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

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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5

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

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

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

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

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 =

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

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

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.

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 ==

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F P ]F[ N =

dxxvxu ds 221 ++=

) ,f,f( F 210=

ds]F[ N

F

P

•  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

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

•  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

22

PME with m = 1

23

PME with m = 3

24

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)

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.”

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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32

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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