Mustafa Kuzuoglu

Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers

Perfectly matched layers (PMLs), which are employed for mesh truncation in the finite-difference time-domain (FDTD) or in finite element methods (FEMs), can be realized by artificial anisotropic materials with properly chosen permittivity and permeability tensors. The tensor constitutive parameters must satisfy the Kramers-Kronig relationships, so that the law of causality holds. These relations are used to relate the real and imaginary parts of the constitutive parameters of the PML media to deduce the asymptotic behaviors of these parameters at low and high frequencies.

Investigation of nonplanar perfectly matched absorbers for finite-element mesh truncation

We present a detailed theoretical and numerical investigation of the perfectly matched layer (PML) concept as applied to the problem of mesh truncation in the finite-element method (FEM). We show that it is possible to extend the Cartesian PML concepts involving half-spaces to cylindrical and spherical geometries appropriate for closed boundaries in two and three dimensions by defining lossy anisotropic layers in the relevant coordinate systems. By using the method of separation of variables, it is possible to solve the boundary value problems in these geometries. The analytical solutions demonstrate that under certain conditions, outgoing waves are absorbed with negligible reflection, and the transmitted wave is attenuated within the PML. To reduce the white space in radiation or scattering problems, conformal PMLs are constructed via parametric mappings. It is also verified that the PML concept, which was originally introduced for problems governed by Maxwells equations, can be extended to cases governed by the scalar Helmholtz equation. Finally, numerical results are presented to demonstrate the use of the PML in FEM mesh truncation.

Electrical impedance tomography using induced currents

The mathematical basis of a new imaging modality, induced current electrical impedance tomography (EIT), is investigated, The ultimate aim of this technique is the reconstruction of conductivity distribution of the human body, from voltage measurements made between electrodes placed on the surface, when currents are induced inside the body by applied time varying magnetic fields. In this study the two-dimensional problem is analyzed. A specific 9-coil system for generating nine different exciting magnetic fields (50 kHz) and 16 measurement electrodes around the object are assumed, The partial differential equation for the scaler potential function in the conductive medium is derived and finite element method (FEM) is used for its solution. Sensitivity matrix, which relates the perturbation in measurements to the conductivity perturbations, is calculated. Singular value decomposition of the sensitivity matrix shows that there are 135 independent measurements. It is found that measurements are less sensitive to changes in conductivity of the objects interior. While in this respect induced current EIT is slightly inferior to the technique of injected current EIT (using Sheffield protocol), its sensitivity matrix is better conditioned. The images obtained are found to be comparable to injected current EIT images In resolution. Design of a coil system for which parameters such as sensitivity to inner regions and condition number of the sensitivity matrix are optimum, remains to be made.

Non-Maxwellian Locally-Conformal PML Absorbers for Finite Element Mesh Truncation

We introduce the locally-conformal perfectly matched layer (PML) approach, which is an easy and straightforward PML implementation, to the problem of mesh truncation in the finite element method (FEM). This method is based on a locally-defined complex coordinate transformation which has no explicit dependence on the differential geometric characteristics of the PML-free space interface. As a result, it is possible to handle challenging PML geometries with interfaces having arbitrary curvature, especially those with curvature discontinuities. In order to implement this approach, we also introduce the concept of complex space FEM using elements with complex nodal coordinates. After developing the analytical background of this method, we present some numerical results to demonstrate the performance of this method in three-dimensional electromagnetic scattering problems

Utilization of Anisotropic Metamaterial Layers in Waveguide Miniaturization and Transitions

We introduce a new approach which enables a waveguide to support propagation of electromagnetic waves below the cutoff frequency, as well as which avoids undesirable reflections in a waveguide. These are achieved through the usage of an anisotropic metamaterial layer by employing the concept of coordinate transformation. The proposed method can be utilized for the fabrication of miniaturized waveguides, and for the elimination of discontinuities in abrupt waveguide transitions. We demonstrate some numerical experiments for finite element simulations of parallel-plate and dielectric slab waveguides.

Finite element solution of electromagnetic problems over a wide frequency range via the Padé approximation

In electromagnetic wave propagation problems, it is usually necessary to calculate the field quantities over a wide band of frequencies. In this paper, we develop a computationally-efficient scheme, which combines the finite element method (FEM) with the Pade approximation procedure, to derive the power series expansion of the unknown solution vector in terms of the frequency. Explicit power series expressions of the matrix operator are obtained for boundary value problems that are defined, not only over bounded spatial domains, but also over unbounded domains truncated either by an absorbing boundary condition (ABC) or by a perfectly matched layer (PML). It is shown that the FEM matrix is always a polynomial function of the frequency variable, even with the ABC or PML mesh truncations. The coefficients of the power series expansion are obtained iteratively, and certain a priori estimates are derived for the radius of convergence of this series expansion. Finally, Pade approximants are utilized to extend the region of convergence of the power series, enabling us to cover the frequency band with a minimum number of LU decompositions.

Monte Carlo-Based Characteristic Basis Finite-Element Method (MC-CBFEM) for Numerical Analysis of Scattering From Objects On/Above Rough Sea Surfaces

The Monte Carlo-based Characteristic Basis Finite-Element Method (MC-CBFEM) is developed for predicting the statistical properties of the 2-D electromagnetic scattering from objects (such as ship- and decoy-like objects) on or above random rough sea surfaces. At each realization of the Monte Carlo technique, the 1-D rough sea surface is randomly generated by using the Pierson-Moskowitz spectrum, and the bistatic radar cross section (RCS) is computed by employing the CBFEM approach. The CBFEM is a noniterative domain decomposition finite-element algorithm, which is designed to alleviate the challenges of the conventional finite-element method in solving large-scale electromagnetic problems. The CBFEM partitions the problem into a number of nonoverlapping subdomains and generates physics-based characteristic basis functions for the representation of the fields in each subdomain. Since this approach reduces the matrix size and lends itself to convenient parallelization, it is attractive for efficiently solving large-scale problems many times in the Monte Carlo simulation with the use of direct solvers and small-sized matrices. For a number of surface realizations, each of which can be considered as a sample from the random process specifying the surface, a family of bistatic RCS values is obtained as a function of incidence angle and surface roughness (or wind speed). The coherent (mean) and incoherent (variance) components of the RCS are illustrated with particular emphasis on the effects of surface roughness and the angles near grazing. Statistical characterization is also achieved by other means, such as correlation coefficient and density functions represented by histograms.

A Novel Two-Way Finite-Element Parabolic Equation Groundwave Propagation Tool: Tests With Canonical Structures and Calibration

A novel two-way finite-element parabolic equation (PE) (2W-FEMPE) propagation model which handles both forward and backward scattering effects of the groundwave propagation above the Earths surface over irregular terrain paths through inhomogeneous atmosphere is introduced. A Matlab-based propagation tool for 2W-FEMPE is developed and tested against mathematical exact and asymptotic solutions as well as the recently introduced two-way split-step PE model through a canonical validation, verification, and calibration process for the first time in literature.

A Transformation Media Based Approach for Efficient Monte Carlo Analysis of Scattering From Rough Surfaces With Objects

This paper presents a computational model that utilizes transformation-based metamaterials to enhance the performance of numerical modeling methods for achieving the statistical characterization of two-dimensional electromagnetic scattering from objects on or above one-dimensional rough sea surfaces. Monte Carlo simulation of the rough surface scattering problem by means of differential equation-based finite methods (such as finite element or finite difference methods) usually places a heavy burden on computational resources because at each realization of the Monte Carlo technique, a mesh must be generated anew for each surface realization. The main purpose of the proposed approach in this paper is to create a single mesh, without repeating mesh generation at each step, by introducing a transformation medium above the rough surface in the computational domain of the finite methods. Material parameters of the medium are obtained by the coordinate transformation technique, which is based on the form-invariance property of Maxwells equations. At each realization, only the material parameters are modified with respect to the geometry of surface without changing the mesh. In this manner, a great reduction in CPU time is achieved. The proposed technique is analyzed and validated via various finite element simulations.

Form Invariance of Maxwell's Equations: The Pathway to Novel Metamaterial Specifications for Electromagnetic Reshaping

We present spatial-coordinate transformation techniques to control the propagation of electromagnetic fields in several surprising and useful applications. The implementation of this approach is based on the fact that Maxwells equations are form-invariant under coordinate transformations. Specifically, the effect of a general coordinate transformation can be realized by means of an equivalent anisotropic material, in which the original forms of Maxwells equations are still preserved in the transformed space. Constitutive parameters of the anisotropic material are determined to appropriately reflect the consequences of the coordinate transformation on the electromagnetic fields. In this paper, we introduce novel implementations and interpretations of the coordinate-transformation approach for the purpose of “reshaping” objects in electromagnetic scattering, and for reshaping and miniaturizing waveguides. We demonstrate the applications of the proposed techniques via several finite-element simulations.