moving finite elements; from shallow water equations to ... · • pattern formation, and in...
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Moving Finite Elements; From Shallow Water equations to Aggregation of microglia in
Alzheimer’s disease ___________________________
Abigail Wacher
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Past collaborators on this topic: Ian Sobey (Oxford Computing Laboratory) Keith Miller (Berkeley Department of Mathematics) Dan Givoli (Technion Department of Aerospace Engineering) Current collaborator: Simon Kaja (University of Missouri Kansas City School of Medicine )
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Outline
• Why Moving Mesh Methods?
• Finite Elements to Moving Finite Elements (MFE)
• Gradient Weighted MFE to String Gradient Weighted MFE
• Applications in 2D
• Aggregation of Microglia
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Why Moving Mesh Methods instead of classical methods?
1. Solve moving boundary problems efficiently,
some moving meshes can resolve the solution and the location of the moving boundary in a single step.
2. Resolve moving shocks or fine structures with a fixed number of degrees of freedom: which can save up to a factor of 10 (in 1D) or 100 (in 2D) nodes for problems with steep moving fronts. 3
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6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11
1.2
1.4
1.6
1.8
2
2.2
2.4
x
u
Solution to 1D scalar combustion model problem, N = 18 nodes
€
ut = uxx + e30
6(2 - u)e(-30/u), 0 < x <1, t > 0
ux (0,t) = 0, u(1,t) = 1, t > 0 u(x,0) = 1, 0 < x <1
1D combustion model
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Examples of Moving Mesh Methods:
• Adaptive Mesh Redistribution
• Moving Mesh Partial Differential Equations • Moving Finite Elements (not to be confused with
other moving mesh techniques which use finite elements)
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• 1981 MFE developed by R. Miller & K. Miller in 1D, K. Miller in 2D. Some subsequent authors: Carlson, Wathen, Gelinas, Doss, Herbst, Baines, Kuprat.
• 2006 A. Wacher and D. Givoli combined re-meshing &
refining with SGWMFE. 2007 studied dispersive shallow water equations.
• 2007 A. Wacher and I. Sobey developed a generalized
SGWMFE formulation, with applications to the gray scott equations, shallow water equations and the porous medium equation.
Publications on MFE
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Class of problems solved with piecewise linear GW Moving Finite Element methods
• Systems of time-dependent partial differential equations (so far of 1st and 2nd order).
• The methods assume the problem to be solved is well posed.
• The method is most suitable for problems with sharp moving fronts, where one needs to resolve fine scale structures of the front to compute accurate results.
• However, the methods are not very successful for certain steady-state convection problems, nor is it likely competitive with methods designed for problems in pure conservation form. 9
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Consider a system of time-dependent partial differential equations
• Consider the following system of PDEs, with two
unknown variables : are 1st or 2nd order non-linear differential operators.
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2 1 LvLu tt ==
vu,
),( and ),( 21 vuLvuL
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A Finite Element approach: one PDE in 1D
• On a fixed grid, using a piecewise linear basis function, take a piecewise linear approximation of
(x)j(t)αj jUU(x,t) ∑=
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U u
,,...,1 Nj =
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Taking the residual: define a functional that is the integral of the square
of the residual: Minimize with respect to obtain:
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Ω−=Ψ ∫Ω dULUt2))((
)(ULUt −
Ψ
}N,...,U {UU
R(U).
UA
1=
=
⋅
iU
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Using the same functional as before, but now minimizing with respect to and we now
obtain:
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A Moving Finite Element approach
}N,UN,...,X,U {XU
R(U).
UA
11=
=
⋅
iX⋅
iU
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GWMFE Each minimizing functional is weighted by its corresponding
arc-length.
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SGWMFE Single functional and the
weight is the arc-length of the string from the (x, u, v)
manifold.
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Outline of SGWMFE in 1-D • Defining:
• We can interpret the solution “string” to have imposed forces of per unit length.
• We are interested in the normal part of which arises from subtracting out the tangential part using a projection
matrix
• The normal ‘force’ on an arc length of the string: is .
2211 - L v , f - L u f tt ==
15
F P ]F[ N =
dxxvxu ds 221 ++=
) ,f,f( F 210=
ds]F[ N
F
P
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• The discretization of SGWMFE is obtained by letting the approximate solution graph
be piecewise linear in
• One then concentrates the distributed normal ‘forces’ onto each ith node:
• This can be interpreted as a normal ‘force’ balance equation at each node. (This can also be derived using a variational approach)
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N]F[
0=∫ dsiαN]F[
)),(),,(,( yxvyxuxx
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SGWMFE for systems of PDEs in 2-D • For 3 PDEs for example:
• The solution graph is a single piecewise linear graph embedded in 5-D. The 5-D normal part of the force on
the graph: • is the normal force acting on a surface area:
where and are tangent vectors defined at each point of the graph.
332211 - L w , f - L v f, - L u f ttt ===
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FP]F[ N =
dxdyDdxdy)Y X -(|Y||X| dS =⋅= 222
X
dS]F[ N
Y
),,,,( wvuyx
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• The theory for SGWMFE reduces/extends between 1D and 2D. Also it is easy to add/eliminate PDEs using the definition of the projection matrix P.
• Implementation in 2D was more difficult than 1D
since it involved considering triangular meshes rather than nodes on a line.
• The integrator used is the same for 1D and 2D. The method used is a BDF2 for stiff ODEs provided by Neil Carlson and Keith Miller.
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0)0,,()0,,(2.0)0,,(
)2
,(),,2
(),,(
2)(
2222
==
+=
+=+==
=⋅∇+
=⋅∇+
=⋅∇+
+−
yx v yxu e yxh
hhv
huv H
huvh
hu GvuF
εΔvHt v
εΔu, Gt u
εΔh, Ft h
yx
19 No slip boundary conditions
2D Shallow Water Equations
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2D Porous Medium Equation Self-similar solution, Barenblatt(2003)
Arises as a model for physical phenomena such as spreading of a thin film of liquid under gravity, or the percolation of gas through a porous medium.
021,)41()0,,(
)(
12
=
≤−=
∇⋅∇=
boundary
m
m
u
rr yxu
uutu
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PME with m = 1
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PME with m = 3
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elsewhere0
70302500
elsewhere1703050
0
0600240610410:constants
212
21
21
⎩⎨⎧ ≤≤
=
⎩⎨⎧ ≤≤
=
====
+++=−+=
, .x., .
) v(x,y,
,
.x., . ) u(x,y,
. , k . , f -, ε- ε
k)v(fuvΔvεtu), vf(Δu-uvεtu
25 Fixed boundary conditions
2D Gray Scott Equations Chemical Concentrations u and v
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• Pattern formation, and in particular cell aggregation, is an important phenomenon within the fields of Biology and Chemistry.
• The application motivating a current paper is that of chemotactic cells, known as microglia, in Alzheimer's disease.
• Of particular relevance to this study is a paper by Luca et.al., 2003, where the authors study a chemotaxis model analytically as well as with Moving Mesh Partial Differential Equations (MMPDEs) in 1D.
Aggregation of Microglia (in 2D)
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Senile plaques http://library.med.utah.edu/
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“The characteristic microscopic findings of Alzheimer's disease include "senile plaques" which are collections of degenerative presynaptic endings along with astrocytes and microglia. These plaques are best seen with a silver stain, as seen here in a case with many plaques of varying size.”
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Chemoattraction-chemorepulsion Model Equations
• The unknown variables ( ), are the cell density and the chemical
concentrations of attractant and repellent. • The non-dimensional constants are defined in the paper by M. Luca, A.
Chavez-Ross, L. Edelstein-Keshet, A. Mogliner, 2003 derived from Biology research literature.
• The equations are defined on a real and bounded domain. • The boundary conditions which hold are zero flux through the
boundary.
,
),(
),()(
2
21
21
ψψψ
ε
φφφ
ε
ψφ
−+Δ=∂
∂
−+Δ=∂
∂
∇⋅∇+∇⋅∇−Δ=∂
∂
mt
mat
mAmAmtm
ψφ,,m
1.1,0367.0,27,14.37 2121 ===== aAA εε
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))(
())(
( 21 ψψ
φφ
∇+
⋅∇+∇+
⋅∇−Δ=∂
∂
kmA
kmAm
tm
Changing the first model equation (Michaelis-Menten receptor kinetics):
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The End