefficients of binary and ternary alloys.Finally,a series of results on the diffusion of multicomponent alloys are applied to the development of a general program for the calculation of thermal physical properties of materials.In order to make the relevant staff can use the efficient algorithms related to multicomponent alloy diffusion problems,designed the diffusion in the material thermal physical property parameters calculation of general program module,using the structure characteristics of the database file parameters design the related structure system,based on traditional optimization algorithm and heuristic algorithm established initial low dependence of basic process parameters optimization.As the core and main body of the general program for thermophysical properties calculation,these algorithms improve the matching degree with diffusion related models,provide efficient numerical simulation algorithms that preserve physical properties for the transformation process of diffusion control,provide convenient and reliable evaluation means for the establishment of diffusion dynamics database,and better support the domestic development of material computing software with independent intellectual property rights.