Electric and magnetic losses modeled by a stable hybrid with explicit–implicit time-stepping for Maxwell’s equations
Introduction
Radar absorbing materials (RAM) with electric and magnetic losses are important for the reduction of the radar cross section (RCS) in stealth applications [1]. Optimized designs may involve geometries with sharp edges and inhomogeneous materials. Thus, local mesh refinement may be necessary to resolve rapid variations in the electromagnetic field and/or the material parameters.
Ampère’s law with electric losses modeled by an electric conductivity and Faraday’s law with a magnetic conductivity that models the magnetic losses can be time-stepped as a system of first order differential equations. Several electromagnetic field solvers in the time-domain exploit this form of Maxwell’s equations. Taflove and Hagness [2] describe a conventional finite-difference time-domain (FDTD) scheme based on leap-frog time integration, which suffers from the staircase approximation inherent to the FDTD scheme formulated on structured (Cartesian) grids. Rodrigue and White [3] use hexahedral finite elements for the spatial discretization and time step the coupled Maxwell’s equations with a leap-frog scheme, which gives a method that reduces to the FDTD scheme on rectilinear grids. The algorithm presented by Rodrigue and White [3] does not allow for unconditionally stable time-stepping and they do not consider tetrahedral meshes that are useful for local mesh refinement. Rieben, Rodrigue and White also published a similar method [4] that exploits higher order approximations for the spatial and temporal discretization. Riley and Jin [5] use finite-element (FE) techniques to discretize with respect to space and they arrive at a wave equation for the electric field that also involves the magnetic field. They use an explicit update algorithm of leap-frog type to compute the magnetic field by means of integration of Faraday’s law. However, the time-stepping scheme used for the updating of the wave equation is not mentioned or described in their paper. Furthermore, their paper does not provide any information on the stability properties for their time-domain method.
A combination of the FDTD scheme applied to large homogeneous regions for efficiency and the finite-element method (FEM) for regions with complicated geometry and materials is attractive for many scattering problems. Wu and Itoh proposed FEM–FDTD hybridizations for both two [6] and three [7] dimensions. These schemes suffer from late-time instabilities that may be damped by temporal filtering [8]. Abenius et al. [9] combine the FDTD scheme with an implicit FEM and numerical studies indicate that it is stable, although no formal proof of stability is given. Monorchio et al. [10], [11] proposed a hybrid that suffers from late-time instabilities and some attempts to mitigate this problem involve averaging or extrapolation techniques. Marrone and Mittra describe a way of interfacing triangles [12] and tetrahedrals [13] to FDTD cells but no explicit Courant criterion is derived. Rylander and Bondeson presented a stable FEM–FDTD hybrids [14], [15] for 3D problems that are stable up to the Courant condition of the FDTD scheme, where convergence studies [15] and proofs of stability [15], [16] are available in the literature. We emphasize that the treatment of magnetically lossy materials is not considered in any of these articles on hybrid FEM–FDTD algorithms.
There is a broad selection of numerical techniques that are formulated in the frequency domain [17], [18], such as the method of moments (MoM) that may be accelerated by the multi-level fast multipole method (MLFMM) for electrically large scattering problems. The MoM is particularly efficient for scattering problems with many different incident angles. However, it provides the response for only one single frequency in contrast to time-domain field solvers that yield the response in a broad frequency-interval. The MoM can handle inhomogeneous materials and it may be formulated in the time domain, but such methods are computationally expensive and difficult to program. For problems that feature non-linear media, frequency domain methods in general and the MoM in particular are inappropriate, if not impossible, to use.
In this paper, we present a stable FEM–FDTD hybrid method for electromagnetic problems in two dimensions that feature complex geometry with materials that have both electric and magnetic losses. In contrast to what is available in the open literature on the FEM treatment of magnetic losses in combination with electric losses, our method is distinguished by a number of unique features: (i) an unconditionally stable time-stepping scheme based on FE techniques and Galerkin’s method applied to the first order system of Ampère’s and Faraday’s law; (ii) a proof of stability for this type of implicit FEM; and (iii) a generalization of the stable FEM–FDTD hybrid [14] for 2D problems that is stable up to the Courant condition of the FDTD scheme. The unconditionally stable time-stepping scheme that we present in this paper, reduces to a special case of the conventional Newmark scheme [19] when it is applied to problems without magnetic losses. The FE techniques used to construct the unconditionally stable time-integration scheme offers the possibility to also treat dispersive materials [20]. In addition, we demonstrate that it is feasible to use the FEM–FDTD interface in regions that have both electric and magnetic losses. We would like to stress that the two-dimensional case is important for the design of wing profiles intended for stealth aircraft, and we will consider the corresponding 3D formulation in a future publication since the 3D Maxwell problem is significantly different from the 2D problem. For the hybrid applied to the transverse electric (TE) case, the electric field is expanded in terms of linear edge elements and the magnetic field by piecewise constants – a construction that is feasible in 2D but yields an incorrect formulation for 3D problems since the 3D problem requires that the magnetic flux density is expanded in terms of divergence conforming elements [3], [4], [21]. The hybrid applied to the transverse magnetic (TM) case follows by duality.
We find it advantageous to treat the electric and magnetic losses by the implicitly time-stepped FEM for a number of reasons: (i) local mesh refinement can be used in the unstructured FEM region to resolve rapid variations in the electromagnetic field and/or the radar absorber’s material parameters; (ii) the FDTD scheme is highly efficient for the free-space region; (iii) conventional FDTD techniques [2] such as the perfectly matched layer, Huygen’s surface and near-to-far-field transformations can be directly applied in the FDTD region; (iv) it is possible to prove stability for this type of hybrid scheme; (v) the global time step is related to the highest frequency of interest and the Courant condition specifies an appropriate cell size for the free-space FDTD region; and (vi) the choice of the global time step does not depend on the cell size in the unstructured mesh. To summarize, our hybrid method combines the efficiency of the FDTD scheme with the excellent capabilities of the FEM to model curved boundaries, lossy inhomogeneous materials, rapid field variations and other challenges that are difficult to handle on structured grids with explicit time-stepping schemes. To the best of our knowledge, the hybrid algorithm for Maxwell equations with electric and magnetic losses that we propose has not been published in the open literature and it provides a unique combination of characteristic features that can not be offered by other existing methods.
Section snippets
The hybrid method
The two-dimensional cylinder scatterer typically consists of metal structures that are partially or completely covered by RAM for the purpose of reducing the RCS. This allows for an outer boundary designed with respect to e.g. aerodynamical performance. The metal is modeled as a perfect electric conductor (PEC) and the losses of the RAM by an electric conductivity and a corresponding magnetic conductivity . The conductivities and can be functions of the space coordinate and,
Numerical results
In order to verify the accuracy and convergence properties of the hybrid method, we apply the method to three different test problems. The first problem consists of a circular metal cylinder with a lossy coating. The second problem is a homogeneous lossy circular cylinder. The RCS can be calculated analytically for both these problems. The third problem is a metal airfoil with its leading edge covered by a lossy material. In all problems, the scatterer is illuminated by a plane wave that is
Conclusions
We have presented a stable FEM–FDTD hybrid in two dimensions that allows for both electric and magnetic losses. For the transverse electric (TE) case, the electric field is expanded in linear edge elements on the computational domain, and the magnetic field is expressed in terms of piecewise constant basis functions. The transverse magnetic (TM) case follows from duality. The computational domain is divided in two parts: (i) homogeneous space discretized by a structured grid of rectangles; and
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