Moving Finite Elements; From Shallow Water equations to Aggregation of microglia in Alzheimer s disease

Transcription

1 Moving Finite Elements; From Shallow Water equations to Aggregation of microglia in Alzheimer s disease Abigail Wacher 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 ) 1

2 Outline Why Moving Mesh Methods? Finite Elements to Moving Finite Elements (MFE) Gradient Weighted MFE to String Gradient Weighted MFE Applications in D Aggregation of Microglia

3 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.. 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 D) nodes for problems with steep moving fronts. 3

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6 1D combustion model u t u xx + e30 6 ( - u)e(-30/u), 0 < x <1, t > 0 u x (0,t) 0, u(1,t) 1, t > 0 u(x,0) 1, 0 < x <1.4 Solution to 1D scalar combustion model problem, N 18 nodes. 1.8 u x 6

7 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) 7

8 Publications on MFE 1981 MFE developed by R. Miller & K. Miller in 1D, K. Miller in D. Some subsequent authors: Carlson, Wathen, Gelinas, Doss, Herbst, Baines, Kuprat. 006 A. Wacher and D. Givoli combined re-meshing & refining with SGWMFE. 007 studied dispersive shallow water equations. 007 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. 8

9 Class of problems solved with piecewise linear GW Moving Finite Element methods Systems of time-dependent partial differential equations (so far of 1 st and nd 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 9 conservation form.

10 Consider a system of time-dependent partial differential equations Consider the following system of PDEs, with two unknown variables : u,v u L v L t 1 t L are 1 st or nd 1( u, v) and L ( u, v) order non-linear differential operators. 10

11 A Finite Element approach: On a fixed grid, one PDE in 1D j 1,..., N using a piecewise linear basis function, take a piecewise linear approximation of, U u U(x,t) U j (t)α j (x) j 11

12 Taking the residual: U L(U) t define a functional that is the integral of the square of the residual: Ψ ( L( U)) dω Ω U t Minimize to obtain: Ψ with respect. AU R(U) U i U {U,...,U } 1 N 1

13 Using the same functional as before, but now minimizing with respect to and we now obtain: A Moving Finite Element approach. AU R(U) U i X i U {X,U,...,X,U } 1 1 N N 13

14 GWMFE Each minimizing functional is weighted by its corresponding arc-length. SGWMFE Single functional and the weight is the arc-length of the string from the (x, u, v) manifold. 14

15 Outline of SGWMFE in 1-D Defining: f 1 ut - L 1, f vt - L We can interpret the solution string to have imposed forces of ( 0,f,f ) per unit length. F 1 We are interested in the normal part of F which arises from subtracting out the tangential part using a projection matrix P [ F ] N PF The normal force on an arc length of the string: is [ F ] ds. ds 1 + u + x v x dx N 15

16 The discretization of SGWMFE is obtained by letting the approximate solution graph ( x, u( x, y), v( x, y)) be piecewise linear in One then concentrates the distributed normal forces [ F ] onto each ith node: N x [ F ] i α ds N 0 This can be interpreted as a normal force balance equation at each node. (This can also be derived using a variational approach) 16

17 SGWMFE for systems of PDEs in -D For 3 PDEs for example: f The solution graph is a single piecewise linear graph ( x, y, u, v, w) the graph: [ F ] ds 1 ut - L 1, f vt - L, f3 wt - L3 embedded in 5-D. The 5-D normal part of the force on [ F ] N PF N is the normal force acting on a surface area: ds X Y -( X Y ) dxdy Ddxdy X Y where and are tangent vectors defined at each point of the graph. 17

18 The theory for SGWMFE reduces/extends between 1D and D. Also it is easy to add/ eliminate PDEs using the definition of the projection matrix P. Implementation in D 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 D. The method used is a BDF for stiff ODEs provided by Neil Carlson and Keith Miller. 18

19 0,0), (,0), ( 0.,0), ( ), ( ),, ( ),, ( ) ( y x v y x u e y x h h h v h uv H h uv h h u G v u F εδv H t v εδu, G t u εδh, F t h y x 19 No slip boundary conditions D Shallow Water Equations

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21 D Porous Medium Equation Self-similar solution, Barenblatt(003) u t ( u m u) u( x, y,0) (1 4r ) 1 m, r 1 u boundary 0 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. 1

22 PME with m 1

23 PME with m 3 3

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25 u t ε 1 Δu-uv D Gray Scott Equations Chemical Concentrations u and v + f( 1 u), v t ε Δv + uv + (f + k)v constants : ε , ε 10-6, f 0. 04, k u(x,y, 0 ) v(x,y, 0 ) 0. 5, 1, 0. 5, 0, 0. 3 x 0. 7 elsewhere 0. 3 x 0. 7 elsewhere Fixed boundary conditions 5

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28 Aggregation of Microglia (in D) 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., 003, where the authors study a chemotaxis model analytically as well as with Moving Mesh Partial Differential Equations (MMPDEs) in 1D.

29 Senile plaques 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, 9 as seen here in a case with many plaques of varying size.

30 Chemoattraction-chemorepulsion Model Equations m Δm A1 ( m φ) + A t φ ε1 Δφ + a ( m φ), t ψ ε Δψ + m ψ, t ( m ψ ), The unknown variables ( m,φ,ψ ), 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, 003 derived from Biology research literature. A , A 7, ε1 ε , a 1.1 The equations are defined on a real and bounded domain. The boundary conditions which hold are zero flux through the boundary.

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33 Changing the first model equation (Michaelis- Menten receptor kinetics): m t ( m Δm A φ) + A ( ( k + φ) ( k m + ψ ) 1 ψ ) 33

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