New Finite Difference Time Domain (νFDTD) Electromagnetic Field Solver

Chapter

Abstract

With the advent of sub-micron technologies and increasing awareness of Electromagnetic Interference and Compatibility (EMI/EMC) issues, designers are often interested in full-wave simulations of complete systems, and of their possible environments. Such simulations can be very complex, especially when the problems of interest involve multi-scale geometries with very fine features. Under these circumstances, even the well-established 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 Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

9.1 Introduction

With the advent of sub-micron technologies and increasing awareness of Electromagnetic Interference and Compatibility (EMI/EMC) issues, designers are often interested in full-wave simulations of complete systems, and of their possible environments. Such simulations can be very complex, especially when the problems of interest involve multi-scale geometries with very fine features. Under these circumstances, even the well-established 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 multi-scale 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 low-frequency 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 general-purpose 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

The νFDTD solver is a hybridized version of conformal FDTD (CFDTD) [11], and a novel frequency domain technique called the Dipole Moment Approach (DM Approach) [2, 5, 7]. This blend of time domain and frequency domain techniques empowers the solver with potential to solve problems that involve:
  • Calculating low frequency response accurately and numerically efficiently

  • Handling non-Cartesian geometries such as curved surfaces (see Fig. 9.1) accurately without staircasing

  • Handling thin structures, with or without finite losses (see Fig. 9.2)

  • Dealing with multi-scale geometries (see Fig. 9.3)

Fig. 9.1

An elliptical geometry

Fig. 9.2

A very thin sheet

Fig. 9.3

PEC loop over a finite ground plane

9.2.1 Advantages

Some of the notable features of νFDTD are:
  • Unlike the conventional FDTD, the mesh-size 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 post-processing 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 integral-equation-based 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 ill-conditioned matrices at these frequencies. On the other hand, while the time-domain-based 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 non-physical 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.

As an example, let us consider a 32 port connector circuit example shown in Fig. 9.4. This connector geometry has been analyzed by using a commercial FDTD solver and the variation of the transmission co-efficient S 21 is plotted in Fig. 9.5 as a function of the frequency, and we observe that the results shows ripples that are numerical artifacts. Table 9.1 compares the number of time steps required for the solution to converge at different frequencies for the connector geometry. It can be inferred from this Table that the number of time steps required for the convergence increases as we go down in frequency, and eventually it becomes totally impractical to solve the problem at very low frequencies. Accurate calculation of the low frequency response becomes especially critical in the area of RF and digital circuits, since inaccurate results can affect the causality behavior of the overall system. The νFDTD utilizes a new technique, which is based on analytic continuation of the results derived at higher frequencies, and which is implemented by using the DM Approach and related techniques. This new technique is universal in nature, and it covers the entire range of frequencies, including the limiting case of f → 0. Also, the νFDTD can handle both the RF/Digital circuit problems as well as the radiation/scattering problems with same ease by employing unique methodologies tailored for each of these categories. We present these methodologies in detail in the sections that follow.
Fig. 9.4

A 32 port connector with a overall dimension of 5.6  × 11.88  × 27.35 mm (Housing not shown here)

Fig. 9.5

Variation of the transmission co-efficient S 21 for the 32 port connector shown in Fig. 9.4

Table 9.1

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

Consider the variation of the isolation co-efficient S 31 shown in Fig. 9.6 for the connector geometry (Fig. 9.4). This plot is divided into three regions, namely:
  • Region-1: Low-frequency regime

  • Region-2: Validation region

  • Region-3: High-frequency regime

Fig. 9.6

Variation of the isolation co-efficient S 31 for the 32 port connector shown in Fig. 9.4

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 cut-off 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. 1.

    Smooth the “DC Gaussian” Results.

     
  2. 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 fine-tuned based on the quality of the resulting fit.

     
  3. 3.

    Validate the smoothed “DC Gaussian” results in region-2 by comparing them with those generated by “single frequency” simulations at a few points (typically 2 or 3).

     
We have recalculated the results for the 32-port connector geometry, shown in Fig. 9.4, by using the above method. The new results for the variation of the transmission co-efficient S 21 and the isolation co-efficient S 31 are shown in Figs. 9.7 and 9.8. From these figures we can clearly see that the conventional FDTD simulation utilizing the DC Gaussian pulse does not generate an accurate low frequency response and has numerical artifacts, while the νFDTD does not suffer from the same.
Fig. 9.7

Variation of the transmission co-efficient S 21 for the 32 port connector shown in Fig. 9.4 calculated using νFDTD

Fig. 9.8

Variation of the isolation co-efficient S 31 for the 32 port connector shown in Fig. 9.4 calculated using νFDTD

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. 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. 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. 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. 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).

     
As an example application of the procedure just outlined, we consider a sphere with a diameter of \(\frac{\lambda }{20}\), with λ defined at 10 GHz (Fig. 9.9). The sphere is illuminated by a plane wave traveling in the negative-z direction, with its E-field polarized along y. Figure 9.10 compares the fields calculated by the proposed technique, in the frequency range of 1 Hz to 30 GHz, with those derived analytically. We find that the fields calculated by the extracted DM using the proposed technique exhibits good agreement with those calculated by using the analytical expression. The small deviation between the two curves is attributable to the staircase modeling of the sphere in the conventional FDTD, and it can be corrected by using an effective radius in the analytical expression. It is important to recognize the fact that we have used the same technique to calculate the response over the entire frequency range, including frequencies as low as 1 Hz, without using either the quasi-static approximation or other special treatments that are employed in the conventional computational electromagnetic (CEM) techniques. Even after the use of special treatments in the existing techniques, such as the FEM and MoM, the accuracy of the low-frequency solution is often questionable because of the large condition numbers of the associated matrix. Thus, despite all the special treatments implemented in these methods to address the low frequency breakdown problem, it is totally impractical to go down to frequencies as low as 1 Hz in the existing techniques.
Fig. 9.9

A PEC sphere of diameter \(\frac{\lambda }{20}\) @ 10 GHz

Fig. 9.10

Amplitude variation of the scattered E y at a point z = 0. 25 cm with frequencies from 1 Hz to 30 GHz

The amplitude variation of the scattered field with the distance along z, calculated by using the proposed technique, is shown in Fig. 9.11 for a frequency of 1.8 GHz. This plot also compares the results with those calculated by using analytical expressions. Again we find good agreement between the νFDTD results and those generated from the analytical expression along z from \(\frac{\lambda }{67}\) to \(\frac{\lambda }{10}\), for the frequency of 1.8 GHz. The field variation derived by using the νFDTD matches well with that generated from the analytical expression, both in the near and far field regions.
Fig. 9.11

Amplitude variation of the scattered E y with distance along z from \(\frac{\lambda }{67}\) to \(\frac{\lambda }{10}\), at 1.8 GHz

Based on the illustrative examples presented above, we can list some of the advantages of the proposed method, as shown below.
  • RF and Digital Circuit Problems:

    Efficient for constructing low frequency solution, compared to the long runs in FDTD.

  • Scattering Problems:
    1. (a)

      Can be used for an arbitrary geometry.

       
    2. (b)

      Can be used to efficiently calculate not only the frequency response, but the near and far fields as well.

       

9.4 Non-Cartesian Geometries

The conventional FDTD uses a staircase-approximation to model non-Cartesian geometries, as shown in Fig. 9.12, and requires the use of a very fine mesh to mitigate the effects of this staircase approximation when dealing with curved objects. This, in turn, makes the simulation computationally expensive, both in terms of memory and CPU time. Even though methods such as FEM and MoM can handle curved geometries with much ease because they do not restrict themselves to a Cartesian type of meshing, often they are not necessarily the most computationally efficient when dealing with inhomogeneous media. Hence, it would be a great advantage to modify the existing FDTD algorithm so that it can handle curved geometries, enabling us to conveniently handle arbitrary objects. In the past, a generalization of the conventional FDTD, namely the CFDTD algorithm [11], has been developed for this purpose. In CFDTD, the magnetic field update equations are modified by using the areas of the partially-filled cells, as opposed to those of the entire cells.
Fig. 9.12

Meshing of a non-Cartesian geometry by the conventional FDTD. (a) A PEC wedge geometry. (b) A PEC wedge with staircase approximation

To explain the CFDTD concept, we consider a partially filled cell, shown in Fig. 9.13. The equation for this partially-filled cell is derived by using Farady’s law, to get:
$$\displaystyle{ \oint _{C_{1}}\mathbf{E} \cdot \mathbf{dl} = -\mu \frac{\partial } {\partial t}\int _{S_{1}} \mathbf{H} \cdot \mathbf{ds} }$$
(9.1)
Fig. 9.13

A partially-filled cell

where C 1 is the loop ABCDA and S 1 is the area enclosed by loop C 1. Upon discretizing this equation, we obtain:
$$\displaystyle\begin{array}{rcl} H_{z}^{n+\frac{1} {2} }(i,j,k)& =& H_{z}^{n-\frac{1} {2} }(i,j,k) - \frac{dt} {\mu S_{1}}[-E_{y}^{n}(i,j,k) \cdot l_{ AB} + E_{x}^{n}(i,j,k) \cdot dh \\ & & \qquad \qquad \qquad \qquad \ \qquad + E_{y}^{n}(i + 1,j,k) \cdot l_{ CD}] {}\end{array}$$
(9.2)

The update magnetic equation for the partially-filled 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 H-fields 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 H-fields employed in the CFDTD approach.

Because we use the asymptotic method to compute the reflection or diffraction coefficients, the proposed technique requires a “single frequency” simulation. However as shown in [8], this technique can be extended to “DC Gaussian” simulations with a slight modification, as shown, for instance in the examples presented in Figs. 9.14 and 9.15. Also, the proposed technique can be extended to dielectrics and inhomogeneous geometries without any modification, while the CFDTD cannot handle either of them without compromising the accuracy.
Fig. 9.14

A curved surface (with a height of 4λ)

Fig. 9.15

Amplitude variation of the scattered E y at 10 GHz

Let us consider the case of a square PEC sheet whose sides are approximately 4λ (λ referenced at 10 GHz) and inclined at an angle of 0. 72 ∘  with respect to the x-axis, as shown in Fig. 9.16. The tilt angle chosen is 0. 72 ∘  so that the edges of the sheet are offset only by ± λ ∕ 40 above or below the x-axis, i.e., half the FDTD cell size of λ ∕ 20. We calculate the amplitude variation of the scattered E x field at a frequency of 10 GHz, when the plate is illuminated by a plane wave traveling along the negative-y direction and is polarized along x. Figure 9.17 compares the results obtained by using the proposed technique, with those returned by the CFDTD, and a commercial MoM code for the same problem. The results generated by using the proposed technique show good agreement with the ones from the commercial MoM results, while the CFDTD results show spurious ripples in the lit region because of the instability problem encountered in the CFDTD algorithm when the area S 1 → 0. What is more, this is even true when a fine mesh size of \(\frac{\lambda }{160}\) is used in the CFDTD, in contrast to the \(\frac{\lambda }{20}\) mesh size used in νFDTD. Table 9.2, presents a comparison of the mesh size and the memory requirements, and shows that the proposed technique easily out-performs the CFDTD.
Fig. 9.16

A inclined PEC sheet (not to scale)

Fig. 9.17

Amplitude variation of the scattered E x with distance along y at 10 GHz

Table 9.2

Comparison of mesh size and memory required for convergence for PEC geometry shown in Fig. 9.16

Parameter

νFDTD

CFDTDa

 

Mesh size used

\(\frac{\lambda }{20}\)

\(\frac{\lambda }{160}\)

 

Memory required

413 MB

31 GB

 

a Results still have numerical artifacts

We now summarize below some of the advantages of the proposed method. They are:
  1. (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.

     
  2. (b)

    More accurate than the conventional Conformal FDTD.

     
  3. (c)

    Retains λ/20 cell size even for thin, slanted and curved bodies, offering memory advantage and computational efficiency over conventional conformal FDTD.

     
  4. (d)

    Free of instability problems even when the fractional area of the partially filled cell is very small, even when it tends to zero.

     
  5. (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 fine-featured objects that can cause spurious reflections introduced at the interfaces of non-uniform FDTD grids, the system matrix generated by either the FEM or the MoM algorithm can become highly ill-conditioned. 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 small-scale effects.

Consider a thin-wire dipole located over a PEC plate, coated with a dielectric whose permitivity ε r  = 6, as shown in Fig. 9.18. To solve the problem by using the conventional FDTD, we need to use a very fine mesh size which, for this problem, turns out to \(\frac{\lambda }{400}\) if we follow the rule of thumb that says that we should nominally use at least two cells to model the thickness of a thin-wire whose radius is λ/200. This causes over-discretization of the coated plate, which is large in terms of the wavelength, and increases the number of DoFs significantly over that needed with a \(\frac{\lambda }{20}\) discretion. Hence, a better approach would be to solve the large and the small problem separately with different methods, since this would help reduce the CPU time and memory requirements by a large factor. Hence, to solve such a multiscale problem, we propose a new technique, based on the steps outlined below:
Fig. 9.18

An example for a multiscale problem (not to scale)

  1. 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. 2.

    Using these field, as sources, solve the plate problem alone by using the FDTD on a fine mesh.

     
  3. 3.

    Insert these converged fields in to the coarse-grid FDTD algorithm, and find the field scattered on the dipole surface.

     
  4. 4.

    Finally, update the right hand side of the DM formulation using these fields; update and solve for the current distribution.

     
The problem geometry, shown in Fig. 9.18, was solved at 10 GHz by using the technique proposed above. Because the length of the wire in this problem traverses through multiple cells, each with a length of \(\frac{\lambda }{20}\), we use the “Single Frequency” simulations in steps 2 and 3. However if the geometry of the small object is contained within a single \(\frac{\lambda }{20}\) cell, we can use a “Gaussian” simulation with just a few modifications in the proposed technique [8] to derive the solution over a wide frequency band. Figure 9.19 compares the amplitude variation of the E y fields along z, with those calculated from a commercial MoM code, for the problem shown in Fig. 9.18. Table 9.3 compares the memory and simulation time required by the νFDTD and commercial MoM codes. The fields calculated by using the νFDTD, which uses only a fraction (1/3) of the computational resources, shows good agreement with those generated by the commercial MoM code (see Fig. 9.19). Even though the commercial MoM code was able to handle this multiscale problem, the greater advantage of νFDTD becomes apparent when the spacing between the wire and the plate is decreased from \(\frac{\lambda }{5}\) to \(\frac{\lambda }{10}\). In this case, the commercial MoM code was unable to solve the problem because of the highly ill-conditioned nature of the impedance matrix which we need to invert for the problem at hand. In contrast to the MoM, the νFDTD was able to solve the above problem relatively easily without requiring any modifications in the solution procedure.
Fig. 9.19

Amplitude variation of E y with distance along z at 10 GHz

Table 9.3

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

 
Some of the advantages of the proposed method are listed below:
  1. (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 ill-conditioned matrices, nor do we have to search for preconditioners to improve their condition numbers.

     
  2. (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].

Another important factor that affects the accuracy and efficiency of the FDTD simulations is the boundary conditions used to truncate the computational domain. Even though there are many boundary conditions that can be used, for mesh truncation, the most widely used and effective one is the Convoluted Perfectly Matched Layer or more commonly known as CPML [12]. Even though the CPML is effective, it is computationally expensive. It is possible to reduce the computational expense, with little loss of accuracy by using a new algorithm, which is based on the impedance boundary condition (IBC) [4, 8]. Here the tangential E-Fields at the end of the computational domain are calculated from the H-fields based on the impedance relationship:
$$\displaystyle{ E_{tan} =\eta \hat{ n} \times \mathbf{H} }$$
(9.3)
where η is the intrinsic impedance of the medium. The H-fields at the end of the computational domain are updated by using the conventional FDTD update equations, though the E-fields are derived by using the IBC. The results presented in Figs. 9.20 and 9.21 for the dipole geometry (see Fig. 9.22) illustrate the accuracy of the proposed algorithm, which requires much less CPU time and memory than those required by the CPML.
Fig. 9.20

Variation of input resistance with frequency for the PEC dipole

Fig. 9.21

Variation of input reactance with frequency for the PEC dipole

Fig. 9.22

Geometry of a PEC dipole (not to scale)

Another area of interest where νFDTD outperforms the conventional CEM algorithms is well-logging application. Consider the stratified medium shown in Fig. 9.23, which is typical the case in well logging problems with the frequency range of interest being 1 KHz–1 MHz. Figure 9.24 shows the incident pulse received at the observation point, while the Fig. 9.25 shows the reflected pulse from the interface received at the observation point calculated using νFDTD. Because of the low frequency range of interest, other commercial solvers typically fail to solve for the reflected pulse, while νFDTD is able to handle it with ease without requiring any modifications.
Fig. 9.23

Geometry of a stratified medium with oil (not to scale)

Fig. 9.24

Variation of E x component of the incident pulse

Fig. 9.25

Variation of E x component of the reflected pulse

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 non-Cartesian geometries accurately without any instability issues that are often encountered in the conventional CFDTD; model multi-scale 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 well-logging 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 well-logging problem efficiently whereas the commercial solvers are unable to handle the problem because the frequency of interest for this problem is very low.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.EMC Lab, Department of Electrical Engineering, State CollegeThe Pennsylvania State UniversityUniversity ParkUSA

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