C Poussot-Vassal

O Sename

L Dugard

P Gáspár

Z Szabó

J Bokor

Multi-objective qLPV H ∞ / H 2 control of a half vehicle

Keywords: Half vehicle, LMI based multi-objective synthesis, qLPV systems, Mixed qLPV polytopic H∞/ H2 control, Pareto limit

In this paper we compare LTI and qLPV H∞/ H2 controllers. The Pareto limit is used to show the compromise that has to be done when a mixed synthesis is achieved. Simulations on a nonlinear half vehicle model, with multiple objectives, are performed to show the efficiency of the method.

I. INTRODUCTION

The main role of suspensions is to improve comfort by isolating the vehicle chassis to an uneven ground and to provide a good road holding to ensure passenger safety. Suspension control of quarter vehicle have been widely explored the past few years to improve vertical movements either by applying LQ [START_REF] Hrovat | Survey of advanced suspension developments and related optimal control application[END_REF], Skyhook [START_REF] Sammier | Skyhook and H∞ control of active vehicle suspensions: some practical aspects[END_REF], H ∞ control [START_REF] Gaspar | Active suspension design using LPV control[END_REF], [START_REF] Zin | An LPV/H∞ active suspension control for global chassis technology: Design and performance analysis[END_REF], LPV [START_REF] Fialho | Road adaptive active suspension design using linear parameter varying gain scheduling[END_REF] or mixed synthesis [START_REF] Abdellahi | On the design of active suspension system by H∞ and mixed H 2 /H∞: An LMI approach[END_REF], [START_REF] Takahashi | A multiobjective approach for H 2 and H∞ active suspension control[END_REF]. Roll dynamic is catched by the half vehicle model and is directly linked to suspension behavior. Separated synthesis on each suspension can't guarantee global performances. The aim of the mixed H ∞ / H 2 control synthesis is to treat the standard H 2 and H ∞ optimal control problems as separate problems but in a unified state-space framework. This method yields a compensator that combines the H 2 quadratic performance criterion for disturbance rejection with the H ∞ performance criterion for maximum robustness against destabilizing uncertainties. It means, the controller which minimizes the H 2 performance index is selected from the suitable H ∞ controllers, thus the desired criteria are met by creating a balance between H 2 and H ∞ norms [START_REF] Doyle | Mixed H 2 and H∞ performance objectives: Optimal control[END_REF].

The mixed qLPV H ∞ / H 2 method is proposed here for the design of active suspension system, in which different optimization criteria are applied to guarantee the performance specifications and the nonlinearity of the suspension system. The nonlinearity in the suspension system is caused by the changes in the spring and damping coefficients. It is assumed that the nonlinear dynamics of road vehicles is approximated by LPV (qLPV) models, in which nonlinear terms are hidden with newly defined scheduling variables and they are available from calculated signals. The active suspension based on the LPV model takes the nonlinear dynamics of the system into consideration. Performance limitations according to the importance given between the H ∞ and the H 2 criteria is shown with the Pareto limit.

The paper is organized as follows: in Section II we introduce a linear and a nonlinear model of the half vehicle. LTI and qLPV polytopic H ∞ / H 2 control, based on LMIs, are presented in Section III. The Pareto limit, applied to the half vehicle provides smart indications in the way to choose H ∞ / H 2 attenuation parameters according to the desired performances. In Section IV, validations are done on the nonlinear model presented.

II. HALF VEHICLE MODEL

A. Model description

Roll dynamic is the main movement that enters when a driver turns. The half vehicle model involved here is a chassis model that catches vertical and roll dynamics [START_REF] Sammier | Skyhook and H∞ control of active vehicle suspensions: some practical aspects[END_REF] (Figure 1). It models the left/right vehicle load transfers that appears during a steering situation. The model is composed of two suspensions, each of them modeled by a spring (F k {f l,f r} ), a damper (u {f l,f r} = F c {f l,f r} , in the passive case) or an actuator (u {f l,f r} = u {f l,f r} H∞/H 2 + c {f l,f r} , in the active case) linked to a common suspended mass (m s ) and to a specific unsprung mass m us f l and m us f r . Tires k t f l and k t f r are linked to the ground and to the unsprung mass m us f l and m us f r respectively. Note that u {f l,f r} H∞/H 2 comes from the controller synthesized in Section III. The movements taken into account are the vertical displacement of the suspended mass (z s ), the unsprung masses (z us {f l,f r} ), the suspension deflections (z def {f l,f r} ) and the roll angle (θ) of the center of gravity of the suspended mass. The nonlinear effects of the actuator dynamics are ignored for simplicity.

F k f l F kt f l z s z r f l m us f l m us f r z us f l z r f r z us f r m s , I x k t f r k f r θ 7 7 - t f t f - u f r u f l

B. Model dynamical equations

The model is obtained by simply adding two suspensions and tires equations with the dynamical equation of the chassis as follows. First we derive the suspension (F szi ) and tire (F tzi ) forces,

         F tz f l = k t f l (z us f l -z r f l ) F tz f r = k t f r (z us f r -z r f r ) F sz f l = F k f l (z s f l -z us f l ) -k b θ 2l f l + u f l F sz f r = F k f r (z s f r -z us f r ) + k b θ 2l f r + u f r where,        z s f l = z s -t f sin(θ) z s f r = z s + t f sin(θ) żs f l = żs -θt f cos(θ) żs f r = żs + θt f cos(θ)
then the dynamic of the chassis and unsprung masses (bounce and roll) are given by,

       m s zs = -(F sz f l + F tz f r + F dz ) m us f l zus f l = F sz f l -F tz f l m us f r zus f r = F sz f r -F tz f r I x θ = F sz f l t f -F tz f r t f + M dx (1)
where F ki and F ci , {i = f l, f r}, represent the force delivered by the spring and by the damper (either linear or nonlinear), k ti is the stiffness of the tire and k b models the influence of an anti-roll bar. I x is the chassis inertia on the roll axis, t f is the distances of the unsprung masses to the center of gravity of the suspended mass (table I). Finally, θ, z s , z us f l and z us f r represent the roll angle and the chassis, unsprung mass left and right bounce. Then z r f l and z r f r represent the road disturbances on the wheels. F dz , M dx represent the load and inertia disturbances. Note that when the passive system is considered,

u f r = F c f r and u f l = F c f l .
Then, the state space vector of the linear model is defined by x = z us f l żus f l z us f r żus f r z s żs θ θ , the input are given by w = z r f l z r f r F dz M dx u f l u f r and the measured signal used for control y In the nonlinear model, we assume that the spring (F ki ) and the damper (F ci ) forces are nonlinear functions of z defi and żdefi respectively [START_REF] Zin | Robust automotive suspension control toward global chassis control[END_REF] (Figure 2). For all the simulations we will consider the nonlinear model.

= z def f l z def f r . Symbol Value Description ms 315 × 2 kg sprung mass mus f l , mus f r 37.5 kg unsprung masses k f l , k f r 29500 N/m suspensions linearized stiffness c f l , c f r 1500 N/m/

A. A LTI multi-objective controller

The multi-objective synthesis consists of giving different kind of constraints on the output of a system. With this formulation (for the case of H ∞ / H 2 ), let describe the system as follows

    ẋ z ∞ z 2 y     =     A B ∞ B 2 B C ∞ D ∞1 D ∞2 E ∞ C 2 D 21 D 22 E 2 C F ∞ F 2 0         x w ∞ w 2 u     (2) the controller, ẋc u = A c B c C c D c x y = S (3)
and the closed loop,

  ẋ z ∞ z 2   =   A B ∞ B 2 C ∞ D ∞1 D ∞2 C 2 D 21 0     x w ∞ w 2   (4)
The

H ∞ / H 2 synthesis consists of, imposing T ∞ = ||z ∞ /w ∞ || ∞ < γ ∞ and T 2 = ||z 2 /w 2 || 2 < γ 2 .
                         A T K + KA KB ∞ C T ∞ B T ∞ K -γ 2 ∞ I D T ∞1 C ∞ D ∞1 -I   < 0 A T K + KA KB 2 B T 2 K -I < 0 , K C T 2 C 2 Z > 0 , T race(Z) < γ 2 , D 22 = 0 (5)
Then solving this problem gives the LTI controller that achieves the desired performances. Note that to relax BMIs (5) into LMIs we use the transformation given in [START_REF] Scherer | Multiobjective output-feedback control via LMI optimization[END_REF].

B. A qLPV multi-objective controller

Linear parameter varying theory is useful to tackle measurable and bounded nonlinearities. We talk about qLPV when the varying parameters only enter in the dynamic matrix A of the system. In the suspension system, the measure of the deflection (used as a controller input) can also be used to reconstruct the stiffness coefficient [START_REF] Zin | An LPV/H∞ active suspension control for global chassis technology: Design and performance analysis[END_REF]. To build a qLPV controller, we use the polytopic approach which consists of building a controller to the k-corners of the polytope (formed by all the possible combinations of the upper and lower bounds of each varying parameters) and to schedule theses k-controllers by the measure of the varying variables [START_REF] Gaspar | Mixed H 2 /H∞ control design for active suspension structures[END_REF], [START_REF] Gaspar | Rollover stability control for heavy vehicles by using LPV model[END_REF]. The qLPV system is described as follows, with p a varying parameter,

    ẋ z ∞ z 2 y     =     A(p) B ∞ B 2 B C ∞ D ∞1 D ∞2 E ∞ C 2 D 21 D 22 E 2 C F ∞ F 2 0         x w ∞ w 2 u     (6)
the parameter dependent controller,

ẋc u = A c (p) B c (p) C c (p) D c (p) x c y = S(p) (7)
and the parameter dependent closed loop,

  ẋ z ∞ z 2   =   A(p) B ∞ B 2 C ∞ D ∞1 D ∞2 C 2 D 21 0     x w ∞ w 2   (8)
Then the corresponding mixed problem is similar to the LTI one: minimize γ 2 and γ ∞ subject to K and Z.

                         A(p) T K + KA(p) KB ∞ C T ∞ B T ∞ K -γ 2 ∞ I D ∞ 1 T C ∞ D ∞ 1 -I   < 0 A(p) T K + KA(p) KB 2 B T 2 K -I < 0 , K C T 2 C 2 Z > 0 , T race(Z) < γ 2 , D 22 = 0 (9)
The polytopic approach consists of finding a common Lyapunov function K and a γ ∞ , γ 2 and Z that solve the previous LMIs ( 9) problem at each corners of the polytope. Then the control to apply is a convex combination of theses kcontrollers expressed as follows,

S(p) = 2 i k=1 α k (p) A c k B c k C c k D c k
where,

α k (p) = i j=1 |p(j) -compl(coinΘ k ) j | i j=1 (p(j) -p(j))
, and,

2 i k=1 α k (p) = 1 , α k (p) > 0
where i is the number of varying parameters and k = 2 i , the number of corners of the polytope. Let note p and p the upper and lower bounds of a parameter respectively. Finally, compl(coinΘ k ) represents the complementary of Θ k , which is simply the k th corner of the polytope [START_REF] Zin | Robust automotive suspension control toward global chassis control[END_REF].

C. Design characteristics and performances on the half vehicle model

In the case of a half vehicle, the measure is the suspension deflection and the selected varying variables are the stiffness of the suspension spring, i.e. k f l and k f r . The associated polytope is then formed by k = 4 corners (10) and kcontrollers.

Θ =      k f l k f r k f l k f r k f l k f r k f l k f r      , k f l ∈ [k f l , k f l ] k f r ∈ [k f r , k f r ] (10)
According to the dissipative theory, each constraint can be expressed as a supply function, then translated into an LMI (5, 9) [START_REF] Scherer | Multiobjective output-feedback control via LMI optimization[END_REF], [START_REF] Scherer | LMI in control[END_REF].

• The H ∞ performance is used to enforce robustness to model uncertainties and to express frequency-domain performance specifications • The H 2 performance can be used to minimize energy of the signal (note the equivalence of these norms in the frequency domain, but not in the time one)

Hence, coupled together, these specifications should improve the single H ∞ constraint. On a half vehicle model the performances we want to reach are multiple. As exposed in [START_REF] Poussot-Vassal | Optimal skyhook control for semi-active suspensions[END_REF], [START_REF] Sammier | Skyhook and H∞ control of active vehicle suspensions: some practical aspects[END_REF], some frequency specifications have to be specified concerning the suspension deflection, suspended mass and the unsprung masses (to reduce gain around sensitive low frequencies). Also, weight on the control signal prevent actuator saturation. To these frequency specifications, expressed by the H ∞ theory, the addition of H 2 constrain is used to minimize energy of time signals. Hence, the applied weights can be the following,

           W zs = 5 1 1/(2π5)s+1 W θ = 1 1/(2π)s+1 W u {f l,f r} = 2.10 -3 W zr {f l,f r} = 0.07 W n {f l,f r} = 0.001
Then, the resulting generalized plant is (Figure 4),

W zr {f l,f r} W u {f l,f r} - - - + ? W n {f l,f r} S(ρ) z s y u z r z u w 1 w 2 n u żdef - -z θ W θ θ - -z s W zs

Fig. 4. Generalized plant

In the mixed synthesis the considered H ∞ and H 2 controlled output, noted z ∞ and z 2 respectively, are the following,

z ∞ = z s z θ z u f l z u f l z 2 = z s z θ z u f l z u f l
Note that when we will compare the mixed synthesis to the H ∞ one, the controlled outputs of the H ∞ controller are, z ∞ = z s z θ z u f r z u f l .

D. Pareto limit

It is impossible to minimize both γ ∞ and γ 2 . In the literature, the mixed problem is generally solved by minimizing a convex combination of H ∞ and H 2 that represents a compromise between the two performances. Such a minimization can take the following form,

min{α 1 T ∞ + α 2 T 2 }, where {α 1 , α 2 } ∈ [0, 1] × [0, 1], α 1 + α 2 = 1
Hence a natural problem raises, how to choose in a smart way α 1 and α 2 . The concept of non-inferiority (also called Pareto optimality) is used here to characterize the objectives. A non-inferior solution is one in which an improvement in one objective requires a degradation of an other. In our case the objectives are H ∞ and H 2 . To plot the Pareto optimum, applied to our problem, we iteratively fix the γ ∞ and minimize the γ 2 . The corresponding results are given in Figure 5.

k {f l,f r} ∈ knom × [1, 1.2], [1, 1.5], [1, 2]
The achievable combinations {γ ∞ ,γ 2 } is the set of couples located over the Pareto limit. The Pareto limit is also useful to measure the conservatism of a method and to exhibit how much one can decrease the performances with a qLPV approach compared to the LTI one. Such a Figure can also motivate researches on polytope reduction. In effect, the more you increase the size of your polytope (bounds of the parameters), the more far you go from the LTI Pareto optimum (Figure 5), and loose performance.

IV. SIMULATION RESULTS

To validate the control design, first, simulations are done in order to show the advantages of mixed synthesis compared to single H ∞ objective, then, we study the influence of the choice of the couple {γ ∞ , γ 2 } on the reached performances. Finally, we compare the LTI mixed approach with the qLPV one. Note that on theses simulations, when a control law is considered, the damper is removed so that the considered suspension simulated is a real semi-active one i.e. a spring (nonlinear) plus an active actuator. In such a way we explicitly model the fact that the damper is replaced by the actuator. Such a control also justify the choice of {k f l , k f r } as varying parameters in the qLPV synthesis.

A. The LTI case

First we show the advantages of the mixed H ∞ / H 2 compared to H ∞ synthesis (for the same γ ∞ ). In this simulation we generate a step road disturbance on the first then on the second wheel, then a roll moment disturbance and we compare controllers performances according to the passive suspension. By using the mixed synthesis instead of single H ∞ , we reduce the roll angle due to the roll energy minimization (Figure 6). Then, we compare the performances of the mixed synthesis for different couples {γ ∞ , γ 2 } (Figure 7). If one decrease the γ 2 attenuation value, then it increases the γ ∞ one (see Pareto limit Figure 5). The H 2 criteria's aim is to minimize the energy and variations of a signal (here, z s but also the control input are limited). Hence we observe that the z s variations are smoother by using a smaller attenuation gain on the H 2 criteria (less oscillations, i.e. ameliorate vertical comfort).

B. The qLPV case

LTI and the qLPV controllers are here (Figure 8, 9 and 10) investigated for parameters k {f l,f r} varying between k nom × [1, 1.95] (i.e. [k f l,f r , k f l,f r ]) and for a fixed γ ∞ = 0.25.

Here, we assume bigger road step disturbance to reach the nonlinear area of the suspension deflection. The qLPV synthesis improves the performances achieved by the LTI one. Then, such a control tackle the nonlinearities, hence enforce robustness [START_REF] Zin | Robust automotive suspension control toward global chassis control[END_REF]. The α variations shows the scheduling done according to the parameters variations. Note that a qLPV approach, even if enforces robustness, exhibits more complexity than the LTI one because it increases the number of controllers to be synthesized (4 in our case) and requires to schedule them in real-time. Then, the control signal looks sensitive to the parameters variations (Figure 10). Nevertheless, we use in both synthesis (LTI and qLPV) the same number of measures.

V. CONCLUSION AND FUTURE WORKS

In this paper we investigate a multi-objective mixed qLPV H ∞ / H 2 control applied to a half vehicle model. A special interest is made on the advantages of such a synthesis and on the compromises that have to be done in multi-objective applications. By using the Pareto limit (non-inferior solution) we expose a smart way to select the objectives and show the influence on significative driving situations.

In future works we aim to extend the mixed synthesis to a full vehicle model, in a global chassis control framework. Effort have to be made to express precise frequency specifications to each controlled output (roll, pitch. . . ). LPV theory, in the global strategy involving steering and braking actuators, can also be extended to prevent emergency rollover or braking situations [START_REF] Gaspar | The design of an integrated control system in heavy vehicles based on an LPV method[END_REF]. Investigations on polytope reduction (reduce conservatism) should also be explored.

Fig. 1 .

 1 Fig. 1. Half vehicle model

Fig. 2 .

 2 Fig. 2. Linear (dashed) vs. Non-Linear (solid) stiffness (left) and damping (right) coefficient

Fig. 3 .

 3 Fig. 3. Linear (dashed) vs. Non-Linear (solid) half vehicle model open loop frequency behavior

2 Fig. 5 .

 25 Fig.[START_REF] Gaspar | The design of an integrated control system in heavy vehicles based on an LPV method[END_REF]. LTI (solid) vs qLPV (dashed) Pareto limit fork {f l,f r} ∈ knom × [1, 1.2], [1, 1.5],[START_REF] Abdellahi | On the design of active suspension system by H∞ and mixed H 2 /H∞: An LMI approach[END_REF][START_REF] Chilali | H∞ design with pole placement constraints: an LMI approach[END_REF]

Fig. 6 .

 6 Fig. 6. Comparison between H∞ (dashed) and Mixed (solid) design with Passive (solid slim)

Fig. 7 .

 7 Fig. 7. Comparison of mixed synthesis performances according to different {γ∞, γ 2 }

Fig. 8 .1

 8 Fig. 8. Comparison of LTI (dashed) and qLPV (solid) mixed synthesis performances

Fig. 9 .

 9 Fig. 9. Variations of k f l = k 1 and k f r = k 2 (down) and the α (up) computed by the LPV controller

Fig. 10 .

 10 Fig. 10. Control signal u f l (up) and u f r (down) for the LTI (dashed) and qLPV (solid) controller

TABLE I PARAMETERS

 I

OF THE MODEL

  Hence the LMI based problem formulation is the following: minimize γ 2 and γ ∞ subject to K and Z.[START_REF] Chilali | H∞ design with pole placement constraints: an LMI approach[END_REF],[START_REF] Scherer | Multiobjective output-feedback control via LMI optimization[END_REF]

Laboratoire d'Automatique de Grenoble, ENSIEG -Domaine Universitaire -BP46, 38402 Saint Martin d'Hères -Cedex FRANCE, e-mail : {charles.poussot, olivier.sename, luc.dugard}@lag.ensieg.inpg.fr

Computer and Automation Research Institute, Hungarian Academy of Sciences, Kende u. 13-17, H-1111, Budapest, HUNGARY, e-mail : {gaspar,szabo,bokor}@sztaki.hu


est destinée au dépôt


Benoît Le Lepvrier

Renaud Loison

Raphaël Gillard

Philippe Pouliguen

Patrick Potier

Laurent Patier

Benoit Le Lepvrier

Member, IEEE Raphael Gillard

A new hybrid method for the analysis of surrounded antennas mounted on large platforms

Keywords: hybrid methods, DG-FDTD, IPO, multi-scale method, antenna on structure

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

A NTENNA placement on platforms such as aircraft, spacecraft or motor vehicles is becoming ever more constrained. Therefore, antennas are frequently placed close to complex elements on the structure. For this kind of problem, the complex vicinity of the antenna and the platform may have a significant effect on its behavior. So, it is of great interest to develop appropriate methods to estimate the distortion of the performance of the on-board antennas [START_REF] Zhang | Space antenna modeling[END_REF].

EM analysis of modern antenna-on-platform problems is very challenging for several reasons. Fig. 1 depicts the different regions which have to be considered when analyzing such problems. Obviously, the transmitting antenna (subdomain A) has to be simulated rigorously with a very fine mesh. Likewise, a potentially large subdomain including all complex elements in the vicinity of the antenna (subdomain B) must also be considered rigorously with a fine mesh. Although this caution requires large computational resources, it is necessary to correctly take into account the strong interactions between the transmitting antenna and the complex elements in its vicinity. The computational burden is compounded by the simulation of the platform (subdomain C), which represents a very large object electrically. The last challenge deals with the wide-frequency band over which antennas usually have to be analysed nowadays. This is mainly due to the evolution of applications toward increased bandwidth and frequency reconfigurability.

B. Le Lepvrier, R. Loison, and R. Gillard are with the European University of Brittany, France, INSA, IETR, UMR 6164, F-35708 Rennes, France (e-mail:Benoit.le-lepvrier@insa-rennes.fr;

Renaud.Loison@insa-rennes.fr; Raphael.Gillard@insa-rennes.fr).

P. Pouliguen is with the Direction Générale de l'Armement / Direction de la Stratégie / Mission pour la Recherche et l'Innovation Scientifique (DGA/DS/MRIS), 92221 Bagneux, France (e-mail: philippe.pouliguen@dga.defense.gouv.fr).

P. Potier is with the Direction Générale de l'Armement (DGA), 35174 Bruz cedex, France (e-mail: patrick.potier@dga.defense.gouv.fr).

L. Patier is with the Centre National d'Etudes Spatiales (CNES), Toulouse 31401, France (e-mail: laurent.patier@cnes.fr).

A B C

Fig. 1. A: Transmitting antenna, B: complex antenna vicinity, C: large supporting platform without any complex details.

With improvements in numerical techniques and computer performance over recent decades, direct Full Wave Methods (FWM) such as FDTD, FEM or MoM have appeared in the computation of antenna-platform problems. For example, in [START_REF] Lei | EMC analysis of antennas mounted on electrically large platforms with parallel FDTD method[END_REF], [START_REF] Riley | Analysis of airframe-mounted antennas using parallel and hybridized finite-element time-domain methods[END_REF], full wave time domain methods are used with parallelization schemes to perform the computation. These approaches have the great benefit of giving exact solutions. However, the necessity to finely mesh a complex and potentially large subdomain around the antenna and to simulate an electrically very large structure (platform) still require massive computational resources. Finally, regarding the problem illustrated in Fig. 1, direct FWM are properly suited for the subdomain A.

More efficient approaches, based on advanced FWM, have been proposed to analyze antenna-on-platform problems. For example, in [START_REF] Barka | Domain decomposition method based on generalized scattering matrix for installed performance of antennas on aircraft[END_REF], [START_REF] Wang | Multisolver domain decomposition method for modeling EMC effects of multiple antennas on a large air platform[END_REF], the authors use Domain Decomposition Methods (DDM) with frequency domain solvers, decomposing the entire problem into several subregions. Subsequently, each of the subregions is analyzed separately and an iteration process is then started, whose aim is to link the different subregions. DDM based methods working in the time domain have also been proposed. For example, in [START_REF] Johnson | MR/FDTD: A multiple-region finite-difference-time-domain method[END_REF] the authors use the FDTD method to analyze each independent subregion of the EM problem. Advanced FWM also include multi-scale methods, among which the Multi-Level Fast Multipole Method (MLFMM) is probably the most popular. For example, in [START_REF] Zhao | The multilevel fast multipole algorithm for EMC analysis of multiple antennas on electrically large platforms[END_REF], [START_REF] Jakobus | Advanced EMC modeling by means of a parallel MLFMM and coupling with network theory[END_REF], MLFMM is applied to the Method of Moments (MoM) to analyze antennas mounted on aircraft. Time-domain multiscale methods, such as the Dual-Grid FDTD (DG-FDTD) [START_REF] Pascaud | Dualgrid finite-difference time-domain scheme for the fast simulation of surrounded antennas[END_REF], have also been proposed to analyze problems presented by antennas on platforms, especially when wide-band characterizations are required. Finally, all the advanced approaches cited above allow rigorous and efficient simulation of subdomains A and B. Although more efficient than direct FWM, advanced FWM do not completely succeed in cutting the computational burden, especially when very large platforms must be considered (such as domain C).

The most efficient way to deal with such large structures is certainly to use Asymptotic Methods (AM) such as Physical-Optics (PO) or Uniform Theory of Diffraction (UTD) [START_REF] Pathak | High-frequency techniques for antenna analysis[END_REF], [START_REF] Bhattacharyya | High-Frequency Electromagnetic Techniques : Recent Advances And Applications[END_REF]. However, regarding the problem illustrated in Fig. 1, these methods are not properly suited for subdomains A and B.

Hybrid method coupling a direct FWM with an AM has frequently been written about in published material [START_REF] Jakobus | Improved PO-MM hybrid formulation for scattering from three-dimensional perfectly conducting bodies of arbitrary shape[END_REF]- [START_REF] Chia | The application of FDTD in hybrid methods for cavity scattering analysis[END_REF], however, this hybridization scheme does not give a satisfactory answer to our problem. Indeed, the complex antenna vicinity is both too complex to be analyzed with AM and too large to be computed efficiently with direct FWM.

As a consequence, a judicious solution consists of hybridizing an advanced FWM with an AM. One example of hybridization in the frequency domain is given in [START_REF] Burkholder | A hybrid framework for antenna/platform analysis[END_REF]. The authors use the FE-BI method, enhanced with domain decomposition and fast algorithms, to efficiently compute large antenna arrays. The platform is then analyzed with UTD or Iterative Physical Optics (IPO). But full frequency domain hybrid methods, such as the latter, are not appropriate for wide-band characterizations: thus time-domain methods have to be preferred. However, full time-domain hybrid methods experience difficulties in handling electrically large structures. In [START_REF] Yang | FDTD/TDPO hybrid approach for analysis of the EM scattering of combinative objects[END_REF], a specific algorithm is proposed to improve efficiency and reduce memory storage in computation, but it has only been applied to small structures.

Hence, we propose here a new hybrid method combining a time-domain AFWM, DG-FDTD [START_REF] Pascaud | Dualgrid finite-difference time-domain scheme for the fast simulation of surrounded antennas[END_REF], and a frequency-domain AM, IPO [START_REF] Obelleiro-Basteiro | An iterative physical optics approach for analyzing the electromagnetic scattering by large open-ended cavities[END_REF]- [START_REF] Obelleiro | Iterative physicaloptics formulation for analyzing large waveguides with lossy wall[END_REF]. DG-FDTD is a multi-scale time domain method based on FDTD which enables efficient and rigorous wide-band simulations of antennas and their complex vicinity [START_REF] Pascaud | Numerical modeling of an integrated multi-sector antenna using the DG-FDTD method[END_REF]. The hybridization with IPO allows the limitation on the size of the structure to be overcome. In other words, the application domain of the DG-FDTD is extended to the accurate and efficient computation of surrounded antennas mounted on electrically large structures thanks to its combination with the IPO method. To our best knowledge it is the first time that DG-FDTD has been hybridized with IPO.

This paper is organised as follows. DG-FDTD/IPO is presented in section II. Section III is dedicated to the validation of the method. Examples of its application and associated numerical results are presented in Section IV. Finally, conclusions are drawn in the last section.

II. DG-FDTD/IPO FORMULATION

Let us consider the problem described in Fig. 2 in order to illustrate the DG-FDTD/IPO principle. The figure presents a basic configuration of a surrounded antenna which is mounted on a large metallic platform. More precisely, the antenna is placed close to a complex element (involving fine details) and mounted on a vehicle. The DG-FDTD/IPO approach consists in breaking down the initial EM problem into two successive simulations (Fig. 3). First, the DG-FDTD method is used to

⃗ J 0 ⃗ J 0 ⃗ J eq ⃗ M eq ⃗ J N ⃗ J N ⃗ J N ⃗ J N
Second step : Iterative computation of the electric currents on the structure Metallic structure (Sc)

⃗ J eq ⃗ M eq
Scattered field domain Huygens surface (Sa)

⃗ 0 N>0 (b) (a)
Huygens surface (Sa) analyze the antenna and its complex vicinity rigorously and efficiently. Then, an IPO simulation of the metallic structure hosting the antenna is performed. DG-FDTD and IPO simulations are respectively described in sections II-A and II-C, whereas the hybridization technique is discussed in II-B.

A. DG-FDTD simulation

As mentioned above, DG-FDTD is a multi-scale time domain method based on FDTD which enables efficient and rigorous wide-band simulations of antennas and their complex vicinity. Unfortunately, this method becomes inapropriate for the simulation of electrically very large structures due to the amount of resources needed. Only a brief description of the DG-FDTD principle is given here. For further details, please refer to the full description given in [START_REF] Pascaud | Dualgrid finite-difference time-domain scheme for the fast simulation of surrounded antennas[END_REF].

The DG-FDTD simulation of the antenna and its complex vicinity is performed using two FDTD simulations, which are run sequentially (Fig. 3 (a)). The objective of the first FDTD simulation (first step) is to very accurately characterize the isolated antenna. So, a very fine FDTD mesh can be used during this simulation and the FDTD volume is terminated by Absorbing Boundary Conditions (ABCs) in order to simulate an infinite problem. More precisely, in the general case presented in Fig. 3 (a), five faces of this volume are terminated by perfectly matched layers, whereas a metallic boundary condition is applied on the sixth one (bottom face) to account for the ground plane. A near field surface is placed around the antenna in order to accurately store the primary radiation of the isolated antenna. In the second FDTD simulation (second step), the antenna and its vicinity are described using a coarser FDTD mesh. The primary radiation stored in the previous step is used as the excitation. ABCs are used again to simulate an infinite problem.

B. Interfacing of the DG-FDTD and IPO simulations

The hybrid DG-FDTD/IPO approach uses the surface equivalence principle [START_REF] Jin | Electromagnetic theorems and principles[END_REF] to interface DG-FDTD and IPO. Time domain equivalent currents representing the antenna and its vicinity are first derived from the DG-FDTD simulation. After a conversion in the frequency domain, these currents serve as excitation for the IPO simulation. Note that a time domain version of IPO also exists [START_REF] Li | Time-domain iterative physical optics method for analysis of EM scattering from the target half burried in rough surface: Pec case[END_REF] but it has not been used here as it experiences difficulties in handling large structures.

More precisely, during the second DG-FDTD step, the tangential fields over a closed Huygens surface, including the antenna and its vicinity, are stored (Fig. 3 (a)). Then, Fast Fourier Transform (FFT) is applied to calculate the time harmonic tangential fields for any point within the DG-FDTD excitation spectrum. Note that two compression techniques are applied to interface the DG-FDTD simulation with the IPO code more efficiently. First, the number of time domain field samples on the Huygens surface is reduced according to the Nyquist sampling theorem. This time domain compression allows a reduction of the FFT burden, which enables faster calculation of the time harmonic form of the tangential fields. The second aspect of the compression deals with the reduction of the number of equivalent sources on the Huygens surface. With this objective, subapertures grouping several FDTD cells are defined and averaged tangential field components are calculated (linear interpolation). The equivalent sources J eq and M eq are finally obtained thanks to the equivalence principle formulas.

Next, a Huygens surface with the same location and dimensions as the one used in the DG-FDTD simulation is defined in the IPO simulation (Fig. 3 (b)). It is worth mentioning that the elements analyzed during the DG-FDTD simulation are not redescribed in the IPO simulation. As a consequence, the volume defined by the equivalent surface and closed by a flat metallic plate is empty. Finally, the equivalent currents calculated previously are positioned on this virtual surface. They will be used later as excitation for the IPO simulation.

C. IPO simulation

IPO [START_REF] Obelleiro-Basteiro | An iterative physical optics approach for analyzing the electromagnetic scattering by large open-ended cavities[END_REF]- [START_REF] Obelleiro | Iterative physicaloptics formulation for analyzing large waveguides with lossy wall[END_REF] is an asymptotic method which consists of an iterative resolution of the Magnetic Field Integral Equation (MFIE). This method is used here to compute the currents over the metallic structure hosting the antenna. IPO is given preference over the traditional PO because it provides more accurate results when the shape of the metallic structures causes multiple reflections [START_REF] Burkholder | Iterative physical optics: it's not just for cavities anymore[END_REF]. The IPO computation of the EM field radiated by the antenna on its platform is divided into 3 steps.

Firstly, IPO simulation starts by the computation of the magnetic fields H Sa Sc , on the structure (S c ), radiated by equivalent currents J eq and M eq over the Huygens surface (S a ) (Fig. 3 (b)). This first step represents the excitation phase of the IPO simulation: excitation technique used here is based on the total-field/scattered-field decomposition principle. As indicated in Fig. 3 (b), the Huygens surface plays the role of a frontier between the total-field and the scattered-field domain. Following this excitation principle, the equivalent currents do not radiate in the scattered field domain. For the rest of the structure (e.g. in the total field domain) the magnetic field H Sa Sc is computed using the Kirchhoff approximation:

H Sa Sc ( r c ) = Sa J eq ( r a ′ ) × ∇ ′ G( r c -r a ′ )dS ′ a + 1 jkZ 0 × ∇ Sa M eq ( r a ′ ) × ∇ ′ G( r c -r a ′ )dS ′ a ( 1
)
where r a ′ and r c refer respectively to points over the equivalent Huygens surface S a and over the structure S c outside the Huygens surface. ∇ ′ G is the gradient of the free-space Green's function. The prime on the gradient operator indicates that the differentiation is performed on the source coordinates. Z 0 is the free space impedance and k is the wavenumber. The electric currents J 0 ( r c ) on the structure S c after the excitation phase are given by

J 0 ( r c ) = 2 n × H Sa Sc ( r c ) ( 2
)
where n is the unit vector normal to the surface Sc at the point r c . Note that, because of the total-field/scattered-field decomposition principle, J 0 ( r c ) is equal to 0 for r c in the scattered field area. The second step of the IPO simulation consists of an iterative computation of the electric currents on the structure. These currents are induced by the radiation of both equivalent currents over the Huygens surface and currents on the other facets of the structure. The electric current J N ( r c ) at a point r c at iteration N (N > 0) is given by where r c and r c ′ are two points on the structure S c . Note that point r c can belong to the scattered field domain since N > 0. This iterative process is repeated until a stable value for the electric currents is reached. It is worth noting that an IPO simulation with 0 IPO iteration (N = 0) corresponds to the traditional PO simulation. Consequently, J 0 ( r c ) are equal to the traditional PO currents.

J N ( r c ) = J 0 ( r c ) + 2 n × Sc J N -1 ( r c ′ ) × ∇ ′ G( r c -r c ′ )dS ′ c (3)
The integrations presented in equations ( 1) and ( 3) are performed taking into account shadowing effects. For instance, in equation ( 1), if point r c is not visible from point r a ′ , the contribution from point r a ′ is set to zero. The last step aims at computing the far field radiated at observation point r by the equivalent currents, and the currents on the structure. The Kirchhoff approximation is used again :

H( r) = Sa J eq ( r a ′ ) × ∇ ′ G( r -r a ′ )dS ′ a + 1 jkZ 0 × ∇ Sa M eq ( r a ′ ) × ∇ ′ G( r -r a ′ )dS ′ a + Sc J N ( r c ′ ) × ∇ ′ G( r -r ′ )dS ′ c (4)
E( r) = 1 jkY 0 × ∇ Sa J eq ( r a ′ ) × ∇ ′ G( r -r a ′ )dS ′ a + 1 jkY 0 × ∇ Sc J N ( r c ′ ) × ∇ ′ G( r -r c ′ )dS ′ c - Sa M eq ( r a ′ ) × ∇ ′ G( r -r a ′ )dS ′ a ( 5
)
where Y 0 is equal to 1/Z 0 .

III. VALIDATION OF DG-FDTD/IPO

This section is dedicated to the validation of the DG-FDTD/IPO approach presented in section II. To do so, the radiation of a canonical antenna-on-structure is computed with both DG-FDTD/IPO and advanced FWMs.

A. Description of the validation scenario

The EM problem (Fig. 4) includes a monopole antenna surrounded by a dielectric element, mounted on an electrically large metallic structure. The monopole antenna, whose central frequency is 1 GHz, is placed at the center of a rectangular finite ground plane. Moreover, a vertical metallic plate loads this ground plane so as to create a dihedral structure. Finally, a lossless dielectric cube is placed close to the monopole in order to create a scenario requiring a rigorous description of the antenna and its vicinity.

It must be noted that the dimensions of this scenario are smaller than the ones encountered in real antenna-on-platform problems. This choice is justified by the need to get, in a reasonable amount of time, full wave computation of the overall problem as a reference.

B. Reference simulations

The complete validation scenario is first computed with two multi-scale approaches: MLFMM and Multi-Level DG-FDTD (ML DG-FDTD) [START_REF] Miry | Multi-level dual-grid finitedifference time-domain approach for the analysis of body implanted antennas[END_REF]. Note that these two approaches are based on rigorous methods (respectively MoM and FDTD). They allow multi-scale problems such as the validation test case to be computed both rigorously and efficiently. Moreover, the use of a time-domain volume method (ML DG-FDTD) and a frequency-domain surface method (MLFMM), increases the reliability of the validation.

MLFMM computation is performed with FEKO [34] software using a λ 1GHz /12 standard mesh to describe the elements (146,537 triangles). MLFMM is combined with the Surface Equivalence Principle (SEP) to analyze the dielectric region close to the antenna.

As mentioned above, the validation scenario was also simulated with Multi-Level DG-FDTD. This method is an extension of DG-FDTD where the number of successive FDTD steps is not limited to two (as is the case in Fig. 3 (b)). This time domain multi-scale method is well suited to the simulation of highly multi-scale EM problems. Here, the simulation of the overall scenario is broken down into three FDTD simulations (Fig. 5). The first step, aiming at precisely simulating the monopole, uses a uniform λ 1GHz /60 cubic mesh, which results in an FDTD volume of 60×60×56 unit cells. Then, the antenna and the dielectric block are analyzed using a coarser Fig. 6. DG-FDTD/IPO breakdown of the validation structure.

FDTD λ 1GHz /30 mesh (volume size : 60 × 80 × 40 unit cells). Finally, the third step takes into account the influence of the metallic structure using a λ 1GHz /10 cubic mesh (FDTD volume size : 170 × 170 × 65 unit cells). Radiation patterns computed by MLFMM and multi-level DG-FDTD are presented in Fig. 7 and Fig. 8. The reference results obtained with the two full wave multi-scale methods agree.

C. DG-FDTD/IPO simulation of the validation scenario 1) Decomposition of the EM problem:

The DG-FDTD/IPO simulation of the validation scenario is based on two successive simulations (Fig. 6). First, the DG-FDTD method is used to analyze the domain including the monopole and the dielectric block. This DG-FDTD simulation is performed in two steps: the monopole lying on an infinite ground plane is first simulated using a fine (λ 1GHz /60) cubic mesh. Then, this monopole is computed in the presence of the dielectric material using a coarser FDTD mesh (λ 1GHz /30). In this second step, the antenna and the dielectric are both placed on an infinite ground plane. Once the DG-FDTD simulation has been completed, an IPO simulation is launched in order to take into account the influence of the metallic structure on the antenna radiation. This simulation uses a coarse (λ 1GHz /6) mesh, which results in a structure with 7, 878 facets. As indicated in section II, the antenna and the dielectric block are not described in this simulation.

It is worth noting that in the DG-FDTD/IPO simulation, the two DG-FDTD steps are the same as the first two steps in the ML DG-FDTD reference simulation (Fig. 5). Moreover, IPO simulation substitutes to the third step of the ML DG-FDTD.

2) Numerical results: The results presented in this section were computed with one IPO iteration (N = 1). A parametric study of the IPO iteration number has shown that far-field results reach a good convergence with only one iteration. However, as it is not within the scope of this article, the study is not presented here.

In order to quantify the accuracy of DG-FDTD/IPO, the radiated field is compared with the MLFMM results by eva- luating the normalised mean absolute error ǫ given by where X is either the E θ (co-polar) or the E φ (cross-polar) far-field complex component, θ i = (-90 + (i -1)) degrees, and N θ = 181. Subscript ref denotes the reference fields (MLFMM solution). ML DG-FDTD is also compared to MLFMM in order to better estimate the loss of accuracy involved by the IPO simulation (indeed, ML DG-FDTD and DG-FDTD/IPO only differ in this IPO simulation). All the results are presented in Table I. It shows that the results coming from DG-FDTD/IPO are almost as accurate as the ones coming from ML DG-FDTD when compared with the reference far-field (MLFMM). Indeed, in the (x0z) plane, an error of 2.51% is observed with DG-FDTD/IPO compared to 1.77% with ML DG-FDTD. In the (y0z) plane, DG-FDTD/IPO gives even more accurate results than ML DG-FDTD (1.56% against 2.20%) for the co-polar component. However, the hybrid method presents a slight loss of accuracy for the cross-polar in this plane (5.97% against 3.53%). This can be explained by the fact that edge and wedge diffractions, which strongly determine the cross-polar radiated field, are not taken into account in the IPO simulation. Fig. 7 and Fig. 8a present the E θ (co-polar) far-field radiation pattern computed by DG-FDTD/IPO, and the reference methods in the (x0z) and (y0z) planes. These figures show that the DG-FDTD/IPO results are in very good agreement with those obtained by the two reference methods. Fig. 8b shows the E φ (cross-polar) far-field radiation pattern in the (y0z) plane. Once again, results are in good agreement. It is worth noting that the simulation of the surrounded monopole gives a radiation pattern quite different from that of a monopole over an infinite ground plane. In particular, the traditional monopole radiation pattern does not show oscillations as in Fig. 8a or a cross polarisation component as in Fig. 8b.

ǫ = N θ i=1 |X(θ i ) -X ref (θ i )| N θ | max i [X ref (θ i )]| (6)
So, these results indicate that the DG-FDTD/IPO method is capable of performing a reliable computation of the co-and the cross-polarisation components radiated by a surrounded antenna mounted on a metallic structure.

3) Computation time: Table II presents the computation time required by the new hybrid approach and the two ref- erence methods. It must be noted that none of the three simulations compared here were performed using a parallelization scheme.

The left column refers to the computation of one frequency point (at 1 GHz), whereas the right column considers the computation of 11 frequency points taken over the [0.8;1]GHz band. First, DG-FDTD/IPO turns out to be faster than the reference methods in the two frequency configurations. In particular, it proves to be very efficient for the computation of 11 frequency points. This result indicates that the new hybrid method makes good use of the wide-band characteristic of DG-FDTD and the rapidity of IPO computation.

IV. APPLICATION OF DG-FDTD/IPO

In section III, the DG-FDTD/IPO method was validated for a canonical case. The aim of this section is to demonstrate its ability to analyze more realistic situations.

A. Description of the EM problem

A description of the studied problem is given in Fig. 9. A wide-band diamond antenna, identical to the one described in [START_REF] Pascaud | Dualgrid finite-difference time-domain scheme for the fast simulation of surrounded antennas[END_REF], is mounted on a metallic structure representing a vehicle. The antenna presents a reflection coefficient below -10 dB over the [5 ; 9.5] GHz band. A metallic box that could contain its electronic system (metallic parallelepiped) is placed close to the antenna. Considering its electrical dimensions (7.8 λ 0 × 5.3 λ 0 × 4.56 λ 0 at 9.5 GHz), this element may already be seen as an electrically large element. Finally, the dimensions (55 λ 0 × 50 λ 0 × 32 λ 0 at 9.5 GHz) of the metallic structure provide a highly multi-scale feature to this scenario, and they may be considered to be representative of a terrestrial drone. Note that the structure considered in the IPO simulation only corresponds to the top side of the platform (light blue part in Fig. 9) because of the visibility considerations in equations ( 1) and (3). The DG-FDTD/IPO computation of this EM problem is based on the breakdown proposed in Fig. 3 (section II). Hence, The initial EM problem is split into two simulations. First, DG-FDTD is used to simulate the diamond antenna, taking into account the metallic element in the vicinity. In the first step of DG-FDTD, the antenna is analyzed alone on an infinite ground plane. This simulation uses a very fine FDTD mesh (λ 9.5GHz /105) so as to correctly model the triangular geometry of the antenna and in so doing recover the measured performances as presented in [START_REF] Pascaud | Dualgrid finite-difference time-domain scheme for the fast simulation of surrounded antennas[END_REF]. Then, an FDTD simulation of the antenna with the metallic element in its vicinity is performed. A coarser mesh (λ 9.5GHz /26.25) is used during this step to reduce the computational resources. Note that the spatial discretization used in this second step of the DG-FDTD must both be a multiple of the one used in the first step (×4), and comply with the classic FDTD dispersion criteria. The DG-FDTD simulation is followed by an IPO simulation of the metallic structure hosting the antenna. The structure is discretized with λ 9.5GHz /4.4 square facets, about 19 facets per square wavelength. This sampling density is in agreement with the recommendation given in [START_REF] Obelleiro-Basteiro | An iterative physical optics approach for analyzing the electromagnetic scattering by large open-ended cavities[END_REF]. Finally, note that to compute the structure only one iteration is used in the IPO code.

B. Numerical results

The comparison of the radiation patterns computed with DG-FDTD/IPO and FEKO at 9.5 GHz is first presented. Then, the results of a wide-band computation are proposed.

1) comparison with FEKO at 9.5 GHz: Fig. 10 shows the radiation patterns computed with DG-FDTD/IPO and FEKO software at 9.5 GHz. To generate the FEKO results, the overall structure is simulated with the MLFMM method. The computation, on a 48 Gbit RAM workstation, requires an appropriate parametrization of the FEKO simulation. First, an SPAI (SParse Approximate Inverse) iterative preconditioner is used to reduce the memory. Moreover, the overall structure is discretized using a coarse mesh (λ 9.5GHz /8) with the excep- tion of the antenna which is meshed more finely (λ 9.5GHz /20). So, the structure to be studied contains 1,324,360 triangles.

The error criteria given in equation 6 is used again to evaluate the accuracy of the DG-FDTD/IPO taking the MLFMM as reference. Note that the angular increment is reduced (here N theta = 361) in order to follow the quick oscillations encountered at this frequency. The error ǫ on the E θ and E φ far-field components (respectively the co-polar and crosspolar) is presented in Table III.

The E θ (co-polar) results (solid line in Fig. 10a and Fig. 10b) show a good agreement, especially in the (y0z) plane. Indeed, an error of 5.93% is observed with the hybrid method in the (x0z) plane. This error is even reduced to 0.825% in the (y0z) plane. This demonstrates that the new hybrid approach is capable of accurately computing large and highly multi-scale problems.

Two reasons may be put forward to explain the differences on the cross-polar component. First, the antenna vicinity is not redescribed in the IPO simulation. Then, it is well known that the classic (I)PO method does not take into account the edge/wedge diffractions. But these diffractions may have a significant effect on the cross-polar component.

Finally, as can be seen from Table IV, time-saving provided by DG-FDTD/IPO is very significant for such realistic problems.

2) Wide-band computation: One of the benefits of the DG-FDTD/IPO method is its wide-band characteristic. In order to illustrate this aspect a wide-band computation was performed. As an example, the electric far-field in the (y0z) plane for θ = 45 degrees over the [6 ; 9.5] GHz band is shown in Fig. 11. Note that a 100 MHz step was used.

Table IV also presents the computation time associated with the simulation of the far-field over the 3.5 GHz bandwidth with DG-FDTD/IPO and MLFMM (FEKO). These results clearly show the efficiency of DG-FDTD/IPO to compute wide-band parameters. Indeed, the wider the analysis bandwith used, the more attractive the DG-FDTD/IPO method becomes. As indicated in section III-C3, the wide-band efficiency of the new hybrid method is both linked to the wide-band characteristic of the DG-FDTD and the low computational cost required by the IPO simulation.

3) Antenna placement: We have seen before that DG-FDTD/IPO could provide an efficient way to accurately simulate surrounded antennas mounted on electrically large structures. This hybrid approach may also be very useful to quickly optimize antenna placement. Indeed, the DG-FDTD/IPO method allows the simulation to be performed for several positions without computing the overall problem for each position. In order to illustrate this point, the simulation of the diamond antenna and the metallic box, including its electronic system, was carried out for three different positions on the structure (Fig. 12). The results of these three simulations are presented in Fig. 13.

In table IV, the computation time needed to perform those three simulations with the DG-FDTD/IPO approach is compared to the one required to perform the same simulations with MLFMM (FEKO). It shows that the DG-FDTD/IPO approach greatly reduces the computation time associated with the simulation of antenna in different positions on the struc- ture. This is mainly due to the fact that the DG-FDTD/IPO approach can reuse the simulation of the monopole in the presence of the metallic scatterer. So, once the antenna and its complex surrounding environment have been simulated (DG-FDTD), only IPO simulation has to be repeated to compute the radiation patterns associated with a new position.

V. CONCLUSION

A new hybrid approach associating DG-FDTD and IPO is proposed. This method, called DG-FDTD/IPO, allows accurate and efficient simulation of wide-band surrounded antennas mounted on large metallic platforms.

This approach was first validated on a canonical test case by comparison with two multi-scale reference methods (MLFMM and ML DG-FDTD). The rapidity and the accuracy of the proposed method are shown. The new hybrid method was next used to analyze a real-life problem regarding a wide-band Finally, the new approach proved to be faster than the commercial code, especially for the computation of wide-band parameters, and the optimization of antenna placement.

Fig. 2 .

 2 Fig. 2. Antenna mounted on a vehicle.

Fig. 3 .

 3 Fig. 3. DG-FDTD/IPO principle.

Fig. 4 .

 4 Fig. 4. Antenna mounted on a metallic structure. l = 3.9 m = 13 λ1GHz, h = 1.15 m = 3.83 λ1GHz, d = 0.15 m = 0.5 λ1GHz, C dielec = 0.18 m = 0.6 λ1GHz, ǫr = 10.

Fig. 5 .

 5 Fig. 5. Multi-Level DG-FDTD computation of the validation scenario.

Fig. 7 .

 7 Fig. 7. E θ far-field (CO-polar) in the (x0z) plane at 1GHz for the validation scenario.

  Fig. 8. Far-field in the (y0z) plane at 1GHz for the validation scenario.

Fig. 9 .

 9 Fig. 9. Wide-band antenna and its vicinity mounted on a vehicle. l = 1728 mm; w = 1584 mm; h = 1008 mm; l 2 = 246 mm; w 2 = 168 mm; h 2 = 144 mm; d 1 = 78 mm; d 2 = 25.2 mm.

Fig. 10 .

 10 Fig. 10. Far-field at 9.5GHz with DG-FDTD/IPO and MLFMM (FEKO) in the (x0z) and the (y0z) planes, Fig. 10a and Fig. 10b respectively .

Fig. 11 .

 11 Fig. 11. Computation of the electric far field for θ = + 45 degrees in the (y0z) plane over the [6 ; 9.5] GHz band.

Fig. 12 .

 12 Fig. 12. Antenna placement : three positions studied for the antenna and its vicinity on the vehicle (top side view).

Fig. 13 .

 13 Fig. 13. E θ Radiation pattern in the (x0z) plane for three positions.

ACKNOWLEDGMENT

This work was supported by the French Defense Agency (Direction Générale de l'Armement, DGA) and the French Space Agency (Centre National des Etudes Spatiales, CNES). The authors thank Mr J. Burdon for its valuable help during the proofreading of this paper.

Philippe Pouliguen received the M.S. degree in signal processing and telecommunications, the Doctoral degree in electronic and the "Habilitation à Diriger des Recherches" degree from the University of Rennes 1, France, in 1986, 1990 and 2000. In 1990, he joined the Direction Générale de l'Armement (DGA) at the Centre d'Electronique de l'Armement (CELAR), now DGA Information Superiority (DGA/IS), in Bruz, France, where he was a "DGA expert" in electromagnetic radiation and radar signatures analysis. He was also in charge of the EMC (Expertise and ElectoMagnetism Computation) laboratory of CELAR. Since December 2009, he has been the head of "acoustic and radio-electric waves" scientific domain at the office for advanced research and innovation of the strategy directorate, DGA, Bagneux, France. His research interests include electromagnetic scattering and diffraction, Radar Cross Section (RCS) measurement and modeling, asymptotic high frequency methods, radar signal processing and analysis, antenna scattering problems and Electronic Band Gap Materials. In these fields, he has published more than 30 articles in refereed journals and more than 100 conference papers.


B Le Lepvrier
email: benoit.le-lepvrier@insa-rennes.fr

R Loison

R Gillard

L Patier

P Potier

P Pouliguen
email: philippe.pouliguen@dga.defense.gouv.fr

Analysis of Surrounded Antennas Mounted on Large and Complex Structures Using a Hybrid Method

Keywords: radiation analysis, antenna on structure, DG-FDTD, IPO, hybrid methods

A new hybrid method, the DG-FDTD/IPO, has recently been proposed to analyze surrounded antennas mounted on large platforms. In this paper the capabilities of this method to go through such large and complex problems are shown. More precisely, a deep analysis of the far-field radiated for a canonical antenna-on-platform scenario is proposed.

I. INTRODUCTION

It is well known that the radiation characteristics of antennas can be strongly altered when they have to operate in a complex environment. In the specific context of antennas mounted on large platforms, such as satellites, planes or vehicles, the radiation characteristics are influenced by both complex elements close to the antenna and by the platform itself. For these reasons, antennas mounted on complex structures must be analyzed in their complete environment of integration [START_REF] Zhang | Space antenna modeling[END_REF].

To do so, many methods are available that have been developed over the years. These methods can be separated into two groups, rigorous methods [START_REF] Lei | Emc analysis of antennas mounted on electrically large platforms with parallel fdtd method[END_REF], [START_REF] Barka | Domain decomposition method based on generalized scattering matrix for installed performance of antennas on aircraft[END_REF], [START_REF] Zhao | The multilevel fast multipole algorithm for emc analysis of multiple antennas on electrically large platforms[END_REF] and hybrid methods [START_REF] Manyas | Hybridizing physical optics with mlfma for efficient scattering computations of three-dimensional complex targets[END_REF], [START_REF] Burkholder | A hybrid framework for antenna/platform analysis[END_REF]. A hybrid method allowing modular analysis has been proposed recently [START_REF] Lepvrier | Hybridization of the dual-grid fdtd with the physical optics to analyze antennas mounted on large and complex platforms[END_REF]. This method, called DG-FDTD/PO, consists of a hybridization of the Dual-grid FDTD (DG-FDTD) [START_REF] Pascaud | Dualgrid finite-difference time-domain scheme for the fast simulation of surrounded antennas[END_REF] with Physical Optics (PO). The antenna and its complex vicinity are first analyzed in two successive FDTD steps. Then a PO simulation of the platform is performed.

In this paper, we present the deep analysis of a canonical structure by the DG-FDTD/IPO. The Iterative Physical Optics (IPO) [START_REF] Obelleiro-Basteiro | An iterative physical optics approach for analyzing the electromagnetic scattering by large open-ended cavities[END_REF] is used instead of the traditional PO in order account for multiple reflections. This paper aims at demonstrating that this new modular method provides an efficient tool to separate the contributions of the different elements participating in the EM radiation.

This detailed analysis is presented in section II using three sub-cases of increasing complexity. Finally, conclusions are drawn in the last section. In order to show the possibilities offered by this new hybrid method the scenario presented in Fig. 1 is considered. Note that the analyzed structure is identical to the one used to validate the DG-FDTD/PO in [START_REF] Lepvrier | Hybridization of the dual-grid fdtd with the physical optics to analyze antennas mounted on large and complex platforms[END_REF]. The complete scenario is first analyzed with the DG-FDTD/IPO method in II-A. Then, it is decomposed into three sub-cases (Fig. 2) in order to identify the influence of the different involved elements and thus to illustrate the capabilities of the method. The analysis of the three sub-cases with the DG-FDTD/IPO are presented in II-B, II-C and II-D. All the results shown in this paper corresponds to simulations performed at 1 GHz (operating frequency of the monopole).

A. Complete structure

The DG-FDTD/IPO simulation of the complete scenario (Fig. 1) is performed using two simulations steps as presented in Fig. 3 (see [START_REF] Lepvrier | Hybridization of the dual-grid fdtd with the physical optics to analyze antennas mounted on large and complex platforms[END_REF] for more details). The antenna and its complex vicinity are first simulated rigorously and efficiently using the DG-FDTD method. Then, an asymptotic simulation of the metallic structure hosting the antenna is performed with the IPO method. A parametrical study on the IPO iteration number N , not presented here because of space limitation, has shown that a single iteration (N = 1) is sufficient for the convergence of far-field. The classical PO simulation would be consisting of an IPO simulation without iteration (N = 0). Fig. 4 and Fig. 5 show the far-field radiation results in the (x0z) and (y0z) planes respectively.

As described in [START_REF] Lepvrier | Hybridization of the dual-grid fdtd with the physical optics to analyze antennas mounted on large and complex platforms[END_REF] and shown in Fig. 3, DG-FDTD/IPO approach considers successively larger and larger subdomains. This hybridization scheme is very modular and thus allows identifying the contribution of the different elements participating in the radiation. To do so, we now analyze three subcases, each of which highlighting a specific effect in the overall radiation.

B. Subcase 1 : antenna on a square ground plane

As presented in Fig. 6a, the simulation of this scenario is decomposed into two steps. First, the fine FDTD simulation of the isolated monopole is carried out. Note that this simulation has already been performed as the first step of the DG-FDTD simulation of the complete scenario. It provides equivalent currents on an equivalent surface enclosing the monopole. Then, these currents serve as the excitation for the IPO simulation of the square ground plane. No iterative computation of these currents is necessary here given the planar shape of the structure which means IPO simply reduces to classical PO (N = 0). The computation time for the two steps are reported in Table I.

Fig. 5a (comparison of sub-case 1 and complete structure) clearly shows that the oscillations for the E θ far-field in the (y0z) plane are mainly due to the metallic square plate. Note that no result is given for E φ (Fig. 5b) since the symmetry of

⃗ J 0 ⃗ J 0 ⃗ J eq ⃗ M eq ⃗ J N ⃗ J N ⃗ J N ⃗ J N

Second step : Iterative computation of the electric currents on the structure

Metallic structure (Sc)

⃗ J eq ⃗ M eq
Scattered field domain

Equivalent surface ⃗ 0 N>0 Fig. 3. DG-FDTD/IPO decomposition of the complete scenario. sub-case 1 leads to zero cross-polar.

C. Subcase 2 : antenna and dielectrick block on a square ground plane

Sub-case 2 is set-up to analyze the influence of the dielectric block. As described in Fig. 6a, the simulation of this scenario is based on two simulations. The first one, corresponding to the DG-FDTD simulation of the antenna in presence of the dielectric bloc, has already been performed during the DG-FDTD/IPO simulation of the complete structure. Hence, the equivalent currents representing the monopole and its complex vicinity can be reused to excite the asymptotic simulation of the square ground plane. As for sub-case 1, the asymptotic simulation of the metallic square plate is performed with a traditional PO method. Table I indicates the computation time for the two simulations steps.

In Fig. 4, the results of subcase 2 show that the dielectric block strongly influences the far-field radiation pattern of the complete structure for directions defined by -45 • < θ < +45 • . Indeed, the results are in very good agreement in this angular section of the diagram.

D. Subcase 3 : antenna mounted on the metallic dihedral structure

The analysis of the third sub-case aims at characterizing the influence of the vertical metallic plate on the radiation pattern. The decomposition scheme used to simulate this scenario is Equivalent surface

DG-FDTD : first step PO simulation ⃗ J eq ⃗ M eq
Equivalent surface very close to the one presented for sub-case 1 (Fig. 6). The first step is rigorously the same. Regarding the asymptotic simulation, the square metallic plate is replaced by the dihedral structure hosting the antenna. Because of the dihedral shape of the metallic structure the IPO method is preferred (with N = 1). Finally, only a fast asymptotic simulation (Table I) is required to analyze sub-case 3.

For directions defined by θ < -45 • or θ > +45 • in the (x0z) plane, Fig. 4 shows that the results associated to subcase 3 and the complete structure are in good agreement. This means that the vertical plane strongly affects the radiation pattern of the complete structure for the grazing angles. More precisely, for negative value of θ angle, the vertical metallic plate generates oscillations due to recombination of the direct and the scattered field. Then, for positive value, the results indicate that the metallic plate is responsible for the strong attenuation observed on the radiation patten.

In the (y0z) plane, Fig. 5b shows that the cross-polar (E φ component) are very close for sub-case 2 and the complete structure. This figure also indicates that cross-polar results associated to sub-case 3 and complete structure are very different. Moreover, as mentioned during the analysis of subcase 1, the cross-polar in this plane cannot originate from the square ground plane. So, the cross-polar in the radiation diagram of the complete structure can be attributed to the presence of the dielectric block.

III. CONCLUSION

In this paper, we have shown the possibilities offered by a recently proposed method, the DG-FDTD/IPO, to carry out the deep analysis of surrounded antennas mounted on large metallic structures. The modularity of the method has been taken up to separate and identify the contribution of different elements in a canonical antenna-on-platform problem. In particular, we have shown that the different steps used to perform the DG-FDTD/IPO simulation of the complete structure can be reused directly or combined with fast asymptotic simulation to analyze the influence of the different elements of the structure.

Fig. 1 .

 1 Fig. 1. Antenna mounted on a metallic structure. l = 3.9 m = 13 λ1GHz, h = 1.15 m = 3.83 λ1GHz, d = 0.15 m = 0.5 λ1GHz, C dielec = 0.18 m = 0.6 λ1GHz, ǫr = 10

Fig. 2 .

 2 Fig. 2. (a) sub-case 1 : monopole on the square metallic plate; (b) sub-case 2 : monopole and dielectric block on the square metallic plate; (c) sub-case 3 : monopole mounted on the metallic dihedral structure.

Fig. 4 .

 4 Fig. 4. Far-field results (E θ ) in the (x0z) plane at 1 GHz.

Fig. 5 .

 5 Fig. 5. Far-field results (E θ and E φ respectively 5a and 5b) in the (y0z) plane at 1 GHz.

Fig. 6 .

 6 Fig. 6. Decomposition schemes used to simulate the sub-cases : (a) sub-case 1; (b) sub-case 2; (c) sub-case 3.

TABLE I COMPUTATION

 I TIME ASSOCIATED TO THE DIFFERENT STRUCTURES

	Simulated	Method	Computation
	structures		time
	isolated monopole	FDTD	2 min 30s
	monopole + dielectric block	DG-FDTD	13 min
	square metallic plate	PO	50s
	dihedral metallic structure	IPO	1 min 15s
	complete structure	DG-FDTD/IPO	14 min 15s

ACKNOWLEDGMENT

This work was supported by the French Defense Agency (Direction Générale de l'Armement, DGA) and the French Space Agency (Centre National des Etudes Spatiales, CNES).