PORTUGALIAE MATHEMATICA
Vol. 53 Fasc. 2 – 1996
ASYMPTOTIC STUDY OF LATTICE STRUCTURES
WITH DAMPING *
H.T. Banks, D. Cioranescu and R.E. Miller
Abstract: This paper considers a time-dependent system with damping and general
(non-zero) initial conditions on a perforated domain and presents a careful derivation of
the homogenized system.
1 – Introduction
Applications of partial differential equations often involve equations of the
form
(1)
A ² u² = f
in Ω
where A² is a family of operators depending on the small parameter ² and Ω is a
bounded open subset of IRN . For example, the coefficients in A² may be periodic
functions with period ² (due, e.g., to a periodic mixing of two different materials).
For small ² it can be quite difficult to obtain an accurate approximation to u ² by
standard numerical methods. Thus we seek a simpler problem
(2)
Au = f
in Ω
with the property that for sufficiently small ², the solution u of (2) accurately
approximates the solution u² of (1). Unfortunately, one cannot obtain system (2)
simply by averaging the coefficients in (1) over one period. Instead, one can use
the homogenization techniques discussed in [5] and [14].
Received : January 25, 1995; Revised : January 27, 1996.
* This research was supported in part by the Air Force Office of Scientific Research
under grant AFOSR-90-0091 and in part by the National Science Foundation under grant
DMS-8818530.
210
H.T. BANKS, D. CIORANESCU and R.E. MILLER
One can also use these techniques for problems of the form of (1) defined on
a perforated domain Ω² obtained by periodically removing material from Ω (see
[7], [8]) In this case, the goal is to obtain a limit problem defined on all of Ω
(which may be simply connected) rather than on the original domain with many,
possibly thousands, of small holes. In this paper we consider time dependent
systems defined on a perforated domain Ω²µ depending on two small parameters
² and µ. The domain Ω²µ , which is that part of Ω covered by material, is obtained
Q
as follows. Set Y = N
i=1 [0, li ] and let Tµ ⊂ Y be such that the boundary ∂Tµ
of Tµ does not meet the boundary ∂Y of Y . Let χ∪Y ∗ denote the characteristic
µ
function of Yµ∗ extended by periodicity to all of IRN . Then we define Ω²µ as
½
Ω²µ = x ∈ Ω | χ∪Y ∗
µ
µ ¶
x
²
=1
¾
.
We shall assume throughout our presentation that the holes do not meet the
boundary ∂Ω. This assumption restricts the geometry of Ω (e.g., Ω can be a
finite union of rectangular cells homothetic to the representative cell Y ) and the
values taken by ² (e.g., ² ∈ {n−1 } or ² ∈ {2−n }). Physically, this assumption
means that the material is distributed along the faces of Y rather than along the
edges. This assumption is needed for the construction of the extension operators
given by Lemmas 2.1 and 2.2.
An example of such a domain is the grid Ω²µ depicted in Fig. la. A natural
period of the grid is depicted in Fig. lb. This grid is typical of actual engineering
structures.
Fig. 1 – The grid Ω²µ and the representative cell Y .
ASYMPTOTIC STUDY OF LATTICE STRUCTURES WITH DAMPING
211
Such structures are characterized by a periodic distribution of holes and a
relatively small amount of material (i.e., µ is small compared to l1 and l2 ).
We note that Ω²µ is composed of layers of thickness ²µ with holes of dimension
²(l1 − µ) × ²(l2 − µ). If we take the transformation y = x² , we are in IR2 covered
periodically by cells of Y -type. We denote by T²µ the union of all the holes of
Ω²µ .
Consider the following problem (here and throughout the paper we adopt the
summation convention on repeated indices unless explicitly stated otherwise):
µ
µ ¶
∂ 2 u²µ
∂
x ∂u²µ
ρ
−
a
ij
∂t2
∂xi
² ∂xj
¶
=f
u²µ = 0
(3)
aij
u²µ (0) = u²µ
0 ,
∂u²µ
ni = 0
∂xj
in Ω²µ ,
on ∂Ω
on ∂T²µ ,
²µ
u²µ
t (0) = v0 ,
where n = (ni ) is the outward unit normal. We make the following assumptions:
1.I. The nonhomogeneous forcing function f satisfies f ∈ L2 (0, T, L2 (Ω));
1.II. The coefficients satisfy aij (·) ∈ L∞ (IR2 );
1.III. There exists A > 0 such that aij ξi ξj ≥ A ξi ξi for all ξ ∈ IR2 ;
1.IV. The coefficients aij are Y -periodic;
²µ
²µ
1
1.V. The initial conditions satisfy u²µ
0 ∈ H (Ω²µ ), u0 = 0 on ∂Ω and v0 ∈
L2 (Ω²µ ).
By Theorem 29.1 in [17], we have that (3) has a unique solution u²µ . Our
goal is to study the dependence of u²µ on the parameters ² and µ. The equation
we obtain by passing to the limit as ² → 0 is called the “homogenized” equation
and is defined on all of Ω. The techniques used here to take the limit are similar
to those used in problems involving composite media (see [5] and [14]). The
homogenized coefficients are expressed in terms of functions defined on Yµ∗ , hence
the homogenized system depends on µ. For the static case, it is proved in [1], [8]
and [13] that when letting µ → 0 in the homogenized system, one recovers the
simplicity of the original coefficients. We prove here that a similar result holds
for systems with damping. We remark that Francfort et al. (see [9] and [10]) have
studied similar problems for non-perforated domains.
We close this section with a summary of notation and conventions adopted
throughout the paper. In Sections 2 and 3 we consider our problem with µ fixed
212
H.T. BANKS, D. CIORANESCU and R.E. MILLER
and take the limit as ² → 0. Hence, for now we suppress the µ in order to simplify
the notation.
We will make use of the following function spaces throughout the paper: Let
n
o
VY ∗ = ϕ ∈ H 1 (Y ∗ ) | ϕ is Y -periodic ,
where “Y -periodic” means that the function has equal values on opposite edges
of Y . We define
n
o
V² = ϕ ∈ H 1 (Ω² ) | ϕ = 0 on ∂Ω ,
and
V = H01 (Ω) ,
H² = L2 (Ω² ) ,
H = L2 (Ω) .
Note that ϕ ∈ V² vanishes only on the external boundary of Ω² . The embeddings
V² ,→ H² ,→ V²∗ and V ,→ H ,→ V ∗ define Gelfand triples. We denote by
h·, ·iV²∗ ,V² and h·, ·iV ∗ ,V , respectively, the corresponding duality pairings. The
inner products on the spaces H and H² will be denoted h·, ·iH and h·, ·iH² . Also,
∂ϕ
.
for a function ϕ ∈ V or ϕ ∈ V² , we use the symbol ∂i ϕ to denote ∂x
i
2
For any function g ∈ L (Ω² ) we will denote by ge the extension by zero of g to
the whole domain Ω. For any measurable set E, |E| denotes the measure of E,
and χE denotes the characteristic function of E; i.e.,
χE (z) =
(
1 for z ∈ E,
0 for z ∈ IR2 \E .
If f ∈ L1 (E), we denote the mean value of f by ME (f ):
1
ME (f ) =
|E|
Z
f (x) dx .
E
The symbol C will be used interchangeably for different constants which are
independent of ².
We shall make frequent use of the following lemma (see [14, p. 57]).
Lemma 1.1. Suppose the Y -periodic function f ∈ L2 (Y ) is extended periodically to all of IR2 . If we define F² (x) = f ( x² ), then as ² → 0, F² → MY (f ) in
L2 (Ω) weakly.
Remark 1.1. Let χ∪Y ∗ denote the extension by periodicity of the characteristic function of Y ∗ to all of IR2 . Then χΩ² (x) = χ∪Y ∗ ( x² ), so by Lemma
1.1,
χΩ² → MY (χY ∗ ) = θ in L2 (Ω) weakly .
ASYMPTOTIC STUDY OF LATTICE STRUCTURES WITH DAMPING
213
Remark 1.2. If f ∈ L∞ (Y ), then as ² → 0, F² → MY (f ) in L∞ (Ω) weak*.
2 – Homogenization of time-dependent systems
Set a²ij (x) = aij (x/²). The weak form of (3) is
(4)
hρ u²tt (t), ϕiV²∗ ,V² + σ1² (u² (t), ϕ) = hf (t), ϕiH²
u² (0) = u²0 ,
for all ϕ ∈ V² ,
u²t (0) = v0² ,
where the sesquilinear form σ1² (·, ·) is defined by
σ1² (ϕ, ψ) =
Z
Ω²
a²ij ∂j ϕ ∂i ψ dx
for ϕ, ψ ∈ V² .
We assume that u²0 ∈ V² and v0² ∈ H² and moreover that
ku²0 kV² ≤ C
(5)
and
kv0² kH² ≤ C ,
where C is independent of ². By Theorem 29.1 in [17], we have that (4) has
a unique solution u² in the extended V²∗ sense with u² ∈ L2 (0, T ; V² ), u²t ∈
L2 (0, T ; H² ) and u²tt ∈ L2 (0, T ; V²∗ ). A careful examination of the proof of this
theorem reveals that in fact ku² (t)kV² ≤ C and ku²t (t)kH² ≤ C for almost every
t ∈ [0, T ] where C is independent of ² (see [3] and [11, p. 268]). Hence, we in fact
obtain u² ∈ L∞ (0, T ; V² ) and u²t ∈ L∞ (0, T ; H² ). We make use of the following
extension results (see [6], [7]).
Lemma 2.1. There exists an extension operator
P ² ∈ L(V² , V )
such that
kP ² u² kV ≤ C ku² kV²
for all u² ∈ V² .
Remark 2.1. The above lemma can be generalized to produce extensionoperators P`² ∈ L(H ` (Ω² ), H ` (Ω)) (where Ω² , Ω ⊂ IRN with `, N ∈ IN) which
preserve derivative bounds independently of ². For details, see [12].
Lemma 2.2. There exists an extension operator
³
Q² ∈ L L∞ (0, T ; V² ), L∞ (0, T ; V )
´
214
H.T. BANKS, D. CIORANESCU and R.E. MILLER
such that
X
kDα Q² ukL∞ (0,T ;L2 (Ω)) ≤ C
|α|=1
X
kDα ukL∞ (0,T ;L2 (Ω² )) ,
|α|=1
and, if ut ∈ L∞ (0, T ; V² ), then Q² ut = (Q² u)t in Ω × [0, T ].
Remark 2.2. If u ∈ L∞ (0, T ; V² ), then from the construction of P ² and Q² ,
we have that Q² u(t, x) = [P ² u(t, ·)](x) (see [2], [12]).
Thus, we have the existence of a constant C independent of ² such that
kQ² u² kL∞ (0,T ;V ) ≤ C ;
hence, there exists a subsequence of {²} and a function u ∈ L∞ (0, T ; V ) such that
Q ² u² → u
(6)
in L∞ (0, T ; V ) weak* .
We also need the following lemma.
Lemma 2.3. Suppose {ϕ² } ⊂ V² with kϕ² kV² ≤ C where C is independent of
². Then there exists a subsequence, still denoted {ϕ² }, such that ϕe² → ϕ weakly
in H as ² → 0. Moreover, ϕ ∈ V .
Proof: First of all, notice that kϕe² kH = kϕ² kH² ≤ kϕ² kV² so there exists
ϕ ∈ H and a subsequence such that ϕe² → ϕ weakly in H; i.e.,
Z
²
Ω
ϕe ψ dx →
Z
ϕ ψ dx
for all ψ ∈ H .
Ω
But kP ² ϕ² kV ≤ C, so there exists ϕ ∈ V such that P ² ϕ² → ϕ weakly in V ; hence,
P ² ϕ² → ϕ strongly in H. Thus, for all ψ ∈ H
Z
Ω
so ϕ = θϕ ∈ V .
ϕe² ψ dx =
Z
Ω
χΩ² P ² ϕ² ψ dx
Z
Ω
χΩ ² →
Z
θ ϕ ψ dx ,
Ω
Remark 2.3. Observe that ϕ, hence ϕ, is independent of P ² .
By the assumption that the initial data are bounded independently of ², (see
(5)) there exist u0 ∈ V and v0 ∈ H such that

e²0 → u0 
u
ve0²
→ v0 
weakly in L2 (Ω) .
ASYMPTOTIC STUDY OF LATTICE STRUCTURES WITH DAMPING
215
Consider the equation
hρ θ utt , ϕiV ∗ ,V +
(7)
Z
Ω
qij ∂j u ∂i ϕ dx = hθf, ϕiH
u(0) = u0 /θ ,
for all ϕ ∈ V ,
ut (0) = v0 /θ ,
where the qij are the homogenized coefficients obtained for the static Neumann
problem (see [7]) and are given by
1
qij =
|Y |
Z
Y∗
µ
¶
∂χj
aij − ail
dy
∂yl
with χj the Y -periodic solutions of
Z
Y∗
∂(χj − yj ) ∂ϕ
dy = 0
∂yk
∂yl
alk
for all ϕ ∈ VY ∗ .
The qij satisfy the ellipticity condition 1.III, so by standard results (see [17]),
Eq. (7) also has a unique solution uh . We are now ready to prove the following
theorem.
Theorem 2.4. The limit u in (6) is the unique solution of (7).
Proof: We must show that u = uh . First we extend all functions defined on
[0, T ] by zero for t > T . Taking the Laplace transform of (4), we obtain
2
²
b (s), ϕiH² +
(8) hρ s u
Z
b² (s) ∂i ϕ dx = hfb(s), ϕiH ² + hρ(s u²0 + v0² ), ϕiH²
a²ij ∂j u
Ω²
R
b² (s) = L[u² ](s) = 0∞ e−st u² (t) dt is the Laplace transform
for all ϕ ∈ V² , where u
b² (s) satisfying
of u² (t). For fixed real s > 0, this equation has a unique solution u
b² (s)kV² ≤ C. By Lemma 2.3, there exists u(s) ∈ V and a subsequence such
ku
that
b² (s) → u(s)
P ²u
(9)
²
weakly in V ,
e
b (s) → θ u(s) weakly in H. Set ξi² (s) = a²ij ∂j u
b² (s). Then kξi² (s)kH² ≤ C, so
and u
there exists ξi∗ (s) ∈ H such that ξei² (s) → ξi∗ (s) weakly in H. We extend equation
(8) to all of Ω, obtaining
²
e
b (s), ϕiH +
hρ s u
2
Z
Ω
e²0 + ve0² ), ϕiH
ξei² (s) ∂i ϕ dx = hχΩ² fb(s), ϕiH + hρ(s u
for all ϕ ∈ V . Letting ² → 0 we find
hρ s2 θ u(s), ϕiH +
Z
Ω
ξi∗ (s) ∂i ϕ dx = hθ fb(s), ϕiH + hρ(s u0 + v0 ), ϕiH .
216
H.T. BANKS, D. CIORANESCU and R.E. MILLER
The energy method of Tartar gives now ξi∗ (s) = qij ∂j u(s) (for details see for
instance [2], [7] and [16]). Thus, u(s) satisfies
(10) hρ s2 θ u(s), ϕiH +
Z
Ω
qij ∂j u(s) ∂i ϕ dx = hθ fb(s), ϕiH + hρ(s u0 + v0 ), ϕiH
for all ϕ ∈ V . Again, by the ellipticity of the qij , this equation has a unique
solution for fixed s > 0. Next, taking the Laplace transform of equation (7), we
bh (s) satisfies for all ϕ ∈ V
see that u
2
h
b (s), ϕiH +
hρ s θ u
Z
Ω
bh (s) ∂i ϕ dx = hθ fb(s), ϕiH + hρ(s u0 + v0 ), ϕiH .
qij ∂j u
bh (s) for real s > 0. Now
Since this is exactly the same as equation (10), u(s) = u
taking e−st ϕ as a test function in (6), we see that
² u² (s) → u
b(s)
Qd
(11)
in V weakly
for all s > 0. Finally, observe that for s ∈ C+ (i.e., Re s > 0)
² u² (s)(x) =
Qd
=
Z
∞
Z0 ∞
e−st Q² u² (t, x) dt
e−st [P ² u² (t, ·)](x) dt = P ²
0
hZ
∞
0
i
b² (s)(x) .
e−st u² (t) dt (x) = P ² u
b(s) = u(s) = u
bh (s) for all real s > 0,
Thus, combining (9) and (11), we see that u
hence for all s ∈ C+ since the Laplace transform is an analytic function of s.
Therefore, by the uniqueness of the inverse Laplace transform, u = uh .
3 – Homogenization of systems with damping
We now extend equation (4) to include a damping term. We assume that bij ∈
are Y -periodic and that there exists B > 0 such that bij ξi ξj ≥ B ξi ξi
for all ξ ∈ IR2 ; i.e., we assume that 1.II–1.IV hold for the bij . We also assume
that bij = bji . Define b²ij by b²ij (x) = bij ( x² ), and the sesquilinear form σ2² (·, ·) by
L∞ (IR2 )
σ2² (ϕ, ψ) =
Z
Ω²
b²ij ∂j ϕ ∂i ψ dx
for all ϕ, ψ ∈ V² .
We consider the problem
hρ u²tt (t), ϕiV²∗ ,V² + σ1² (u² (t), ϕ) + σ2² (u²t (t), ϕ) = hf (t), ϕiV²∗ ,V² for all ϕ ∈ V² ,
(12)
u² (0) = u²0 ∈ V² ,
u²t (0) = v0² ∈ H² ,
ASYMPTOTIC STUDY OF LATTICE STRUCTURES WITH DAMPING
217
with f ∈ L2 (0, T ; V²∗ ) and u²0 , v0² satisfying bounds as in (5). Again, a unique
solution u² exists with u² ∈ L2 (0, T ; V² ), u²t ∈ L2 (0, T ; V² ) and u²tt ∈ L2 (0, T ; V²∗ )
(e.g., see [3] for details). Moreover, we still have the bounds ku² (t)kV² ≤ C
and ku²t (t)kH² ≤ C for almost every t ∈ [0, T ] for a constant C independent
of ². Hence, just as before, there exists a subsequence of {²} and a function
u ∈ L∞ (0, T ; V ) (this time with ut ∈ L∞ (0, T ; V )) such that
Q ² u² → u
in L∞ (0, T ; V ) weak*
and
Q² u²t → ut
in L∞ (0, T ; H) weak* .
In order to obtain the homogenized equation satisfied by u, we again use
Laplace transforms and apply standard results. Taking the Laplace transform of
(12) we obtain
b² (s), ϕ) + σ2² (s u
b² (s), ϕ) =
b² (s), ϕiH² + σ1² (u
hρ s2 u
= hfb(s), ϕiV²∗ ,V² + hρ(s u²0 + v0² ), ϕiH² + σ2² (u²0 , ϕ)
for all ϕ ∈ V² .
We rewrite this equation as
(13)
b² (s), ϕ) =
b² (s), ϕiH² + σ ² (s)(u
hρ s2 u
= hfb(s), ϕiV²∗ ,V² + hρ(s u²0 + v0² ), ϕiH² + σ2² (u²0 , ϕ)
for all ϕ ∈ V² , where σ ² (s) is defined by
σ ² (s)(ϕ, ψ) =
Z
Ω²
(a²ij + s b²ij ) ∂j ϕ ∂i ψ dx
for all ϕ, ψ ∈ V² .
Set c²ij (s) = a²ij + s b²ij . For fixed s > 0 (real), c²ij (s) ∈ L∞ (IR2 ) and c²ij (s) ξi ξj ≥
b² (s) ∈ V² with ku
b² (s)kV² ≤ C.
(A + sB) ξi ξi . Thus, (13) has a unique solution u
Just as in the proof of Theorem 2.4, we can extend (13) to all of Ω and let ² → 0,
applying the standard homogenization formulas to the sesquilinear forms σ ² (s)
and σ2² . If we define
b² (s)
ξi² (s) = c²ij (s) ∂j u
ζi² = b²ij ∂j u²0 ,
and
then when we extend (13) to Ω, the two terms involving these forms become
Z
Ω
ξe² (s) ∂i ϕ dx
i
and
Z
Ω
ζei² ∂i ϕ dx ,
respectively. There exist u(s) ∈ V and u0 ∈ V such that
b² (s) → u(s)
P ²u
and
P ² u²0 → u0
218
H.T. BANKS, D. CIORANESCU and R.E. MILLER
(both convergences are weak in V ); hence, applying the energy method, these
terms become, in the limit,
Z
Ω
q ij (s) ∂j u(s) ∂i ϕ dx
and
Z
Ω
b
qij
∂j u0 ∂i ϕ dx ,
respectively. The coefficients q ij (s) are defined by
q ij (s) =
1
|Y |
Z
Y∗
·
aij + s bij − (ail + s bil )
¸
∂χj (s)
dy
∂yl
where the functions χj (s) are the Y -periodic solutions of
Z
Y∗
(akl + s bkl )
∂(χj (s) − yj ) ∂ϕ
dy = 0
∂yl
∂yk
for all ϕ ∈ VY ∗ ,
b are given by
and the qij
b
qij
1
=
|Y |
Z
Y∗
µ
¶
∂χjb
dy
bij − bil
∂yl
where the χjb are the Y -periodic solutions of
Z
(14)
Y∗
bkl
∂(χjb − yj ) ∂ϕ
dy = 0
∂yl
∂yk
for all ϕ ∈ VY ∗ .
e²0 → u0 weakly in H where u0 = θ u0 . Thus,
From Lemma 2.3 we also have u
1
replacing u0 by θ u0 , we obtain
(15)
2
hρ s θ u(s), ϕiH +
Z
Ω
q ij (s) ∂j u(s) ∂i ϕ dx =
Z
= hθ fb(s), ϕiV ∗ ,V + hρ(s u0 + v0 ), ϕiH +
1
θ
µ ¶
¶
Ω
b
qij
∂j u0 ∂i ϕ dx .
Using standard arguments we can show that the coefficients q ij (s) satisfy an
ellipticity condition, so that equation (15) has a unique solution u(s). Just as in
b(s) = u(s).
the proof of Theorem 2.4, we can show that u
Rather than taking the inverse Laplace transform of equation (15), we derive
the homogenized equation for u using the multiple-scale method. First we write
equation (12) in strong form as follows:
ρ u²tt (t, x) −
(16)
µ
¶
µ
µ ¶
∂
x ∂u²
x ∂ ∂u²
∂
aij
bij
(t, x) −
(t, x) = f (t, x) in Ω² ,
∂xi
² ∂xj
∂xi
² ∂t ∂xj
µ
¶
∂ ∂u²
∂u²
a
+ b²ij
ni = 0
∂xj
∂t ∂xj
²
u² (0, x) = u²0 ,
on ∂T² ,
u²t (0, x) = v0² .
ASYMPTOTIC STUDY OF LATTICE STRUCTURES WITH DAMPING
219
Now we seek u² (t, x) of the form
u² (t, x) = u0 (t, x, y) + ² u1 (t, x, y) + ²2 u2 (t, x, y) + ...
where y = x/², and each ui is defined for all x ∈ Ω and all y ∈ Y ∗ and is
Y -periodic in y. We also need to specify initial conditions. Based on our experience in Section 2, we take for u0 :
u0 (0, x, y) = u0 (x)/θ (independent of y)
u0t (0, x, y) = v0 (x)/θ .
We will specify other initial conditions later as needed. Let A² be defined by
µ ¶
µ
x ∂
∂
aij
A² = −
∂xi
² ∂xj
¶
.
Since aij depends only on x/² = y, we see that A² can be written as
A² = ²−2 A0 + ²−1 A1 + A2
where
A0 = −
µ
∂
∂
aij (y)
∂yi
∂yj
¶
,
µ
∂2
∂
∂
A1 = −aij (y)
aij (y)
−
∂xi ∂yj
∂yi
∂xj
A2 = −aij (y)
¶
,
¶
,
∂2
.
∂xi ∂xj
Similarly define B² = ²−2 B0 + ²−1 B1 + B2 where
µ
∂
∂
bij (y)
B0 = −
∂yi
∂yj
¶
,
µ
∂
∂2
∂
B1 = −bij (y)
bij (y)
−
∂xi ∂yj
∂yi
∂xj
B2 = −bij (y)
∂2
.
∂xi ∂xj
Substituting into equation (16) and matching powers of ², we obtain first of all
A 0 u0 + B 0
µ
∂u0
=0
∂t
¶
in Y ∗ ,
∂u0
∂ ∂u0
ni = 0
aij
+ bij
∂yj
∂t ∂yj
on ∂T ,
220
H.T. BANKS, D. CIORANESCU and R.E. MILLER
or, in weak form, u0 ∈ VY ∗ , and
Z
(17)
Y∗
µ
¶
∂ ∂u0 ∂ϕ
dy = 0
aij + bij
∂t ∂yj ∂yi
for all ϕ ∈ VY ∗ .
Take ϕ = u0 (t) in equation (17). We have
Z
Y∗
aij
∂u0 (t) ∂u0 (t)
1 ∂
dy +
∂yj
∂yi
2 ∂t
Z
Y∗
bij
∂u0 (t) ∂u0 (t)
dy = 0
∂yj
∂yi
by the symmetry of the bij . Integrating from 0 to t we find
Z tZ
0
Y∗
aij
∂u0 (τ ) ∂u0 (τ )
dy dτ +
∂yj
∂yi
Z
Z
1
∂u0 (t) ∂u0 (t)
1
∂u0 ∂u0
+
bij
dy − 2
bij
dy = 0 .
∗
∗
2 Y
∂yj
∂yi
2θ Y
∂yj ∂yi
But u0 is independent of y, so the last term vanishes. Hence, by the coercivity
of the aij and bij we have
0≤A
Z tZ
0
Thus,
∂u0 (t)
∂yi
Y∗
∂u0 (τ ) ∂u0 (τ )
B
dy dτ +
∂yi
∂yi
2
Z
Y∗
∂u0 (t) ∂u0 (t)
dy ≤ 0 .
∂yi
∂yi
= 0 for all t, i = 1, 2.
Remark 3.1. Sanchez–Palencia assumes a priori that u0 is independent of
y (see [14, p. 99]).
Matching the next powers of ² we obtain
A 0 u1 + A 1 u0 + B 0
µ
¶
∂u0
∂u1
+ B1
=0
∂t
∂t
in Ω × Y ∗ ,
¶
µ
∂ ∂u1
∂ ∂u0
ni = − aij + bij
ni
∂t ∂yj
∂t ∂xj
aij + bij
on Ω × ∂T .
Since u0 is independent of y, we can write the above equation as
∂
−
∂yi
·µ
¶
¸
¶
µ
∂ ∂u1
∂
∂ ∂u0
=0,
aij + bij
−
aij + bij
∂t ∂yj
∂yi
∂t ∂xj
or in weak form as
(18)
Z
Y∗
·µ
¶
µ
¶
¸
∂ ∂u0 ∂ϕ
∂ ∂u1
+ aij + bij
dy = 0
aij + bij
∂t ∂yj
∂t ∂xj ∂yi
ASYMPTOTIC STUDY OF LATTICE STRUCTURES WITH DAMPING
221
for all ϕ ∈ VY ∗ . We want to express u1 in terms of u0 . In order to make u1
unique, we introduce the space
n
V Y ∗ = ϕ ∈ VY ∗ | MY ∗ (ϕ) = 0
with inner product
hϕ, ψiV Y ∗ =
Z
Y
∂ϕ ∂ψ
dy .
∂yj ∂yi
bij
∗
o
This inner product induces a norm k · kV Y ∗ on V Y ∗ equivalent to the usual norm
k · kH 1 (Y ∗ ) . Now we define A ∈ L(V Y ∗ , V Y ∗ ) by
(19)
hAϕ, ψiV Y ∗ =
Z
Y
∗
aij
∂ϕ ∂ψ
dy .
∂yj ∂yi
The right side of equation (19) is a bounded coercive sesquilinear form on
V Y ∗ × V Y ∗ , so A is defined uniquely and is bijective and bicontinuous (by the
Lax–Milgram theorem — see [17, p. 272]).
Now define Fja : V Y ∗ → C by
Fja (ϕ)
=
Z
Y
∗
aij
∂ϕ
dy .
∂yi
Since Fja is a bounded linear functional on V Y ∗ , there exists a unique fja ∈ V Y ∗
such that
hfja , ϕiV Y ∗
(20)
=
Fja (ϕ)
=
Z
Y
∗
aij
∂ϕ
dy .
∂yi
Similarly, we define fjb ∈ V Y ∗ by
hfjb , ϕiV Y ∗ =
(21)
Z
Y∗
bij
∂ϕ
dy .
∂yi
Remark 3.2. If we take χj defined in (14) so that MY ∗ (χj ) = 0, then
b
b
χj = fjb .
b
Using A, fja and fjb we can rewrite equation (18) as
¿
Au1 +
µ
¶
∂u1
∂u0
∂ ∂u0
, ϕ
+ fja
+ fjb
∂t
∂xj
∂t ∂xj
À
=0
VY∗
which implies that
(22)
∂u1
∂u0
∂ ∂u0
+ Au1 = −fja
− fjb
.
∂t
∂xj
∂t ∂xj
for all ϕ ∈ V Y ∗
222
H.T. BANKS, D. CIORANESCU and R.E. MILLER
Set w = u1 + fjb
∂u0
∂xj .
Then
∂u1
∂ ∂u0
∂w
=
+ fjb
.
∂t
∂t
∂t ∂xj
If we define fj = Afjb − fja , then we can rewrite equation (22) as
∂w
∂u0
+ Aw = fj
.
∂t
∂xj
(23)
Since A is bounded it generates a uniformly continuous semigroup (in fact a
group) of operators. Thus, once we specify an initial value for w, we can solve
equation (23) to obtain w, hence u1 , in terms of u0 . It will be convenient to take
w(0) = 0. Hence, we take as the initial condition on u1 :
1
∂u0
u1 (0) = − fjb
.
θ
∂xj
The unique solution to equation (23) with w(0) = 0 is
w(t) =
Z
t
0
e−(t−σ)A fj
∂u0 (σ)
dσ .
∂xj
Hence u1 is given by
u1 (t) =
(24)
Z
t
0
e−(t−σ)A fj
∂u0 (σ)
∂u0 (t)
dσ − fjb
.
∂xj
∂xj
We will need the following lemma to prove the main result of this section.
Lemma 3.1. If we take χj (s) as defined above to be in V Y ∗ , then
χj (s) = fjb − (sI + A)−1 fj .
Proof: The χj (s) for which MY ∗ (χj (s)) = 0 is the unique solution of
Z
Y∗
·
(akl + s bkl )
¸
∂ϕ
∂χj (s)
− (akj + s bkj )
dy = 0
∂yl
∂yk
for all ϕ ∈ V Y ∗ .
We can rewrite this equation as
D
(A + sI) χj (s) − (fja + s fjb ), ϕ
E
VY∗
=0,
ASYMPTOTIC STUDY OF LATTICE STRUCTURES WITH DAMPING
223
which implies that
(sI + A) χj (s) = fja + s fjb
= (sI + A) fjb − A fjb + fja
= (sI + A) fjb − fj .
Hence, χj (s) = fjb − (sI + A)−1 fj .
We obtain the homogenized equation by matching the next powers of ², integrating over Y ∗ and substituting in the expression for u1 in terms of u0 (equation
(24)). We find
(25)
−
−
1
|Y |
1
|Y |
∂ 2 u0 (t)
1
ρθ
−
2
∂t
|Y |
Z
Z
Y∗
µ
bij − bil
Z t·
Y∗ 0
ail
Z
b¶
∂fj
∂yl
Y∗
µ
dy
∂fjb
∂fj
aij − ail
+ bil
∂yl
∂yl
¶
∂ 2 u0 (t)
−
∂xi ∂xj
dy
∂ ∂ 2 u0 (t)
∂t ∂xi ∂xj
¸
∂ 2 u0 (σ)
∂ −(t−σ)A
∂
(e
fj ) − bil
(A e−(t−σ)A fj )
dσ dy = θ f (t).
∂yl
∂yl
∂xi ∂xj
The following theorem is the main result of this section.
Theorem 3.2.
Equation (25) along with initial values u0 (0) = u0 /θ,
u0t (0) = v0 /θ is the homogenized system satisfied by u.
Proof: Define the coefficients
αij =
1
|Y |
βij =
1
|Y |
1
γij (t) =
|Y |
Z
Z
Z
Y∗
Y∗
Y∗
µ
µ
·
aij − ail
∂fjb
∂fj
+ bil
∂yl
∂yl
bij − bil
∂fjb
dy
∂yl
¶
¶
dy ,
b
(= qij
),
¸
∂ −tA
∂
ail
(e
fj ) − bil
(A e−tA fj ) dy .
∂yl
∂yl
With these definitions, we can rewrite equation (25) as
ρ θ u0tt (t)
∂ 2 u0 (t)
∂ ∂ 2 u0 (t)
− αij
− βij
−
∂xi ∂xj
∂t ∂xi ∂xj
Z
t
0
γij (t − σ)
∂ 2 u0 (σ)
dσ = θ f (t) .
∂xi ∂xj
We then multiply by ϕ ∈ V and integrate over Ω to obtain
(26)
Z
∂u0 (t) ∂ϕ
dx +
∂xj ∂xi
Ω
¶
Z
Z µ
∂ϕ
∂u0
∂ ∂u0 (t) ∂ϕ
(t)
+
dx +
γij ∗
dx = hθ f (t), ϕiV ∗ ,V .
βij
∂t ∂xj ∂xi
∂xj
∂xi
Ω
Ω
hρ θ u0tt (t), ϕiV ∗ ,V
+
αij
224
H.T. BANKS, D. CIORANESCU and R.E. MILLER
Taking the Laplace transform of (26) we find
b0 (s), ϕiH +
hρ θ s2 u
Z
Ω
(αij + s βij + γbij (s))
b0 (s) ∂ϕ
∂u
dx =
∂xj ∂xi
Z
D
E
1
∂u0 ∂ϕ
b
= hθ f (s), ϕiV ∗ ,V + ρ θ(s u0 /θ + v0 /θ), ϕ +
dx .
βij
H
θ
Ω
∂xj ∂xi
b , this equation will be the same as (15) if we can show that
Since βij = qij
(27)
Observe that
Z
Z
q ij (s) = αij + s βij + γbij (s) .
·
¸
∞
∂ −tA
1
∂
(e
fj ) − bil
(A e−tA fj ) dt dy
e−st ail
|Y | Y ∗ 0
∂yl
∂yl
Z ½
i
i¾
∂ h
1
∂ h
−1
−1
ail
(sI + A) fj − bil
A(sI + A) fj dy
=
|Y | Y ∗
∂yl
∂yl
Z ½
i
i¾
∂ h
1
∂ h
ail
=
(sI + A)−1 fj + bil
s(sI + A)−1 fj − fj dy .
|Y | Y ∗
∂yl
∂yl
γbij (s) =
Thus, by Lemma 3.1,
Z
¾
½
i
∂ h b
1
(aij +sbij )−(ail +sbil )
fj −(sI +A)−1 fj dy
αij + sβij + γbij (s) =
|Y | Y ∗
∂yl
¸
Z ·
1
∂χj (s)
=
(aij +sbij )−(ail +sbil )
dy .
|Y | Y ∗
∂yl
Hence, we have established equation (27), and so, by the uniqueness of the inverse
Laplace transform, equation (25) (equivalently (26)) is the equation satisfied by
u.
An important special case is when aij = κ bij for all i, j where κ 6= 0 is a
constant (this is the case for Kelvin–Voigt damping investigated in [2]). In this
case
hAϕ, ψiV Y ∗ = κhϕ, ψiV Y ∗ for all ϕ, ψ ∈ V Y ∗ ,
which implies that A = κ I. We also have
hfja , ϕiV Y ∗ = κhfjb , ϕiV Y ∗
for all ϕ ∈ V Y ∗ ,
so fja = κ fjb . Hence, fj = 0 which implies that αij = κ βij and γij (t) = 0 for all
t. Thus, the form of equation (25) simplifies to
ρ θ u0tt (t)
− βij
µ
∂
κ+
∂t
¶
∂ 2 u0 (t)
= θ f (t) .
∂xi ∂xj
ASYMPTOTIC STUDY OF LATTICE STRUCTURES WITH DAMPING
225
4 – The limit in µ for constant coefficients
In this section we shall investigate the dependence of the homogenized coefficients on the parameter µ. Investigations of this nature were first carried out
in [1], [7] and [13]. We began with a simple equation (system (3)) defined on the
domain Ω²µ which has a complicated geometry. Taking the limit as ² → 0, we
obtained a homogenized equation on all of Ω, but in order to obtain the coefficients, we must solve a differential equation on the representative cell. Since the
parameter µ is small (compared to the dimensions of Ω), we next let µ → 0 and
obtain a limit problem. We shall see that in this limit problem, we retrieve the
simplicity in the coefficients, and in fact we shall compute them explicitly. In
order to simplify the computations, we assume that the coefficients aij and bij
are constant and that l1 = l2 = 1 (hence θ = 2µ(1 − µ/2)). We now display the
µ
dependence on µ. In particular, we write Aµ for A, αij
for αij , etc.
From systems (20) and (21) one can derive the a priori estimates
k∇fjaµ k[L2 (Yµ∗ )]2 ≤ C µ1/2
and
k∇fjbµ k[L2 (Yµ∗ )]2 ≤ C µ1/2 .
Using the fact that Aµ is bounded independently of µ, we also obtain
k∇fjµ k[L2 (Yµ∗ )]2 ≤ C µ1/2 .
µ
µ
Thus, αij
≤ C µ, βij
≤ C µ and
µ
γij
(0) =
1
|Y |
Z
Yµ∗
·
ail
¸
∂(Aµ fjµ )
∂fjµ
− bil
dy ≤ C µ .
∂yl
∂yl
µ
µ
Since Aµ is positive, γij
decays exponentially as t → ∞, so γij
(t) ≤ C µ for all t.
Thus, as µ → 0,
µ
∗
µ−1 αij
→ αij
,
µ
∗
µ−1 βij
→ βij
,
and
µ
∗
µ−1 γij
(t) → γij
(t) ,
with the last convergence being uniform in t on bounded intervals. Rather than
∗ , β ∗ and γ ∗ (t) however, we treat the transformed equation
seeking the limits αij
ij
ij
(15) so that we can apply the results of [2] to obtain that
µ−1 q µij (s) → q ∗ij (s) = 2(aij + s bij ) −
2
X
(ail + s bil ) (alj + s blj )
l=1
all + s bll
226
H.T. BANKS, D. CIORANESCU and R.E. MILLER
and
bµ
b∗
µ−1 qij
→ qij
= 2 bij −
2
X
bil blj
bll
l=1
.
b∗ = 0 if i 6= j. Thus, multiplying equation (15) by µ−1
Observe that q ∗ij (s) = qij
and letting µ → 0, we obtain
2
b(s), ϕiH +
h2 ρ s u
Z
Ω
b(s) ∂i ϕ dx =
q ∗ii (s) ∂i u
= h2fb(s), ϕiV ∗ ,V + hρ(s u0 + v0 ), ϕiH +
1
2
Z
Ω
qiib∗ ∂i u0 ∂i ϕ dx .
Since the coefficients appearing in this equation are rational functions in s, it is
straightforward to invert the Laplace transform. The limit equation is
(28)
h2 ρ utt (t), ϕiV
−
∗ ,V
µ
+ 2 aii −
(·
2
X
ail ali
+
bll
l=1
l=1
b2ll
¸
all (ail bli + bil ali ) a2ll bil bli
−
+
·
b2ll
b3ll
bll
l=1
¶Z
2
2
X
ail bli + bil ali X
all bil bli
·
Z Z
t
e
all (t−σ)/bll
Ω 0
µ
+ 2 bii −
)
∂i u(σ) ∂i ϕ dσ dx +
¶Z
2
X
bil bli
l=1
∂i u(t) ∂i ϕ dx−
Ω
bll
Ω
∂i ut (t) ∂i ϕ dx = h2f (t), ϕiV ∗ ,V
with initial data u(0) = u0 /2 and ut (0) = v0 /2.
For the special case that aij = κ bij , the convolution term vanishes, and the
equation simplifies to
µ
h2 ρ utt (t), ϕiV ∗ ,V + 2 bii −
¶µ
2
X
bil bli
l=1
bll
κ+
∂
∂t
¶Z
Ω
∂i u(t) ∂i ϕ dx = h2f (t), ϕiV ∗ ,V .
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H.T. Banks,
Center for Research in Scientific Computation,
North Carolina State University, Raleigh, NC 27695 – USA
and
D. Cioranescu,
Laboratoire D’Analyse Numérique,
Université Pierre et Marie Curie, 75005 Paris – France
and
R.E. Miller,
Department of Mathematical Sciences,
University of Arkansas, Fayetteville, AR 72701 – USA