18.086, Spring 2016
Introduction
Syllabus, Psets etc.
•
Course website: http://math.mit.edu/~stoopn/
18.086/
•
3 PSets (50% grade)
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Computational project & presentation (50% grade)
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Office hours: TBA
Schedule
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Check updated detailed schedule on the course website
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Rough schedule:
February
Finite differences, stability/accuracy, conservation laws
March
Conservation laws, finite volume, levelset methods
April
Solving large linear (and maybe nonlinear) systems,
Newton-Krylov methods etc., Optimization
May
Optimization/minimzation problems, N-body problems,
project presentations
Course book
Prerequisites
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Calculus: Gradients/Divergence, Volume/Area/Line
integrals, Stokes/Gauss-Theorem
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Linear Algebra: Matrices/Vectors, Eigenvalues and
-vectors
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Basic differential equations (ODE/PDE) (18.08)
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Programming: C/C++, Matlab, Mathematica,
Python…
Turbulent mixing
https://www.youtube.com/watch?v=OM0l2YPVMf8
Rayleigh-Bernard convection
https://www.youtube.com/watch?v=X5h6hbCxjz8
Flow in porous media
simulation 3D
simulation 2D
0
r=1.0
fluid speed
r=0.0
umax
experiment
c energy of a freestandrmations of its central
on). Although the Koiully used in computatheir nonlinear tensot beyond linear stabilhat substantial analythen a sti↵ film (Young
substrate
with Young
a
Information). Functional variation of the elastic energy
with respect to u then yields a nonlinear partial di↵er
ential equation for the wrinkled equilibrium state of th
film. Assuming overdamped relaxation dynamics, one
thus obtains the following GSH equation (Supplementary
Information)
Surface patterns
Film
i
Air
channel
e
(1
Pattern selection in the
b
c wrinkling regime ⌃e
Here, a 4 denotes the Laplace-Beltrami
operator, involv
nonlinear process and, therefore, cannot be inferr
ing the surface metric
tensor
of analysis.
the sphere
and parameter
Christofs
linear
stability
Numerical
2
Eq. (1) yield
a variety
qualitatively
fel symbols of the second
kind,
andof4
is the di↵erent
surfac
states that can be classified as representativ
biharmonic operatorary
[33].
The ( , )-terms describe
hexagonal phase (Fig.0 1a),2 labyrinth phase (Fig.
stress and bending, the
(a, b, c)-terms
film
intermediate
coexistencecomprise
phase (Fig.local
1b). Qualit
R/h=50
R/h=200 to labyrinths can be
theR/h=75
transition
from hexagons
substrate
interactions
and
stretching
contributions,
and
stood
through a symmetry
argument: The (b, 1
d
e
f
the ( 1 , 2 )-terms account
for the
higher-order
in Eq. (1) break
radial reflection stretching
invariance o
forces. For 1 = 2lutions
= 0,under
Eq.the(1)
reduces to
theu.stan
transformation
u!
Sinc
Theory
, which are described
herical geometryR with
at film and psubstrate
Generalizations
to nonp
by replacing the metc
nuity across the filmations that are domiu (Fig. 2; from now
2
2
3
4
u
au
bu
cu
+
2
⇥
⇤
2 Pattern selection
2 u) · (ru) + 2u4u
0 4u
( 1+
Experiment
Substrate
@t u =
D. Terwange et. al., Adv. Mat. 2014
FIG. 2: Notation and experimental system.
a,
Schematic of a curved thin film adhering
a soft umax
spheri/h
umin/h to u/h
are controlled by  = h/R (Table I), we ex
curvature-induced SB transition at some critica
of . Furthermore, recalling that the inclusion of
SB terms causes a transition from labyrinths to h
nal patterns in the classical SH model [34], it is p
to expect a hexagonal phase at large curvatures
Increasing e ective radius R/h
labyrinths at
smaller values of  in our system.
1
2D vs. 3D flagellar dynamics
of left- and right-turning sperm cells is shown in Fig. S4
(head-on view from the front).
Woolley [10] reports characteristic flagelloid curves for
mouse sperm in the rolling beat mode. These curves were
found for head-fixated sperm pointing towards and being oriented perpendicular towards the coverslip. By focussing at a focal plane ⇠ 15 µm behind the sperm head,
Woolley could directly observe the out-of-plane motion
of a single point along the flagellum (Fig. S3A). Tracking
3D flagellar beat reconstruction of
human sperm cells: optimization!
We first evaluate whether the beat pattern is planar
constrained to the focal plane, as observed and studin previous high viscosity experiments [6–9]. To this
we compare dynamics of the 2D projected tant angle ↵(s⇤ ) at arclength s⇤ ⇠ 15µm, corresponding
bout half of the 3D-reconstructed flagellum length
. S1A) [6], with the 3D angle
between the maFocal
• inertia
z˜
x
2D
microscopy:
(hologram)
axes of two
ellipsoids,
of
which
the
first
is
a)
b)
plane
structed for s 2 [0, s⇤ ], and the second from the
z
y˜
aining flagellum, s > s⇤ (Fig. S2A). Note that by
struction, ↵(s⇤ ) Condenser
attains positive and negative values,
Aperture
eters
= 0.505
µ m,For
a=
µ m,
the particle
and surrounding
θ
reas were
(s⇤ ) is λ
always
positive.
the0.1
same
periodic
E0
Condenser
al,
(s⇤1.55
) will and
thus noscillate
at twice
the frequency
respectively,
and the sampling frequency
np =
m = 1.33
Lens
(s
As Based
expected,
are periodic
⇤ ) and (s⇤ )
onboth
this↵(ssimulated
data,
we reconstruct the light
s/⇤µ).m.
ES field
Time (s)
0
0.26
. S2B). However, the ↵-time series exhibits a spurile
using
the Rayleigh-Sommerfeld
scheme. This method has been
mode
(highlighted
gray in Fig. S2B) that disappears
′
ere
[11, 12],
but briefly,
we -signal
recover
theS2C).
electric
field
at
a
height
−z
he power
spectrum
of the 3D
(Fig.
Fig.
S3:
(A)
Flagelloid
curve
for mouse sperm, reproduced
c)
Sample
Propagation
of scatteredthat 3D beat re- from [10]. (B) Flagelloid curves for human sperm, obtained
s qualitative
di↵erence
demonstrates
gh-Sommerfeld
propagator
by observing the sperm head-on and tracking the motion of a
struction
is essential for understanding human sperm
field:
single point located at s = 15 µm.
motion. ′ ′ ′ Objective1 ∂ exp(ikr′ )
LED
A
B
Camera
(6)
)1/2 . Note the use of primed coordinates to indicate a position in the
Wherevolume.
to put scatterers
to
to physical)
We can reconstruct
the field (and from there,
best
“fit”layout;
observed
scattering
bove
the(a)
hologram
planeanby
theisconvolution
Fig. 1.
Optical
LED
placed behind the condenser aperture, which is closed
pattern?
optimization
as far as
possible to=>
approximate
a point
source. (b) The scattering geometry. (c) Calculated
′ ′ ′
′ ′ ′
,problem!
y , z )with
= Epixel
0) ⊗rescaled
h(x , y ,toz ).
(7)
Es (x
s (x, y,
RG
hologram
values
the range 0-255. The scale bar indicates
10µ m.
0.3
r′
0.2
2π
∂ z′
0.1
0
h(x , y , z ) =