1 of 117

Computation, discrete geometry, and soft mechanics in non-Euclidean elasticity

Ken Yamamoto

Program in Applied Mathematics, University of Arizona

PhD Final Oral Dissertation Defense

August 11, 2020

2 of 117

Introduction and Motivation

Thin sheets are omnipresent in nature, technology, and everyday life. Being much thinner in one dimension than in the other two, they exhibit an unparalleled, rich variety of deformations when subjected to external forces, spatial constraints, or intrinsic growth
They buckle easily and create striking examples of periodic, self-similar patterns along the edges of growing leaves, flowers, marine invertebrate, torn plastic sheets.

PAST

the kinds of problems i’m interested in are inspired by hyperbolic sheets we see in nature
what do we see in nature? - division process of cells along the edge naturally results in hyperbolic surfaces when growing due to this differential growth
thin sheet is very hard to stretch but very easy to bend - nondimensional measure of how easy to bend compared to stretching is given by the ratio of the thickness to the other lengths in the problem - if this ratio is small than it’s much easier to bend
rich variety of deformations when subjected to external forces, spatial constraints, or intrinsic growth
understand the interesting mechanical properties of these systems and perhaps exploit them to engineer active/smart/programmable materials and structures that predictably and functionally respond to various stimuli - optimal design for self-propelling sea-slug-like soft robots

3 of 117

Introduction and Motivation

E. Sharon, B. Roman, and H. L. Swinney. Geometrically driven wrinkling observed in free plastic sheets and leaves. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), 75(4):046211, 2007.

Thin sheets are omnipresent in nature, technology, and everyday life. Being much thinner in one dimension than in the other two, they exhibit an unparalleled, rich variety of deformations when subjected to external forces, spatial constraints, or intrinsic growth
They buckle easily and create striking examples of periodic, self-similar patterns along the edges of growing leaves, flowers, marine invertebrate, torn plastic sheets.

PAST

the kinds of problems i’m interested in are inspired by hyperbolic sheets we see in nature
what do we see in nature? - division process of cells along the edge naturally results in hyperbolic surfaces when growing due to this differential growth
thin sheet is very hard to stretch but very easy to bend - nondimensional measure of how easy to bend compared to stretching is given by the ratio of the thickness to the other lengths in the problem - if this ratio is small than it’s much easier to bend
rich variety of deformations when subjected to external forces, spatial constraints, or intrinsic growth
understand the interesting mechanical properties of these systems and perhaps exploit them to engineer active/smart/programmable materials and structures that predictably and functionally respond to various stimuli - optimal design for self-propelling sea-slug-like soft robots

4 of 117

O. Albarra ́n, D. V. Todorova, E. Katifori, and L. Goehring. arXiv:1806.03718. 2018

J. Hure, B. Roman, and J. Bico. Phys. Rev. Lett., 109:054302, Aug 2012.

B. Davidovitch, Y. Sun, and G. M. Grason. PNAS. 2019.

An experimental system that spontaneously forms wrinkled patterns is a \emph{non-Euclidean} shell floating atop a \emph{Euclidean} substrate, e.g., a flat body of water which represent two soft bodies that are geometrically incompatible with each other
Upon perfect wetting with the flat liquid surface, the intrinsically non-flat shell is driven to wrinkle as it seeks to flatten itself to minimize the gravitational potential energy associated with displaced liquid adhered to the shell. While the shell could stretch in-plane to become flat, this requires a lot of energy. The thin shell would rather form wrinkles that cause the sheet to confine its vertical deflection, i.e., flatten, without stretching
There are also studies that examine wrinkles arising in the reverse situation, i.e., when a flat sheet is confined onto the surface of a sphere

5 of 117

Hyperbolic art from the “The Crochet Coral Reef Project” (M. Wertheim and C. Wertheim) and jewelry from “Floraform” (J. Louis-Rosenberg. http://n-e-r-v-o-u-s.com/blog/?p=6721)

6 of 117

Dynamics: Morphogenesis, Biomechanics, and Shape Control

7 of 117

Outline

Variational analysis and elastic energy functional
Discrete differential geometry and branch point defects
Modeling and computations with hyperbolic sheets
Energy minimization and optimization
Numerical results and scaling laws
Connections to experiments and data analysis
Dynamics: Biological and engineering applications

8 of 117

Outline

PART 1: Non-Euclidean model of elasticity, energy functional, and constrained variational problem for weak external forces

PART 2: Small-slopes isometry, energetics, scaling laws, and smoothness

PART 3: Experiments and data analysis

PART 4: Discrete Differential Geometry (DDG) of hyperbolic sheets

9 of 117

PART 1

Non-Euclidean model of elasticity, energy functional, and constrained variational problem for weak external forces

10 of 117

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

August Föppl and Theodore von Kármán. 1910.

E. Efrati, E. Sharon, and R. Kupferman. 2009.

11 of 117

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.

Bending and Gravity are comparable.

12 of 117

Constrained variational problem

Minimize bending and gravity subject to zero-stretching constraint.
For zero-stretching, the intrinsic and extrinsic Gauss curvatures must equal, by Gauss’s Theorema Egregium. So, we constrain ourselves to immersions that satisfy zero-stretching, i.e., isometric immersions.

Note: There are different asymptotic regimes for the problem. We assume the external force is weak and both bending/gravity scale similarly. This is more appropriate for free rather than stamped sheets. When external forces are strong, e.g., in stamping, there is a different asymptotic regime that looks at so-called asymptotic rather than exact isometries (B. Davidovitch, I. Tobasco et al.).

13 of 117

Föppl-von Kármán ansatz

Zero-stretching constraint PDE: Monge-Ampère Equation

Small-slopes approximation for the immersion F : Ω⊂ ℝ² → ℝ³

Monge-Ampère Equation

Say we take the sheet to have constant intrinsic (i.e., target) Gauss curvature

K(g̃) = –ε²

Then, the zero-stretching constraint is...

14 of 117

Föppl-von Kármán ansatz

Scaled dimensionless energy functional

Small-slopes approximation for the immersion F : Ω⊂ ℝ² → ℝ³

Bending

Gravity

Minimize this energy functional over isometric immersions

15 of 117

Föppl-von Kármán ansatz

Small-slopes approximation ⇒ approximate isometries

16 of 117

Energy minimization: Euler Lagrange equations

System of 3 nonlinear PDEs
Difficult to solve

17 of 117

PART 2

Small-slopes isometry, energetics, scaling laws, and smoothness

18 of 117

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.

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

in gravity dominant there are non-smooth branch points that make only c1,1 solutions that cannot be approximated by c2 - c2 is a good approximation of minimizers in bending dominant regime
surfaces are floppy due to branch point singularities
allowance of c1,1 defects expands the space of possible geometries greatly
furthermore, many (even significantly different) c1,1 geometries turn out to be energetically similar - that is, we can introduce and move branch points to cause large changes in geometry without much change in energy
in particular, weak external forces are already strong enough to easily generate and manipulate these geometric defects since it costs relatively little energy to do so
this is the essence of floppiness, which is characterized by a relatively flat energy landscape, where large changes in geometry are possible with little changes in energy
hence, inherent floppiness of non-Euclidean elastic sheets is governed by and may, in turn, be quantified by these localized geometric defects

19 of 117

Review: Smoothness classes

C²

Twice differentiable

Second derivative is continuous

C^1,1

Second derivative may be discontinuous but it’s bounded

C^0,1

Not even once differentiable

y = x²

y = |x| x

y = |x|

20 of 117

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

But, formulation of constrained variational problem involving Lagrange multipliers is nontrivial
Nonetheless, 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

21 of 117

Zero-stretching constraint PDE: Monge-Ampère Equation

Solutions are...

Monge-Ampère Equation

Say we take the sheet to have intrinsic (i.e., target) Gauss curvature K(g̃) = -1.

Then, the zero-stretching constraint is...

quadratic, saddle-shaped, and doubly-ruled surfaces.

such that

22 of 117

Construction of C^1,1 Isometries

Red lines are inflection/asymptotic lines

Intersection is branch point

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

Gemmer et al. 2009

This surface is C²

This surface is C^1,1

23 of 117

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)

24 of 117

Constrained variational problem

In vanishing thickness limit, energy-minimizing immersions necessarily have zero stretching.
Constrained variational system: Minimize bending and gravity subject to zero-stretching constraint PDE and w(x,y)≥0.
For zero-stretching, the intrinsic and extrinsic metrics must equal.

25 of 117

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

𝜃_+/-=𝜋/2 → number of waves = 4

This solution surface is smooth

26 of 117

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

27 of 117

C^1,1 Isometries: Single branch point at origin

𝜃_+/-=𝜋/3 ⇒ number of waves = 6

𝜃_+/-=𝜋/4 ⇒ number of waves = 8

𝜃_+/-=𝜋/2 ⇒ number of waves = 4

This surface is C²

These surfaces are C^1,1

28 of 117

Constrained variational problem

In vanishing thickness limit, energy-minimizing immersions necessarily have zero stretching.
Constrained variational system: Minimize bending and gravity subject to zero-stretching constraint PDE and w(x,y)≥0.
For zero-stretching, the intrinsic and extrinsic metrics must equal.

29 of 117

Numerical investigation: Branch point at the origin

Radius = 5 → Lowest energy = 8 wrinkles

Minimal Energy

Red = Total Energy

Blue = Bending

Green = Gravity

30 of 117

Numerical investigation: Branch point at the origin

Minimal Energy

Red = Total Energy

Blue = Bending

Green = Gravity

Radius = 60 → Lowest energy = 26 wrinkles

31 of 117

Numerical investigation: Single branch point at the origin

Minimal Energy

Red = Total Energy

Blue = Bending

Green = Gravity

λ/R²= 0.0003 ⇒ Lowest energy = 26 wrinkles

Bending

Gravity

32 of 117

Numerical investigation: Branch point at the origin

Scaling analysis of our system gives the black line

Numerical simulations are represented by red dots

33 of 117

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

34 of 117

Also Optimize Up/Down Angular Ratio

18 wrinkles

Total Energy = 0.4288

Bending = 0.1618 | Gravity = 0.2770

16 wrinkles

Up/Down angular ratio = 1.93

Total Energy = 0.4205

Bending = 0.1419 | Gravity = 0.2786

λ/R²= 0.0016

UPWARD Sectors LARGER due to GRAVITY

35 of 117

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

36 of 117

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)

there’s nothing special about the origin - i go to any other point and do the same process relative to the local tangent plane and call that the origin - so i can do this multiple times - cut a piece out of a sector and do essentially what i did before with odd reflections
has tangent plane everywhere - further, the tangent is lipschitz - but the surface is not twice differentiable - since you’re patching a surface that curves up with one that curves down, the second derivative is not continuous although it’s bounded
the c1,1 surfaces are isometries because they are isometries on each patch and across the lines of inflection the metrics agree
general message is that we can make a new class of non-smooth isometric immersions - but, it’s a hard a combinatorial problem with two pieces - enumerate all the possible quadmeshes that give different topologies (and there’s a lot of them!) and there’s geometric data on top of that

37 of 117

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

38 of 117

Optimize 6 Degrees of Freedom

Single Origin Branch Point

16 wrinkles

Energy = 164256

UPWARD Sectors LARGER due to GRAVITY

NO Branch Points placed in DOWNWARD Parent Sectors

5 Distributed Branch Points

16 wrinkles

Energy = 156424

Radius = 25

39 of 117

Optimize 6 Degrees of Freedom

Single Origin Branch Point

16 wrinkles

Energy = 0.4205

UPWARD Sectors LARGER due to GRAVITY

NO Branch Points placed in DOWNWARD Parent Sectors

5 Distributed Branch Points

16 wrinkles

Energy = 0.4004

λ/R²= 0.0016

40 of 117

Optimize 6 Degrees of Freedom

5 Distributed Branch Points

16 wrinkles

Energy = 156424

Wrinkles originating from center is HALVED.

Radius = 25

41 of 117

Optimize 6 Degrees of Freedom

Single Origin Branch Point

16 wrinkles

Energy = 164256

Bending = 55430 | Gravity = 108826

5 Distributed Branch Points

16 wrinkles

Energy = 156424

Bending = 52132 | Gravity = 104292

Radius = 25

UPWARD Sectors LARGER due to GRAVITY

NO Branch Points placed in DOWNWARD Parent Sectors

42 of 117

Optimize 6 Degrees of Freedom

Branch point defects lower the energy.

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

43 of 117

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.

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

in gravity dominant there are non-smooth branch points that make only c1,1 solutions that cannot be approximated by c2 - c2 is a good approximation of minimizers in bending dominant regime
surfaces are floppy due to branch point singularities
allowance of c1,1 defects expands the space of possible geometries greatly
furthermore, many (even significantly different) c1,1 geometries turn out to be energetically similar - that is, we can introduce and move branch points to cause large changes in geometry without much change in energy
in particular, weak external forces are already strong enough to easily generate and manipulate these geometric defects since it costs relatively little energy to do so
this is the essence of floppiness, which is characterized by a relatively flat energy landscape, where large changes in geometry are possible with little changes in energy
hence, inherent floppiness of non-Euclidean elastic sheets is governed by and may, in turn, be quantified by these localized geometric defects

44 of 117

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

Gravity Dominant

Bending Dominant

2.04

1.85

1.47

1.78

1.19

1.01

C^1,1 isometries allow for wrinkles to form in the gravity-dominant regime and substantial decreases in gravity energy.

45 of 117

PART 3

Experiments and data analysis

46 of 117

Experiments: Forces in Hyperbolic hydrogels

Thanks to Prof. Eran Sharon and Ido Levin

vertical extent of hydrogel as a function of number of wrinkles n

buckling transitions of hyperbolic sheets undergoing spatial confinement
compressed a hyperbolic sample in between two glass plates, measuring the force exerted by the sample on the plates and observing its morphological changes
each additional wrinkle, created when compressing the sample further, was accompanied by a sharp decrease in the force against the plates
nonconservative forces are significant (e.g., friction, adhesion)
minima force curves we expect this is correlated with where the transition occurs - this is a natural candidate
samples are not purely hyperbolic - strong compresion… is asymptotic regime is even valid? - friction and adhesion - just a start to quantify how floppy these things are
although the force curves and wrinkle transitions were not exactly the same as in the forward direction due to hysteresis and friction, repeating the experiment backwards, i.e., increasing the displacement between the plates, qualitatively reproduced the same observations in reverse: the hyperbolic gel transitioned from a plane-stress to a highly wrinkled state, before losing wrinkles one at a time, and the force trend was retraced backwards
work became even more exciting when I began to analyze and model this experiment mathematically which, in turn, prompted me to collect additional data and try other tests - his back-and-forth between theoretical and experimental investigations was a scientific thrill that I had not experienced before - collected a lot of good data that I will continue to analyze, as I seek to derive a theoretical model that reproduces the forces and buckling transitions of confined sheets
it’s thick enough to not be able to ignore stretching - but it’s thin enough that we can do no stretching with boundary layers seems promising

47 of 117

Experiments: Forces in Hyperbolic Sheets

Thanks to Prof. Eran Sharon and Ido Levin

buckling transitions of hyperbolic sheets undergoing spatial confinement
compressed a hyperbolic sample in between two glass plates, measuring the force exerted by the sample on the plates and observing its morphological changes
each additional wrinkle, created when compressing the sample further, was accompanied by a sharp decrease in the force against the plates
although the force curves and wrinkle transitions were not exactly the same as in the forward direction due to hysteresis and friction, repeating the experiment backwards, i.e., increasing the displacement between the plates, qualitatively reproduced the same observations in reverse: the hyperbolic gel transitioned from a plane-stress to a highly wrinkled state, before losing wrinkles one at a time, and the force trend was retraced backwards
work became even more exciting when I began to analyze and model this experiment mathematically which, in turn, prompted me to collect additional data and try other tests - his back-and-forth between theoretical and experimental investigations was a scientific thrill that I had not experienced before - collected a lot of good data that I will continue to analyze, as I seek to derive a theoretical model that reproduces the forces and buckling transitions of confined sheets
it’s thick enough to not be able to ignore stretching - but it’s thin enough that we can do no stretching with boundary layers seems promising

48 of 117

Experiments: Forces in Hyperbolic Sheets

Thanks to Prof. Eran Sharon and Ido Levin

buckling transitions of hyperbolic sheets undergoing spatial confinement
compressed a hyperbolic sample in between two glass plates, measuring the force exerted by the sample on the plates and observing its morphological changes
each additional wrinkle, created when compressing the sample further, was accompanied by a sharp decrease in the force against the plates
although the force curves and wrinkle transitions were not exactly the same as in the forward direction due to hysteresis and friction, repeating the experiment backwards, i.e., increasing the displacement between the plates, qualitatively reproduced the same observations in reverse: the hyperbolic gel transitioned from a plane-stress to a highly wrinkled state, before losing wrinkles one at a time, and the force trend was retraced backwards
work became even more exciting when I began to analyze and model this experiment mathematically which, in turn, prompted me to collect additional data and try other tests - his back-and-forth between theoretical and experimental investigations was a scientific thrill that I had not experienced before - collected a lot of good data that I will continue to analyze, as I seek to derive a theoretical model that reproduces the forces and buckling transitions of confined sheets
it’s thick enough to not be able to ignore stretching - but it’s thin enough that we can do no stretching with boundary layers seems promising

49 of 117

Experiments: Forces in Hyperbolic Sheets

Thanks to Prof. Eran Sharon and Ido Levin

buckling transitions of hyperbolic sheets undergoing spatial confinement
compressed a hyperbolic sample in between two glass plates, measuring the force exerted by the sample on the plates and observing its morphological changes
each additional wrinkle, created when compressing the sample further, was accompanied by a sharp decrease in the force against the plates
although the force curves and wrinkle transitions were not exactly the same as in the forward direction due to hysteresis and friction, repeating the experiment backwards, i.e., increasing the displacement between the plates, qualitatively reproduced the same observations in reverse: the hyperbolic gel transitioned from a plane-stress to a highly wrinkled state, before losing wrinkles one at a time, and the force trend was retraced backwards
work became even more exciting when I began to analyze and model this experiment mathematically which, in turn, prompted me to collect additional data and try other tests - his back-and-forth between theoretical and experimental investigations was a scientific thrill that I had not experienced before - collected a lot of good data that I will continue to analyze, as I seek to derive a theoretical model that reproduces the forces and buckling transitions of confined sheets
it’s thick enough to not be able to ignore stretching - but it’s thin enough that we can do no stretching with boundary layers seems promising

50 of 117

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

51 of 117

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

52 of 117

PART 4

Discrete Differential Geometry (DDG) of hyperbolic sheets

53 of 117

What is a hyperbolic metric?

54 of 117

DDG: Small-slopes Isometry Constraint

Monge-Ampère Equation

Small-slopes approximation of a constant Gauss curvature metric.

This is a nonlinear hyperbolic PDE ⇒ Solve along characteristics.

55 of 117

DDG: Small-slopes Isometry Constraint

Monge-Ampère equations on charactersitics...

Darboux (1870)

Goursat (1922)

Sauer (1950)

Wunderlich (1951)

Kirchheim (2001)

56 of 117

DDG: Small-slopes Isometry Constraint

57 of 117

DDG: Small-slopes Isometry Constraint

58 of 117

DDG: No-stretching Constraint

59 of 117

DDG: No-stretching Constraint

KEY IDEAS

Discrete versions of these equations retain compatibility at any discretization size.
r and w data along the u and v characteristics do not “talk to” each other. This allows patching together of sectors along characteristics where each sector may be constructed independently.

60 of 117

DDG: No-stretching Constraint

KEY IDEAS

Discrete versions of these equations retain compatibility at any discretization size.
Displacement boundary data along the u and v characteristics do not “talk to” each other. This allows patching together of sectors along characteristics where each sector may be constructed independently.

61 of 117

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)

62 of 117

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

we’re computing on a lattice - the discrete equations relates points on quadrilaterals - so we can use any underlying quadmesh - this allows us to defined meshes with nontrivial topology - there are points that are not special with four neighbors - but this point is actually different; it has six neighbors
diff asymp meshes ⇒ diff surfaces in the continuous limit due to nontrivial topology built in to the surface
idea from hyperbolic problems that there’s not information transfer across a characteristic - so you can patch things together so long as your putting a characteristic with a characteristic
monkey saddle is not smooth because any c^2 surface can only have two local asymptotic directions - the monkey saddle has branch point with 3 local asymptotic directions
(on a computer - how do you decide if something is smooth)
(high level ddg - quads and closing/compatibility)
topology of quad mesh corresponds to the smoothness of the corresponding continuous solution
asymptotic/topological skeleton tells you where are the points of excess degree are and how they are distributed - then there’s also geometric boundary data that tells you how the surface is along the characteristics (i.e., asymptotic lines)
framework where we can put this branch-point structure in / whereas with other discretization we cannot do it - more faithful geometric representation / like appropriating a corner with smooth function / parsimonious - many people do other discretization / but philosophically we try to delineate and characterize branch point and asymptotic - non-smoothness is important in our problem / want discretization that allows us to characterize/quantify the branch points and non-smoothness and the degrees of the non-smoothness and we want to a way to characterize and represent them

63 of 117

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.

64 of 117

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

65 of 117

Experiments: Forces in Hyperbolic Sheets

Thanks to Prof. Eran Sharon and Ido Levin

66 of 117

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.

67 of 117

Evolution of Curved Asymptotic Lines in Minimal-Energy C² Sheets

Gravity Dominant

Bending Dominant

68 of 117

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.

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

in gravity dominant there are non-smooth branch points that make only c1,1 solutions that cannot be approximated by c2 - c2 is a good approximation of minimizers in bending dominant regime
surfaces are floppy due to branch point singularities
allowance of c1,1 defects expands the space of possible geometries greatly
furthermore, many (even significantly different) c1,1 geometries turn out to be energetically similar - that is, we can introduce and move branch points to cause large changes in geometry without much change in energy
in particular, weak external forces are already strong enough to easily generate and manipulate these geometric defects since it costs relatively little energy to do so
this is the essence of floppiness, which is characterized by a relatively flat energy landscape, where large changes in geometry are possible with little changes in energy
hence, inherent floppiness of non-Euclidean elastic sheets is governed by and may, in turn, be quantified by these localized geometric defects

69 of 117

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.

70 of 117

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

71 of 117

In-plane Displacements: Finite Element Method

All previous surface plots were of the (energy-minimizing) out-of-plane displacements w(x,y) as a function of material coordinates (x,y).
To obtain the surface in full physical coordinates (i.e., the immersion), we need to also compute the in-plane displacements (ξ, η) that correspond to zero stretching.

General forcing (f₁₁, f₁₂, f₂₂) ⇒ Minimization is elliptic linear elasticity problem in the in-plane displacements (ξ, η).

72 of 117

In-plane Displacements: Finite Element Method

Out-of-plane displacement w(x,y) as a function of material coordinates (x,y)

The surface in full physical coordinates with in-plane displacements (ξ, η) applied to minimize stretching

73 of 117

In-plane Displacements: Finite Element Method

Out-of-plane displacement w(x,y) as a function of material coordinates (x,y)

The surface in full physical coordinates with in-plane displacements (ξ, η) applied to minimize stretching

74 of 117

75 of 117

Isometry condition ⇒ Specific forcings as a function of the out-of-plane displacement w.
First solve for w that satisfies the isometry condition (i.e., Monge Ampère equation) ⇒

All strain entries can be made to equal zero exactly.
There exists a DDG to solve PDE system locally quadrilateral-by-quadrilateral.

In-plane Displacements: Discrete Differential Geometry

General forcing (f₁₁, f₁₂, f₂₂) ⇒ Minimization is elliptic linear elasticity problem in the in-plane displacements (ξ, η).

Full math and details on DDG in my thesis! ✍️

76 of 117

77 of 117

78 of 117

79 of 117

In-plane Displacements: Discrete Differential Geometry

Compute in-plane displacements (ξ, η) quadrilateral-by-quadrilateral, but...

The solutions for (ξ, η) contain linear and constant terms for in-plane rotation and rigid translation. These need to be set properly for all quadrilaterals to patch together to make a continuous global surface.

Out-of-plane w

Rotate/translate each quad to patch

Compute in-plane (ξ, η) locally (quad-by-quad)

⇒

80 of 117

In-plane Displacements: Discrete Differential Geometry

Out-of-plane displacement w(x,y) as a function of material coordinates (x,y)

The surface in full physical coordinates with in-plane displacements (ξ, η) applied to minimize stretching

81 of 117

82 of 117

In-plane Displacements: Curved Asymptotic Lines

Out-of-plane w

Physical (DDG)

Physical (FEM)

fem is a one tool for all jobs | the ddg is more tailored - different emthods/discretization based on preserving different properties - method respects the geometry / knows the asymptotic line / not fixed mesh for all problem | mesh changes / build surface locally rather than giant global - cannot just triangulate the domain from the beginning
while it has been shown that there exists a DDG for the full, nonlinear geometry of a hyperbolic surface, it is not assured that a corresponding DDG exists for the small-slopes projection, since some information is lost in the projection
it’s remarkable that the ddg exists in the small-slopes setting not just at the level of solving for the out-of-plane displamcents (i.e., a single monge ampere pde) but also for the in-plane displacements (i.e., a system of 3 nonlinear pdes)

83 of 117

84 of 117

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).

85 of 117

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.

86 of 117

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).

87 of 117

Conclusions

Deformation of hyperbolic sheets would be affected by weak external forces that are comparable to the bending energy.
Hyperbolic sheets are inherently floppy.
There is a numerical method based on discrete differential geometry for computing hyperbolic sheets.
Some connections to experiments.
Ongoing work with dynamics.

modeling a hyperbolic sheet with a non-smooth class of $\mathcal{C}^{1,1}$ isometries that exhibit zero stretching but have uniformly bounded curvatures that are not necessarily continuous
These surfaces are characterized by non-smooth defects called branch points that generate fractal-like wrinkling patterns that resembles those in both natural, e.g., leaves, flowers, and marine invertebrates, and artificial, e.g., torn plastic and floating shells, hyperbolic sheets. In particular, we demonstrate how these non-smooth $\mathcal{C}^{1,1}$ have substantially lower elastic energy than their smoother, i.e., $\mathcal{C}^2$, counterparts and confer a considerable amount of flexibility for the wrinkled shape of the sheet without stretching.
while bending prefers smooth saddle surfaces, gravity drives the creation of multiple, higher degree branch points that generate wrinkles and flatten the sheet's vertical profile
The non-neglible effects of weak forces evinces the inherent floppiness of such sheets.

88 of 117

Acknowledgements

Generous support from the

2018-2019 Michael Tabor Graduate Scholarship.

89 of 117

Acknowledgements

Generous support from the

2018-2019 Michael Tabor Graduate Scholarship

2019-2020 Marshall Foundation Dissertation Fellowship

Advisor: Shankar Venkataramani

Committee Members: Moysey Brio and Andrew Gillette

Collaborators: Eran Sharon, Ido Levin, Ian Tobasco, ...

90 of 117

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.

91 of 117

Biomechanics

92 of 117

Backup Slides

93 of 117

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

94 of 117

Experimental context: Non-Euclidean gel discs

Klein et al. 2007

95 of 117

Experimental context: Non-Euclidean gel discs

Klein et al. 2007

96 of 117

Shaping mechanism: Prescription of non-Euclidean metrics

Klein et al. 2007

97 of 117

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

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)

98 of 117

Equilibrium configuration

Stretching energy…

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

Bending energy…

scales as t³
vanishes in flat configurations

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

99 of 117

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

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

100 of 117

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…

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

101 of 117

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…

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

102 of 117

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

103 of 117

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...

104 of 117

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)

105 of 117

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

𝜃_+/-=𝜋/2 → number of waves = 4

This solution surface is smooth

106 of 117

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

107 of 117

Numerical investigation: Branch point at the origin

Radius = 5 → Lowest energy = 8 wrinkles

Minimal Energy

Red = Total Energy

Blue = Bending

Green = Gravity

108 of 117

Numerical investigation: Branch point at the origin

Minimal Energy

Red = Total Energy

Blue = Bending

Green = Gravity

Radius = 60 → Lowest energy = 26 wrinkles

109 of 117

Numerical investigation: Branch point at the origin

Scaling analysis of our system gives the black line

Numerical simulations are represented by red dots

110 of 117

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

111 of 117

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

112 of 117

Optimize 6 Degrees of Freedom

Single Origin Branch Point

16 wrinkles

Energy = 164256

Bending = 55430 | Gravity = 108826

Numerical: 55452 | 106748

5 Distributed Branch Points

16 wrinkles

Energy = 156424

Bending = 52132 | Gravity = 104292

Numerical: 52493 | 107436

Radius = 25

113 of 117

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

114 of 117

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

115 of 117

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.

116 of 117

Optimize 6 Degrees of Freedom

Branch point defects lower the energy.

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

117 of 117

Quantitative Measure of Floppiness

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

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

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

T. Shearman’s Dissertation 2017