How to Solve a Partial Differential Equation on a surface
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How to Solve a Partial Differential Equation on a Surface Tom Ranner University of Warwick [email protected]http://go.warwick.ac.uk/tranner University of Warwick, Graduate Seminar, 3rd November 2010
How to Solve a Partial Differential Equation on a surface
Familiar techniques of separation of variables and Fourier series can be used to solve a variety of pde based on domains in the plane, however these techniques do not extend naturally to surface problems. Instead we look to take a computational approach. The talk will cover the basics of finite difference and finite element approximations of the one dimensional heat equation and show how to extend these ideas on to surfaces. If time allows, we will show numerical results of an optimal partition problem based on a sphere. No background knowledge of pde or computation is required.
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1. How to Solve a Partial Dierential Equation on a Surface Tom
Ranner University of Warwick [email protected]
http://go.warwick.ac.uk/tranner University of Warwick, Graduate
Seminar, 3rd November 2010
2. Where do surface partial dierential equations come from?
Partial dierential equations on surfaces occur naturally in many
dierent applications for example: uid dynamics, materials science,
cell biology, mathematical imaging, several others. . .
3. Example Surfaces Cell Biology Figure: An endoplasmic
reticulum (ER)
4. Example Surfaces Pattern Formation Invited Article R253
Figure 1. Examples of self-organization in biology. Clockwise from
top-left: feather bud patterning, somite formation, jaguar coat
markings, digit patterning. The rst is reprinted from Widelitz et
al 2000, -catenin in epithelial morphogenesis: conversion of part
of avian foot scales into feather buds with a mutated -catenin Dev.
Biol. 219 98114, with permission from Elsevier. The second is
courtesy of the Pourqui Laboratory, Stowers Institute for Medical
Research, and e the remainder are taken from the public image
reference library at http://www.morguele.com/. (a) Activator (b)
Inhibitor 80 80 8 15 Taken from R E Baker, E A Ganey and P K Maini,
Partial dierential equations for self-organisation in cellular 60
60 Distance (y) Distance (y) and development biology. Nonlinearity
21 (2008) R251R290. 40 10 40 4 20 20
5. Example Surfaces Dealloyed surface The surface can be the
interface of two materials in an alloy. 9738 C. Eilks, C.M. Elliott
/ Journal of Computational Physics 227 (2008) 97279741 Fig. 3.
Simulation on a large square, t 0:04; t 0:1 and t 0:2. Taken from C
The geometric motion ofNumerical simulation of dealloying by
surface nonvanishing right hand evolving surface plane. Eilks, C M
Elliott, the surface has little inuence, except for providing for a
dissolution via the side for the conservation law of gold on the
surface, since gold from the bulk is accumulating on the surface.
While the concrete nite element method. Journal of Computational
Physics 227 random distribution in the bulk, the lengthscales of
the appearance of the structure obviously depends on the particular
(2008) 97279741. structure depend only on the particular values of
the parameters in the equation. After the phases have separated,
etching still continues in the areas with a small concentration of
gold, while the motion is negligible in regions covered by gold
yielding a maze like structure of the surface. The origins of this
shape can still be explained by the initial phase separation which
xed the gold covered regions that proceeded to move into the bulk.
So at this stage the simulation does not necessarily show the
mechanisms for the emergence of a nanoporous structure. By
undercutting the gold rich portion of the surface, the area of the
surface that is not covered by gold increases. In the last stages
of this simulation new components of the gold rich phase emerge at
the bottom of the surface. Additionally the inter- face separating
gold rich and gold poor phases shows no effect of coarsening as for
the planar Cahn-Hilliard equation, but instead becomes more
complicated. These two effects can be seen as signs that the model
shows increasing formation of morphological complexity. We explore
them in more detail in the following examples. Note however that
due to self-inter- sections the surface is not embedded at later
stages, as can be seen in Fig. 4, where the cross sections along
the plane parallel
6. 1D Heat Equation The one dimensional heat equation is given
by ut = uxx on (0, 1) (0, T ) u(0, t) = u(1, t) = 0 for t (0, T )
u(x, t) = u0 (x) for x (0, 1). Joseph Fourier solved this in 1822
using separation of variables. The idea is to write u(x, t) = X
(x)T (t) and derive a system of decoupled odes for X and T , which
can then be solved simply.
7. 2D Heat Equations In two dimensions the heat equation on a
square = (0, 1) (0, 1) becomes ut = u := uxx + uyy in (0, T ) u = 0
on [0, T ] u(x, 0) = u0 (x) on {0}. This can be solved using
Fourier analysis. We rewrite u(x, t) = uk (t)e ikx , kZ2 and derive
a system of odes of uk .
8. What is the surface Heat Equation? For a d-dimensional
hypersurface we would like to write down something like ut = u but
u is only dene on . So instead we have ut = u, where is the
Laplace-Beltrami operator.
9. Surface gradients For a function : R we dene the surface
gradient of by = ( ) where is the outward pointing normal on and is
the gradient in ambient coordinates.
10. Surface gradients For a function : R we dene the surface
gradient of by = ( ) where is the outward pointing normal on and is
the gradient in ambient coordinates. The Laplace-Beltrami operator
is given by the surface divergence of the surface gradient = .
11. Surface Heat Equation The surface heat equation on a closed
surface is given by ut = u on [0, T ] u(x, 0) = u(x) on .
12. Surface Heat Equation The surface heat equation on a closed
surface is given by ut = u on [0, T ] u(x, 0) = u(x) on . How do we
solve this equation? separation of variables Fourier analysis
parametrisation approximation. . .
13. Approximation of the 1D Heat Equation using nitedierences
Lets rst go back to the 1D problem is demonstrate some possible
methods. Take the 1D heat equation ut = uxx , and lets approximate
the derivatives by nite dierences. We divide (0, 1) into N
intervals of length x, (xj , xj+1 ) for j = 0, . . . N. Our
approximate solution u h solves for j = 1, . . . , N 1 and t (0, T
) h u h (xj1 , t) 2u h (xj , t) + u h (xj+1 , t) ut (xj , t) = . x
2
14. Approximation of the 1D Heat Equation using nitedierences
From, for j = 1, . . . , N 1 and t (0, T ) h u h (xj1 , t) 2u h (xj
, t) + u h (xj+1 , t) ut (xj , t) = , x 2 we wish to discretize in
time. We approximate the time derivative on the left hand side in
the same way, but there is a choice on the right-hand side to
evaluate at t = tk or tk+1 . We choose for numerical reasons the
Backwards Euler method t = tk+1 . So we have a linear system u h
(xj , tk+1 ) u h (xj , tk ) u h (xj1 , tk+1 ) 2u h (xj , tk+1 ) + u
h (xj+1 , tk+1 = t x 2
15. Approximation of the 1D Heat Equation using nitedierences
The solve strategy is then 1. Initialise u h (xj , 0) = u0 (xj )
for j = 0, . . . , N 2. For k = 0, 1, 2, . . . solve the linear
system t h u h (xj , tk+1 ) u (xj1 , tk+1 ) 2u h (xj , tk+1 ) + u h
(xj+1 , tk+1 ) x 2 = u h (xj , tk ) for j = 1, . . . , N 1, and u h
(xj , tk+1 ) = 0 for j = 0, N.
16. Approximation of the 1D Heat Equation using nitedierences
This was implemented in MATLAB with the following result:
17. Approximation of the 2D Heat Equation using nitedierences
The same method can be implemented for the 2D problem, with the
following result:
18. Can nite dierences work on a surface? This method works
best on a regular grid, which is almost always impossible on a
surface. One must parameterise the surface rst! Projections or
embeddings can be used to solve a the surface pde using this type
of method. An example of this type of method is the closest point
method. Instead, we can try to approximate the solution rather than
the problem.
19. Approximation of the 1D Heat Equation using niteelements
Again we split the domain (0, 1) into N intervals of length h. We
dene the space Vh of nite element functions to be Vh = {h C0 (0, 1)
: h |(xj ,xj+1 ) is ane linear for each j}. We would like to solve
h h ut = uxx so that u h (, t) Vh for all t, but the nite element
functions dont have two derivatives in space!
20. Approximation of the 1D Heat Equation using niteelements
The nite element functions do have a rst derivative almost
everywhere, so we put the equations in integral form to remove one
of the derivatives.
21. Approximation of the 1D Heat Equation using niteelements
The nite element functions do have a rst derivative almost
everywhere, so we put the equations in integral form to remove one
of the derivatives. We multiply by a test function and integrate
with respect to x 1 1 ut = uxx , 0 0 we then integrate by parts
using the boundary condition u(0) = u(1) = 0 to get 1 1 ut + ux x =
0 0 0 Now all the terms in the above equation exist for all 1 u, Vh
H0 (0, 1). This is called the weak form of the heat equation.
22. Approximation of the 1D Heat Equation using niteelements We
wish to nd u h (, t) Vh such that for all time t 1 1 h h ut + ux x
= 0. 0 0 We would like this to be true for all Vh , but this is
equivalent to being true for all basis functions j Vh .
23. Approximation of the 1D Heat Equation using niteelements We
wish to nd u h (, t) Vh such that for all time t 1 1 h h ut + ux x
= 0. 0 0 We would like this to be true for all Vh , but this is
equivalent to being true for all basis functions j Vh . We can nd a
basis for Vh by setting j (xi ) = ij . Our problem is to nd u h (,
t) Vh for t such that 1 1 h h ut j + ux (j )x = 0 for j = 1, . . .
, N. 0 0 Notice that the boundary condition is automatically
satised if u h Vh .
24. Approximation of the 1D Heat Equation using niteelements We
can decompose u h in terms of the basis function j to get N u h (x,
t) = Ui (t)j (x). i=0 The equations become 1 1 Ui,t i j + Ui (i )x
(j )x = 0, 0 i 0 i If we write U(t) = (U0 (t), . . . , UN (t)) and
dene the mass matrix M and stiness matrix S by 1 1 Mij = i j Sij (i
)x (j )x , 0 0 we can write a matrix system MUt = SU.
25. Approximation of the 1D Heat Equation using niteelements We
discretize in time using backwards Euler again to get the following
solve strategy: 1. Initialise U 0 as Uj0 = u0 (xj ). 2. For each k
= 0, . . ., solve the linear system (M + tS)U k+1 = MU k .
26. Approximation of the 1D Heat Equation using niteelements
This method was implemented in MATLAB with the following
result:
27. Approximation of the 2D Heat Equation using niteelements
This method can also be used for the 2D problem:
28. tinuously; the method should also be robust enough to
tolerate dierent resolutions and boundaries; for database indexing,
each class index should be small for storageCan we use nite
elements for surface partial dierential and easy to compute.
Conformal mapping has many nice properties to make it suitable for
classication problems. Conformal mapping only depends on the
Riemann metric and is indepen-equations? dent of triangulation.
Conformal mapping is continuously dependent of Riemann metric, so
it works well for dierent resolutions. Conformal invariants can be
repre- sented as a complex matrix, which can be easily stored and
compared. We propose to use conformal structures to classify
general surfaces. For each conformally equivalent class, we can
dene canonical parametrization for the purpose of comparison.
Geometry matching can be formulated to nd an isometry between two
surfaces. We can use what is called a surface nite element method,
in By computing conformal parametrization, the isometry can be
obtained easily. For which we approximate the domain also. surfaces
with close metrics, conformal parametrization can also give the
best geometric matching results. Fig. 1. Surface & mesh with
20000 faces 1.1. Preliminaries. In this section, we give a brief
summary of concepts and Justication of this method including well
posedness, stability and convergence can be found in G. Dziuk and
C. M. notations. Elliott, Surface nite elements for parabolic
equations. J.realization |K| is homeomorphic Let K be a simplicial
complex whose topological Comput. Math. 25(4), (2007) 385407. Image
taken from Xianfeng Gucompact 2-dimensional manifold.
SupposeConformal Structures of surfaces. Communications in to a and
Shing-Tung Yau Computing there is a piecewise linear embedding
Information and Systems. 2(2) (2002), 121146. (1) F : |K| R3 . The
pair (K, F ) is called a triangular mesh and we denote it as M .
The q-cells of K
29. Solving the surface heat equation using surface
niteelements We dene h to be a polyhedral approximation of (made of
triangles) with vertices xi . Vh is the space of piecewise linear
functions on h with basis j given by j (xi ) = ij We look to solve
the weak form of the surface heat equation on h : h h ut + h u h =
0. h h
30. Solving the surface heat equation using surface
niteelements We dene the surface mass matrix M and surface stiness
matrix S by Mij = i j Sij = h i h j h h Using the same notation for
U as before we have MUt + SU = 0.
31. Solving the surface heat equation using surface
niteelements We discretize using backwards Euler to get the solve
strategy 1. Initialise U 0 by Uj0 = u0 (xj ) 2. For k = 0, 1, . .
., solve the linear system (M + tS)U k+1 = MU k .
32. Solving the surface heat equation using surface
niteelements This method was implemented using the ALBERTA nite
element toolbox on S 2 to get:
33. Optimal Partition Problem Given a surface , a positive
integer m and parameter > 0, we wish to minimised the following
energy functional for a vector-valued function u = {uj } (H 1 ())m
: 2 E (u) = (| u| + 2F (u)), where F is in interaction potential of
the form m 1 F (u) = 2 ui2 + uj2 . i=1 j
34. Optimal Partition Problem Gradient Decsent If we
minimised E subject to uj L2 () = 1 for each j we have the
following norm preserving gradient decent equations: uj,t uj = j
(t)uj F,uj (u) on R+ |uj |2 = 1 for j = 1, . . . , m. along with
the initial condition u(0, x) = g (x) H 1 (, ), with gj L2 () = 1.
Here is the singular subset in Rm m = y Rm : yj2 yk = 0, yk 0 for 1
k m . 2 j=1 k
35. Optimal Partition Problem Partition? Since the
vector-valued function u takes values on in , the above system is
equivalent to the following problem For a given surface , partition
into m region j such that the sum j j , with j the rst eigenvalue
of over j with a Dirichlet boundary condition, minimised.
36. Optimal Partition Problem How to solve? To progress from
step tn to tn+1 , given u n we calculate u n+1 by 1. Set u n+1 as
the solution the heat equation at t = tn+1 ut = u for x , tn < t
< tn+1 u(tn , x) = u n (x). 2. use (Gauss-Seidel) solver of
decoupled ODEs: sequentially for j = 1. . . . , m, 2uj uj,t = 2
(in+1 )2 + u (in+1 )2 for x , tn < t < tn+1 u ij uj (tn , x)
= ujn+1 (x) Set u n+1 as the solution u(tn+1 , x) of this system at
t = tn+1 , 3. normalisation: we set u n+1 via ujn+1 ujn+1 = for j =
1, . . . , m, ujn+1 L2 () 37. Optimal Partition Problem Numerical
Results This equation was discretize using a surface nite element
method and solved here for m = 6 on S 2 . 38. Summary Methods such
as Fourier series and separation of variables dont work on general
surface so we must approximate solutions. Finite dierence methods
work by approximating the dierential operator by dierence
quotients, but only work best on rectangular domains. Finite
element methods approximate weak solutions by dividing up the
domain into nitely many subdomains and can be extended to surface
problems. A surface nite element method can be used to solve many
dierent types of problems on surfaces.