物理光学模型中特征值问题的分析、计算及优化

发布时间：2018-02-15 02:29

本文关键词： 特征值问题分片常数水平集方法反散射问题有限元方法敏感度分析　出处：《浙江大学》2016年博士论文　论文类型：学位论文

【摘要】：光学研究在生活生产中具有十分重要的意义。在工业上,它广泛应用于通讯业,计算机业,制造业及数据存储等的产品中。对于与之相关的物理现象。科学家们一直在试图建立有效的模型进行解释,比如说衍射光学中研究的重心在微光子原件上,由于其元件维度与可见光波长在同一单位上,如此小的结构规模通过经典几何光学已经无法解释了,需要通过求解对应方程,对在其中的光波的传递进行预测。光作为一种电磁波,其在介质中的传播,可以用Maxwell方程组来刻画,对于不同类型的材料,可以体现在方程本身的参数中,也可以体现在其方程所满足的边界上。我们希望可以不用实际物理实验就能对光在介质中的传播给出一些判断,比如说,光是否被吸收了?光是何种频率的?对光起作用的介质是什么形状的?这一系列问题都可以通过求解问题所对应的数学模型给出结论,我们关心的就是这一类具有实际物理意义的特征值问题。本文主要讨论两类光学中的数学模型及其中包含的特征值计算、分析及优化问题,分别为光子晶体形状优化问题和散射理论中的传递特征值问题。第一章对此两类问题的数学实质进行了简单介绍。第二章针对光子晶体这类特殊的材料的光子禁带最大化问题,提出了两类不同的数值方法进行优化：一类是分片常数水平集方法,首先将光子晶体结构使用分片常数水平集函数进行表示,其次使用有限元方法求解离散后的特征值问题,再将通过敏感度分析手段得到的目标泛函关于水平集函数的导数作为下降梯度,使用梯度下降法,得到最优结构,给出了TM模下,TE模下的光子禁带和完全光子禁带,其中TE模下的最大光子禁带为0.2922,为目前了解到的最大光子禁带；另一类方法是针对完全光子禁带最大化问题,提出将方程中的密度函数写成Fourier函数展开形式,将对晶体结构的优化问题转化为求解Fourier函数前面的系数的问题,同样得到了不错的数值结果。第三章是针对散射理论中具有重要意义的内／外传递特征值问题进行的研究,对于这一大类具有特殊结构的特征值问题,提出了一类新的数值求解算法。我们首先将问题转化为四阶形式,然后在变分意义下给出其问题的等价弱解形式,同时在变分内部进行变量替换,得到补充边界条件,使用有限元离散化后,问题转化为多项式特征值问题,最后将其线性化,最终数值结果显示算法能够高效,精确的一次解出多个特征值。第四章对全文的工作做了一下总结,并提出了未来的研究方向。
[Abstract]:Optical research is of great significance in the production of life. In industry, it is widely used in the communications industry, the computer industry, In products such as manufacturing and data storage, scientists have been trying to build effective models to explain the physical phenomena associated with it, for example, in diffraction optics, where the focus of research is on the original photons. Since the element dimension is in the same unit as the visible wave length, such a small structure can no longer be explained by classical geometrical optics, so it is necessary to solve the corresponding equation. The propagation of light in the medium as an electromagnetic wave can be described by Maxwell equations, and for different types of materials, it can be reflected in the parameters of the equation itself. We hope that we can give some judgment on the propagation of light in the medium without actual physical experiments. For example, is the light absorbed? What is the frequency of light? What is the shape of the medium acting on light? This series of problems can be obtained by solving the corresponding mathematical model of the problem. We are concerned with this kind of eigenvalue problems of practical physical significance. This paper mainly discusses the mathematical models of two kinds of optics and the calculation, analysis and optimization of the eigenvalues contained therein. In the first chapter, the mathematical essence of these two kinds of problems are briefly introduced. In the second chapter, we focus on the maximization of photonic bandgap for special materials such as photonic crystals. Two kinds of numerical methods are proposed for optimization: one is the level set method of piecewise constants. Firstly, the structure of photonic crystals is represented by the level set function of piecewise constants, and then the finite element method is used to solve the discrete eigenvalue problem. Then the derivative of the level set function of the target functional obtained by sensitivity analysis is taken as the descending gradient, and the optimal structure is obtained by using the gradient descent method. The photonic bandgap and the complete photonic band gap in te mode under TM mode are given. The maximum photonic bandgap in te mode is 0.2922, which is the largest photon gap known at present. Another method is to write the density function in the equation as Fourier function expansion for the problem of maximization of complete photonic bandgap. The optimization problem of crystal structure is transformed into the problem of solving the coefficients in front of Fourier function, and good numerical results are obtained. For this class of eigenvalue problems with special structure, a new numerical algorithm is proposed. Firstly, we transform the problem into fourth-order form, then give the equivalent weak solution form of the problem in the variational sense. At the same time, the variables are replaced inside the variation to obtain the supplementary boundary conditions. After discretization of finite element, the problem is transformed into polynomial eigenvalue problem. Finally, it is linearized. The final numerical results show that the algorithm is efficient. In chapter 4th, the work of this paper is summarized, and the future research direction is proposed.
【学位授予单位】：浙江大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O436

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