Modeling and two-scale behavior of the Harmonic Maxwell equations in and near composite material

Transcription

1 Modeling and two-scale behavior of the Harmonic Maxwell equations in and near composite material Emmanuel Frénod 1 Hélène Canot 1 Abstract - The main purpose of this article is to study the behavior of the two-scale electromagnetic field in 3D in and near the composite material. For this, the problem of Harmonic Maxwell equations, for a conducting twophase composite and the air above, is considered. Technique of a two-scale asymptotic expansion is used to obtain the homogenized problem and rigorously justify by two-scale convergence between the microscopic scale and the macroscopic scale. Keywords - Maxwell equations; Electromagnetic pulses, Homogenization; Asymptotic analysis; two-scale convergence; frequencies. 1 Université de Bretagne-Sud, UMR 6205, LMBA, F Vannes, France 1

2 1 Introduction We are interested in the evolution of Harmonic Maxwell equations in and near a composite material with boundary conditions modeling electromagnetic field radiated by an electromagnetic pulse (EMP). This is the first step of a larger study which goal is to understand the behavior of the electromagnetic field and its interaction with a skin fuselage which is made of composite material. The paper focuses on what happens over a period of time of a millisecond during the peak of the first return stroke. An electromagnetic pulse is a short burst of electromagnetic energy. Such a pulse may be generated by a natural occurrence such that a lightning strike, meteoric EMP, EMP caused by geomagnetic Storm or military EMP such that nuclear EMP and can generated an electromagnetic wave, depending of the source. It is this electromagnetic wave that we study. EMP interference is generally damaging to electronic equipment. A lightning strike can damage physical objects such as aircraft structures, either through heating effects or the disruptive effects of the very large magnetic field generated by the current. Structures and systems require some form of protection against lightning. On average, every commercial aircraft is struck by lightning once a year. Lightning protection for aircraft is a major concern for aircraft manufacturers. With the increasing of use of composite materials, up to 53% for the latest Airbus A350, and 50% the Boeing B787, aircrafts offer increased vulnerability facing lightning. Earlier generation aircrafts, whose fuselages were predominantly constructed from aluminum, behave like a Faraday cage and offer maximum protection for the internal equipment. Currently, composite materials consisting of a matrix enclosing fibers have significant advantages in terms of weight gain and therefore fuel saving. Yet they are much less electrical conductors with loss of the Faraday effect: 100 to 1000 times lower than aluminum. Modern aircrafts have seen also the increasing reliance on electronic avionics systems instead of mechanical controls and electromechanical instrumentation. For these reasons, aircraft manufacturers are very sensitive to lightning protection and pay special attention to aircraft certification through testing and analysis. There are two types of lightning strikes to aircraft: the first one is the interception by the aircraft of a lightning leader. The second one, which makes about 90% of the cases, is when the aircraft initiates the lightning discharge by emitting two leaders as it is found in the intense electric field region produced by a thundercloud, our approach applies this case. When the aircraft flies through a cloud region where the atmospheric electric field is large enough, an ionized channel, called a positive leader, merges from the aircraft in the direction of the ambient electric field. Laroche et al at an altitude of 6000m, observed an atmospheric electric field close to 50 kv/m inside the storm clouds, 100kV/m to the ground. When upward leader connects with the downward negative leader of the cloud, a return stroke is produced and a bright return stroked wave travels from aircraft to cloud. The lightning return strokes radiate powerful 2

3 electromagnetic fields which may cause damage to aircraft electronic equipment. Our work is devoted to the study of the electromagnetic waves propagation in the air and in the composite material. In this artificial periodic material, the electromagnetic field verifies the Maxwell equations. In this paper we evaluate the electromagnetic field within and near a periodic structure when the period of this microstructure is small compared to the wavelength of the electromagnetic wave. Our model is composed by air above the composite fuselage and we study the evolution and the behavior of the electromagnetic wave in the domain filled by the composite material representing the skin aircraft and the air. We build the 3D model, under simplifying assumptions, using linear time-harmonic Maxwell equations and constitutive relations for electric and magnetic fields. Composite materials consist of conducting carbon fibers, distributed as periodic inclusions in a matrix, which can be in epoxy. The matrix may be doped or not. By hypothesis, to simplify our model, we impose in this paper that the composite is characterized by the magnetic permeability µ 0 uniform and the electrical permittivity ɛ = ɛ 0 ɛ r, where ɛ r is the relative permittivity depending of the medium. In the future, we will enrich this model by adding complexity, we will consider tensors magnetic permeability and electrical permittivity. Now, we focus on the boundary conditions as we consider them as the source, then we take on the upper frontier, the magnetic field induced by the peak of the current of the first return stroke H d = I with I = 200 ka, this is the worst aggression that 2πr can suffer an aircraft, and we deduce a characteristic electric field E = 300 kv/m. Generally, the boundary conditions used are that of perfectly conducting walls, but in our model we consider that we have very conductive carbon fibers but not perfectly and a matrix whose conduction depends on his doping rate. The conductivity of the air is non-linear. Air is a strong insulator [17] on the order of S.m 1 but at some threshold, the air loses its insulating nature and locally becomes suddenly conductive. The ionization phenomenon is the only cause that can make the air conductor of electricity. The ionized channel becomes very conductive. In this paper, we consider that air is in the ionized phase and, carbon fibers are conductive. This paper is motivated by the recent studies [9], [5], [3] describing the skin effect, formed in areas of the dielectric materials other conductor. It is inspired by the article by V. Shelukin and S. Terentev [14] which characterizes the two-scale homogenization of Maxwell equations, and like them, our equations in electric field and magnetic field take into account the characteristic sizes of skin thickness and wavelength around a material made of dielectric and conductive carbon fibers. The skin effect quantifies the penetration of the current flowing in a conductor. The characteristic skin length, 1 which appears in our model, δ = ω µ0, is close to the definition of theoretical 2 thickness skin δ = The thickness skin is the depth at which the surface ωµ 0. current moves to a factor of 1/e. Indeed, at a certain frequency, the skin effect 3

4 phenomenon appears because the current tends to concentrate at the periphery of the conductor. On the contrary, at low frequencies the penetration depth is much greater than the thickness of the plate which means that a part of the electric field penetrates the composite plate. Our mathematical context is periodic homogenization. We consider a microscopic scale ε, which represents the ratio between the diameter of then fiber and thickness of the composite material. So, we are interested by the understanding of how the microscopic structure affects the macroscopic electromagnetic field behavior radiated by the return stroke. First, we use the formal two-scale asymptotic expansion method. Secondly, we rigorously justify the homogenization problem by using the theory of two-scale convergence introduced by G. Nguetseng [12] and developed by G. Allaire [2]. We show that Harmonic Maxwell equations in a heterogeneous environment has a unique limit 2 scales when the small parameter ɛ 0. Many authors studied the homogenization of time-harmonic Maxwell equations in heterogeneous environments. N. Wellander became interested in the homogenization of linear Maxwell equations in [15] and at fixed frequencies in [16] but without taking into account the skin effect. Our homogenate model takes into account the induced effects such as skin effect for consistent waves and a high conductivity fibers. The paper is organized as follows : in Section 2 we specify the geometry of the model, after dimensionless equations converting the problem into an equivalent with which we work in the following sections. In Section 3, we make the mathematical analysis of the models en particular we introduce the weak formulation of the problem of the electric field. We establish the existence and uniqueness result for the solution to this weak formulation, thanks to the regularized Maxwell equation and Lax-Milgram theorem. We conclude this section by estimate of the electric and magnetic fields. In Section 4, we give the definitions and results that we needed for the two-scale convergence in Section 5. The last Section is devoted to the homogenization of the problems for electric and magnetic fields : We give in this section the results we get by a formal asymptotic expansion and convergence results we achieve by two scales convergence method. 2 Modeling We now build the mathematical model we will study in this paper. In a first place, we consider a problem that seems relevant with the perspective of propagation of the electromagnetic field in the air and in the skin of aircraft fuselage made of composite material. Secondly, we make a scaling of this model and finally we operate simplifications. 4

5 2.1 Notations and setting of the problem We consider set = {( x, ỹ, z) R 3, ỹ ( L, d)} for L and d two positive constants, with two open subsets a and m (see fig 1, left). The air fills a and we consider that the composite material, with two materials periodically distributed, stands in domain m. We assume that the thickness L of the composite material is much smaller than its horizontal size. We denote by e the lateral size of the basic cell Ỹ e of the periodic microstructure of the material. The cell is composed of a fiber in a matrix. We define now more precisely the material, introducing: P = {( x, ỹ, z) R 3 / L < ỹ < 0}, (1) which is the domain containing the material. Now we describe precisely the basic cell. For this we first introduce the following cylinder with square base: Then considering α, 0 < α < 1, and R e = α e. We set 2 e = [ e 2, e ] [ e, 0] R, (2) 2 D e = {( x, ỹ) R 2 /( x 2 + (ỹ + e 2 )2 ) < ( R e ) 2 }. (3) We define the cylinder containing the fiber as (see fig 1): Then the part of the basic cell containing the matrix is and by definition, the basic cell Ỹ e is the couple C e = D e R. (4) Ỹ e R = e \ C e, (5) (Ỹ e R, C e ). (6) The composite material results from a periodic extension of the basic cell. More precisely the part of the material that contains the fibers is c = P {(ie, je, 0) + C e, i, j }, (7) where the intersection with P limits the periodic extension to the area where stands the material. Set {(ie, je, 0) + C e, i, j } is a short notation for {( x, ỹ, z) R 3, i, j, (x b, y b, z b ) C e ; x = x b + ie, ỹ = y b + je, z = z b }. (8) 5

6 In the same way the part of the material that contains the matrix is or equivalently r = P {(ie, je, 0) + Ỹ e R}, (9) r = P {(ie, je, 0) + e \ C e } = (R ( L, 0) R)\ c. (10) So the geometrical model of our composite material is couple ( c, r ). Now, it remains to set the domain that contains the air: a = {( x, ỹ, z)/0 ỹ < d}. (11) We consider that d is of the same order as L and we introduce the upper frontier Γ d = {( x, ỹ, z)/ỹ = d} of domain. On this frontier we will consider that the electric field and magnetic field are given. We also introduce the lower frontier Γ L = {( x, ỹ, z)/ỹ = L} and for any ( x, ỹ, z) = Γ d Γ L and, we write ñ, the unit vector, orthogonal to and pointing outside. We have : ñ = e 3 on Γ d and ñ = e 3 on Γ L. In the following we need to describe what happen at the interfaces between resin and carbon, and resin and air. So we introduced Γ ra = {( x, ỹ, z) / ỹ = 0} and Γ cr on the boundary of the set defined by (8). 2.2 Maxwell equations In, we now write a PDE model that has to do with electromagnetic waves radiated from return stroke. We are well aware that the model we write is a simplified one. Nonetheless, it seems to be well dimensioned for our problem which consists in making homogenization. It is well known (see Maxwell [10]) the propagation of the electromagnetic field is described by the Maxwell equations which write: D + t = J, (12) B + t = 0, (13) D = ρ, (14) B = 0, (15) in R In (12)-(15), and are the curl and divergence operators. Ẽ (t, x, y, z) is the electric field, H (t, x, y, z) the magnetic field, D (t, x, y, z) the electric induction and B (t, x, y, z) the magnetic induction. (see T. Abboud and I. Terrasse [1]). We 6

7 also introduce the charge density ρ and the current density J. They are linked by the charge conservation equation: ρ t + J = 0, (16) which is obtained taking the divergence of (12) and using (14). System of Maxwell equations (12), (13), (14) and (15) is completed by the constitutive laws which are given in R by : D = ɛ 0 ɛ rẽ, (17) B = µ 0 H. (18) where µ 0 and ɛ 0 are the permeability and permittivity of free space. ɛ r is the relative permittivity of the domains defined by ɛ a = 1, ɛ r = ɛ r, ɛ c = ɛ c, (19) where ɛ r and ɛ c are positives constants. In order to account for energy transfer between the electromagnetic compartment and the propagation of the electric charges, we take for granted the Ohmic law, in R J = Ẽ, (20) where is the electric conductivity. Its value depends on the location: where a, r and c were defined in (11), (9) and (7). a = a, r = r, c = c, (21) 2.3 Boundary conditions For mathematical as well as physical reasons we have to set boundary conditions on Γ d and Γ L. On Γ d we will write conditions that translate that Ẽ and H are given by the source located in ỹ = d. The way we chose consists in setting: Ẽ ñ = Ẽ d ñ; H ñ = H d ñ on R Γ d, (22) where Ẽ d, H d are functions defined on Γ d for any t R. On Γ L, we chose something simple, i.e : Ẽ ñ = H ñ = 0 on R Γ L, (23) that translate that Ẽ and H do not vary in the ỹ-direction near Γ L. Problem (12)-(20) provided with (22) and (23), is considered as the reference model in the sense that it contains all physics we want to account for. In the following we will consider simplifications of this reference model. 7

8 Figure 1: Left: The global microstructure in 2D. Right: Y ɛ -cell of the periodic structure. 8

9 2.4 Time-harmonic Maxwell equations The first simplification we make, consists in considering the harmonic version of the Maxwell equations (12)-(18). This simplification is used in many references studying electromagnetic phenomena and especially for lightning applications [7], in spite of the fact that it considers implicitly that every fields and currents are waves of the form : a( x, ỹ, z) cos( ωt + φ( x, ỹ, z)) = Re[a( x, ỹ, z) exp i ωt exp iφ( x,ỹ, z) ]. (24) In (24), ω stands for the pulsation, φ( x, ỹ, z) is the phase shift of the wave and a( x, ỹ, z) is its amplitude. In particular, it supposes Ẽ d, H d, ρ d in (22) are of the form, for all w R + : Ẽd(t, x, z) = Re(Ẽd( x, z) exp i ωt ), (25) H d(t, x, z) = Re( H d ( x, z) exp i ωt ), (26) ρ d (t, x, z) = Re( ρ d( x, z) exp i ωt ), (27) where Ẽd and H d take into account the amplitude and the phase shift of their corresponding fields. Taking (20) into account, the time-harmonic Maxwell equations, which describe the electromagnetic radiation, are written: H i ωɛ 0 ɛ rẽ = Ẽ, Maxwell - Ampere equation (28) Ẽ + i ωµ H 0 = 0, Maxwell - Faraday equation (29) Ẽ = ρ ɛ 0 ɛ, (30) H = 0, (31) where Ẽ (t, x, ỹ, z) = Re(Ẽ( x, ỹ, z) expi ωt ) and H (t, x, ỹ, z) = Re( H( x, ỹ, z) exp i ωt ), t R, ( x, ỹ, z). We also write continuity equation (16) in this framework. For this, we define ρ (t, x, ỹ, z) such that ρ (t, x, ỹ, z) = Re( ρ( x, ỹ, z) exp i ωt ) and we have that ρ is solution to: i ω ρ + (Ẽ) = 0. (32) Notice that equation (32) is redundant with (30) except when ω = 0. The magnetic field H can be directly computed from the electric field Ẽ H = 1 Ẽ. (33) iωµ 0 9

10 Now, for the electric approach, taking the curl of equation (28) yields an expression of H in term of Ẽ. Replacing H by this expression in (29) we get the following equation for the electric field: Ẽ + ( ω2 µ 0 ɛ 0 ɛ r + i ωµ 0 )Ẽ = 0 in R. (34) Taking the divergence of the equation (28) yields the natural gauge condition: [(i ωɛ 0 ɛ r + )Ẽ] = 0 in R. (35) Notice that i ωɛ 0 + is equal to i ωɛ 0 + a in a, to i ωɛ 0 ɛ r + r in r and to i ωɛ 0 ɛ c + c in c, those quantities being all nonzero. Then (35) is equivalent to: Ẽ a = 0 in a, Ẽ r = 0 in r, Ẽ c = 0 in c. (36) with the transmission conditions (i ωɛ 0 + a )Ẽ a.ñ = (i ωɛ 0 ɛ r + r )Ẽ r.ñ on (i ωɛ 0 ɛ r + r )Ẽ r.ñ = (i ωɛ 0 ɛ c + c )Ẽ c.ñ on Γ (37) cr Summarizing, we finally obtain the PDE model : Ẽ + ( ω2 µ 0 ɛ 0 ɛ r + i ωµ 0 )Ẽ = 0 in R. (38) According to the tangential trace of the Maxwell-Faraday equation (29) we obviously obtain that using boundary condition (22), is equivalent to using : Ẽ ñ( x, ỹ, z) = i ωµ 0 H d ( x, z) ñ( x, ỹ, z) on Γ d R (39) where H d is defined in (25). With also the following boundary condition : Ẽ ñ( x, ỹ, z) = 0 on Γ L R, (40) where Ẽd is defined by (25). The magnetic field can be computed from Maxwell- Ampere equation by Γ ra H = 1 Ẽ. (41) i ωµ 0 Similarly, replacing E by the expression in (28) we get the equation of the magnetic field: H + ( ω 2 µ 0 ɛ 0 ɛ r + i ωµ 0 ) H = 0 in R, (42) 10

11 with H = 0, (43) According to the tangential trace of the Maxwell-Ampere equation (28) we obviously obtain that using boundary condition (22), is equivalent to using : H ñ( x, ỹ, z) = (i ωɛ 0 + )Ẽd( x, z) ñ( x, ỹ, z) on Γ d R. (44) and the boundary condition: H ñ( x, ỹ, z) = 0 on Γ L R. (45) And the electric field can be computed from the Maxwell-Faraday equation : Ẽ = 1 i ωε 0 + H. (46) Conclusion: the collections of equations ((38) - (46)) is our reference model. 2.5 Scaling In this subsection we propose a rescaling of system (38)-(40), we will consider a set of characteristic sizes related to our problem. Physical factors are then rewritten using those values leading a new set of dimensionless and unitless variables and fields in which the system is rewritten. The considered characteristic sizes are : ω the characteristic pulsation, the characteristic electric conductivity, E the characteristic electric magnitude, H the characteristic magnetic magnitude. We also use the already introduced thickness L of the plate P. We then introduce the dimensionless variables: x = (x, y, z) with x = x, y = ỹ, z = z and fields E, H and that are such that L L L E(x, ω) = 1 Ẽ(Lx, Ly, Lz, ωω), E H(x, ω) = 1 H(Lx, Ly, Lz, ωω), (47) H (x) = 1 (Lx, Ly, Lz), Taking (21) into account, also reads: (x) = a (x) = r (x) = c if 0 Ly d, (48) if (Lx, Ly, Lz) r, (49) if (Lx, Ly, Lz) c. (50) 11

12 Doing this gives the status of units to the characteristic sizes. Since, for instance: E x (x, ω) = L Ẽ E x (Lx, Ly, Lz, ωω), (51) using those dimensionless variables and fields and taking (48)-(50) into account, the first equation of (38) gives: E E(x, ω) ( L 2 ω 2 c 2 ɛ rω 2 + i ω ωl 2 µ 0 (x, ω) ) EE(x, y, z, ω) = 0, (52) for any (x, ω) such that (Lx, Ly, Lz, ωω) R. Now we exhibit λ = 2πc ω, (53) which is the characteristic wave length and 1 δ =, (54) ω µ0 which is the characteristic skin thickness. Using those quantities equation (52) reads, for any (x, ω) R 4 : E(x, ω) + ( 4π2 L 2 λ 2 ω 2 + i L 2 a δ 2 ω)e(x, ω) = 0 when 0 Ly d, E(x, ω) + ( 4π2 L 2 λ 2 ɛ r ω 2 + i L 2 r δ 2 ω)e(x, ω) = 0 when (Lx, Ly, Lz) r, E(x, ω) + ( 4π2 L 2 λ 2 ɛ c ω 2 + i L 2 c δ 2 ω)e(x, ω) = 0 when (Lx, Ly, Lz) c. The the boundary conditions are written L E(x, ω) n(x) = iωωµ 0 H d (Lx, Lz) n(x) (Lx, Ly, Lz) Γ d R, E E(x, ω) n(x) = 0 (Lx, Ly, Lz) Γ L R, where n(x) = ñ(lx, Ly, Lz). In the same way we have for the magnetic field: H(x, ω) + ( 4π2 L 2 λ 2 ω 2 + i L 2 a δ 2 ω)h(x, ω) = 0 when 0 Ly d, H(x, ω) + ( 4π2 L 2 λ 2 ɛ r ω 2 + i L 2 r δ 2 ω)h(x, ω) = 0 when (Lx, Ly, Lz) r, H(x, ω) + ( 4π2 L 2 λ 2 ɛ c ω 2 + i L 2 c δ 2 ω)h(x, ω) = 0 when (Lx, Ly, Lz) c. (55) (56) (57) 12

13 And the boundary conditions are written H(x, ω) n(x) = (iωωε 0 + a ) L Ẽ d (Lx, Lz) n(x) (Lx, Ly, Lz) Γ d R, H H(x, ω) n(x) = 0 (Lx, Ly, Lz) Γ L R. (58) The characteristic thickness of the plate L is about 10 3 m and the size of the basic cell e is about 10 5 m. Since e is much smaller than the thickness of the plate L, it is pertinent to assume the ratio e equals a small parameter ε: L e L 10 2 = ε. (59) Caution, do not confuse the small parameter with the permittivity. Then, in what concerns the characteristic pulsation ω, in the tables we consider and calcule for several values. The lightning is seen as a low frequency phenomenon. Indeed, energy associated with radiation tracers and return stroke are mainly burn by low and very low frequencies (from 1kHz to 300kHz). Components of the frequency spectrum are however observed beyond 1GHz (see [8]). So, in the case when we want to catch low frequency ie we consider ω = 100 rad/s. In our study we will consider ω = 10 6 rad/s and for high frequency phenomena we set ω = rad/s and ω = rad/s. Then, concerning the characteristic electric conductivity it seems to be reasonable to take for the value of the effective electric conductivity of the composite material. Yet this choice implies compute a coarse estimate of this effective conductivity at this level. For this we take into account that the composite material is made of carbon fibers and resin. The resin is doped which increases strongly its conductivity or not. In the tables bellow that summarize every situation we consider, cases 1, 3, 5 and 7 are cases when the resin is doped. Cases 2, 4, 6 and 8 are when the resin is not doped. We also account for fact there is not one effective electric conductivity but a first one in the fiber direction which is taken for in cases 1, 2, 5 and 6 with worth the arithmetic average, taking into account the volume proportion of carbon and resin, of the two conductivities: A( c, r ) = π α2 4 c + (1 π α2 4 ) r, (60) and a second effective electric conductivity, in the direction orthogonal to the fibers considered in cases 3, 4, 7 and 8 with worth, the harmonic average of the two conductivities: H( c, r ) = π α2 4 c (1 π α2 4 ) r. (61)

14 The characteristic values are computed for the four frequencies of waves. For the computation, we take values close to reality. We consider composite materials with close proportions of carbon and resin, this means that α is close to 1. When the 2 resin is not doped r S.m 1 is much smaller than c 40000S.m 1. Then, A( c, r ) is close to π α2 4 c c and H( c, r ) is close to r r. (1 π α2 4 ) Now we express the electric conductivity of the air in terms of, we consider two possibilities. The first one is relevant for a situation with a ionized channel. The second one of situation with a strong atmospheric electric field but without a ionized channel. In this situation air is not ionized and has a low conductivity. Cases 5 to 8 are associated with the first situation with air conductivity a being lightning = 4242S.m 1 for an ionized lightning channel see [6]. Cases 1 to 4 to the second one with a = S.m 1. Now we detail the cases of tables bellow. - The case 1 corresponds to the air not ionized, a resin not doped and the effective electric conductivity in the direction of the carbon fibers. According to (60) we have for the effective electric conductivity = c 40000S.m 1, the resin conductivity is about r S.m 1 and the conductivity in the air is about S.m 1. So when we want to calculate the ratio in (48)-(50) depending on ε we get: r ε7 and a ε 9. - In case 2, the air is not ionized, the resin is doped and the effective conductivity is in direction of carbon fibers. According to (60) we have like the case 1 = c 40000S.m 1. The resin conductivity is about r 10 3 S.m 1 and the conductivity in the air is about S.m 1. So r ε4 and a ε9. - In case 3, the air is not ionized, the resin is not doped and the effective conductivity is orthogonal to the fibers. According to (61), = r S.m 1. The carbon fiber conductivity is about c 10 4 S.m 1 and the conductivity in the air is about S.m 1. c 1 and a ε 7 ε2. - Case 4 corresponds to the air non ionized, the resin doped and the effective conductivity orthogonal to the fibers. According to (61) the effective electric conductivity is = r 10 3 S.m 1. The carbon fiber conductivity is about c 40000S.m 1 and the conductivity in the air is about S.m 1. c 1 and a ε 4 ε6. - In case 5, the air is ionized, the resin is not doped and the effective conductivity is in the direction of the carbon fibers. This one is equal = c 40000S.m 1, the resin conductivity is about r S.m 1 and the conductivity in the air is now about 4242S.m 1. r ε7 and a ε. - Case 6 corresponds to the air ionized, the resin doped and the effective conductivity in direction of the carbon fibers. This one is equal = c 40000S.m 1, the resin conductivity is about r 10 3 S.m 1 and the conductivity in the air is now about 4242S.m 1. r ε4 and a ε. - Case 7 corresponds to the air ionized, the resin not doped and the effective conductivity orthogonal to the fibers. The effective conductivity is = r S.m 1, the carbon fibers conductivity is about c 40000S.m 1 and the conductivity in the 14

15 case 1 case 2 case 3 case 4 case 5 case 6 case 7 case 8 L(m) e(m) λ(m) (S.m 1 ) δ(m) 0, 1 0, , 1 0, c (S.m 1 ) ε 7 ε 4 ε 7 r (S.m 1 ) ε 7 ε 4 ε 7 ε 4 a (S.m 1 ) ε 9 ε 9 ε 2 ε 6 ε ε ε 5 4πL 2 λ 2 ε 9 ε 9 ε 9 ε 9 ε 9 ε 9 ε 9 ε 9 L 2 δ 2 ε 2 ε 2 ε 10 ε 7 ε 2 ε 2 ε 10 ε 7 ε 4 ε 2 Figure 2: Tableau 1: for ω = 100rad.s 1. air is now about 4242S.m 1. c 1 and a 1. ε 7 ε 6 - Case 8 corresponds to the air ionized, the resin doped and the effective conductivity orthogonal to the fibers. The effective conductivity is according to (61) = r 10 3 S.m 1, the carbon fibers conductivity is about c 40000S.m 1 and the conductivity in the air is now about 4242S.m 1. c 1 and a 1. ε 4 ε 2 As it is well known and as the scale analyse we just led seems to confirm at high frequencies the thickness of the plate is very much greater than the skin depth δ. This one depends on and ω and decreases strongly for high conductivity or high frequencies. For ω = rad.s 1 and = 10 4, an effective conductivity in the direction of the carbon fibers the skin effect phenomenon appears. Indeed, in the high frequencies ω = rad.s 1 and when the effective conductivity is in direction of the carbon fibers i.e. in high conductivity δ = 10 4 S.m 1. In such a case, a well-known phenomenon called skin effect appears and creates a thin layer. In low frequencies and low conductivity δ is important so the electromagnetic wave can penetrate the composite material. The high conductivity limits the penetration of the electromagnetic wave to a boundary layer whose depth is equals to δ. Now, we will discuss on the values of E, H and ρ that are involved in equations (30) and (32). It seems that the density of electrons in a ionized channel is about part.m 3. Hence we take ρ = When the air is not ionized, the charge density is much smaller, and we choose: ρ = 1. We emphasize the following points: we choose for the boundary conditions, in the context of the case 6 ω = 10 6 rad/s, we take the peak of the current of the first return stroke then the magnetic field H d = I with I = 200 ka, and r = 2πr 10 4 m, which 15

16 case 1 case 2 case 3 case 4 case 5 case 6 case 7 case 8 L(m) e(m) λ(m) (S.m 1 ) δ(m) c (S.m 1 ) ε 7 ε 5 ε 7 r (S.m 1 ) ε 7 ε 4 ε 7 ε 4 a (S.m 1 ) ε 9 ε 9 ε 2 ε 6 ε ε ε 5 4πL 2 λ 2 ε 5 ε 5 ε 5 ε 5 ε 5 ε 5 ε 5 ε 5 L 2 δ ε 8 ε ε 8 ε 5 ε 5 ε 2 Figure 3: Tableau 2: for ω = 10 6 rad.s 1. case 1 case 2 case 3 case 4 case 5 case 6 case 7 case 8 L(m) e(m) λ(m) (S.m 1 ) δ(m) / /2 c (S.m 1 ) ε 7 ε 4 ε 7 r (S.m 1 ) ε 7 ε 4 ε 7 ε 4 a (S.m 1 ) ε 9 ε 9 ε 2 ε 6 ε ε ε 5 4πL 2 ε ε ε ε ε ε ε ε λ 2 L ε 6 ε ɛ 6 ɛ 3 δ 2 ε 2 ε 2 ε 2 ε 2 ε 4 ε 2 Figure 4: Tableau 3: for ω = rad.s 1. 16

17 case 1 case 2 case 3 case 4 case 5 case 6 case 7 case 8 L(m) e(m) λ(m) (S.m 1 ) δ(m) / /2 c (S.m 1 ) ε 7 ε 4 ε 7 r (S.m 1 ) ε 7 ε 4 ε 7 ε 4 a (S.m 1 ) ε 9 ε 9 ε 2 ε 6 ε ε ε 5 4πL λ 2 L ε ε ɛ 3 ε δ 2 ε 3 ε 3 ε 3 ε 3 ε 4 ε 2 Figure 5: Tableau 3: for ω = rad.s 1. results concerning the dimensionless boundary condition (39) at the surface Γ d : E(x, ω) n(x) = iω H d (Lx, Lz) n(x), (62) L where H d = ωµ 0 H E d being of order 1, with the characteristic electric field E = 300 kv/m. In the same way, considering the boundary conditions of the magnetic field problem, Ẽd = cb = 1 cε 0 H d on the surface Γ d gives: where E d = a H(x, ω) n = ( iωε 3 + 1) E d (Lx, Lz) n(x). (63) L Ẽ H d being of order 1, with H = H d. In the following we will focus on electromagnetic field behavior that is fundamental. From the physical spatial coordinates ( x, ỹ, z) we define y = (ξ, ν, ζ) with ξ = x, ν = ỹ, ζ = z or equivalently ξ = x, ν = y, ζ = z. And we now introduce Y the e e e ε ε ε basic cell. It is built from: = [ 1, 1 ] [ 1, 0] R and the set C = D R with the 2 2 disc D defined by: and R = α 2. The set c is then defined as: D = {(ξ, ν) R 2 /ξ 2 + (ν )2 < R 2 }, (64) c = {(i, j, 0) + C, i, j }. (65) 17

18 We denote Y r as Y r = \C and then the set r = {(i, j, 0) + Y r, i, j }. (66) Then unit cell Y is defined as Y = (Y r, C). Finally, we define the domain a : a = {y = (ξ, ν, ζ) / ν > 0}. (67) Using this, we will give a new expression of the sets in which the variables range in equations (55). We see the following: { (Lx, Ly, Lz) (Lx, Ly, Lz) r P, ( Lx, Ly, Lz) (68) e e e r, i.e. (Lx, Ly, Lz) r { (Lx, Ly, Lz) P, ( x ε, y ε, z ε ) r. (69) In the same way: (Lx, Ly, Lz) c { (Lx, Ly, Lz) P, ( x ε, y ε, z ε ) c, (70) and: { y R 2 0 Ly d Ly d, (71) or (Lx, Ly, Lz) a { Ly d ( x ε, y ε, z ε ) a. (72) We define: Σ ε Σ ε (y) = Σ ε a in a, (ξ, ν, ζ) = Σ ε r in r, Σ ε c in c, (73) 18

19 where Σ ε a = a L 2, Σ ε δ 2 r = r L 2 and Σ ε δ 2 c = c L 2 have their expression in term of ε δ 2 given from Table 1, 2 or 3 depending on the case we are interested in. Defining also mapping ψ ε : R 3 R 3 (x, y, z) ( x ε, y ε, z ε ), (74) we can set ε a as ψε 1 ( a ) R [0, d ] R, L ε r as ψε 1 ( r ) P and ε c as ψε 1 ( c ) P. We also define the boundaries Γ d = {x R 3, y = d } and Γ L L = {x R 3, y = L} and interfaces Γ ra = {x R 3, y = 0} and Γ ε cr = c. Hence equation (55) reads: E ε + ( ω 2 ε η ɛ r + i ω ε (x, y, z))e ε = 0 in, (75) and (57) for the magnetic field reads: H ε + ( ω 2 ε η ɛ r + i ω ε (x, y, z))h ε = 0 in, (76) where = ε a ε r ε c = {x R 3, 1 < y < d } does not depend on ε. Only its L partition in ε a, ε r and ε c is ε-dependant where and ε (x, y, z) = Σ ε ( x ε, y ε, z ), (77) ε ε η = 4π2 L 2 λ 2, (78) with the value of η 0 extracted from table 1, 2, 3 or 4, and where we place E, H by E ε, H ε, to clearly state that they depend ε. Equation (75) can also read: E ε + ( ω 2 ε η + i ω Σ ε a)e ε = 0 in ε a, E ε + ( ω 2 ε η ɛ r + i ω Σ ε r)e ε = 0 in ε r, E ε + ( ω 2 ε η ɛ c + i ω Σ ε c)e ε = 0 in ε c. (79) E ε n ε a ε = a Eε n ε r ε on Γ r ra, E ε n ε r ε = r Eε n ε c ε on Γ ε c cr, where n ε a is the unit outward normal to ε a, n ε r is the unit outward normal to ε r and n ε c is the unit outward normal to ε c. Those equations are provided with the following boundary conditions: E ε n ε = iωh d ( x ε, z ε ) nε on R Γ d, (80) 19

20 E ε n ε = 0 on R Γ L, (81) where n ε = n( x, y, z ). From (75) we can deduce a use-full condition on the divergence ε ε ε of E ε which can be written in two ways. As previously in (35), (36) and (37) we obtain : [( ω 2 ε η ɛ r + iω ε )E ε ] = 0 in, (82) which will be preferentially used with (75) and its second one is E ε ε a = 0 in ε a, E ε ε r = 0 in ε r, E ε ε c = 0 in ε c, (83) with the transmission conditions on the interfaces Γ ra and Γ ε cr ( ω 2 ε η + iωσ ε a)e ε ε a n ε a = ( ω2 ε η ɛ r + iωσ ε r)e ε ε r n ε r on Γ ra, ( ω 2 ε η ɛ r + iωσ ε r)e ε ε r n ε r = ( ω2 ε η ɛ c + iωσ ε c)e ε ε c n ε c on Γ ε cr. (84) The expressions of Σ ε a, Σ ε r and Σ ε c are given in the various cases bellow. Concerning the magnetic field, equation (76) reads: H ε + ( ω 2 ε η + i ω Σ ε a)h ε = 0 in ε a, H ε + ( ω 2 ε η ɛ r + i ω Σ ε r)h ε = 0 in ε r, H ε + ( ω 2 ε η ɛ c + i ω Σ ε c)h ε = 0 in ε c. (85) H ε n = ε Hε a n on Γ ra, ε r H ε n = ε Hε r n on ε Γε c cr, with the boundary conditions: H ε n ε = (iωε 3 + 1)E d ( x ε, z ε ) nε on R Γ d, (86) H ε n ε = 0 on R Γ L. (87) The strong formulation of the magnetic filed problem in the sub-domains ε a, ε r and ε c is similar to the strong electric field formulation. However, the transmission conditions of the magnetic field H at the interface are different from those of the electric field E. { H ε ε (88) a n = Hε ε r n on Γ ra, H ε ε r n = Hε ε c n on Γε cr. Moreover, H ε = 0 in. (89) 20

21 For ω = 100rad.s 1, we have Case 1 and case 3 η = 9 and Σ ε a = ε 12, Σ ε r = ε 9, Σ ε c = ε 3. (90) Case 2 η = 9 and Σ ε a = ε 12, Σ ε r = ε 7, Σ ε c = ε 3. (91) Case 4 η = 9 and Σ ε a = ε 13, Σ ε r = ε 7, Σ ε c = ε 3. (92) Case 5 η = 9 and Σ ε a = ε 4, Σ ε r = ε 10, Σ ε c = ε 3. (93) Case 6 and 8 η = 9 and Σ ε a = ε 4, Σ ε r = ε 7, Σ ε c = ε 3. (94) Case 7 η = 9 and Σ ε a = ε 5, Σ ε r = ε 10, Σ ε c = ε 3. (95) Case 8 η = 9 and Σ ε a = ε 5, Σ ε r = ε 7, Σ ε c = ε 3. (96) For ω = 10 6 rad.s 1 Case 1 η = 5 and Σ ε a = ε 9, Σ ε r = ε 7, Σ ε c = 1. (97) Case 2 η = 5 and Σ ε a = ε 9, Σ ε r = ε 4, Σ ε c = 1. (98) 21

22 Case 3 η = 5 and Σ ε a = ε 10, Σ ε r = ε 8, Σ ε c = ε. (99) Case 4 η = 5 and Σ ε a = ε 11, Σ ε r = ε 5, Σ ε c = 1. (100) Case 5 η = 5 and Σ ε a = ε, Σ ε r = ε 7, Σ ε c = 1. (101) Case 6 η = 5 and Σ ε a = ε, Σ ε r = ε 4, Σ ε c = 1. (102) Case 7 η = 5 and Σ ε a = ε 3, Σ ε r = ε 8, Σ ε c = ε. (103) Case 8 η = 5 and Σ ε a = ε 3, Σ ε r = ε 5, Σ ε c = 1. (104) For ω = rad.s 1 Case 1 and case 3 η = 1 and Σ ε a = ε 8, Σ ε r = ε 6, Σ ε c = 1 ε. (105) Case 2 η = 1 and Σ ε a = ε 8, Σ ε r = ε 3, Σ ε c = 1 ε. (106) Case 4 η = 1 and Σ ε a = ε 9, Σ ε r = ε 3, Σ ε c = 1 ε. (107) Case 5 η = 1 and Σ ε a = 1, Σ ε r = ε 6, Σ ε c = 1 ε. (108) 22

23 Case 6 η = 1 and Σ ε a = 1, Σ ε r = ε 5, Σ ε c = 1 ε. (109) Case 7 η = 1 and Σ ε a = ε, Σ ε r = ε 6, Σ ε c = 1 ε. (110) Case 8 η = 1 and Σ ε a = ε, Σ ε r = ε 5, Σ ε c = 1 ε. (111) For ω = rad.s 1 Case 1 η = 0 and Σ ε a = ε 7, Σ ε r = ε 5, Σ ε c = 1 ε 2. (112) Case 2 η = 0 and Σ ε a = ε 7, Σ ε r = ε 2, Σ ε c = 1 ε 2. (113) Case 3 η = 0 and Σ ε a = ε 5, Σ ε r = ε 3, Σ ε c = 1 ε 4. (114) Case 4 η = 0 and Σ ε a = ε 7, Σ ε r = ε, Σ ε c = 1 ε 3. (115) Case 5 η = 0 and Σ ε a = 1 ε, Σε r = ε 5, Σ ε c = 1 ε 2. (116) Case 6 η = 0 and Σ ε a = 1 ε, Σε r = ε 2, Σ ε c = 1 ε 2. (117) Case 7 η = 0 and Σ ε a = 1 ε 2, Σ ε r = ε 3, Σ ε c = 1 ε 4. (118) 23

24 Case 8 η = 0 and Σ ε a = 1 ε 2, Σ ε r = ε, Σ ε c = 1 ε 3. (119) Before treating mathematically the question we are interested in a last simplification. Since it seems clear that physical relevant phenomena occur in the upper part of the plate and that the boundary condition on the lower boundary of the plate has very little influence on the physics of what happens in the upper part, we consider that the lower boundary of is located in y = in place of y = 1, making the second boundary condition useless. Besides, as L and d are of the same order it seems reasonable to set Γ d = {x R 3, y = 1} and consequently = {x R 3, y < 1} = ε a ε r ε c with ε a = ψε 1 ( a ) ε r = ψε 1 (120) ( r ) ε c = ψε 1 ( c ) with ψ defined in (74). We have that the border of is Γ d. And, for which we will establish existence, uniqueness and asymptotic results are ((75), (80), (82)) or ((75), (80), (83), (84)) or ((79), (80), (82)) or ((79), (80), (83), (84)), that are all equivalent, for electric field E ε. Concerning magnetic field H ε the equivalent problems we consider are : ((76), (86), (88), (89)) and ((85), (86), (88), (89)). 3 Mathematical analysis of the models 3.1 Preliminaries We are going to make precise the variational formulation. First of all, we need to introduce the following functional spaces. We have the standard function spaces L 2 ( ε ) = [L 2 ( ε )] 3 with the usual norms: H(curl, ) = {u L 2 () : H(div, ) = {u L 2 () : u L 2 ()}, u L 2 ()}, (121) u 2 H(curl,) = u 2 L 2 () + u 2 L 2 (), u 2 H(div,) = u 2 L 2 () + u 2 L 2 (). (122) They are well known Hilbert spaces. H 1 2 (Γ d ) denotes the space of traces on Γ d of distributions in H 1 and H 1 2 (Γ d ) its dual space. The duality product between 24

25 H 1 2 (Γ d ) and H 1 2 (Γ d ) is denoted by.,. Γd. For a bounded Lipschitz domain, for any u H(curl, ) and V H(curl, ) u V u V dx = γ t (u), γ T (V ) Γd (123) with dx = dxdydz and where the tangential traces was defined by It is known that the trace operators γ t (u) = n u and γ T (u) = n (u n). (124) γ t : H(curl, ) H 1/2 (div, ), γ T : H(curl, ) H 1/2 (curl, ) (125) are surjective on image spaces defined by H 1 2 (div, Γd ) = {u H 1 2 (Γd, R 3 ), n u = 0, div Γd u H 1 2 (Γd )} (126) and its dual space H 1 2 (curl, Γd ) = {u H 1 2 (Γd, R 3 ), n u = 0, curl Γd u H 1 2 (Γd, R 3 )} (127) where the surface divergence div Γd u and the surface rotation curl Γd u are defined are defined by duality and restriction (div Γd u, V ) = (u, Γd V ) V C 1 (Γ d ) curl Γd u = n ( u Γd ) (128) and Γd V is the projection of on Γ d. We consider also the spaces: H 0 (curl, ) = D() H(curl,) = {u L 2 () : u L 2 () u n = 0 on }, H 0 (div, ) = D() H(div,) = {u L 2 () : u L 2 () u n = 0 on }. (129) Above, the curl and the divergence are defined in the weak sense as: ( u, V ) = (u, V ), V C 1 (), ( u, V ) = (u, V ), V C 1 (). (130) The space of fields with zero divergence is: H(div 0, ) = {u L 2 () u = 0 in }. (131) We define the next spaces: X ε N () = H(curl, ) H(div, ), (132) 25

26 and X ε () = {u H(curl, ) u ε a L 2 ( ε a), u ε r L 2 ( ε r), u ε c L 2 ( ε c), [ ω 2 ε η ɛ r + ε u ε n] Γ ε ra = 0, [ ω 2 ε η ɛ r + ε u ε n] Γ ε rc = 0} (133) where [ ω 2 ε η ɛ r + ε u ε n] Γ ε ra = [( ω 2 ε η + Σ ε au ε ) ε ( ω2 ε η ɛ a r + Σ ε ru ε )] n, ε r [ ω 2 ε η ɛ r + ε u ε n] Γ ε rc = [( ω 2 ε η ɛ r + Σ ε ru ε ) ε ( ω2 ε η ɛ r c + Σ ε cu ε )] n. (134) ε c This space is equipped with the norm u 2 X ε () = u 2 L 2 () + u ε a 2 L 2 ( ε a) + u ε r 2 L 2 ( ε r) + u ε c 2 L 2 ( ε c) + u 2 L 2 (). 3.2 Weak formulation Now, we introduce the variational formulation of problem ((75), (80)) for the electric field and we show the rigorous equivalence of (75) - (80)) and (79), (80)). As a by product of this equivalence proof, we obtain additional transmission conditions of E ε over interfaces Γ ra and Γ cr. The existence and uniqueness of the solution for this problem are established by a variational approach in space X ε (). Integrating (75) over and using the Green s formula and (80) we obtain ( E ε V dx + ( ω 2 ε η + iωσ ε a)e ε V dx ε a + ( ω 2 ε η ɛ c + iωσ ε c)e ε V dx + ( ω 2 ε η ɛ r + iωσ ε r)e ε V dx ε c = ( E ε n ε ) V t d Γ d ε r (135) = iωh d n ε V T d Γ d where V is the complex conjugate of V. We introduce the sesquilinear form depending on parameters η and ε: For E ε, V X ε (), a ε,η (E ε, V ) = ( E ε V dx + i=a,r,c ε i ω 2 ε η ɛ i + iωσ ε i E ε V dx. (136) Hence, the weak formulation of ((75), (80)) that we will use is the following: 26

27 Find E ε X ε () such as V X ε () we have : a ε,η (E ε, V ) = iω H d n ε V T d. Γ d (137) We will now prove that weak formulation (137) contains additional transmission conditions on interfaces Γ ε ra and Γ ε cr: { [ E ε n] Γra = 0, [ E ε (138) n] Γ ε cr = 0, meaning that problem (137) is equivalent to problem ((75), (80)). Moreover we will also show that (137) is equivalent ((79), (80)). This finally mean that ((75), (80)) and ((79), (80)) are equivalent problems. More precisely, we have the following proposition: Lemma 3.1. For any ε positive and non negative η, E ε X ε () is solution to ((75), (80)), if and only if E ε is solution to (137), if and only if E ε is solution to ((79), (80)) if and only if E ε is solution to ((79), (80), (138)). Proof. The first equivalence of this Lemma is obvious. Similarly as in V. Peron s thesis [13], if E ε is solution to (137) a test function in weak formulation (137) V C 1 () is taken with compact support in ε c. Since E ε V dx = ε Eε c, V ε c ε, (139) c we deduce the third equation in (79). Similarly, taking V C 1 () with compact support respectively in ε r and ε a, we obtain the first and the second equation in (79). Now, since E ε a H(curl, ε a) and E ε r H(curl, ε r), let V C 1 ( ε a ε r ) integrating by parts we get E V dx = E ε ε a V dx + E ε r V dx a ε r ε a ε r = E ε a V dx + E ε r V dx (140) ε a ε r + (E ε a n ε a + E ε r n ε r ) V ds. Γ ra Since on every point of Γ ra n ε a = n ε r the assumed continuity require E ε a n ε a = E ε r n ε r, (141) we obtain the fourth relation in (79). With the same argument on Γ ε cr, we obtain the last relation in (79). This shows that (137) implies (79). And since (80)) are obtain 27

28 from (137) as obviously as the proof of the first equivalence of the Lemma, we deduce that (137) implies ((79), (80)). As the reverse is obvious the second equivalence is obvious. Now to obtain the last equivalence it just remain to prove that if E ε X ε () is solution to ((79), (80)), (138)). And, if E ε is solution to (79) following that for any regular set in the Stockes s formula gives, for more details see p 57, 58 of P. Monk s book [11]: E, V H(curl, ) E V E V dx = E n, V T (142) H(curl, ) has the same definition as H(curl, ) with replaced by and where V T = (n V ) n, and n is the unit outward normal of. For all V H(curl, ), V ε r H(curl, ε r), V ε a H(curl, ε a) and V ε c H(curl, ε c). Hence, fixing any E H(curl, ) according to the second equation in (79), we have E ε H(curl, ε ε r r) then applying (142) in ε r with E = E ε and V we get ε r E ε V dx = E ε ε r ε V dx + ε Eε n r ε r ε ε r, V T Γra r r (143) + E ε n ε ε r, V T Γ ε cr. c Doing the same for ε c, we have E ε V dx = ε c ε c Finally for ε a, we have E ε V dx = ε a ε a ε c ε a E ε ε c V dx + Eε ε c n, V T Γ ε cr. (144) E ε ε a V dx + Eε ε a nε ε a, V T Γd. (145) Summing the relations above since in every point of Γ ra n ε r = n ε a and in every point of Γ ε cr n ε c = n ε r, it comes E ε V dx = E ε V dx+ < [ E ε n], V T > Γra + [ E ε n], V T Γcr iω H d n ε V T d. Γ d (146) According to (137) and the equations in (79) 1, (79) 2, (79) 3 we have [ E ε n], V T Γra + [ E ε n], V T Γcr = 0, (147) for all V H(curl, ) which causes the last two equalities in (138) and concludes the proof of the first part of the proposition. 28

29 From (85), integrating by parts we have: V X ε (), E ε V dx + ( ω 2 ε η + iωσ ε a)e ε V dx ε a = iω H d n ε V T d, Γ d (148) and V X ε (), and V X ε (), E ε V dx + ε r E ε V dx + ε c ( ω 2 ε η ɛ r + i ωσ ε r)e ε V dx = 0, (149) ( ω 2 ε η ɛ c + i ωσ ε c)e ε V dx = 0. (150) By adding these three integrals, we get (137). Incorporating now the condition concerning the divergence we can state the next lemma. Lemma 3.2. For any ε > 0 and any η 0, E ε X ε is solution to ((75), (80), (83), (84)) if and only if it is solution to ((79), (80), (83), (84)) if and only if it is solution to ((79), (80), (83), (84), (138)) if and only if it is solution ((137), (84), (138)). The condition on the divergence of E ε (see (82)) is more complicated as the one for H ε (see (89)). As a consequence, setting out the existence and uniqueness is more complicated for E ε that for H ε. Hence, we only focus on the details concerning E ε and for H ε we will only give the result. Concerning magnetic field H ε, we can state the following weak formulation For H ε, V X ε N (), ( H ε V dx + ε i=a,r,c = (iωε 3 + 1) E d n V T d. Γ d ε i ( ω 2 ε η ɛ i + iωσ ε i )H ε V dx (151) which coupled with divergence condition (89) is equivalent to ((85), (86), (88), (89)). This ends the proof of the lemma. 3.3 Regularized Maxwell equations for the electric field Now that proposition 2.1 is established, our goal is to prove existence and uniqueness result for the solution to weak formulation (137). As the sesquilinear form a ε,η is not coercive on X ε () due to the term ω 2 ε η ɛ i (E ε, V ), it is necessary to regularize it. So, we add regularization terms involving the divergence of E ε in ε a, ε r and ε c. 29

30 Thanks to the additional terms, existence and uniqueness of the weak solution will be established by the Lax-Milgram theory. Let s > 0 be an arbitrary real number, we define the regularized formulation of problem (137): Find E ε X ε ()such that for any V X ε () a ε,η R (Eε, V ) = ( E ε V dx + s E ε V dx ε ε a + s E ε V dx + s E ε V dx + ( ω 2 ε η ɛ i + iωσ ε i )E ε V dx ε r = iωh d n ε V T d. Γ d ε c i=a,r,c ε i (152) For any ε > 0 and ε > 0 and any η 0, sesquilinear form a ε,η R (.,.) is continuous over X ε () thanks to the jump condition its definition (130) embed. We will show that it is also coercive. Proposition 3.1. For any η 0, taking ω < min(σε a,σ ε r,σ ε c), there exists a positive ε η constant C 0 such that E ε X ε (), R[α a ε,η R (Eε, E ε )] C 0 E ε X ε () (153) Proof. Begin by evaluating R[αa ε,η R (Eε, E ε )], E ε X ε (). Taking α = exp( i π) 4 we have: R[exp( i π 4 ) aε,η R (Eε, E ε )] = ( E ε E ε dx + s E ε E ε dx ε a + s E ε E ε dx + s E ε E ε dx ε r ε c (154) + ω 2 ε η + ωσ ε ae ε E ε dx + ω 2 ε η ɛ r + ωσ ε re ε E ε dx ε a ε r + ω 2 ε η ɛ c + ωσ ε ce ε E ε dx. ε c We consider that Σ ε a L ( ε a), Σ ε r L ( ε r) and Σ ε c L ( ε c). By taking into account we obtain the following estimate: R[αa ε,η R (Eε, E ε )] E ε 2 L 2 () + Eε 2 L 2 ( ε a) + Eε 2 L 2 ( ε r) + Eε 2 L 2 ( ε c) ω 2 ε η max(1, ɛ r, ɛ c ) E ε 2 L 2 () + ω min(σε a, Σ ε r, Σ ε c) E ε 2 L 2 (). (155) Now let s check the term ω 2 ε η E ε 2 L 2 () by the other term min(σε a, Σ ε r, Σ ε c) E ε 2 L 2 (). The right-hand side in the above inequality as an equivalent norm E ε X ε (), when 30

31 ε 0, and: ω 2 ε η max(1, ɛ r, ɛ c ) + ω min(σ ε a, Σ ε r, Σ ε c) > 0, (156) which gives It can be concluded on the coercivity of the form ω < min(σε a, Σ ε r, Σ ε c) ε η max(1, ɛ r, ɛ c ). (157) R[αa ε,η R (Eε, E ε )] C 0 E ε X ε (). (158) where the positive constant C 0 depending of s and ω ending the proof of the proposition. Thanks to proposition 2.1 we can state the existence of the solution to problem (152). Lemma 3.3. For any ε > 0 and η 0 and taking ω < min(σε a,σ ε r,σ ε c) ε η max(1,ɛ r,ɛ c) unique solution E ε to the regularized problem (152). Proof. Since the sesquilinear form a ε,η R there exists a is continuous, bounded, coercive and the right hand side is continuous on X( ε ), problem (152) has a unique solution in X ε () thanks to the Lax-Milgram theorem. Now we show that the weak formulation and the regularized weak formulation are equivalent. Lemma 3.4. For any ε > 0 and η 0, and if ω < min(σε a,σε r,σε c ) ε η max(1,ɛ r,ɛ c) there exists at least a positive s such that E ε is solution to regularized formulation (152) if and only if E ε is solution to ((137), (83), (84)). Proof. In a first place it is easy to see that if E ε is solution to ((137), (83), (84)) the second, third and fourth terms of the left hand side of equality (152). Secondly, if E ε is solution to (152), we define the Laplace operator = ( ): : H 1 ( ε c) H 1 ( ε c) ψ (ψ). (159) The domain of Laplace operator is D( ) = {ψ H 1 0( ε c) L 2 ( ε c)} (160) The operator is invertible on its domain onto L 2 () and has a discrete spectrum. 31

32 Let E ε be a solution of the regularized formulation (152). First we take a test function V = ψ where ψ D( ). Thus from (152) we obtain: s E ε ( ψ) dx + ( ω 2 ε η ɛ c + iωσ ε c)e ε ψ dx = iω H d n ε ψ d. Γ d ε c ε c (161) Since iω H d n ε ψ d = Γ d E ε n ψ d = 0, because div(curl) = 0. Γ d By Green s formula, ψ D( ), we arrive at: E ε ( ϕ + ω2 ε η ɛ c iωσ ε c )ψ dx = 0 s (162) ε c Thus if we chose s such that ω2 ε η ɛ c iωσ ε c is not an eigenvalue of the operator ( ), s which is possible because of the discrete nature of its spectrum, we find that E ε = 0 in ε c. Thus E ε solves: E ε X ε (), V X ε () ( E ε V dx + ( ω 2 ε η ɛ c + iωσ ε c)e ε V dx ε c = iωh d n ε V T d. Γ d (163) Thus we get equation in (83). We use similar argument with E ε = 0 in ε r and ε a. Now we take a test function V = ψ in (152) through the domain composed by the air and the resin ε a and ε r: ( ω 2 ε η + iωσ ε a)e ε ψ dx + ( ω 2 ε η ɛ r + iωσ ε r)e ε ϕ dx ε a ε r + ( ω 2 ε η ɛ c + iωσ ε c)e ε ψ dx = iω H d n ε (164) ψ d Γ d = 0. ε c Using the Green s formula, we get [( ω 2 ε η ω 2 ε η ɛ r )E ε n]ϕ dx + Γ ra iωσ ε a(e ε n)ϕ dx + ε a Γ ra iωσ ε r(e ε n)ϕ dx = 0. ε r Γ ra (165) And we use the same argument through the composite domain composed by the resin and the carbon ε c and ε r: [( ω 2 ε η ɛ r ω 2 ε η ɛ c )E ε n]ϕ dx + Γ cr iωσ ε r(e ε n)ϕ dx + ε r Γ cr iωσ ε c(e ε n)ϕ dx = 0. ε c Γ cr 32 (166)