Huseyin Arda Ulku

Application of Analytical Retarded-Time Potential Expressions to the Solution of Time Domain Integral Equations

Recently, exact closed-form expressions of the electric and magnetic fields and potentials due to impulsively excited Rao-Wilton-Glisson basis functions have been presented. In this work, the application of these expressions in the solution of the combined field integral equation (CFIE) is presented. Solutions via analytical expressions of the electric and magnetic fields are verified and compared with the conventional marching-on-in time (MOT) algorithm solutions that employ numerical basis integrations. It is shown that the accuracy and stability of the solutions obtained with the proposed (analytical) approach are better when compared to those obtained with the conventional (numerical) method. In addition, a discussion section about the effect of the proposed approach in solving the electric field integral equation (EFIE) is added. In this section, it is demonstrated that the dependency of the solution via analytical expressions of the fields are less sensitive to the time step size in comparison to the conventional solution, and that the increase in accuracy is a necessary but not sufficient condition for stability.

Analytical Evaluation of Transient Magnetic Fields Due to RWG Current Bases

A new analytical approach for obtaining the time samples of the magnetic field intensity due to an impulsively excited Rao-Wilton-Glisson (RWG) basis function is presented. The approach is formulated directly in the time domain. It is shown that the magnetic field is related to the arc segments formed by the intersection of the triangular patch of the RWG basis with the sphere that is centered at the observation point and that has a radius of , where is the speed of light. In particular, the magnetic field can be expressed as the variations of two quantities with respect to . The first quantity is the arc segment length, and the second quantity is the bisecting vector of the arc segment. Analytical representations of these quantities are presented. Contrary to previous studies, these representations do not require the calculation of the intersection points of the sphere with the boundaries of the bases. The validity of the obtained time domain formulae is demonstrated through comparison of the results with those obtained in the frequency domain by using numerical quadrature. Finally, it is demonstrated that the derived formulae yield closed-form expressions when convolved with piecewise polynomial temporal basis functions.

Marching On-In-Time Solution of the Time Domain Magnetic Field Integral Equation Using a Predictor-Corrector Scheme

An explicit marching on-in-time (MOT) scheme for solving the time-domain magnetic field integral equation (TD-MFIE) is presented. The proposed MOT-TD-MFIE solver uses Rao-Wilton-Glisson basis functions for spatial discretization and a PE(CE)m-type linear multistep method for time marching. Unlike previous explicit MOT-TD-MFIE solvers, the time step size can be chosen as large as that of the implicit MOT-TD-MFIE solvers without adversely affecting accuracy or stability. An algebraic stability analysis demonstrates the stability of the proposed explicit solver; its accuracy and efficiency are established via numerical examples.

A Stable Marching On-In-Time Scheme for Solving the Time-Domain Electric Field Volume Integral Equation on High-Contrast Scatterers

A time-domain electric field volume integral equation (TD-EFVIE) solver is proposed for characterizing transient electromagnetic wave interactions on high-contrast dielectric scatterers. The TD-EFVIE is discretized using the Schaubert-Wilton-Glisson (SWG) and approximate prolate spherical wave (APSW) functions in space and time, respectively. The resulting system of equations cannot be solved by a straightforward application of the marching on-in-time (MOT) scheme since the two-sided APSW interpolation functions require the knowledge of unknown “future” field samples during time marching. Causality of the MOT scheme is restored using an extrapolation technique that predicts the future samples from known “past” ones. Unlike the extrapolation techniques developed for MOT schemes that are used in solving time-domain surface integral equations, this scheme trains the extrapolation coefficients using samples of exponentials with exponents on the complex frequency plane. This increases the stability of the MOT-TD-EFVIE solver significantly, since the temporal behavior of decaying and oscillating electromagnetic modes induced inside the scatterers is very accurately taken into account by this new extrapolation scheme. Numerical results demonstrate that the proposed MOT solver maintains its stability even when applied to analyzing wave interactions on high-contrast scatterers.

Application of analytical expressions of transient potentials to the MOT solution of integral equations

As the numerical results demonstrate, it is shown that the developed analytical evaluation of potentials and fields due to RWG basis functions improve the accuracy and stability of the MOT solutions of the MFIE.

On the mixed discretization of the time domain magnetic field integral equation

Time domain magnetic field integral equation (MFIE) is discretized using divergence-conforming Rao-Wilton-Glisson (RWG) and curl-conforming Buffa-Christiansen (BC) functions as spatial basis and testing functions, respectively. The resulting mixed discretization scheme, unlike the classical scheme which uses RWG functions as both basis and testing functions, is “proper”: Testing functions belong to dual space of the basis functions. Numerical results demonstrate that the marching on-in-time (MOT) solution of the mixed discretized MFIE yields more accurate results than that of classically discretized MFIE.

Explicit solution of the time domain magnetic field integral equation using a predictor-corrector scheme

An explicit yet stable marching-on-in-time (MOT) scheme for solving the time domain magnetic field integral equation (TD-MFIE) is presented. The stability of the explicit scheme is achieved via (i) accurate evaluation of the MOT matrix elements using closed form expressions and (ii) a PE(CE)m type linear multistep method for time marching. Numerical results demonstrate the accuracy and stability of the proposed explicit MOT-TD-MFIE solver.

Recent advances in marching-on-in-time schemes for solving time domain volume integral equations

Transient electromagnetic field interactions on inhomogeneous penetrable scatterers can be analyzed by solving time domain volume integral equations (TDVIEs). TDVIEs are constructed by setting the summation of the incident and scattered field intensities to the total field intensity on the volumetric support of the scatterer. The unknown can be the field intensity or flux/current density. Representing the total field intensity in terms of the unknown using the relevant constitutive relation and the scattered field intensity in terms of the spatiotemporal convolution of the unknown with the Green function yield the final form of the TDVIE. The unknown is expanded in terms of local spatial and temporal basis functions. Inserting this expansion into the TDVIE and testing the resulting equation at discrete times yield a system of equations that is solved by the marching on-in-time (MOT) scheme. At each time step, a smaller system of equations, termed MOT system is solved for the coefficients of the expansion. The right-hand side of this system consists of the tested incident field and discretized spatio-temporal convolution of the unknown samples computed at the previous time steps with the Green function.

An explicit marching on-in-time solver for the time domain volume magnetic field integral equation

Summary form only given. Transient scattering from inhomogeneous dielectric objects can be modeled using time domain volume integral equations (TDVIEs). TDVIEs are oftentimes solved using marching on-in-time (MOT) techniques. Classical MOT-TDVIE solvers expand the field induced on the scatterer using local spatio-temporal basis functions. Inserting this expansion into the TDVIE and testing the resulting equation in space and time yields a system of equations that is solved by time marching. Depending on the type of the basis and testing functions and the time step, the time marching scheme can be implicit (N. T. Gres, et al., Radio Sci., 36(3), 379-386, 2001) or explicit (A. Al-Jarro, et al., IEEE Trans. Antennas Propag., 60(11), 5203-5214, 2012). Implicit MOT schemes are known to be more stable and accurate. However, under low-frequency excitation, i.e., when the time step size is large, they call for inversion of a full matrix system at very time step.One can expect that an explicit scheme would be more efficient at low frequencies if it uses the same time step as its implicit counterpart while maintaining its stability and accuracy. Indeed, recently a novel explicit MOT solver, which satisfies this criterion, has been developed for solving the time domain surface magnetic field integral equation (H. A. Ulku, et al., IEEE Trans. Antennas Propag., 61(8), 4120-4131, 2013). In this work, this explicit MOT scheme is applied in solving the time domain volume magnetic field integral equation (TDVMFIE). The proposed solver expands the unknown fields using curl-conforming basis functions in space. Inserting this expansion into the TDVMFIE and using Galerkin testing yield a semi-discrete system of equations. This system is integrated in time using a predictor-corrector scheme (A. Glaser and V. Rokhlin, J. Sci. Comput., 38(3), 368-399, 2009) to obtain the coefficients of the expansion. The resulting scheme calls for inversion of a matrix system at every time step but this can be carried out very efficiently since the pertinent Gram matrix is well conditioned and sparse regardless of the time step. The stability of the resulting MOT scheme is maintained using successive over relaxation (SOR) applied at the corrector step(S. B. Sayed, et al., Pr. Electromagn. Res. S., 378, Stockholm, 2013). Numerical results, which demonstrate that the proposed MOT-TDMVIE solver (i) uses time step as large as those of its implicit counterparts without sacrificing accuracy or stability, (ii) is faster than the implicit solver for low-frequency excitations, and (iii) maintains its efficiency, accuracy, and stability even when applied on high contrast scatterers, will be presented.

Analysis of transient plasmonic interactions using an MOT-PMCHWT integral equation solver

In this work, a marching on-in-time (MOT) scheme is proposed to solve the Poggio-Miller-Chan-Harrington-Wu-Tsai (PMCHWT) (L. N. MedgyesiMitschang et al., J. Opt. Soc. Am. A, 11(4), 1383-1398, 1994) IE for analyzing transient plasmonic interactions. The MOT-PMCHWT solver calls for convolutions of the spatio-temporal basis functions with the time domain Green function of the dispersive medium. These convolutions are carried out using a semi-numerical procedure. It is shown that Green function consists of a Dirac delta term and a temporal tail. The convolution with the delta term is analytically evaluated. Samples of the temporal tail are computed from frequency domain samples using the Fast Relaxed Vector Fitting (FRVF) algorithm (B. Gustavsen, IEEE Trans. Power Delivery, 21(3), 1587-1592, 2006). FRVF generates a rational function fit to frequency domain samples, which is used in time domain to represent the tail of the Green function in terms of shifted exponentials. Applying this procedure to every source-observer pair during the computation of MOT matrix entries is computationally costly. Therefore, a look-up table consisting of Green function samples at discrete distances and times is generated. Then, an interpolation scheme is used to fill the MOT matrix elements.