Applications of the periodic unfolding method to multi-scale problems

Transcription

1 Applications of the periodic unfolding method to multi-scale problems Doina Cioranescu Université Paris VI Santiago de Compostela December 14, 2009 Periodic unfolding method and multi-scale problems 1/56

2 Introduction Homogenization - Introduction Let A ε (x) = (a ε ij(x)) 1 i,j n a.e. on Ω R n, satisfying for any λ R n, { (A ε (x)λ, λ) α λ 2, Set Consider the equation A ε (x)λ β λ, A ε = div (A ε ) = A ε u ε = f, n i,j=1 with, for instance, a Dirichlet condition on Ω. ( a ε ) x ij. i x j Periodic unfolding method and multi-scale problems 2/56

3 Introduction One has to solve { div (A ε u ε ) = f in Ω u ε = 0 on Ω. with f H 1 (Ω). By the Lax-Milgram theorem there exists a unique u ε H0 1 (Ω) such that A ε u ε v dx = f, v H 1 (Ω),H0 1(Ω), v H1 0 (Ω). Ω Moreover, since u ε H 1 0 (Ω) 1 α f H 1 (Ω), there exists u 0 H0 1 (Ω) such that u ε u 0 weakly in H 1 0 (Ω). Periodic unfolding method and multi-scale problems 3/56

4 Set Introduction ξ ε = (ξ ε 1,..., ξ ε n) = ( n j=1 a1j ε u ε,..., x j n j=1 which satisfies ξ ε v dx = f, v H 1 (Ω),H0 1(Ω), Ω anj ε u ε ) = A ε u ε, x j v H1 0 (Ω). There exists ξ 0 L 2 (Ω), such that (up to a subsequence) ξ ε ξ 0 weakly in (L 2 (Ω)) n. Moreover, i.e. Ω ξ 0 v dx = f, v H 1 (Ω),H 1 0 (Ω), div ξ 0 = f in Ω. v H1 0 (Ω), Periodic unfolding method and multi-scale problems 4/56

5 Introduction Question : the relation between ξ 0 and u 0? Several methods give the answer : the multiple-scale method, the Tartar s oscillating test functions method, the two-scale convergence, the periodic unfolding method. Periodic unfolding method and multi-scale problems 5/56

6 Introduction Multiple-scale method Suppose now that ( x ) A ε (x) = A, A(y) = (a ij (y)) 1 i,j n, a ij Y periodic. ε One looks for ( u ε (x) = u 0 x, x ) ( + εu 1 x, x ) ( + ε 2 u 2 x, x ) + ε ε ε with u j (x, y) for j = 1, 2,... such that { uj (x, y) is defined for x Ω and y Y u j (, y) is Y periodic. Periodic unfolding method and multi-scale problems 6/56

7 Introduction Tartar s oscillating test function method Let ϕ D(Ω) et z ε a sequence of functions such that z ε ϕ H0 1(Ω). With z ε as test fuctions, x ) u ε z ε x ) u ε ϕ ϕ a ij( + z ε a ij( = f z ε ϕ dx. Ω ε x j x i Ω ε x j x i Ω Idea : construction of z ε as the solution of the adjoint system = ( x ) ϕ aij zε u ε ( x ) + u ε a zε ϕ ij = 0. ε x j x i ε x j x i Ω = The bad terms cancel and one can pass to the limit in all the others. Ω Periodic unfolding method and multi-scale problems 7/56

8 Introduction Two-scale method (G. Nguetseng et G. Allaire Let {v ε } a sequence of functions in L 2 (Ω). One says that {v ε } is two-scale convergent to v 0 = v 0 (x, y) with v 0 L 2 (Ω Y ) if for any ψ = ψ(x, y) D(Ω; C per(y )), one has lim v ε (x) ψ(x, x ε 0 Ω ε )dx = 1 v 0 (x, y) ψ(x, y) dydx. Y Ω Y Theorem. Let {v ε } be a bounded sequence in L 2 (Ω). There exists {v ε } and a function v 0 L 2 (Ω Y ) such that {v ε } two-scale converges to v 0. Periodic unfolding method and multi-scale problems 8/56

9 The periodic unfolding method Periodic unfolding Joint works with Alain Damlamian, Patrizia Donato, Georges Griso and Rachad Zaki. Main topics I. The periodic unfolding method in fixed domains II. Multi-scale domains III. Unfolding in perforated domains IV. Application to multi-structures... Periodic unfolding method and multi-scale problems 9/56

10 The periodic unfolding method I. The periodic unfolding method in fixed domains The periodic unfolding method is a fixed domain method, well suited to treat periodic homogenization problems. The basic idea is that if the proper scaling is used, oscillatory behaviour can be turned into weak or even strong convergence, at the price of an increase in the dimension of the problem, but with significant simplifications in the proofs and explicit formulas. It also gives a most elementary proof for the results of the theory of two-scale convergence due to G. Nguetseng and G. Allaire. Originally, the periodic unfolding method was applied for linear problems in the standard periodic case of homogenization with applications to multi-scale problems. Then it was extended to monotone operators, perforated domains, nonlinear integral energies,... Periodic unfolding method and multi-scale problems 10/56

11 The periodic unfolding method References D. C, Alain Damlamian and Georges Griso, Periodic unfolding and homogenization, C. R. Acad. Sci. Paris, 335 (2002), D. C, Alain Damlamian and Georges Griso, The periodic unfolding method in homogenization, SIAM J. of Math. Anal. Vol. 40, 4 (2008), D. C, P. Donato and R. Zaki, The periodic unfolding method in perforated domains, Portugaliae Mathematica, 63, 4 (2006), D. C, P. Donato and R. Zaki, Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions, Asymptotic Analysis, 53, 4 (2007), Periodic unfolding method and multi-scale problems 11/56

12 The periodic unfolding method Plan 1. The periodic unfolding operator 2. First convergence results 3. Scale decomposition 4. Main convergence result 5. Homogenization of the model problem 6. Application to multi-scale problems Periodic unfolding method and multi-scale problems 12/56

13 The periodic unfolding method 1. The periodic unfolding operator 1. The periodic unfolding operator T e Ω R n, Y a reference cell (ex. ]0, 1[ n ), or more generally a set having the paving property with respect to a basis (b 1,, b n ). For z R n, [z] Y denotes the unique integer combination n j=1 k jb j such that z [z] Y belongs to Y, {z} Y = z [z] Y Y a.e. for z R n, = ([ x ] { x } ) x = ε ε + a.e. for x R n. Y ε Y Periodic unfolding method and multi-scale problems 13/56

14 The periodic unfolding method 1. The periodic unfolding operator { } Ξ ε = ξ Z N, ε(ξ + Y ) Ω, { ( ) Ω } ε = interior ξ + Y, Λ ε = Ω \ Ω ε. ξ Ξ ε ε Ω ε is the largest union of ε(ξ + Y ) cells (ξ Z n ) included in Ω. Periodic unfolding method and multi-scale problems 14/56

15 The periodic unfolding method 1. The periodic unfolding operator Definition For φ Lebesgue-measurable on Ω, T ε is given by ( [ x ) φ ε T ε (φ)(x, y) = ε ]Y + εy a.e. for (x, y) Ω ε Y, 0 a.e. for (x, y) Λ ε Y. T ε (φ) is Lebesgue-measurable, T ε is linear and continuous from L p (Ω) to L p (Ω Y ), T ε (vw) = T ε (v) T ε (w), Let f measurable on Y, extend it by Y periodicity to R n, and set ( x ) f ε (x) = f a.e. for x R n = ε { f (y) a.e. for (x, y) T ε (f ε Ω )(x, y) = Ω ε Y, 0 a.e. for (x, y) Λ ε Y. Periodic unfolding method and multi-scale problems 15/56

16 The periodic unfolding method 1. The periodic unfolding operator Example 1. ( f (x) = sin 2π x ). ε Periodic unfolding method and multi-scale problems 16/56

17 The periodic unfolding method 1. The periodic unfolding operator Example 2. g(x) = x 2 x. Periodic unfolding method and multi-scale problems 17/56

18 The periodic unfolding method 1. The periodic unfolding operator Example 3. { ( 1 for y 0, 1 ) / 2, f (y) = 2 for y ( 1 / 2, 1 ), ({ x } ) f ε = f. T ε (f ε ) ε Y Periodic unfolding method and multi-scale problems 18/56

19 The periodic unfolding method 1. The periodic unfolding operator Example 4. f ε (x) = 1 ( 4 sin 2π x ) + x; ε = 1 ε 6 T ε (f ε ) Periodic unfolding method and multi-scale problems 19/56

20 The periodic unfolding method 1. The periodic unfolding operator Other property. Any integral of a function on Ω, is almost equivalent to the integral of its unfolded on Ω Y, the integration defect is due to the cells intersecting the boundary Ω. Indeed, for φ in L 1 (Ω) and w in L p (Ω) 1 T ε (φ)(x, y) dx dy = φ(x) dx φ(x) dx = φ(x) dx, Y Ω Y Ω Λ ε bω ε φ dx 1 T ε (φ) dxdy Y Λ φ dx. ε Ω Ω Y If {φ ε } is a sequence in L 1 (Ω) satisfying Λ ε φ ε dx 0, then Ω φ ε dx 1 T ε (φ ε ) dxdy 0. Y Ω Y Notation Ω φ ε dx Tε 1 T ε (φ ε ) dxdy. Y Ω Y Periodic unfolding method and multi-scale problems 20/56

21 The periodic unfolding method 2. First convergence results 2. First convergence results Let p belong to [1, + [. T ε (w) w strongly in L p (Ω Y ), w L p (Ω). Let {w ε } be a sequence in L p (Ω) such that w ε w strongly in L p (Ω). Then T ε (w ε ) w strongly in L p (Ω Y ). For every relatively weakly compact sequence {w ε } L p (Ω), T ε (w ε ) is relatively weakly compact in L p (Ω Y ). Furthermore, if T ε (w ε ) ŵ weakly in L p (Ω Y ), then w ε M Y (ŵ) = 1 Y Y ŵ dy weakly in L p (Ω). Periodic unfolding method and multi-scale problems 21/56

22 The periodic unfolding method 2. First convergence results Two-scale convergence. Let p ]1, + [. A bounded sequence {w ε } in L p (Ω), two-scale converges to some w belonging to L p (Ω Y ), whenever, for every smooth function φ on Ω Y, the following convergence holds : Ω ( w ε (x)φ x, x ) dx 1 ε Y Ω Y w(x, y) φ(x, y) dxdy. The following result holds true : If {w ε } is a bounded sequence in L p (Ω) with p ]1, + [, then the following assertions are equivalent : {T ε (w ε )} converges weakly to w in L p (Ω Y ). {w ε } two-scale converges to w. Remark. To check the two-scale convergence one has to use special test functions. To check a weak convergence in the space L p (Ω Y ) due to density, one can use of functions smooth functions from D(Ω Y ). Periodic unfolding method and multi-scale problems 22/56

23 The periodic unfolding method 3. Macro-micro decomposition 3. Macro-micro decomposition : the scale-splitting operators Q ε and R ε Every φ in W 1,p 0 (Ω) in the form φ = Q ε (φ) + R ε (φ), where Q ε (φ) is an approximation of φ having the same behavior as φ, while R ε (φ) is a remainder of order ε. Suppose that Ω is smooth enough so that there exists a continuous extension operator P : W 1,p (Ω) W 1,p (R n ) satisfying P(φ) W 1,p (R n ) C φ W 1,p (Ω), φ W 1,p (Ω), where C is a constant depending on p and Ω only. The construction of Q ε is based on the Q 1 interpolate of a discrete approximation, as in the finite element method. Periodic unfolding method and multi-scale problems 23/56

24 The periodic unfolding method 3. Macro-micro decomposition The operator Q ε : L p (R n ) W 1, (R n ), for p [1, + ], is defined as follows Q ε (φ)(εξ) = M ε (φ)(εξ) for ξ εz n, where the local average operator M ε is defined by 1 M ε (φ)(x) = ε n Y ε [ x ] φ(ζ) dζ, if x Ω ε, + εy ε 0 if x Λ ε. Observe that for every φ L p (Ω), M ε (φ) φ strongly in L p (Ω). Periodic unfolding method and multi-scale problems 24/56

25 The periodic unfolding method 3. Macro-micro decomposition For any x R n, Q ε (φ)(x) is the Q 1 interpolate of the values of Q ε (φ) at the vertices [ x ] of the cell ε ε + εy. Y Then the operator Q ε : W 1,p (Ω) W 1, (Ω) is defined by Q ε (φ) = Q ε (P(φ)) Ω. Well known (from finite elements method) Q ε (φ) φ strongly in L p (R n ), ε Q ε (φ) 0 strongly in (L p (R n )) n, Q ε (φ) W 1,p (Ω) C P φ W 1,p (Ω). Clear : Q ε is designed NOT to capture any oscillation of order ε. Periodic unfolding method and multi-scale problems 25/56

26 The periodic unfolding method 3. Macro-micro decomposition The remainder R ε (φ) is given by R ε (φ) = φ Q ε (φ) for any φ W 1,p (Ω), and has the following properties : R ε (φ) L p (Ω) ε C P φ W 1,p (Ω), R ε (φ) L p (Ω) C P φ L p (Ω) Let {w ε } be a sequence converging weakly in W 1,p (Ω) to w. Then, R ε (w ε ) 0 Q ε (w ε ) w strongly in L p (Ω), weakly in W 1,p (Ω), T ε ( Q ε (w ε )) w weakly in L p (Ω Y ). Periodic unfolding method and multi-scale problems 26/56

27 The periodic unfolding method 4. Main convergence result 4. Main convergence result Theorem Let {w ε } W 1,p (Ω) satisfying w ε w weakly in W 1,p (Ω). Then, up to a subsequence there exists some ŵ in the space L p (Ω; W 1,p per (Y )) such that 1 ε T ε( Rε (w ε ) ) ŵ weakly in L p (Ω; W 1,p (Y )), T ε ( Rε (w ε ) ) y ŵ weakly in L p (Ω Y ), T ε ( w ε ) w + y ŵ weakly in L p (Ω Y ). Periodic unfolding method and multi-scale problems 27/56

28 The periodic unfolding method 5. Homogenization of the model problem 5. Periodic unfolding and the standard homogenization Let A ε = (a ε ij ) 1 i,j n be a sequence of matrices satisfying a.e. on Ω (A ε λ, λ) α λ 2, A ε (x)λ β λ, λ R n. For f given in H 1 (Ω), consider the Dirichlet problem { div (A ε u ε ) = f in Ω, u ε = 0 on Ω. By the Lax-Milgram theorem, there exists a unique u ε H0 1(Ω) A ε u ε v dx = f, v H 1 (Ω),H0 1(Ω), v H1 0 (Ω). Ω Moreover, one has the apriori estimate u ε H 1 0 (Ω) 1 α f H 1 (Ω) = u ε u 0 weakly in H 1 0 (Ω). Periodic unfolding method and multi-scale problems 28/56

29 The periodic unfolding method 5. Homogenization of the model problem Classical periodic homogenization result. Let a ij = a ij (y) for 1 i, j n, be Y periodic, and let A ε (x) = (a ij (x/ε)) 1 i,j n a.e. on Ω. Then u 0 is the unique solution of n div (A 0 u 0 ) = a 0 2 u 0 ij = f in Ω, x i x j i,j=1 u 0 = 0 on Ω, where the constant matrix A 0 = (aij 0) 1 i,j n is elliptic and given by ( n aij 0 χ j ) = M Y (a ij ) M Y a ik. y k k=1 The corrector function χ j for j {1,..., n}, is solution of n ( ( χ j y j ) ) a ik = 0 in Y, y i y k i,k=1 χ j Y -periodic and M Y ( χ j ) = 0. Periodic unfolding method and multi-scale problems 29/56

30 The periodic unfolding method 5. Homogenization of the model problem Periodic homogenization via unfolding. Suppose that there exists a matrix B such that B ε = T ε ( A ε ) B strongly in [L 1 (Ω Y )] n n. Then there exists u 0 H 1 0 (Ω) and û L2 (Ω; H 1 per(y )) such that { uε u 0 weakly in H 1 0 (Ω), T ε ( u ε ) u 0 + y û weakly in L 2 (Ω Y ), and the pair (u 0, û) is the unique solution of the problem Ψ H0 1 (Ω), Φ L 2 (Ω; Hper 1 (Y )), 1 B(x, y) [ u 0 (x) + y û(x, y) ][ Ψ(x) + y Φ(x, y) ] dxdy Y Ω Y = f, Ψ H 1 (Ω),H0 1(Ω) If A ε (x) = (A(x/ε), then B(x, y) = A(y). Periodic unfolding method and multi-scale problems 30/56

31 The periodic unfolding method 5. Homogenization of the model problem The proof by the periodic unfolding method is elementary! Integration formula = With Ψ H0 1(Ω), f, Ψ H 1 (Ω),H0 1(Ω) = A ε u ε Ψ dx Ω Tε 1 T ε (A ε ( ) ( ) ) T ε uε Tε Ψ dx dy Y Ω Y 1 ( ) ( ) B ε T ε uε Tε Ψ dx dy Y Ω Y and the last integral gives at the limit 1 B(x, y) [ u 0 (x)+ y û(x, y) ] Ψ(x) dxdy = f, Ψ Y H 1 (Ω),H0 1(Ω). Ω Y Periodic unfolding method and multi-scale problems 31/56

32 The periodic unfolding method 5. Homogenization of the model problem Take now as test function ( x ) v ε (x) = εψ(x)ψ, Ψ D(Ω), ψ H 1 ε per(y ). 1 Y Ω Y + 1 Y B ε ( ) ( ) T ε uε εψ(y)tε Ψ dxdy Ω Y B ε ( ) ( ) Tε T ε uε y ψ(y)t ε Ψ dxdy f, v ε H 1 (Ω),H0 1(Ω). Since v ε 0 in H0 1 (Ω), passing to the limit one has 1 B(x, y) [ u 0 (x) + y û(x, y) ] Ψ(x) y ψ(y) dxdy = 0. Y Ω Y By the density of the tensor product D(Ω) H 1 per(y ) in L 2 (Ω; H 1 per(y )), this holds for all Φ in L 2 (Ω; H 1 per(y )). Periodic unfolding method and multi-scale problems 32/56

33 II. Application to multi-scale problems II. Periodic unfolding and multiscales The periodic unfolding method turns out to be particularly well-adapted to multi-scales problems. Consider two periodicity cells Y and Z. Suppose that Y is partitioned in two non-empty disjoint open subsets Y 1 and Y 2, i.e. such that Y 1 Y 2 = and Y = Y 1 Y 2. Let A εδ be a matrix field defined by ({ x } ) { x } A 1 for A εδ ε Y ε Y 1 Y (x) = ({ { } x } ) { ε Y x } A 2 for δ Z ε Y 2, Y where A 1 and A 2 have the same properties as before. Here there are two small scales, ε and εδ, associated respectively to the cells Y and Z. Periodic unfolding method and multi-scale problems 33/56

34 II. Application to multi-scale problems Z δ Ω ε Y 1 Y Periodic unfolding method and multi-scale problems 34/56

35 II. Application to multi-scale problems Consider the problem A εδ u εδ w dx = Ω Ω f w dx, w H 1 0 (Ω), with f in L 2 (Ω). The Lax-Milgram theorem gives the existence and uniqueness of u εδ in H0 1 (Ω) satisfying the estimate u εδ H 1 0 (Ω) 1 α f L 2 (Ω). So, as above there is some u 0 H0 1(Ω) and û L2 (Ω; Hper(Y 1 )) such that, u εδ u 0 weakly in H0 1 (Ω), and ( ) T ε uεδ u0 + y û weakly in L 2 (Ω Y ). These convergences do not see the oscillations at the scale εδ. Periodic unfolding method and multi-scale problems 35/56

36 II. Application to multi-scale problems In order to capture them, one considers the restrictions to the set Ω Y 2 v εδ(x, y) =. 1 ε T ( ε Rε (u εδ ) ) Y2. Obviously, T y δ v εδ û Y2 weakly in L 2 (Ω; H 1 (Y 2 )). Apply to v εδ a similar unfolding operation, denoted T y δ, for the variable y, thus adding a new variable z Z. T y δ (v ( [ y ] εδ)(x, y, z) = v εδ x, δ δ +δz) for x Ω, y Y 2 and z Z. Z All the estimates and weak convergence properties for the original unfolding T ε still hold for T y δ with x being a mere parameter. = There is ũ L 2 (Ω Y 2, Hper(Z)/R) 1 such that T y ( ) δ y v εδ y û Ω2 + z ũ weakly in L 2 (Ω Y 2 Z), ( ( ) ) T ε uεδ u 0 + y û + z ũ weakly in L 2 (Ω Y 2 Z). Periodic unfolding method and multi-scale problems 36/56

37 II. Application to multi-scale problems Theorem. The functions u 0 H 1 0 (Ω), û L 2 (Ω, H 1 per(y )/R), ũ L 2 (Ω Y 2, H 1 per(z)/r) are the unique solutions of the variational problem 1 [ ][ ] A 2 (z) u 0 + y û + z ũ Ψ + y Φ + z Θ dxdydz Y Z Ω Y Y Ω Z Y 1 [ ][ ] A 1 (y) u 0 + y û Ψ + y Φ dxdy = Ψ H0 1 (Ω), Φ L 2 (Ω; Hper(Y 1 )/R), Θ L 2 (Ω Ω 2, Hper(Z)/R). 1 Ω f Ψ dx, Periodic unfolding method and multi-scale problems 37/56

38 III. Periodic unfolding in perforated domains The unfolding operator in perforated domains Perforated domains Same notations { } Ξ ε = ξ Z N, ε(ξ + Y ) Ω, { ( ) Ω } ε = interior ξ + Y, Λ ε = Ω \ Ω ε. ξ Ξ ε ε Periodic unfolding method and multi-scale problems 38/56

39 III. Periodic unfolding in perforated domains The unfolding operator in perforated domains The notion of periodig unfolding is well adapted for perforated domains, almost all the results of concerning fixed domains still hold. Let S be a strict closed subset of Y and set Y = Y \ S (the part occupied by the material). Here Y happens to be a parallelotop, but the definition holds for a general Y. Periodic unfolding method and multi-scale problems 39/56

40 III. Periodic unfolding in perforated domains The unfolding operator in perforated domains The perforated domain Ω ε is obtained by removing from Ω the set of holes S ε Ω ε = Ω \ S ε, where S ε = ξ G ε ( ξ + S ). The sets Ω ε (blue and green), Ω ε (blue) and Λ ε (green). Periodic unfolding method and multi-scale problems 40/56

41 III. Periodic unfolding in perforated domains The unfolding operator in perforated domains Definition For any function φ Lebesgue-measurable on Ω ε, the unfolding operator Tε is defined by ( [ x ) Tε φ ε (φ)(x, y) = ε ]Y + εy a.e. for (x, y) Ω ε Y, 0 a.e. for (x, y) Λ ε Y. The relationship between T ε and Tε is given for any w defined on Ω ε, by Tε (w) = T ε ( w) Ω Y. = The operator Tε those of T ε. enjoys properties which follow directly from Periodic unfolding method and multi-scale problems 41/56

42 III. Periodic unfolding in perforated domains The unfolding operator in perforated domains The main convergence result requires that the perforated cell Y satisfy the following geometrical hypothesis : The open set Y satisfies the Poincaré-Wirtinger inequality for the exponent p (p [1, + ]) and for all vector b i, i {1,..., n} of the periodicity basis, the open set interior(y (b i + Y )) is connected. The bounded open set O satisfies the Poincaré-Wirtinger inequality for the exponent p [1, + ] if there exists a constant C p such that u W 1,p (O), u M O (u) L p (O) C p u L p (O). If the reference hole S is compact in a parallelotop Y, and if Y satisfies the Poincaré-Wirtinger inequality for p [1, + ], then the geometrical hypothesis holds true. Periodic unfolding method and multi-scale problems 42/56

43 III. Periodic unfolding in perforated domains The unfolding operator in perforated domains In general, the cell Y is not a parallelotop, and this requires the use of another reference cell with a more general shape. No choice of parallelotop gives a connected Y, while there are many possible Y s that give a connected Y. Periodic unfolding method and multi-scale problems 43/56

44 III. Periodic unfolding in perforated domains The unfolding operator in perforated domains Main theorem. Let w ε in W 1,p (Ω ε) satisfying w ε (L p (Ω ε)) n C. Then (up tp a subsequence), there exist w in W 1,p (Ω) and ŵ in L p (Ω; W 1,p per (Y )), Tε (w ε ) w strongly in L p loc (Ω; W 1,p (Y )), Tε (w ε ) w weakly in L p (Ω; W 1,p (Y )), Tε ( w ε ) w + y ŵ weakly in L p (Ω Y ). Periodic unfolding method and multi-scale problems 44/56

45 Application to the classical periodic homogenization in perforated domains Dirichlet and Neumann conditions Classical homogenization in periodically perforated domains One can treat a full range of problems without asking nor that the holes do not intersect Ω, neither that S be sufficiently smooth (necessary for having extension operators to the whole of Ω). It is again elementary to homogenize the Dirichlet-Neumann problem div (A ε u ε ) = f in Ω ε, u ε = 0 on Ω Ω ε, A ε u ε n ε = 0 on S ε Ω ε. and the Neumann problem { div (A ε u ε ) + u ε = f in Ω ε, For both problems one has A ε u ε n ε = 0 on Ω ε. ũ ε Y Y u 0 weakly in L 2 (Ω), with u 0 solution of the corresponding homogenized problem. Periodic unfolding method and multi-scale problems 45/56

46 Application to the classical periodic homogenization in perforated domains Robin boundary conditions Robin boundary conditions on the boundary of the holes In the same framework, consider the following non-homogeneous Neumann problem, : div (A ε u ε ) = f in Ω ε, u ε = 0 on Ω ε Ω, (6.1) A ε u ε n ε = g ε on S ε Ω. The domain Ω is bounded with Lipschitz boundary Ω. The function g ε is given in L 2 ( S ε Ω). The variational formulation is Find u ε H0 1 (Ω ε; Ω Ω ε) such that A ε u ε v dx = f v dx + Ω ε Ω ε v H0 1 (Ω ε; Ω Ω ε). S ε Ω Difficulty : the treatment of the surface integral. g ε v dσ(x), Periodic unfolding method and multi-scale problems 46/56

47 Application to the classical periodic homogenization in perforated domains Robin boundary conditions The boundary unfolding operator For p be in (1, + ), suppose that S has a finite number of connected components and that S is Lipschitz. The boundary of the set of holes in Ω is S ε Ω. Then a well-defined trace operator exists from W 1,p ( Ω ε) to W 1 1/p,p ( S ε ). Definition For any function ϕ Lebesgue-measurable on Ω ε S ε, the boundary unfolding operator Tε b is defined by ( [ x ) Tε b φ ε (φ)(x, y) = ε ]Y + εy a.e. for (x, y) Ω ε S, 0 a.e. for (x, y) Λ ε S. If ϕ W 1,p (Ω ε), T b ε (ϕ) is just the trace on S of T ε (ϕ). Periodic unfolding method and multi-scale problems 47/56

48 Application to the classical periodic homogenization in perforated domains Robin boundary conditions The operator Tε b has similar properties as the previous unfolding operators. In particular, the integration formula reads 1 ε Y Ω S T b ε (ϕ)(x, y) dx dσ(y) = c S ε ϕ(x) dσ(x). One can now give several convergence results allowing to give the limit of the integral g ε v dσ(x), with v H0 1 (Ω). S ε Ω To do so, one has to make some hypothese on the behaviour of g ε. Periodic unfolding method and multi-scale problems 48/56

49 Application to the classical periodic homogenization in perforated domains Robin boundary conditions Example. If there exist g in L 2 (Ω S) and G in L 2 (Ω) satisfying T b ε (g ε ) g 1 ε M S(T b ε (g ε )) G then the limit of S ε Ω g εv dσ(x) is S Y Ω weakly in L 2 (Ω S), weakly in L 2 (Ω), M S (y M g)(x) v(x) dx + S Y where y M = y M Y (y). Ω G(x)v(x) dx, Periodic unfolding method and multi-scale problems 49/56

50 Multi-scale problems Multi-scale domains The unfolding methods for fixed domains and for perforated domains, can be combined to consider mixed situations. Let Y be a subset of Y, and Y 2 be given an open subset of Y with Lipschitz boundary. Denote Y \ Y 2 by Y 1. Let Z be another periodicity cell, and ε, be two small parameters. For x in (R n ) ε, set A εδ be a matrix field defined by ({ x } ) { x } A 1 for A εδ ε Y ε Y 1, Y (x) = ({ { } x } ) { ε Y x } A 2 for δ Z ε Y 2. Y Periodic unfolding method and multi-scale problems 50/56

51 Multi-scale problems As before, the perforated domain Ω ε is defined by Ω ε = Ω \ S ε. So, Ω ε has ε periodic holes (the set S ε ) and a ε periodic set of a composite material corresponding to the set Y 2,ε. Z e d Y2 B Y 1 W e * Y * Periodic unfolding method and multi-scale problems 51/56

52 Multi-scale problems The homogenized problem corresponding to A εδ u εδ v dx = f v dx, v H0 1 (Ω ε; Ω Ω ε), Ω ε is of the form 1 Y Z Ω Y 2 Z + 1 [ ][ ] A 1 (y) u 0 + y û Ψ + y Φ Y Ω Y 1 Ω ε ][ ] A 2 (z) [ u 0 + y û + z û 1 Ψ + y Φ + z Θ dxdydz for Ψ H 1 0 (Ω), Φ L2 (Ω; H 1 per(y )/R) and Θ L 2 (Ω Y 2 ; H 1 per(z)/r). dxdy = Y Y Ω f Ψ dx, Periodic unfolding method and multi-scale problems 52/56

53 Multi-scale problems Ω ε,δ1 δ 2 Small and big holes Domains with small holes δ 1 B.. B δ 2 T T Yδ 1 δ 2 To treat such a situation one defines an appropriate unfolding operator depending on several small parameters. Periodic unfolding method and multi-scale problems 53/56

54 Multi-scale problems Small and big holes Reticulated structures Periodic unfolding method and multi-scale problems 54/56

55 Multi-scale problems Small and big holes Periodic unfolding method and multi-scale problems 55/56

56 Multi-scale problems Small and big holes Periodic unfolding method and multi-scale problems 56/56