Heterogeneous media: Case studies in non-periodic averaging and perspectives in designing packaging materials

Transcription

1 Heterogeneous media: Case studies in non-periodic averaging and perspectives in designing packaging materials Department of Mathematics and Computer Science, Karlstad University, Sweden Karlstad, January 2015

2 Outline of the Talk Microstructure models of heterogeneous media Distributed microstructures Structured transport Generic microscopic model Derivation via formal homogenization of a micro-macro model Analysis of the micro-macro model Weak formulation. Basic estimates Global existence and uniqueness of weak solutions Numerical illustration Justification of the formal homogenization Outline of the proof for the corrector estimate Open issues

3 Heterogeneous media: Diffusion in a chessboard domain Heterogeneous media and their homogeneous representation Is an equivalent formulation posed in an homogeneous domain possible?

4 Asymptotic homogenization for chessboard-type microstructures Here A ɛ(x) = A(x, x ɛ ). Idea 1: Use tu ɛ div(a ɛ(x) u ɛ) = f ɛ in Ω ɛ u ɛ = g ɛ at black/white inner boundaries n A ɛ u ɛ = 0 at the outer boundary + i.c. u ɛ(t, x) = u 0 (t, x) + ɛu 1 (t, x, x ɛ ) + ɛ2 u 2 (t, x, x ɛ ) + O(ɛ3 ) to get PDEs for u 0. Idea 2: Along the same line: show the corrector estimate u ɛ u 0 c(u 1,... )ɛ α, α > 0.

5 Macroscopic equation for chessboard diffusion. Range of validity? t(φu 0 ) div(φd(x, w(x)) u 0 ) = φf 0 in Ω n φd(x, w(x)) u 0 = 0 at the outer boundary φ is the chessboard porosity, i.e. φ = 1 16 D(, w( )) is an effective transport coefficient w is a parameter depending on the chessboard s microstructure (cell functions) Rate of convergence (correctors): u ɛ u 0 L 2 c 1 ɛ tu ɛ tu 0 L 2 c 2 ɛ u ɛ u 0 oscillations L 2 c 3 ɛ

6 Averaging techniques in action Towards understanding mixed transport fluxes: Estimating speeds of macroscopic corrosion fronts in concrete [large-time behavior for free boundary problems] Ionic transport in porous media [homogenization of the Stokes-Nernst-Planck-Poisson system] Multiscale combustion [smoldering thin sheets, fast-reaction limits] Cross-diffusion effects [averaging the Becker-Döring system for hot colloids] Thermo-diffusion effects [estimating permeability tensors for mass and heat transport through porous membranes]

7 Bridging length scales in heterogeneous media Averaging techniques (periodic homogenization, renormalization,...) PDE models with distributed microstructure 1. two-scale models A. Friedman, A. Tzavaras, P. Knabner 2. distributed-microstructure models R. E. Showalter and co-workers (Walkington, Cook, Clark, Visarraga,...) + M. Böhm, S. Meier, D. Treutler, J. Esher 3. dual- or double-porosity models U. Hornung, W. Jäger, T. Arbogast, two-scale models with freely evolving micro-interfaces C. Eck., H. Emmerich, P. Knabner, A. Muntean (2 scale phase-field models), S. Meier, A. Muntean (2 scale fast-reaction asymptotics) 5. coupling multi-physics/discrete-to-continuum etc.

8 Double-porosity structure of materials Barenblatt, Zheltov, Kochina, PMM, 24(1960), 5, pp

9 Locally-periodic distributions of perforations!

10 Generic micro-model ut ɛ = (D h u ɛ q ɛ u ɛ ) q ɛ = κ p ɛ q ɛ = 0 { vt ɛ = ɛ 2 (D l v ɛ ) in Ω ɛ l, ν ɛ (D h u ɛ ) = ɛ 2 ν ɛ (D l v ɛ ) u ɛ = v ɛ q ɛ = 0 { u ɛ (x, t) = u b (x, t) on Γ, q ɛ (x, t) = q b (x, t) { u ɛ (x, 0) = ui ɛ (x) in Ω ɛ h, v ɛ (x, 0) = vi ɛ (x) in Ω ɛ l, in Ω ɛ h, on Γ ɛ,

11 Basic ideas of the (formal) two-scale homogenisation asymptotics u ɛ (x, t) = u 0 (x, x/ɛ, t) + ɛu 1 (x, x/ɛ, t) + ɛ 2 u 2 (x, x/ɛ, t) +... v ɛ (x, t) = v 0 (x, x/ɛ, t) + ɛv 1 (x, x/ɛ, t) + ɛ 2 v 2 (x, x/ɛ, t) +... q ɛ (x, t) = q 0 (x, x/ɛ, t) + ɛq 1 (x, x/ɛ, t) + ɛ 2 q 2 (x, x/ɛ, t) +... p ɛ (x, t) = p 0 (x, x/ɛ, t) + ɛp 1 (x, x/ɛ, t) + ɛ 2 p 2 (x, x/ɛ, t) +... S ɛ = 1 ɛ ys + O(ɛ0 ) ν ɛ = ν 0 + ɛν 1 + O(ɛ 2 ), Ω ɛ l = {S(x, x/ɛ) < 0 : x Ω}, Ω ɛ h = {S(x, x/ɛ) > 0 : x Ω}

12 Micro-macro model tv 0 (x, y, t) = D l yv 0 (x, y, t) y < r(x), x Ω, t (θ(x)u 0 + ) v y <r(x) 0 dy = div x(d h A(x) xu 0 qu 0 ) for x Ω, q = K(x) xp 0 for x Ω, x q = 0 for x Ω, v 0 (x, y, t) = u 0 (x, t) for y = r(x), u 0 (x, t) = u b (x, t) for x Γ, q(x, t) = q b (x, t) for x Γ, { u 0 (x, 0) = u I (x) for x Ω, v 0 (x, y, 0) = v I (x, y) for y < r(x), x Ω.

13 The porosity θ(x) of the medium is given by θ(x) := 1 πr 2 (x), while the effective diffusivity D(x) := (a ij (x)) i,j and the effective permeability K(x) := (k ij (x)) i,j are defined by a ij (x) := D h δ ij + yi U j (x, y, t) dy, {y U y >r(x)} and k ij (x) := V ji (x, y, t) dy. {y U y >r(x)}

14 x-dependent cell problems yu j (x, y) = 0 ν 0 yu j (x, y) = ν 0 e j U j (x, y) y-periodic, for all x Ω, y Y, y > r(x), for all x Ω, y = r(x), and V j (x, y) = yπ j (x, y) + e j y V j (x, y) = 0 V j = 0 V j (x, y) and π j (x, y) y-periodic, for all x Ω, y Y, y > r(x), for all x Ω, y Y, y > r(x), for all x Ω, y = r(x), for j = 1, 2.

15 Reduced micro macro model θ(x) tu x (D(x) xu qu) = B(x) νy (D l yv) dσ in Ω, tv D l yv = 0 in Ω B(x), u(x, t) = v(x, y, t) at (x, y) Ω B(x), u(x, t) = u b (x, t) at x Ω, u(x, 0) = u I (x) in Ω, v(x, y, 0) = v I (x, y) at (x, y) Ω B(x)

16 Memory effects?

17 Assumptions on data and parameters (A1) S 0 : Ω U R, which defines B(x) and also the 1-dimensional boundary Ω B(x) of Ω B(x) as (x, y) Ω B(x) if and only if S 0 (x, y) = 0, is an element of C 2 (Ω U). Additionally, the Clarke gradient ys 0 (x, y) is regular for all choices of (x, y) Ω U. (A2) θ, D L + (Ω), q L (Ω; R d ) with q = 0, u b L + (Ω S) H 1 (S; L 2 (Ω)), tu b 0 a.e. (x, t) Ω S, u I L + (Ω) H 1, v I (x, ) L + (B(x)) H 2 for a.e. x Ω.

18 Functional setting V 1 := H0 1 (Ω), V 2 := L 2 (Ω; H 2 (B(x))), H 1 := L 2 θ(ω), H 2 := L 2 (Ω; L 2 (B(x))). If 0 < B(x), B(x) <, then the direct Hilbert integrals L 2 (Ω; H 1 (B(x))) := {u L 2 (Ω; L 2 (B(x))) : yu L 2 (Ω; L 2 (B(x)))} L 2 (Ω; H 1 ( B(x))) := {u : Ω B(x) R meas. s. t. u(x) 2 L 2 ( B(x)) < } are separable Hilbert spaces with distributed trace: γ : L 2 (Ω; H 1 (B(x))) L 2 (Ω, L 2 ( B(x))) given by γu(x, s) := (γ xu(x))(s), x Ω, s B(x), u L 2 (Ω; H 1 (B(x))) Ω

19 Definition (Weak formulation) Assume (A1) and (A2). The pair (u, v), with u = U + u b and where (U, v) V, is a weak solution if the following identities hold θ t(u + u b )φ dx + (D x(u + u b ) q(u + u b )) xφ dx = Ω Ω Ω B(x) Ω B(x) ν y (D l yv)φ d tvψ dydx + D l y yψ dydx = ν y (D l yv)φ dσdx, Ω B(x) Ω B(x) for all (φ, ψ) V and t S.

20 Basic estimates Lemma Let (A1) and (A2) be satisfied. Then any weak solution (u, v) of problem (P) has the following properties: (i) u 0 for a.e. x Ω and for all t S; (ii) v 0 for a.e. (x, y) Ω B(x) and for all t S; (iii) u M 1 for a.e. x Ω and for all t S; (iv) v M 2 for a.e. (x, y) Ω B(x) and for all t S; (v) The following energy inequality holds: u 2 L 2 (S;V 1 ) L (S;H 1 ) + v 2 L 2 (S;L 2 (Ω,V 2 )) L (S;H 2 ) + xu 2 L 2 (S;H 1 ) + yv 2 L 2 (S Ω B(x)) c 1. N.B. Assume (A1), (A2). Then uniqueness of weak solutions holds.

21 Global existence Theorem There exists at least a weak solution of the micro-macro model. Proof. (Sketch) Schauder fixed-point argument in L 2 (S; L 2 (Ω)) framework X 1 := L 2 (S; L 2 (Ω)), X 2 := L 2 (S; H0 1 (Ω)) H 1 (S; L 2 (Ω)), X 3 := L 2 (S; V 2 ) H 1 (S; L 2 (Ω; L 2 (B(x)))). T : X 1 X 1 with T := T 3 T 2 T 1.

22 T 1 maps a f X 1 to the solution w X 2 of θ t(u + u b )φ dx + (D x(u + u b ) q(u + u b )) xφ dx = Ω for all φ H0 1 (Ω). T 2 maps a w X 2 to a solution v X 3 of t(v + w)ψ dydx + Ω B(x) for all ψ V 2 and t S. T 3 maps a v X 3 to f X 1 by f = Lemma T is well defined. Ω Ω Ω B(x) B(x) B(x) D l y(v + w) yψ dydx = ν y (D l y(v + w))ψ dσdx, ν y yv dσ. Ω f φ dx,

23 Compactness step Lemma The operator T is compact. Step 1. Use Ψ : Ω B(0) Ω B(x). We call Ψ a regular C 2 -motion if Ψ C 2 (Ω B(0)) with the property that for each x Ω Ψ(x, ) : B(0) B(x) := Ψ(x, B(0)) is bijective, and if there exist constants c, C > 0 such that c det yψ(x, y) C, for all (x, y) Ω B(0). The existence of such a mapping is ensured by the fact that S 0 C 2 (Ω U), by (A1). If Ψ is a regular C 2 -motion, then are continuous functions of x and y. F := yψ and J := det F yv = F T ŷ ˆv, tv = t ˆv, ν y j dσ = JF T ˆνŷ ĵ dσ. B(x) Γ 0

24 The transformed equation can be now written as: Let w X 2 be given. Find ˆV L 2 (S; L 2 (Ω; H 1 0 (B(0)))) H 1 (S; L 2 (Ω; L 2 (B(0)))) such that Ω B(0) t( ˆV + w)ψj dydx + Ω Ω B(0) JF 1 D l F T y( ˆV + w) yψ dydx = Γ 0 ˆν y (JF 1 D l F T y( ˆV + w))ψ dσdx, for all ψ L 2 (Ω; H 1 0 (B(0))) and t S. Denote by Γ 0 the boundary of B(0).

25 Step 2. Interior and boundary regularity (in y) Assume (A1) and (A2). Then Γ 0 is C 2 and ˆV L 2 (S; L 2 (Ω; H 2 (B(0)) H 1 0 (B(0)))). Step 3. Additional two-scale regularity (in y) Assume (A1) and (A2). Then ˆV L 2 (S; H 1 (Ω; H 2 (B(0)) H 1 0 (B(0)))). Step 4. Apply Lions-Aubin Lemma

26 Concentration profiles of the two-scale model Solution profiles of the two-scale model at different times: Up: R = 0.1; Bottom: R = 0.9.

27 Numerical approximation of RD systems on two scales Convergent two-scale Galerkin approximations A. Muntean, M. Neuss-Radu: A multiscale Galerkin approach for a class of nonlinear coupled reaction-diffusion systems in complex media. JMAA 371 (2010), (2), Convergence rates (a priori) A. Muntean, O. Lakkis (2010). Rate of convergence for a Galerkin scheme approximating a two-scale reaction-diffusion system with nonlinear transmission condition. RIMS Kokyuroku (Kyoto), 1693, pp Numerical example in 1D 1D

28 Corrector estimates How goed is the averaging method? (u ɛ, v ɛ) solution vector for the micro problem (u 0, v 0 ) solution vector for the macro problem u ɛ 0, v ɛ 0, u ɛ 1 macroscopic reconstructions u ɛ 0(x, t) := u 0 (x, t) v ɛ 0 (x, t) := v 0 (x, x/ɛ, t) u ɛ 1(x, t) := u ɛ 0(x, t) + ɛu(t, x, x/ɛ) u ɛ 0(x, t)

29 Justification of the formal asymptotics Theorem Assume (A1) and (A2). Then the following convergence rate holds u ɛ u ɛ 0 L (S,L 2 (Ω ɛ)) + v ɛ v ɛ 0 L (S,L 2 (Ω Ω ɛ)) + u ɛ u ɛ 1 L (S,H 1 (Ω ɛ)) + ɛ v ɛ v ɛ 0 L (S,H 1 (Ω Ω ɛ)) c ɛ

30 Outline of the proof for the corrector estimate Step 1. Write weak formulations for both micro and macro pbs. (the later in terms of macro reconstructions) Step 2. Subtract the 2 weak formulations and choose suitable test functions ϕ := u ɛ u ɛ 0(x, t) + ɛu(t, x, x/ɛ) u ɛ 0(x, t) ψ := v ɛ v ɛ 0 Step 3. A technical lemma: Prove that 1 ɛ ɛd l v ɛ ν ɛψdσ ν y D l yv0 ɛ ψdσ cɛ ψ S ɛ Y B(x) H 1 (Ω ɛ). B(x) Step 4. Bookkeeping of ɛ

31 Open issues 1. Computability in 2D (including a priori / a posteriori error estimates) 2. Free micro interfaces? 3. Two-scale coupling between macro PDEs and micro phase transitions [deterministic/stochastic ODEs]. Analysis + computability issues + PDEs in measures(?) 4. Applications to harvesting geothermal energy NWO-MPE project (ongoing) 5. Applications to design of smart packaging Preparation phase

32 Intermezzo: Multiscale smoldering combustion of thin sheets O. Zik, Z. Olami, and E. Moses (1998) Fingering instability in combustion, Phys. Rev. Lett. 81, E. Ijioma, T. Ogawa, A. Muntean (2013). Pattern formation in reverse smoldering combustion : a homogenization approach. Combustion Theory and Modelling, 17(2), Ijioma, E.R., Muntean, A. Ogawa, T. (2015). Effect of material anisotropy on the fingering instability in reverse smouldering combustion. International Journal of Heat and Mass Transfer, 81, Fatima, T., Ijioma, E.R., Ogawa, T. Muntean, A. (2014). Homogenization and dimension reduction of filtration combustion in heterogeneous thin layers.. Networks and Heterogeneous Media, 9(4),

33 Background of this work: A. Muntean, T. van Noorden: Corrector estimates for the homogenization of a locally-periodic medium with areas of low and high diffusivity. Eur. J. Appl. Math. 24 (2013), (5), T. van Noorden, A. Muntean: Homogenization of a locally-periodic medium with areas of low and high diffusivity. Eur. J. Appl. Math. 22 (2011), (5), A. Muntean, M. Neuss-Radu: A multiscale Galerkin approach for a class of nonlinear coupled reaction-diffusion systems in complex media. JMAA 371 (2010), (2), T. Fatima, N. Arab, E. Zemskov, A. Muntean: Homogenization of a reaction-diffusion system modeling sulfate corrosion in locally-periodic perforated domains. J. Engng. Math. 69(2011), (2), V. Chalupecky, T. Fatima, A. Muntean: Multiscale sulfate attack on sewer pipes: Numerical study of a fast micro-macro mass transfer limit. J. Math-for-Industry, 2(2010) (B-7), A. Muntean, O. Lakkis (2010). Rate of convergence for a Galerkin scheme approximating a two-scale reaction-diffusion system with nonlinear transmission condition. RIMS Kokyuroku (Kyoto), 1693, pp S. Meier, M. Peter, A. Muntean, M. Böhm, J. Kropp: A two-scale approach to concrete carbonation in Proc. Int. RILEM Workshop on Integral Service Life Modeling of Concrete Structures, Guimares, Portugal, 2007, 3 10.