PDE Modeling And Inversion Algorithms For Nucleation-Growth Behavior During The Preparation Process Of Iron-based Catalyst

Multiscale modeling of the crystallization process is an important mathematical problem in the preparation of iron-based catalysts.It has attracted many materials scientists and mathematicians.The main concern was the multi-scale model of the crystallization process and the corresponding inverse problems with nucleation rate and growth rate.In this paper,we first studied the mesoscale modeling in the preparation of iron-based catalysts under dilute concentrations.Then,we studied the inverse problem of source terms based on the mesoscale model.Subsequently,we studied the inverse problem of growth rate and nucleation rate based on the first-kind Volterra integral equations.Finally,we gave the numerical simulations of direct and inverse problems.In the first chapter,the research background of the preparation of iron-based catalysts by precipitation method were briefly introduced.The previous research results on mathematical models of crystallization processes were illustrated,and some concepts such as inverse problems and regularization were given.In the second chapter,the mesoscale partial differential equation model of nucleation and growth during the preparation of iron-based catalysts in the dilute solution concentration was given.And a macro-microscopic coupled mesoscale mathematical model was presented.The model is a three-dimensional parabolic equation of space obtained by a random space marker process and a causal cone method.It is calculated by a finite difference method.Moreover,the numerical results showed the rationality of the proposed mathematical model and the effectiveness of the algorithm.In the third chapter,we studied the inverse problem of nucleation rate and growth rate by solving the inverse problem of source term and the first-kind Volterra integral equations on the basis of direct problem.For the inverse problem of the source term,a modified regularization method was proposed.By this way,the inverse problem was transformed a minimum value problem of a function.The variational adjoint method was adopted to solve the gradient of the function.Then an iterative method was constructed.An inverse problem of nucleation rate and growth rate of the integral equation by the determined source term was studied.The inverse problem can be illustrated as a least squares problem and solved by the alternate direction search method.The uniqueness of the inverse problem was proven by a priori hypothesis.Finally,the numerical examples of inverse problems were compared with the real solutions.The effectiveness of the algorithm was shown.And some mathematical principles of crystal nucleation and growth during catalyst preparation were revealed.In the fourth chapter,the main content and innovation of this paper were summarized and given.That is,the crystallization process was modeled,the inversion of source terms and the first-kind Volterra integral equation was solved.And the goal of solving the nucleation rate and growth rate was achieved.