Optimization and data analysis for non-Euclidean elastic sheets

Ken Yamamoto and Shankar Venkataramani

Program in Applied Mathematics, University of Arizona

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Arizona – Los Alamos Days

Optimization, Inference, & Learning (OIL)

April 20, 2019

Introduction and Motivation

Introduction and Motivation

• Thin elastic sheets with lateral swelling and shrinking ubiquitous in nature and industry
• Complex, self-similar buckling patterns
• Differential growth law permanently deforms intrinsic distance
• How can we understand the reason for these complex geometries?
• Can weak forces other than stretching or bending play a role?

Notice…

• Buckling cascades
• Patterns repeating a various scales
• Fractal-like patterns (typically produced by nonlinear dynamical processes)

Experimental context: Non-Euclidean gel discs

Klein et al. 2007

Dimensionless energy functional with gravity

• Differential growth encoded by 2-D Riemannian reference metric g̃ defined on mid-surface Ω ⊂ ℝ²
• Equilibrium conformation of the sheet is immersion F : Ω → ℝ³ that minimizes energy functional

• Y = Young’s modulus
• t = thickness of sheet
• R = radius/length of sheet
• L = vertical deflection of sheet
• ⍴ = density of sheet
• g ∼ 10 m/s² is standard acceleration due to gravity
• H = mean curvature
• K = Gaussian curvature

Föppl-von Kármán ansatz

• Small-slopes approximation ⇒ approximate isometries

Energy minimization: Euler Lagrange equations

• System of 3 nonlinear PDEs
• Difficult to solve

Physical scales and units

• Y ∼ 10⁶
• Y(t/R)² ∼ 10³
• 𝞺gL ∼ 10²

• Y ∼ 10⁶ Pa
• t ∼ 10^-3 m
• R ∼ 3×10^-2 m
• L ∼ 10^-2 m
• 𝞺 ∼ 10³ kg/m³
• g ∼ 10 m/s²

Stretching is dominant.

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Bending and Gravity are comparable.

Constrained variational problem

• In vanishing thickness limit, energy-minimizing immersions necessarily have zero stretching
• Constrained variational system
• Variational PDE from bending and gravity energies
• Zero-stretching constraint PDE
• w(x,y)≥0
• We can solve the zero-stretching constraint PDE analytically by method of characteristics. By Gauss’s Theorema Egregium, extrinsic K(F) and intrinsic K(g̃) curvatures are equal for no stretching. Then, say for K(g̃) = -1:

Monge-Ampère Equation

Solutions to No-Stretching Constraint PDE

Monge-Ampère Equation

w is a quadratic, saddle-shaped, and a doubly-ruled surface

Solve for minimizing out-of-plane displacement function w(x,y)

Construction of No-Stretching C^1,1 Surfaces

Red lines are inflection/asymptotic lines

Intersection is branch point

Cut and glue quadratic, saddle-shaped, and ruled surfaces

Gemmer et al. 2009

Solutions to no-stretching constraint PDE: One branch point at origin

• 𝜃_+/-=𝜋/2 → number of waves = 4

• This solution surface is smooth

Solutions to no-stretching constraint PDE: One branch point at origin

• 𝜃_+/-=𝜋/3 → number of waves = 6

• 𝜃_+/-=𝜋/4 → number of waves = 8

Topology of a Branch Point

Shearman 2017

Smooth Saddle (C²)

Monkey Saddle (C^1,1)

Degree = -1

Degree = -2

Graph Degree = 6

(6 asymptotic lines extend from it)

Projection of normal field onto xy-plane.

Asymptotic Skeleton: Quadmesh

Shearman 2017

Smooth Saddle (C²)

Monkey Saddle (C^1,1)

Center node is a branch point and has

Graph Degree = 6

(6 asymptotic lines extend from it)

Asymptotic skeleton describes the network of branch points and lines of inflection in potentially non-smooth hyperbolic surfaces

C^1,1 Solutions to No-Stretching Constraint PDE: Multiple, offsetted branch points

Gemmer et al. 2016

Branch points

Inflection/Asymptotic lines

Graph Degree = 6

(6 asymptotic lines extend from it)

Optimize 6 Degrees of Freedom

Angular Extent of Up/Down Parent Sector Pair = 𝜱

Upward Curving Parent Sector

Downward Curving Parent Sector

𝞺_u

𝞺_d

𝜃_u

Angular Extent of Upward Parent Sector = 𝝋

All solid lines are inflection/asymptotic lines, which intersect at a branch point.

𝜃_d

Optimize 6 Degrees of Freedom

Single Origin Branch Point

16 wrinkles

Energy = 164256

UPWARD Sectors LARGER due to GRAVITY

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NO Branch Points placed in DOWNWARD Parent Sectors

5 Distributed Branch Points

16 wrinkles

Energy = 156424

Radius = 25

Optimize 6 Degrees of Freedom

5 Distributed Branch Points

16 wrinkles

Energy = 156424

Wrinkles originating from center is HALVED.

Radius = 25

Optimize 6 Degrees of Freedom

Single Origin Branch Point

16 wrinkles

Energy = 164256

Bending = 55430 | Gravity = 108826

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5 Distributed Branch Points

16 wrinkles

Energy = 156424

Bending = 52132 | Gravity = 104292

Radius = 25

UPWARD Sectors LARGER due to GRAVITY

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NO Branch Points placed in DOWNWARD Parent Sectors

Evolution of Curved Asymptotic Lines in Minimal-Energy C² Sheets

Gravity Dominant

Bending Dominant

Gravity vs Bending Dominant Scaling & C^1,1 vs C²

Mechanics of Static Hyperbolic Sheets

Orange = Same angle for up/down sectors | Blue = Optimized angular ratio

There is a significant energy gap between C^1,1 vs C²

surfaces in the gravity-dominant regime.

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Branch points allow for dramatic decreases in gravity energy.

Gravity Dominant

Bending Dominant

1.01

1.12

1.19

1.78

1.47

1.85

2.04

1.97

Up/Down Angular Ratio

Geometric Discretization of Hyperbolic Sheets

Discretizations of C^1,1 solutions to no-stretching constraint PDE. Discrete Differential Geometry framework for numerical optimization with multiple branch points and curved asymptotic lines.

Discretizations of C^1,1 solutions to no-stretching constraint PDE. Discrete Differential Geometry framework for numerical optimization with multiple branch points and curved asymptotic lines.

Geometric Discretization of Hyperbolic Sheets

Inverse Problem: Given Geometry (i.e., w)… Find λ

Inverse Problem: MCMC Results

TRUE λ

Mean of MCMC Result

Future: Extract asymptotic skeleton from noisy profilometric data for real-world sheets.

TRUE SURFACE

MCMC RESULT

Dynamics: Morphogenesis, Biomechanics, and Shape Control

A saddle surface whose asymptotic lines are rotating with respect to the material points satisfying a no-slip contact condition with the table below (left). The material coordinates represented by colored sectors are rotating at a slower rate than the frame in which the shape of the surface is fixed. The asymptotic frame is indicated by the white ball which rolls on the surface to remain at the minimum. A “mathematical” sea slug (right) with merging and splitting branch points is a cartoon for the motion of a true sea slug (middle).

Conclusions

• Branch point defects result in…
• much lower energy surfaces.
• floppy sheets with geometric defects easily manipulated by weak forces (resulting in large changes in geometry).
• Gravity is the primary driver for wrinkles to form. With only bending, there is no energetic incentive for wrinkles to form in small-slopes setting.
• The exact number, location, and property of branch point defects (i.e., wrinkles), however, is influenced by bending energy...
• Bending energy limits the number of wrinkles.
• Asymmetry between upward- and downward-curving wrinkles is preferred to decrease gravity energy rather than introducing another wrinkle (up to a certain point).

Ongoing and Future Work

• Study forces and moments near the non-C² branch points.
• Investigate numerics of branched surfaces including methods that converge to non-smooth minimizers by accounting for the asymptotic skeleton (with multigeneration branch points).
• Analysis of physical experiments of thin sheets, including detection of asymptotic skeletons using topological data analysis.
• Develop an elasticity theory for hyperbolic sheets with weak effects.
• Dynamics, including morphogenesis of naturally growing biological tissue (e.g., leaves and flowers) as well as the biomechanics of marine invertebrates (e.g., sea slugs).
• Engineering applications, e.g., shape manipulation and locomotion.

Main References

• Y. Klein, E. Efrati, and E. Sharon. Shaping of elastic sheets by prescription of non-euclidean metrics. Science, 315(5815):1116–1120, 2007.
• E. Efrati, E. Sharon, and R. Kupferman. Elastic theory of unconstrained non-euclidean plates. Journal of the Mechanics and Physics of Solids, 57(4):762–775, 2009.
• J. Gemmer, E. Sharon, T. Shearman, and S. C. Venkataramani. Isometric immersions, energy minimization and self-similar buckling in non-euclidean elastic sheets. EPL (Europhysics Letters), 114(2):24003, 2016.

Biomechanics

Backup Slides

Outline

• Introduction and motivation
• Energy functional with gravity (an example of a weak force)
• Föppl-von Kármán ansatz
• Energy minimization
• Euler Lagrange equations
• Physical scales
• Constrained variational problem
• Branched isometric immersions
• Numerical investigations
• Ongoing and future work

Experimental context: Non-Euclidean gel discs

• Construct gel discs with heat-induced reversible shrinkage
• Program radially symmetric shrinkage-inhibiting chemical gradient in gel disc ⇒ radially symmetric shrinkage upon heat ⇒ radially symmetric non-Euclidean target metric g_tar

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• Known shrinkage ratio 𝜂(r) across surface from programmed chemical concentration
• Closed circle of radius r on unheated disc, upon heat, will have…
• new perimeter:
• new radius:
• So… perimeter of circle radius 𝞺 on shrunken disc is where f(𝞺) is a function of 𝜂(r)

Equilibrium configuration

• Stretching energy…
• scales linearly with sheet thickness t
• vanishes in (isometric) embeddings that fully follow g_tar

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• Bending energy…
• scales as t³
• vanishes in flat configurations

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• As t→0, stretching term dominates and bending is favorable. We expect to see isometric embeddings for very thin sheets.

Equilibrium configuration

• Configuration metric g is close to target metric g_tar when averaged over 𝜽 for both K_tar>0 and K_tar<0

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• But…
• K_tar>0: preserves radial symmetry of g_tar (by forming surface of revolution)
• K_tar<0: breaks this symmetry forming wavy surfaces

Dotted is from g_tar

Solid is actual g

Dash black is for flat disc with radius 2𝜋𝞺

Equilibrium configuration

• By Gauss’s Theorema Egregium, local differences in configuration K and K_tar indicate nonzero stretching energy density.
• Bending energy density is proportional to…

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• K_tar>0: radial symmetry of K ⇒ g_tar obeyed locally (not just on average over 𝜽)
• K_tar<0: K oscillates ⇒ periodic deviations from g_tar ⇒ large modulations of stretching over 𝜽

Equilibrium configuration

• By Gauss’s Theorema Egregium, local differences in configuration K and K_tar indicate nonzero stretching energy density.
• Bending energy density is proportional to…

​

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• K_tar>0: radial symmetry of K ⇒ g_tar obeyed locally (not just on average over 𝜽
• K_tar<0: K oscillates ⇒ periodic deviations from g_tar ⇒ large modulations of stretching over 𝜽

Dotted is K_tar

Solid is actual K

Red is K_tar>0

Blue is K_tar<0

Interpretation of experimental results

• Intrinsically differ only by sign on K_tar…��So, what is the mechanism causing discs with K_tar<0 to break symmetry, bend a lot, and localize in-plane strain?

thicker sheet

thinner sheet

Interpretation of experimental results

• K_tar>0
• Selects (smooth) isometric embedding of g_tar which is minimal in both stretching and bending energies. Stretching and bending not competing.
• K_tar<0
• Does not select isometric embedding…?
• Perhaps bending energy too large? Introduce localized in-plane strain to reduce some bending energy? We’ve provoked stretching and bending to compete. The conflict produces interesting shapes?
• Some balance between stretching and bending leading to wrinkling behavior?
• Precise shaping mechanism is unclear...

Constrained variational problem

• a, b, c are constants
• (x,y) and (r,𝜃) are Cartesian and polar coordinates in ℝ²
• 0 < 𝜃_+/- < 𝜋
• continuous tangent plane along 𝜽 = 0 (finite bending energy there) -- this is called an inflection line
• point where multiple inflection lines intersect is called a branch point

• Solve for minimizing out-of-plane displacement function w(x,y)

Solutions to no-stretching constraint PDE: One branch point at origin

• 𝜃_+/-=𝜋/2 → number of waves = 4

• This solution surface is smooth

Solutions to no-stretching constraint PDE: One branch point at origin

• 𝜃_+/-=𝜋/3 → number of waves = 6

• 𝜃_+/-=𝜋/4 → number of waves = 8

Numerical investigation: Branch point at the origin

Radius = 5 → Lowest energy = 8 wrinkles

Minimal Energy

Red = Total Energy

Blue = Bending

Green = Gravity

Numerical investigation: Branch point at the origin

Minimal Energy

Red = Total Energy

Blue = Bending

Green = Gravity

Radius = 60 → Lowest energy = 26 wrinkles

Numerical investigation: Branch point at the origin

• Scaling analysis of our system gives the black line

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• Numerical simulations are represented by red dots

Also Optimize Up/Down Angular Ratio

18 wrinkles

Total Energy = 171406

Bending = 63213 | Gravity = 108193

16 wrinkles

Up/Down angular ratio = 1.93

Total Energy = 164256

Bending = 55430 | Gravity = 108826

Radius = 25

UPWARD Sectors LARGER due to GRAVITY

Also Optimize Up/Down Angular Ratio

18 wrinkles

Total Energy = 171406

Bending = 63213 | Gravity = 108193

Numerical: 63270 | 107370

16 wrinkles

Up/Down angular ratio = 1.93

Total Energy = 164256

Bending = 55430 | Gravity = 108826

Numerical: 55452 | 106748

Radius = 25

Optimize 6 Degrees of Freedom

Single Origin Branch Point

16 wrinkles

Energy = 164256

Bending = 55430 | Gravity = 108826

Numerical: 55452 | 106748

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5 Distributed Branch Points

16 wrinkles

Energy = 156424

Bending = 52132 | Gravity = 104292

Numerical: 52493 | 107436

Radius = 25

Also Optimize Up/Down Angular Ratio

Orange = Same angle for up/down sectors | Blue = Optimized angular ratio

Tune Up/Down Angular Ratio to avoid additional Wrinkles

UPWARD Sectors LARGER due to GRAVITY

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Also Optimize Up/Down Angular Ratio

For very large radius, the optimal up/down angular ratio tends to 2.

Limiting scaling behavior suggests self-similar mechanism if we have multigeneration branch points.

Optimize 6 Degrees of Freedom

Wrinkles are distributed across branch points. Half at the origin and rest across offsetted branch points.

Blue = Optimized angular ratio | Green = 6 DOF optimization

Optimize 6 Degrees of Freedom

Blue = Optimized angular ratio | Green = 6 DOF optimization

UPWARD Sectors LARGER due to GRAVITY

NO Branch Points placed in DOWNWARD Parent Sectors

Optimize 6 Degrees of Freedom

Blue = Optimized angular ratio | Green = 6 DOF optimization

UPWARD Sectors LARGER due to GRAVITY

Limiting scaling behavior suggests self-similar mechanism if we have multigeneration branch points.

Optimize 6 Degrees of Freedom

Branch point defects lower the energy.

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Branch points are easily manipulated by weak forces (e.g., gravity) which is a manifestation of how floppy these surfaces are.

Orange = Same angle for up/down sectors | Blue = Optimized angular ratio | Green = 6 DOF optimization

Quantitative Measure of Floppiness

Floppiness is the degree in which large changes in geometry can occur without much change in energy.

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Inherent floppiness of non-Euclidean elastic sheets is governed by and may, in turn, be quantified by localized geometric defects.

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These defects may be introduced and moved with little energetic cost while significantly altering the geometry.

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T. Shearman’s Dissertation 2017

Inverse Problem: Given Geometry (i.e., w)… Find λ

Inverse Problem: MCMC Results

TRUE λ

Mean of MCMC Result

Future: Extract asymptotic skeleton from noisy profilometric data for real-world sheets.

TRUE SURFACE

MCMC RESULT