Modelling Deforming Interfaces using Level Sets
byHans Mühlhaus, Laurent Bourgouin and
Lutz Gross
The Australian Computational Earth Systems Simulator
(ACcESS)
Overview
Introducing Example What’s needed to model interfaces
• Constitutive models• Surface tracking, level set, stress advection
EScript & Finley Separation of Physics from Linear Algebra and Parallel Computing
Applications• Advection, upwinding, implicit vs. explicit• Lava dome simulation , Subduction, Rayleigh-Taylor Instabilities
Moving Interface: a 1D Example
2xh
0arg
0
ifel
ifsmall
02,2, vt
v2
x1
x2h large
small
We define: so that
Update: Note that:
2tvttt Time integration:
12,
Governing equations
0))(( ,,,
izijlkjkiljlik gpv
ij
jjjjtp kTTvTc ,,,, )()(
))()(1( 00
00 CCTT
Temperature and concentration dependence of density:
Heat Equation
Stress Equilibrium
Concentration advection:
0,, iit CvC
else
materialheavierinC
0
1
Example for Rayleigh – Taylor Instabilities using level sets: Mantle Plumes
The General Case
• Implicit representation of the interface by the zero level set of a smooth function φ
• φ is usually chosen as a “signed” distance function ( )
• At each time step, φ is updated solving the (hyperbolic) advection equation:
0,, kkt v
1
Problems……
1
x
TT ii
1
1. Symmetric difference expressions like
(symm.) (non-symm.)
don’t work well in hyperbolic problems (upwinding etc!)
2. Inhomogeneous velocity field causes lossof distance function property ( ) of
x
TT ii
2
11
Problems……(cont.)
),(
)23262
(
11
111111
,,,
1
1
ii
iiiiiiiiiiii
x
x
x
x
x
xxx
wwO
h
vvh
h
vvh
hhvw
wdxvwdxvv
i
i
i
i
i
hhvv
ii
xx i2
11
,
1. Symmetric difference expressions don’t work well in hyperbolic problems (upwinding!)
1. Upwinding
If v is constant:
Problems……
hhv
hhv
hhv
iiiiiii 11111
2
2
2
This can be transformed into a non-symmetric expression by adding….
We expect that the FE approx. of the PDE:
02 ,,, xxxt
hvv
is better conditioned than the original Hyperbolic problem
Generalisations…..
)2
32
2
tO(+t
Δt+
tΔt+=
t2tttt
.2
hotx
vvx
Δt+
xvΔt=
j
t
jii
2
j
t
jttt
j
t
jt
t+t
xv
Δt=
2
2
Taylor-Galerkin:
2-step alternative to Taylor-Galerkin upwinding (very effective in the presence of diffusionterms….):
j
tt
jttt
xvΔt=
2
2 Gaussians
1 Gaussian
The Level Set Method: Solving the advection equation
• Explicit
• Implicit
• Taylor Galerkin
)( ,tjj
ttt vt
tttjj
tt vt )( ,,
ktjjk
tjj
ttt vvt
vt ,,
2
,, )(2
)(
Test:A Gaussian is advected in a constant 1D velocity field.
Formulation
Finley PDE:
jijijijjkijkjkijkjlkijkl XYvDvCvBvA ,,,,, )()(
Example : Momentum and Heat equation
0))
2(
0)(
,,,
/1
)(
,,,
,
jti
X
ijijtj
C
j
Y
t
tD
tt
Y
i
p
ijlk
A
ijkl
Tvvt
Tvt
T
t
T
RaTgpvE
jj
i
ijX
jijijkl
Software can be downloaded fromwww.esscc.uq.edu.au, contact Ken Steube ([email protected]) If you need instructions re libraries etc
LinearPDE class
General form (as relevant here):
Y=Du+uAj,i,ij
duy=uAn i,ijj
PDE:
natural boundary condition
g=yη=d
f=Yω=Dκδ=A ijij
Kronecker symbol: δij=0 for i=j and 0 otherwise
f=κuωui,i,
Helmholtz Class in mytools.py
from esys.linearPDEs import LinearPDEimport numarrayclass Helmholtz(LinearPDE): def setValue(self,kappa,omega,f,eta,g): ndim=self.getDim() # spatial dimension kronecker=numarray.identity(ndim) self._setValue(A=kappa*kronecker,\ D=omega,Y=f,d=eta,\
y=g)
Use the Helmholtz Class
# Helmholtz class defined in mytools.py
from mytools import Helmholtzmydomain=...mypde=Helmholtz(mydomain)mypde.setValue(kappa=10,omega=0.1,\ f=12,eta=0,g=0)u=mypde.getSolution()
2. Problem: Inhomogeneous velocity field causes loss of distance function property of
Previous test:No topological change in the
solution
Need for a new test with:
0x
v0
y
vand
New test: shear flow
)cos()sin(
)sin()cos(
xyv
xyv
y
x
Mesh: 100x100Courant Number: 0.25•1000 steps forward•1000 steps with -v
The Level Set Method: Solving the advection equation The shape gets “noisy”…
Problem:φ
looses its distance function property
Reinitialisation needed!
The Level Set Method: Reinitialisation Idea:
Rebuild a “signed” distance function ψ from the distorted function φ
Requirements:• The interface must not be changed
• ψ must represent a distance function
Solution:Solve to steady state the equation:
Rewritten as:
00
1
)1)(( 0
sign
)( 0
sign
w
wwith
Interpretation:The “distance information” is carried by w, a unit vector
pointing away from the interface.
Remarks on re-initialisation…..
• During iteration (pseudo time integration) the vector w is established once and then kept constant
• In the explicit solution of the advection problem for we found that only alumped mass matrix discretisation works
The Level Set Method: Reinitialisation (2/3)
1D2D
3D
The Level Set Method: Reinitialisation
Same test as before, with
reinitialisation
Level set cont. : Merger of small and large bubbles
4
222
411
10
10,1800
10,3160
11,
121 )()( jjjjijij nnn
0
1 /
gradgradn
Parameters:
Surface tension:
10
1121 )()( nnnσσ divnT
Calculation, includes inertia, CourantNumber=0.5, msh:30 by 458 node quad’s
Level set cont. : Calculation of curvature for C_0 continuity
grad
gradN Ndiv
RRS
11
jijijijjkijkjkijkjlkijkl XYvDvCvBvA ,,,,, )()(
Projection:
and
Representation of surface tension b.c. as volume force:
NY )11
(2
2
RRl
en
s
l
x
T
n
l smoothing length, related to the element size
=distance in the direction of the normal of
nx0 at
Level set cont. : Merger of small and large bubbles
Surface Tension: Benchmark
Level set: Surface membrane shell, surface tension
R
n
R
np
s
ssn ),(,, zrjinnp jiijn where
Tss nnn )11
(RR
nps
Tn Inserting yields
whereTnpR 2 at equilibrium.
Collapsing Cylinder
Lava Dome
Remarks
• Escript & Finley: Rapid development of simulation software; parallelised assembly and solution phase; separation of physics from linear algebra
• Level set modelling of interfaces: distance function property crucial
• Modelling of surface tension; example of higher order b.c.’s• Upwinding strategy dependent on element
type• Re-initialisation strategy has an (undesirable)
element of mystique…..
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