Homogenization of linear spatially periodic electronic
circuits
Michel Lenczner
∗January 20, 2006
Abstract: In this paper we establish a simplified model of general spatially periodic linear electronic analog networks. It has a two-scale structure. At the macro level it is an algebro-differential equation and a circuit equation at the micro level. Its construction is based on the concept of two-scale convergence, introduced by the author in the framework of partial differ-ential equations, adapted to vectors and matrices. Simple illustrative examples are detailled by hand calculation and a numerical simulation is reported.
1
Introduction
It is well known that when the size of an analog electronic network increases too much, the size of the unknown vectors, namely the voltages, the currents and the electric node0s voltage, become very large and the system of equation becomes impossible to solve on existing computers. In this paper, we are concerned by such large systems of electronic equations arising in the case of spatially periodic architectures of analog electronic circuits. Among the applications that we have in mind, some of them are for purely analog electronic systems or for Micro-Electro-Mechanical Systems (MEMS) arrays which have always a periodic structure and include or will include in a near future an electronic network. The MEMS arrays are used for a wide range of applications in various scientific or technological areas as biology, medicine, communications, aeronautics, etc... . Due to the small place available in those architectures, analog circuits are preferred in comparison with digital circuits. Other motivations of using arrays of analog circuits are their good computing power per unit area (when moderated resolutions are required) accompanied with a low energy consumption. Some applications to Smart Structures may also be found in the cases where the actuators and sensors are numerous and distributed in a periodic way in their host structure, see for example [7] and [6].
The method for the simplified model derivation that we present here refers to the general homogenization which has been intensively developed in mechanics for composite materials
∗Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695. E-mail
modelling. Various approaches have been investigated under various denominations. We will not make a comparison of them, we only mention that the more general and rigorous one was based on an asymptotic expansion with respect to the vanishing cells size (or equivalently to the number of cells that is assumed to tend to infinity). It was introduced by E. Sanchez-Palencia and then widely developed in the reference book [11]. This theory has been rigorously justified in [2] and later its domain of applications has been expanded and the proofs significantly simplified by the introduction of the two-scale convergence in [1] and later by the introduction of the two-scale transform and a new two-scale convergence in [8]. This last improvement has allowed the treatment of network equations which was not encompassed by the other approaches. Furthermore, it has led to a so simple and natural technique that later it has been rediscovered independently by two other groups [3] and [5] in the context of partial differential equations.
In our first works on the electronic networks homogenization [8], [9] and [10], we have formulated the electric network equations under the form of partial differential equations under variational form. Its well posedness has been proved by a combination of functional analysis arguments commonly used in the field of partial differential equations and some graph theory properties. Then, the two-scale limit of the transposed incidence matrix, which was expressed as a spatial derivative along the network, has been carefully formulated. This was the corner-stone of the two-scale models construction from which the homogenized models have been built. This program has been achieved for general network topologies but limited to static problems and to some particular linear devices, passive devices in [8], passive devices plus linear VCVS in [9] and passive devices plus linear VCCS in [10].
Let us turn to this paper contributions. First, the two-scale transform and convergence which was formulated in the context of functions and partial differential equations are now rewritten for vectors and matrices which is the usual framework in electronics. It is the first time that the fundamental properties of the two-scale transform of vector and of matrices are stated and proved. Second, the asymptotics of the Kirchhoff voltage law is carefully analyzed. This is the more difficult and technical part. The technic of this proof is new. It is more general and adaptable that the former thus it may be easier to extend to complex systems including electronics as well as thermal or mechanical effects for instance. Third, this paper covers general linear multi-port devices under the condition that all their ports belong to a same cell. We say that they are local. Fourth, the condition under which the model is justified and its solution exists are made in details. Fifth, three illustrative examples are presented. They have been chosen very simple so that to allow hand calculations with the hope that they are sufficiently illustratives. The solution of the third example was numerically simulated so we report a comparison between the complete solution and the solution of our simplified model. Through this example we also underline the interest of the simplified model in term of computing time.
three examples are stated. The third section if devoted to the statement of the assumptions and of the simplified model itself. Then the definition and the properties of the two-scale transform for vectors and for matrices are stated and partially proved in the fourth section. Some technical points are postponed in annex. The derivation of the model is detailed in the fifth section. Finally, in the sixth section, the simplified model is applied to the three examples and the numerical results are reported.
2
Presentation of the problem
In this section we start by introducing standard circuit equations in § 2.1, then we describe what is called a periodic circuit § 2.2 and we end by three examples of such circuits in § 2.3-2.5.
2.1
Circuit equations
A graph associated with an electrical circuits is denoted by G = (E,N) where E is the branch set and N the node set. We denote by ϕ∈ R|N |, v ∈ R|E| andi ∈ R|E| the nodal voltages (or
electric potential), the branch voltages and the currents where |Z| represents the number of elements belonging to a set Z. The circuit equations used in this paper are:
• the Kirchhoff voltage law:
v=ATϕ, (1)
• the Tellegen theorem:
iTATψ = 0 for all ψ ∈Ψ, (2)
• the branch equations characterizing the circuit devices:
Mv+Ri=us, (3)
• and the ground node equations:
ϕi = 0 for all nodeni ∈N0 (4)
whereA ∈R|N |×R|E| is the incidence matrix, N
0 ⊂N is the subset of ground nodes,
Ψ={ψ ∈R|N | such that ψi = 0for allni ∈N0}
is the set of admissible potentials,MandRare two square matrices with|E|rows and columns and us ∈R|E| represents voltages and currents sources regrouped in a single vector.
In the following, we reformulate this set of equations in a condensed form:
ϕ∈Ψ,v =ATϕ,
These equations may take into account general multi-port linear devices in statics. Linear circuit equations of evolution may also be written on this form when applying the Laplace transform.
2.2
Periodic circuit
Now let us consider the class of circuits that are distributed in d ≥1 space directions so that their graph is periodic in all these directions. Electrical devices are assumed to be periodically distributed excepted on the boundary where specific devices may be installed so that to realize specific boundary conditions. Each branch is assumed to belong entirely to one and only one cell. If it is not the case, the circuit must be rearranged in a convenient manner.
We assume that the circuit is confined in a bounded set Ω ⊂ Rd and that the number of its periods is large in all the d directions. For simplicity, it is assumed that Ω is an unit square Ω= (0,1)d and that in all directions the period lengths are equal to an identical small parameter ε.
A unit graph is built by picking one cell of the complete graph, expanding it by a factor
1/ε and shifting it so that to occupy the unit cell Y = (−12,12)d. The unit graph is denoted by G = (E, N). From the above assumption, it turns out that E is a set of entire branches. Because N is εY−periodic, each noden∈N located on the boundary ofY has its counterpart
n0 on the opposite side. We assume that n andn0 are linked by at least one path (a sequence of connected branches) that does not include any ground node. Such a path is called a crossing path. Let us introduce the subset EC of 00crossing branches00.
Criterion 1 The subset EC ⊂ E is constituted of all branches of some of the crossing paths. For each n and n0 defined as above the branches of at least one crossing path linking n to n0
among many must belong to EC.
It may be noticed that the criterion 1 do not determine totally EC. A complementary criterion is given in the remark 6. The complementary set E −EC is denoted by EN C. The subset EC is partitioned in itsnc connected componentsEC =∪nk=1c ECk.
The subsetsNC andNN C of N are defined as the set of nodes involved in at least one of the branch of EC andEN C respectively. It is worth pointing out that these two subsets are not a partition of N because in general NC ∩NN C =6 ∅ as soon as EC have EN C common nodes.
The set N0 of ground nodes is shared in two parts, the first N0Γ referring to ground nodes
located on the boundaryΓ of the whole domainΩ and the other being distributed periodically in the graph. The corresponding set of this later in N is denoted byN0. The ground nodes in
N0Γ correspond to some nodes inN located on the cell boundary. Therefore they belong toNC which have been separated in many connected componentsNCk which in turn define a partition of N0Γ =∪nkc=1N0Γk. We denote by Γ0k the part ofΓ where the nodesN0Γk are distributed.
2.3
Example 1: A regular grid of resistors
The first example of periodic circuit has been extensively studied in the literature. It is a two-dimensional regular mesh of resistors. The elementary cell is made of four resistor (with the same resistance in all cells) that realize two crossing paths in the two directions and one source of current that may vary from one cell to the other. Thus EC is made of the resistors andEN C of the current source. The nodes located on the part Γ0 of the boundary are connected to the
earth. The complementary part of the boundary is denoted by Γ1. Making an adapted choice
of resistance r = εr0 and of current sources is = εi0s, this circuit realizes the discretization by the finite differences method of the Laplace equation with mixed (Dirichlet and Neumann) boundary conditions:
−∆ϕ0 =f inΩ
ϕ0 = 0 on Γ0 and ∇ϕ0.n= 0 onΓ1
where f = 2r0i0s. It turns out that the components of the nodal voltage ϕat the center of the cells are a approximations ofϕ0 at those points. The model presented in this paper recover this
result and in addition provides the expressions of the currents and voltages in all branches of the circuit. Evidently, our model is very general so it encompasses much more general situations.
Example 1
2.4
Example 2: Disconnected circuits
In that example the sub-circuits of a cell is disconnected from the sub-circuit of the other so
vs may take different values in different cells but not the resistor. The circuit equations can be solved independently in each cell, it comes that
ϕ=−vs andi=−
vs
r .
Example 2
If the vector vs is an approximation of a continuous field vs0 for ε << 1 then the vectors ϕ and i are some approximations of the continuous fields −v0
s and −
v0
s
R. This trivial result is encompassed by our model that can represent general periodic disconnected circuits.
2.5
Example 3: Active and passive devices
Example 3
3
Statement of the simplified model
Before to state the model in § 3.3 we introduce in § 3.1 the concept of two-scale transform and in § 3.2 the assumption on which the model is justified.
3.1
Two-scale transform
The multi-integer µ = (µ1, .., µd) enumerates all the cells Yε
µ in Ω and takes its values in
{1, .., m}d. The center of a cell Yε
µ is denoted by xεµ. We define the concept of two-scale transform relatively to a set Z of objects being distributed εY−periodically inΩ. It must be understood that Z may represent either N or E. Similarly, Z represents either N or E. The objects of Z are indexed by I ∈ {1, ..,|Z|} and those of Z by j ∈ {1, ..|Z|}. Each object is referenced by an unique index I, but it can also be referred by a multi-integer µ referring to the cell which it belongs and by an index j inZ. This correspondence is denoted by I ∼(µ, j)
and is not one to one in general. Using this correspondence, for each vector u ∈R|Z| one may
define a unique tensor Uµj with (µ, j)∈{1, .., m}d× {1, ..,|Z|}by Uµj =uI for I ∼(µ, j).
By another way, we introduce the setP0(Ω) of piecewise constant functions on each cell of
Ω: f(x) =PiχYε
µ(x)fi, each fi being a scalar coefficient and χYµε(x) being the characteristic
function of the set Yε
µ equal to 1when x∈Yµε and0 otherwise. We denote by P0(Ω)|Z| the set of vectors having |Z| components, each of them being in P0(Ω). It is easy to verify that
P0(Ω)
is included in L2(Ω) =
{f such that RΩf2(x) dx <+
Definition 2 The two-scale transform of a vector u ∈ R|Z| is the vector of functions ub ∈
P0(Ω)|Z| defined by
b
uj(x) =
X
µ∈{1,..,m}d
χYε
µ(x)Uµj for all x∈Ω and j ∈{1, ..,|Z|}
where Uµj = uI with I ∼ (µ, j). The linear map u 7→ bu from R|Z| to P(Ω)|Z| ⊂ L2(Ω)|Z| is
denoted by TZ.
Let us illustrate this concept on the example 2 where d = 2, |E| = 2 and |E| = 2m2.
The components of the two-scale transform bv(x) = (bv1(x),bv2(x)) ∈ P0(Ω)2 of branch voltages
v∈R2m2
have the formbvj(x) = Pµ∈Yε
µ χYµε(x)Vµj for j = 1,2where Vµj represent the voltages
in the 2 branches of the cell Yε µ.
In the following, we will constantly refer to the concept of local matricesB∈R|Z1|×R|Z2|,Z
1
andZ2 being two periodic sets, which transform a vector having its non vanishing components
in one cell into a vector having also its non vanishing components in the same cell.
Definition 3 (i) B is said to be local ifBIJ = 0for all I ∼(µ, j) and J ∼(λ, l)when µ6=λ.
(ii) LetB ∈R|Z1|×R|Z2| be a local matrix, if there exist a matrix B ∈R|Z1|×R|Z2| such that
BIJ =δµλBjl for allI ∼(µ, j)and J ∼(λ, l) thenB is said to be local and εY−periodic. The matrix B is called the reduced matrix of B.
Example 4 Since we have assumed that each branch belong to only one cell it comes that the incidence matrix A is local and εY−periodic. Its reduced matrix is the incidence matrix of the graph G= (E, N) denoted by A. The transpose AT is also local and εY
−periodic with reduced matrix AT.
Here δµλ is the Kronecker symbol equal to 1 when µ = λ and equal to zero otherwise. In
other words, a local εY−periodic matrix is a bloc diagonal matrix so that all its blocs are identical.
The linear spaceL2(Ω)|Z| admits a scalar product and a norm
(u, v) = Z
Ω×Z
uj(x)vj(x)dx and||u||= (u, u)1/2
where we use the notation
Z
Ω×Z
fj(x) dx= X
j/zj∈Z
Z
Ω
fj(x)dx
zj describing an element of Z. This notation is constantly used in this paper for Z being E,
N or one of their parts. The proposition 8 shows that for Z = E and Z = E the two-scale transform preserves the norm,
εduT.u=εdX
I
The linear space L2(Ω)|Z| being normed and ε being a parameter tending to zero, one will say that a sequence uε ∈ L2(Ω)|Z| indexed by ε converges strongly in L2(Ω)|Z| towards a limit u0,
which necessarily belongs toL2(Ω)|Z|,if||uε−u0||vanishes whenε tends to zero. The sequence is said to be weakly convergent inL2(Ω)|Z|towardsu0 if the scalar product(uε
−u0, v)vanishes
when ε tends to zero for all v ∈ L2(Ω)|Z|, see [12] for more details. The strong convergence implies the weak convergence but the converse is generally false. For example, the sequence
sin(xε)∈L2(Ω)is bounded inL2(Ω),it is weakly convergent towards0,but it does not converge strongly towards any limit in L2(Ω).
The weak convergence plays an important role in our approach because the model is stated on the weak limits of the voltage0s and current0s two-scale transforms. The existence of such weak limits comes from the following lemma (see [12]).
Lemma 5 From any bounded sequence inL2(Ω) one may extract a subsequence that is weakly
convergent in L2(Ω).
3.2
Assumptions
Before to state further assumptions, let us summarize those made in the past sections. (H0) A branche∈E can intersect the boundary of a cell only with its tips.
(H1) Each opposite nodesnandn0 are linked by at least one crossing path that do not come
across the ground. Furthermore, they do not belong to any corner of the cell.
The next assumptions state that not only the graph is periodic but also the distribution of devices in the circuit as well as their coefficients.
(H2) The matricesMandRare local andεY−periodic. Their reduced matrices are denoted by M andR.
The next assumption says that the voltages and the currents are respectively of the order of ε and 1 in EC and of the order of 1 and ε in EN C. We formulate this by using the scaling matrices Sv, Sc and Ss applied to the two-scale transforms
biε=Scbi, bvε=Svbv, ubsε=Ssubs, bϕε =ϕb. (7)
(H3) The norms ||biε
||, ||bvε
||, ||ϕbε||, ||ubsε|| are bounded and the dataubsε converges weakly in
L2(Ω)|E| towards a limit u0
s.
The |E| × |E| scaling matrices are
Sv =ε−1IEC +IEN C, Sc=IEC +ε
−1I
EN C andSs=ΠcSc+ΠvSv
where for any subset E1 of E the|E| × |E| matrix IE1 is the projector on E1:
(IE1)jk = δjk if ej ∈E1
= 0 otherwise.
The reduced matricesM andR of MandR are scaled in a consistent manner
Mε =SsM Sv−1 and R ε
=SsRSc−1. (8)
(H4) The scaled reduced matrices Mε andRε converge towards some limit M0 andR0.
Remark 6 As indicated in the criterion 1, EC is made of all the branches of some crossing paths and for each couple (n, n0) at least one crossing path linking nandn0 must be part of EC. In the case where many crossing paths are linking nand n0 the designer is free to decide which
are included in EC and which are not, with regard to the assumption (H2).
Let us introduce the so-called cell problem (or problem micro). For two given vectorsη∈Rnc, us ∈ R|E| and a given matrix θ ∈Rd×Rnc the vectors i, v ∈R|E| and(ϕC,ϕN C)∈Ψm(η) are solutions of the cell problem
v=IECA
Tϕ
C +IEN CA
Tϕ N C
R0i+M0v=us−M0(τ θ+I0η) (9)
andiTw= 0
for all vector w=IECA
Tψ
C+IEN CA
Tψ
N C with(ψC,ψN C)∈Ψm, the admissible nodal voltage set for the cell problem being
Ψm={(ψC,ψN C)∈R|perN|×R|
N| such that I
N0
C∪N−NCψC = 0, INC∪N0ψN C = 0}.
The tensor τ is defined by
τlkp =
|N|
X
j=1
yk(nj)AjlIjp0 , (10)
wherey(n)∈Rdis the coordinates vector of a noden
∈N. For computational purpose, it may be remarked that τlkp = 0for el 6∈ECp because Ajl= 0when nj ∈NCp andel6∈ECp. Then
τlkp =
X
js.t. nj∈NCp
yk(nj)Ajl for el ∈ECp
= 0 otherwise.
Throughout this paper, we will use the tensorial product notation
(τ θ)l = d X
k=1
nc
X
p=1
τlkpθkp. (11)
Moreover, I0 is a matrix in
R|N|×Rnc defined by
Ijp0 = 1 if nj ∈NCp (12)
NC0 is the set ofnc nodes constituted of one arbitrary node of each connected componentNCp. Finally
R|perN|={φ∈R|
N| such that φ
j =φj0 for all couple (nj, nj0)of opposite nodes}. (13)
(H5) For each η ∈Rnc, θ
∈Rd
×Rnc and u
s ∈ R|E| the cell problem (9) has a unique solution (ϕC,ϕN C, i, v).
From (H5) and the map (η,θ, us)→7 (i, v,ϕN C) being linear there exists some matrices Lx,
Hx and a third order tensor Px such that
Liη+Piθ+Hius =i, Lϕη+Pϕθ+Hϕus =ϕN C andLvη+Pvθ+Hvus=v (14)
where Pxθ is defined according to (11).
3.3
The model
From the assumption (H3) and the lemma 5 there exists at least one extracted subsequence of
(biε,bvε,ϕbε
) that is weakly converging towards some limits (i0, v0,ϕ0) in L2. The model satisfied
by the latter is stated in this subsection. It constitutes the main result of the paper.
Theorem 7 (i) If the assumptions (H0-H3) are fulfilled then IECA
Tϕ0 = 0 or equivalently
there exist ϕ0
C ∈L2(Ω)nc such that
ϕ0 =I0ϕ0C +ϕ0N C (15)
where ϕ0N C :=IN−NCϕ
0. Moreover there existsϕ1
C ∈L2(Ω;R| E|
per) such that
v0 =∂τϕ0C +IECA
Tϕ1
C +IEN CA
Tϕ0 and I
N−NCϕ
1
C = 0. (16)
This is the asymptotic Kirchhoff Voltage Law.
(ii) Furthermore, if the assumptions (H4) and (H5) are satisfied then ϕ0
C ∈ΨH is solution of the algebro-differential equation, so-called homogenized circuit equations,
QH∇ϕ0C+S H
ϕ0C =F H
u0s and A H
(Pi∇ϕ0C +Liϕ0C) =−A H
Hiu0s in Ω (17)
with the boundary conditions
ϕ0Cp = 0on Γ0p and (Pi∇ϕ0C +Liϕ0C)nτ = 0 on Γ−Γ0p.
(iii) Finally (ϕ1C,ϕN C0 , i0, v = IECA
Tϕ1
C +IEN CA
Tϕ0
N C) is solution of the cell problem (9) with (η,θ, us) = (ϕC0,∇ϕ0C, u0s).
The homogenized matricesQH, SH, FH and operatorAH are defined by
QH = R0Pi+M0(τ +Lv), SH =R0Li+M0(IECA
TI0+
Lv),
where ∂τ∗i=τ∗∇i with τ∗pkl =τlkp and the use of notation (11). The derivative ∂τϕ0C and the normal nτ are defined by
∂τϕ0C =τ∇ϕ
0
C and(nτ)lp = d X
k=1
τlkpnk
∇ being the gradient (∂xk)k=1..d and n = (nk)k=1..d being the outward normal vector to the
boundary Γ of Ω.
4
Properties of the two-scale transform
We prove the fundamental properties of the two-scale transform which are useful for the model derivation.
4.1
Adjoint of
T
Eand norm preservation
First the adjointT∗
E of the two-scale transformTE is established. Then the relationship between the scalar product [., .]and the norm |.|in R|E|, defined by
[u, v] =ε−duT.v and|v|= [v, v]1/2 for allu, v ∈R|E|,
and the scalar product and the norm in L2(Ω)|E| is derived.
Proposition 8 (i) Under the assumption (H0) the adjoint TE∗ is equal to
(TE∗u)I =ε−d
Z
Yε µ
uj(x) dx for I ∼(µ, j) (18)
for all u∈L2(Ω)|E|.
(ii) Furthermore, the restriction TE∗ to P(Ω)|E| is
(TE∗u)I =Uµj for I ∼(µ, j)
for all u∈P(Ω)|E| so that uj(x) =Pµ∈{1,..,m}dχYε
µ(x)Uµj.
(iii)TE∗TE =IE on R|E|.
(iv) TETE∗ =IE on P0(Ω)|E|.
(v)TE is one to one fromR|E| to P0(Ω)|E| and TE∗ is its inverse.
(vi) The scalar product as well as the norm are conserved through the two-scale transform
(TEu, TEv) = [u, v] and ||TEu||=|u| for all u, v ∈R|E|. (19)
Proof. (i) For u ∈ L2(Ω)|E| T∗
Eu is defined through the equality [TE∗u, v] = (TEv, u) for all v∈ R|E|. But (T
Ev, u) = R
Ω(TEv).u(x) dx=ε
dP
µ∈{1,..,m}d
P|E|
j=1ε−d
R Yε
µ uj(x) dx Vµj which
leads to the characterization of T∗
(iii) Letu∈R|Eε|andI ∼(µ, j) (TE∗TEu)I =TI∗(
P
λ∈{1,..,m}dUλjχYε
λ(x)) =ε
−dR Yε
µ
P
λ∈{1,..,m}dχYε λ(x dx Uλj =Uµj =uI.
(iv) Let u∈P0(Ω;R|E|) so thatuj(x) = P
µ∈{1,..,m}dUµjχYε
µ(x) then
(TETE∗u)j(x) = (TE(ε−d R
Yε µuj(x
0)dx0))
j(x) = P
µ∈{1,..,m}dε−d
R Yε
µuj(x 0)dx0χ
Yε
µ(x). Replacing u by its expression yields
=Pµ∈{1,..,m}d
P
λ∈{1,..,m}dε−d
R Yε
µχλ(x
0)dx0U
λjχYε
µ(x).Finally we use the fact thatε
−dR Yε
µχλ(x 0)
dx0 =δµλ to conclude that (TETE∗u)j(x) = P
µ∈{1,..,m}d UµjχYε
µ(x) =uj(x).
(v) is just a consequence of (iii) and of (iv).
The proof of (vi) is straightforward(TEu, TEv) = [TE∗TEu, v] = [u, v]from which the equality of norms follows by posing v=u.
4.2
Two-scale transform of matrices
We start this section by providing the definition of the two-scale transform of a matrix operating on R|E| providing that the assumption H0 holds. We continue by stating some of its properties
in the particular case of |E| × |E| matrices. Since we wish to apply the two-scale transform to the incidence matrix AT which operate on
R|N | we end this section by defining the two-scale
transform of general local εY−periodic matrices which evidently applies toAT.
Definition 9 Assuming that H0 holds, then the two-scale transform of a matrixB∈R|Z|×R|E|
is the linear operator defined from L2(Ω)|E| to P0(Ω)|Z| ⊂L2(Ω)|Z| by
b
B =TZBTE∗.
Let us focus on matricesB ∈R|E|×R|E|. Its two-scale transformBb=T
EBTE∗ is a linear oper-ator fromL2(Ω)|E|toP0(Ω)|E| ⊂L2(Ω)|E|, however, in the following statement we consider only
its restriction defined fromP0(Ω)|E| to itself. The norm of such a matrix is|B|= sup
u∈R|E| |B|uu||.
Proposition 10 For B a |E| × |E| matrix and u, v ∈R|E| the following properties hold:
(i) ([Bu) =Bbbu;
(ii) (Bbu,b bv) = [Bu, v]; (iii)BcT =Bb∗;
(iv) ||B||b =|B|.
Proof. (i)([Bu) =TEBu=TEBTE∗TEu=Bbu.b
(ii)(BbTEu, TEv) = (TEBTE∗TEu, TEv) = [Bu, TE∗TEv] = [Bu, v]. (iii) BcT =T
EBTTE∗ = (TEBTE∗)∗ =Bb∗.
(iv) ||B||b = supu∈P0(Ω;R|E|) ||
b
Bu||
||u||E, ||Bbu|| = ||TEBT
∗
Eu|| = |BTE∗u| and ||u||E = |TE∗u| yield
||B||b = supu∈P0(Ω;R|E|)
|BT∗
Eu|
|T∗
Eu|
= supv∈R|E||B v|
|v| =|B| because T
∗
E is one to one from P0(Ω;R|E|) to R|E|.
We cannot define the two-scale transform of a general matrixB ∈R|Z1|
×R|Z2| but we can
Definition 11 If B ∈R|Z1|
×R|Z2| is a local εY
−periodic matrix and B is its reduced matrix then the two-scale transform BbofB is defined by
(Bbφ)j(x) =
|Z2| X
k=1
Bjkφk(x) for all φ ∈P
0(Ω)|Z2|.
Based on this definition the following property holds.
Lemma 12 If B∈R|Z1|
×R|Z2| is a local εY
−periodic matrix then
c
Bφ=Bbφb.
4.3
Two-scale convergence of Kirchho
ff
Voltage Law
This section is devoted to the derivation of the point (i) of the main theorem 7. The proof is a little technical, so it has been decomposed. One part requiring detailed explanation regarding the two-scale transform for nodes as well as tricky operations is postponed in annex.
Let us recall that the setsN andE of nodes and of branches depend on the number of cells in the circuit or equivalently depend on the parameter ε.For a given ε we consider a vector of nodal voltages ϕ ∈ R|N | andv = ATϕ
∈ R|E| the branch voltages. By doing so, ϕ and v are
also depending on ε and all together constitute a sequence indexed by ε. The same thing can be said about their scaled two-scale transformsϕbε=ϕb and bvε =S
vbvwhere the dependence on ε is made more visible. When their norms ||ϕbε|| and||bvε|| are bounded, thanks to the lemma 5, one may extract a subsequence of the couple still denoted by(ϕbε,bvε)which converges weakly
in L2(Ω)|N|×L2(Ω)|E| towards a limit (ϕ0, v0).
Lemma 13 The weak limits(ϕ0, v0) satisfyI
ECA
Tϕ0 = 0or equivalently there existsϕ0
C ∈Rnc such that
INCϕ
0 =I0ϕ0
C
then
ϕ0 =I0ϕ0C+ϕN C0 where ϕ0N C =IN−NCϕ
0
and there exists ϕ1
C ∈L2(Ω;R
|E|
per) such that
v0 =∂τϕ0C +IECA
Tϕ1
C +IEN CA
T(ϕ0
N C +I
0ϕ0
C).
Proof. (i) We start by proving that INCϕ
0 = I0ϕ0
C. The fact that ||vbε|| is bounded and the lemma 12 imply together that ε−1
||IECA
Tϕbε
|| is bounded and by passing to the limit in
(IECA
Tϕbε
, w) ≤Cε||w|| for all w∈L2(Ω)|E| that I
ECA
Tϕ0 = 0. This is equivalent to say that
ϕ0 is constant on each connected component E
Ck of EC or equivalently that there exists a vector ϕ0
C ∈L2(Ω)|nc| such thatINCϕ
0 =I0ϕ0
(ii) Let us establish thatIEN Cv
0 =I
EN CA
T(ϕ0
N C+I
0ϕ0
C) where ϕ
0
N C =IN−NCϕ
0. The fact
thatϕbε andbvεconverge weakly towardsϕ0 andv0 implies that the equalityIEN Cbv
ε =I EN CA
T b
ϕε converges weakly towards IEN CA
Tv0 =I
EN CA
Tϕ0 =I
EN CA
T(ϕ0
N C +I
0ϕ0
C). The end of the proof is devoted to the derivation of the expression of v0 inE
C :
IECv
0 =∂
τϕ0+IECA
Tϕ1
C.
(iii) We prove that the two following statements are equivalent: (A)v=ATφ with φ ∈R|perN|;
(B)v ∈R|E|, (v, µ) = 0 for allµ∈R|E| such thatIN−∂NAµ= 0andAµ∈R| N|
per. Let us introduce the matrixB ∈R|N|×R|E| defined by
(Bµ)j = (Aµ)j−(Aµ)j0 for nj ∈∂N
= (Aµ)j for for nj ∈N −∂N.
The statement (B) is equivalent to(v, µ) = 0for allµ∈KerB which means thatvis orthogonal to KerB = ImBT.But
(BTφ)j =
|N|
X
k=1
Akj((IN−∂Nφ)k+ (I∂Nφ)k+ (I∂Nφ)k0)
which is equivalent to (A).
(iv) We prove thatIECv=∂τφC+IECA
Tϕ1 withϕ1
∈R|perN|andINN Cϕ
1 = 0.From the lemma
14 we know that v = IECv
0
−∂τϕ0C satisfies the statement (B) or equivalently the statement (A) and we pose φ=ϕ1
∈R|perN|. This ends the proof.
The subset ofN of nodes belonging to the boundary of the cellY is denoted by∂N.Consider
µ ∈ C1(Ω)|E| such that Aµ(x) ∈ R|perN|, µ(x) vanishes in E
N C (IEN Cµ = 0) for all x ∈ Ω and IN−∂NAµ= 0 where ∂N is the subset of nodes belonging to the boundary of the cell (remark that ∂N ⊂ NC).
Lemma 14 If µ∈C1(Ω)|E| satisfies
Aµ(x)∈R|perN|, IEN Cµ(x) = 0 and IN−∂NAµ= 0 for all x∈Ω
then
(IECv
0, µ) = (∂
τϕ0C, µ).
Proof. Forψ =Aµ,
(IECv
0, µ) = lim
ε→0(IECbv
ε, µ).
Noticing that (IECbv
ε, µ) = ( b
vε, µ) = 1ε(ϕbε, Aµ) = 1ε(bϕε,ψ)∂N and using the lemma 20,
(IECv
0, µ) = lim
ε→0
1
ε
Z
Ω×∂N b
ϕεj(x)ψj(x) dx= lim ε→0−
Z
Ω×∂N b
ϕεj(x)(y.∇xψ)j(x)dx+b(ϕbε,ψ)
= −
Z
Ω×∂N
ϕ0j(x)(y.∇xψ)j(x) dx+ Z
∂(Ω,N)
where
∂(Ω, N) = {(x, n)∈Γ×∂N such that nY(n) =nΩ(x)}.
(i) Let us prove that
Z
Ω×∂N
ϕ0j(x) (y.∇xψ)j(x) dx= (ϕ0C,∂τ∗µ)
with
(∂τ∗µ)p(x) = d X
k=1
|E|
X
l=1
τlkp∂xkµl(x) for p∈{1, .., nc}.
From∂N ⊂NC andINCϕ
0 =I0ϕ0
C comes Z
Ω×∂N
ϕ0j(x) (y.∇xψ)j(x) dx= Z
Ω×∂N
(I0ϕ0C)j(x)(y.∇xψ)j(x) dx.
Combined with the facts thatINC−∂N(τ.∇xψ) = 0 and IN−NCI
0ϕ0
C = 0 yields
= (I0ϕ0C, y.∇xψ) = (ϕ0C, I
0T(y.
∇xAµ)) = (ϕ0C,∂τ∗µ).
(ii) For all ν∈R|antiperN| and x∈Γ let us prove that:
X
j/(x,nj)∈∂(Ω,N)
νj = X
j/nj∈∂N
νjy(nj)nΩ(x)
where
R|antiperN| ={φ∈L
2(Ω;
R|N|)such that φj(x) =−φj0(x) a.e. x∈Ωfor allnj ∈∂Y}.
We remark that if(x, nj)∈∂(Ω, N)thennY(nj).nΩ(x) = 1andνj =νjnY(nj).nΩ(x).Moreover
nY(nj0) =−nY(nj) andνj0 =−νj imply thatνjnY(nj) =νj0nY(nj0). Thus νj = 12(νjnY(nj) + νj0nY(nj0))nΩ(x) = (νjy(nj) +νj0y(nj0))nΩ(x)and
X
j/(x,nj)∈∂(Ω,N)
νj =
X
j/(x,nj)∈∂(Ω,N)
(νjy(nj) +νj0y(nj0))nΩ(x) =
X
j/nj∈∂N
νjy(nj)nΩ(x).
(iii) Let us derive the formula:
Z
∂(Ω,N)
ϕ0j(x)ψj(x) ds(x) = Z
Γ×N
(I0ϕ0C)j(y.nΩAµ)j(x) ds(x).
with (y.nΩAµ)j(x) =
Pd
k=1
P|E|
l=1yk(nj)nΩk(x)Ajlµl(x). Since IECϕ
0 =I0ϕ0
C Z
∂(Ω,N)
ϕ0j(x)ψj(x) ds(x) = nc
X
k=1
Z
Γ
ϕ0Ck(x) X j /(x,nj)∈∂(Ω,N)
But (ii) withνj =Ijk0 ψj(x) (k being frozen), providing that ψ =Aµ, says that X
j/(x,nj)∈∂(Ω,N)
Ijk0 ψj(x) = X
j/nj∈∂N
Ijk0 (Aµ)j(x)y(nj).nΩ(x).
Thus
Z
∂(Ω,N)
ϕ0j(x)ψj(x)ds(x) = Z
Γ×∂N
(I0ϕ0C)j(x)(y.nΩAµ)j(x) ds(x).
A reasoning similar to this made in (i) yields
= Z
Γ×N
(I0ϕ0C)j(y.nΩAµ)j(x)ds(x).
(iv) The end of the proof is done using (i), (iii) and the Green like formula:
−
nc
X
p=1
Z
Ω
ϕ0Cp(x)(∂τ∗µ)p(x) dx+
|N|
X
j=1
Z
Γ
(I0ϕ0C)j(x)(y.nΩAµ)j(x) ds(x)
=
|E|
X
l=1
Z
Ω
(∂τϕ0C)l(x)µl(x)dx.
4.4
Convergence of test functions
Let us introduce the set of admissible two-scale potentials
Ψ={(ψ0C,ψ1C,ψ0N C)∈L2(Ω)nc
×L2(Ω;R|perN|)×L2(Ω)|N| s.t. ∂τψ0C ∈L
2(Ω)|E|, (20)
IN−NCψ
1
C = 0, INC∪N0ψ
0
N C = 0, ψ
0
Cp(x) = 0 ∀x∈Γ0p for all p= 1..nc}.
For (ψ0C,ψ
1
C,ψ
0
N C)∈Ψ∩C1(Ω)nc× C1(Ω;R| N|
per)× C1(Ω)|N| let us define ψ0 andψ1 in R|N | by
ψ0I = ψ0Cp(x ε
µ+εy(nj)) for nj ∈NCp for p= 1, .., nc = ψ0N Cj(xεµ) for nj ∈N −NC,
ψ1I = ψ1Cj(x ε
µ)for nj ∈N −∂N
= 1
2(ψCj(x ε
µ) +ψCj0(xµε0)) fornj ∈∂N
where I ∼(µ, j)and(µ0, j0), (see in annex for details regarding(µ0, j0)).
Lemma 15 (i) ψb0 =I0ψ0
C+ψ
0
N C+O(ε). (ii) SvATψb
0
=∂τψ0C +IEN CA
T(ψ0
N C +I0ψ
0
C) +O(ε). (iii)ψb1 =ψ1C +O(ε).
(iv) IECA
Tψb1 =I ECA
Tψ1
Proof. (i) Let us prove successively that INCψb
0
(x) = I0ψ0C(x) +O(ε) and IN−NCψb
0
=
ψ0N C +O(ε). Let us start with INCψb
0
. For nj ∈NCp,
b
ψ0j(x) = X
µ
ψ0Cp(x ε
µ+εy(nj))χYε µ(x) =
X
µ
ψ0Cp(x ε µ)χYε
µ(x) +O(ε)
= X
µ
ψ0Cp(x)χYε
µ(x) +O(ε) =ψ
0
Cp(x) +O(ε).
Now we continue with IN−NCbψ
0
.For nj ∈N −NC :
(IN−NCψb
0
)j(x) = X
µ
ψ0N Cj(xεµ)χYε µ(x) =
X
µ
ψ0N Cj(x)χYε
µ(x) +O(ε) =ψ
0
N Cj(x) +O(ε).
(ii) Let us establish successively that IECSvA
Tψb0 = ∂
τψ0C + O(ε) and IEN CSvA
Tψb0 =
IEN CA
T(ψ0
N C +I0ψ
0
C) +O(ε). UsingIECSv =
1
εIEC,for el∈EC :
(IECSvA
Tψb0)
l(x) = 1
ε
|N|
X
j=1
Ajlψb
0
j(x)
= 1 ε X µ nc X p=1
|N|
X
j=1s.t. nj∈NCp
Ajlψ0j(x ε
µ+εy(nj))χYε µ(x).
But ψ0j(xεµ+εy(nj)) =ψ0j(xεµ) +
Pd
k=1∂xkψ
0
j(xεµ)εyk(nj) +εO(ε)then
= 1 ε X µ nc X p=1
|N|
X
j=1s.t. nj∈NCp
Ajlψ0j(x ε µ)χYε
µ(x) +
d X
k=1
Ajl∂xkψ
0
j(x ε
µ)εyk(nj)χYε
µ(x) +εO(ε).
Since P|jN=1| s.t.nj∈NCpAjl = 1−1 = 0 for all l and ∂xkψ
0
j(xεµ) = ∂xkψ
0
j(x) +O(ε) for x ∈ Yµε it remains 1 ε nc X p=1
|N|
X
j=1s.t. nj∈NCp
d X
k=1
Ajl∂xkψ
0
j(x)εyk(nj) +εO(ε) = (∂τψ0C)l+O(ε).
Now, IEN CSvA
Tψb0 =I EN CA
Tψb0 =I EN CA
Tψb0 =I EN CA
TI NN Cψb
0
.But
(INN Cψb
0
)j(xεµ) = ψ
0
N Cj(x ε
µ)for nj ∈N −NC
= ψ0Cp(xεµ+εy(nj)) fornj ∈NCp∩NN C.
But ψ0Cp(xε
µ+εy(nj)) = ψ0Cp(xεµ) +O(ε) (INN Cψb
0
)(xε µ) =ψ
0
N C(xεµ) +I0ψ
0
C +O(ε). Thus
(IEN CSvA
Tψb0)(x) = I
EN CSvA
T(ψ0
N C +I
0ψ0
C)(x) +O(ε).
This complete the proof of (ii). (iii) Fornj ∈NC −∂N
b
ψ1j(x) =X µ
ψ1Cj(xεµ)χYε µ(x) =
X
µ
ψ1Cj(x)χYε
µ(x) +O(ε) =ψ
1
For nj ∈∂N
b
ψ1j(x) = X
µ 1 2(ψ
1
Cj(xεµ) +ψ
1
Cj0(xεµ0))χYε µ(x)
but ψ1Cj0(xεµ0) =ψ1Cj0(xεµ) +O(ε) =ψ
1
Cj(xεµ) +O(ε) due to periodicity. Then
=X
µ
ψ1Cj(xεµ)χYε
µ(x) +O(ε) =ψ
1
Cj(x) +O(ε).
The global result ψb1j =ψ1Cj+O(ε) follows. (iv) comes from (iii) by applying IECA
T on each side of the equality.
5
Proof of the theorem 7
The point (i) has been established in the lemma 13. In order to state (ii) and (iii), we establish the so called two-scale model which is posed on both the cell circuit and the macroscopic domain
Ω. From (i) we know thatϕ0 and ofv0 can be expressed with respect to the fieldsϕ0
C, ϕ1C and ϕ0
N C so that they satisfy the expression (15) and (16).
Lemma 16 Under the assumptions (H0-H4), (ϕ0
C,ϕ1C,ϕ0N C) ∈ Ψ and i0 ∈ L2(Ω)|E| are solu-tions of the two-scale circuit equasolu-tions
R0i0(x) +M0v0(x) = u0s(x) for all x∈Ω (21) (i0,∂τψ0C +IECA
Tψ1
C+IEN CA
T(ψ0
N C +I
0ψ0
C) = 0 for all (ψ
0
C,ψ
1
C,ψ
0
N C)∈Ψ.
with Ψdefined in (20).
In order to prove (iii), we replacev0 by its expression and pose ψ0C = 0 :
v=IECA
T
ϕ1C +IEN CA
T ϕ0N C
R0i0(x) +M0v(x) =u0s(x)−M0(∂τϕ0C +IEN CA
TI0ϕ0
C)(x) for all x∈Ω (i0, IECA
T
ψ1C+IEN CA
T
ψ0N C) = 0.
This proves that(ϕ1C,ϕN C0 , i0, v)is solution of the cell problem (9) with(η,θ, us) = (ϕ0C,∇ϕ0C, u0s)(x) at a given x and (ψC,ψN C) := (ψ1C,ψ0N C). Remark that INC∪N0ψ
0
N C = 0 has been replaced by
IN0
C∪N−NCψC = 0for the sake of uniqueness of ϕC.
(ii) Thanks to the assumption (H5) and to (iii) we know that
i0 =Liϕ0C+Pi∇ϕ0C +Hiu0s,
ϕN C =Lϕϕ0C +Pϕ∇ϕ0C +Hϕu0s,
Replacing in the two-scale branch equations leads to
(R0Pi+M0(τ+Lv))∇ϕ0C+ (R
0
Li+M0(IEN CA
TI0+
Lv))ϕ0C = (I−R
0
Hi−M0Hv)u0s
or equivalently to QH∇ϕ0C +SHϕ0C =FHu0s. Now, posing ψ
1
C =ψ
0
N C = 0 it follows that Z
Ω×E
(Pi∇ϕ0C +Liϕ0C)j(x)(∂τψ0C +IEN CA
TI0ψ0
C)j(x)dx
= −
Z
Ω×E
(Hiu0s)j(x)(∂τψ0C+IEN CA
TI0ψ0
C)j(x) dx for all ψ0C ∈Ψ H.
Applying standard argument in related to variational formulations of partial differential equa-tions yields to the partial differential equation (172) and its associated boundary conditions.
It remains to prove the lemma 16.
Proof. The fact that(ϕ0
C,ϕ1C,ϕ0N C)∈Ψcomes from the lemma 13. It remains to derive the equations (21). We start from the circuit equations (5). Let us apply the two-scale transform and the lemma 10 to the first equation and the scalar product preservation (19) and the lemma 12 to the second equation:
Mbv+Rbi=ubs and(bi, ATψb) = 0.
Introducing the scaled two-scale transforms (7) and (8) of vectors and matrices
MεSvATϕbε+Rεbiε =ubsε and(biε, SvATψb) = 0.
The scalar product between the first equation and a test function j ∈L2(Ω)|E| yields
(MεSvATϕbε, j) + (Rεbiε, j) = (ubsε, j)and (biε, SvATψb) = 0
or equivalently
(SvATbϕε, MεTj) + (biε, RεTj) = (ubsε, j) and(biε, SvATψb) = 0.
Thanks to (H3) and (H4) and the lemma 15 one may pass to the limit ε→0
(v0, M0Tj) + (i0, R0Tj) = (u0s, j) and(i0, w0) = 0.
The first equation being valid for all j ∈ L2(Ω)|E| is also equivalent to R0i0 +M0v0 = u0s. According to the lemma 15 for each (ψ0C,ψ
1
C,ψ
0
N C)∈Ψ, there exists such aw0 with
w0 =∂τψ0C +IECA
Tψ1
C+IEN CA
T(I0ψ0
C +ψ
0
N C).
Plugging this expression in the second equation ends the proof.
6
Examples
6.1
Example 1
The nodes and branches are numbered according to the figure, nc = 1, EC = {e1, e2, e3, e4},
EN C = {e5}, NC = {n1, n2, n3, n4, n5}, NN C = {n2, n6}, N0 = {n6}, NC0 = {n2} (arbitrary
choice in NC). The local matrices are
R = Ã
rI4 04×1
01×4 1
!
, M =Mε =M0 = Ã
−I4 04×1
01×4 0
!
, ubs = Ã
04
bis !
,
Sv =
à 1
εI4 04×1 01×4 1
!
, Sc= Ã
I4 04×1
01×4 1ε
!
, Πc= Ã
04×4 04×1
01×4 1ε
!
,
Πv = Ã
I4 04×1
01×4 0
!
, Ss = 1
εI5, R
ε= Ã
1
εrI4 04×1 01×4 1
!
, ubεs = Ã
04 1
εbis !
.
So we pose r = εr0 andbis = ε(i0s +O(ε)) then R0 = Ã
r0I4 04×1
01×4 1
!
, u0
s = Ã 04 i0 s ! . The
incidence matrix is
AT =
1 −1 0 0 0 0
0 1 −1 0 0 0
0 1 0 −1 0 0
0 −1 0 0 1 0
0 1 0 0 0 −1
.
HereψN C = 0, thenΨm =
{ψC ∈R6/ψ
C =Jψ∗Cwhereψ∗C ∈R2}withJ = Ã
1 0 1 0 0 0 0 0 0 1 1 0
!T
.
Moreovery(n) = 1 2
Ã
−1 0 1 0 0 0
0 0 0 1 −1 0
!
,τ =−1 2
Ã
1 1 0 0 0 0 0 1 1 0
!T
,I0 =³ 1 1 1 1 0 ´T
u0
s =
³
0 0 0 0 i0s
´T
. The problem micro has the form K(i,ϕ∗
C)T =L(θ,η, is)T (here we prefer to work withi0
s in place of the wholeu0s).An explicit calculation shows thatG=K−1L= Ã
G11 04×2
03×2 G22
!
withG11 =−21r0
Ã
1 1 0 0 0 0 1 1
!T
andG22=
Ã
0 0 0 1 0 0
!T
.ThusLi,Pv and
Hv vanish, Pi = 21r0τ, Hi = Lv = ³
0 0 0 0 1
´T
. Then, QH, SH and FH vanish and ϕ0
C is governed by the Laplace equation
∆ϕ0C = 2r0i0s inΩ
with the boundary conditions ϕ0C = 0 onΓ0 and∇ϕ0C.n = 0 onΓ−Γ0. Finally, the two-scale
current and voltages are given by
i0 = ³ ∂x1ϕ
0
C ∂x1ϕ
0
C ∂x2ϕ
0
C ∂x2ϕ
0
C is0
´T
v0 = ³ ∂x1ϕ
0
C ∂x1ϕ
0
C ∂x2ϕ
0
C ∂x2ϕ
0
C ϕ0C
´T
6.2
Example 2
Here nc = 0, EC =∅, EN C ={e1, e2}, NC =∅, NN C ={n1, n2}. The local matrices are
R = Ã
r 0 0 0
!
, M =Mε =M0 = Ã
−1 0
0 1
!
, bus=buεs= Ã 0 b vs ! ,
Sv = I2, Sc= 1
εI2, Πc= 0, Πv =I2, Ss=I2, R
ε = Ã
εr 0
0 0
!
, AT = Ã
1 −1
−1 1
!
.
So we poser= 1εr0 andbvs =vs0+O(ε)thenR0 = Ã
r0 0
0 0
!
, u0
s =
à 0
v0s !
.SinceEC =∅there
is no macroscopic model and (i0, v,ϕ0
N C) solves only the cell problem with Ψm = {ψN C ∈ R2 / ψN C = Jψ∗N C where ψ∗N C ∈ R} with J =
à 1 0
!
. So The problem micro has the form
K(i,ϕ0∗
N C)T =L(vs)T which leads toi0 =− v0
s
r0(1,1) T ϕ0
N C =−v0s and v =v0s(−1,1)T.
6.3
Example 3
Here nc = 1, EC = {e1, e2, e3, e4, e5}, EN C = {e6, e7}, NC = {n1, n2, n3, n4, n5}, NN C =
{n6, n7}, N0 ={n7}, NC0 ={n2} (arbitrary choice inNC),
R = Ã
rI4 04×3
03×4 δ11+δ33
!
, M =Mε= Ã
−I4 04×3
03×4 kδ13+δ22
!
, ubs = 05 b vs 0 ,
Sv = Ã
1
εI5 05×2 02×5 I2
!
, Sc= Ã
I5 05×2
02×5 1εI2
!
, Πc= Ã
04×4 04×2
02×4 1ε(δ11+δ33)
!
,
Πv = Ã
I4 04×3
03×4 δ22
!
, Ss = Ã
1
εrI4 04×3
03×4 δ11+δ22+ 1εδ33
!
,
Rε = Ã
1
εrI4 04×3 03×4 δ11+δ33
!
, buεs= 05 b vs 0
where we used the submatrix δij having all its entries vanishing excepted the entry(i, j). The size of such a submatrix is known by its surrounding submatrices. So we pose r= εr0, k =k0
and bvs =v0s +O(ε) then R0 = Ã
r0I4 04×3
03×4 δ11+δ33
!
, u0
s = (05, vs0,0)T. The incidence matrix is
AT = Ã
X11 X12
X21 X22
!
where
X11=
1 −1 0 0 0
0 1 −1 0 0
0 1 0 −1 0
0 −1 0 0 1
, X12= 04×2, X22 =δ11−δ12, X22=
0 0
1 −1
−1 1
y(n) is the same than in example 1, τ = −1 2
Ã
1 1 0 0 1 0 0 0 0 1 1 0 0 0
!T
and I0 = INC. Here
Ψm ={(ψC,ψN C) = (JCψ∗C, JN Cψ∗N C) where(ψ∗C,ψ∗N C)∈R2×R}with
JC = Ã
1 0 1 0 0 0 0 0 0 0 1 1 0 0
!T
and JN C = ³
0 0 0 0 0 1 0
´T
. The problem
mi-cro has the form K(i,ϕ∗
C,ϕ∗N C)T = L(θ,η, v0s)T (here we prefer to work with v0s in place of
the whole u0s). An explicit calculation shows that G = K−1L = Ã
G11 G12
06×2 G22
!
with G11 =
−21r0 Ã
1 1 0 0 0 0 1 1
!T
, G12= 21k
0 Ã
0 0 0 0
−1 1 0 0
!
andG22=
Ã
0 0 0 0 0 0
k0 0 0 −12k0r0 0 1
!
.
Since
i=Ti(θ,η, vs)T,
v=IECA
TJ
Cϕ∗C +IEN CA
TJ
N Cϕ∗N C =Tv(θ,η, vs)T, andϕN C =JN Cϕ∗N C =Tϕ(θ,η, vs)T
with Ti = [K−1L]{1,..,7}×., Tv =IECA
TJ
C[K−1L]{8,9}×. +IEN CA
TJ
N C[K−1L]{10}×.,
Tϕ = JN C[K−1L]{10}×.. Then Pi = [Ti].×{1,2} = −
1 2r0
Ã
1 1 0 0 0 0 0 0 0 1 1 0 0 0
!T
, Li =
[Ti].×{3} = 0, Hi = [Ti].×{4} = k20(−1,1,0,0,2,0,0)
T,
Pv = [Tv].×{1,2} = 0, Lv = [Tv].×{3} = 0
and
Hv = [Tv].×{4} = (−r02k0,
r0k0
2 ,0,0,
r0k0
2 ,1,−1)
T,
Pϕ = [Tϕ].×{1,2} = 0, Lϕ = [Tϕ].×{3} = 0
and Hϕ = [Tϕ].×{4} = (0,0,0,0,0,1,0)T. QH, SH and FH vanishes and ϕ0C is governed by the Laplace equation
−∆ϕ0C =r0∂x1(k0v
0
s) inΩ
with the boundary conditions ϕ0C = 0 onΓ0 and∇ϕ0C.n = 0 onΓ−Γ0. Finally, the two-scale
current and voltages are given by
i0 =−1 2(
∂x1ϕ
0
C
r0
+k0v
0
s 2 ,
∂x1ϕ
0
C
r0 −
k0v0s 2 ,
∂x2ϕ
0
C
r0
,∂x2ϕ
0
C
r0
,2k0,0,0)T,
v0 =−1 2(∂x1ϕ
0
C−r0k0vs0,∂x1ϕ
0
C −r0k0v0s,∂x2ϕ
0
C,∂x2ϕ
0
C,∂x1ϕ
0
C −r0k0,1,−1)T,
and ϕ0N C = (0,0,0,0,0,1,0)Tvs0.
6.4
Numerical validation
Let us report the result of our simulation in the third example. First, we compare the two solutions computed on the one hand on the periodic network of 15×15cells and on the other hand by using the homogenized model. The calculation have been carried out for the values
counterpart bvsε for the periodic circuit is taken equal to vs(xεi), the xεi being the centres of the cells . The node0s voltage ϕ0C is computed using a P1 finite elements method with 15× 15 elements.
The first component of the two-scale transformbϕε1(xεi)is compared with the first component of the approximation, with the finite elements method, of the limitϕ0at the pointxεi.The results are presented respectively to the left and to the right of the first row of the figure below. The second figure presents the first component of the two-scale transform of the branch s voltages
b
v1ε(xεi)and the approximation of the first component of the limit v1 at the point xεi. They are placed respectively to the left and to the right of the second row. The results show a good qualitative agreement between the two models.
-1 -0.8 -0.6 -0.4 -0.2 0
0 0.5 1 0
0.2 0.4 0.6 0.8 1
φε : node =1
-1 -0.8 -0.6 -0.4 -0.2 0
0 0.5 1 0
0.2 0.4 0.6 0.8 1
φ0 : node =1
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
0 0.5 1 0
0.2 0.4 0.6 0.8 1
vε : edge =1
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
0 0.5 1 0
0.2 0.4 0.6 0.8 1
vH : edge =1
Comparison of the complete model and the two-scale model
Quantitative comparisons are detailed on the next figure. Global relative errors, inL2 norm, for
node0s voltages and branch s voltages are compared when the periodic network has 10, 15 or 20
cells in each direction and when the finite elements methods is with N = 10,15 or20 elements in each direction. It shows that, in this case, the errors diminish with the increase of the cell number but is not so much in! uenced by the number of finite elements. The observation of the
ratio t H
tε of the simulation times of the two models yields to the conclusion that the homogenized
10 15 20 0
0.05 0.1 0.15
0.2 Potential relative errors
N 10 cells
15 cells 20 cells
10 15 20
0 0.05 0.1 0.15
0.2 Voltage relative errors
N 10 cells
15 cells 20 cells
10 15 20
0 0.5 1
1.5 Ratio tH/teps
N 10 cells
15 cells
20 cells
Errors and simulation times
7
Annex
The proof of the lemma 14 necessitates the fundamental lemma 20 stated and proved hereafter. It requires additional results on two-scale transform for nodes that we establish at first. The proof includes quite long calculations that we do not want to see in the core of the paper. However, we must emphasis that it constitutes an important part of our work.
7.1
Properties of the two-scale transform
T
NLet us recall that the set N is made of nodes nI with I∈ {1, ..,|N |} and the set N of nodes
nj with j ∈ {1, ..,|N|}. The subset of N of nodes belonging to the boundary of the cell Y is denoted by ∂N. Because N is εY−periodic, it turns out that each node n ∈ ∂N has its counterpart n0 ∈ ∂N on the opposite side. If the former0s index is j then the latter0s one is
denoted by j0.The outward normal vector to the boundary of Y atn being denoted byn
Y(n), it turns out that nY(n0) = −nY(n). For a given multi-integer µ ∈ {1, .., m}d we define the multi-integer
µ0 =µ+nY(n)
associated tonandµ.Let I and(µ, j)be linked through the relation I∼(µ, j).IfnI belongs to only one cell then there exists an unique such (µ, j). IfnI is located at the interface between two cells then I is associated to two couples (µ, j) and (µ0, j0) with µ0 and j0 derived as above from µ and j. In short we say that I∼ (µ, j) and I∼ (µ0, j0). Conversely if two couples (µ, j)
and (λ, l) correspond to the same I then j is the index of a node located on the boundary of the cell and (λ, l) = (µ0, j0).These statements are condensed in the next proposition.
Proposition 17 Two couples(µ, j) and(λ, l)come from a same index I if and only ifnj ∈∂N and (λ, l) = (µ0, j0).
The map that send a vector u ∈ R|Nε| towards a tensor U ∈ Rmd × R|N| has been well defined in § 3.1. From the above discussion, it is clear that it is not onto. There exist some
Proposition 18 A tensor U ∈ Rmd×R|N| is the image of u ∈ R|Nε| if and only if Uµj = Ukl for all couples (µ, j), (λ, l) such that nj ∈∂N and (λ, l) = (µ0, j0).
Proof. We must prove that Uµj = Ukl for all couples (µ, j) and (λ, l) associated a same index I. The proposition 17 yields the conclusion.
It becomes clear that if∂N 6=∅thenTN is not onto inP0(Ω)|N|.Let us state the compatibility conditions on v ∈P0(Ω)|N| insuring that it has a counter-image byT
E. For a given x∈Ω and node n∈∂N we define
x0 =x+εnY(n).
Proposition 19 A function v∈P0(Ω;R|N|) is the two-scale transform of a vectorR|Nε| if and only if
vj0(x0) =vj(x)
for all (x, nj)∈Ω×∂N and for x0 =x+εnY(nj).
Proof. Since v ∈ P0(Ω)|N| it may be written vj(x) = P
µ∈{1,..,m}dVµjχYε
µ(x). From the
proposition 17 there exists u ∈ R|Nε|
such that Uµj = Vµj if and only if Vµj = Vµ0j0 for
all µ ∈ {1, .., m}d and j ∈ ∂N. In other words vj(xµε) = vj0(xεµ0) or equivalently vj(xεµ) =
vj0(xεµ+εnY(nj))because xµε0 =xεµ+εnY(nj).The result follows remarking that v is piecewise constant with respect to x.
For a given noden∈∂N,the largest subset of x∈Ωsuch that x0 ∈Ω is denoted byΩ(n) :
Ω(n) ={x∈Ω such thatx0 ∈Ω}.
Because nY(n0) = −nY(n) one may observe that x = x0 +nY(n0), so x ∈ Ω(n) if and only if
x0 ∈Ω(n0).
The outward normal vector to the boundary Γ of Ω in a point x ∈ Γ is denoted by nΩ(x)
and the subset of couples(x, n)∈Γ×∂N having the same normalnY(n) andnΩ(x) is denoted
by
∂(Ω, N) = {(x, n)∈Γ×∂N such that nY(n) =nΩ(x)}.
A straightforward characterization of the complementary set Ω−Ω(n) of Ω(n)follows:
Ω−Ω(n) ={x=x−εθnY(n)where θ ∈(0,1) and(x, n)∈∂(Ω, N)}. (22)
7.2
Fundamental lemma
Lemma 20 For φ∈P0(Ω;R|N|) belonging to the range ofTN and ψ ∈C1(Ω;R|antiperN| ) then
1
ε
Z
Ω×∂N
φj(x)ψj(x) dx=− Z
Ω×∂N
more precisely
|1ε
Z
Ω×∂N
φj(x)ψj(x) dx+ Z
Ω×∂N
φj(x)(y.∇xψ)j(x) dx−b(φ,ψ)|
≤ε(e1(φ,ψ) +e2(φ, y.∇xψ) +e2(φ,ψ)).
Here
y.∇xψ ∈ C0(Ω)|N|,(y.∇xψ)j(x) = d X
l=1
yl(nj)∂xlψj(x),
b(φ,ψ) = X j∈∂N
Z
Γ
χ∂(Ω,N)(x, j)φj(x)
Z 1
0
ψ(x−εθnY(nj))dθ dx
e1(φ,ψ) =
1
2||φ||∂N(||
∆εnYψ−nY.∇xψ
ε ||∂N +||(y−
nY
2 )∇x∆εnYψ||∂N)
e2(φ,ψ) = (
Z
∂(Ω,N)
φ2j(x) dx)1/2( Z
∂(Ω,N)
Z 1
0
ψ2j(x−εθnY) dθdx)1/2
where nY(n)is set to zero for n /∈∂N,(∆εnYψ)j(x) =
ψj(x+εnY(nj))−ψj(x)
ε ,
R|antiperN| ={φ ∈L
2(Ω;
R|N|) such thatφj(x) = −φj0(x) a.e. x∈Ω for all nj ∈∂N}
and for shortness we have used the notations
(φ,ψ)∂N =
Z
Ω×∂N
φj(x)ψj(x) dx and||φ||∂N = (φ,φ)1∂N/2.
Proof. For each n∈∂N we use the partition of Ωin Ω(n) and its complementary so that
(φ,ψ)∂N =aint(φ,ψ) +ab(φ,ψ)
where
aint(φ,ψ) = X
nj∈∂N
Z
Ω(nj)
φj(x)ψj(x)dx et ab(φ,ψ) = X
nj∈∂N
Z
Ω−Ω(nj)
φj(x)ψj(x)dx.
(i) The characterization (22) ofΩ−Ω(n) yields|ab(φ,ψ)|=|εb(φ,ψ)|≤εe2(φ,ψ).
(ii) Let us prove that
aint(φ,ψ) =−aint(φ,ψ(x+εnY(n))).
In the one side φbelongs to the range of TE and the proposition 19 tell us thatφj(x) =φj0(x0)
and in the other side ψj =−ψj0. Then
aint(φ,ψ) =− X
nj∈∂N
Z
Ω(nj)
For a given j let us first apply the variable change x→x0 =x+εnY(nj) which maps Ω(nj) to
Ω(nj0)and in a second step let us replace the numbering by j with a numbering byj0 it comes
=− X
nj0∈∂N
Z
Ω(nj0)
φj0(x0) ψj0(x) dx0 =−aint(φ,ψ(x0)) = −aint(φ,ψ(x+εnY(n))).
(iii) Let us deduce that
|1
εaint(φ,ψ) +aint(φ,(y.∇x)ψ)| ≤εe1(φ,ψ). (23)
Thanks to (ii),
1
εaint(φ,ψ) =
aint(φ,ψ)−aint(φ,ψ(x0))
2ε =−
1
2aint(φ,∆εnYψ). (24)
For n ∈ ∂N, we make use of the decomposition y(n) = [y(n)] +{y(n)} in its periodic part
{y(n)} = (y(n) +y(n0))/2 and its counter-periodic parts [y(n)] = (y(n)−y(n0))/2. For n ∈ N −∂N, [y(n)]and{y(n)}are set to0. From the triangular inequality,
|1
εaint(φ,ψ) +aint(φ, y.∇xψ)| ≤ |
1
εaint(φ,ψ) +aint(φ,[y].∇xψ)|
+|aint(φ,[y].∇xψ)−aint(φ, y.∇xψ)|
combined with (24) and the fact that nY(y) = 2[y] :
≤|1
2aint(φ,∆εnYψ−nY.∇xψ)|+|aint(φ,{y}.∇xψ)|.
Applying (24) with ψj :={y(nj)}∇xψj:
aint(φ,{y}∇xψ) =− ε
2aint(φ,{y}.∇x∆εnYψ)≤
ε
2||φ||∂N||{y}.∇x∆εnYψ||∂N
thus
≤ ε2||φ||∂N(||∆εnYψ−nY.∇xψ
ε ||∂N +||{y}.∇x∆εnYψ||∂N)
which is the wanted result (23). (iv) Thus
|1
εaint(φ,ψ) + (φ, y.∇xψ)∂N|≤ε(e1(φ,ψ) +e2(φ, y.∇xψ)).
after remarking that
(φ, y.∇xψ)∂N −aint(φ, y.∇xψ) =ab(φ, y.∇xψ)
and by using (i).
(v) The conclusion comes from
|1ε(φ,ψ)∂N + (φ, y.∇xψ)∂N −b(φ,ψ)|≤|
1
εaint(φ,ψ) + (φ, y.∇xψ)∂N|+|
1
