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Part 2: Homogenization
Helene Canot, Emmanuel Frénod
To cite this version:
Helene Canot, Emmanuel Frénod. Method of Homogenization for the Study of the Propagation of Electromagnetic Waves in a Composite Part 2: Homogenization. Proceedings of the World Congress on Engineering 2017, Jul 2017, London, United Kingdom. �hal-01572327�
Method of Homogenization for the Study of the Propagation of Electromagnetic Waves in a
Composite Part 2: Homogenization
Helene Canot, Emmanuel Frenod
Abstract—In this paper we study the two-scale behavior of the electromagnetic field in 3D in and near composite material. It is the continuation of the paper [6] in which we obtain existence and uniqueness results for the problem, we performed an estimate that allows us to approach homogenization. Technique of two-scale convergence is used to obtain the homogenized problem.
Index Terms—Harmonic Maxwell Equations; Electromag- netism; Homogenization; Asymptotic Analysis; Asymptotic Ex- pansion; Two-scale Convergence; Frequencies; Composite Ma- terial.
I. INTRODUCTION
We are interested in the time-harmonic Maxwell equations in and near a composite material with boundary conditions modeling electromagnetic field radiated by an electromag- netic pulse (EMP). In the first part, we have presented the model and proved the existence of a unique solution of the problem. Our mathematical context is periodic homogeniza- tion. We consider a microscopic scale ε, which represents the ratio between the diameter of the fiber and thickness of the composite material. So, we are trying to understand how the microscopic structure affects the macroscopic electro- magnetic field behavior. Homogenization of Maxwell equa- tions with periodically oscillating coefficients was studied in many papers. N. Wellander homogenized linear and non- linear Maxwell equations with perfect conducting boundary conditions using two-scale convergence in [16] and [17]. N.
Wellander and B. Kristensson homogenized the full time- harmonic Maxwell equation with penetrable boundary condi- tions and at fixed frequency in [18]. The homogenized time- harmonic Maxwell equation for the scattering problem was done in F. Guenneau, S. Zolla and A. Nicolet [10]. Y. Amirat and V. Shelukhin perform two-scale homogenization time- harmonic Maxwell equations for a periodical structure in [4].
They calculate the effective dielectricεand effective electric conductivity σ. They proved that homogenized Maxwell equations are different in low and high frequencies. The result obtained by two-scale convergence approach takes into account the characteristic sizes of skin thickness and wave- length around the material. We use the Asymptotic expansion and the theory of two-scale convergence introduced by G.
Nguetseng [12] and developed by G. Allaire [2].
Helene Canot and Emmanuel Frenod are with the Department of Mathe- matics of University of Bretagne Sud (LMBA), Centre Yves Coppens, Bat.
B, 1er et., Campus de Tohannic BP 573, 56017 VANNES, FRANCE e-mail:
helene.canot@univ-ubs.fr, emmanuel.frenod@univ-ubs.fr
II. HOMOGENIZATION
We recall that our problem is:
∇×∇×Eε+ (−ω2εη?+i ω σε(x, y, z))Eε= 0inΩ.
Equation (1) is provided with the following boundary conditions:
∇×Eε×e2=−iωHd(x, z)×e2 on R×Γd, (2) and
∇×Eε×e2= 0 on R×ΓL. (3) We propose an approach based on two-scale convergence.
This concept was introduced by G. Nguetseng [13] and specified by G. Allaire [3] which studied properties of the two-scale convergence. M. Neuss-Radu in [11] presented an extension of two-scale convergence method to the periodic surfaces. Many authors applied two-scale convergence ap- proach D. Cionarescu and P. Donato [8], N. Crouseilles, E. Fr´enod, S. Hirstoaga and A. Mouton [9], Y. Amirat, K.
Hamdache and A. Ziani [1] and also A. Back, E. Fr´enod [5].
This mathematical concept were applied to homogenize the time-harmonic Maxwell equations S. Ouchetto, O. Zouhdi and A. Bossavit [14], H.E. Pak[15].
In our model, the parallel carbon cylinders are periodically distributed in direction x and z, as the material is homoge- nous in the y direction, we can consider that the material is periodic with a three directional cell of periodicity. In other words, introducingZ= [−12,12]×[−1,0]2, functionΣεgiven in [7] is naturally periodic with respect to(ξ, ζ)with period [−12,12]×[−1,0] but it is also periodic with respect to y with periodZ.
Now, we review some basis definitions and results about two-scale convergence.
A. Two-scale convergence
We first define the function spaces
H(curl,Ω) ={u∈L2(Ω) : ∇×u∈L2(Ω)}, H(div,Ω) ={u∈L2(Ω) : ∇·u∈L2(Ω)}, (4) with the usual norms:
kuk2
H(curl,Ω)=kuk2
L2(Ω)+k∇×uk2 L2(Ω), kuk2
H(div,Ω)=kuk2
L2(Ω)+k∇·uk2L2(Ω). (5) They are well known Hilbert spaces.
H#(curl,Z) ={u∈H(curl, R3) :uisZ-periodic}
H#(div,Z) ={u∈H(div, R3) :uisZ-periodic}
(6)
We introduce
L2#(Z) ={u∈L2(R3), u isZ-periodic}, (7) and
H1#(Z) ={u∈H1(R3), u isZ-periodic}, (8) where H1(R3) is the usual Sobolev space on R3. First, denoting by C0#(Z)the space of functions in C0(R3)and Z-periodic,C00(R3) the space of continuous functions over R3with compact support, we have the following definitions:
Definition 2.1: A sequence uε(x) in L2(Ω) two-scale converges to u0(x,y) ∈ L2(Ω,L2#(Z)) if for every V(x,y)∈C00(Ω,C#0(Z))
ε→0lim R
Ωuε(x)·V(x,x/ε)dx
=R
Ω
R
Zu0(x,y)·V(x,y)dxdy. (9) Proposition 2.2: If uε(x) two-scale converges to u0(x,y) ∈ L2(Ω,L2#(Z)), we have for all v(x) ∈ C0(Ω) and allw(y)∈L2#(Z)
ε→0lim R
Ωuε(x)·v(x)w(xε)dx
=R
Ω
R
Zu0(x,y)·v(x)w(y)dxdy. (10) Theorem 2.3: (Nguetseng). Let uε(x)∈L2(Ω). Suppose there exists a constantc >0such that for allε
kuεkL2(Ω)≤c. (11) Then there exists a subsequence of ε (still denoted ε) and u0(x,y)∈L2(Ω,L2#(Z))such that:
uε(x)two-scale converges tou0(x,y). (12) Proposition 2.4: Let uε(x) be a sequence of functions in L2(Ω), which two-scale converges to a limit u0(x,y)∈L2(Ω,L2#(Z)).
Then uε(x) converges also to u(x) = Z
Z
u0(x,y)dy in L2(Ω) weakly.
Furthermore, we have
ε→0limkuεkL2(Ω)≥ ku0kL2(Ω×Y)≥ kukL2(Ω). (13) Proposition 2.5: Let uε(x) be bounded inL2(Ω). Up to a subsequence, uε(x) two-scale converges to u0(x,y) ∈ L2(Ω,L2#(Z))such that:
u0(x,y) =u(x) +fu0(x,y), (14) wherefu0(x,y)∈L2(Ω,L2#(Z))satisfies
R
Zfu0(x,y)dy= 0, (15) andu(x) =
Z
Z
u0(x,y)dyis a weak limit inL2(Ω).
Proof: Due to the a priori estimates (32), uε(x) is bounded in L2(Ω), then by application of Theorem 2.3, uε we get the first part of the proposition. Furthermore by defininguf0 as
fu0(x,y) =u0(x,y)−R
Zu0(x,y)dy, (16) we obtain the decomposition 14 ofu0.
Proposition 2.6: Let any two-scale limit u0(x,y), given by Proposition (2.5), can be decomposed as
u0(x,y) =u(x) +∇yΦ(x,y). (17) where Φ∈ L2(Ω,H1#(Z)) is a scalar-valued function and whereu∈L2(Ω).
Proof: Proof of (17), integrating by parts, for any V(x,y)∈C10(Ω,C1#(Z)), we have
ε R
Ω∇×uε(x)·V(x,xε)dx
=εR
Ωuε(x)· ∇×V(x,xε)dx
=R
Ωuε(x){ε∇x×V(x,xε) +∇y×V(x,xε)} dx.
(18) Taking the two-scale limit asε→0 we obtain
0 =R
Ω
R
Zu0(x,y)· ∇y×V(x,y)dxdy, (19) which implies that∇y×u0(x,y) = 0. To end the proof of the proposition we use the following result:
Proposition 2.7: Ifu0∈L2(Ω)satisfies
∇y×u0(x,y) = 0, (20) then there exists u∈L2(Ω) and Φ∈L2(Ω,H1#(Z)) such thatu0(x,y) =u(x) +∇yΦ(x,y).
Applying this proposition we obtain equality (17) ending the proof of Proposition (2.6).
These results are important properties of the two-scales convergence. We note that the usual concepts of convergence do not preserve information concerning the micro-scale of the function. However, the two-scale convergence preserves information on the micro-scale.
III. HOMOGENIZED PROBLEM
We will explore in this section the behavior of electromag- netic fieldEε using the asymptotic expansion and the two- scale convergence to determine the homogenized problem.
We place in the context of the case 6 with δ > L and ω= 106rad.s−1, then we haveη= 5andΣεa =ε,Σεr=ε4, Σεc= 1 which gives the following equation:
∇×∇×Eε−ω2ε5k()Eε+iω[(1εC(xε)
+ε41εR(xε))1{y<0}+ε1{y>0}]Eε= 0, (21) where for a given set A, 1A stands for the characteristic function of A and where 1εA(x) = 1A(x
ε), hence 1εC and 1εR are the characteristic functions of the sets filled by carbon fibers and by resin. And where k() = (c1εC(x) +r1εR(x))1{y<0} +1{y>0}. First, we will use the classical method of the asymptotic expansion.
A. Asymptotic expansion
We assume that (Eε, Hε) satisfies the following asymp- totic expansion, asε→0:
Eε(x) =E0(x,x
ε) +εE1(x,x
ε) +ε2E2(x,x
ε) +...., (22) where for any k ∈ N Ek = Ek(x,y) are considered as Z-periodic functions with respect toy. Applied to functions Ek(x,xε)the curl operator becomes∇x×Ek(x,xε)+1ε∇y× Ek(x,xε). Plugging (22) in the formulations (21), gathering the coefficients with the same power ofε, we get:
1
ε2∇y× ∇y×E0(x,xε) +1ε[∇y× ∇y×E1(x,xε)
+∇y× ∇x×E0(x,xε) +∇x× ∇y×E0(x,xε)]
+ε0[∇x× ∇y×E1(x,xε) +∇x× ∇x×E0(x,xε) +∇y× ∇y×E2(x,xε) +∇y× ∇x×E1(x,xε) +iω1εC(x)1{y<0}E0(x,xε)]
+ε[∇x× ∇y×E2(x,xε)
+∇x× ∇x×E1(x,xε)) +∇y× ∇x×E2(x,xε) +∇y× ∇y×E3(x,xε) +iω(1εC(x)1{y<0}E1(x,xε) +1{y>0})E0(x,xε)] +...) = 0.
(23) In order to write what is in factor of ε in the last equation we used that : 1{y<0} =1{y
ε<0}. Since (23) is considered as true for any smallεit gives a cascade of equations, from which we extract the four first equations
∇y× ∇y×E0(x,y) = 0
∇y× ∇y×E1(x,y)
+∇y× ∇x×E0(x,y) +∇x× ∇y×E0(x,y) = 0
∇x× ∇y×E1(x,y) +∇x× ∇x×E0(x,y) +∇y× ∇y×E2(x,y)
+∇y× ∇x×E1(x,y) +iω1C(y)1{ν<0}E0(x,y) = 0
∇x× ∇y×E2(x,y) +∇x× ∇x×E1(x,y) +∇y× ∇x×E2(x,y) +∇y× ∇y×E3(x,y) +iω 1C(y)1{ν<0}E1(x,y) +1{ν>0}
E0(x,y) = 0 (24) Applying divy in the last two equations in (24), we obtain
∇y· iω1C(y)1{ν<0}E0(x,y)
= 0, (25) and
∇y· iω(1C(y)1{ν<0}E1(x,y) +1{ν>0})E0(x,y)
= 0.
(26) The boundary condition in (2) write:
(1ε∇y×E0(x,y) +∇x×E0(x,y) +∇y×E1(x,y+...)×n
=−iωHd×n, x∈R3, y∈ Z.
(27) Now we take the first equation of (24) and the equation (25) to obtain :
∇y× ∇y×E0(x,y) = 0,
∇y· {iω(1C(y)1{ν<0})E0(x,y)}= 0. (28) Multiplying the first equation in (28) byE0 and integrating by parts overZ leads to
R
Z∇y× ∇y×E0(x,y)E0(x,y) dy
=R
Z|∇y×E0(x,y)|2 dy
= 0.
(29) We deduce that the equation (28) is equivalent to
∇y×E0(x,y) = 0, (30) for any y∈ Z. Hence from Proposition (2.7) we conclude that E0(x,y)can be decomposed as
E0(x,y) =E(x) +∇yΦ0(x,y), (31) whereΦ0(x,y)∈L2(Ω;H1#(Z))andE(x)∈L2(Ω).
B. Mathematical justification
Now we will show rigorously with two-scale convergence that the solution of problems (1),(2) and (3) converge to the solution of the homogenized problem whenεtends to 0. We recall the following Theorem, we give a proof in [7]:
Theorem 3.1: For anyε >0, for anyη ≥0, there exists a positive constantω0 which does not depend onεand such that for allω∈(0, ω0),Eε∈Xε(Ω)solution of (1), (2), (3) satisfies
kEεkXε(Ω)≤C (32) withC= CγtCCγT
0 kHdkH(curl,Ω).
Then, we have the following Theorem:
Theorem 3.2: Under assumptions of Theorem (32), se- quence Eε is solution of ((1), (2), (3)). Eε two-scale converges to E(x) +∇yΦ0(x,y), where E ∈ L2(Ω) and Φ0 ∈ L2(Ω,H1#(Z)), the unique solution of the homoge- nized problem:
∇x× ∇x×E(x) +iωθE(x) = 0 inΩ,
−∇y· {∇x× ∇x× ∇yΦ0(x,y)}
−iω∇y· {1C(y)∇yΦ0(x,y)}
=iωE(x)∇y· {1C(y)} inΩ× Z,
∇x×E(x)×e2=−iωHd×e2 onΓd,
∇x×E(x)×e2= 0 onΓL,
(33)
where θ = R
Z1C(y) dy is the volume fraction of carbon fiber.
Proof: Step 1: Two-scale convergence. Due to the estimate (32), Eε is bounded in L2(Ω). Hence, up to a subsequence, Eε two-scale converges to E0(x,y) belong- ing to L2(Ω,L2#(Z)). That means for any V(x,y) ∈ C10(Ω,C1#(Z)), we have:
ε→0lim R
ΩEε(x)·V(x,xε)dx=R
Ω
R
ZE0(x,y)·V(x,y)dydx.
(34) Step 2: Deduction of the constraint equation. We multiply Equation (21) by oscillating test functionVε(x) = V(x,xε)whereV(x,y)∈C10(Ω,C1#(Z)):
R
Ω∇×Eε(x)·(∇x×Vε(x,xε)) +1ε∇y×Vε(x,xε +[−ω2ε5k() +iω 1εC(xε) +ε41εR(xε)
1{y<0}
+ε1{y>0}]Eε·Vε(x,xε) dx
=−iωR
ΓdHd×e2·(e2×V(x,1, z, ξ,1ε, ζ))×e2 dσ.
(35) Integrating by parts, we get:
R
ΩEε(x)·(∇x× ∇x×Vε(x,xε)
+1ε∇y× ∇x×Vε(x,xε) +1ε∇x× ∇y×Vε(x,xε) +ε12∇y× ∇y×Vε(x,xε)) + [−ω2ε5k()
+iω 1εC(xε) +ε41εR(xε) 1{y<0}
+ε1{y>0}]Eε(x)·Vε(x,xε)dx
=−iωR
ΓdHd×e2·(e2×V(x,1, z, ξ,1ε, ζ))×e2 dσ.
(36) Now we multiply (36) byε2 and we pass to the two-scale limit, applying Theorem 2.3, using (34) we obtain:
R
Ω
R
ZE0(x,y) ∇y× ∇y×V(x,y)
dydx= 0. (37) We deduce the constraint equation for the profileE0:
∇y× ∇y×E0(x,y) = 0. (38)
Step 3. Looking for the solutions to the constraint equation.Multiplying Equation (38) by the conjugate ofE0 and integrating by parts over Z leads to
R
Z∇y× ∇y×E0(x,y)E0(x,y)dy
=R
Z|∇y×E0(x,y)|2 dy
= 0.
(39) We deduce that equation (39) is equivalent to
∇y×E0(x,y) = 0, (40) Moreover since a solution of (40) is also solution of (38), (38) and (40) are equivalent. Hence, from Proposition 2.7 we conclude thatE0(x,y)can be decomposed as
E0(x,y) =E(x) +∇yΦ0(x,y). (41) Step 4. Equations forE(x)andΦ0(x,y).Now, we seek whatE satisfies. For this, we build oscillating test functions satisfying constraint (41) and use them in weak formulation (36). We define test functionV(x,y) =α(x) +∇yβ(x,y), V(x,y) ∈ C10(Ω)×C10(Ω,C1#(Z)) and we inject in (36) test functionVε=V(x,xε), which gives:
R
ΩEε(x)· ∇x× ∇x×V(x,xε) +2ε∇x× ∇y×V(x,xε) +ε12∇y× ∇y×V(x,xε)
+ [−ω2ε5k() +iω 1εC(xε) +ε41εR(xε)
1{y<0}+ε1{y>0}]Eε(x)·V(x,xε)dx
=−iωR
ΓdHd×e2·(e2×V‡(x,1, z, ξ, ζ))×e2 dσ, (42) with V(x,1, z, ξ, ν, ζ) = V‡(x,1, z, ξ, ζ) the restriction on V which does not depend on ν. The term containing the constraint, the third one, disappears. Passing to the limitε→ 0 and replacing the expression of V by the term α(x) +
∇yβ(x,y), we have
∇x× ∇y×V(x,y) =∇x× ∇y×[α(x) +∇yβ(x,y)]
=∇x× ∇y×(α(x)) +∇x× ∇y×(∇yβ(x,y))
=∇x× ∇y×(∇yβ(x,y)).
(43) Since ∇y ×(∇y) = 0, the term 2ε∇x × ∇y ×V(x,y) vanishes. Therefore, (42) becomes:
R
Ω
R
ZE0(x,y)· ∇x× ∇x×(α(x) +∇yβ(x,y) +iω 1C(y)E0(x,y)·(α(x) +∇yβ(x,y))dydx
=−iωR
ΓdHd×e2·(e2×(α(x,1, z) +∇yβ(x,1, z, ξ, ζ)))×e2 dσ.
(44) Now in (44) we replace ExpressionE0 giving by (41). We obtain
R
Ω
R
Z(E(x) +∇yΦ0(x,y))· ∇x× ∇x×(α(x) +∇yβ(x,y)) +iω1C(y)(E(x)
+∇yΦ0(x,y))·(α(x) +∇yβ(x,y))dydx
=−iωR
ΓdHd×e2·(e2×(α(x,1, z) +∇yβ(x,1, z, ξ, ζ)))×e2 dσ.
(45) Now using the fact that R
Ω
R
Z∇yΦ0(x,y)· ∇x × ∇x × α(x)dydx= 0we have:
R
Ω
R
ZE(x)· ∇x× ∇x×α(x) +E(x)· ∇x× ∇x× ∇yβ(x,y) +∇yΦ0(x,y)· ∇x× ∇x× ∇yβ(x,y) +iω1C(y)(E(x)α(x) +∇yΦ0(x,y)α(x)
+E(x)∇yβ(x,y) +∇yΦ0(x,y)∇yβ(x,y))dydx
=−iωR
ΓdHd×e2·(e2×α(x,1, z))×e2 dσ.
(46)
Then takingβ(x,y) = 0in (46) and integrating by parts, we get
R
Ω ∇x× ∇x×E(x)·α(x) +iω R
Z1C(y)dy
E(x)α(x) dx
=−iωR
ΓdHd×e2·(e2×α(x,1, z))×e2 dσ.
(47) which gives the following well posed problem forE(x)
∇x× ∇x×E(x) +iω(R
Z1C(y)dy)E(x) = 0 inΩ,
∇x×E(x)×e2=−iωHd×e2 onΓd,
∇x×E(x)×e2= 0 on ΓL.
(48) Now takingα(x) = 0 in (46), we obtain
R
Ω
R
Z∇x× ∇x× ∇yΦ0(x,y)∇yβ(x,y) +iω1C(y)(E(x)· ∇yβ(x,y)
+∇yΦ0(x,y)∇yβ(x,y))dydx= 0.
(49) Integrating by parts
R
Ω
R
Z−∇y· {∇x× ∇x× ∇yΦ0(x,y)}β(x,y)
−iω∇y· {1C(y)∇yΦ0(x,y)}β(x,y)dydx
=iωR
Ω
R
Z∇y· {1C(y)}E(x)β(x,y)dydx.
(50) which gives the microscopic problem forΦ0(x,y)
−∇y· {∇x× ∇x× ∇yΦ0(x,y)}
−iω∇y· {1C(y)∇yΦ0(x,y)}
=iωE(x)∇y· {1C(y)}.
(51) This concludes the proof of Theorem 3.2.
IV. CONCLUSION
We presented in this paper the homogenization of time harmonic Maxwell equation by the method of two-scale con- vergence. We started by studying the time harmonic Maxwell equations with coefficients depending ofε. We remind that λ is the wave length, δ is the skin length, L is thickness of the medium and e the size of the basic cell and then ε= Le is the small parameter. We find for low frequencies the macroscopic homogenized Maxwell equations depending on the volume fraction of the carbon fibers and we find also the microscopic equation.
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@articleWellanderKris, author = Wellander, N. and Kristensson, B., title
= Homogenization of the Maxwell equations at fixed frequency., journal
= Technical Report, publisher = Lund Institute of Technology, year = 2002, volume = LUTEDX/TEAT-7103/1-37/(2002), number = , pages = , note = Departement of Electroscience, P.O. Box 118, S-211 00, Lund, Sweden,
[18] N. Wellander and B. Kristensson. “Homogenization of the Maxwell equations at fixed frequency.” Technical Report, 2002, vol.
LUTEDX/TEAT-7103/1-37.
