Computational Electromagnetics pp 317338  Cite as
New Finite Difference Time Domain (νFDTD) Electromagnetic Field Solver
Abstract
With the advent of submicron technologies and increasing awareness of Electromagnetic Interference and Compatibility (EMI/EMC) issues, designers are often interested in fullwave simulations of complete systems, and of their possible environments. Such simulations can be very complex, especially when the problems of interest involve multiscale geometries with very fine features. Under these circumstances, even the wellestablished methods either in the time or frequency domains, such as the Finite Difference Time Domain (FDTD), Finite Element Method (FEM), or the Method of Moments (MoM), are often challenged to the limits of their capabilities. The nu (use symbol for nu) FDTD solver is an approach which is being proposed to handle such challenges. The nu (use Symbol for nu) FDTD is a hybridized version of the conformal FDTD (CFDTD) and a novel frequency domain technique called the Dipole Moment Approach (DM Approach). We show that this blend of time domain and frequency domain techniques empowers the solver to solve a wide variety of problems in a numerically efficient way.
Keywords
Finite Difference Time Domain Impedance Boundary Condition Frequency Domain Technique Finite Difference Time Domain Simulation Multiscale Problem9.1 Introduction
With the advent of submicron technologies and increasing awareness of Electromagnetic Interference and Compatibility (EMI/EMC) issues, designers are often interested in fullwave simulations of complete systems, and of their possible environments. Such simulations can be very complex, especially when the problems of interest involve multiscale geometries with very fine features. Under these circumstances, even the wellestablished methods [10] either in the time or frequency domains, such as the Finite Difference Time Domain (FDTD), Finite Element Method (FEM), or the Method of Moments (MoM), are often challenged to the limits of their capabilities. On the basis of our experience with the conventional frequency domain methods, we can identify the following areas of concern:

Handling thin wires and/or sheets, with or without finite losses

Deriving a universal approach for PEC, dielectric and inhomogeneous objects

Accurate modeling of multiscale geometries

Accurately integrating the Green’s function for curved geometries

Dealing with singular and hypersingular behavior of the Green’s function when generating the MoM matrix

Dealing with the lowfrequency breakdown problem introduced by the dominance of the scalar potential term over the vector potential as the frequency approaches zero.
The conventional time domain technique FDTD also demands extensive computational resources when solving low frequency problems, or when dealing with dispersive media. To tackle some of these challenges, the conventional techniques are often modified in a manner that is tailored to solve a particular problem of interest. However, more often than not, these tailored methods turn out to be computationally expensive, and they often lead to instabilities. Hence, it is useful to develop techniques that can overcome the above limitations, while preserving the advantages of the existing methods. The νFDTD (New FDTD) technique, which is described in this chapter, is a new generalpurpose field solver, which is designed to tackle the above issues using some novel approaches, that deviate significantly from the legacy methods that only rely on minor modifications of the FDTD update algorithm.
9.2 νFDTD Solver
9.2.1 Advantages

Unlike the conventional FDTD, the meshsize utilized by the νFDTD is not dictated by the finest feature of the geometry, and this size is usually maintained at the conventional \(\frac{\lambda }{20}\) level. This helps to reduce the computational burden by a large factor.

The νFDTD algorithm incorporates a novel postprocessing technique which requires relatively few time steps, in comparison to the number of steps required by the conventional FDTD.
9.3 Low Frequency Response
Despite many advances in finite methods, such as the FEM and the FDTD, as well as in integralequationbased techniques such as the MoM, it still remains a challenge to accurately calculate the low frequency response for radiation and scattering problems [6]. The frequency domain techniques, such as the FEM and MoM both experience difficulties at low frequencies, because they have to deal with illconditioned matrices at these frequencies. On the other hand, while the timedomainbased techniques, such as the FDTD, can accurately generate results at high frequencies, usually above 1 GHz, the same cannot be said about their performance at low frequencies. This is not only because the FDTD results are often corrupted by the presence of nonphysical artifacts at low frequencies, but also because the FDTD requires exorbitantly large number of time steps for accurate calculation of the response. The required number of time steps can exceed a few million in some cases before convergence is achieved.
Comparison of time steps required for convergence for the circuit shown in Fig. 9.4
Frequency  10 MHz  1 MHz  1 Hz  
Time steps in millions  0.7  7  70 
9.3.1 RF and Digital Circuits

Region1: Lowfrequency regime

Region2: Validation region

Region3: Highfrequency regime
There are four frequency values which delimit the above three regions. The frequency f _{ L } describes the lowest frequency of interest defined by the user. The frequency f _{1}, which divides the regions 1 and 2, is typically chosen to be between 500 and 1,000 MHz, while the frequency f _{2} dividing the regions 2 and 3 is chosen to be on the order of 2f _{1} or 3f _{1}. The frequency f _{ H } is the user input indicating the highest frequency of interest. In each of these three regions the results are calculated by using a different method. The results in the high frequency regime are generated by using the conventional FDTD, using a DC Gaussian pulse as the excitation source, whose 3 dB cutoff frequency is set to be f _{ H }. In the low frequency regime, the results are generated using the proposed new technique, which involves the following steps:
 1.
Smooth the “DC Gaussian” Results.
 2.
Fit the curve from f _{ L } to f _{1} with the DC values, using a quadratic, for instance. The choice of f _{1} can be finetuned based on the quality of the resulting fit.
 3.
Validate the smoothed “DC Gaussian” results in region2 by comparing them with those generated by “single frequency” simulations at a few points (typically 2 or 3).
9.3.2 Scattering Problems
In this section we turn to the solution of scattering problems by using the νFDTD. The methodology for handling the radiation and scattering problems are different from those used for the RF/Digital circuits, as we will explain below. For the high frequency regime, we use the conventional FDTD, and use a Gaussian excitation source to generate the results. However, we employ a different procedure, as outlined below, in the low frequency regime:
 1.
Run a “Single Frequency” simulation at a frequency f _{1} where the largest dimension of the geometry is \(\frac{\lambda }{100}\) to calculate the fields at a point located \(\frac{\lambda }{20}\) from the surface of the object.
 2.
Extract the dipole moment by using the analytical expressions [7] for the field radiated by an infinitesimal dipole [1] from the field values calculated above.
 3.
Use the extracted dipole moment to calculate the results from f _{ L } to f _{2}, where f _{ L } is the lowest frequency of interest, and f _{2} is typically chosen to be 2f _{1} or 3f _{1}. It has been found that the results generated by using this dipole moment is not only valid for frequencies as low as 0, but also up to frequencies where the largest dimension of the geometry becomes \(\frac{\lambda }{10}\); hence it enables us to dovetail the low frequency results, seamlessly, with the lower end of the high frequency response.
 4.
Validate the “DC Gaussian” results in region between f _{1} and f _{2} by comparing them with those calculated by using the analytical expression at a few points (typically 2 or 3).

RF and Digital Circuit Problems:
Efficient for constructing low frequency solution, compared to the long runs in FDTD.
 Scattering Problems:
 (a)
Can be used for an arbitrary geometry.
 (b)
Can be used to efficiently calculate not only the frequency response, but the near and far fields as well.
 (a)
9.4 NonCartesian Geometries
The update magnetic equation for the partiallyfilled cell is shown above in (9.2). But, as S _{1} → 0, this modified update equation becomes unstable since, as we see from (9.2), the expression for the updated H contains S _{1} in the denominator. The update equation can be modified to circumvent this instability problem that arises when the partial area is small, albeit at the cost of compromising the accuracy. This motivates us to develop a new approach in which the field values, as opposed to the update equations, are modified by using the local field solution. The proposed new technique is described below:

For the partially filled cells with a fill factor ≤ 50 %, the E fields are updated by using the Hfields calculated by the modified CFDTD Eq. 9.2.

For the partially filled cells with a fill factor > 50 %, the E fields are updated by using local solutions generated based on the concepts of reflection or diffraction, rather than using the Hfields employed in the CFDTD approach.
Comparison of mesh size and memory required for convergence for PEC geometry shown in Fig. 9.16
Parameter  νFDTD  CFDTD^{a}  

Mesh size used  \(\frac{\lambda }{20}\)  \(\frac{\lambda }{160}\)  
Memory required  413 MB  31 GB 
 (a)
Usable for arbitrary geometries, even if the surfaces do not coincide with the Cartesian mesh, e.g., thin sheets, with or without a slant.
 (b)
More accurate than the conventional Conformal FDTD.
 (c)
Retains λ/20 cell size even for thin, slanted and curved bodies, offering memory advantage and computational efficiency over conventional conformal FDTD.
 (d)
Free of instability problems even when the fractional area of the partially filled cell is very small, even when it tends to zero.
 (e)
Proposed method can be extended to dielectric objects, with just a few modifications.
9.5 Multiscale Geometries
Regardless of which of the conventional computational methods we use to solve multiscale problems, whether it is FEM, FDTD or MoM, direct solution of multiscale electromagnetic problems remain highly challenging [3, 9]. Multiscale problems involve combinations of objects whose dimensions range from small to large in terms of the wavelength. To model them accurately, we need to work with a large number of Degrees of Freedom (DoFs) since such problems require a fine mesh to capture all the nuances of their fine features. Despite substantial advancements in our computational capabilities in recent times, handling a large number of DoFs still presents a huge challenge. Also, because of the difference in the mesh sizes required to effectively capture both the large and finefeatured objects that can cause spurious reflections introduced at the interfaces of nonuniform FDTD grids, the system matrix generated by either the FEM or the MoM algorithm can become highly illconditioned. As a result, direct solution of such multiscale problems requires a large CPU time and memory, since we need to handle a large number of DoFs to accurately capture the smallscale effects.
 1.
Use the DM approach [2, 5, 7] to find the fields on the surface of the plate, when illuminated by a dipole carrying a unit current.
 2.
Using these field, as sources, solve the plate problem alone by using the FDTD on a fine mesh.
 3.
Insert these converged fields in to the coarsegrid FDTD algorithm, and find the field scattered on the dipole surface.
 4.
Finally, update the right hand side of the DM formulation using these fields; update and solve for the current distribution.
Comparison of memory requirements and simulation time for the multiscale geometry shown in Fig. 9.18
Parameter  νFDTD  Commercial MoM  

Memory required (MB)  271  646.2  
Simulation time (s)  97.35  276.19 
 (a)
Since the FDTD is an explicit recursive algorithm, rather than one that requires matrix inversion, we neither have to concern ourselves with the problems associated with illconditioned matrices, nor do we have to search for preconditioners to improve their condition numbers.
 (b)
We are able to solve multiscale problems efficiently and accurately in comparison to the “brute force” methods. In fact, for the example in Fig. 9.18, the νFDTD outperforms the MoM, rather than the other way round. As is well known, the MoM normally outperforms the conventional FDTD algorithm, in terms of CPU time and memory, often by a large factor.
9.6 Enhancements
Since νFDTD relies upon the conventional FDTD to solve different types of problems, its performance can be further enhanced by parallelizing the algorithm [12]. Time advantage can also be gained by using signal processing to determine where to terminate the FDTD simulations by checking its convergence in the frequency domain instead of in the time domain [8].
9.7 Conclusions
In this chapter, we have introduced the νFDTD solver, which is a blend of time and frequency domain techniques that can generate accurate electromagnetic responses at low frequencies; handle nonCartesian geometries accurately without any instability issues that are often encountered in the conventional CFDTD; model multiscale geometries accurately; and, handle lossy/lossless thin structures with ease. In all the cases for which we have carried out comparison studies with the existing algorithms and commercial codes, the νFDTD was not only accurate but also computationally the most efficient. We have also introduced a new boundary condition for the mesh truncation, which is numerically efficient both from the points of view of CPU time and memory as compared to the widely used CPML algorithm, without a noticeable compromise in the relative accuracy of the computed results. We have also demonstrated the efficacy of the νFDTD when used to solve welllogging problems that are typically computationally expensive not only because of the large problem size, but also because of low frequency range of interest. We have shown that the νFDTD is able to solve the welllogging problem efficiently whereas the commercial solvers are unable to handle the problem because the frequency of interest for this problem is very low.
References
 1.Balanis CA (2005) Antenna theory: analysis and design, 3rd edn. Wiley, Hoboken, New JerseyGoogle Scholar
 2.Mittra R, Panayappan K, Pelletti C, Monorchio A (2009) A universal dipolemomentbased approach for formulating MoMtype problems without the use of Greens functions. In: Proceedings of the 4th European conference on antennas and propagation, Barcelona, SpainGoogle Scholar
 3.Mittra R, Bringuier J, Pelletti C, Panayappan K, Ozgun O, Monorchio A (2011) On the hybridization of dipole moment (DM) and finite methods for efficient solution of multiscale problems. In: Proceedings of the 5th European conference on antennas and propagation, Rome, ItalyGoogle Scholar
 4.Panayappan K, Mittra R (2013) A new impedance boundary condition for FDTD mesh truncation. In: IEEE international APS and UNSC/URSI national radio science meeting, Orlando, FloridaGoogle Scholar
 5.Panayappan K, Bringuier JN, Mittra R, Yoo K, Mehta N (2009) A newdipolemomentbased MoM approach for solving electromagnetic radiation and scattering problems. In: Proceedings of IEEE international APS and UNSC/URSI national radio science meeting, North Charleston, SCGoogle Scholar
 6.Panayappan K, Mittra R, Arya RK (2011) A universal approach for generating electromagnetic response over a wide band including very low frequencies. In: Proceedings of IEEE international APS and UNSC/URSI national radio science meetingGoogle Scholar
 7.Panayappan K, Pelletti C, Mittra R. An Efficient DipoleMomentbased Method of Moments (MoM) formulation In: Computational Electromagnetics. Springer (in print)Google Scholar
 8.Panayappan K (2013) Novel frequency domain techniques and advances in finite difference time domain (FDTD) method for efficient solution of multiscale electromagnetic problems. Pennsylvania State University, University ParkGoogle Scholar
 9.Pelletti C, Panayappan K, Mittra R, Monorchio A (2010) On the hybridization of RUFD algorithm with the DM approach for solving multiscale problems. In: Proceedings of EMTS 20th international symposium on electromagnetic theoryGoogle Scholar
 10.Peterson AF, Ray SL, Mittra R (1998) Computational methods for electromagnetics. IEEE Press, New JerseyGoogle Scholar
 11.Yu W, Mittra R (2000) A conformal FDTD software package modeling antennas and microstrip circuit components. IEEE Antennas Propag Mag 8:28–39Google Scholar
 12.Yu W, Mittra R, Su T, Liu Y, Yang X (2006) Parallel finitedifference timedomain method. Artech House, BostonMATHGoogle Scholar