基于有限元方法的金属-介质周期结构的数学和计算研究

Metamaterials, artificial composite structures with exotic material properties, have attracted the attention of many investigators and secured an important position as a general research paradigm in today’s electromagnetics. This project will focus on the mathematical theory and numerical methods for some problems which emerge from practical applications of some metamaterials, such as dispersion medium and complex materials, in optical communication system. Because metamaterial structures include not only dielectrics but also metals, their extension to the optical domain also participates to the strong revival of plasmonics with the hope of truly sub-wavelength photonics. The research on mathematical theory and numerical methods for metamaterial structures has important theoretical significance and many practical applications..In this research, we will investigate the theoretical properties, mathematical models, numerical methods and optimization methods for the electromagnetic transmission in several metallic-dielectric and periodic nanostructures. In this project, we will investigate several nanostructures, include metal-dielectric photonic crystals, low loss metallic photonic crystal waveguide structures, metal-dielectric photonic crystal slabs, Gundu diffraction grating and metallic grating waveguide structures. Two typical mathematical problems are considered: one is a new eigenvalue problem for the two-dimensional dispersion medium in which the phase velocity of a wave depends on its frequency, and another is a scattering problem posed on the entire two-dimensional plane with metal-dielectric optical materials where the refractive index for an electromagnetic wave can be determined has a negative value over some frequency range. The research includes mathematical models, finite element analysis, adaptive finite element methods based on a posteriori error estimates, absorbing boundary conditions and perfect matching layers techniques, optimization theory, iteration algorithms, genetic algorithms and parallel algorithms. .All in all, the aim of this project is to provide the mathematical models, numerical methods and structure design guidance for some metallic-dielectric optical devices. Much of our research, however, is directed at constructing more efficient numerical methods and obtaining their theoretical properties, so that we can make new development for mathematics and other related subjects.

本项目拟致力于几类金属-介质周期结构在光通信系统实际应用中所产生问题的数学模型、有限元理论和算法、最优化算法和应用研究，包括：构建固定频率下色散介质周期结构传输特性的特征值微分方程系统，以金属-介质光子晶体为对象进行特征值问题的有限元理论分析和算法设计，构造后验误差估计子，设计自适应有限元算法，运用最优化理论和算法进行晶格结构设计研究，给出宽带隙结构与参数关系。开放性区域金属周期微纳结构散射问题的数学模型和数值模拟算法研究，以金属光子晶体波导、金属衍射光栅和金属光栅波导为研究对象进行有限元理论分析和算法设计，进行负数介电常数散射问题的自适应完美匹配层技术研究，自适应有限元算法的理论分析和算法实现，基于自适应有限元算法框架研究传输效率与波导结构参数的关系。本项目研究目的是为这几种金属-介质周期结构在光学器件提供新数值计算模型、高效数值计算方法、数学理论基础和结构优化设计的指导。

本项目研究内容包括理论和应用两个方面，涉及有限元方法、最优化理论与算法、以及若干基于光子晶体的光学元件的数值计算模型、数值计算方法、数学理论基础和结构优化设计等。..我们从金属光子晶体的数学模型和晶格结构的拓扑优化出发，研究了基于残量型后验误差估计子的自适应有限元方法、失败记忆驱动的自适应差分进化算法ATBDE；并基于求解带椭圆约束的PDE最优控制问题的理论分析设计了进行晶格结构优化问题求解的SIADMM算法。从金属光子晶体波导数学建模和数值模拟出发，研究了周期介质中弹性波散射问题的自适应有限元方法和两层介质中声波散射问题的数值方法，对于无界问题构造了基于PML技术的自适应 PML 有限元方法；并进行了多种弯曲波导光学传输性能的数值模拟和结构最优设计，实现了高传输效率的弯曲光子晶体波导的设计。针对双周期光栅，研究了时谐弹性波散射问题的PML方法，通过DtN算子与截断DtN算子之间的误差分析，证明了PML问题的适定性和指数收敛性，用设计的数值算法进行了光子晶体金属波导光栅中一些物理特性的数值模拟和优化设计，得到了优化结构。我们给出的一些新算法在精度或计算效率上优于原有算法，数值实验也表明了这些算法是有效的。本项目取得的主要学术成果在 “周期结构的有限元算法和智能算法及其在结构设计研究中的应用”、“基于PML技术的有限元算法及其在光子晶体波导中的应用”、“光子晶体光栅的有限元数学模型和算法研究”、“二次锥松弛约束的内在机理和拉格朗日对偶型算法研究”、“Maxwell方程和热方程耦合的非线性模型和阻抗边界条件的半平面声波散射问题的有限元方法”和“其它PDE问题的理论分析和算法设计及应用研究”等几个方面，上述部分结果曾获得2017年北京运筹学会年会青年优秀论文一等奖、2017年北京计算数学学会青年优秀论文二等奖，2020年北京运筹学会青年优秀论文奖。