N C H C96 / 12 / 20

T.-I TsengT.-I Tseng

National Center for High-performance ComputingNational Center for High-performance Computing

The Development and Application of the Space-Time Conservation Element and Solution Element Method

Institute of Physics, NCTUDec 20, 2007

22

Background of the Space-Time CE/SE Method

33

Transport Theory

Mass conservation in continuum mechanics

Transport theory 0

0

0

)(

)(

u

su

t

ddVt

dVdt

d

V VS

tV

x

y

z

44

The Finite Volume Method

The rate of change of conserved properties in a finite volume (FV) is equal to its flux across cell boundaries

The FV methods focus on calculating spatial flux (a temporal evolving spatial flux).

V VS

t

t

t

t

V VS

ddtdV

ddVt

)(

)(

2

1

2

1su

su

x

t

FV

55

The Space-Time CE/SE Method

The convection equation is a space-time divergence free condition

Let x1 = x and x2 = t be the coordinates of a 2D Euclidean space E2

Using Gauss’ divergence theorem, one obtains the space-time flux balance equation

)(

21

0

0

where,0

0)(

Rs

R

sdh

dVh

uhh

xx

u

r + d r

r

d r

d s

R

S ( R )

t

x

66

The Space-Time CE/SE Method

The Euler equations are three space-time divergence free conditions

Using Gauss’ divergence theorem, one obtains the space-time flux balance equation

0

0

),(

3,2,1,0

)(

0

)(

2

3

2

1

3

2

1

12

sd

dR

uf

m

upe

pu

u

f

f

f

e

u

u

u

u

xx

RS

m

R

m

Tmmm

m

h

h

h

h

FU

FU

77

Space-Time CE/SE Method

A staggered space-time mesh

the discretization step Space time region is divide into n

on- overlapping Conservation Elements (CEs) and Solution Elements (SEs).

88

The Solution Element

Flow properties are assumed continuous inside each SE.

The 1st order Taylor series expansion is used inside a SE

Inside a SE,

U and Ux are the unknowns to be solved; all other properties can be expressed by them.

(j,n)

(j-1/2,n-1/2) (j+1/2,n-1/2)

)()()()(),;,(

)()()()(),;,(*

*

nnjtj

njx

nj

nnjtj

njx

nj

ttxxnjtx

ttxxnjtx

FFFF

UUUU

njx

nj

nj

njt

nj

njt

njx

nj

njx

njt

)()()(

)()()(

UAAUAF

UAFU

(j,n)

Solution element; SE(j, n)

99

The Conservation Element

The space-time region is divided into non-overlapping CEs

The space-time flux conservation is imposed over CE- and CE+

For one conservation equation, CE- and CE+ provide two conditions.

For the 1D Euler equations, CE- and CE+ provide 6 conditions for the 6 components of U and Ux at point (j, n) CE- CE+

(j,n)

(j-1/2,n-1/2) (j+1/2,n-1/2)

0)(

sdVS

m

h

x

t

B A F

C D E

(j,n)

(j-1/2,n-1/2) (j+1/2,n-1/2)

(j-1/2,n-1/2)

(j,n)

B A

C D

(j,n)

(j+1/2,n-1/2)

A F

D E

Conservation element; CE(j, n)

Basic Conservation element; BCE(j, n)

CE-CE+

3,2,1],)()()()[(2

1)( 2/1

2/12/12/1

2/12/1

2/12/1

mssuuu njm

njm

njm

njm

njm

3,2,1,)(4

)()(4

)( 2/12/1

22/12/1

2/12/1

2/12/1

mf

x

tf

x

tu

xs n

jmtnjm

njmx

njm

1010

2D Space-Time Mesh

Triangular unstructured mesh Quad cylinders for CEs 3 CEs between A, E, C and G’, for the 3 unknown

s U, Ux and Uy.

G

AB

C

DE

F

x

y

x

y

t

A

CE

G

G’

Time marching 0

)(

sdVS

m

h

1111

The CE and SE in 2D

G'

F' n-1/2

A

B

C

D E

F

A'

B'

D' E'

G

C'

n

tCE(1)

CE(2)

CE(3)

G''

B''

D''

F''

D'

G'

F'

A

C

D E

F

n-1/2

n

n+1/2

t

B'

G

B

Three CEs: Quadrilateral cylinder EFGDEFGD(1) CDGBCDGB(2), and ABGFABGF ( 3 ).

One SE: Four planes ABCDEF + GGBB + GGDD + GGFF + their immediate neighborhood.

1212

3D Space-Time Mesh

Tetrahedrons are used as the basic shape

Every mesh node has 4 neighboring nodes

The projection of a space-time CE on the 3D space is a 6-surface polygons

Flux conservation over 4 CEs determine the 4 unknowns: U, Ux, Uy, and Uz

x

z

y

1313

Special Features of the CE/SE Method

Space and time are unified and treated as a single entity. Separation of conservation element and solution element. No flux function or characteristics-based techniques and n

o reconstruction step. Numerical dissipation doesn’t overwhelm physical dissipat

ion Use the simplest mesh stencil -- triangles for 2D and tetrah

edrons for 3D. 1,2, and 3 D Euler/NS codes for structured/unstructured m

eshes running on serial and parallel platforms. Many application in aero acoustics and combustion.

1414

A Space-Time CE/SE Method with Moving Mesh Scheme for One-Dimensional Hyperbolic Conservation Laws

1515

Motivating idea

A suitable computational grid is important for solving the hyperbolic systems.

Major challenge of numerical scheme is to capture the discontinuous solution with sufficient accuracy.

The characteristic of the discontinuous is non-stationary and consequently the fixed uniform grid may not be the best suited.

The idea of an adaptive grid is to add, remove, or move the grid concentrated to enhance accurate and achieve efficiency.

1616

Adaptive Mesh

Adaptive Mesh Refinement (AMF):

automatic refinement or coarsening of the spatial mesh.

Adaptive Mesh Redistribution (AMF):

relocates the grid points with a fixed number of nodes.

it’s also known as moving mesh method (MMM).

Key ingredients of the moving mesh method include: Mesh equation Monitor function Interpolations

n

n’

n+1/2

Interpolation Free MMM :

Interpolation of dependent variables from the old mesh to the new mesh is unnecessary.

1717

Mesh Equation

)(),.....,,(

)(),.....,,(

21

21

x

xxxxx

d

d

1,.....,1,0,)),((1

)),((1

JMJdxtxu

JMdxtxu

R

L

j

j

x

x

x

x

The logical and physical coordinates:

Equidistribution principles

)(),(),(0

),(

0tdxtxdxtx

Mxtx

0)( x

x :

:

(Quasi-Static equidistribution principles;QSEPs)

1818

Monitor Function

22/1

22/1 )(1 jxj u

)()(

12/1

1

jj

x

x x

jx xx

dxuu

j

j

pj

pjk

jk

pj

pjk

jkk

r

r

r

r

tx)

1(

)1

(

),(~

Scaled solution arc-length

where is a scaling parameter and the cell average of the solution gradient over the interval [xj, xj+1]

= 0, uniform mesh

>> 1, adapted grid

Smoothing the monitor function

1919

MMCESE

x j-1/2n+1/2

wj-1/2

x nx j-1n+1/2 x n+1/2

j+1/2

wj+1/2

x jnx n

j-1 x nj+1

x jn+1/2 x n+1/2

j+1

FAB

C

D

E

Moving mesh strategy:

use the Gauss-Seidel iteration to solve the mesh equation

0)(~)(~12/112/1

vj

vjj

vj

vjj xxxx

Moving mesh CE/SE (MMCESE)method

2020

MMCESE Algorithm

Step 1: Given a uniform partition of logical and physical domains, then specified the initial conditions.

Step 2 : Calculate the monitor function. Step 3 : Move the grid point by mesh equation. Step 4 : Evolve the underlying PDEs by CE/SE method on the new me

sh system to obtain the flow variables at new time level. Step 5 : If tn+1 < T, go to Step 2.

2121

Burger Equation

0)2

1( 2 xt uu

Initial condition :

)sin(2

1)2sin()0,( xxxu

X

U

0 0.5 1 1.5 2-1.5

-1

-0.5

0

0.5

1

1.5

2

X

Tim

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

t = 0.85 Sec.t = 0.35 Sec.

XU

0 0.5 1 1.5 2-1.5

-1

-0.5

0

0.5

1

1.5

2

t = 0.85 Sec.t = 0.35 Sec.

X

Tim

e

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

XU

0 0.5 1 1.5 2-1.5

-1

-0.5

0

0.5

1

1.5

Exact solution

Original CESE solution

MMCESE solution

Monitor function :2

2/12

2/1 )(1 jxj u

2222

Sod Problem

X-0.5 -0.25 0 0.25 0.5

0

0.3

0.6

0.9

1.2

Fig. 3. Solutions of Sod problem at t = 0.2. (a) trajectory of mesh. (b) flow variables distribution(dots and solid lines denote the moving and fixed grid solutions).

X

Tim

e

-0.5 -0.25 0 0.25 0.50

0.05

0.1

0.15

0.2

(a)

density

velocity

pressure

X-0.5 -0.25 0 0.25 0.5

0

0.3

0.6

0.9

1.2

Fig. 3. Solutions of Sod problem at t = 0.2. (a) trajectory of mesh. (b) flow variables distribution(dots and solid lines denote the moving and fixed grid solutions).

density

velocity

pressure

X

Tim

e

-0.5 -0.25 0 0.25 0.50

0.05

0.1

0.15

0.2

X-0.4 -0.2 0 0.2 0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

X

De

nsi

ty

-0.4 -0.2 0 0.2 0.4-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0.1 0.15 0.2 0.25 0.3 0.35 0.40.1

0.2

0.3

0.4

0.5

X

D

0.1 0.15 0.2 0.25 0.3 0.35 0.40.1

0.2

0.3

0.4

0.5

X

D

0.1 0.15 0.2 0.25 0.3 0.35 0.40.1

0.2

0.3

0.4

0.5

Analytical solution

MMCESE solution

Original CE/SE solution

H

H

P

L

L

P

2max

2 ])/()[(1 xx

0

0

,1.0,0,125.0,,

,0.1,0,0.1,,

x

x

pv

pv

Initial conditions :

Monitor function :

2323

Piston Problem

X

Tim

e

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

X

Tim

e

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

(a)

Piston Locus

Simplecompression waves

Tail

HeadX

De

nsi

ty

0 1 2 3 4 5

1

1.1

1.2

1.3

1.4

t = 0.02

t = 0.14t = 0.06 t = 0.1

0)(~)(~12/112/1

vj

vjj

vj

vjj xxxx

t

brJM

t

bl dtvxJMxdtvxx0

0

0

00 )(,)0(

Moving Boundary

2424

Conclusion

MMCESE not only maintains the essential features of the original CE/SE method but also clusters the mesh space at the locations where large variation in physical quantities exists.

Current approach can be extend to moving boundary problem easily.

MMCESE is an interpolation free MMM. Computation accuracy and efficiency can be improved by

this approach.

Tseng, T.I, and Yang R.J.. “A Space-Time Conservation Element and Solution Element Method with Moving Mesh Scheme for One-Dimensional Hyperbolic Conservation Laws,” the 6th Asian Computational Fluid Dynamics Conference, 2005.

2525

Applications in Shallow Water Equations

2626

Shallow Water Equations

Depth averaging of the free surface flow equations under the shallow-water hypothesis leads to a common version of the shallow-water equations (SWEs)

,)(

0,

2/,

022

fmmm ssgh

Sghhv

hvf

hv

hu

where s0 and sf are bed slop and friction slop, respectively.

The friction slop is determined by the Manning formula

n is the Manning roughness coefficient.

Zs 0

3/10

2 )(

h

huhuns f

2,1,

mSx

f

t

um

mm

2727

CE/SE method with Source terms

By using Gauss’ divergence theorem in space-time region, the SWEs can be written in integral form

)(

)(VS V mmm dVuSdsh

For any point belong the solution element njmt

nnjmxj

njmm uttuxxunjtxu ))(())(()(),;,(

njmt

nnjmxj

njmm fttfxxfnjtxf ))(())(()(),;,(

From SWEs, one can get))(()()( n

jmmnjmx

njmt uSfu

As a result, there are two independent marching variables and associated with in each solution elements. Furthermore

njmu )(

njmxu )(

)),;,(),,;,((),;,( njtxunjtxfnjtxh mm

2828

CE/SE method with Source terms

2,1],)()()()[(2

1))((

2)( 2/1

2/12/12/1

2/12/1

2/12/1

mssuuuSt

u njm

njm

njm

njm

njmm

njm

2,1,)(4

)()(4

)( 2/12/1

22/12/1

2/12/1

2/12/1

mf

x

tf

x

tu

xs n

jmtnjm

njmx

njm

We employ local space-time flux balance over conservation element to solve the unknowns, i.e.,

)),(( ),(

)(njCES njCE mmm dVuSdsh

because the boundary of CE(j, n) is a subset of the union of SE(j, n), SE(j-1/2, t-1/2), and SE(j+1/2, t-1/2), the conservation laws imply that

The space derivatives of flow variables are evaluated using the α scheme.

Yu, S.T., and Chang, S.C. “Treatments of Stiff Source Terms in Conservation Law by the Method of Space-Time Conservation Element and Solution Element,”, AIAA Paper 97-0435,1997

B A F

C D E

(j,n)

(j-1/2,n-1/2) (j+1/2,n-1/2)

2929

1D Dam-Break Problem with Finite Downstream Water Depth

Two different initial water depths are assigned to the upstream and downstream parts of a horizontal, frictionless, infinitely wide rectangular channel including a dam. The upstream water depth is 100 m and the downstream one is 1 m. Spatial domain is 2000 m length and it is discretized with 200 elements.

X

H

-1000 -500 0 500 1000

0

20

40

60

80

100

Analytica l solu tionN um erica l result

x

tCCFL

3030

Propagation and Reflection

It is assumed that a dam located in the middle of a 1000 m long channel separates reservoir with depth hr and tailwater depth ht. At the ends of the channel are the walls th

at the wave cannot surmount. The boundary conditions with free-slip and zero discharge are satisfied at the ends of the channel. At time t = 30 s and t = 75 s after dame break, the results with different ratios of water depth (R = 0.15 and 0.001)

X

H

0 200 400 600 800 10000

2

4

6

8

10 T = 30 sec .T = 75 sec .

X

H

0 200 400 600 800 1000

0

2

4

6

8

10

T = 3 0 sec.T = 7 5 sec.

3131

Propagation and Reflection

It is assumed that a dam located in the middle of a 1000 m long channel separates reservoir with depth hr and tailwater depth ht. Comparisons the numerical results with an

d without the bed friction at time t = 30 s after dame with bed slop and different ratios of water depth (R = 0.15 and 0.001).

X

H

0 200 400 600 800 10000

2

4

6

8

10

12n = 0.03, S0 = 0.003n = 0, S0 = 0.003

X

H

0 200 400 600 800 10000

2

4

6

8

10

12n = 0.03, S0 = 0.003n = 0, S0 = 0.003

3232

Interaction of 1D Bore Waves

It is supposed that in two 1000 m long channels there are two dams located at 300 m and 600 m, respectively. Each channel is divided into three parts by both dames. The initial clam water depths are h01, h02, and h03.

(1) h01 = 20 m, h02 = 3 m, and h03 = 10 m;

(2) h01 = 20 m, h02 = 10 m, and h03 = 2 m;

X

H

0 200 400 600 800 10000

5

10

15

20

25

T = 1 0 sec .T = 3 0 sec .

X

H

0 200 400 600 800 10000

5

10

15

20

25

T = 8 sec .T = 2 4 sec.

3333

Steady flow over a bump with hydraulic jump

A steady-state transcritical flow over a bump, with a smooth transition followed by a hydraulic jump is simulated. The channel is infinitely large, horizontal, frictionless, 25 m long.

m.,t)h(

/sm.,t)q(

3300

1800 3

otherwise

xxxz

,0

128,)10(05.02.0)(

2

X

H,

Z

5 10 15 20 250

0 .1

0 .2

0 .3

0 .4 HBed le ve l

3434

2D Dam-Break Problem

X

Y

0 50 100 150 2000

50

100

150

200

X

0

50

100

150

200

Y

0

50

100

150

200

D

4

6

8

10

X

0

50

100

150

200

Y

0

50

100

150

200

D

-2

0

2

4

6

8

10

12

A square box of 200╳200 m2 with a horizontal bed is divided into two equal compartments. The initial still water depth is 10 m on one side and 5m and 0.01m on the other side of the dividing wall for the wet bed and dry test cases, respectively. The breach is 75 m in length, and the dame is 15 m in thickness.

Tseng, T.I, and Yang R.J.. “Solution of Shallow Water Equations Using Space-Time Conservation Element and Solution Element Method,” the 14th National Computational Fluid Dynamics Conference, 2007.

3535

3636

Acknowledgement

• Dr. S.-C., Chang NASA Gleen Research Center

• Prof. S.-T., Yu Ohio State University

• Dr. C.-L., Chang NASA Langley Research Center

• Prof. R.-J., Yang NCKU

• Dr. Z.-C., Zhang Livermore Software Technology Co.

• Prof. W.-Y., Sun NCTFR

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3737

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