Bilayer Plates: From Model Reduction to Γ-Convergent Finite Element Approximations and More

Transcription

1 Bilayer Plates: From Model Reduction to Γ-Convergent Finite Element Approximations and More Department of Mathematics Texas A&M University Joint work with: Soeren Bartels, University of Freiburg (Germany) Ricardo H. Nochetto, University of Maryland (USA). June 8, 2018 Seminaire du Laboratoire Jacques-Louis Lions Paris

2 FRAC Centre Orléans (see ; Achim Menges)

3 Bilayer Bending General setting: two thin sheets attached to each other thermal, electrical or moisture stimuli the two materials expand/compress differently small energies, large deformations bending Applications: climate regulation, thermostats, nanotubes, microrobots, micro-switches, micro-grippers, micro-scanners, micro-probes,... Goals: effective mathematical description convergent discretization reliable (and efficient) solution technique applications

4 Experiment 1: Selfassembling Microcube Conducting layers of polymer and Au were used as hinges (30µm) to connect rigid plates (300µm each side) to each other and to a Si substrate. The bending of the hinges was electrically controlled. E. W. H. Jager, E. Smela, and O. Inganäs, Microfabricating conjugated polymer actuators, Science, 290 (2000),

5 Experiment 3: Microactuator Moving Silicon Plates with Bilayer Hinges. The actuator holds a couple of fixed positions and is robust. E. Smela, M. Kallenbach, and J. Holdenried, Electrochemically Driven Polypyrrole Bilayers for Moving and Positioning Bulk Micromachined Silicon Plates, J. Microelectromechanical Systems, 8(4), (1999),

6 Experiment 4: Microhand and Hair The actuators move from completely flat to fully curled and back (to/from fully oxidized to/from fully reduced) in about 1 second (the bilayer is 0.5 µm thick). E. Smela, O. Inganäs, and I. Lundström, Controlled folding of micrometer-size structures, Science, 268 (1995),

7 Nonlinear Kirchhoff Models: Additional References Theory: G. Friesecke, R.D. James, and S.Müller, A Theorem on Geometric Rigidity and the Derivation of Nonlinear Plate Theory from Three-Dimensional Elasticity, Comm. Pure Appl. Math., Vol. LV, (2002), B. Schmidt, Plate theory for stressed heterogeneous multilayers of finite bending energy, J. Math. Pures Appl. 88 (2007) B. Schmidt, Minimal energy configurations of strained multi-layers, Calc. Var. 30, (2007), Applications: N. Bassik, B. T. Abebe, K. E. Laflin, D. H. Gracias, Photolithographically patterned smart hydrogel based bilayer actuators, Polymer 51 (2010), G. Stoychev, N. Puretskiy, and L. Ionov, Self-folding all-polymer thermoresponsive microcapsules, Soft Matter, 7 (2011), Also E. Efrati, E. Sharon, R. Kupferman (2009), J-N. Kuo, G-B. Lee, W-F. Pan and H-H. Lee (2005), M. Wardetzky, M. Bergou, D. Harmon, D. Zorin, and E. Grinspun (2006). Most Material in this talk: S. Bartels, A.B., R.H. Nochetto, Bilayer Plates: Model Reduction, Γ Convergent Finite Element Approximation and Discrete Gradient Flow, Comm. Pure Appl. Math. (2017).

8 Outline Reduced Model Energy Gradient Flow: Equilibrium Shapes Space Discretization Numerical Experiments - Equilibrium Shapes Thermal Actuation Conclusions

9 OUTLINE Reduced Model Energy Gradient Flow: Equilibrium Shapes Space Discretization Numerical Experiments - Equilibrium Shapes Thermal Actuation Conclusions

10 Mathematical Model s DΩ L y Ω x Ω θr = L(1 δs) θ R R+ = R + s L(1 + δs) = θr+ Domain: Ω s = Ω (0, s) R 3 with thickness s and midplane Ω R 2 ; Plate deformation: u : Ω s R 3 ; Surface parametrization: y : Ω R 3, Γ = y(ω); x := (x 1, x 2) Ω Unit normal to Γ: ν : Ω R 3, ν = 1y 1 y 2y 2 y ; Intuitive Deformation Assumption: u(x, x 3) := y(x ) + x 3β(s)ν(x ); Energy: Hyper-Elastic (Non-Linear) Plate Theory W (u) := s Ω 3 dist 2 ( u, (1+δ(s)N)SO 3) s ( ) Z m where N(x ) := m T R 3 3, δ(s), characterizes the material n mismatch at x 3 = 0.

11 Reduced Model Goal 1: Characterizing the asymptotic bending behavior of the plate Ω s as s 0 upon assuming that the energy remains bounded in this limit. Reduced Model (see also Schmidt): y H 2 (Ω) 3 s.t y D Ω = y D, y D Ω = Φ D (clamped), y T y = I 2 2 (isometries) and J(y) = 1 II + Z 2, 2 where II i,j := ν i jy is second fundamental form, ν is the normal of the surface Γ = y(γ) parametrized by y. Spontaneous Curvature Z : Ω R 2 2 acts as a spontaneous curvature and encodes (the possibly anisotropic) properties of the bilayer material. Ω

12 Simulation: Partially Clamped Plate Domain: Ω = ( 2, 2) (0, 10) Boundary Condition: DΩ = ( 1, 1) {0}. Goals: effective mathematical description convergent discretization reliable solution technique applications

13 Equivalent Energies: Taking advantage of the Geometric structure Isometries: ( y) T y = I y : Ω R 3 isometry parametrizing the surface: iy = 1, II i,j = i jy ν, II = h = y J(y) = 1 II + Z 2 2 Ω = 1 2 Notes for Later y 2 }{{} Ω variational + 2 i,j=1 Ω 1y i jy 2y 1y 2y Zi,j }{{} ν The discrete scheme does not guarantee the isometries properties (e.g. iy = 1). A gradient flow (De Giorgi Minimization of Movement) is set up to compute the relaxation to minima (equilibrium plate shapes). Ω Z 2.

14 OUTLINE Reduced Model Energy Gradient Flow: Equilibrium Shapes Space Discretization Numerical Experiments - Equilibrium Shapes Thermal Actuation Conclusions

15 Semi-Discrete H 2 Gradient Flow: Equilibrium Shapes H 2 gradient flow: Let τ > 0 and set k = 0. Choose y 0 H 2 (Ω) 3 such that y 0 D Ω = y D, y 0 D Ω = Φ D and [ y 0 ] T [ y 0 ] = Id 2. (1) Compute y k+1 H 2 (Ω) 3 which is minimal for Gradient Flow Algorithm / Minimization of Movement / Nonlinear minimize y 1 2τ (y yk ) 2 + J[y] subject to y D Ω = y D, y D Ω = Φ D and the Linearized Isometry Constraint [ (y y k )] T [ y k ] + [ y k ] T [ (y y k )] = 0. (2) Stop if D 2 (y k+1 y k ) ɛ stop; otherwise increase k k + 1 and go to (1).

16 Key Property of Linearized Isometry Constraint δn n n = 1 If y H 1 (Ω, R 3 2 ) satisfies [ (y y k )] T [ y k ] + [ y k ] T [ (y y k )] = 0 then [ y] T y I 2 [ y k ] T y k I 2... [ y 0 ] T y 0 I 2 = 0; Hence y 1 at each vertex and the discrete scheme is well defined. Iterative scheme for each time step (1 + τ) 2 y k = 2 y k 1 + nonlinear terms u.c (y k y k 1 ) T y k 1 + ( y k ) T (y k y k 1 ) = 0. + boundary conditions

17 Key Property of Linearized Isometry Constraint δn n If y H 1 (Ω, R 3 2 ) satisfies then n = 1 [ (y y k )] T [ y k ] + [ y k ] T [ (y y k )] = 0 [ y] T y I 2 [ y k ] T y k I 2... [ y 0 ] T y 0 I 2 = 0; Hence y 1 at each vertex and the discrete scheme is well defined. Iterative Fixed Point Algorithm (each pseudo-time step) The subiterations y k,l are well defined and converge to the unique solution in H 2 (Ω) 3 of the nonlinear problem provided τ 1. Moreover, [ y k ] T y k I 2 L 1 (Ω) 2 2 τ.

18 OUTLINE Reduced Model Energy Gradient Flow: Equilibrium Shapes Space Discretization Numerical Experiments - Equilibrium Shapes Thermal Actuation Conclusions

19 Choice of Finite Element Space Subdivision: T h is a subdivision of Ω made of rectangles of diameters h. We denote by N h the set of vertices. Kirchhoff quadrilaterals: gradients at vertices. W h continuous p.w. Q 3 with continuous h W h Θ h, 2 copies 4-th Order: y [H 2 (Ω)] 3 y h [W h ] 3 H 1 (Ω) 3 nonconforming space. Gradient operator: h : [W h ] 3 [Θ h ] 3 2 H 1 (Ω) 3 2 matching the usual gradient on N h, i.e. w h = h w h on N h for w h [W h ] 3. Θ h continuous p.w. Q 2. Isometry constraint: [ y] T [ y] = I 2 (or linearized version) h y h must satisfy constraints on N h.

20 Discrete Energy J h [y h ] = 1 2 Ω Φ h 2 dx + Ω Isometry Constraints: 2 i,j=1 ( ( Φh,1 iφ h,j Φ h,1 Φ ) h,2 ), Z ij Φ h,2 h Z 2 dx, y h W 3 h Φ h := h y h Θ 3 2 h. Gradient Flow: The pairs (y h, Φ h := h y h ) are limits of k-th iterates (y k h, Φ k h) of a discrete H 2 gradient flow based on a linearized isometry constraint enforced at the vertices [ (y h (z) y k h(z))] T [ y k h(z)]+[ y k h(z)] T [ (y h (z) y k h(z))] = 0 z N h Nodal Constraints Consequences: The minimizers y h [W h ] 3 of J h [y h ] obtained via the gradient flow (τ h) satisfy the inexact isometry constraint [ h y h (z)] T h y h (z) I 2, [ h y h (z)] T h y h (z) I 2 h z N h ; Implementation: Only the implementations of Q 1 and Q 2 elements are required. In particular, the basis function of W h are not needed.

21 Convergence of Minimizers J h J Question Does the sequence of almost absolute minimizers of J h (.) converges to an absolute minimizer of J(.)? Continuous Problem: 1 II 2 Ω Z 2, when y H 2 (Ω; R 3 ) with y T y = I, J(y) := y D Ω = y D, y D Ω = Φ D; +, otherwise. Discrete Problem: J h [y h ] = 1 2 ( ( Φ h 2 Φh,1 dx + iφ h,j 2 Ω Φ h,1 Φ ) h,2 ), Z ij Φ h,2 h i,j=1 + 1 Z 2 dx, y h Wh, 3 Φ h = h y h 2 Ω when y h D Ω = y D,h, Φ h D Ω = Φ D,h and [Φ h (z)] T Φ h (z) I 2, [Φh (z)] T Φ h (z) I 2 Ch and J h (y h ) = + otherwise. z Nh

22 De Giorgi Γ Convergence Theorem (Γ-Convergence) We have J Γ h J, i.e. Attainment: Given any y Dom(J), there exists a sequence {y h } h>0 Dom(J h ) such that y h y in H 1 (Ω; R 3 ), h y h y in H 1 (Ω; R 3 2 ), J h [y h ] J[y] as h 0. Lower bound property: If {y h } h>0 Dom(J h ) is a sequence satisfying J h [y h ] C, then there exist y Dom(J) such that as well as y h y in H 1 (Ω; R 3 ), h y h y in H 1 (Ω) 3 2, J[y] lim inf h 0 J h[y h ]. Corollary (Convergence of almost minimizers without smoothness) Any {y h } h sequence of almost global discrete minimizers of J h is precompact in H 1 (Ω) 3 and every cluster point y of y h is a global minimizer of J. Moreover, there exists a subsequence of {y h } h (not relabeled) such that lim y y h H h 0 1 (Ω) = 0 and lim J h [y h ] = J[y]. h 0

23 OUTLINE Reduced Model Energy Gradient Flow: Equilibrium Shapes Space Discretization Numerical Experiments - Equilibrium Shapes Thermal Actuation Conclusions

24 Experiment 4: Unexpected Relaxation The bilayers are 1 mm long and 30 µm wide, and they are attached on the bottom edge. The left actuators do not move from completely flat to fully curled. E. Smela, O. Inganäs, and I. Lundström, Controlled folding of micrometer-size structures, Science, 268 (1995),

25 Clamped Plate From Right to Left: aspect ratio (length/clamped side) = 0.5, 1.0, 7/4, 10/4. From Bottom to Top: spontaneous curvature Z = ai, a = 1, 2, 5.

26 Numerical Experiment: Corner Clamped Plate

27 Numerical Experiment: Clamped Plate with Tensor Spontaneous curvatures Alben et al. (2011) Simpson et al. (2010) (left) µ 1 = 5 e 1 = [1, 0] T ; µ 2 = 1 e 2 = [0, 1] T ; (right) µ 1 = 5 e 1 = [1, 1] T ; µ 2 = 1 e 2 = [1, 1] T ;

28 Numerical Experiment: DNA (using dg + trick for BC)

29 Numerical Experiment: Alternating (using dg + trick for BC)

30 OUTLINE Reduced Model Energy Gradient Flow: Equilibrium Shapes Space Discretization Numerical Experiments - Equilibrium Shapes Thermal Actuation Conclusions

31 Simplified Mathematical Model (+ A. Muliana MEEN-TAMU) Full Elastic Energy: include variable rigidity coefficient µ and plate temperature Θ. Reduced Elastic Energy: J[y, t] = 1 2 Ω µ(x ) II ΘI2 2 dx }{{} =Z obtained using a linear constitutive relation Z(Θ) = ΘI 2. Fourier law (Heat): tθ κ Θ = f supplemented with boundary conditions such as on Ω κ Θ µ = η(θ Ext Θ) or Θ = Θ Ext. No gradient flow, no subiterations, very fast but no mathematical proof... yet

32 Dog-Ears: Effect of Diffusivity. Newton Cooling Law κ Θ µ = 2(Θ }{{ Ext Θ). } =100 κ = 1 κ = 0.1 Diffusivity coefficient ratio of 10.

33 Self-Assembling Composite-Material Box Domain: 6 squares of size 1x1; Hinges: width π/24 Rigidity coefficient: µ = 1 in the hinges and µ = 20 otherwise Temperature source: f = 5 until t = 28.2 then f = 5 afterwards Spontaneous curvature: Z = ΘI in the hinges, 0 otherwise Heat diffusion coefficient: κ = 5

34 Self folding airfoils Initial temperatue T = 0 Outside boundaries of Hinges heated up to temperature 20 (inner) and 24 (outer). Diffusion coefficients: 10 (hinges), 1/4 plates Bending coefficients: 1 (hinges), 40 plates

35 Encapsulation with Self-Folding Microcapsules: Drug Delivery G. Stoychev, N. Puretskiy, and L. Ionov, Self-folding all-polymer thermoresponsive microcapsules, Soft Matter, 7 (2011),

36 Encapsulation with Self-Folding Microcapsules: Simulation Domain: center 1x1 square; sides: trapezoidal base 1 top 3/5 height 1 Rigidity coefficient: µ = 1 Spontaneous curvature: 12I. Obstacle problem: not discussed.

37 OUTLINE Reduced Model Energy Gradient Flow: Equilibrium Shapes Space Discretization Numerical Experiments - Equilibrium Shapes Thermal Actuation Conclusions

38 Conclusions Geometric PDEs: No general recipes but numerical methods must take advantage of the geometric structure. Bilayer model: Nonlinear Kirchhoff model that allows for bending but not stretching or shearing (isometry constraint). The model account for a spontaneous curvature tensor and large deformations. Kirchhoff Quadrilaterals: Nonconforming FEM of H 2 (ω) and key properties of discrete gradients h. Alternative using dg (with D. Guignard, D. Ntogkas and R.H Nochetto). Discrete Gradient Flow: H 2 gradient flow for a modified energy J h (.); constructive existence of every sub-iterate; convergence to the discrete problem; control of violation of isometry constraint. Γ convergence: Convergence of inexact discrete minimizers to minimizers of the continuous energy J(.). Temperature Actuator: Very fast, very simple, and predicts the dynamics. Simulations: Exhibit presence of local minimizers (other than cylinders) and interesting interplay between geometry and bending patterns. Obtained with deal.ii. Supports: National Science Foundation and Air force of scientific research grant.