Computational Fluid Dynamics CFD

Solving system of equations, Grid generation

2

Basic steps of CFD

Problem

?

•Gov. Eq. •BC •Init. Cond.

...,, ++ jijti uu

•Discretization •Result

•Solution •OK?

Solving system of equations

P Domain of dependence Region of influence

Region of influence Domain of dependence

P

P

Every point influences all other points

Parabolic

Hyperbolic

Elliptic

The type of equations decides solution strategy

Marching problems Equilibrium problems

Solving system of equations

Parabolic Hyperbolic Elliptic

Marching methods may be used since the solution only depends on previous data.

Has to be solved for the whole domain simultaneously, since all points depend on each other. Relaxation techniques.

Examples:

•Inviscid supersonic flow

Examples:

•Steady incompressible flow

Note! Time dependent incompressible flow has a mixed character: elliptic in space and parabolic in time.

Marching methods

1st order

Consider the inviscid Burger equation 0=∂∂

+∂∂

xuu

tu

Conserved form )( 0 uFFxF

tu

==∂∂

+∂∂

Start with a Taylor expansion around (x,t+∆t)

( ) ( ) HOT,,,

+

∂∂

∆+=∆+txt

uttxuttxuu1,1 ui,1 ui+1,1 ui-1,1 uN,1

u1,n+1 ui,n+1 ui+1,n+1 ui-1,n+1 uN,n+1

u1,n ui,n ui+1,n ui-1,n uN,n

∆t

∆x

Marching methods

1st order

The idea is to replace the time derivatives in the expansion by spacial ones

First derivative: xF

tu

∂∂

−=∂∂

Apply 2nd order central differencing:

2111

ni

nin

ini

FFxtuu −++ −

∆∆

−=

u1,1 ui,1 ui+1,1 ui-1,1 uN,1

u1,n+1 ui,n+1 ui+1,n+1 ui-1,n+1 uN,n+1

u1,n ui,n ui+1,n ui-1,n uN,n

∆t

∆x

Marching methods

Lax-Wendroff scheme

Consider the inviscid Burger equation 0=∂∂

+∂∂

xuu

tu

Conserved form )( 0 uFFxF

tu

==∂∂

+∂∂

Start with a Taylor expansion around (x,t+∆t)

( ) ( ) HOT2

,,,

2

22

,+

∂∂∆

+

∂∂

∆+=∆+txtx t

uttuttxuttxu

Marching methods

Lax-Wendroff scheme

The idea is to replace the time derivatives in the expansion by spacial ones, which gives a scheme that is 2nd order accurate in space and time.

First derivative: xF

tu

∂∂

−=∂∂

∂∂

∂∂

−=∂∂

∂−=

∂∂

tF

xxtF

tu 2

2

2Second derivative:

Since F is a function of u we can write

xuA

xu

uF

xF

tu

∂∂

−=∂∂

∂∂

−=∂∂

−=∂∂

Jacobian

tuA

tu

uF

tF

∂∂

=∂∂

∂∂

=∂∂

Marching methods

Lax-Wendroff scheme

Hence,

∂∂

∂∂

=∂∂

xFA

xtu2

2

The Taylor expansion can now be written as:

( ) ( ) HOT2

,,,

2

,+

∂∂

∂∂∆

+

∂∂

∆−=∆+txtx x

FAx

txFttxuttxu

Marching methods

Lax-Wendroff scheme

Apply 2nd order central differencing:

( ) ( )[ ]nj

nj

nj

nj

nj

nj

nj

njn

jnj FFAFFA

xtFF

xtuu 12/112/1

2111

21

2 −−++−++ −+−

∆∆

+−

∆∆

−=

Since uuFAuF =

∂∂

=⇒=2

2the Jacobian is calculated as

21

2/1jj

juu

A+

= ++

A stability analysis gives ( ) θθ sin2cos1212

AxtiA

xtG

∆∆

−−

∆∆

−=

Stable if 1≤∆∆xtu the CFL-condition

Marching methods

MacCormack scheme

This is a two step version of the L-W with the advantage that no Jacobians are needed. Otherwise it has identical properties to the L-W

( )

( )

−

∆∆

−+=

−∆∆

−=

+−

+++

++

11

111

11

21 n

jnj

nj

nj

nj

nj

nj

nj

nj

FFxtuuu

FFxtuu Predictor

Corrector

Solving system of equations

P Domain of dependence Region of influence

Region of influence Domain of dependence

P

P

Every point influences all other points

Parabolic

Hyperbolic

Elliptic

Marching problems Equilibrium problems

13

Relaxation techniques

SxT

=∂∂

2

2ni

ni

ni

ni SxO

xTTT

=∆+∆

+− +−

+++ )(2 2

2

11

111

T1,n+1 Ti,n+1 Ti+1,n+1 Ti-1,n+1 TN,n+1

T1,n Ti,n Ti+1,n Ti-1,n TN,n

∆t

ni

ni

ni

ni SxTTT 21

111

1 2 ∆=+− +−

+++

nnnn SxTTT 221

11

21

3 2 ∆=+− +++

nnnn SxTTT 321

21

31

4 2 ∆=+− +++

nnnn SxTTT 421

31

41

5 2 ∆=+− +++

Relaxation techniques Basic techniques for solving a system of equations

bAx =System of equations

=

NNNNN

N

b

bb

x

xx

aa

aaaa

2

1

2

1

1

21

11211

Direct methods •Cramer •Gauss elimination

•Heavy •Error accumulation

•Thomas algorithm •Tri-diagonal systems

Iterative methods

Thomas algorithm

=

−

NNNN

N

c

cc

x

xx

dba

badb

ad

2

1

2

1

1

3

222

11

0

0

00

11

11

−−

−−

−=

−=

jj

jjj

jj

jjj

cdb

cc

adb

ddPut on upper triangular form:

Unknowns computed using back-substitution:

11

−−

−=

=

j

jjjj

N

NN

axdc

x

dcx

Nj ,.......3,2=

1.........2,1 −−= NNj

Jacobi

Easy but slow

bAx = ∑=

=N

jijij bxa

1

ii

ij

kjiji

ki a

xab

x∑

≠

−−

=

1

In interation step k:

ni

ni

ni

ni SxTTT 21

111

1 2 ∆=+− +−

+++

2

2111

ni

ki

kik

iSxTTT ∆−+

= −++

T1,k+1 Ti,k+1 Ti+1,k+1 Ti-1,k+1 TN,k+1

T1,k Ti,k Ti+1,k Ti-1,k TN,k

∆t

Gauss-Seidel bAx = ∑

=

=N

jijij bxa

1

ii

ij

kjij

ij

kjiji

ki a

xaxab

x∑∑

>

−

<

−−

=

1

In interation step k:

Always uses the best value available, gives faster solution

ni

ni

ni

ni SxTTT 21

111

1 2 ∆=+− +−

+++

2

21111

ni

ki

kik

iSxTTT ∆−+

=+

−++

T1,k+1 Ti,k+1 Ti+1,k+1 Ti-1,k+1 TN,k+1

T1,k Ti,k Ti+1,k Ti-1,k TN,k

∆t

18

Successive Over-Relaxation (SOR) • Accelerate convergence

• ω > 1 overrelaxation • ω < 1 underrelaxation

(for stability)

k

T Texact

∆T

k+1

Tk

Tk+1

k

T Texact

ω∆T

k+1

T*k

T*k+1

)( 11 ki

ki

ki

ki TTTT −+= ++ ω

2

21111

ni

ki

kik

iSxTTT ∆−+

=+

−++

2221

11

1

ki

ni

ki

ki

ki

ki

TSxTTTT

−∆−+

+=+

−+

+

ω

Residuals When should we stop the iterations?

mji

mji RLu ,, =

Iteration Nr.

R

Jacobi

Gauss-Seidel SOR

Relaxation techniques Point relaxation

y

x i i+1 i-1

j

j+1

j-1

02

2

2

2=

∂∂

+∂∂

yxφφ

022

21,1,

2,1,1 =

∆

+−+

∆

+− −+−+

yxjiijjijiijji φφφφφφ

( )( )2

11,1,

21,1,11

12 βφφβφφ

φ−

+++=

+−+

+−++

kji

kji

kji

kjik

ij

Example: Potential flow

Gauss-Seidel, point relaxation:

22

∆∆

=yxβ

Relaxation techniques Line relaxation

y

x i i+1 i-1

j

j+1

j-1

( )( )2

11,1,

21,1

1,11

12 βφφβφφ

φ−

+++=

+−+

+−

+++

kji

kji

kji

kjik

ij

Gauss-Seidel, line relaxation in x:

In line relaxation a whole line is solved at once using a direct method, for example the Thomas algoritm.

Relaxation techniques ADI, alternating direction implicit

y

x i i+1 i-1

j

j+1

j-1

( )( )2

2/11,1,

22/1,1

2/1,12/1

12 βφφβφφ

φ−

+++=

+−+

+−

+++

kji

kji

kji

kjik

ij

Gauss-Seidel, ADI

Further improvement of numerical convergence speed. Computational time can be reduced with up to 20-40 % as compared to Gauss-Seidel with SOR

( )( )2

11,

11,

21,1

2/1,11

12 βφφβφφ

φ−

+++=

+−

++

+−

+++

kji

kji

kji

kjik

ij

First along x-direction

then along y-direction

Multigrid methods

• Accelerate convergence

Multigrid methods

• Accelerate convergence

Multigrid methods

• Accelerate convergence

Multigrid methods

• Accelerate convergence

Multigrid methods Multigrid methods are used to increase the computational efficiency of an implicit method

Consider the equation: ( )xfdx

ud=− 2

2

10 ≤≤ x

Periodic boundary conditions

Create a grid: jhx j = 120 +≤≤ njn

h21

=

Discretise j

jjj fh

uuu=

−+− +−

211 2

Gauss-Seidel jmj

mj

mj fhuuu 21

11 2 +=+− −+−

nj 21 ≤≤

nj 21 ≤≤

( ) ( )01 uu =

Multigrid methods von Neumann stability analysis

Use the numerical error *m

jmj

mj uu −=ξ

111 2 −

+− =+− mj

mj

mj ξξξto rewrite the equation

αθ

ααξ ij

nmm

j ec∑−

=

=12

0

Fourier modes of the error:

hn

παπαθα 2==

( ) 1lim 10=

→θG

h

What does this tell us?

Amplification factor

Remember:

))sin()(cos( bibee aiba +=+

Multigrid methods ( ) 1lim 10

=→

θGh

Short wavelength (high frequency) errors damps faster

Create grids with different resolutions

Low frequency errors on fine grids are high(er) frequency errors on coarser grids (damps faster when relaxed on coarse

grids)

Multigrid methods Example of a linear problem, the Laplace equation

02

2

2

2

=∂∂

+∂∂

yu

xu

21,,1,

2,1,,1

,

22y

uuux

uuuLu jijijijijiji

ji ∆+−

+∆

+−= +−+−

On each grid, m, we solve: mji

mji RLu ,, =

Procedure for the Correction Storage (CS) scheme:

1. On the finest grid, M, do a few relaxations (iterations) of to reduce the short wave length error modes.

0, =MjiLu

2. Calculate the residual and transfer it to the next coarser grid, restriction: M

jiMM

Mji RIR ,

11,

−− =

Residual

Multigrid methods Example of a linear problem, the Laplace equation

3. On the coarser grid solve

6. Transfer the correction back to finer grid, prolongation, and do a few relaxations on each grid until the finest grid is reached

0,, =+∆ mji

mji RuL

mji

mji

mji uuu ,,, ˆ −=∆

4. Repeat steps 2 and 3 until the coarsest grid is reached

5. On the coarsest grid, solve the problem exactly.

correction Previous solution on grid m

mji

mm

mji

mji uIuu ,

11,

1,ˆ ∆+= +++

Multigrid methods Multigrid cycles

V-cycle:

= relaxation

restriction prolongation

m=M

m=1

Geometric multigrid

• Several grids explicitly generated

• Suitable for structured grids

• Several type of cycles: – V, W, ...

33

Algebraic multigrid

• Coarser levels built ’on-line’

• Can be used for unstructured meshes

• Mostly for elliptic problems

• Too many/coarse levels not neccessarily help

34

Discretisation and grid

Questions:

•How complex is the geometry?

•What accuracy is required? Grid quality?

•What about stability?

•Grid refinement?

Grid generation-classification

36

Structured

Number of blocks

Mono Multi

Hierarchy

Cartesian

Shape

H C O

Orthogonality

Orthogonal

Body-fitted

Oct-tree

Unstructured

Tetra-hedral

Poly-hedral

Cell types

2-dimensional 3-dimensional

Triangular (tri)

Quadrilateral (quad)

Tetrahedron (tet)

Hexahedron (hex)

pyramid wedge

Grid types Structured grid Unstructured grid

Unstructured grid

Grid types Structured grid Multi-block

Grid types

Hybrid grid

Grid types Hybrid grid

42

Structured

• Easy to generate • Inbuilt topology

• Low memory footprint

• Fast solution algorythms

• Easy to use high-order schemes

• Difficult for complex geometries

Unstructured

• Difficult to generate • Topology has to be

stored • Lower mesh quality • Slow algorythms • Difficult for high-

order • Suitable for

complex geometries

Mesh quality measures

43

• Stretching • dy2/dy1 • Best: 1

• Aspect ratio • dy1/dx • Best: 1

• Skewness �α • Best: equi-angle

α

dx

dy1

dy2

Efficient grids – grid stretching

44

Large gradients

Fine resolution

Efficient grids – grid stretching

45

i

i+1

i-1

...62 3

33

2

22

1iii

ii xTx

xTx

xTxTT

∂∂∆

−

∂∂∆

+

∂∂

∆−=−

...6

)(2

)(3

33

2

22

1iii

ii xTxr

xTxr

xTxrTT

∂∂∆

+

∂∂∆

+

∂∂

∆+=+

)(2

)1()1(

22

211 xO

xTxr

xrTT

xT

i

ii ∆+

∂∂∆−

+∆+

−=

∂∂ −+

• Not second order any more!

• Can be corrected • Extra work

Grid stretching

Grid stretching

Efficient grids – local refinement

48

Efficient grids – adaptive refinement

49

• Right marker parameter • P, u, T, dt/dx, etc.

• Not too often • Comp. Time • Errors introduced

• Not too late • Follow the flow

features

Refinement techniques

50

• Easier to implement • Mesh quality

decreases

• Neighbours affected • Better quality

Other grid related issues

• Grid generation strategies • Moving boundaries