eikonal equation for buckling and beyond Sergei Nechaev Interdisciplinsary Scientific Center Poncelet CNRS Moscow Lebedev Physical Institute RAS Moscow How to describe the profile ID: 648180
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Slide1
From geometric optics to plants: eikonal equation for buckling… and beyond
Sergei Nechaev
Interdisciplinsary Scientific Center Poncelet (CNRS, Moscow)Lebedev Physical Institute (RAS, Moscow) Slide2
How to describe the profile? Slide3
J
upe
à godets(R. Voituriez, S.N., J. Phys. A, 2001)Slide4
Growth in a discGrowth in a stripExponential proliferation of cells in a thin slit Slide5
When we open the slit, the material relaxes into the 3D structureConjecture: buckling occurs as a conflict between internal geometry of object and geometry of space of embeddingSlide6
Growth induces an increasing strain in a tissue near its edge and results in: in-plane tissue compression and/or redistribution of layer cells accompanied by the in-plane circumference instability (“stretching”)out-of-plane tissue buckling with the formation of saddle-like surface regions
(“bending”) Energetic approach to growing patterns is based on a competition between the bending and stretching energies of elastic membranes
For bending rigidity of a thin membrane B ~ h3, while for stretching rigidity, S ~ h, where h is the membrane thickness. Thin enough tissues, with h <<1
, prefer to bend, i.e. to be negatively
curved under
relatively small critical
strainSlide7
Formulation of the problemExponentially growing colony, being the hyperbolic structure,
admits Cayley trees as possible discretizations. The Cayley
trees cover the hyperbolic surface isometrically, i.e. without gaps and selfintersections, preserving angles and distances. Our goal is an embedding a Cayley tree into a 3D Euclidean space with a signature {+1,+1,+1}. Hilbert theorem prohibits embedding of unbounded Hyperbolic surface into Euclidean space smoothlySlide8
a
b
a
b
a
b
a
b
The
relief
of
the
surface
is
encoded
in
the
coefficient
of
deformation
,
coinciding
with the
Jacobian
J
(
z
)
of the conformal
transform
z
(
z
), whereSlide9
Isometric embedding of a Cayley tree into Poincare disc and a strip Slide10
Optimal profile – is the surface in which we can embed the exponentially growing graph isometricallyThe metric ds2 of a 2D surface parametrized by (u,v), is given by the coefficients
of the first quadratic form of this surface
The surface area then readsGF(S. N., K. Polovnikov, Soft Matter, 2017)Slide11
If z(w) is holomorphic, the Cauchy-Riemann conditions provide Surface
embedded in 3D has the same metric as Poincare discSlide12
Surface embedded in 3D has the same metric as Poincare discIf we impose the condition for a surface to be a function above (u,v), then we could write the surface element in curvilinear coordinatesSlide13
Relief of the surface f (u,v
) is defined by the eikonal equation
In our particular case we have to solve the
following
equation
for the function
f
(
u,v
)
where
is the Dedekind
h
-function
a
-2Slide14
Comparing equationto the standard eikonal equation for the rays in optically inhomogeneous media
W
e conclude that the rays propagate along optimal Fermat paths in Euclidean domain. They are projections of geodesics of corresponding “eikonal surface”. The refraction coefficient in this case reads
Geometric optic analogy
a
-2Slide15
Fermat principleTime dt for a ray propagating along g between points M(x) and N(x+dx) should be minimal
where is the refraction coefficient and c and v(
x) are the light speed in vacuum and in the medium, d|x| = ds is the spatial increment along the ray . Supposing that the “optic length”, S=cT, is the action, we have with the Lagrangian
Minimizing the action , we get the equation
. Due to Huygens principle we findSlide16
Details of the conformal mapping construction z → ζ
2. ζ → r
of Picard-Fuchs type3. r → w Slide17
Solution of the eikonal equation 1) In polar coordinates (for a disc): and2) In rectangular coordinates (for a strip): andwhere Slide18
The rigidity in our geometric approach is controlled by the parameter a – the size of elementary flat domain1) If a ≥1, the surface is rigid2) If 0< a <<1, the surface is flexible
Rigid and flexible circular surfaces
a=0.07
a=0.14Slide19
Buckling in a stripFormation of hierarchical folds due to ultrametricity of Dedekind h-functionSlide20
Speculation: how to interpret the complex-valued refraction index?In geometric optics means the existence of absorption mediaPropagating wavefront of a moving particle dissipates the energy in areas where the refraction index is complex-valued. For graphs of high rigidity (a >> 1) the profile is flat (no real solution of eikonal
equation exists.Applying Cauchy-Riemann conditions to the complex solution of eikonal equation
f(u,v) = fR(u,v) + i fI(
u,v
)
,
we get ∂
u
f
R
(
u,v
) =
∂
v
f
I
(
u,v
);
∂
v
f
R
(
u,v
) =
-∂
u
f
I
(
u,v
).
The real solution of
eikonal
equation can be analytically continued along the curve
G
in plane (
u,v
), where the
r.h.s
. of
eikonal
equation nullifies.
Slide21
“Energetic view” on conformal approach
Metric tensor
from conformal approach
Metric tensor
of deformed sample
Internal tensions
Flexible material buckles without internal tensions, i.e.
F
= 0:
eikonal
equationSlide22
Some constructions involving Dedekind h
-functionSlide23Slide24
Modular
h-function is invariant with respect of SL(2,Z) transformations
xSlide25
Phyllotaxis
Energetic approach to phyllotaxis
, L. Levitov, 1991Slide26Slide27
Static and Dynamical
Phyllotaxis in Magnetic Cactus
C. Nisoli et al: ArXiv: cond-mat/0702335
Thomae (Dirichlet) function