Interior penalty DG methods for Maxwell’s equations in dispersive media

In this paper, we develop a fully-discrete interior penalty discontinuous Galerkin method for solving the time-dependent Maxwell’s equations in dispersive media. The model is described by a vector integral–differential equation. Our scheme is proved to be unconditionally stable and achieve optimal error estimates in both L² norm and energy norm. The scheme is implemented and numerical results supporting our analysis are presented.     

Spatial contraction of the Poincaré group and Maxwell’s equations in the electric limit

The contraction of the Poincaré group with respect to the space translations subgroup gives rise to a group that bears a certain duality relation to the Galilei group, that is, the contraction limit of the Poincaré group with respect to the time translations subgroup. In view of this duality, we call the former the dual Galilei group. A rather remarkable feature of the dual Galilei group is that the time translations constitute a central subgroup. Therewith, in unitary irreducible representations (UIRs) of the group, the Hamiltonian appears as a Casimir operator proportional to the identity H = EI, with E (and a spin value s) uniquely characterizing the representation. Hence, a physical system characterized by a UIR of the dual Galilei group displays no non-trivial time evolution. Moreover, the combined U(1) gauge group and the dual Galilei group underlie a non-relativistic limit of Maxwell’s equations known as the electric limit. The analysis presented here shows that only electrostatics is possible for the electric limit, wholly in harmony with the trivial nature of time evolution governed by the dual Galilei group.     

A linear nonconforming finite element method for Maxwell’s equations in two dimensions. Part I: Frequency domain

We suggest a linear nonconforming triangular element for Maxwell’s equations and test it in the context of the vector Helmholtz equation. The element uses discontinuous normal fields and tangential fields with continuity at the midpoint of the element sides, an approximation related to the Crouzeix–Raviart element for Stokes. The element is stabilized using the jump of the tangential fields, giving us a free parameter to decide. We give dispersion relations for different stability parameters and give some numerical examples, where the results converge quadratically with the mesh size for problems with smooth boundaries. The proposed element is free from spurious solutions and, for cavity eigenvalue problems, the eigenfrequencies that correspond to well-resolved eigenmodes are reproduced with the correct multiplicity.     

Effective coefficients of quasi-steady Maxwell’s equations with multiscale isotropic random conductivity

The effective coefficients in the quasi-steady Maxwell’s equations are calculated for a multiscale isotropic medium by using a subgrid modeling approach. The conductivity is mathematically represented by a Kolmogorov multiplicative continuous cascade with a lognormal probability distribution. The scale of the solution domain is assumed to be large as compared with the scale of heterogeneities of the medium. The theoretical results obtained in the paper are compared with the results of a direct 3D numerical simulation and the results of the conventional perturbation theory.     

Hybridizable discontinuous Galerkin methods for the time-harmonic Maxwell’s equations

We present two hybridizable discontinuous Galerkin (HDG) methods for the numerical solution of the time-harmonic Maxwell’s equations. The first HDG method explicitly enforces the divergence-free condition and thus necessitates the introduction of a Lagrange multiplier. It produces a linear system for the degrees of freedom of the approximate traces of both the tangential component of the vector field and the Lagrange multiplier. The second HDG method does not explicitly enforce the divergence-free condition and thus results in a linear system for the degrees of freedom of the approximate trace of the tangential component of the vector field only. For both HDG methods, the approximate vector field converges with the optimal order of k + 1 in the L²-norm, when polynomials of degree k are used to represent all the approximate variables. We propose elementwise postprocessing to obtain a new H^curl-conforming approximate vector field which converges with order k + 1 in the H^curl-norm. We present extensive numerical examples to demonstrate and compare the performance of the HDG methods.     

A fully coupled 3-D mixed finite element model of Biot consolidation

The numerical solution to the Biot equations of 3-D consolidation is still a challenging task because of the ill-conditioning of the resulting algebraic system and the instabilities that may affect the pore pressure solution. Recently new approaches have been advanced based on mixed formulations. In the present paper a fully coupled 3-D mixed finite element model is developed with the aim at alleviating the pore pressure numerical oscillations at the interface between materials with different permeabilities. A solution algorithm is implemented that takes advantage of the block structure of the discretized problem. The proposed model is verified against well-known analytical solutions and successfully experimented with in realistic applications of soil consolidation.     

Sufficient uniqueness conditions for the solution of the time harmonic Maxwell’s equations associated with surface impedance boundary conditions

We consider the time-harmonic Maxwell’s equations for the scattering or radiating problem from a 3-D object that is either a metallic surface coated with material layers (MCS) or a dichroic structure (DS) made up of multiple frequency selective surfaces (FSS) embedded in materials. Low or high order impedance boundary conditions (IBC) are employed to reduce the numerical complexity of the solution of this problem via an integral equation or a finite element formulation. An IBC links the tangential components of the electric field to those of the magnetic field on the outer surface of the MCS, or on the FSSs, and avoids the solution of Maxwell’s equations inside the inhomogeneous domain for a MCS or, for a DS, the meshing of the numerous unit cells in a FSS. Sufficient uniqueness conditions (SUC) are established for the solutions of Maxwell’s equations associated with these IBCs, the performances of which, when constrained by the corresponding SUCs, are numerically evaluated for an infinite or finite planar structure.     

Revisiting Maxwell’s accommodation coefficient: A study of nitrogen flow in a silica microtube across all flow regimes

Gas flows have been studied quantitatively for more than a hundred years and have relevance in modern fields such as the control of gas inputs to processes, the measurement of leak rates and the separation of gaseous species. Cha and McCoy have derived a convenient formula for the flow of an ideal gas applicable across a wide range of Knudsen numbers (Kn) that approaches the Navier–Stokes equations at small Kn and the Smoluchowski extension of the Knudsen flow equation at large Kn  . Smoluchowski’s result relies on the Maxwell definition of the tangential momentum accommodation coefficient α$\alpha$, recently challenged by Arya et al. We measure the flow rate of nitrogen gas in a smooth walled silica tube across a wide range of Knudsen numbers from 0.0048 to 12.4583. We find that the nitrogen flow obeys the Cha and McCoy equation with a large value of α$\alpha$, unlike carbon nanotubes which show flows consistent with a small value of α$\alpha$. Silica capillaries are therefore not atomically smooth. The flow at small Kn   has α=0.91$\alpha =0.91$ and at large Kn   has α$\alpha$ close to one, consistent with the redefinition of accommodation coefficient by Arya et al., which also resolves a problem in the literature where there are many observations of α$\alpha$ of less than one at small Kn and many equal to one at large Kn. Silica capillaries are an excellent choice for an accurate flow control system.     

The adaptive immersed interface finite element method for elliptic and Maxwell interface problems

We propose self-adaptive finite element methods with error control for solving elliptic and electromagnetic problems with discontinuous coefficients. The meshes in the methods do not need to fit the interfaces. New error indicators are introduced to control the error due to non-body-fitted meshes. Flexible h-adaptive strategies are developed, which can be systematically extended to a large class of interface problems. Extensive numerical experiments are performed to support the theoretical results and to show the competitive behavior of the adaptive algorithm even for interfaces involving corner or tip singularities.     

Solving Maxwell’s equations in singular domains with a Nitsche type method

In this paper, we propose and analyze a method derived from a Nitsche approach for handling boundary conditions in the Maxwell equations. Several years ago, the Nitsche method was introduced to impose weakly essential boundary conditions in the scalar Laplace operator. Then, it has been worked out more generally and transferred to continuity conditions. We propose here an extension to vector div–curl problems. This allows us to solve the Maxwell equations, particularly in domains with reentrant corners, where the solution can be singular. We formulate the method for both the electric and magnetic fields and report some numerical experiments.     

Generalization of the FDTD algorithm for simulations of hydrodynamic nonlinear Drude model

In this paper we present a numerical method for solving a three-dimensional cold-plasma system that describes electron gas dynamics driven by an external electromagnetic wave excitation. The nonlinear Drude dispersion model is derived from the cold-plasma fluid equations and is coupled to the Maxwell’s field equations. The Finite-Difference Time-Domain (FDTD) method is applied for solving the Maxwell’s equations in conjunction with the time-split semi-implicit numerical method for the nonlinear dispersion and a physics based treatment of the discontinuity of the electric field component normal to the dielectric-metal interface. The application of the proposed algorithm is illustrated by modeling light pulse propagation and second-harmonic generation (SHG) in metallic metamaterials (MMs), showing good agreement between computed and published experimental results.     

A hybrid finite/boundary element method for periodic structures on non-periodic meshes using an interior penalty formulation for Maxwell’s equations

This paper presents a hybrid finite element/boundary element (FEBE) method for periodic structures. Periodic structures have been efficiently analyzed by solving for a single unit cell utilizing Floquet’s theorem. However, most of the previous works require periodic meshes to properly impose the boundary conditions on the outer surfaces of the unit cell. To alleviate this restriction, the interior penalty method is adopted and implemented in this work. Also, the proper treatment of the boundary element part is addressed to account for the non-conformity of the boundary element mesh. Another ingredient of this work is the use of the efficient boundary element computation, accelerated by the Ewald transformation for the calculation of the periodic Green’s function. Finally, the method is validated through examples which are discretized without the constraint of a periodic mesh.     

The Discontinuous Galerkin Time-Domain method for Maxwell’s equations with anisotropic materials

The Discontinuous Galerkin method is an accurate and efficient way to numerically solve the time-dependent Maxwell equations. In this paper, we extend the basic, two-dimensional formulation for isotropic materials to allow anisotropic permittivity tensors. Using a reference system with an analytical solution, we demonstrate that our extensions do not alter the superior convergence characteristics of the fundamental algorithm. We further apply our method to cylindrical invisibility cloaks to investigate the performance which can be achieved in experiments.     

A stochastic mixed finite element heterogeneous multiscale method for flow in porous media

A computational methodology is developed to efficiently perform uncertainty quantification for fluid transport in porous media in the presence of both stochastic permeability and multiple scales. In order to capture the small scale heterogeneity, a new mixed multiscale finite element method is developed within the framework of the heterogeneous multiscale method (HMM) in the spatial domain. This new method ensures both local and global mass conservation. Starting from a specified covariance function, the stochastic log-permeability is discretized in the stochastic space using a truncated Karhunen–Loève expansion with several random variables. Due to the small correlation length of the covariance function, this often results in a high stochastic dimensionality. Therefore, a newly developed adaptive high dimensional stochastic model representation technique (HDMR) is used in the stochastic space. This results in a set of low stochastic dimensional subproblems which are efficiently solved using the adaptive sparse grid collocation method (ASGC). Numerical examples are presented for both deterministic and stochastic permeability to show the accuracy and efficiency of the developed stochastic multiscale method.     

Modeling light propagation in liquid crystal devices with a 3-D full-vector finite-element beam propagation method

A full-vector finite-element beam propagation method in 3-D is introduced for the simulation of light propagation in liquid crystal (LC) devices. The three electric field components are expressed in terms of mixed finite elements, providing the correct enforcement of boundary conditions. Moreover, the optical dielectric tensor of the medium can have all its nine elements nonzero, thus allowing the LC director to have an arbitrary orientation. A photonic crystal fiber with a LC infiltrated core and a homeotropic to multi-domain cell are analyzed. Comparison with other existing simulation techniques is provided, in order to validate the accuracy of the proposed method.     

统一坐标系下二维气动方程组的Runge-Kutta间断Galerkin有限元方法

构造了统一坐标系下二维可压缩气动方程组的Runge-Kutta 间断Galerkin(RKDG)有限元格式. 文中将流体力学方程组和几何守恒律统一求解, 所有计算都在固定的网格上进行, 在计算过程中不需要网格节点的速度信息. 文中对几个数值算例进行了数值模拟, 得到了较好的数值模拟结果.     

A RKDG finite element method for the one-dimensional inviscid compressible gas dynamics equations in Lagrangian coordinate

In this paper, Runge-Kutta Discontinuous Galerkin (RKDG) finite element method is presented to solve the one-dimensional inviscid compressible gas dynamic equations in Lagrangian coordinate. The equations are discretized by the DG method in space and the temporal discretization is accomplished by the total variation diminishing Runge-Kutta method. A limiter based on the characteristic field decomposition is applied to maintain stability and non-oscillatory property of the RKDG method. For multi-medium fluid simulation, the two cells adjacent to the interface are treated differently from other cells. At first, a linear Riemann solver is applied to calculate the numerical flux at the interface. Numerical examples show that there is some oscillation in the vicinity of the interface. Then a nonlinear Riemann solver based on the characteristic formulation of the equation and the discontinuity relations is adopted to calculate the numerical flux at the interface, which suppress the oscillation successfully. Several single-medium and multi-medium fluid examples are given to demonstrate the reliability and efficiency of the algorithm.