Department of Mathematics Texas A&M University ... · Bilayer Plates: From Model Reduction to...

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

Transcript of Department of Mathematics Texas A&M University ... · Bilayer Plates: From Model Reduction to...

Page 1: Department of Mathematics Texas A&M University ... · Bilayer Plates: From Model Reduction to -Convergent Finite Element Approximations and More Andrea Bonito Department of Mathematics

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

Andrea Bonito

Department of MathematicsTexas 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

Page 2: Department of Mathematics Texas A&M University ... · Bilayer Plates: From Model Reduction to -Convergent Finite Element Approximations and More Andrea Bonito Department of Mathematics

Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

FRAC Centre Orleans (see http://icd.uni-stuttgart.de ; Achim Menges)

Bilayers Plates Andrea Bonito

Page 3: Department of Mathematics Texas A&M University ... · Bilayer Plates: From Model Reduction to -Convergent Finite Element Approximations and More Andrea Bonito Department of Mathematics

Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Bilayer Bending

General setting:

• two thin sheets attached to each other

• thermal, electrical or moisture stimuli

• the two materials expand/compressdifferently

• 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

Bilayers Plates Andrea Bonito

Page 4: Department of Mathematics Texas A&M University ... · Bilayer Plates: From Model Reduction to -Convergent Finite Element Approximations and More Andrea Bonito Department of Mathematics

Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Experiment 1: Selfassembling Microcube

Conducting layers of polymer and Au were used as hinges (30µm) to connectrigid plates (300µm each side) to each other and to a Si substrate. Thebending of the hinges was electrically controlled.

E. W. H. Jager, E. Smela, and O. Inganas, Microfabricating conjugatedpolymer actuators, Science, 290 (2000), 1540–1545.

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Experiment 3: Microactuator

Moving Silicon Plates with Bilayer Hinges. The actuator holds a couple of fixedpositions and is robust.

E. Smela, M. Kallenbach, and J. Holdenried, Electrochemically DrivenPolypyrrole Bilayers for Moving and Positioning Bulk Micromachined SiliconPlates, J. Microelectromechanical Systems, 8(4), (1999), 373–383.

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Experiment 4: Microhand and Hair

The actuators move from completely flat to fully curled and back (to/from fullyoxidized to/from fully reduced) in about 1 second (the bilayer is 0.5 µm thick).

E. Smela, O. Inganas, and I. Lundstrom, Controlled folding ofmicrometer-size structures, Science, 268 (1995), 1735–1738.

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Nonlinear Kirchhoff Models: Additional References

Theory:

• G. Friesecke, R.D. James, and S.Muller, A Theorem on Geometric Rigidityand the Derivation of Nonlinear Plate Theory from Three-Dimensional Elasticity,Comm. Pure Appl. Math., Vol. LV, (2002), 1461–1506.

• B. Schmidt, Plate theory for stressed heterogeneous multilayers of finite bendingenergy, J. Math. Pures Appl. 88 (2007) 107-122.

• B. Schmidt, Minimal energy configurations of strained multi-layers, Calc. Var. 30,(2007), 477-497.

Applications:

• N. Bassik, B. T. Abebe, K. E. Laflin, D. H. Gracias, Photolithographicallypatterned smart hydrogel based bilayer actuators, Polymer 51 (2010), 6093–6098.

• G. Stoychev, N. Puretskiy, and L. Ionov, Self-folding all-polymerthermoresponsive microcapsules, Soft Matter, 7 (2011), 3277–3279.

• 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, BilayerPlates: Model Reduction, Γ−Convergent Finite Element Approximation andDiscrete Gradient Flow, Comm. Pure Appl. Math. (2017).

Bilayers Plates Andrea Bonito

Page 8: Department of Mathematics Texas A&M University ... · Bilayer Plates: From Model Reduction to -Convergent Finite Element Approximations and More Andrea Bonito Department of Mathematics

Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Outline

Reduced Model Energy

Gradient Flow: Equilibrium Shapes

Space Discretization

Numerical Experiments - Equilibrium Shapes

Thermal Actuation

Conclusions

Bilayers Plates Andrea Bonito

Page 9: Department of Mathematics Texas A&M University ... · Bilayer Plates: From Model Reduction to -Convergent Finite Element Approximations and More Andrea Bonito Department of Mathematics

Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

OUTLINE

Reduced Model Energy

Gradient Flow: Equilibrium Shapes

Space Discretization

Numerical Experiments - Equilibrium Shapes

Thermal Actuation

Conclusions

Bilayers Plates Andrea Bonito

Page 10: Department of Mathematics Texas A&M University ... · Bilayer Plates: From Model Reduction to -Convergent Finite Element Approximations and More Andrea Bonito Department of Mathematics

Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Mathematical Model

R+ = R− + s

θy Ω

x′ ∈ ΩR−

θR− = L(1− δs)L(1 + δs) = θR+

s

L

∂DΩ

• Domain: Ωs = Ω× (0, s) ⊂ R3 with thickness s and midplane Ω ⊂ R2;

• Plate deformation: u : Ωs → R3;

• Surface parametrization: y : Ω→ R3, Γ = y(Ω); x′ := (x1, x2) ∈ Ω

• Unit normal to Γ: ν : Ω→ R3,ν = ∂1y|∂1y|

× ∂2y|∂2y|

;

• Intuitive Deformation Assumption: u(x′, x3) := y(x′) + x3β(s)ν(x′);

• Energy: Hyper-Elastic (Non-Linear) Plate Theory

W (u) := s−3

∫Ωs

dist2(∇u, (1+δ(s)N)SO3)

where N(x′) :=

(Z mmT n

)∈ R3×3, δ(s), characterizes the material

mismatch at x3 = 0.Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Reduced Model

Goal 1: Characterizing the asymptotic bending behavior of the plate Ωs ass→ 0 upon assuming that the energy remains bounded in this limit.

Reduced Model (see also Schmidt):

y ∈ H2(Ω)3 s.t y|∂DΩ = yD, ∇y|∂DΩ = ΦD (clamped),∇yT∇y = I2×2 (isometries) and

J(y) =1

2

∫Ω

|II + Z|2,

where IIi,j := ν · ∂i∂jy is second fundamental form, ν is the normal of thesurface Γ = y(Γ) parametrized by y.

Spontaneous Curvature

Z : Ω→ R2×2 acts as a spontaneous curvature and encodes (the possiblyanisotropic) properties of the bilayer material.

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Simulation: Partially Clamped Plate

• Domain: Ω = (−2, 2)× (0, 10)

• Boundary Condition: ∂DΩ = (−1, 1)× 0.

Goals:

• effectivemathematicaldescription

• convergentdiscretization

• reliable solutiontechnique

• applications

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Equivalent Energies: Taking advantage of the Geometric structure

Isometries: (∇y)T∇y = I

y : Ω→ R3 isometry parametrizing the surface:

|∂iy| = 1, IIi,j = ∂i∂jy · ν, |II| = |h| = |∆y|

J(y) =1

2

∫Ω

|II + Z|2

=1

2

∫Ω

|∆y|2︸ ︷︷ ︸variational

+2∑

i,j=1

∫Ω

∂i∂jy ·

∂1y

|∂1y|× ∂2y

|∂2y|︸ ︷︷ ︸ν

Zi,j +1

2

∫Ω

|Z|2.

Notes for Later

• The discrete scheme does not guarantee the isometries properties (e.g.|∂iy| = 1).

• A gradient flow (De Giorgi Minimization of Movement) is set up tocompute the relaxation to minima (equilibrium plate shapes).

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

OUTLINE

Reduced Model Energy

Gradient Flow: Equilibrium Shapes

Space Discretization

Numerical Experiments - Equilibrium Shapes

Thermal Actuation

Conclusions

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Semi-Discrete H2 Gradient Flow: Equilibrium Shapes

H2 gradient flow: Let τ > 0 and set k = 0. Choose y0 ∈ H2(Ω)3 such thaty0|∂DΩ = yD,∇y0|∂DΩ = ΦD and

[∇y0]T [∇y0] = Id2.

(1) Compute yk+1 ∈ H2(Ω)3 which is minimal for

Gradient Flow Algorithm / Minimization of Movement / Nonlinear

minimize y 7→ 1

2τ‖∆(y − yk)‖2 + J [y]

subject to y|∂DΩ = yD,∇y|∂DΩ = ΦD and the

Linearized Isometry Constraint

[∇(y − yk)]T [∇yk] + [∇yk]T [∇(y − yk)] = 0.

(2) Stop if ‖D2(yk+1 − yk)‖ ≤ εstop; otherwise increase k → k + 1 and goto (1).

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Key Property of Linearized Isometry Constraint

|n| = 1

n

δn

If ∇y ∈ H1(Ω,R3×2) satisfies

[∇(y − yk)]T [∇yk] + [∇yk]T [∇(y − yk)] = 0

then

[∇y]T∇y − I2 ≥ [∇yk]T∇yk − I2 ≥ ... ≥ [∇y0]T∇y0 − I2 = 0;

Hence |∇y| ≥ 1 at each vertex and the discrete scheme is well defined.

Iterative scheme for each time step

(1 + τ)∆2yk = ∆2yk−1 + nonlinear terms

u.c ∇(yk − yk−1)T∇yk−1 + (∇yk)T∇(yk − yk−1) = 0.

+ boundary conditions

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Key Property of Linearized Isometry Constraint

|n| = 1

n

δn

If ∇y ∈ H1(Ω,R3×2) satisfies

[∇(y − yk)]T [∇yk] + [∇yk]T [∇(y − yk)] = 0

then

[∇y]T∇y − I2 ≥ [∇yk]T∇yk − I2 ≥ ... ≥ [∇y0]T∇y0 − I2 = 0;

Hence |∇y| ≥ 1 at each vertex and the discrete scheme is well defined.

Iterative Fixed Point Algorithm (each pseudo-time step)

The subiterations yk,l are well defined and converge to the unique solution inH2(Ω)3 of the nonlinear problem provided τ 1. Moreover,

‖[∇yk]T∇yk − I2‖L1(Ω)2×2 τ.

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

OUTLINE

Reduced Model Energy

Gradient Flow: Equilibrium Shapes

Space Discretization

Numerical Experiments - Equilibrium Shapes

Thermal Actuation

Conclusions

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Choice of Finite Element Space

• Subdivision: Th is a subdivision of Ω made of rectangles of diameters ∼ h.We denote by Nh the set of vertices.

• Kirchhoff quadrilaterals: Wh continuous p.w. Q3 with continuousgradients at vertices.

h

Wh Θh, 2 copies

• 4-th Order: y ∈ [H2(Ω)]3 ⇒ yh ∈ [Wh]3 ⊂ H1(Ω)3 nonconformingspace.

• Gradient operator: ∇h : [Wh]3 → [Θh]3×2 ⊂ H1(Ω)3×2 matching the usualgradient on Nh, i.e. ∇wh = ∇hwh on Nh for wh ∈ [Wh]3.Θh continuous p.w. Q2.

• Isometry constraint: [∇y]T [∇y] = I2 (or linearized version) ⇒ ∇hyh

must satisfy constraints on Nh.

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Discrete Energy

Jh[yh] =1

2

∫Ω

|∇Φh|2dx+

2∑i,j=1

(∂iΦh,j ·

( Φh,1

|Φh,1|× Φh,2

|Φh,2|

), Zij

)h

+1

2

∫Ω

|Z|2dx, yh ∈W 3h Φh := ∇hyh ∈ Θ3×2

h .

Isometry Constraints:

• Gradient Flow: The pairs (yh,Φh := ∇hyh) are limits of k-th iterates(yk

h,Φkh) of a discrete H2 gradient flow based on a linearized isometry

constraint enforced at the vertices

[∇(yh(z)−ykh(z))]T [∇yk

h(z)]+[∇ykh(z)]T [∇(yh(z)−yk

h(z))] = 0 ∀z ∈ Nh

• Nodal Constraints Consequences: The minimizers yh ∈ [Wh]3 of Jh[yh]obtained via the gradient flow (τ ≈ h) satisfy the inexact isometryconstraint

[∇hyh(z)]T∇hyh(z) ≥ I2,∣∣[∇hyh(z)]T∇hyh(z)− I2

∣∣ h ∀z ∈ Nh;

Implementation: Only the implementations of Q1 and Q2 elements arerequired. In particular, the basis function of Wh are not needed.

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Convergence of Minimizers

Question

Does the sequence of almost absolute minimizers of Jh(.) converges to anabsolute minimizer of J(.)?

Jh

J

Continuous Problem:

J(y) :=

12

∫Ω|II− Z|2, when y ∈ H2(Ω;R3) with ∇yT∇y = I,

y|∂DΩ = yD,∇y|∂DΩ = ΦD;+∞, otherwise.

Discrete Problem:

Jh[yh] =1

2

∫Ω

|∇Φh|2dx+

2∑i,j=1

(∂iΦh,j ·

( Φh,1

|Φh,1|× Φh,2

|Φh,2|

), Zij

)h

+1

2

∫Ω

|Z|2dx, yh ∈W 3h , Φh = ∇hyh

when yh|∂DΩ = yD,h,Φh|∂DΩ = ΦD,h and

[Φh(z)]T Φh(z) ≥ I2,∣∣[Φh(z)]T Φh(z)− I2

∣∣ ≤ Ch ∀z ∈ Nh

and Jh(yh) = +∞ otherwise.Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

De Giorgi Γ−Convergence

Theorem (Γ-Convergence) We have JhΓ−→ J , i.e.

• Attainment: Given any y ∈ Dom(J), there exists a sequenceyhh>0 ⊂ Dom(Jh) such that

yh → y in H1(Ω;R3), ∇hyh → ∇y in H1(Ω;R3×2), Jh[yh]→ J [y]

as h→ 0.

• Lower bound property: If yhh>0 ⊂ Dom(Jh) is a sequence satisfyingJh[yh] ≤ C, then there exist y ∈ Dom(J) such that

yh → y in H1(Ω;R3), ∇hyh ∇y in H1(Ω)3×2,

as well asJ [y] ≤ lim inf

h→0Jh[yh].

Corollary (Convergence of almost minimizers without smoothness)

Any yhh sequence of almost global discrete minimizers of Jh is precompactin H1(Ω)3 and every cluster point y of yh is a global minimizer of J .Moreover, there exists a subsequence of yhh (not relabeled) such that

limh→0‖y − yh‖H1(Ω) = 0 and lim

h→0Jh[yh] = J [y].

Bilayers Plates Andrea Bonito

Page 23: Department of Mathematics Texas A&M University ... · Bilayer Plates: From Model Reduction to -Convergent Finite Element Approximations and More Andrea Bonito Department of Mathematics

Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

OUTLINE

Reduced Model Energy

Gradient Flow: Equilibrium Shapes

Space Discretization

Numerical Experiments - Equilibrium Shapes

Thermal Actuation

Conclusions

Bilayers Plates Andrea Bonito

Page 24: Department of Mathematics Texas A&M University ... · Bilayer Plates: From Model Reduction to -Convergent Finite Element Approximations and More Andrea Bonito Department of Mathematics

Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Experiment 4: Unexpected Relaxation

The bilayers are 1 mm long and 30 µm wide, and they are attached on thebottom edge. The left actuators do not move from completely flat to fullycurled.

E. Smela, O. Inganas, and I. Lundstrom, Controlled folding ofmicrometer-size structures, Science, 268 (1995), 1735–1738.

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

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.

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Numerical Experiment: Corner Clamped Plate

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Numerical Experiment: Clamped Plate with Tensor Spontaneouscurvatures

Alben et al. (2011) Simpson et al. (2010)

(left) µ1 = 5 e1 = [1, 0]T ; µ2 = 1 e2 = [0, 1]T ;

(right) µ1 = 5 e1 = [1,−1]T ; µ2 = 1 e2 = [1, 1]T ;

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Numerical Experiment: DNA (using dG + trick for BC)

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Numerical Experiment: Alternating (using dG + trick for BC)

Bilayers Plates Andrea Bonito

Page 30: Department of Mathematics Texas A&M University ... · Bilayer Plates: From Model Reduction to -Convergent Finite Element Approximations and More Andrea Bonito Department of Mathematics

Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

OUTLINE

Reduced Model Energy

Gradient Flow: Equilibrium Shapes

Space Discretization

Numerical Experiments - Equilibrium Shapes

Thermal Actuation

Conclusions

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Simplified Mathematical Model (+ A. Muliana MEEN-TAMU)

• Full Elastic Energy: include variable rigidity coefficient µ and platetemperature Θ.

• Reduced Elastic Energy:

J [y, t] =1

2

∫Ω

µ(x′)∣∣II− ΘI2︸︷︷︸

=Z

∣∣2dx′obtained using a linear constitutive relation Z(Θ) = ΘI2.

• Fourier law (Heat):∂tΘ− κ∆Θ = f on Ω

supplemented with boundary conditions such as

κ∇Θ · µ = η(ΘExt −Θ) or Θ = ΘExt.

No gradient flow, no subiterations, very fast but no mathematical proof... yet

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Dog-Ears: Effect of Diffusivity.

Newton Cooling Law

κ∇Θ · µ = 2(ΘExt︸ ︷︷ ︸=100

−Θ).

κ = 1 κ = 0.1

Diffusivity coefficient ratio of 10.

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

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

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

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

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Encapsulation with Self-Folding Microcapsules: Drug Delivery

G. Stoychev, N. Puretskiy, and L. Ionov, Self-folding all-polymerthermoresponsive microcapsules, Soft Matter, 7 (2011), 3277–3279.

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

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.

Bilayers Plates Andrea Bonito

Page 37: Department of Mathematics Texas A&M University ... · Bilayer Plates: From Model Reduction to -Convergent Finite Element Approximations and More Andrea Bonito Department of Mathematics

Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

OUTLINE

Reduced Model Energy

Gradient Flow: Equilibrium Shapes

Space Discretization

Numerical Experiments - Equilibrium Shapes

Thermal Actuation

Conclusions

Bilayers Plates Andrea Bonito

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Motivation Energy Gradient Flow Space Discretization Numerical Experiments Thermal Actuation

Conclusions

• Geometric PDEs: No general recipes but numerical methods must takeadvantage of the geometric structure.

• Bilayer model: Nonlinear Kirchhoff model that allows for bending but notstretching or shearing (isometry constraint). The model account for aspontaneous curvature tensor and large deformations.

• Kirchhoff Quadrilaterals: Nonconforming FEM of H2(ω) and keyproperties of discrete gradients ∇h. Alternative using dG (with D.Guignard, D. Ntogkas and R.H Nochetto).

• Discrete Gradient Flow: H2 gradient flow for a modified energy Jh(.);constructive existence of every sub-iterate; convergence to the discreteproblem; control of violation of isometry constraint.

• Γ−convergence: Convergence of inexact discrete minimizers tominimizers 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 researchgrant.

Bilayers Plates Andrea Bonito