Optimal Method Of Photonic Crystal Shape Based On Phase Field Model

In this paper,we discuss the shape optimization of photonic crystal problem.In the past 30 years,the shape optimization problem had obtained rapid devel-opment and application and there had been some researches on the problem.We use the phase field method to solve the shape optimization of photonic crystal problem.But there is much room for improvement in solving the model.This artical will focus on these problems,the main research content is as follows:First,we introduce the basis of shape optimization problem.We present the basic thought of the phase field method and the evolution equation of phase field method.Further more,we also deduce the solution space,base function and Galerkin method of the finite element method through Laplace eigenvalue problem.Then we introduce the process of FreeFem++ and use it to solve the shape optimization of a two-density drum problem by phase field method.Next,we describe the photonic crystal optimization problem in mathematics and deduce the finite element formate of state equation and evolution equation.In order to reduce the time of calculation,we use the multi-mesh method and adaptive mesh.We propose an essential of the multi-mesh approach about the interpolation and the projection between the fine mesh and the coarse one.In the numerial examples,the multi-mesh method is right and it can reduce 50V%time of the computation time of one mesh.Finally,because of the node element having physical defects in solving Maxwell equation,we consider to use the edge element.We deduce the base function of the edge element and give the edge element formulation of the typical finite element method of Maxwell eqaution.In the end,we solve the eigenvalue problem and scatter problem of Maxwell equation by edge element to verify its correction.