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Moving grid methods and Multi-mesh methods Tao Tang Hong Kong Baptist Hong Kong Baptist University University International Workshop on Frontiers in Scientific International Workshop on Frontiers in Scientific Computing Computing June 10-13, 2008, Wuyi Mountain, China June 10-13, 2008, Wuyi Mountain, China

Moving grid methods and Multi-mesh methods Hong Kong Baptist University Tao Tang Hong Kong Baptist University International Workshop on Frontiers in Scientific

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Page 1: Moving grid methods and Multi-mesh methods Hong Kong Baptist University Tao Tang Hong Kong Baptist University International Workshop on Frontiers in Scientific

Moving grid methods and Multi-mesh methods

Tao Tang Hong Kong Baptist UniversityHong Kong Baptist University

International Workshop on Frontiers in Scientific ComputingInternational Workshop on Frontiers in Scientific Computing

June 10-13, 2008, Wuyi Mountain, ChinaJune 10-13, 2008, Wuyi Mountain, China

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Outline:

Basic ideas of moving mesh method Applications and results Why using multi-mesh methods? Conclusions and future work

Collaborators:Ruo Li, Huazhong Tang, Pingwen Zhang (Peking University)

Yana Di (Chinese Academy of Sciences)

Heyu Wang, Xianliang Hu (Zhejiang University)

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Different types of adaptive grid refinement

h-refinement: # of grid points not constant adds (or deletes) grid points grid equations are not coupled to physical PDEs )=>

extra interpolation procedure needed p-refinement:

varies degree of piecewise polynomials (FE’s) often in combination with h-refinement

r -refinement: # of grid points constant r e-locates (moves) grid points grid equations uncoupled or coupled with physical PDE

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Physical vs. computational coordinates

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Viscous Shock Problem

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Key ingredients of the moving mesh methods

Mesh equation -- determine a one-to-one mapping from a parameter space to

a physical space.

Monitor function -- used to guide the mesh redistribution.

Interpolation -- may be required to pass the solution information on the old

mesh to the newly generated mesh.

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The ‘grid-energy ’

Can be taken to represent the energy of a system of springs with spring constants spanning each interval. The non-uniform grid point distribution resulting from the equidistribution principle thus represents the equilibrium state of the spring system, i.e., the state of minimum energy.

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The adaptive grid seen as a system of springs

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Winslow’s method (1960s)

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Regularity of the transformation in 2D

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Harmonic mapping

Let dij and rαβ be metric tensors in some local coordinates . Define the energy for a map as

(1)

where d=det(dij), (dij)=(dij)-1. The Euler-Lagrange equations, whose solution minimizes the above energy, are given by

(2)a. The inverse of (Gij) is called monitor functions.b. Solutions to (2) are harmonic functions giving a continuous and one-to-one mapping.c. Solutions to (2) minimizes the energy (1).

Dvinsky (JCP, 1991): (2) may provide a general framework for mesh generation.

,2

1)( xd

xxrddE

jiij

.0

j

kij

i xG

x

andx

)(x

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We solve the constrained optimization problem:

(3)

Note that the boundary values ξb are not fixed, instead they are unknowns in the same way as the interior points.

...

min

K

b

kj

k

i

kij

ts

xdxx

G

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The whole moving mesh algorithm can be packed in a black box which requires the following inputs:

the current solution of the underlying PDEs,the current solution of the underlying PDEs,

the algorithm for solving the mesh PDEs, the algorithm for solving the mesh PDEs,

and an interpolation algorithmand an interpolation algorithm. .

Such a black box has been implemented in the adaptive finite element package AFEPack (Ruo Li and Wenbin Liu)

http://circus.math.pku.edu.cn/AFEPackhttp://circus.math.pku.edu.cn/AFEPack

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Level set / Moving mesh Approach

Recently, methods that couple two different schemes have been developed for simulating fluid flows with moving interfaces. Examples are

The coupled level set and volume-of-fluid (VOF) method The hybrid particle level set method The mixed markers and VOF method

A coupled method takes advantage of the strengths of each of the two methods, and is therefore superior to either method alone.

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The 2D incompressible Navier-Stokes equations separated by a free surface:

After a standard non-dimensionalization procedure,

.0u t

. ,n00

0u

,g)(D)2(u)uu(

pt

.)(We

1g

Fr

1D)2(

Re

1u)uu( pt

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./)(

,each For .3

).(meas )),(meas(

),/ ,0(max/)/ ,0max(

),/)( , ,2

, ,meas2 ,)(meas

1

, ,3 ,2 ,1 ,)( ,)(

,3 ,2 ,1 T, gleeach trianFor .2

.,,1 ,0 Initialize .1

*1

**

3

1

3

1

3

1

3

1

3

1

1

*

ni

ni

ni

ni

i

iiiii

ll

ii

llli

llii

lll

ii

jjjiiT

ijjii

ii

Vv

TTF

KKKKnF

K

NNTndxST

F

TxjxNPxN

i

Vi

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Solve the N-S eqns:

.0,

),Fr

1(),

We

1(),u(),(

1

,,2,Re

1),(

1

),We

1(),u(),(

1

,,,2Re

1),(

1

11

11111

11111

ny

nx

ny

nnnnn

yn

yny

nx

nx

nny

nnn

xnnnnn

xn

ynx

nny

nx

nx

nnn

vuq

vvt

pvvuvt

uut

pvuuut

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Solution interpolation can be realized by solving the system

The following monitor function is proposed:

.0

0,u

,uu

x

px

x

x

x

,max/1

max/1

G

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Impact of water drop with an 1002 moving grid. The parameters used are Re = 28144, Fr = 204, We = 1760, g / l =1 / 816, g /l = 1; t = 0.25, 0.3, 0.46, 0.67.

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Multiphase flows in 3D

Dimension-independent moving mesh method Multi-grid speed up Grid redistribution Solution interpolation Monitor function

21),( txm

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A phase field model for two incompressible fluids

with initial conditions

and appropriate boundary conditions. (t) is the Lagrange multiplier corresponding to the constant volume constraint:

))()((u

0,u

),(g)(uuuu

t tf

xvpt

0000 ,uu tt

.0 dxdt

d

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Finite element scheme

The discretization of the time derivative is given by the backward difference formula:

resulting in a semi-implicit second-order approximation. Introduce the notation

N(u, )=uu ( )+g(x).

v ()2 and qH1(),

Then update k+1 by: H1()

),)((2

uu4u3u 211

tOtt

nnn

10H

).,N(n,u,

, ,,NN2,u,uu4u3

11n1n

n11n11

qqvqp

vpvvvvt

n

nnnnn

. ),)(1())()(2(),2(u

,,2

43

21111

1n11

nnnnnnn

nnn

ff

t

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Coalescence of two kissing bubbles, = 0.04, = 0.1, = 0.1, = 0.1, 32 32 64 grid in an 1 1 2 domain. t = 0, 0.1, 0.3, 0.6, 0.8. The bottom figures are the computational meshes (t = 0.8). The box regions are magnified in the next figure.

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Rising bubble example: (a) the bubble evolution: t = 1, 4, 7 (b) the rising Reynolds number, obtained with two fixed grids (16 16 32, 32 32 64) and one moving grid (16 16 32).

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Nonaxisymmetric merging of two viscous gas bubbles ( = 0.04, = / We = 0.02, = 1/503/4, = 0.01 and Fr = 1). The moving tetrahedral mesh contains 32 32 64 nodes. t = 0.0, 1.0, 2.0, 3.0, 4.0.

Nonaxisymmetric merging of two viscous gas bubbles: the moving tetrahedral mesh contains 32 32 64 nodes. t = 3.0.

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We compared the efficiency of the moving mesh methods for the nonaxisymmetric merging of two bubbles.

To reach similar accuracy, the fixed mesh simulation requires at least 64 64 128 cells to match that for a moving mesh with 32 32 64 cells. To reach t = 4, the simulation on the moving grid with 32 32 64 cells takes about 20h 30min, while the simulation on the 64 64 128 fixed mesh takes about 106h 20min.

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Dentritic Growth with MovingFinite Element Methods

The mesh redistrbution is realized by solving an elliptic boundary control problem a nonlinear multigrid algorithm

With a particularly designed monitor function, the qualitiy of the redistributed mesh grids is improved significantly.

Dendritic growth has been a central problem in pattern formation and metallurgy

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The quantitative phase field model of the dendritic crystallization of a pure melt in two and three dimensions takes the form

222 11)()(

,2

1

nWn

uD

t

tt

s

i ii

nWnW

1

2

)(

)()(

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The weak formulation of the phase field equations mentioned in the previous page takes the form

),)((),)((

),,(2

1),(),(

2

nWn

vvuDvu

t

tt

),()),1)](1(([ 22

u

where , are the test functions and v

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T

nWnWnW

)(

)(

)(

)(

)(

)(

321

2

))()((

))()((

))()((16

322

2312

13

212

1232

32

132

3122

21

40

SS

SS

SSW

with22 )()( jiijS

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Based on the interaction between the phase field variable and the thermal field, we defined a new variable which satisfies

~

21~

)1(

and reformulated the monitor function of the form

)(22 um

where~

)(

*

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Multi-mesh adaptation(motivation …)

The heuristic error indicator can be selected as

where [] means the jump on the interface and h is the length of edge e. The fixed threshold rule is adopted as the strategy for mesh adaptation: ensuring

the element error indicators T satisfying

where tol is the prescribed tolerance and are two constants. Practically, the tolerance is often selected as

The difference between the system and a scalar equation;

, )(2/1

23

Tee ehhT dnuhu

, toltol T

1 , 0

;/2

t

T NN

tol

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The solving procedure

Algorithm: multi-mesh adaptive finite element method for the phase field model.

Prepare the background mesh 0;Set the initial value for and u and obtain the initial mesh for both variables and u;while t < T do

Solve PDEs;t = t + t;If the meshes have not been updated for N (being constant)

stepsthen

Update mesh and _h;Update mesh u and u_h;

endend

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2D complex dendritic structures

Figure: The simulation result with parameters = 0.70, D = 2, t=0.1, = 0.05 and is chosen to simulate = 0. Clockwisely from upper right quarter in both figures are the contour of u, contour of = 0, and u at 104 time steps. The right figure is zoomed in from the part inside the black box in left figure.

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2D complex dendritic structures

Figure: The meshes and the profiles for complex dendritic structures when = 0.65, D = 3 at 1000 time steps(left). The right one is the magnification at the tip show in black box.

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2D complex dendritic structures

Figure: The comparison of the meshes for u (left part) and (right part) when = 0.65, D = 3 at 1000 time steps. It is the magnification of the left plot in Fig. 5 within center black box.

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3D simulations: DOF comparison

Figure: The comparison of degree of freedoms between and u as the function of the time. The left figure is for the case of = 0.55, D = 1, and the right one is for the case of = 0.45, D = 1.4.

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Concluding Remark

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http://www.math.hkbu.edu.hk/~ttang/MMmovie

Collaborators:Collaborators:Yana Di, Chinese Academy of Sciences

Ruo Li, Pingwen Zhang of Peking University;

Heyu Wang, Hong Kong Baptist University

Thanks

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Oran & Boris (Numerical Simulation of Reactive Flow, 2nd ed., Cambridge University Press, 2001; p.183):

Adaptive griding techniques fall into two broad classes, adaptive adaptive mesh redistribution and adaptive mesh refinementmesh redistribution and adaptive mesh refinement, both contained in the acronym AMR. Techniques for adaptive mesh redistribution continuously Techniques for adaptive mesh redistribution continuously reposition a fixed number of cells, and so they reposition a fixed number of cells, and so they improve the improve the resolution in particular locationsresolution in particular locations of the computational domain of the computational domain. Because of the great potential of AMR for Because of the great potential of AMR for reducing reducing computational costs without reducing the overall level of computational costs without reducing the overall level of accuracy,accuracy, it is a forefront area in scientific computation. it is a forefront area in scientific computation.

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Example: Reactive flow calculations on moving meshes

In our computations, the flow parameters are updated in whole without splitting the system into a hydrodynamical part and an ODE part. The algorithm is based on the Godunov’s scheme on deformed meshes with some modification to increase the scheme order in time and space.

The governing system of the differential equations relating to 2D reactive gas flow is

energy. totalthe)](5.0[,components velocity theandwith

))(,0,0,0,0(,)),(,,,(

,)),(,,,(,),,,,(

where

,

022

2

2

ZqvueEvu

,ZTKvZpEvpvuvv

uZpEuuvpuuZEvu

ttt

TT

TT

cb

a

cba

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Figure 11: Reactive flow computations: pressure contours at t=61.59 computed on the uniform (a) and adapted (b) mesh. Close-up for the uniform (c) and adapted (d) mesh.

(a) (b)

(c)

(d)

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ALGORITHM 1: Mesh-redistribution algorithm

(a): solve the optimization problem (3) and compute the L∞-difference between the solution of (3) and the fixed (initial) mesh in the logical domain. If the difference is not small, then do

(b): obtain the direction and the magnitude of the movement for by using the difference obtained in part (a),

(4)

and then move the mesh using X* = X + τδX (5)

where τ is a parameter in [0,1] and is used to prevent mesh tangling. (c): update on the new grid by solving a system of ODEs; (d): update the monitor function by using obtained in part (c), and go to

part (a).

u

u

x

i

i

TE

TEi

Ei E

Ax

E

X

in

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Solution-updating

Each element of with X as its nodes corresponds uniquely to an element of *(τ) with X + τδX as its nodes. The surface of on Ω will not move, though the nodes of the mesh may be moved to new locations. Then , as the function of at time tn, is independent of the parameter τ. That is

(7)

is expressed as

(8)

where is the basis function of the finite element space at its node Xi+τδXi. Using (7) and (8) gives

(9)

We obtain a system of (linear) ODEs for Ui:

(10)

.0u

),()( xUu ii

.0),(

xuxU

xii

.1).()( NjUxdxx

Uxd i

jkk

iiji

),( xi

:),( xuu

u

u

x

u

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Numerical validation in 2D

Figure: Typical shapes of the dendritic tips on domain [-800,800] [-800,800], from the left to the right are for = 0.45 , 0.55 and 0.65, respectively. The contours of = 0 are shown on the top and the corresponding meshes for lie in the bottom.

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Numerical validation in 2D: High undercooling

Figure: The time evolution of the tip-velocity for high undercoolings = 0.45 , 0.55 , 0.65. The data for case = 0.55 and 0.65 have been shifted up by 0.025 for clarity.

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Numerical validation in 2D: Low undercooling

Figure: The time evolution of the tip-velocity for low undercoolings = 0.25 , 0.3. The data are shifted up by 0.01 for case = 0.25 and 0.025 for case = 0.3.

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3D simulations

Figure: The formatted pattern when = 0.55, D = 1 at three different time steps.

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3D simulations

Figure: The formatted patterns for = 0.45, D = 1.4 at three different time steps.

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3D simulations

Figure: A typical very complex patterns with parameters as = 0.45, D = 3.5.

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Double Shear Flow

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Some applications:

The above numerical procedure was proposed in Li, Tang & Zhang (JCP, 01, 02)

To solve the incompressible Navier-Stokes equations (Di-Li-Tang-Zhang, SISC, 05)

To solve incompressible interface problems (Li and Tang, JSC, 06)

To solve the elliptic optimal control problems (Li-Liu-Tang, SIAM Control Optim, 02).

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Merging of two bubbles in moving 80 80 mesh. The density ratio between the bubbles and the background is 1: 10, and l = 0.0005, g = 0.00025, = 0. t = 0.1, 0.3, 0.4, 0.5.

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Azarenok & Tang (JCP, 2005)

Reactive flow computations: adapted mesh (a) and close-up (b) at t=61.59.

(a) (b)

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Nonaxisymmetric merging of two viscous gas bubbles: the interface in a slice (y = 1, n = (0, 1, 0)) obtained by the uniform tetrahedral mesh with 64 64 128 cells (red) and the moving tetrahedral mesh with 32 32 64 cells (green). t = 2.0 (left) and 3.0 (right).