A new finite element-FDTD-boundary integral equation technique with biconjugate gradient solver for modeling electromagnetic problems in the frequency domain



[1] A new hybrid formalism in the frequency domain is presented, based on a combination of the boundary integral equation (BIE) technique, the finite element (FE) method and the finite difference time domain (FDTD) algorithm. An FE method and FDTD are used to describe the inhomogeneous parts of the simulation domain, whereas large homogeneous portions of the simulation space are modeled with a BIE. To solve the global matrix system efficiently, a biconjugate gradient solver is used. To calculate the matrix-vector products in each iteration step of the solution process, the system matrix is required for the FE and the BIE subregions. For FDTD subregions, we will either construct a system matrix or we will calculate matrix-vector products directly relying on FDTD. Guidelines will be given to choose the most efficient approach for each FDTD subdomain. To illustrate the efficiency of the new method and to compare the new approach with other (hybrid) methods, three representative configurations are studied.

1. Introduction

[2] Complex electromagnetic configurations, which combine inhomogeneous features with an open three-dimensional simulation space, can be described efficiently with hybrid techniques that combine a rigorous boundary integral equation (BIE) for the open domain with the finite element method (FE) or the finite difference-time domain (FDTD) technique for the inhomogeneous part of the problem. On the one hand, the FE-BIE approaches have become mature by now. In electromagnetic (EM) literature, they have proven their usefulness for a wide range of applications [Volakis et al., 1997; Eibert et al., 1999; Gedney et al., 1992; McGrath and Pyati, 1996; Rogier et al., 1996, 2000a]. On the other hand, FDTD-BIE formalisms proved to be more involved. When modeling multiple subregions with a combination of FDTD with a time domain integral equation [Johnson and Rahmat-Samii, 1997; Bretones et al., 1998; Naishadham and Esselle, 1998], a careful implementation is required in order to avoid late-time instability and time dispersion, linked with the dependency of the integral equation formulation on the time history of the fields. Yet, combining FDTD with other full-wave modeling techniques can give rise to additional benefits, since the FDTD is the most memory-efficient simulation technique. Moreover, the FDTD approach allows an easy incorporation of small details in its Yee-cell mesh, through the use of subcell techniques, and the easy incorporation of discrete element models, e.g., for resistors, inductors or capacitors. Finally, FDTD even allows to include materials with time-dependent constitutive relations.

[3] In Rogier et al. [1998b, 2000d, 1998a, 2000b, 2000c], we presented a new hybrid formalism that combines the FDTD and BIE techniques by mainly working in the frequency domain and by constructing interaction matrices equation imageBIE, for each homogeneous subregion, and equation imageFDTD, for each inhomogeneous and bounded subdomain. More specifically, equation imageBIE is found by applying the Poggio and Miller integral expressions, whereas the excitation of a rooftop basis function on the boundary of an inhomogeneous subdomain in an FDTD simulation yields one column of that subregion's system matrix. Obviously, the construction of equation imageFDTD for each inhomogeneous subregion requires a number of FDTD simulations that is equal to the number of basis functions that expand the tangential electric and magnetic fields on the boundary of that subdomain, which results in high CPU-times. Different techniques to alleviate this problem have been presented in the past [Rogier et al., 1998b, 2000d, 2000b]. One particular method [Rogier et al., 2000c] to drastically reduce the number of FDTD runs consists of applying a biconjugate gradient (BiCG) solution mechanism in a special way. Instead of calculating equation imageFDTD explicitly for each FDTD subregion, the FDTD calculations are performed to find their contribution to the matrix-vector products needed in each iteration of the BiCG formalism that solves the global matrix system. The required number of FDTD runs per FDTD subregion then equals the number of iterations needed for the BiCG algorithm to converge.

[4] In this paper, we present a new hybrid formalism that combines both the finite element method and FDTD with a BIE approach and an efficient solution scheme by using a BiCG solver. Schematically, the new technique proceeds as follows. In Section 2, the simulation domain is decomposed into a number of subregions, and the tangential fields at the interfaces between the different subdomains are chosen as unknowns. After expanding these unknown fields into rooftop basis functions, for each homogeneous subregion, system matrices equation imageBIE are constructed, following Section 3.1, and for some of the inhomogeneous subdomains either the system matrix equation image is calculated using the FE technique with the frontal elimination algorithm, following Section 3.2, or the system matrix equation imageFDTD is computed by performing FDTD simulations, following Section 3.3. Under the assumption that a similar matrix equation imageFDTD characterizes the remaining inhomogeneous subdomains, it is shown in Section 3.4 how the global matrix system is assembled and inverted by a BiCG algorithm. We then explain in Section 3.5 how one calculates the contribution of the remaining FDTD subregions to the matrix-vector products needed in the BiCG method by performing FDTD simulations. Finally some representative examples are given in Section 4 to elucidate the new technique. In Section 4.2, for example, an IC package in the neighborhood of a multilayered dielectric substrate is modeled in order to show the versatility of the FDTD technique in the new hybrid formalism.

2. Geometry Decomposition

[5] Let us assume a three-dimensional simulation space, filled with EM scatterers and sources. As shown in Figure 1, the simulation geometry is decomposed into a number of subregions which will either be described by an FE approach, an FDTD formalism or a BIE formulation. Now suppose that the bounded inhomogeneous type Ia regions are described by the FE method, the bounded inhomogeneous type Ib subdomains by the FDTD approach and that the BIE formulation is used for homogeneous bounded type II and unbounded type III subregions. In all subregions, source regions JIa, JIb, JII and JIII (Figure 1) can be present, or a plane wave (Ein, Hin) can be incident. Since we study the EM problem in the frequency domain, an ejωt dependence, with ω = 2πf, is assumed for the fields and sources.

Figure 1.

Geometry decomposition.

[6] In order to treat each subregion separately, the field equivalence theorem introduces additional virtual sources Jvirt = equation image × H and Mvirt = E × equation image on all the interfaces between the different subregions, in addition to the actual sources. These virtual sources act as the problem's unknowns. They are determined by first expressing the fields generated in each subregion Vi by virtual sources on the boundary Si = ∂Vi and then constructing a global matrix system by imposing continuity between the tangential components of these fields at all the interfaces ⋃jSj.

3. Hybrid FE-FDTD-BIE Formulation

[7] After decomposing the geometry, the hybrid full-wave techniques allow an efficient description of the simulation domain by applying finite difference techniques such as FE and FDTD to the inhomogeneous but bounded parts of the structure and BIE to the homogeneous part of the configuration. Finite difference methods result in a large number of unknowns due to the volume discretisation, but they rely on a sparse system matrix. Moreover, in FDTD this matrix system is solved through explicit iterations, thereby requiring even less memory than FE techniques. The BIE approach, on the other hand, discretizes only the boundary fields but makes use of a dense matrix. The major challenge of any hybrid formalism is how to solve a global matrix system that is partially dense and partially sparse. In order to solve this problem, we follow a two-stage approach in our formulation. During the first stage, all interior unknowns are eliminated for subregions described by finite difference schemes. This can be done efficiently in an FE scheme by applying a frontal elimination technique and in the FDTD approach by calculating a system matrix or by calculating matrix-vector products, as described in Sections 3.3 and 3.5. In the second stage the global dense matrix system, only consisting of unknowns at the boundaries between the different subregions, is solved by means of a direct or an iterative solver.

[8] At one particular frequency, the hybrid formalism then schematically proceeds as shown in Figure 2. In the first stage, interaction matrices equation image and equation image are calculated, resp. for each type II and III subregion Vj following the BIE approach explained in Section 3.1 and for each type Ia subdomain following the frontal FE method of Section 3.2. For certain type Ib subregions system matrices equation image are constructed by performing FDTD simulations, whereas for others, the description is postponed to the second stage of the formalism. The choice between explicitly constructing the FDTD system matrix for a certain region or to treat that FDTD domain later in the process depends on which of the two options is the most efficient. This in turn depends on the nature of the problem that is treated, as will become clear in the examples presented in Section 4. In the second stage the global matrix system is solved with the BiCG algorithm, as explained in Section 3.4. During each iteration step of the iterative process, a matrix-vector product is calculated, relying on the precomputed system matrices equation image and equation image and equation image. At this time, a description for the remaining type Ib subregions is introduced into the global matrix system for calculating the matrix-vector product equation imageFDTDPm directly by introducing the search vector Pm as the excitation in an FDTD simulation, following the methodology described in Section 3.5. By proceeding in this fashion, the number of FDTD simulations per subregion required to find the global solution equals the number of iterations needed by the BiCG algorithm to converge.

Figure 2.

FE-FDTD-BIE technique combined with the BiCG solver.

3.1. Boundary Integral Equation Technique

[9] In each homogeneous bounded type II or unbounded type III subregion, the Poggio and Miller [Poggio et al., 1971] 3-D integral expressions relate the electric and magnetic fields to the virtual sources on the boundaries Sj:

equation image
equation image


equation image

the free-space Green's function. For the unbounded type III subregion, the integration extends over all the boundaries S = ⋃jSj, present within the configuration. In order to apply the method of moments, all boundaries S are discretized into rectangles and the unknown virtual sources, or equivalently tangential fields, are expanded into rooftop basis functions wr:

equation image

By using the same rooftops to perform Galerkin weighting, a matrix expression obtains for each subregion Vj, of the following form:

equation image

Each interaction matrix equation image characterizes the EM behavior of one subregion Vj.

3.2. FE Technique

[10] A Galerkin FE formulation describes all inhomogeneous Type Ia subregions Vi with boundary Si:

equation image
equation image

Subsequently, a uniform mesh of brick elements is defined inside the volume of the type Ia subregions and the interior and boundary field are expanded into three-dimensional rooftop basis functions wr, similar to the 2-D rooftops applied in (4). The discretisations of the FE subregion and the adjacent BIE subregions are chosen in order to form conformal meshes, meaning that boundary discretisations and the field expansions in the BIE formulation are found as the restriction of the FE mesh to the boundary of the subregion.

[11] Applying the complete straightforward FE procedure results in a large sparse FE system matrix, which contains variables over the complete volume and boundaries of the FE subregion. To increase efficiency and to allow easy coupling with a BIE system of the type (5), the frontal technique is used during the construction of the FE system matrix to condense the sparse matrix to a dense matrix only containing the coefficients of the expansions (4) of the fields on the boundary Si of the Type Ia subregion as unknowns. By choosing uniform grids that rely on conformal boundary meshes, the frontal elimination can be done in a very efficient and robust way. For each type Ia subregion, the process leads to following matrix expression, similar to (5):

equation image

where equation image is the FE system matrix condensed by the frontal algorithm, having a dimension N × N, where N corresponds to the number of unknowns required to expand the fields at the boundary Si, similar to (4). Note that, due to the frontal elimination scheme used, all unknowns pertaining to the interior of the FE-subregion were eliminated, so that the dense matrix equation image exhibits a more complex dependence on frequency than one would expect from expressions (6) and (7).

3.3. Constructing a Subregion's Interaction Matrix Using FDTD

[12] For some of the type Ib inhomogeneous subregions Vi, we construct a matrix expression, similar to (5) and (8), following the FDTD formalism described in Rogier et al. [1998b]:

equation image

For one type Ib subregion Vi, the interaction matrix equation imageFDTD should relate the electric and magnetic fields on the boundary Si to the virtual currents Jvirt and Mvirt on Si. Therefore, we define a mesh of rectangular BIE cells on its boundary Si and expand the unknown fields on Si into rooftop basis functions following (4). In order to describe the subregion using FDTD, its volume Vi should be subdivided into a mesh of Yee cells. The FDTD and BIE discretizations are defined in such a way that one BIE cell comprises an integer number of Yee-cell facets, as shown in Figure 1.

[13] The subregion Vi is then excited by each basis function, implemented as an electric or magnetic current source within a separate FDTD simulation. The resulting electric and magnetic fields on Si yield one column of the FDTD interaction matrix. By performing a number of FDTD simulations equal to the number of basis functions, the complete interaction matrix is formed. To take into account the excitations JI within a type Ia domain, an additional FDTD simulation is required [Rogier et al., 1998a]. The geometry of Vi remains the same as for constructing equation imageFDTD, but now JI acts as source and no virtual currents are present.

[14] Since a large number of simulations are needed to obtain a system matrix, special measures should be taken to optimize the FDTD calculations by reducing transients within the FDTD simulation domain. By invoking field equivalence, fields are zero outside the FDTD subdomain, because of the presence of the virtual currents Jvirt and Mvirt on Si. To reduce long transients, the FDTD grid is therefore terminated in the region outside Vi by an absorbing boundary condition or by absorbing Berenger material. Obviously, the introduction of absorption alters the matrix equation imageFDTD, but it does not affect the accuracy of the final results, because fields are zero outside Vi. However, as the matrix equation imageFDTD depends on the absorbing material, it will influence the condition number of the global matrix system. This can affect the convergence of the iterative technique described in Section 3.4. The examples in Section 4 show that the introduction of absorber does not substantially deteriorate the convergence of the iterative scheme.

[15] As demonstrated in Rogier et al. [1998b], the fields at the boundary due to the excitation of one basis function can be derived at multiple frequencies in one FDTD run, by choosing a suitable time dependence for the current sources (e.g., sum of sinusoids or Gaussian pulse). After exciting all the rooftop basis functions, an FDTD interaction matrix is stored on disk for the different frequencies under consideration. Since geometrically identical subregions have identical interaction matrices, such a matrix can be reused each time a subregion with that specific geometry occurs.

3.4. Solving the Global Matrix System With a BICG Technique

[16] Assume for now that, by performing FDTD simulations in one way or the other, a matrix expression similar to (9) is available for all inhomogeneous type Ib subregions Vi. Then, as in Rogier et al. [1998b], a global problem description is constructed by imposing continuity of the tangential electric and magnetic fields at Si/j, in a weak sense through Galerkin weighting:

equation image
equation image

This leads to a matrix equation of the form

equation image

with X containing the unknown coefficients Er and Hr, and B a source vector due to incoming fields. Note that only the tangential fields at the interfaces between the different subdomains remain as unknowns and that by using both electric and magnetic fields components in the formulation, the problem of internal resonances is avoided.

[17] Now, the matrix system is iteratively solved by the BiCG technique. This method mainly consists in constructing a sequence of approximate solutions Xi, by means of search vectors Pi that span a Krylov subspace and satisfy the biorthogonality condition 〈equation imagePj, equation imagei〉 = 0, ij. In most cases the approximate solution Xi converges to the exact solution X within a number of iterations that is much smaller than the total number of basis functions within the configuration. For complex symmetric system matrices, the algorithm reduces to

equation image
equation image
equation image
equation image
equation image
equation image

A new search direction Pm and a new approximate solution Xm are generated during each iteration m, from the calculation of (Σiequation image − Σkequation image − Σjequation image) · Pm−1. This clearly shows that the knowledge of equation image is not strictly required for a type Ib subregion Vi, provided that one is able to compute equation image · Pm−1 directly, by performing an FDTD simulation.

3.5. Constructing Matrix-Vector Products With FDTD

[18] For some type Ib regions, it is more efficient to skip the construction of equation image by applying a modified FDTD formalism. Focus on one such type Ib subregion Vi. From Section 3.3, we know that a specific column of equation image can be found by introducing the correct rooftop basis function wr with unit amplitude as a well-chosen configuration of current excitations in the FDTD scheme and by observing the tangential fields generated at the surface Si. Hence, by applying superposition, the matrix-vector product at iteration m of the BiCG routine is found in a similar fashion by exciting all rooftop basis functions wr on Si with an amplitude corresponding to element Pm−1r of the search vector at iteration m − 1. This defines a configuration of tangential electric and magnetic current sources along Si within the FDTD simulation and yields the matrix-vector product through observation of tangential electric and magnetic fields on Si. The FDTD simulations to calculate the matrix-vector products can be optimized in a similar way as in Section 3.3, i.e., by applying field equivalence, treating different subregions in parallel, and calculating matrix-vector products at F different frequencies f, f(1 + equation image), …, f(1 + equation image), with F and M being integers. For the latter, one excites each rooftop basis function with a time dependence that consists of a sum of cosines:

equation image

with H(n, t, T) a Hanning window function to alter the shape of the rise of the excitation at t = 0, in order to reduce transients [Presscot and Shuley, 1994].

4. Examples

[19] In order to validate the new FE-FDTD-BIE technique, we will present three examples. The first two illustrate the combination of the FE-FDTD-BIE technique with the BiCG solver, whereas the last example applies a direct solver and calculates the system matrix of the FDTD subregion explicitly.

4.1. Two Multilayered Dielectric Substrates

[20] Consider two multilayered nonmagnetic dielectric substrates, each consisting of three layers with a relative permittivity of either ϵr = 12 or ϵr = 2.33, as shown in Figure 3. All layers in both substrates have a thickness of 2 cm. The first substrate has a size of 6 cm × 6 cm × 6 cm, whereas the second substrate has dimensions 9 cm × 9 cm × 6 cm. As shown in Figure 4, the first substrate is placed into an FE-subregion whereas the second one is embedded within an FDTD-subregion. In total 2296 unknowns represent the field in the FE subregion. After condensing the FE matrix, 1096 unknowns describe the tangential electric and magnetic fields on the boundary. As for the FDTD subregions the Yee-cell mesh represents approximately 21600 unknown field components, whereas 672 unknowns describe the tangential fields on the boundary BIE mesh. As a point of reference, we will also consider the same structure modeled with an FE-BIE, i.e., both substrates enclosed in FE-subregions (Figure 5). Let us now illuminate the structure with an incoming plane wave Ein = e+jkzuy at frequencies f = 0.8 GHz, 1 GHz, 1.2 GHz. The normalized bistatic radar cross section (RCS) in the xz-plane of the configuration (Figure 3, θ = 0 along the x-axis) is shown in Figure 6 for f = 0.8 GHz, in Figure 7 for 1 GHz, and in Figure 8 for f = 1.2 GHz. For the FE-FDTD-BIE approach, the accuracy of the BiCG solver is set to 10−3, whereas the accuracy for the BiCG solver used in the FE-BIE reference technique is set to 10−7. Moreover, we also show results obtained with the FE-FDTD-BIE formalism for a tolerance of 10−5. All figures show a good agreement between the new FE-FDTD-BIE approach and the FE-BIE technique, even when the accuracy of the BiCG solver is set to 10−3. The statistics of the technique are shown in Table 1. The simulations were performed on a 2.4 GHz Pentium IV processor machine. If we would construct a system matrix for the FDTD subregion, the number of FDTD simulations would equal the number of unknowns along the boundary of the FDTD subregion, i.e., 672 (column 4). When we calculate matrix-vector products directly in the BiCG solver, the number of FDTD simulations required per subregion equals the number of iterations needed by the solver to converge to a certain accuracy (columns 2 and 3). It is found that the new technique greatly reduces the number of FDTD simulations required per subregion. Indeed, to reach an accuracy of 10−3, less than equation image of the number of FDTD simulations is required when equation imageFDTD is not calculated explicitly. Even for an accuracy of 10−5, the number of FDTD simulations is reduced by more than equation image.

Figure 3.

Two multilayered substrates.

Figure 4.

Discretisation scheme for the FE-FDTD-BIE technique.

Figure 5.

Discretisation scheme for the FE-BIE technique.

Figure 6.

Normalized bistatic RCS in the xz-plane for f = 0.8 GHz.

Figure 7.

Normalized bistatic RCS in the xz-plane for f = 1 GHz.

Figure 8.

Normalized bistatic RCS in the xz-plane for f = 1.2 GHz.

Table 1. FDTD Runs Required for the FE-FDTD-BIE Technique
FrequencyIterations BiCG, 10−3Iterations BiCG, 10−5Unknowns/SubdomainCPU-TimeRAM FDTDRAM equation image
800 MHz451376723h 16m37.8MB46MB
1000 MHz841716726h 14m37.8MB46MB
1200 MHz1042256726h 39m37.8MB46MB

4.2. An IC Package Besides a Multilayered Dielectric Substrate

[21] In order to demonstrate the versatility of the FDTD method to model circuits with fine details, consider a 12 pin IC package placed on a 2 cm × 2 cm microstrip substrate with thickness 5 mm and ϵr = 2.33, as shown in Figure 9. In the neighborhood of the package, a multilayered dielectric substrate is placed, consisting of three layers, each with dimensions 3 cm × 3 cm × 1 cm. The top and bottom substrate have a dielectric constant ϵr = 12, whereas the middle substrate has ϵr = 2.33. The pins, the die, which has dimensions 5 mm × 5 mm, and the ground plane of the substrate are made of copper, with σ = 5.8e7 S/m. The mold, of size 11 mm × 11 mm × 4 mm, has a dielectric constant of ϵr = 3.0. The structure is illuminated with an incoming plane wave Ein = e+jkzuy at frequencies f = 1.6 GHz, 1.8 GHz and 2 GHz. The IC package is embedded in an FDTD subregion with 6 × 6 × 4 cells on the boundary for the application of the BIE formalism. The FDTD Yee-cell grid is five times denser in order to accommodate the fine details of the IC package. The multilayered substrate is placed in an FE subregion with 6 × 6 × 8 cells.

Figure 9.

IC package besides a multilayered substrate.

[22] We have plotted the normalized bistatic radar cross-sections in the xz-plane and yz-planes in Figure 10 for 1.6 GHz, in Figure 11 for 1.8 GHz and in Figure 12 for 2.0 GHz. The statistics of the technique are shown in Table 2. The simulations were performed on a 2.4 GHz Pentium IV processor machine. Although we restricted ourselves to presenting far-field data for reasons of conciseness, it is possible to perform a detailed analysis of all currents induced in the IC package by observing them in a final FDTD simulation of the FDTD subregion containing the package.

Figure 10.

Normalized bistatic RCS in the xz-plane and yz-plane for f = 1.6 GHz.

Figure 11.

Normalized bistatic RCS in the xz-plane and yz-plane for f = 1.8 GHz.

Figure 12.

Normalized bistatic RCS in the xz-plane and yz-plane for f = 2.0 GHz.

Table 2. FDTD Runs Required for the FE-FDTD-BIE Technique
FrequencyIterations BiCG, 10−3Iterations BiCG, 10−5Unknowns/SubdomainCPU-TimeRAM FDTDRAM equation image
1600 MHz501146725h 26m37.8MB46MB
1800 MHz551206725h 12m37.8MB46MB
2000 MHz611296725h 27m37.8MB46MB

4.3. Dielectric Substrate With Frequency Selective Surface

[23] The previous configurations clearly demonstrated the efficiency of combining the BICG and FE-FDTD-BIE formalisms. In some cases, however, it is more efficient to calculate the FDTD system matrix explicitly. This happens when a large amount of subregions are present within the simulation domain. In this case, the total number of unknowns, which determines the number of BiCG iterations, is much larger than the number of unknowns needed to describe a single FDTD subregion. Moreover, when some of the subregions present in the simulation domain are identical, their system matrix description can be reused. Finally, using a direct matrix solver allows to invert the matrix system for a number of different excitation schemes in one single run. These aspects are now demonstrated by considering the configuration drawn in Figure 13.

Figure 13.

FSS above a substrate with radiating dipole.

[24] The structure consists of a Hertzian dipole, oscillating at 6 GHz, 8 GHz or 10 GHz. The dipole is placed above a substrate consisting of four alternating layers with permittivity ϵr = 2.33 and ϵr = 12. Above this configuration, a frequency selective surface (FSS) is placed, consisting of a 4 by 4 array of dielectric patches with permittivity εr = 12. Some of the dielectric substrates in the FSS are tilted over 45°, as shown in Figure 13.

[25] The discretisation scheme for the FE-FDTD-BIE technique is schematically shown in Figure 14, for a cross section in the xz-plane. The substrate together with the dipole are placed within a FE subregion (resolution 9 × 9 × 12 and cell dimensions 0.1 cm × 0.1 cm × 0.05 cm), whereas each plate of the FSS is embedded within a separate FDTD subregion. For one such region an FDTD mesh is defined with cell dimensions 0.05 cm × 0.05 cm × 0.05 cm together with a surface mesh on the virtual boundary (resolution 6 × 6 × 2 and cell dimensions 0.25 cm × 0.25 cm). In total, 10056 unknowns represent the tangential electric and magnetic fields along the boundaries of all subregions. As a point of comparison, the configurations is also modeled with a FE-BIE approach, following the discretisation scheme shown in Figure 15, for a cross section in the xz-plane.

Figure 14.

Discretisation scheme for the FE-FDTD-BIE technique.

Figure 15.

Discretisation scheme for the FE-BIE technique.

[26] In a first stage of the formalism, we consider one particular FDTD subregion of the configuration, i.e., one dielectric substrate embedded in an FDTD subdomain. Since all FDTD subregions are identical, this will result in sufficient information to model all cells of the FSS. FDTD interaction matrices are generated for this single cell within a frequency range from 6 GHz to 15 GHz on a SUN HPC4000 workstation with 3GB RAM and eight UltraSparc II 250 MHz processors, using the technique described in section 3.3 and performing 74 FDTD simulations in parallel [Rogier et al., 2000b] over eight processors. The statistics for this process are given in Table 3. The resulting matrices at the different frequencies are stored on disk for later use.

Table 3. Statistics of the Parallel FDTD-BIE Simulations
Sun HPC4000CPU-TimeRAM
Construction equation imageFDTD in parallel on 8 processor
f = 6 GHz–15 GHz, 12 frequencies5h 50min123MB × 8
f = 10 GHz, monochromatic1h 35min16MB × 8

[27] In the second stage of the approach we concentrate again on the configuration of Figure 13. The precalculated interaction matrices are retrieved from disk for f = 6, 8 and 10 GHz. They are used for all cells of the FSS and, finally, they are coupled with the remaining FE and BIE descriptions. In Figures 16, 17 and 18 the radiation patterns in the xz- and yz-plane are shown at 6 GHz, 8 GHz and 10 GHz respectively. A good agreement is seen between the hybrid FE-FDTD-BIE technique and FE-BIE approach, with the patches placed in FE subregions as in Figure 15.

Figure 16.

Radiation pattern in the xz-plane and yz-plane for f = 6 GHz.

Figure 17.

Radiation pattern in the xz-plane and yz-plane for f = 8 GHz.

Figure 18.

Radiation pattern in the xz-plane and yz-plane for f = 10 GHz.

[28] To calculate the radiation patterns for a single dipole excitation at different frequencies, the BiCG solver was used. Let us now turn our attention toward the calculation of the normalized monostatic RCS of the configuration in Figure 13 (in absence of the dipole). A direct matrix solver, such as an LU-decomposition, is now more efficient, since it is able to invert the matrix system for a number of different incoming plane waves in one run. In Figure 19, we present the normalized monostatic RCS in the xz-plane (θ = 0 corresponds to the x-axis) for a y-polarized incoming plane wave at f = 10 GHz.

Figure 19.

Normalized monostatic RCS in the xz-plane for f = 10 GHz.

5. Conclusions

[29] We have presented a new hybrid formalism that combines the FE method, the FDTD approach and the BIE technique. By combining the hybrid approach with a BiCG algorithm, a considerable reduction in the number of FDTD runs is achieved. This is done by calculating matrix-vector products with the FDTD-formalism instead of constructing a system matrix for each FDTD subregion. Although clearly requiring more simulation time, the explicit construction of a system matrix can be useful as well. This is the case when inverting the EM problem for a number of different right-hand sides simultaneously by applying a direct solver or when reusing the system matrix if there are multiple identical subregions present within the configuration. These features are lost when applying the BiCG scheme. Note that the proposed formalism can be applied with other Krylov-based iterative solvers as well. The BiCG method for complex symmetric matrices was preferred since it requires only one matrix-vector product per iteration and since it generally converges rapidly.

[30] The main computational cost of the formalism in terms of memory and CPU time is due to the application of the BIE approach, which leads to a dense system matrix. When the global simulation domain exhibits a sufficiently high degree of regularity, schemes like CG-FFT can be used in combination with the proposed hybrid method to accelerate the matrix-vector product. However, acceleration schemes based on the CG-FFT approach cannot be applied in a straightforward way to general global problems, since not all the different interactions exhibit sufficient translation invariance, as the discretisation of one subdomain takes place in various planes and as the orientation between different subregions can be arbitrary. For such irregular configurations, some gain might also be expected by extending the BIE approach with a fast multipole solver or with the adaptive integral method (AIM) in order to treat the far interactions that occur between different subregions.