Time-integration methods for finite element discretisations of the second-order Maxwell equation

This article deals with time integration for the second-order Maxwell equations with possibly non-zero conductivity in the context of the discontinuous Galerkin finite element method DG-FEM) and the $H(\mathrm{curl})$-conforming FEM. For the spatial discretisation, hierarchic $H(\mathrm{curl})$-conforming basis functions are used up to polynomial order $p=3$ over tetrahedral meshes, meaning fourth-order convergence rate. A high-order polynomial basis often warrants the use of high-order time-integration schemes, but many well-known high-order schemes may suffer from a severe time-step stability restriction owing to the conductivity term. We investigate several possible time-integration methods from the point of view of accuracy, stability and computational work. We also carry out a numerical Fourier analysis to study the dispersion and dissipation properties of the semi-discrete DG-FEM scheme as well as the fully-discrete schemes with several of the time-integration methods. The dispersion and dissipation properties of the spatial discretisation and those of the time-integration methods are investigated separately, providing additional insight into the two discretisation steps

Exponential Krylov time integration for modeling multi-frequency optical response with monochromatic sources

Light incident on a layer of scattering material such as a piece of sugar or white paper forms a characteristic speckle pattern in transmission and reflection. The information hidden in the correlations of the speckle pattern with varying frequency, polarization and angle of the incident light can be exploited for applications such as biomedical imaging and high-resolution microscopy. Conventional computational models for multi-frequency optical response involve multiple solution runs of Maxwell's equations with monochromatic sources. Exponential Krylov subspace time solvers are promising candidates for improving efficiency of such models, as single monochromatic solution can be reused for the other frequencies without performing full time-domain computations at each frequency. However, we show that the straightforward implementation appears to have serious limitations. We further propose alternative ways for efficient solution through Krylov subspace methods. Our methods are based on two different splittings of the unknown solution into different parts, each of which can be computed efficiently. Experiments demonstrate a significant gain in computation time with respect to the standard solvers.Comment: 22 pages, 4 figure

Time-integration methods for finite element discretisations of the second-order Maxwell equation

This article deals with time integration for the second-order Maxwell equations with possibly non-zero conductivity in the context of the discontinuous Galerkin finite element method (DG-FEM) and the H(curl)-conforming FEM. For the spatial discretisation, hierarchic H(curl)-conforming basis functions are used up to polynomial order p = 3 over tetrahedral meshes, meaning fourth-order convergence rate. A high-order polynomial basis often warrants the use of high-order time-integration schemes, but many well-known high-order schemes may suffer from a severe time-step stability restriction owing to the conductivity term. We investigate several possible time-integration methods from the point of view of accuracy, stability and computational work. We also carry out a numerical Fourier analysis to study the dispersion and dissipation properties of the semi-discrete DG-FEM scheme as well as the fully-discrete schemes with several of the time-integration methods. The dispersion and dissipation properties of the spatial discretisation and those of the time-integration methods are investigated separately, providing additional insight into the two discretisation steps

Aplicação do método de Galerkin descontínuo para a análise de guias fotônicos

Orientador: Hugo Enrique Hernández FigueroaDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de ComputaçãoResumo: Um novo método de onda completo para realizar a análise modal em guias de onda é introduzido nesta dissertação. A ideia central por trás do método é baseada na discretização da equação de onda vetorial com o Método de Galerkin Descontínuo com Penalidade Interior (IPDG, do inglês Interior Penalty Discontinuous Galerkin). Com uma função de penalidade apropriada, um método de alta precisão e sem modos espúrios é obtido. A eficiência do método proposto é provada em vários guias de onda, incluindo complicados guias de ondas ópticos com modos vazantes e também em guias de onda plasmônicos. Os resultados foram comparados com os métodos do estado-da-arte descritos na literatura. Também é discutida a importância dessa nova abordagem. Além disso, os resultados indicam que o método é mais preciso do que abordagens anteriores baseadas em Elementos Finitos. As principais contribuições deste trabalho são: foi desenvolvido um novo método robusto e de alta precisão para a análise de guias de ondas arbitrários, uma nova função de penalidade para o IPDG foi proposta e aplicações práticas do método proposto são apresentadas. Adicionalmente, no apêndice é apresentado uma aplicação da análise modal em simulação eletromagnética 3D com um método de Galerkin DescontínuoAbstract: A novel full-wave method to perform mode analysis on waveguides is introduced in this dissertation. The core of the method is based on an Interior Penalty Discontinuous Galerkin (IPDG) discretization of the vector wave equation. With an appropriate penalty function a spurious-free and high accuracy method is achieved. The efficiency of the proposed method was proved in several waveguides, including intricate optical waveguides with leaky modes and also on plasmonic waveguides. The obtained results were compared with the state-of-the-art mode solvers described in the literature. Also, a discussion on the importance of this new approach is presented. Moreover, the results indicate that the proposed method is more accurate than the previous approaches based on Finite Elements Methods. The main contributions of this work are: the development of a novel robust and accurate method for the analysis of arbitrary waveguides, a new penalty function for the IPDG was proposed and practical applications of the methods are discussed. In addition, in the appendix an application of modal analysis on 3D electromagnetic simulations with a Discontinuous Galerkin method is detailedMestradoTelecomunicações e TelemáticaMestre em Engenharia ElétricaCAPE