Finite volume approximation of the Maxwell's equations in nonhomogeneous media.

Chung Tsz Shun Eric.Thesis (M.Phil.)--Chinese University of Hong Kong, 2000.Includes bibliographical references (leaves 102-104).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Applications of Maxwell's equations --- p.1Chapter 1.2 --- Introduction to Maxwell's equations --- p.2Chapter 1.3 --- Historical outline of numerical methods --- p.4Chapter 1.4 --- A new approach --- p.5Chapter 2 --- Mathematical Backgrounds --- p.7Chapter 2.1 --- Sobolev spaces --- p.7Chapter 2.2 --- Tools from functional analysis --- p.8Chapter 3 --- Discretization of Vector Fields --- p.10Chapter 3.1 --- Domain triangulation --- p.10Chapter 3.2 --- Mesh dependent norms --- p.11Chapter 3.3 --- Discrete circulation operators --- p.13Chapter 3.4 --- Discrete flux operators --- p.20Chapter 4 --- Spatial Discretization of the Maxwell's Equations --- p.23Chapter 4.1 --- Derivation --- p.23Chapter 4.2 --- Consistency theory --- p.29Chapter 4.3 --- Convergence theory --- p.33Chapter 4.3.1 --- Polyhedral domain --- p.33Chapter 4.3.2 --- Rectangular domain --- p.38Chapter 5 --- Fully Discretization of the Maxwell's Equations --- p.63Chapter 5.1 --- Derivation --- p.63Chapter 5.2 --- Consistency theory --- p.65Chapter 5.3 --- Convergence theory --- p.69Chapter 5.3.1 --- Polyhedral domain --- p.69Chapter 5.3.2 --- Rectangular domain --- p.77Chapter 6 --- Numerical Tests --- p.97Chapter 6.1 --- Convergence test --- p.97Chapter 6.2 --- Electromagnetic scattering --- p.99Bibliography --- p.10

Application of Helmholtz/Hodge Decomposition to Finite Element Methods for Two-Dimensional Maxwell\u27s Equations

In this work we apply the two-dimensional Helmholtz/Hodge decomposition to develop new finite element schemes for two-dimensional Maxwell\u27s equations. We begin with the introduction of Maxwell\u27s equations and a brief survey of finite element methods for Maxwell\u27s equations. Then we review the related fundamentals in Chapter 2. In Chapter 3, we discuss the related vector function spaces and the Helmholtz/Hodge decomposition which are used in Chapter 4 and 5. The new results in this dissertation are presented in Chapter 4 and Chapter 5. In Chapter 4, we propose a new numerical approach for two-dimensional Maxwell\u27s equations that is based on the Helmholtz/Hodge decomposition for divergence-free vector fields. In this approach an approximate solution for Maxwell\u27s equations can be obtained by solving standard second order scalar elliptic boundary value problems. This new approach is illustrated by a P1 finite element method. In Chapter 5, we further extend the new approach described in Chapter 4 to the interface problem for Maxwell\u27s equations. We use the extraction formulas and multigrid method to overcome the low regularity of the solution for the Maxwell interface problem. The theoretical results obtained in this dissertation are confirmed by numerical experiments

Staggered discontinuous Galerkin methods for the three-dimensional Maxwell's equations on Cartesian grids.

在本文中，我們為了三維空間的馬克士威方程組(Maxwell’s equation)制定和分析了一套新種類的交錯間斷伽遼金(discontinuous Galerkin)方法，同時考慮了時間依賴性和時間諧波的馬克士威方程組。我們用了空間離散上交錯笛卡兒網格，這種方法具有許多良好的性質。首先，我們的方法所得出的數值解保留了電磁能量，並自動符合了高斯定律的離散版本。第二，質量矩陣是對角矩陣，從而時間推進是顯式和非常有效的。第三，我們的方法是高階準確，最佳收斂性在這裏會被嚴格地證明。第四，基於笛卡兒網格，它也很容易被執行，並可視為是典型的Yee’s Scheme的以及四邊形的邊有限元的推廣。最後，超收斂結果也會在這裏被證明。在本文中，我們還提供了幾個數值結果驗證了理論的陳述。我們計算了時間依賴性和時間諧波的馬克士威方程組數值收斂結果。此外，我們計算時間諧波馬克士威方程組特徵值問題的數值特徵值，並與理論特徵值比較結果。最後，完美匹配層(Perfect Matching Layer)吸收邊界的問題也有實行其數值結果。We develop and analyze a new type of staggered discontinuous Galerkin methods for the three dimensional Maxwell’s equations in this paper. Both time-dependent and time-harmonic Maxwell’s equations are considered. The spatial discretization is based on staggered Cartesian grids which possess many good properties. First of all, our method has the advantages that the numerical solution preserves the electromagnetic energy and automatically fulfills a discrete version of the Gauss law. Second, the mass matrices are diagonal, thus time marching is explicit and is very efficient. Third, our method is high order accurate and the optimal order of convergence is rigorously proved. Fourth, it is also very easy to implement due to its Cartesian structure and can be regarded as a generalization of the classical Yee’s scheme as well as the quadrilateral edge finite elements. Lastly, a superconvergence result, that is the convergence rate is one order higher at interpolation nodes, is proved.In this paper, we also provide several numerical results to verify the theoretical statements. We compute the numerical convergence order using L2-norm and discrete-norm respectively for both the time-dependent and time-harmonic Maxwell’s equations. Also, we compute the numerical eigenvalues for the time-harmonic eigenvalue problem and compare the result with the theoretical eigenvalues. Lastly, applications to problems in unbounded domains with the use of PML are also presented.Detailed summary in vernacular field only.Detailed summary in vernacular field only.Yu, Tang Fei.Thesis (M.Phil.)--Chinese University of Hong Kong, 2012.Includes bibliographical references (leaves 46-49).Abstracts also in Chinese.Chapter 1 --- Introduction and Model Problems --- p.1Chapter 2 --- Staggered DG Spaces --- p.4Chapter 2.1 --- Review on Gauss-Radau and Gaussisan points --- p.5Chapter 2.2 --- Basis functions --- p.6Chapter 2.3 --- Finite Elements space --- p.7Chapter 3 --- Method derivation --- p.14Chapter 3.1 --- Method --- p.14Chapter 3.2 --- Time discretization --- p.17Chapter 4 --- Energy conservation and Discrete Gauss law --- p.19Chapter 4.1 --- Energy conservation --- p.19Chapter 4.2 --- Discrete Gauss law --- p.22Chapter 5 --- Error analysis --- p.24Chapter 6 --- Numerical examples --- p.29Chapter 6.1 --- Convergence tests --- p.30Chapter 6.2 --- Diffraction by a perfectly conducting object --- p.30Chapter 6.3 --- Perfectly matched layers --- p.37Chapter 7 --- Time Harmonic Maxwell’s equations --- p.40Chapter 7.1 --- Model Problems --- p.40Chapter 7.2 --- Numerical examples --- p.40Chapter 7.2.1 --- Convergence tests --- p.41Chapter 7.2.2 --- Eigenvalues tests --- p.41Chapter 8 --- Conclusion --- p.45Bibliography --- p.4

A finite-difference method for the one-dimensional time-dependent schrödinger equation on unbounded domain

AbstractA finite-difference scheme is proposed for the one-dimensional time-dependent Schrödinger equation. We introduce an artificial boundary condition to reduce the original problem into an initial-boundary value problem in a finite-computational domain, and then construct a finite-difference scheme by the method of reduction of order to solve this reduced problem. This scheme has been proved to be uniquely solvable, unconditionally stable, and convergent. Some numerical examples are given to show the effectiveness of the scheme

Staggered discontinuous Galerkin method for the curl-curl operator and convection-diffusion equation.

Lee, Chak Shing."August 2011."Thesis (M.Phil.)--Chinese University of Hong Kong, 2011.Includes bibliographical references (leaves 60-62).Abstracts in English and Chinese.Chapter 1 --- Model Problems --- p.1Chapter 1.1 --- Introduction --- p.1Chapter 1.2 --- The curl-curl operator --- p.2Chapter 1.3 --- The convection-diffusion equation --- p.6Chapter 2 --- Staggered DG method for the Curl-Curl operator --- p.8Chapter 2.1 --- Introduction --- p.8Chapter 2.2 --- Discontinuous Galerkin discretization --- p.8Chapter 2.3 --- Stability for aligned fields --- p.14Chapter 2.4 --- Error estimates --- p.17Chapter 2.5 --- Numerical experiments --- p.21Chapter 2.6 --- Concluding Remarks --- p.32Chapter 3 --- Staggered DG method for the convection-diffusion equation --- p.33Chapter 3.1 --- Introduction --- p.33Chapter 3.2 --- Method description --- p.33Chapter 3.3 --- Preservation of physical structures --- p.38Chapter 3.4 --- Stability and convergence --- p.42Chapter 3.4.1 --- Static problem --- p.42Chapter 3.4.2 --- Time-dependent problem --- p.46Chapter 3.5 --- Fully discrete scheme --- p.49Chapter 3.6 --- Numerical examples --- p.55Chapter 3.6.1 --- The static problem --- p.55Chapter 3.6.2 --- Time dependent problem --- p.56Chapter 3.7 --- Concluding Remark --- p.59Bibliography --- p.6

Coupled structural, thermal, phase-change and electromagnetic analysis for superconductors, volume 2

Two families of parametrized mixed variational principles for linear electromagnetodynamics are constructed. The first family is applicable when the current density distribution is known a priori. Its six independent fields are magnetic intensity and flux density, magnetic potential, electric intensity and flux density and electric potential. Through appropriate specialization of parameters the first principle reduces to more conventional principles proposed in the literature. The second family is appropriate when the current density distribution and a conjugate Lagrange multiplier field are adjoined, giving a total of eight independently varied fields. In this case it is shown that a conventional variational principle exists only in the time-independent (static) case. Several static functionals with reduced number of varied fields are presented. The application of one of these principles to construct finite elements with current prediction capabilities is illustrated with a numerical example

Contribution to the characterization of stratified structures : electromagnetic analysis of a coaxial cell and a microstrip line

The objective of this dissertation is the development of electromagnetic modelling software specific to the cells of microwave material characterization. This development is based on numerical methods that are alternative to finite element method which is widely used in commercial software. For the need to extract the properties of materials by inverse modelling methods, research into the numerical efficiency of direct analysis is the focus in this thesis. The characterization targeted cells in this work concern a coaxial cell and a planar line. The presence of an unknown material is modelled by a stratified heterogeneous transmission structure. The application of the transverse operator method (TOM) on the multi-layered coaxial cell allowed the determination of the propagation constant of fundamental mode and its corresponding field distribution of the electromagnetic fields, and the characteristics of higher-order modes for the need of the characterization of discontinuities between empty line and loaded line. In the case of the microstrip line, the use of the modified transverse resonance method (MTRM) allowed the determination of characteristics of the fundamental and higher order modes. Since each cell consists of several different sections, the matrix S of the set will be determined by the use of the several modal methods, such as modal connection method (''mode matching'') and multimodal variational method (MVM). The direct analysis codes are coupled with several optimization programs to constitute the software for extracting the material parameters. These are applied to material samples in cylinder form holed by the coaxial cell, or thin rectangular wafer by the microstrip line. Broadband extraction results were obtained, values are comparable with those published. Both high-loss dielectrics and nanostructured materials have been studied by our method

Modeling EMI Resulting from a Signal Via Transition Through Power/Ground Layers

Signal transitioning through layers on vias are very common in multi-layer printed circuit board (PCB) design. For a signal via transitioning through the internal power and ground planes, the return current must switch from one reference plane to another reference plane. The discontinuity of the return current at the via excites the power and ground planes, and results in noise on the power bus that can lead to signal integrity, as well as EMI problems. Numerical methods, such as the finite-difference time-domain (FDTD), Moment of Methods (MoM), and partial element equivalent circuit (PEEC) method, were employed herein to study this problem. The modeled results are supported by measurements. In addition, a common EMI mitigation approach of adding a decoupling capacitor was investigated with the FDTD method

Singularities and Concentration Phenomena in Nonlinear Elliptic and Parabolic PDE's

[no abstract available