We propose transmission conditions of order $1$, $2$ and $3$ approximating the shielding behaviour of thin conducting curved sheets for the magneto-quasistatic eddy current model in 2D. This model reduction applies to sheets whose thicknesses $\eps$ are at the order of the skin depth or essentially smaller. The sheet has itself not to be resolved, only its midline is represented by an interface. The computation is directly in one step with almost no additional cost. We prove the well-posedness w.r.t.~to the small parameter $\eps$ and obtain optimal bound for the modelling error outside the sheet of order $\eps^{N+1}$ for the condition of order $N$. We end the paper with numerical experiments involving high order finite elements for sheets with varying curvature.
In this paper a high-order finite element method with curvilinear elements is proposed for the simulation of plasmonic structures. Most finite element packages use low order basis functions and non-curved elements, which is very costly for demanding problems such as the simulation of nano-antennas. To enhance the performance of finite elements, we use curvilinear quadrilateral elements to calculate the near-field from an impinging plane wave with
second order absorbing boundary conditions. The magnetic field amplitude on the surface of one object is compared with a computation based on a multiple multipole expansion. Moreover, the convergence behavior of p-FEM with absorbing boundary conditions motivate an adaptive strategy of polynomial degree enhancement and enlargement of the domain.
Three families of transmission conditions of different order are proposed for thin conducting sheets in the eddy current model. Resolving the thin sheet by a finite element mesh is often not possible. With these transmission conditions only the middle curve, but not the thin sheet itself, has not to be resolved by a finite element mesh. The families of transmission conditions are derived by an asymptotic expansion for small sheet thicknesses $\eps$, where each family results from a different asymptotic framework. In the first asymptotic framework the conductivity remains constant, scales with $1/\eps$ in the second and with $1/\eps^2$ in the third. The different asymptotics lead to different limit conditions, namely the vanishing sheet, a non-trivial borderline case, and the impermeable sheet, as well as different transmission conditions of higher orders. We investigated the stability, the convergence of the transmission conditions as well as their robustness. We call transmission conditions robust, if they provide accurate approximation for a wide range of sheet thicknesses and conductivities. We introduce an ordering of transmission conditions for the same sheet with respect to the robustness, and observe that the condition derived for the $1/\eps$ asymptotics is the most robust limit condition, contrary to order 1 and higher, where the transmission conditions derived for the $1/\eps^2$ asymptotics turn out to be most robust.
The computation of guided modes in photonic crystal wave-guides is a key issue in the process of designing devices in photonic communications. Existing methods, such as the super-cell method, provide an efficient computation of well-confined modes. However, if the modes are not well-confined, the modelling error of the super-cell method becomes prohibitive and advanced methods applying transparent boundary conditions for periodic media are needed. In this work we demonstrate the numerical realization of a recently proposed Dirichlet-to-Neumann approach and compare the results with those of the super-cell method. For the resulting non-linear eigenvalue problem we propose an iterative solution based on Newton's method and a direct solution using Chebyshev interpolation of the non-linear operator. Based on the Dirichlet-to-Neumann approach, we present a formula for the group velocity of guided modes that can serve as an objective function in the optimization of photonic crystal wave-guides.
Resolving thin conducting sheets for shielding or even skin layers inside by the mesh of numerical methods like the finite element method (FEM) can be avoided by using impedance transmission conditions (ITCs). Those ITCs shall provide an accurate approximation for small sheet thicknesses $d$, where the accuracy is best possible independent of the conductivity or the frequency being small or large -- this we will call robustness. We investigate the accuracy and robustness of popular and recently developed ITCs, and propose robust ITCs which are accurate up to $O(d^2)$.
In this paper we present an efficient algorithm for the calculation of photonic crystal band structures and band structures of photonic crystal waveguides. Our method relies on the fact that the dispersion curves of the band structure are smooth functions of the quasi-momentum in the one-dimensional Brillouin zone. We show the derivation and computation of the group velocity, the group velocity dispersion, and any higher derivative of the dispersion curves. These derivatives are then employed in a Taylor expansion of the dispersion curves. We control the error of the Taylor expansion with the help of a residual estimate and introduce an adaptive scheme for the selection of nodes in the one-dimensional Brillouin zone at which we solve the underlying eigenvalue problem and compute the derivatives of the dispersion curves. The proposed algorithm is not only advantageous as it decreases the computational effort to compute the band structure but also because it allows
for the identification of crossings and anti-crossings of dispersion curves, respectively. This identification is not possible with the standard approach of solving the underlying eigenvalue problem at a discrete set of values of the quasi-momentum without taking the mode parity into account.
The efficient and reliable computation of guided modes in photonic crystal wave-guides is of great importance for designing optical devices. Transparent boundary conditions based on Dirichlet-to-Neumann operators allow for an exact computation of well-confined modes and modes close to the band edge in the sense that no modelling error is introduced. The well-known super-cell method, on the other hand, introduces a modelling error which may become prohibitively large for guided modes that are not well-confined. The Dirichlet-to-Neumann transparent boundary conditions are, however, not applicable for all frequencies as they are not uniquely defined and their computation is unstable for a countable set of frequencies that correspond to so called Dirichlet eigenvalues. In this work we describe how to overcome this theoretical difficulty introducing Robin-to-Robin transparent boundary conditions whose construction do not exhibit those forbidden frequencies. They seem, hence, well suited for an exact and reliable computation of guided modes in photonic crystal wave-guides.
With symmetric local absorbing boundary conditions for the Helmholtz equation scattering problems can be solved on a truncated domain, where the outgoing radiation condition is approximated by a Dirichlet-to-Neumann map with higher tangential derivatives on its outer boundary. Feng's conditions are symmetric local absorbing boundary conditions, which are based on an asymptotic expansion of the coefficients of the exact Dirichlet-to-Neumann map for large radia of the circular outer boundary. In this article we analyse the well-posedness of variational formulations with symmetric local absorbing boundary conditions in general and show how the modelling error introduced by Feng's conditions depends on the radius of the truncated domain.
Shielding sheets are commonly used in the protection of electronic devices. With their large aspect ratios they become a serious issue for the direct application of boundary element methods due to the occurring almost singular integrals. Impedance transmission conditions allow for boundary element formulations where only sheet mid-line has to be triangulated. We propose and analyse boundary element methods of second kind for impedance transmission conditions of two different types for the time-harmonic eddy current problem in two dimensions. In numerical experiments we study the effect on the magnetic shielding efficiency of the discretisation error due the boundary element discretisation and the modelling error due to the
impedance transmission condition.
In this paper, we introduce a high order finite element (FEM) implementation using perfectly matched layer (PML) for the scattering by plasmonic structures inside layered media. The PML is proven to be very accurate and efficient by a comparative analysis with a commercial FEM software and the Multiple Multipole Program (MMP). A convergence analysis using hp-adaptive refinement inside the PML layer
shows that adaptive mesh refinement inside the PML layer is most efficient. Based on this convergence analysis an hp-strategy is proposed, which shows a remarkable error reduction for small additional computational costs.