基于间断有限元方法的光子晶体计算

发布时间：2018-04-09 05:00

本文选题：Maxwell方程　切入点：光子晶体　出处：《中国科学技术大学》2017年博士论文

【摘要】：本文主要研究与光子晶体有关的微分算子谱的逼近理论与应用。光子晶体是具有折射率的材料在空间中周期排布形成的结构。在适当的材料特性和晶体的几何形状的条件下,某些频率的光是不能在里面传播的,而且反射率可达百分之百,而其他电磁波可以几乎无损耗地传播,这一特性可以用来控制电磁波的传播和辐射。Maxwell方程可以用来描述电磁波在其中传播时的行为,而且模型精确性很高,其模拟结果和实验观测相差较小,是研究光子晶体电磁特性的一种重要途径。首先,我们分析了带周期系数的Maxwell方程的间断有限元离散。全空间中的Maxwell算子的谱是连续的,无法通过数值计算来逼近。通过Bloch/Floquet理论,将R3中的带周期系数的Maxwell算子转化为具有周期边界的问题,这样就可以采用修正的Nedelec基函数来构造间断有限元格式进行数值逼近。我们证明了混合间断有限元格式的收敛性。然而,这种收敛只是点点收敛,而对于有界算子,数值特征值的正确收敛与其数值算子的一致收敛是充要条件。为此,我们分析数值解空间的离散紧性,证明了该数值算子是一列相对紧算子,继而证明了其收敛是一致收敛。我们还证明了数值特征值的收敛速度是基函数次数与解的正则性最小值的两倍,该结果在二维和三维的数值例子上得到了验证。最后我们应用这些方法计算了几种常见的光子晶体模型的能带结构。当光子晶体的材料性质为非线性时,会导出非线性特征值问题。我们主要研究了 Drude模型和Lorentz模型等有理型非线性模型所产生的多项式特征值问题。我们将多项式特征值问题处理成与其具有相同谱的线性化算子来分析。而线性化算子通常不是有界算子,不能再用范数来描述其数值算子的收敛性。为此我们引入gap作为衡量算子间距离的工具,将紧算子或有界算子谱的逼近理论推广到了一般的线性算子,然后应用其构造了多项式特征值问题的逼近理论。我们研究了多项式特征值问题中的本质谱,证明了本质谱在相对紧摄动下的稳定性。对于用具有有理型非线性介电常数的材料制成的晶体,即使其几何形状不同,其本质谱保持不变。这样我们通过分析就能精确地知道其本质谱的位置。在数值计算相应的离散形式的多项式矩阵特征值问题时,可以避开本质谱附近聚集了大量数值特征值的小区域,从而只需要计算一小部分数值特征值,就能知道谱的分布情况。计算量的降低使计算机可以相对容易地处理大型离散的多项式特征值问题,从而允许网格进一步细化,提高数值特征值的精度。
[Abstract]:In this paper, the approximation theory and application of differential operator spectrum related to photonic crystals are studied.Photonic crystals are structures of materials with refractive index arranged periodically in space.With proper material properties and the geometrical shape of the crystal, light at certain frequencies cannot propagate inside, and the reflectivity can reach 100 percent, while other electromagnetic waves can propagate almost without loss.This characteristic can be used to control the propagation of electromagnetic wave and radiation. Maxwell equation can be used to describe the behavior of electromagnetic wave propagating in it.It is an important way to study the electromagnetic properties of photonic crystals.First, we analyze the discontinuous finite element discretization of the Maxwell equation with periodic coefficients.The spectrum of Maxwell operators in the whole space is continuous and can not be approximated by numerical calculation.By means of Bloch/Floquet theory, the Maxwell operator with periodic coefficients in R3 is transformed into a problem with periodic boundary, so the modified Nedelec basis function can be used to construct the discontinuous finite element scheme for numerical approximation.We prove the convergence of the mixed discontinuous finite element scheme.However, this convergence is only a point convergence, and for bounded operators, the correct convergence of numerical eigenvalues and the uniform convergence of its numerical operators are sufficient and necessary conditions.Therefore, we analyze the discrete compactness of numerical solution space, prove that the numerical operator is a series of relative compact operators, and then prove that its convergence is uniformly convergent.We also prove that the convergence rate of the numerical eigenvalue is twice as fast as the minimum value of regularity of the order and solution of the basis function. The results are verified in two and three dimensional numerical examples.Finally, we use these methods to calculate the band structures of several common photonic crystal models.When the material property of photonic crystal is nonlinear, the nonlinear eigenvalue problem is derived.We mainly study the polynomial eigenvalue problems generated by rational nonlinear models such as Drude model and Lorentz model.We treat the polynomial eigenvalue problem as a linearized operator with the same spectrum as the polynomial eigenvalue problem.The linearization operator is usually not a bounded operator, and the convergence of its numerical operator can no longer be described by norm.In this paper, gap is introduced as a tool to measure the distance between operators. The approximation theory of the spectrum of compact or bounded operators is extended to general linear operators, and then the approximation theory of polynomial eigenvalue problems is constructed.We study the essential spectrum of polynomial eigenvalue problems and prove the stability of this mass spectrum under relatively compact perturbation.For a crystal made of a material with a rational nonlinear dielectric constant, the essential spectrum remains unchanged even if its geometry is different.In this way, we can accurately know the location of its essential spectrum through analysis.In the numerical calculation of the corresponding discrete polynomial matrix eigenvalue problem, we can avoid the small region where a large number of numerical eigenvalues are gathered around the mass spectrometry, so that only a small part of the numerical eigenvalues need to be calculated to know the distribution of the spectrum.The reduction of computational complexity makes it relatively easy for the computer to deal with large discrete polynomial eigenvalue problems, thus allowing the mesh to be further refined and to improve the accuracy of numerical eigenvalues.
【学位授予单位】：中国科学技术大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O241.82

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