### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Multidomain Decomposition
- 3. Algorithm
- 4. Numerical Results
- 5. Range of Applications
- 6. Conclusion
- References
- Supporting Information

[1] This paper is devoted to the numerical treatment wide electromagnetic scattering scenes by a decomposition into disjointed subdomains to solve time domain Maxwell's equations. It presents a way to adapt existing time volumic scheme in order to simulate interactions between multiple sources and/or scatterers when either computational costs or the accumulation of numerical errors become prohibitive because of important volumes of free-space meshes. This is done by an automatic implementation of the proposed method which leads to a parallel algorithm with an adaptive criterion on the accuracy, introducing by the way some very significant gains on the computational effort. Numerical studies are performed to show the efficiency of this method in comparison with mono-domain resolutions. Improvements of the approximations, relevant of the introduction of the adaptive parameter, are also quantified and then demonstrate their benefits with respect to a straightforward multidomain scheme involving integral formulas.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Multidomain Decomposition
- 3. Algorithm
- 4. Numerical Results
- 5. Range of Applications
- 6. Conclusion
- References
- Supporting Information

[2] Differential volumic methods in time domain have for long proven their usefulness and efficiency for solving a large variety of electromagnetic problems of interest in radiation, diffraction, compatibility, especially with wide band or ultra wide band illumination. It is the case for Finite Differences in Time Domain (FDTD) [see, e.g., *Taflove and Hagness*, 2000] and also Finite Volumes in Time Domain (FV) [e.g., *Bonnet et al.*, 1997], Discontinuous Galerkin [e.g., *Cohen et al.*, 2006], Finite Elements [*Pernet et al.*, 2005; *Jiao et al.*, 2001, 2002]. When the computational volume grows, the corresponding necessary computational effort rises as well up to limits imposed by the computing power available, imposing thereby practical limits to the use of these methods. Furthermore, according to the numerical scheme used, numerical errors, either dispersion or dissipation ones, propagate inside the computational volume, all the more so as this volume is large. To counter this effect meshing has to be refined, worsening these size limitations. When several separated sources and/or diffracting objects are present in the so-considered electromagnetic scattering scene, some variable amount of the computational power is dedicated to treat the surrounding free space volume. This is even more prejudicial and prohibits the treatment by time domain volumic methods of a substantial number of cases within this class of problems.

[3] To bypass these limitations, the idea of a stand alone simulation of the close neighborhood for the different sources/scatterers coupled with some time domain radiation integral formulas rendering their mutual interaction has been proposed [*Johnson and Rhamat-Samii*, 1997; *Bernardi et al.*, 2002; *Xu and Hong*, 2004]. It leads to multidomain approaches. However these mixed approaches present specific difficulties inherent to their implementation. Indeed, for each time step calculus are done both over all pick-up surfaces for radiation part and over all Huygens surfaces [*Mur*, 1981] for reception part. A brute force implementation of this corresponding algorithm consists in using time domain radiation formulas [*Craddock and Railton*, 1992; *Shlager and Smith*, 1994, 1995; *Johnson and Rhamat-Samii*, 1997]. Then, the evaluation of these interactions prove to be very costly, when moreover the global scheme may be unstable [*Johnson and Rhamat-Samii*, 1997]. To diminish the costs, one way would consist in evaluating these mutual interactions through fast algorithms like the multilevel plane-wave time-domain algorithm [*Jiao et al.*, 2002]. To our best knowledge, such kind of implementation has not been reported yet. So, at the moment, it is not possible to forecast which benefits might be withdrawn by such an approach in the most general case of domains with arbitrary sizes and shapes, and with some wide band incident illumination. Another way rests in compressing the data to be transferred by introducing approximations. In *Bernardi et al.* [2002], it is proposed to divide the radiation and Huygens surfaces in patches of arbitrary size and assume the field to be constant over each of them. As a matter of fact, there is no criterion to determine up to which patch size this approach is correct. In a 2D implementation, *Xu and Hong* [2004] has proposed for this a far field criterion by switching into the frequency domain.

[4] In *Mouysset et al.* [2007], a fully consistent method, exclusively in time domain, has been proposed. It rests on the use, for the mutual interaction, of the approximate radiation formula proposed by *Mouysset* [2005] which accuracy is controlled by an adaptive parameter [*Mouysset et al.*, 2006] and may be rendered consistent with the accuracy of the volume numerical methods applied in the various subdomains. Setting a value to this parameter permits the creation of groupings on pick-up and Huygens surfaces. It still remains consistent with this accuracy, and thereby realizes a consistent space compression. The present paper aims first at presenting the automatic and parallel implementation of this method in the general case of multiple separated domains of arbitrary sizes, shapes, places, and with wide band incident pulse. Through numerical studies we will exhibit performances and benefits of this method.

[5] For sake of consistency in the paper we will present in section 2 a brief recall of the method. Then, section 3 describes the algorithm for its further implementation. The numerical study are thus proposed in section 4 using FDTD into each domain, and so incorporating complexity evaluation and situating its performances on one hand with respect to mono-domain computations and on the other hand to brute force implementation of the mutual interaction. And at last, before concluding (section 6), we will detail in section 5 its range of applications.

### 2. Multidomain Decomposition

- Top of page
- Abstract
- 1. Introduction
- 2. Multidomain Decomposition
- 3. Algorithm
- 4. Numerical Results
- 5. Range of Applications
- 6. Conclusion
- References
- Supporting Information

[6] This section is devoted to briefly recall the principle of the multidomain method such as it was introduced with our previous work [*Mouysset et al.*, 2007]. This will be the starting point of the present paper. From this we will be able to derive an efficient multidomain algorithm leading to the hybrid and parallel scheme, and study its accuracy and costs.

[7] Basically, the principle of the multidomain decomposition is to enclose each element (i.e., a source or a scatterer), or group of elements if they are considered to be too close one each other, into a computational domain noted ^{i} (*i* = 1.*n*). All these domains are assumed to be disjointed one from the others. Then, as Figure 1 illustrates, we introduce in each domain ^{i} a couple of surfaces ^{i} and ^{i}, such that ℋ^{i} is embedded into ^{i}. Coupling between the domains is taken into account propagating electric and magnetic currents *J*_{j} and *M*_{j} picked on ^{j}, for *j* ≠ *i*, to ℋ^{i}. So ^{i} (*i* = 1.*n*) are called pick up surfaces, and ℋ^{i} (*i* = 1.*n*) Huygens' ones.

[8] Thus, in *Mouysset et al.* [2007] we proposed to replace the system of Maxwell's equations by the following coupled one

where ɛ_{i}(*x*), *μ*_{i}(*x*) and *σ*_{i}(*x*) stand for the classical electromagnetic parameters describing the media and the element into ��^{i}; (*E*^{inc},*H*^{inc}) plays the part of any incident field (e.g., plane wave); *J*_{j} and *M*_{j} are given by

[9] At last, terms with a tilde are coming from some approximations.

[10] 1. For any function *f*, *f* gives a piecewise function on a determined clustering of ℋ^{i} into a set of subfaces ∪_{k}ℋ_{k}^{i}. For each ℋ_{k}^{i} a reference point *X*_{k}^{i} is associate and this piecewise function verifies

[12] Note that we assume the electric and magnetic permittivities, ɛ_{i}(*x*) and *μ*_{i}(*x*), to be constant out of ℋ^{i}, while *σ*_{i}(*x*) has to vanish out of ℋ^{i}.

[13] These two approximations, on the currents along the Huygens'surface (2) and on the propagation of the electromagnetic field (3)–(4), are the key points of the method. Let *δ*^{j}_{k} be a characteristic length for the subface _{k}^{j} (e.g., its diameter). Then, according to *Mouysset* [2005], at any observatory *X* situated out of the volume delimited by ℋ^{i}, the difference between the fields (*E*_{j}, *H*_{j}) and the propagation of the currents (*J*_{j}, *M*_{j}) by formulas (3)–(4) is bounded by

On the same time, error on approximating by (2) the currents on the Huygens' surface ℋ^{i}, trace of the electric and magnetic currents (*J*_{j}, *M*_{j}) propagated with (3)–(4) verifies [*Mouysset et al.*, 2007]

where Δ_{k}^{i} is a characteristic length of ℋ_{k}^{i}, and *D*_{k}^{i} stands for the distance between the reference point *X*_{k}^{i} on ℋ_{k}^{i} to the pick up surface ^{j}.

[15] 1. For the interior part of the problem: *i*, ∀(*t*,*X*) ∈ ^{+} × *V*(ℋ_{i})

[16] 2. For the exterior part: (*t*, *X*) ∈ ^{+} × (∪_{i} ��_{i})^{c}

[18] The main matter of this paper is then to study the influence of this parameter *δ*/*d* on the global costs (in terms of CPU time and memory storage) and also on the precision. Note that *δ*/*d* is an adaptive parameter according to the freedom let on the choice of *δ*. Our goal is to put into light a suitable range of applications for the multidomain method which gives both an interesting balance between costs and accuracy, and a wide class of simulations interesting to perform.

[19] Till the end of the paper, we will refer to the two sets of approximations (2) and (3)–(4) as anterpolation and interpolation phases in reference to the Fast Multipole Methods [see *Darve*, 2000, and references therein]. As for the FMM, we define (1) an anterpolation (Figure 2) phase which consists in computing the contribution from a given domain ��^{i} to the others, and (2) an interpolation (Figure 3) phase which is the reconstruction of the coupled problem by adding all contributions coming from the other domains ��^{j} (*j* ≠ *i*) to a given one ��^{i}.

[20] This terminology is chosen in regard of the constructive process which is quite similar to FMM's one, and because these two phases are introduced in order to simplify calculations and to diminish storage and CPU time costs according like for the FMM.

### 4. Numerical Results

- Top of page
- Abstract
- 1. Introduction
- 2. Multidomain Decomposition
- 3. Algorithm
- 4. Numerical Results
- 5. Range of Applications
- 6. Conclusion
- References
- Supporting Information

[34] All numerical results presented in the following are computed by treating all domains ��^{i} with separated processor, on the multidomain (denoted by MD) parallelized scheme previously presented (see Figure 8), and using FDTD [see *Yee*, 1966] on each domain. Nevertheless, we want to precise that our multidomain method is not specific to FDTD (in opposition with *Johnson and Rhamat-Samii* [1997], *Bernardi et al.* [2002], and *Xu and Hong* [2004]). We only chose FDTD as the numerical scheme to perform some significant comparisons with mono-domain computations and indicate the global behavior of the method. In all the test cases we present, the time step is chosen in agreement with the CFL condition of the FDTD scheme according to the mesh used. Moreover, implementations of the NTRF and the interpolation stage over the Huygens surface are led according the quadrature formulas inherent to the FDTD. However, as in *Martin* [1998], for each domain, we have chosen two staggered pickup surfaces (one centered on *E*-fields and the other one on *H*-fields) and then computation of the NTRF is straightforward. All computations are made onto a cluster of workstations, in a local area network, with one 2.6 Ghz Pentium 4 processor and 2GB of memory per computer. Parallelization is led using the standard MPI library (Message Passing Interface).

[35] At last, to give a suitable presentation of all the numerical cases for the multidomain method in the sequel, we will draw the following elements on all the meshes (Figures 10, 13, and 20): bounds of each computational domain, the pickup surface (embedded into the bounds of the computational domain), and the Huygens surface (embedded into the pickup surface, and containing some scatterer/source). By the same way, for the mono-domain mesh (Figure 9), we will also represent the bounds of the whole computation domain used and the Huygens surface if an incident field is introduced.

#### 4.1. Two Metallic Boxes

[36] As a first numerical test we consider two metallic cubes of edges two meters long in presence of the following plane wave

This case is represented in Figure 9 for mono-domain computations, and in Figure 10 for multidomain ones.

[37] We use a *λ*/10 regular mesh to compute both with FDTD as a single method, and with the multidomain decomposition using FDTD, with *λ* ≈ 2 *m*. The observation is located at the middle of the line segment from center to center of the boxes. Each box is surrounded successively by a Huygens surface and a pick up one respectively at 3 and 6 cells of the boxes. Figures 11 and 12 present a comparison between the result obtained with the single FDTD domain, and the multidomain decomposition using the three following configurations of grouping: a regular grid of 2 cell by 2 cell is used on each face of both Huygens and Pick-up surfaces (it is denoted as the *reg*. 2 × 2 × 2 curve), a proportional grouping using the criteria *δ*/*d* ≤ 0.1 (denoted by *prop*. 0.1) as described in section 3.2, and a proportional grouping using the finer criteria *δ*/*d* ≤ 0.05 (denoted by *prop*. 0.05).

[38] On these curves, the mono-domain FDTD is taken as a reference since we have checked on this example its accuracy.

[39] One can observe on Figures 11 and 12 that whatever the choice of grouping we made, the results are pretty good. Moreover, a sensible result is shown: the smaller the groupings are, the finer the result is. This is in appropriateness with convergence of the multidomain method when the parameter *δ*/*d* (where *δ* describes the maximum diameter of the surface of the groupings) tends to 0.

[40] As we can be ensured that results are still pretty good when we choose a quite big *δ*/*d* maximum value we are interested in comparing what kind of profit one can take from such a choice. This results are detailed in Table 1.

Table 1. Comparison of Costs Between One Domain and Multidomain MethodsTest Case | Domain | Time Used | *N*_{inj} |
---|

One Domain | - | 104s | - |

MD reg. 2 × 2 × 2 | 1 | 4155s | 486 |

| 2 | 4316s | 486 |

MD prop. 0.1 | 1 | 165s | 36 |

| 2 | 150s | 40 |

MD prop. 0.05 | 1 | 360s | 97 |

| 2 | 329s | 100 |

[41] First remark is that FDTD method in the single domain is faster than the multidomain one in all the cases presented here. It seems to be predictable since the test case is very simple and one of those which make FDTD have the best results. Further, in section 4.2, we will study evolution of CPU time and memory costs between FDTD and the multidomain method on a very similar case but leading to pretty good description of what kind of applications are best treated with our method.

[42] The most important result is that increasing precision on the proportional grouping makes bigger the number of coupling points needed (*N*_{inj}) and numerical costs. Therefore, one has to choose maximum value of the parameter *δ*/*d* as big as possible (for a given precision on the result). Moreover, the kind of proportional groupings introduced in section 3.2 appears to lead to a less expensive decomposition that we will have with a regular splitting of the surfaces.

[43] Hence, as we expressed it previously, a fine result can be obtained with a quite big value of *δ*/*d*, and so this leads to the following alternative: be very accurate and increase costs of the method (by increasing number of coupling points); decrease number of coupling points as still we have an efficient scheme, loosing some accuracy (given by *O*(*δ*/*d*)).

[44] Both approaches lead to some realizable and stable numerical cases. In the sequel we will only consider the solving by the multidomain method using the kind of proportional groupings introduced in 3.2 and then deal with their accuracy and performances.

#### 4.2. Efficiency of the Multidomain Method Over Mono-Domain Computation

[45] In this section we propose a study of the efficiency of the multidomain method in terms of precision, memory costs, and CPU time. This is performed on the (simple) following test case made of the same two metallic boxes as previously but there placed diagonally one each other (see Figure 13).

[46] Before studying this case and discuss about the efficiency of the method, we first can give the general evaluation of costs for each domain *i*, per iteration in time, in terms of memory costs:

and of CPU time:

where *M*_{Dom}^{i} and *T*_{Dom}^{i} respectively stand for Memory and Time required by scheme used to compute the domain *i* (there: FDTD) without coupling, and *N*_{inj}^{i} and *N*_{gr}^{i} are respectively the number of injection points and grouping points obtained after the preprocessing phase of grouping; *N*_{huyg}^{i} and *N*_{pick}^{i} are respectively the number of cells composing the Huygens surface and the pick up surface; *N*_{sync} is the number of iterations between two synchronization phases.

[47] According to equations (11) and (12) it is easy to observe that memory and CPU costs are raising with number of grouped points on both pick up and Huygens surfaces. Hence we can notice two main cases in which such a remark leads to pertinent use of the multidomain method:

[48] 1. For a given scattering scene, one has to take grouping as big as possible to obtain better performances in terms of CPU time and memory costs, while results become less accurate (see Figures 14 and 15).

[49] 2. For a fixed precision, here a maximum ratio *δ*/*d* fixed to 0.1, in a “moving” scattering scene (e.g., when modeling the coupling between antennas and scatterers at various distances one each other), the multidomain method will have a better efficiency than mono-domain methods when moving away elements one from each other (see Figures 16 and 17).

[50] Both propositions can a priori seem to be very naive, but according to expressions (11) and (12), we can see that the multidomain method makes all domains (almost) independent one from each other. Indeed, terms *M*_{Dom}^{i} and *T*_{Dom}^{i} are only depending on the numerical scheme used into the domain ��^{i}. Hence, as numerical description of the domains are independent, we can choose the most adequate one for each domain. So, even if the term in “big Oh” into (11) and (12) are inducing extra costs per domain, we can balance them with the needed cost for the numerical scheme to solve each ��^{i}. This naturally leads to some different space and time descriptions for the numerical scheme.

[51] Note that both memory costs and CPU time are tending to asymptotic values on Figures 16 and 17. For the amount of memory involved, this is due to the opposite trends of *N*_{inj} and *N*_{sync} which tend to an equilibrium when distance *d* is growing. Meanwhile, decrease of *N*_{inj} makes computation time smoothly tend to a small value.

[52] Moreover, to illustrate it, we refer in Figures 18 and 19 to a result of the case presented in Figure 13, where the distance between the computational domains is 16*m*. We use the plane wave previously introduced, and give *E*_{x} field versus time at a point located at half-distance of each domain. Then, we compare the four following cases: (1) mono-domain computation with FDTD and a *λ*/10 mesh, (2) mono-domain computation with FDTD and a *λ*/20 mesh, (3) multidomain method with FDTD using *δ*/*d* ≤ 0.1 and a *λ*/10 mesh, and (4) multidomain method with FDTD using *δ*/*d* ≤ 0.05 and a *λ*/10 mesh.

[53] Such an example permits us to show that with a quite large *δ*/*d* (here *δ*/*d* ≤ 0.1) on a *λ*/10 mesh we can improve the results, lowering significantly the dispersion introduced by the scheme. Indeed, this can be easily seen by comparing the mono-domain FDTD computations using successively a *λ*/10 and a *λ*/20 meshes. This leads to a less than half computational time and a fourth of memory costs. Furthermore, a finer precision on *δ*/*d* (taking *δ*/*d* ≤ 0.05) leads to results as good as FDTD on the *λ*/20 mesh, with barely half memory costs and equal time as used for the *λ*/10 FDTD calculus.

[54] Note that even if in the test case presented here, domains are far one from each other (at almost 8*λ*) coupling is weak but still present so they can not be treated without taking it into account. Moreover, according to results presented in Figures 16 and 17 for distance between the domains from 5*λ* to 8*λ*, where coupling remains significant, the multidomain method leads to results as accurate as for the mono-domain computations, but with less computational time and lower memory costs.

#### 4.3. Influences of the *δ*/*d* Approximations Over Costs and Accuracy

[55] Even if we have stated in section 2 that the global error introduced by the two approximations in *δ*/*d* (in the anterpolation and interpolation stages), we propose in this section some test case to verify it numerically. To do so, we consider the scattering scene compounded of two elements: a thin wire taken as a dipole source in the first domain, and a vacuum domain wherein we are going to measure the EM-fields (see Figure 20). According to the principle of the multidomain method, on the point placed in the second domain we are going to evaluate the error introduced successively in the anterpolation stage in the first element, and in the interpolation phase for the second domain. Hence we will be able to quantify the total error inherent to the method. At last, to grant that the global error is independent of the distance between the domains, regardless to the parameter *δ*/*d*, different locations for the wire domain are chosen. Though, we place the two domains at 4.5*λ*, 6.5*λ*, 8.5*λ* and 10.5*λ* one each other, where *λ* ≈ 2 m stands for the wavelength of the source. Numerical comparisons are led with a reference mono-domain FDTD computation for each configuration. We plot on Figure 21 the time-quadratic error for *E* and *H* components versus *δ*/*d* for these distances.

[56] First, we can check in the middle part of the curves a linear comportment for the error according to *δ*/*d*. Moreover, this behavior appears to be independent of the lonely parameter *d*. We are in agreement with the theoretical estimates of section 2.

[57] Then, we also notice two plateaus associated with the two phenomenon of saturations on the groupings. The first one, for *δ*/*d* close to 0, corresponds to the lower limit of *δ* (characteristic length of the groupings) which is the mesh size. Moreover, as it is proved in *Mouysset* [2005] that *δ*/*d* = 0 gives the exact integral transport, it follows that a vanishing parameter *δ*/*d* traduces the numerical error introduced by a descretization of the multidomain method with exact Stratton-Chu formulas (MR/FDTD [*Johnson and Rhamat-Samii*, 1997]). Thus, results obtained for *δ*/*d* = 0 give common reference values to situate the efficiency of our scheme with any further comparisons between other multidomain method and MR/FDTD for example.

[58] The second plateau is due to the reverse phenomena, where *δ* is up-bounded by the size of the face on Huygens and pick up surfaces. Of course, upper and lower saturations of *δ* value are only linked to specific geometrical properties of the domains. Hence, for a fixed value of *d* and a value ɛ for the parameter *δ*/*d*, global trends of the *E* and *H* errors seem to be only due to *δ*, but as *δ* is chosen lower or equal to ɛ *d* then realization of its limits depends on *d*. This explains the differences in the happening of the plateaus in Figure 21.

[59] At last, we propose on Figure 22 the evolution of the CPU time used to compute the free-space domain versus *δ*/*d* on each configuration. It decreases on a parabolic curve and presents two plateaus linked to the saturation of *δ* as observed previously. Moreover we can refine this analysis by plotting the evolution of the number of groupings on the Huygens' surface *N*_{inj}^{2} on Figure 23. Very similar ones are obtained for the other quantities *N*_{inj}^{1}, *N*_{reg}^{1} and *N*_{reg}^{2}. It then appears, in agreement with (12), that the offset of the plateau obtained for large values of *δ*/*d* is essentially due to the time needed by the FDTD to compute the domain alone. Hence, depending on the value we choose for *δ*/*d* we can have a significant benefit up to four orders of magnitude on the numerical treatment of the coupling between domains.

[60] Remark: In *Mouysset* [2005] it is demonstrated that the NTRF tends to exact integral formula when *δ*/*d* comes to 0. Moreover, as expression of the NTRF is very close to the far field introduced by *Yee* [1991], its numerical implementation leads to computational costs of the same order [*Mouysset et al.*, 2006]. Hence, we can state that for *δ*/*d* ≈ 0 CPU time costs on Figure 22 represent an optimistic estimate of those obtained performing coupling by some point-to-point integral formula as for the MR/FDTD [*Johnson and Rhamat-Samii*, 1997] typically. From this, one can observe that relaxing the constrain on *δ*/*d* makes computational costs drop very fast while the global error (on Figure 21) is controlled. Obviously, as the numerical scheme on each domain (for instance here the FDTD) has an error not equal to zero, we can easily choose some *δ*/*d* value granting the evaluation of the coupling terms to be of the same order of accuracy while reducing drastically the computational costs. Indeed, in this paper we present examples where the choice of *δ*/*d* leads to accurate and efficient computations.