# 2D FDTLM hybridization with modal method.

1. INTRODUCTION

Hybridizing techniques, i.e., using each technique in a part of the problem where it is the most appropriate, is an adequate way to improve the modeling of microwaves circuits. Hybrid methods have been successfully considered to study complex configurations [1-3] like inhomogeneous problems [4-7].

In this work, a hybrid technique that combines two different numerical methods, the two-dimensional Frequency Domain Transmission Line Matrix Method (2D FDTLM) [8-11] and the Wave Concept Iterative Procedure method (WCIP) [12-14] is presented. The FDTLM is based on the theory of transmission lines and in the principle of propagation of light described by Huygens [8, 9, 15, 16]. The medium is discretized by transmission lines and circuits' elements in a manner that the arrangement of transmission lines and circuits' elements give equations that are isomorphic with equations of the problem, i.e., in this case, Maxwell's equations. The WCIP is an integral method based on waves' concept that solves Maxwell's equations in enclosed waveguide [17]. This technique takes advantage of the WCIP to model homogeneous media with less computational requirement (use of surface mesh and not volumic mesh) and of the ability of the FDTLM to characterize inhomogeneous media. One of the advantages of the WCIP is its ability to deal with spectral or local (spatial) sources. Furthermore, open boundary conditions are addressed by this method without adding PML conditions or radiation conditions. The hybrid algorithm is therefore based on the partition of the structure under study into two domains: the WCIP is used for the homogeneous media while the FDTLM is used for the inhomogeneous media. To simplify the hybridization, the 2D FDTLM node is reformulated using waves as used in the WCIP. In 2D, two independent orientations corresponding to TE and TM modes are distinguished. Thus, two scattering nodes are defined by either a shunt or a series matrix. Their formulation is analytically derived in this paper from Maxwell's equations using centered differencing and averaging [10,11] as in [18] in 3D case. It is noteworthy that 2D FDTLM TM and TE matrices cannot be immediately obtained from the 3D matrix, since in each 3D development [18, 19] replacing the propagation constant along z by zero does not lead to the scattering matrix obtained in this paper.

The goal of our study is to validate 2D hybridization principles to know the properties of the method so that 3D results can be improved in accuracy compared to [19]. We want to study more complex cases as presented in Fig. 1, where the FDTLM will be applied in inhomogeneous domain containing [[epsilon].sub.r,3] and [[epsilon].sub.r,4] and the WCIP elsewhere. Hybrid method formulation is explained and applied in several test cases: validation of the operator spectrum, comparison with an analytical case and with diffraction by a microstrip line with homogeneous and inhomogeneous substrate. As we do not want to compete with commercial software (and moreover 2D cases cannot be dealt with them), computation times are not given.

2. HYBRID METHOD THEORY

In this section, the 2D scattering FDTLM node formulated with waves is detailed in the TE ([E.sub.z] = 0) and TM ([H.sub.z] = 0) cases (Transverse to z-direction). It leads to the presentation of the hybridization principle between the WCIP method and the FDTLM in Subsection 2.4.

2.1. Maxwell's Equations

Time-harmonic Maxwell's equations for invariant problem along y-axis are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where [omega] is the pulsation. [epsilon] and [mu] are the permittivity and the permeability of the media. [[sigma].sub.e] and [[sigma].sub.m] are the electric and magnetic conductivities. [e.sub.i] and [h.sub.i],i [member of] [x,y,z], are the electromagnetic field components.

Let us consider an elementary cell of dimensions u, v, w respectively along the x, y, z directions. Fields are normalized according to the following transformations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Normalized Maxwell's equations are obtained

[G.sub.ex][E.sub.x] = [partial derivative][H.sub.y]/[partial derivative]Z, (2a)

[G.sub.ey][E.sub.y] = [[partial derivative][H.sub.x]/[partial derivative]Z] - [[partial derivative][H.sub.z]/[partial derivative]X], (2b)

[G.sub.ez][E.sub.z] = [[partial derivative][H.sub.y]/[partial derivative]X], (2c)

[G.sub.mx] [H.sub.x] = [[partial derivative][E.sub.y]/[partial derivative]Z], (2d)

[G.sub.my] [H.sub.y] = [[partial derivative][E.sub.z]/[partial derivative]X] - [[partial derivative][E.sub.x]/[partial derivative]Z]. (2e)

[G.sub.mz] [H.sub.z] = [[partial derivative][E.sub.y]/[partial derivative]X], (2f)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

System (2) indicates that Equations (2a), (2c) and (2e) are independent of Equations (2b), (2d) and (2f). Two independent propagation orientations are therefore studied in 2D respectively the TM modes ([H.sub.y], [E.sub.x] and [E.sub.z]) and the TE modes ([E.sub.y], [H.sub.x] and [H.sub.z]). The elementary FDTLM node is represented in Fig. 2. TE and TM indices on each node port, numbered from 1 to 4, should be considered separately since they do not coexist in 2D propagation.

2.2. TM Modes

In this case, we deal with the field components [E.sub.x], [E.sub.z] and [H.sub.y]. The arrangement of TM transmission lines shown in Fig. 3 can be assimilated to a series node since all waves depend on the magnetic field component [H.sub.y], but differ from the electric field components [E.sub.x] and [E.sub.z]. Maxwell's equations concerned in the TM case are Equations (2a), (2c) and (2e). Centered differencing at point (i' k) is achieved on these equations, leading to finite difference equations

[G.sub.ex][E.sub.x](i, k) = [H.sub.y](i,k - [1/2]) - [H.sub.y] (i,k + [1/2]), (4)

[G.sub.ez][E.sub.x](i, k) = [H.sub.y](i + [1/2], k) - [H.sub.y] (i - [1/2], k), (5)

[G.sub.my][H.sub.y](i, k) = [E.sub.z](i + [1/2], k) - [E.sub.z] (i - [1/2], k) + [E.sub.x](i,k - [1/2]) - [E.sub.x](i,k + [1/2]). (6)

2.2.1. Wave Definition on a Series FDTLM Node

Incident and reflected waves, noted [A.sub.l] and [B.sub.l], are defined on the FDTLM node port numbered 1 at the sampling points defined according to notation of Fig. 2. Writing waves on port 1 means considering fields at mesh point (i,k + [1/2]). Given waves definitions provided in [13], we obtain

[A.sub.1] = [1/2[square root of [Z.sub.0]] [E (i,k + [1/2]) + [Z.sub.0]J (i,k + [1/2])], [B.sub.1] = [1/2[square root of [Z.sub.0]] [E (i,k + [1/2]) + [Z.sub.0]J (i,k + [1/2])]. (7)

Projecting on x-axis, we obtain

[A.sub.1] = [1/2[square root of [Z.sub.0]] [[E.sub.x] (i,k + [1/2]) + [Z.sub.0][J.sub.x] (i,k + [1/2])], [B.sub.1] = [1/2[square root of [Z.sub.0]] [[E.sub.x] (i,k + [1/2]) + [Z.sub.0][J.sub.x] (i,k + [1/2])], (8)

where [Z.sub.0] is the free space impedance and

J = H [conjunction] (-z),

with [conjunction] the vector product and thus

[J.sub.x](i,k + [1/2]) = -[H.sub.y](i,k + [1/2]),s

which entails

[A.sub.1] = [1/2[square root of [Z.sub.0]] [[E.sub.x] (i,k + [1/2]) + [Z.sub.0][H.sub.y] (i,k + [1/2])], [B.sub.1] = [1/2[square root of [Z.sub.0]] [[E.sub.x] (i,k + [1/2]) + [Z.sub.0][H.sub.y] (i,k + [1/2])]. (9)

Relations (9) transform the normalized field components to incident and reflected waves at port 1. Waves at the other ports are found in a similar way, they are reported in Appendix A.1.

2.2.2. Field Definition on Series FDTLM Node Electromagnetic fields are expressed by combining Equation (9)

[E.sub.x](i,k + [1/2]) = [square root of [Z.sub.0]] ([A.sub.1] + [B.sub.1]), [H.sub.y](i,k + [1/2]) = [1/[square root of [Z.sub.0]]] ([B.sub.1] + [A.sub.1]). (10)

Fields at the other ports are found similarly, they are provided in Appendix A.2. By substituting (10) into (4)-(6), the following equations are deduced:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

2.2.3. Averaging the Field Component Combinations

The next step is to express the field component combinations at the central point (i, k) as a combination of the incident and reflected waves of surrounding ports. Since z-axis is orthogonal to both [E.sub.x] and [H.sub.y], the combination ([E.sub.x] - [Z.sub.0] [H.sub.y]) is centered averaged at the central point (i, k) with respect to this direction:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

Substituting field definitions of Appendix A.2 into (14) leads to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

Similarly, the centered average for other combinations at the central point (i, k), namely ([E.sub.x] + [Z.sub.0] [H.sub.y]), ([E.sub.z] - [Z.sub.0] [H.sub.y]) and ([E.sub.z] + [Z.sub.0] [H.sub.y]) are reported in Appendix A.3. From these relations, reflected waves B at each port are deduced from incident waves A and electromagnetic fields at the center of the node

[B.sub.1] + [B.sub.2] = [2[E.sub.x] (i, k)/[square root of ([Z.sub.0])]] - ([A.sub.1] + [A.sub.2]) (17)

This definition is introduced in (11) to obtain Ex at the central point (i, k)

[E.sub.x] (i, k) = [2[square root of ([Z.sub.0])]/([Z.sub.0] [G.sub.ex] + 2)] ([A.sub.1] + [A.sub.2]). (18)

By the same way, we obtain field components [E.sub.z] and [H.sub.y] at the central point (i, k)

[E.sub.z] (i, k) = [2[square root of ([Z.sub.0])]/([Z.sub.0] [G.sub.ez] + 2)] ([A.sub.3] + [A.sub.4]), (19)

[H.sub.y] (i, k) = [2[square root of ([Z.sub.0])]/(4[Z.sub.0] + [G.sub.my])] ([A.sub.4] - [A.sub.3] + [A.sub.2] - [A.sub.1]). (20)

Finally, the scattering matrix of the series node is a 4 x 4 matrix

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (21)

where index e stands for the element, i.e., the FDTLM node and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (22)

with [alpha] = x, z and [beta] = y.

2.3. TE Modes

In this case, we deal with the field components [H.sub.x], [H.sub.z] and [E.sub.y]. The arrangement of TE transmission lines shown in Fig. 4 can be assimilated to a shunt node since all waves depend on the electric field component [E.sub.y], but differ from the magnetic field components [H.sub.x] and [H.sub.z]. Maxwell's equations concerned in the TE case are Equations (2b), (2d) and (2f). Centered differencing at point (i, k) is achieved on these equations, leading to finite difference equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (23)

[G.sub.mx] [H.sub.x] (i, k) = [E.sub.y] (i, k + [1/2]) - [E.sub.y] (i, k - [1/2]), (24)

[G.sub.mz] [H.sub.z] (i, k) = [E.sub.y] (i - [1/2], k) - [E.sub.y] (i + [1/2], k). (25)

Wave and field definitions are detailed in Appendix B.1 and B.2. By substituting these equations into (23)-(25), the following equations are deduced:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

[G.sub.mx] [H.sub.x] (i, k) = [square root of ([Z.sub.0])] ([A.sub.1] + [B.sub.1] - [A.sub.2] - [B.sub.2]), (27)

[G.sub.mz] [H.sub.z] (i, k) = [square root of ([Z.sub.0])] ([A.sub.3] + [B.sub.3] - [A.sub.4] - [B.sub.4]). (28)

Centered averaging electromagnetic field combinations at the center of the node is detailed in Appendix B.3. Expressions of field components at the central point (i, k) are obtained similarly as TM case

[E.sub.y] (i, k) = [2[square root of ([Z.sub.0])]/([Z.sub.0][G.sub.ey] + 4)] ([A.sub.1] + [A.sub.2] + [A.sub.3] + [A.sub.4]), (29)

[H.sub.x] (i, k) = [2[square root of [Z.sub.0]]/(2[Z.sub.0] + [G.sub.mx])] ([A.sub.1] - [A.sub.2]), (30)

[H.sub.z] (i, k) = [2[square root of ([Z.sub.0])]/(2[Z.sub.0] + [G.sub.mx])] ([A.sub.3] - [A.sub.4]). (31)

Consequently, the scattering matrix of the shunt node is a 4 x 4 matrix

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

where

[a.sub.[alpha],[beta]] = [d.sub.[alpha]] - [b.sup.[beta]], [b.sub.[beta]] = [2[Z.sub.0]/(2[Z.sub.0] + [G.sub.m[beta]])] (33)

[c.sub.[alpha],[beta]] = [b.sub.[beta]] + [d.sub.[alpha]] - 1, [d.sub.[alpha]] = [2/(4 + [Z.sub.0][G.sub.e[alpha]])], (33)

with [beta] = z' x and [alpha] = y.

2.4. WCIP/FDTLM Hybrid System

The WCIP is adapted to homogeneous planar circuits and implies only line mesh in this case. If homogeneous substrates are considered, results can be obtained with the WCIP alone. The WCIP method implemented in this particular case may therefore be used as a reference to validate the WCIP/FDTLM hybrid method results. The general case of study is represented in Fig. 5, where an interface ([summation]) separates the two partitions of the domain.

When hybrid method is considered, the media under the surface ([summation]) is discretized by the FDTLM (2D mesh of the whole lower media) while the upper media is always characterized by the WCIP operator. Transfer condition is provided through the S operator of the WCIP. [A.sup.down] corresponds to [A.sub.1] of the upper nodes of the 2D FDTLM mesh and [B.sub.1] corresponds to [B.sub.down] of the system. [B.sub.down] comes from the FDTLM application according to (21) in TM case and (32) in TE case. With a spectral source wave Bo in the upper media, the equations to consider are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

where the exponents W and F stand for WCIP and FDTLM, the operator S contains the transmission/reflection conditions on ([summation]), and [([S.sup.W.sub.i]).sub.i=uv,down] are the scattering operators for homogeneous media in domain i. The discrete operators [([S.sup.W.subi]).sub.i=up,down] are based on the Fast Modal Transform (FMT) [12-14] and on the modal scattering operator ([[??].sub.i]) according to

[S.sup.W.sub.i] = [FMT.sup.-1] [[??].sub.i] FMT. (36)

(FMT) expresses the waves from the spatial domain to modal domain. If B is the incident wave, its associated modal decomposition [({[[??].sub.n]}.sub.n[member of][10;N-1]] and spatial decomposition [({[B.sub.k]}.sub.k[member of][0;N-1])] are expressed through

B = [summation over (n)] [[??].sub.n] [f.sub.n] = [summation over (k)] [B.sub.k] [H.sub.k] (37)

where k is the segment position in the spatial basis, [({[H.sub.k]}.sub.k[member of][0;N-1])], and n the mode order in the modal basis, [({[f.sub.n]}.sub.[member of][0;N-1])].

Its modal decomposition is obtained thanks to FMT writing

[[??].sub.n] = <[f.sub.n], B> = <[f.sub.n], [summation over (k)] [B.sub.k] [H.sub.k]>. (38)

These bases verify

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (39)

[[GAMMA].sub.i] is the modal coefficient of diffraction. Its expression is given by

[[GAMMA].sub.i] = [1 - [Z.sub.0] [Y.sub.i]/1 + [Z.sub.0] [Y.sub.i]], (40)

with [Y.sub.i] detailed in (41) in TE case, in (42) in TM case

[Y.sup.TE.sub.i] = [square root of ([(i[pi]/a).sup.2] - [k.sup.2.sub.0])]/j[omega][[mu].sub.0], (41)

[Y.sup.TM.sub.i] = [j[omega][[epsilon].sub.0]/[square root of [(i[pi]/a).sup.2] - [k.sup.2.sub.0]]], (42)

where i is the index of the mode taken into account in the matrix. S links incident and reflected waves with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (43)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (44)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (45)

Combining (34) and (35), the linear system to solve in the hybrid method is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (46)

where [I.sub.d] is the identity matrix. Note that when the TE case is considered, [B.sub.0] is along the y-axis and in the TM case, [B.sub.0] is along the x-axis. In 2D, TE and TM modes are uncoupled which entails that the generated waves are directed along the same axis as the source wave [B.sub.0].

3. NUMERICAL RESULTS

3.1. Mode Diffraction on a Perfect Conductor Sheet

To validate the method, the propagation of one (TE or TM) mode in vacuum ([[epsilon].sub.r] = 1) between two metallic slabs separated by a distance a ended by a short circuit at distance H is first considered (Fig. 6). The solution is defined by (47) [20].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (47)

where [E.sub.0] is the incident wave amplitude, y and x the unit vectors of the canonical basis, [f.sup.[alpha].sub.1] the function of order 1 of the modal basis of the mode [alpha] ([alpha] = TE or TM), and [p.sub.1] the propagation constant along the z-axis. Here a = 1.27cm, H = 1.27cm and the working frequency is 16 GHz, which implies that the modes in excitation which are either [TE.sub.1], either [TM.sub.1], are propagating. Eigenvalues of [S.sup.W.sub.down] and [S.sup.F.sub.down] operators are compared and represented in Fig. 7 in the TE case, where N denotes the number of segments on the interface ([summation]) and consequently the number of nodes in x direction (see Fig. 2). We take the same number of nodes in z direction. The distance between both operators (norm 2) is presented in Table 1 for the TE and TM cases. Mesh size represents edge length ratio between current and initial mesh. For instance, 1/2 means that the step size is twice smaller than the initial step size in both axes. Initial mesh is characterized by a step size of 794 [micro]m both directions and by N = 16 for the FDTLM.

Since the analytical solution is known, we can calculate relative discretization error, i.e., error between analytical and computed solutions with the hybrid method. This error is evaluated and reported in Table 2 where the exponents H and [E.sub.x] stand for Hybrid and Exact. Convergence order is estimated taking into account the errors from two consecutive mesh sizes, which explains the empty boxes of the tables, and is close to 2. Its definition is given by

Order = [log (Rel_error ([h.sub.1])) - log (Rel_arror ([h.sub.2]))/log ([h.sub.1]) - log ([h.sub.2])] (48)

where Rel_error([h.sub.1]) and Rel_error([h.sub.2]) correspond to the relative discretization errors in [L.sub.2]-norm for mesh step [h.sub.1] and refined mesh step [h.sub.2]. The same study was performed between analytical solution and the WCIP alone and the error was of the order of the machine accuracy, which means that the WCIP is analytical in this case.

The same work was achieved with TM modes. Relative discretization error is represented in Fig. 8. It is calculated while refining the mesh along both axis (x and z-axis). Error is slightly smaller for TM modes than TE ones.

3.2. Mode Diffraction on a Microstrip Line

A second test problem is considered (see Fig. 9). It consists of a centered metallic (perfect conductor) microstrip line of width w = 6. 35 mm, taken without thickness, which is inserted on the surface ([summation]). Domains up and down are vacuum ([[epsilon].sub.r] = 1). Like previously, we have metallic boundary conditions on x = 0, x = a and z = - H, with a = 1.27cm and H = 1.27cm.

A comparison between the electric field and the electric current obtained with the WCIP alone (denoted by the index W) and with the hybrid method is presented as an analytical solution is not known in this particular case. The WCIP alone is meshed enough to consider it as a reference solution (the WCIP is meshed with N = [2.sup.15] where N is the number of segments on ([summation])). Results obtained for [TE.sub.1] and [TM.sub.1] excitation are respectively detailed in Tables 3 and 4. In TE case, convergence orders of 1 and 0.5 are found for the electric field and the electric current, while in TM case convergence orders of 0.5 and 1 are respectively found for the electric field and the electric current.

3.3. Mode Diffraction on a Microstrip Line with Inhomogeneous Substrate

A last test problem is considered (see Fig. 10). It consists of a centered metallic (perfect conductor) microstrip line of width w = 6. 35 mm, taken without thickness, which is inserted on the surface ([summation]) of an inhomogeneous substrate ([[epsilon].sub.r,1] = 1 and [[epsilon].sub.r,2] = 5). Domain up is vacuum ([[epsilon].sub.r] = 1). Like previously, we have metallic boundary conditions on x = 0, x = a and z = --H, with a = 1.27 cm and H = 1.27 cm.

A comparison between the electric field and the electric current obtained with an hybrid WCIP/FEM method [21] taken as our reference is presented, meshing this hybrid method with N = [2.sup.10], since an analytical solution is not known. Results obtained for TE1 excitation are detailed in Table 5. In TE case, convergence orders of 1 and 0.5 are found for the electric field and the electric current, while in TM case convergence orders of 0.5 and 1 are respectively found for the electric field and the electric current.

4. CONCLUSION

The 2D FDTLM node reformulated for hybridization with the WCIP is detailed in this paper in the TE and TM cases. The hybrid FDTLM/WCIP formulation is expressed and tested in an analytical case and in two microstrip line cases (homogeneous and inhomogeneous). In this analytical case (diffraction of one guided mode on a perfect sheet), the spectra of both scattering operators are successfully compared. In the microstrip line case, the method order is 1 (resp. 0.5) on the electric field and 0.5 (resp. 1) on the electric current for TE modes (resp. for TM modes) for homogeneous and inhomogeneous examples. Similar works are under process with other volumic methods such as Finite Element Method (FEM) or a Hybridizable Discontinuous Galerkin (HDG) method [22].

APPENDIX A. DEVELOPMENTS IN TM CASE

A.1. Wave Definition

Incident and reflected waves are detailed at each port of the series FDTLM node:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A.2. Field Definition

Electromagnetic fields are deduced from wave definitions at each port of the series FDTLM node:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A. 3. Centered Average for the Field Component Combinations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

APPENDIX B. DEVELOPMENTS IN TE CASE

B. 1. Wave Definition

Incident and reflected waves are detailed at each port of the shunt FDTLM node:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

B.2. Field Definition

Electromagnetic fields are deduced from wave definitions at each port of the shunt FDTLM node:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

B.3. Centered Average for the Field Component Combinations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

ACKNOWLEDGMENT

The authors would like to thank the Defence Procurement Agency (DGA) which supports the first author.

Received 3 June 2013, Accepted 22 September 2013, Scheduled 23 September 2013

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[17.] Azizi, M., H. Aubert, and H. Baudrand, "A new iterative method for scattering problems," Proceedings of the European Microwave Conference, 255-258, 1995.

[18.] Zugari, A., M. Khalladi, M. I. Yaich, N. Raveu, and H. Baudrand, "New approach: WCIP and FDTLM hybridization," 2009 Mediterranean Microwave Symposium (MMS), 1-4, 2009.

[19.] Glaoui, M., H. Trabelsi, H. Zairi, A. Gharsallah, and H. Baudrand, "A new computationally efficient hybrid FDTLM-WCIP method," International Journal ofElectronics, Vol. 96, No. 5, 537-548, 2009.

[20.] Raveu, N. and O. Pigaglio, "Resolution de problemes hautes frequences par les schemas equivalents cours et exercices corriges," Editions Cepadues, Apr. 2012.

[21.] Girard, C., N. Raveu, R. Perrussel, and S. Lanteri, "Hybridation entre la WCIP et des methodes volumiques," Journees Nationales Microondes (JNM), May 2013 (in French).

[22.] Nguyen, N. C., J. Peraire, and B. Cockburn, "Hybridizable discontinuous Galerkin methods for the time-harmonic Maxwell's equations," Journal of Computational Physics, Vol. 230, No. 19, 7151-7175, Aug. 2011.

Caroline Girard (1,2), *, Asmaa Zugari (3), and Nathalie Raveu (1)

(1) INPT, UPS; CNRS; LAPLACE; ENSEEIHT, Universite de Toulouse, 2 Rue Charles Camichel, Toulouse 31000, France

(2) NACHOS Project-Team, INRIA Sophia Antipolis-Mediterranee, 2004 Route des Lucioles, Sophia Antipolis 06560, France

(3) Electronics and Microwaves Research Group (EMG), Laboratory Systems of Information and Telecommunications (LaSIT), Faculty of Sciences, Abdelmalek Essaadi University, Av. de Sebta, Mhannech II, Tetuan 93002, Morocco

* Corresponding author: Caroline Girard (cgirard@laplace.univ-tlse.fr).
```
Table 1. [[parallel][S.sup.W.sub.down] - [S.sup.F.sub.down]
[parallel].sub.2] in TE and TM cases.

Mesh size         TE case               TM case

1           1.693 x [10.sup.-1]   1.544 x [10.sup.-1]
1/2         8.467 x [10.sup.-2]   8.085 x [10.sup.-2]
1/4         4.233 x [10.sup.-2]   4.137 x [10.sup.-2]
1/8         2.116 x [10.sup.-2]   2.092 x [10.sup.-2]
1/16        1.058 x [10.sup.-2]   1.052 x [10.sup.-2]

Table 2. Relative error on E-fields on ([sigma])
for TE and TM cases.

Mesh   N     [MATHEMATICAL EXPRESSION
size         NOT REPRODUCIBLE IN ASCII]

Error          Order

1      16    5.92 x [10.sup.-3]     --
1/2    32    1.47 x [10.sup.-3]   2.0068
1/4    64    3.68 x [10.sup.-4]   2.0017
1/8    128   9.19 x [10.sup.-5]   2.0004
1/16   256   2.30 x [10.sup.-5]   2.0001

Mesh   N     [MATHEMATICAL EXPRESSION
size         NOT REPRODUCIBLE IN ASCII]

Error          Order

1      16    5.28 x [10.sup.-3]     --
1/2    32    1.32 x [10.sup.-3]   1.9983
1/4    64    3.31 x [10.sup.-4]   1.9996
1/8    128   8.27 x [10.sup.-5]   1.9999
1/16   256   2.07 x [10.sup.-5]   2.0000

Table 3. TE case: relative error on the tangential electric field and
on the current on ([summation]).

Mesh    N         [MATHEMATICAL               [MATHEMATICAL
size             EXPRESSION NOT              EXPRESSION NOT
REPRODUCIBLE IN ASCII]      REPRODUCIBLE IN ASCII]

Error         Order         Error         Order

1      16    2.65.[10.sup.-2]     --     2.73.[10.sup.-2]     --
1/2    32    1.47.[10.sup.-2]   0.8557   1.91.[10.sup.-2]   0.5142
1/4    64    7.72.[10.sup.-3]   0.9237   1.33.[10.sup.-2]   0.5212
1/8    128   4.06.[10.sup.-3]   0.9262   9.22.[10.sup.-3]   0.5308
1/16   256   2.11.[10.sup.-3]   0.9440   6.32.[10.sup.-3]   0.5435

Table 4. TM case: relative error on the tangential electric field
and on the current on ([SIGMA]).

Mesh    N        [MATHEMATICAL               [MATHEMATICAL
size            EXPRESSION NOT              EXPRESSION NOT
REPRODUCIBLE IN ASCII]      REPRODUCIBLE IN ASCII]

Error         Order         Error         Order

1      16    2.73.[10.sup.-2]     --     2.95.[10.sup.-2]     --
1/2    32    1.91.[10.sup.-2]   0.5106   1.55.[10.sup.-2]   0.9252
1/4    64    1.34.[10.sup.-2]   0.5187   8.07.[10.sup.-3]   0.9450
1/8    128   9.26.[10.sup.-3]   0.5287   4.22.[10.sup.-3]   0.9345
1/16   256   6.36.[10.sup.-3]   0.5423   2.19.[10.sup.-3]   0.9486

Table 5. TE case: relative error on the tangential electric field and
on the current on ([SIGMA]).

Mesh     N    [MATHEMATICAL EXPRESSION NOT
size          REPRODUCIBLE IN ASCII]

Error            Order

1       16    2.48.[10.sup.-2]        --
1/2     32    1.34.[10.sup.-2]      0.8824
1/4     64    6.87.[10.sup.-3]      0.9689
1/8     128   3.40.[10.sup.-3]      1.0145
1/16    256   1.56.[10.sup.-3]      1.1222

Mesh    [MATHEMATICAL EXPRESSION NOT
size    REPRODUCIBLE IN ASCII]

Error              Order

1       2.06.[10.sup.-2]         --
1/2     1.39.[10.sup.-2]       0.5725
1/4     8.97.[10.sup.-3]       0.6296
1/8     5.47.[10.sup.-3]       0.7126
1/16    3.07.[10.sup.-3]       0.8366
```