Kersten Schmidt

Estimating the Eddy-Current Modeling Error

The eddy-current model is an approximation of the full Maxwell equations. We will give estimates for the modeling error and show how the constants in the estimates are influenced by the geometry of the problem. Additionally, we analyze the asymptotic behavior of the modeling error when the angular frequency tends to zero. The theoretical results are complemented by numerical examples using high order finite elements. These demonstrate that the estimates are sharp. Hence, this work delivers a mathematical basis for assessing the scope of the eddy-current model.

A Unified Analysis of Transmission Conditions for Thin Conducting Sheets in the Time-Harmonic Eddy Current Model

We introduce tools for a unified analysis and a comparison of impedance transmission conditions (ITCs) for thin conducting sheets within the time-harmonic eddy current model in two dimensions. The first criterion is the robustness with respect to the frequency or skin depth, which means whether they give meaningful results for small and for large frequencies or conductivities. As a second tool we study the accuracy for a range of sheet thicknesses and frequencies for a relevant example and finally analyze their asymptotic order in different asymptotic regimes. For the latter we write all the ITCs in a common form and show how they can be realized within the finite element method. Two new conditions, which we call ITC-2-0 and ITC-2-1, are introduced in this article, which appear in a symmetric form. They are derived by asymptotic expansions in the asymptotic regime of constant ratio between skin depth and thickness like those in [K. Schmidt and A. Chernov, Technical report 1102, Institut for Numerical Simu...

A multiscale hp-FEM for 2D photonic crystal bands

A multiscale generalised hp-finite element method (MSFEM) for time harmonic wave propagation in bands of locally periodic media of large, but finite extent, e.g., photonic crystal (PhC) bands, is presented. The method distinguishes itself by its size robustness, i.e., to achieve a prescribed error its computational effort does not depend on the number of periods. The proposed method shows this property for general incident fields, including plane waves incident at a certain angle to the infinite crystal surface, and at frequencies in and outside of the bandgap of the PhC. The proposed MSFEM is based on a precomputed problem adapted multiscale basis. This basis incorporates a set of complex Bloch modes, the eigenfunctions of the infinite PhC, which are modulated by macroscopic piecewise polynomials on a macroscopic FE mesh. The multiscale basis is shown to be efficient for finite PhC bands of any size, provided that boundary effects are resolved with a simple macroscopic boundary layer mesh. The MSFEM, constructed by combing the multiscale basis inside the crystal with some exterior discretisation, is a special case of the generalised finite element method (g-FEM). For the rapid evaluation of the matrix entries we introduce a size robust algorithm for integrals of quasi-periodic micro functions and polynomial macro functions. Size robustness of the present MSFEM in both, the number of basis functions and the computation time, is verified in extensive numerical experiments.

Robust Transmission Conditions of High Order for Thin Conducting Sheets in Two Dimensions

Resolving thin conducting sheets for shielding or even skin layers inside by the mesh of numerical methods like the finite-element method can be avoided 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(d2).

Numerical realization of Dirichlet-to-Neumann transparent boundary conditions for photonic crystal wave-guides

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 Newtons 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.

High-order asymptotic expansion for the acoustics in viscous gases close to rigid walls

We derive a complete asymptotic expansion for the singularly perturbed problem of acoustic wave propagation inside gases with small viscosity. This derivation is for the non-resonant case in smooth bounded domains in two dimensions. Close to rigid walls the tangential velocity exhibits a boundary layer of size where η is the dynamic viscosity. The asymptotic expansion, which is based on the technique of multiscale expansion is expressed in powers of and takes into account curvature effects. The terms of the velocity and pressure expansion are defined independently by partial differential equations, where the normal component of velocities or the normal derivative of the pressure, respectively, are prescribed on the boundary. The asymptotic expansion is rigorously justified with optimal error estimates.

On the homogenization of thin perforated walls of finite length

The present work deals with the resolution of the Poisson equation in a bounded domain made of a thin and periodic layer of finite length placed into a homogeneous medium. We provide and justify a high order asymptotic expansion which takes into account the boundary layer effect occurring in the vicinity of the periodic layer as well as the corner singularities appearing in the neighborhood of the extremities of the layer. Our approach combines the method of matched asymptotic expansions and the method of periodic surface homogenization.

Non-conforming Galerkin finite element methods for local absorbing boundary conditions of higher order

A new non-conforming finite element discretization methodology for second order elliptic partial differential equations involving higher order local absorbing boundary conditions in 2D and 3D is proposed. The novelty of the approach lies in the application of C 0 -continuous finite element spaces, which is the standard discretization of second order operators, to the discretization of boundary differential operators of order four and higher. For each of these boundary operators, additional terms appear on the boundary nodes in 2D and on the boundary edges in 3D, similarly to interior penalty discontinuous Galerkin methods, which leads to a stable and consistent formulation. In this way, no auxiliary variables on the boundary have to be introduced and trial and test functions of higher smoothness along the boundary are not required. As a consequence, the method leads to lower computational costs for discretizations with higher order elements and is easily integrated in high-order finite element libraries. A priori h -convergence error estimates show that the method does not reduce the order of convergence compared to usual Dirichlet, Neumann or Robin boundary conditions if the polynomial degree on the boundary is increased simultaneously. A series of numerical experiments illustrates the utility of the method and validates the theoretical convergence results.

Asymptotic expansion techniques for singularly perturbed boundary integral equations

For the case of singularly perturbed elliptic transmission problems we demonstrate the use of asymptotic expansion techniques both for establishing regularity results for the solution and for deriving a priori error estimates for boundary element Galerkin discretisation. The dependence of the corresponding bounds on the singular perturbation parameter is studied in detail. This dependence clearly manifests itself in numerical experiments.

Dirichlet-to-Neumann Transparent Boundary Conditions for Photonic Crystal Waveguides

In this paper, we present a complete algorithm for the exact computation of the guided mode band structure in photonic crystal (PhC) waveguides. In contrast to the supercell method, the used approach does not introduce any modeling error and is hence independent of the confinement of the modes. The approach is based on Dirichlet-to-Neumann transparent boundary conditions that yield a nonlinear eigenvalue problem. For the solution of this nonlinear eigenvalue problem, we present a direct technique using Chebyshev interpolation that requires a bandgap calculation of the PhC in advance. For this bandgap calculation, we introduce as a very efficient tool a Taylor expansion of the PhC band structure. We show that our algorithm-like the supercell method-converges exponentially, however, its computational costs-in comparison with the supercell method-only increase moderately since the size of the matrix to be inverted remains constant.