### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Formulation
- 3. Multilevel Plane Wave Time Domain Algorithm for Volume Distributions
- 4. Numerical Results
- 5. Summary
- Acknowledgments
- References

[1] A time domain integral equation (TDIE)-based approach for analyzing transient wave scattering from linear lossy media is proposed. The pertinent TDIEs are cast in terms of a “conduction current corrected flux density” and are solved using a marching-on-in-time (MOT) scheme that incorporates a differential equation update algorithm for the aforementioned flux. The scheme is accelerated by the PWTD algorithm, and it is shown that the computational complexity and memory requirements of the resulting solver scales as (*N*_{t}*N*_{s}) and (*N*_{s}), where *N*_{t} and *N*_{s} denote the number of temporal and spatial degrees of freedom in the flux expansion, respectively. Numerical results that validate the accuracy and efficacy of the proposed method are presented.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Formulation
- 3. Multilevel Plane Wave Time Domain Algorithm for Volume Distributions
- 4. Numerical Results
- 5. Summary
- Acknowledgments
- References

[2] Analysis of transient electromagnetic scattering from, and field propagation through, lossy objects is useful in a plethora of applications including the study of skin effects in high-speed circuits, medical diagnostics, and the assessment of SAR levels in wireless applications [*Ishimaru*, 1978; *Jin*, 1998; *Jensen and Rahmat-Samii*, 1995]. Historically, the analysis of transient field propagation through material media has been performed mostly using differential equation-based schemes, e.g., the finite difference time domain method [*Taflove*, 1995; *Kunz and Luebbers*, 1993]. The popularity of time domain integral equation (TDIE)-based schemes for analyzing such scattering phenomena has lagged that of their differential equation counterparts because, in the past, TDIE-based methods often exhibited instabilities [*Rao and Wilton*, 1991; *Rao and Sarkar*, 1998; *Rynne*, 1985; *Rynne and Smith*, 1990] and were computationally very expensive [*Bluck and Walker*, 1997]. Recent research has provided new insights in, and effective remedies to avoid, instabilities in marching on in time (MOT)-based TDIE solvers [*Bluck and Walker*, 1997; *Weile et al.*, 2001; *Abboud and Sayah*, 1998a, 1998b]. It has been shown that the cost of classical MOT-based TDIE solvers for analyzing surface scattering phenomena can be reduced considerably by augmenting them with Plane Wave Time Domain (PWTD) kernels [*Shanker et al.*, 2000, 2001b; *Ergin et al.*, 1999; *Shanker et al.*, 1999]. Here, use is made of these advancements to produce a robust and fast MOT-based TDIE solver for analyzing scattering from lossy penetrable volumes.

[3] The integral equation-based analysis of transient electromagnetic scattering from inhomogeneous objects is best accomplished by using the volume equivalence theorem. That is, the scatterer is replaced by equivalent polarization currents that reside in free space and an equation is constructed that relates the electric field radiated by these currents, namely, the scattered field, to the total electric field. Because both the polarization currents and the total electric field are intimately related to the electric flux density, this procedure automatically leads to a volume TDIE for the latter [*Gres et al.*, 2001]. This TDIE is typically solved by discretizing the flux density using “volume Rao-Wilton-Glisson” (RWG) basis functions [*Schaubert et al.*, 1994], i.e., divergence conforming basis functions defined on tetrahedra that are assumed to have constant constitutive parameters throughout. These basis functions ensure normal continuity of the discretized flux density across tetrahedral interfaces. In a lossy medium, care should be exercised when defining the flux quantity to be discretized. Indeed, in a lossy object, surface charges accumulate on interfaces between regions with different permittivities and hence ∂_{t}**D**(**r**, *t*) = ɛ(**r**)∂_{t}**E**(**r**, *t*) is discontinuous across material interfaces. One may always model the flux in terms of basis functions that are defined over single tetrahedra, leaving it up to the solver to enforce continuity [*Shanker et al.*, 2001a]. Alternatively, one may define a “conduction current corrected flux density” by ∂_{t}**D**(**r**, *t*) = (σ(**r**) + ɛ(**r**)∂_{t})**E**^{t} (**r**, *t*); this flux accounts for both the polarization and the conduction currents and is continuous across material interfaces. It can, therefore, be represented using the usual volumetric RWG basis, defined over pairs of tetrahedra. It is this last approach that is pursued herein. Lumping the conduction and polarization effects makes the effective permittivity frequency dependent. Therefore, computation of the flux using this procedure requires the convolution of the effective permittivity with the electric field. Incorporation of this convolution procedure into the TDIE is one of the main foci of this paper. Additionally, it will be shown that the computational complexity and the memory requirements of the resulting volume MOT-based TDIE can be reduced from (*N*_{t}*N*_{s}^{2}) and (*N*_{s}^{2}) to (*N*_{t}*N*_{s}) and (*N*_{s}) when accelerated by a multilevel volume PWTD scheme. Here, *N*_{t} and *N*_{s} denote the number of temporal and spatial degrees of freedom of the discretized flux density.

[4] The contributions of this paper therefore are two-fold: (1) A volume TDIE in terms of the conduction current corrected flux density is presented, and the requisite MOT equations for solving this TDIE are derived. (2) The proposed MOT solver is augmented with the PWTD scheme, and it is shown (both theoretically and via numerical experiments) that the resulting algorithm obeys the aforementioned scaling laws.

[5] This paper is organized as follows. Section 2 presents the proposed TDIE for analyzing transient electromagnetic scattering from lossy media in terms of a conduction current corrected flux density. This section also outlines an MOT scheme for solving this equation. Section 3 delineates the implementation of the volume PWTD algorithm within the context of the MOT scheme described in Section 2, focusing on the differences of this scheme to the surface PWTD scheme published in references [*Shanker et al.*, 2000, 2001b; *Ergin et al.*, 1999]. Section 4 presents a set of numerical results that demonstrate the accuracy and efficiency of the solver and Section 5 summarizes the contributions of this paper and outlines directions for future research.

### 2. Formulation

- Top of page
- Abstract
- 1. Introduction
- 2. Formulation
- 3. Multilevel Plane Wave Time Domain Algorithm for Volume Distributions
- 4. Numerical Results
- 5. Summary
- Acknowledgments
- References

[6] Consider a nonmagnetic inhomogeneous dielectric object *V* that resides in free-space (Figure 1): In what follows, *V* will represent both the object and its volume. Frequency independent permittivities and conductivities are denoted by ɛ(**r**) and σ(**r**), respectively; outside *V*, ɛ(**r**) = ɛ_{0} and σ(**r**) = 0. The permeability of both *V* and its surroundings is μ_{0}. An electric field **E**^{i} (**r**, *t*) that is temporally band-limited to ω_{max} is incident upon *V*; it is assumed that **E**^{i} (**r**, *t*) ≈ ∀**r** ∈ *V* for *t* < 0. Scattered and total electric fields are denoted by **E**^{s} (**r**, *t*) and

respectively. The scattered field **E**^{s} (**r**, *t*) is produced by polarization currents **J**(**r**,*t*) in *V* and its temporal derivative is expressed as

where

Here **I** is the idempotent, *R* = ∣**r** − **r**′∣ is the distance from the source to the field point, τ = *t* − *R*/*c* denotes retarded time, and *c* = 1/ is the free-space speed of light. The polarization currents **J** (**r**, *t*) are expressed as

where **D** (**r**, *t*) is the “conduction current-corrected flux density,” which is related to the total electric field as

Substituting (1)–(2) and (4) into (5) yields, ∀**r** ∈ *V*

This constitutes a time domain integral equation for **D** (**r**, *t*); indeed **E**^{t} (**r**, *t*) is related to **D** (**r**, *t*) through (5). To solve (6), **D** (**r**, *t*) is represented by spatial basis functions **f**_{n} (**r**), *n* = 1, …, *N*_{s} and temporal basis functions *T*_{j} (*t*), *j* = 1, …, *N*_{t} as

with unknown expansion coefficients *D*_{n,j}. The spatial basis functions **f**_{n} (**r**) used here are the three dimensional analogues of the RWG basis functions and are detailed in *Gres et al.* [2001]. These basis functions assume that *V* is approximated in terms of a tetrahedral mesh. One spatial basis function is associated with each facet of the mesh. If facet *n* is shared by two tetrahedra *V*_{n}^{+} and *V*_{n}^{−}, the *n*th basis function is defined as **f**_{n} (**r**) = **f**_{n}^{+} (**r**) + **f**_{n}^{−} (**r**), with **f**_{n}^{±} (**r**) = *a*_{n}**ρ**_{n}^{±}/(3*v*_{n}^{±}) for *r* ∈ *V*_{n}^{±} and zero elsewhere; here *a*_{n} is the area of facet *n* and *v*_{n}^{±} is the volume of *V*_{n}^{±}. The vector **ρ**_{n}^{+} is defined from the free vertex of *V*_{n}^{+} to the position vector **r**; **ρ**_{n}^{−} is defined similarly except that it is directed toward the free vertex of *V*_{n}^{−}. It is easily shown that the component of this basis function normal to facet *n* is continuous, consistent with the continuity condition for the conduction current-corrected flux **D** (**r**, *t*) defined by (5). If facet *n* is on the boundary of *V*, then only the + portion of the above definition applies; the function abruptly ends on the facet and the associated current deposits a charge there. The temporal variation of **D** (**r**, *t*) is approximated using shifted Lagrange interpolation functions, i.e., *T*_{j} (*t*) = *T* (*t* − *t*_{j}) = *T* (*t* − *j*Δ*t*) where Δ*t* is the time step size and *T* (*t*) is the quartic analogue of the quadratic basis function used in *Manara et al.* [1997]. It is constructed by expressing the current at time *t* = *t*_{i} + *p*Δ*t*, for −1 ≤ *p* ≤ 0, in terms of a quartic Lagrange interpolant [*Abramowitz and Stegun*, 1972]

where

It is to be noted that this choice of temporal basis function permits the construction of **D** (**r**, *t*) at any time *t* from one future and four past values of the discrete current *D*_{n,j} and accommodates the highest temporal derivative in (6). It should be noted that several authors have proposed alternative basis functions. Those by *Hu et al.* [2001] comprise a weighted sum of continuous exponentials and span only two time steps. The basis functions by *Weile et al.* [2004] are approximate prolate interpolants, namely, band-limited and approximately time-limited functions; compared to the functions by Hu et al., they span more time steps (typically on the order of 6 to 10) but have a much reduced bandwidth. Results from MOT solvers that use these basis functions converge exponentially fast to solutions by frequency domain solvers that use the same spatial basis functions. This said, the polynomial basis functions in (9) are excellent interpolants (for example, they interpolate a band-limited signal with *L*_{2} errors on the order of 10^{−8} provided that the signal is oversampled (from Nyquist) by a factor of 10). Moreover, they are a natural fit to the 4-step backward difference formula (BDF4) [*Lambert*, 1991] used for solving (5) in concert with (6), as explained in the next paragraph. For these reasons, the polynomial interpolants were chosen in this work.

[7] The number of spatial basis *N*_{s} is chosen proportional to *V*ω_{max}^{3}/*c*^{3} to ensure adequate representation of **D** (**r**, *t*) at ω_{max}. Likewise, assuming that the simulation tracks the scattering phenomenon for *T* seconds, the number of time steps is chosen *N*_{t} ∝ *Tf*_{max}. The permittivity and loss are assumed piecewise constant throughout *V*, i.e., ɛ(**r**) = ɛ_{n}^{±} and σ(**r**) = σ_{n}^{±} for **r** ∈ *V*_{n}^{±}. Solving (6) requires **E**^{t} (**r**, *t*) to be expressed in terms of **D** (**r**, *t*) at each time step. Applying BDF4 formula [*Lambert*, 1991] to (5) at *t* = *t*_{j} yields the following relation between time-sampled **E**^{t} (**r**, *t*) and **D**(**r**, *t*):

The α-coefficients in (10) are α_{l} = ∂_{p}β_{l}(*p*)∣_{p=0}; hence, the BDF4 scheme is consistent with the above method for representing/interpolating **D**(**r**, *t*). The BDF4 is zero-stable and has a considerably large region of absolute stability for all Δ*t* [*Lambert*, 1991]. Equation (10) permits the recursive computation of **E**^{t}(**r**, *t*_{j}) in terms of the values of **E**^{t} (**r**, *t*_{j−l}) and **D**(**r**, *t*_{j−l}), *l* = 1, …, 4, and therefore is easily incorporated into the MOT scheme, as will be demonstrated below. It should be noted, however, that spatial variations in **E**^{t} (**r**, *t*_{j}) cannot be represented in terms of the above described **f**_{n} (**r**) as the electric field is not continuous across facet interfaces; **E**^{t} (**r**, *t*_{j}) therefore is represented in terms of “half basis function”, i.e.,

It follows from (10) that

For boundary facets, again only one tetrahedron enters the picture. Substituting (7) into (6) and applying a spatial Galerkin testing procedure results in a set of equations that can be cast in matrix form as

where **I**_{j}^{d} = [*D*_{1,j}, *D*_{2,j}, …, *D*_{N}]^{T}, **I**_{j}^{e+} = [*E*_{1,j}^{+}, *E*_{2,j}^{+}, …, *E*_{N} ^{+}]^{T}, **I**_{j}^{e−} = [*E*_{1,j}^{−}, *E*_{2,j}^{−}, …, *E*_{N}^{−}]^{T},

[8] In the above equations, 〈·, ·〉 denotes the standard inner product. To enable field computation using an MOT scheme, (13) is recast as

Finally, both **I**_{j}^{e+} and **I**_{j}^{e−} can be expressed in terms of the prior values of the electric field and conduction-corrected flux density using (12); hence (17) can be expressed as

where

In (18), *I*_{j}^{d} is expressed in terms of fields and flux densities at times *t* < *t*_{j}. This equation can be solved for *I*_{j}^{d} using a nonstationary iterative solver like (TF)QMR [*Saad*, 1996]. Once *I*_{j}^{d} is computed, *I*_{j}^{e+} and *I*_{j}^{e−} are obtained via (12). In our implementation of the MOT scheme, all inner products are evaluated using a spatial Galerkin scheme with five-point and seven-point Gaussian quadrature rules for volumes and facets, respectively. The time-stepping scheme is implicit and the time step is typically chosen to be Δ*t* = 2πβ/(20ω_{max}), where β is a number between 1.0 and 2.0. The implicitness of the scheme requires that the highly sparse matrix _{0} be inverted to obtain *I*_{j}^{d}; typically, convergence to an error norm of 10^{−6} is obtained within 20 iterations using TFQMR [*Saad*, 1996]. The principal drawback in using the above described MOT scheme for large scale analysis of field propagation though lossy media is the exorbitant time and memory required for computing the RHS of (17). Finally, it should be mentioned that the temporal duration of the basis functions in (9) does not significantly affect the computational cost of the scheme. First, it can be shown that the above scheme is no more than 5/2 times as expensive as that of the same solver using the basis functions by *Hu et al.* [2001] – this upper bound would be reached only if the spatial basis functions were to cover infinitesimally small regions and arises from the fact that the basis functions in (9) and those by Hu et al. span 5 and 2 time steps, respectively. When using spatial basis functions spanning domains of linear dimension of (*c*Δ*t*), the multiplicative factor is closer to 3/2. Furthermore, because the proposed solver is accelerated by the PWTD scheme – see next section – this multiplicative factor only applies to near-field and lowest level ray calculations and therefore the added cost arising from basis functions that span more than two time steps all but vanishes.

[9] In what follows, a succinct derivation and implementation details of the “volume-PWTD algorithm” will be presented. It will be demonstrated that the volume PWTD scheme reduces the computational complexity and memory requirements of the analysis of transients in lossy scatterers to (*N*_{t}*N*_{s}) and (*N*_{s}), respectively.

### 3. Multilevel Plane Wave Time Domain Algorithm for Volume Distributions

- Top of page
- Abstract
- 1. Introduction
- 2. Formulation
- 3. Multilevel Plane Wave Time Domain Algorithm for Volume Distributions
- 4. Numerical Results
- 5. Summary
- Acknowledgments
- References

[10] The PWTD algorithm exploits the fact that the electric field radiated by a temporally band-limited, and approximately time-limited, spatially bounded source can be expressed as a superposition of homogeneous plane waves for any observer that is separated in space-time from the source. This fact allows for the development of a hierarchical scheme for rapidly evaluating the right-hand-side of (17) with an error that can be controlled to arbitrary precision. Our prior PWTD efforts focused on surface source distributions [*Shanker et al.*, 2001b; *Ergin et al.*, 1999, 1998]. The reader is encouraged to consult these references to gain familiarity with the scheme. In what follows, only those details of the PWTD scheme pertinent to its adoption into the above described volume integral equation solver are presented. In addition, the computational complexity of the volume PWTD scheme is discussed as it differs from that of its surface counterpart. In a *N*_{l} level volume PWTD scheme, *V* is enclosed in a hypothetical cube of side length *D* that is partitioned recursively *N*_{l} − 1 times to create a uniform oct-tree. The number of levels is chosen such that the side length of the smallest box thus obtained is approximately 0.2λ_{min} = 0.2(2π)*c*/ω_{max}. The large box containing *V* is said to reside at level *N*_{l}; the smallest boxes are said to reside on level 1. Specifically, starting at the highest (coarsest) level, each level *l* + 1 “parent” box is subdivided into eight equal size level *l* “child” boxes of circumscribing radius *R*_{s}^{l}. A basis function is said to belong to a box if its center of mass resides in that box. Furthermore, a box is said to be empty if it contains no basis functions. In the process of subdividing boxes, empty boxes are immediately discarded. Next, at all levels and starting with the highest one, all far-field box pairs are identified. A pair of boxes constitutes a far-field pair if (1) the distance between their closest points is greater than a fixed constant, typically chosen in the range 3 to 5, times *R*_{s}^{l}, and (2) their parent boxes do not constitute a far-field pair. At the lowest (densest) level, all box pairs (including self-pairs) not identified as far-field pairs are labeled near-field pairs. Contributions to the right-hand side of (17) due to spatial basis functions contained in near-field pairs are evaluated classically. Interactions between basis functions belonging to far-field pairs are computed using the PWTD algorithm. To elucidate the volume PWTD scheme, consider a lowest (densest) level (1) far-field pair comprising boxes labeled *b*_{s} and *b*_{o} with centers **r**_{s}^{c} and **r**_{o}^{c}, respectively. Associated with these boxes are spatial basis functions **f**_{n} (**r**), ∀*n* ∈ *b*_{s} and **f**_{m} (**r**), ∀*m* ∈ *b*_{o}. It is assumed that the temporal signature of all sources *n* ∈ *b*_{s} can be divided into *L* band-limited and approximately time-limited consecutive segments of duration *T*_{s} ∝ *R*_{s}^{1}/*c* such that *T* ∝ *LT*_{s}, i.e.,

Specifically, it is assumed that *g*_{n,l} (*t*) and *h*_{n,l}^{±} (*t*) are band-limited to ω_{s} = χ_{t}ω_{max} where χ_{t} is the temporal oversampling factor, which is typically chosen in the range 1–3. A scheme for subdividing a band-limited source signature into band- and approximately time-limited subsignals is described in *Ergin et al.* [1998]. Provided that *T*_{s} ≤ (*R*_{bo,bs}^{c} − 2*R*_{s}^{1})/*c*, where *R*_{bo,bs}^{c} = ∣**R**_{bo,bs}^{c}∣ = ∣**r**_{o}^{c} − **r**_{s}^{c}∣, the fields due to *g*_{n,l} (*t*) and *h*_{n,l}^{±} (*t*), ∀*n* ∈ *b*_{s} can be recovered at all *m* ∈ *b*_{o} for any time *t* > *lT*_{s} using a superposition of plane waves. Indeed, the field tested by **f**_{m} (**r**) may be expressed as

where

and *x*_{n,l} = *g*_{n,l} (*t*) or *x*_{n,l} (*t*) = −ɛ_{0}*h*_{n,l}^{±} (*t*). The quadrature weights and ray directions are

χ_{s} is the spherical oversampling factor, which is typically chosen in the range 1.0–1.5, and *P*_{ν}(·) is the Legendre polynomial of order ν [*Ergin et al.*, 1999]. The above exposition focused on the mathematical machinery necessary for evaluating lowest level interactions. However, it is apparent, and has been shown elsewhere [*Shanker et al.*, 2001b; *Ergin et al.*, 1999], that this scheme also forms the basis for a hierarchical computational algorithm in which the evaluation of the far-field contribution to the right hand side of (17) is accomplished by executing the following steps:

[11] 1. Evaluate *S*_{n}^{+} (_{qp}) for all applicable sources (*x*_{n,l} = *g*_{n,l} (*t*) or *x*_{n,l} (*t*) = −ɛ_{0}*h*_{n,l}^{±} (*t*)) in all the lowest level boxes. This process is termed constructing outgoing rays and is performed every *M*_{t}^{1} = *T*_{s}/Δ_{t} time steps.

[12] 2. Create rays for boxes at all levels in the tree, using interpolation and resectioning [*Shanker et al.*, 2001b; *Ergin et al.*, 1999]. It follows that the duration of the outgoing rays for the level-2 box is approximately 2*T*_{s}. It becomes therefore apparent that the construction of outgoing rays on the lth level is performed every *M*_{t}^{l} = 2*M*_{t}^{l−1} time steps.

[13] 3. At any level, information is translated between all far-field box pairs active at that level. This is tantamount to convolving each outgoing ray with the level-dependent translation operator *T*(_{qp}). Upon convolving outgoing rays with the translation operator, incoming rays result.

[15] 5. Finally, the incoming rays for each box are projected onto observers. This is equivalent to evaluating *S*_{m}^{−} (_{qp}) over each pair of tetrahedra that constitute a spatial testing function.

[16] The mathematical details of the interpolation, anterpolation, resectioning, and splicing operations for vector fields are elucidated in *Shanker et al.* [2001b] and *Ergin et al.* [1999] and will not be repeated here. The principal difference between the volume PWTD scheme, presented here, and the surface PWTD schemes, presented in *Shanker et al.* [2001b] and *Ergin et al.* [1999], lies in the need to account for all sources *x*_{n,l} = *g*_{n,l} (*t*) and *x*_{n,l} (*t*) = −ɛ_{0}*h*_{n,l}^{±} (*t*).

[17] An added difference between the volume and surface PWTD schemes is their computational complexity. The computational complexity of the volume scheme therefore is analyzed next. The presentation below again assumes some familiarity with the surface PWTD scheme of *Shanker et al.* [2001b] and *Ergin et al.* [1999]. The complexity arguments presented below rest on several assumptions:

[18] 1. The distribution of unknowns is uniform; i.e., *V* is modeled by tetrahedra of approximately equal size and there is no need to finely mesh any regions of *V* to resolve fine geometric details. If this assumption does not hold true, the proposed volume PWTD scheme requires augmentation with the low-frequency PWTD scheme described in *Aygün et al.* [2001] to maintain the below complexity estimate.

[19] 2. *V* is truly volumetric in nature; that is, *V* does not constitute a penetrable wire or thin coating but a bulk object, e.g., a sphere or cube. If this assumption is violated, the complexity analysis below is not valid and the complexity analysis of *Shanker et al.* [2001b] and *Ergin et al.* [1999] applies instead.

[20] 3. The size of the smallest level box is chosen such that it is associated with *O*(1) unknowns; that is, as *V* becomes electromagnetically larger, the number of unknowns in the finest level boxes remains fixed. For low-index scatterers, this assumption is automatically satisfied by the above delineated scheme for constructing the PWTD tree; for high-index scatterers where the discretization density grows to rapidly track possible field changes, this assumption breaks down and augmentation of the proposed scheme with the low-frequency PWTD scheme of *Aygün et al.* [2001] again is called for.

[21] 4. At any level, the number of ray directions associated with a box is proportional to the surface area of the box.

[22] 5. Similarly, at any level, the length of the outgoing/incoming rays is proportional to the linear extent of a box at that level.

[23] 6. Finally, the number of boxes that are in the near-/far-field of a box at any level is of (1).

[24] Assumptions 4–6 are automatically satisfied if the multilevel PWTD scheme is implemented along the lines described in *Shanker et al.* [2001b] and *Ergin et al.* [1999]. With these assumptions, the computational cost of the volume PWTD scheme is estimated as follows:

[25] 1. Near-field evaluation. At the lowest level, there are *O*(*N*_{s}) boxes, each of which belongs to (1) near-field pairs. Because each of these boxes contain (1) sources/observers, the total cost of computing all near-field interactions for one time step scales as

Also, it is apparent the memory required to store the near-field contribution to all matrices in (17) scales as (*N*_{s}).

[26] 2. Far field evaluation: Construction of outgoing rays. At the lowest level, there are (*N*_{s}) boxes. Because each of them supports *D*_{s}^{1} = *O*((*R*_{s}^{1})^{2}) = (1) rays of duration *M*_{t}^{1} = (1) time steps, the total cost of constructing rays at the lowest level is proportional to

When climbing the PWTD tree by one level the number of boxes decreases by eight, the number of directions increases by four, the length of each ray doubles, and the number of times it is translated decreases by two, thereby halving the cost to compute outgoing rays. Thus, the total cost scales for computing outgoing rays across all levels scales as

[27] 3. Far field evaluation: Translation. At any given level *l*, the outgoing rays of a source box *b*_{s}^{l} are translated onto the incoming rays of an observer box *b*_{o}^{l} every *M*_{t}^{l} time steps. Since the size/duration of the outgoing rays and the translation function are approximately the same, because the outgoing ray is time-limited, and because the Fourier transform of the translation function is analytically available, the translation operation is efficiently carried out using a fast Fourier transform (FFT). In actual computations, the translation function is sampled and tabulated, and reconstructed when necessary [*Shanker et al.*, 2001b; *Ergin et al.*, 1999]. The cost of construction of the translation function is smaller than the cost of its convolution with the outgoing rays. At level 1, the cost of convolving translation functions with outgoing rays for all far-field interaction scales as

At the lowest level this cost is proportional to *O*(*N*_{t}*N*_{s}). As in (35), at higher levels the number of boxes decreases by eight, the number of ray directions increases by four, and the duration of the ray increases by two. Summing the translation cost over all levels, it can be verified that the total translation cost scales asymptotically as

The logarithms in the above estimate stem from the use of FFTs in effecting the translation.

[28] 4. Far field evaluation: Incoming rays. It is easily shown that the cost for traversing down the tree scales just like that of climbing the tree. All the above complexity estimates assume the use of efficient algorithms for splicing and interpolation (or resectioning and anterpolation). Typically, a fast exact spherical interpolation scheme [*Shanker et al.*, 2001b] that is a variant of *Jakob-Chien and Alpert* [1997] is used for interpolation (anterpolation). Finally, it is easily shown that the memory required for storing these rays scales as (*N*_{s}). This estimate is in agreement with the fact that only ray data throughout the volume of the scatterer is necessary for computing scattered fields.

[29] Thus, the computational complexity and the memory requirements of the PWTD augmented MOT scheme for volumetric analysis scale as *C*_{NF} + *C*_{FF}^{Rays} + *C*_{FF}^{Trans} = (*N*_{t}*N*_{s}) and (*N*_{s}), respectively. The volume PWTD scheme thus scales more favorably than its surface counterpart. Furthermore, it becomes clear that the computational complexity and memory costs of PWTD-accelerated volume TDIE solvers asymptotically scales as favorably as that of finite difference/element time domain methods (that do not compensate for phase dispersion); for a discussion on the topic, the reader is referred to *Aygün et al.* [2002]. While the multiplicative constant in the TDIE solver cost estimate is larger than that of finite difference/element solvers, we believe that the TDIE solvers will become especially useful in the analysis of coated concave conducting structures, e.g., inlet structures [*Shanker et al.*, 2003].

### 4. Numerical Results

- Top of page
- Abstract
- 1. Introduction
- 2. Formulation
- 3. Multilevel Plane Wave Time Domain Algorithm for Volume Distributions
- 4. Numerical Results
- 5. Summary
- Acknowledgments
- References

[30] This section presents several numerical results that serve to validate, and demonstrate the effectiveness of, the above outlined PWTD-enhanced MOT scheme. In all examples that follow, the incident field is a modulated Gaussian pulse parameterized as

where ^{i}, ^{i}, and *f*_{0} denote the polarization vector, direction of propagation, and center frequency of the incident field, respectively. Also, ς = 6/(2π*f*_{bw}) where *f*_{bw} is the “bandwidth of the signal,” and *t*_{p} = 6ς. At *f*_{max} = *f*_{0} + *f*_{bw} the power in the incident field is down by 160 dB relative to that at *f*_{0}. The mesh is constructed such that the maximum edge-length is about one fifth of the wavelength at *f*_{max} in the densest part in *V*. In what follows, the PWTD-enhanced MOT scheme is used to compute far-field signatures of scattered fields, which are then Fourier transformed to obtain frequency domain Radar Cross Section (RCS) data. These RCS signatures are then compared against analytical computations or data obtained using a fast multipole enhanced frequency domain method of moments (MOM) code.

[32] Next, scattering from compound shells is analyzed. A two-layered shell is considered first. Its inner/outer radii are 0.8m/1.2m and each layer is 0.2m thick. The permittivities of the inner/outer layers are 4ɛ_{0}/2ɛ_{0}, respectively; the conductivity of both layers is 5.5 × 10^{−3}S/m. Polarization currents in the compound shell are discretized using 8626 basis functions. The center frequency of the incidence pulse is *f*_{0} = 75 MHz, and its bandwidth is *f*_{bw} = 55 MHz; all other parameters remain unchanged from the first example. As is apparent from Figures 3a and 3b, the computed RCS data at both 50 MHz and 80 MHz agree very well with the analytical solutions and RCS data obtained using the MOM code. A three-layered shell is studied next. The inner/outer radii of the shell are 0.8 m/1.4 m, and the thickness of each layer is 0.2 m. Polarization currents in the shell are modeled using 15,364 spatial unknowns. The layers, starting with the innermost, have permittivities of 6ɛ_{0}, 4ɛ_{0}, and 2ɛ_{0}, and all layers have a conductivity of 0.0167S/m. The incident pulse has a center frequency of *f*_{0} = 60 MHz, and a bandwidth of *f*_{bw} = 45 MHz; again, all remaining parameters are chosen identical to those in the first example. As is apparent from Figures 3c and 3d, the RCS of this three-layer shell at 90 MHz and 130 MHz computed by the PWTD-enhanced MOT solver is in good agreement with the analytical data. It is also in good agreement with data obtained from the MOM code.

[34] Finally, to verify the theoretically predicted CPU scaling estimate for the PWTD-enhanced MOT code, scattering from a rod of dimensions 1 × 1 × 20 m^{3} in which polarization currents are represented using 1364, 3560, 6406, 11834, and 26684 spatial basis functions, is analyzed. In each simulation, the incident wave is polarized along ^{i} = and traveling along ^{i} = . The center frequency and bandwidth of the incident pulse for each mesh are chosen such that the average edge length is a fifth of the minimum wavelength in the rod medium. The rod's permittivity is always 4ɛ_{0}; its conductivity is varied such that, at the center frequency of the incident pulse, the loss tangent σ/(2π*f*_{0}ɛ_{rod}) = 0.25. Although this analysis is performed primarily for timing purposes, a couple of representative results are presented. For the box in which the polarization current is discretized using 6,406 spatial basis functions, the conductivity, center frequency, and bandwidth are chosen as 2.55 10^{−3}S/m, *f*_{0} = 46 MHz, and *f*_{bw} = 42 MHz, respectively. Figures 5a and 5b compare the RCS data obtained using the time domain scheme against that from the frequency domain MOM code at *f* = 46 and 67 MHz; both results agree well with each other. For the box in which polarization currents are discretized using 11,834 spatial basis functions, the conductivity, center frequency, and bandwidth are 3.61 10^{−3}S/m, *f*_{0} = 65 MHz, and *f*_{bw} = 62.5 MHz, respectively. As is apparent from Figures 5c and 5d, the RCS data obtained from the time domain code agrees very well with that obtained from the frequency domain MOM code at *f* = 33 and 90 GHz. In Figure 6, the CPU time per time step is plotted versus the number of spatial unknowns, and it can be verified that the overall computational complexity indeed scales as (*N*_{t}*N*_{s}). This figure also implies a breakeven point of around 2500 spatial unknowns for; that is, for *N*_{s} < 2500 a classical MOT code will outperform the PWTD-enhanced solver but for larger problems it becomes advantageous to use the accelerator. Note that, in performing this experiment, all the timing data were obtained by running these codes on an SGI Origin 2000 machine. Finally, we note that approach described in this paper is relatively efficient. As an example, consider the box descretized using 11,834 unknowns that was analyzed in Figures 5c and 5d. The total time for the analysis for *N*_{t} = 800 took 56,292 CPUs. A similar analysis using our frequency domain fast multipole code took 17,142 CPUs for one frequency. Thus, obtaining data for a range of frequencies using our frequency domain FMM code would be relatively more expensive that obtaining the same data using our time domain solver. This said, a couple of caveats are in order: we do not claim that our frequency domain code is highly optimized, and note more sophisticated algorithms exist for obtaining broadband data using frequency domain codes than re-running the code for all frequencies. Nonetheless, the timing data presented earlier shows that the time domain analysis scheme presented herein is efficient.