Homogenization of Kirchhoff plate equation

Transcription

1 Homogenization of Kirchhoff plate equation Jelena Jankov J. J. STROSSMAYER UNIVERSITY OF OSIJEK DEPARTMENT OF MATHEMATICS Trg Ljudevita Gaja Osijek, Croatia jjankov@mathos.hr Joint work with: K. Burazin, M. Vrdoljak [VII PARTIAL DIFFERENTIAL EQUATIONS, OPTIMAL DESIGN AND NUMERICS]

2 Properties of The physical idea of homogenization is to average a heterogeneous media in order to derive effective properties. { Au = f in Ω initial/boundary condition Jelena Jankov Homogenization of Kirchhoff plate equation 2/20

3 Properties of The physical idea of homogenization is to average a heterogeneous media in order to derive effective properties. { Au = f in Ω initial/boundary condition Jelena Jankov Homogenization of Kirchhoff plate equation 2/20

4 Properties of Sequence of similar problems { An u n = f in Ω initial/boundary condition. If u n u, A n A the limit (effective) problem is { Au = f in Ω initial/boundary condition... Jelena Jankov Homogenization of Kirchhoff plate equation 3/20

5 Properties of Sequence of similar problems { An u n = f in Ω initial/boundary condition. If u n u, A n A the limit (effective) problem is { Au = f in Ω initial/boundary condition... Jelena Jankov Homogenization of Kirchhoff plate equation 3/20

6 Properties of Kirchhoff plate equation Homogeneous Dirichlet boundary value problem: { divdiv(m u) = f in Ω u H 2 0 (Ω). Ω R 2 bounded domain f H 2 (Ω) external load M M 2 (α, β; Ω) := {M L (Ω; L(Sym, Sym)) : ( S Sym) M(x)S : S αs : S and M 1 S : S 1 β S : S a.e.x} describes properties of material of the given plate u H0 2 (Ω) vertical displacement of the plate Jelena Jankov Homogenization of Kirchhoff plate equation 4/20

7 Properties of Kirchhoff plate equation Homogeneous Dirichlet boundary value problem: { divdiv(m u) = f in Ω u H 2 0 (Ω). Ω R 2 bounded domain f H 2 (Ω) external load M M 2 (α, β; Ω) := {M L (Ω; L(Sym, Sym)) : ( S Sym) M(x)S : S αs : S and M 1 S : S 1 β S : S a.e.x} describes properties of material of the given plate u H0 2 (Ω) vertical displacement of the plate Jelena Jankov Homogenization of Kirchhoff plate equation 4/20

8 Properties of Antonić, Balenović, Definition A sequence of tensor functions (M n ) in M 2 (α, β; Ω) H-converges to M M 2 (α, β; Ω) if for any f H 2 (Ω) the sequence of solutions (u n ) of problems { divdiv(m n u n ) = f in Ω u n H 2 0 (Ω) coverges weakly to a limit u in H 2 0 (Ω), while the sequence (M n u n ) converges to M u weakly in the space L 2 (Ω; Sym). Theorem Let (M n ) be a sequence in M 2 (α, β; Ω). Then there is a subsequence (M n k) and a tensor function M M 2 (α, β; Ω) such that (M n k) H-converges to M. Jelena Jankov Homogenization of Kirchhoff plate equation 5/20

9 Properties of Antonić, Balenović, Definition A sequence of tensor functions (M n ) in M 2 (α, β; Ω) H-converges to M M 2 (α, β; Ω) if for any f H 2 (Ω) the sequence of solutions (u n ) of problems { divdiv(m n u n ) = f in Ω u n H 2 0 (Ω) coverges weakly to a limit u in H 2 0 (Ω), while the sequence (M n u n ) converges to M u weakly in the space L 2 (Ω; Sym). Theorem Let (M n ) be a sequence in M 2 (α, β; Ω). Then there is a subsequence (M n k) and a tensor function M M 2 (α, β; Ω) such that (M n k) H-converges to M. Jelena Jankov Homogenization of Kirchhoff plate equation 5/20

10 Properties of Theorem (Locality of the ) Let (M n ) and (O n ) be two sequences of tensors in M 2 (α, β; Ω), which H-converge to M and O, respectively. Let ω be an open subset compactly embedded in Ω. If M n (x) = O n (x) in ω, then M(x) = O(x) in ω. Theorem (Irrelevance of boundary conditions) Let (M n ) be a sequence of tensors in M 2 (α, β; Ω) that H-converges to M. For any sequence (z n ) such that { divdiv(m n z n ) = f in Ω z n z in H 2 loc (Ω) M n satisfies M n z n M z in L 2 loc (Ω; Sym). Jelena Jankov Homogenization of Kirchhoff plate equation 6/20

11 Properties of Theorem (Locality of the ) Let (M n ) and (O n ) be two sequences of tensors in M 2 (α, β; Ω), which H-converge to M and O, respectively. Let ω be an open subset compactly embedded in Ω. If M n (x) = O n (x) in ω, then M(x) = O(x) in ω. Theorem (Irrelevance of boundary conditions) Let (M n ) be a sequence of tensors in M 2 (α, β; Ω) that H-converges to M. For any sequence (z n ) such that { divdiv(m n z n ) = f in Ω z n z in H 2 loc (Ω) M n satisfies M n z n M z in L 2 loc (Ω; Sym). Jelena Jankov Homogenization of Kirchhoff plate equation 6/20

12 Properties of Theorem (Energy convergence) Let (M n ) be a sequence of tensors in M 2 (α, β; Ω) that H-converges to M. For any f H 2 (Ω), the sequence (u n ) of solutions of { divdiv(m n u n ) = f in Ω u n H 2 0 (Ω). satisfies M n u n : u n M u : u weakly- in the space of Radon measures and M n u n : u n dx M u : u dx, where u is the Ω solution of the homogenized equation { divdiv(m u) = f in Ω Ω u H 2 0 (Ω). Jelena Jankov Homogenization of Kirchhoff plate equation 7/20

13 Properties of Theorem (Ordering property) Let (M n ) and (O n ) be two sequences of tensors in M 2 (α, β; Ω) that H-converge to the homogenized tensors M and O, respectively. Assume that, for any n, M n ξ : ξ O n ξ : ξ, ξ Sym. Then the homogenized limits are also ordered: Mξ : ξ Oξ : ξ, ξ Sym. Theorem Let (M n ) be a sequence of tensors in M 2 (α, β; Ω) that either converges strongly to a limit tensor M in L 1 (Ω; L(Sym, Sym)), or converges to M almost everywhere in Ω. Then, M n also H-converges to M. Jelena Jankov Homogenization of Kirchhoff plate equation 8/20

14 Properties of Theorem (Ordering property) Let (M n ) and (O n ) be two sequences of tensors in M 2 (α, β; Ω) that H-converge to the homogenized tensors M and O, respectively. Assume that, for any n, M n ξ : ξ O n ξ : ξ, ξ Sym. Then the homogenized limits are also ordered: Mξ : ξ Oξ : ξ, ξ Sym. Theorem Let (M n ) be a sequence of tensors in M 2 (α, β; Ω) that either converges strongly to a limit tensor M in L 1 (Ω; L(Sym, Sym)), or converges to M almost everywhere in Ω. Then, M n also H-converges to M. Jelena Jankov Homogenization of Kirchhoff plate equation 8/20

15 Properties of Theorem (Compensated compactness result) Let the following convergences be valid: w n w in Hloc 2 (Ω) and D n D in L 2 loc (Ω; M 2 2) with an additional assumption that the sequence (divdivd n ) is contained in a precompact (for the strong topology) set of the space H 2 loc (Ω). Then we have that E n : D n E : D weakly- in the space of Radon measures, where we denote E n := w n, for n N { }. Can be seen from Tartar s quadratic theorem of compensated compactness... Jelena Jankov Homogenization of Kirchhoff plate equation 9/20

16 Properties of Theorem (Compensated compactness result) Let the following convergences be valid: w n w in Hloc 2 (Ω) and D n D in L 2 loc (Ω; M 2 2) with an additional assumption that the sequence (divdivd n ) is contained in a precompact (for the strong topology) set of the space H 2 loc (Ω). Then we have that E n : D n E : D weakly- in the space of Radon measures, where we denote E n := w n, for n N { }. Can be seen from Tartar s quadratic theorem of compensated compactness... Jelena Jankov Homogenization of Kirchhoff plate equation 9/20

17 Properties of Definition Let (M n ) be a sequence of tensors in M 2 (α, β; Ω) that H-converges to a limit M. Let (w ij n ) 1 i,j N be a family of test functions satisfying w ij n 1 2 x ix j in H 2 (Ω) divdiv(m n w ij n) in H 2 loc (Ω) M n wn ij in L 2 loc (Ω; Sym). The tensor W n defined as [a ijkm ] ij = [ wn km ] ij is called a corrector tensor. Jelena Jankov Homogenization of Kirchhoff plate equation 10/20

18 Properties of Theorem Let (M n ) be a sequence of tensors in M 2 (α, β; Ω) that H-converges to a tensor M. A sequence of correctors (W n ) is unique in the sense that, if there exist two sequences of correctors (W n ) and ( W n ), their difference (W n W n ) converges strongly to zero in L 2 loc (Ω; L(Sym, Sym)). Jelena Jankov Homogenization of Kirchhoff plate equation 11/20

19 Properties of Theorem (Corrector result) Let (M n ) be a sequence of tensors in M 2 (α, β; Ω) which H-converges to M. For f H 2 (Ω), let (u n ) be the solution of { divdiv(m n u n ) = f in Ω u n H 2 0 (Ω). Let u be the weak limit of (u n ) in H0 2 (Ω), i.e., the solution of the homogenized equation { divdiv(m u) = f in Ω u H 2 0 (Ω). Then, r n := u n W n u 0 strongly in L 1 loc (Ω; Sym). Jelena Jankov Homogenization of Kirchhoff plate equation 12/20

20 Properties of A n γ(x) := A 0 + γb n (x), γ R A γ := A 0 + γb 0 + γ 2 C 0 + o(γ 2 ), γ R Jelena Jankov Homogenization of Kirchhoff plate equation 13/20

21 Properties of A n γ(x) := A 0 + γb n (x), γ R A γ := A 0 + γb 0 + γ 2 C 0 + o(γ 2 ), γ R Jelena Jankov Homogenization of Kirchhoff plate equation 13/20

22 Properties of Theorem Let M n : Ω P L(Sym, Sym) be a sequence of tensors, such that M n (, p) M 2 (α, β; Ω), for p P, where P R is an open set. Assume that (for some k N 0 ) a mapping p M n (, p) is of class C k from P to L (Ω; L(Sym, Sym)), with derivatives which are equicontinuous on every compact set K P up to order k: ( K K(P )) ( ε > 0)( δ > 0)( p, q K)( n N) ( i {0,..., k}) p q < δ (M n ) (i) (, p) (M n ) (i) (, q) L (Ω;L(Sym,Sym)) < ε. Then there is a subsequence (M n k) such that for every p P H M n k (, p) M(, p) in M 2 (α, β; Ω) Jelena Jankov Homogenization of Kirchhoff plate equation 14/20

23 Properties of and p M(, p) is a C k mapping from P to L (Ω; L(Sym, Sym)). Jelena Jankov Homogenization of Kirchhoff plate equation 15/20

24 Properties of Periodic case Y = [0, 1] d M n (x) := M(nx), x Ω H 2 # (Y ) := {f H2 loc (Rd ) such that f is Y periodic} with the norm H 2 (Y ) H 2 # (Y )/R equipped with the norm L 2 (Y ) E ij, 1 i, j d are M d d matrices defined as 1, if i = j = k = l 1 [E ij ] kl = 2, if i j, (k, l) {(i, j), (j, i)} 0, otherwise. Jelena Jankov Homogenization of Kirchhoff plate equation 16/20

25 Properties of Theorem Let (M n ) be a sequence of tensors defined by M n (x) := M(nx), x Ω. Then (M n ) H-converges to a constant tensor M M 2 (α, β; Ω) defined as m klij = M(y)(E ij + w ij (y)) : (E kl + w kl (y)) dy, Y where (w ij ) 1 i,j d is the family of unique solutions in H# 2 (Y )/R of boundary value problems { div div (M(y)(Eij + w ij (y))) = 0 in Y, i, j = 1,..., d y w ij (y). Jelena Jankov Homogenization of Kirchhoff plate equation 17/20

26 Properties of A n γ(x) := A 0 + γb n (x) A 0 constant, coercive tensor B n (x) := B(nx) Y-periodic, L tensor function and B(y) dy = 0 Y A γ := A 0 + γb 0 + γ 2 C 0 + o(γ 2 ) Jelena Jankov Homogenization of Kirchhoff plate equation 18/20

27 Properties of C 0 E ij : E kl =(2πi) 2 B 0 E ij : E kl = 0 a k B k k k T : E kl dy Y k J + (2πi) 4 + (2πi) 2 A 0 k k T a k : a k k k T dy Y k J B k E ij : a k k k T dy Y k J Jelena Jankov Homogenization of Kirchhoff plate equation 19/20

28 Properties of Thank you for your attention! Jelena Jankov Homogenization of Kirchhoff plate equation 20/20