Upload
dinhdang
View
216
Download
4
Embed Size (px)
Citation preview
Computational Fluid Dynamics
CFD
Solving system of equations
2
Basic steps of CFD
Problem
?
•Gov. Eq.
•BC
•Init. Cond.
...,, jijti uu
•Discretization •Result
•Solution•OK?
Solving system of equations
PDomain of
dependenceRegion of influence
Region of influenceDomain 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
x
uu
t
u
Conserved form )( 0 uFFx
F
t
u
Start with a Taylor expansion around (x,t)
HOT,,,
txt
uttxuttxu
u1,1 ui,1ui+1,1ui-1,1 uN,1
u1,n+1 ui,n+1ui+1,n+1ui-1,n+1 uN,n+1
u1,n ui,nui+1,nui-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:
x
F
t
u
Apply 2nd order central differencing:
2
111n
i
n
in
i
n
i
FF
x
tuu
u1,1 ui,1ui+1,1ui-1,1 uN,1
u1,n+1 ui,n+1ui+1,n+1ui-1,n+1 uN,n+1
u1,n ui,nui+1,nui-1,n uN,n
t
x
Marching methods
Lax-Wendroff scheme
Consider the inviscid Burger equation 0
x
uu
t
u
Conserved form )( 0 uFFx
F
t
u
Start with a Taylor expansion around (x,t+t)
HOT2
,,
,
2
22
,
txtx t
ut
t
uttxuttxu
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:
x
F
t
u
t
F
xxt
F
t
u 2
2
2
Second derivative:
Since F is a function of u we can write
x
uA
x
u
u
F
x
F
t
u
Jacobian
t
uA
t
u
u
F
t
F
Marching methods
Lax-Wendroff scheme
Hence,
x
FA
xt
u2
2
The Taylor expansion can now be written as:
HOT2
,,,
2
,
txtx x
FA
x
t
x
Fttxuttxu
Marching methods
Lax-Wendroff scheme
Apply 2nd order central differencing:
nj
nj
nj
nj
nj
nj
nj
njn
jnj FFAFFA
x
tFF
x
tuu 12/112/1
2111
2
1
2
Since uu
FA
uF
2
2
the Jacobian is calculated as2
12/1
jjj
uuA
A stability analysis gives sin2cos121
2
Ax
tiA
x
tG
Stable if 1
x
tuthe 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
2
1 nj
nj
nj
nj
nj
nj
nj
nj
nj
FFx
tuuu
FFx
tuu Predictor
Corrector
Solving system of equations
PDomain of
dependenceRegion of influence
Region of influenceDomain of
dependence
P
P
Every point influences all other points
Parabolic
Hyperbolic
Elliptic
Marching problems Equilibrium problems
13
Relaxation techniques
Sx
T
2
2
n
i
n
i
n
i
n
i SxOx
TTT
)(2 2
2
1
1
11
1
T1,k+1 Ti,k+1Ti+1,k+1Ti-1,k+1 TN,k+1
T1,k Ti,kTi+1,kTi-1,k TN,k
n
i
n
i
n
i
n
i SxTTT 21
1
11
1 2
nnnn SxTTT 2
21
1
1
2
1
3 2
nnnn SxTTT 3
21
2
1
3
1
4 2
nnnn SxTTT 4
21
3
1
4
1
5 2
Relaxation techniquesBasic techniques for solving a system of equations
bAx System of equations
NNNNN
N
b
b
b
x
x
x
aa
a
aaa
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
c
c
x
x
x
db
a
b
adb
ad
2
1
2
1
1
3
222
11
0
0
00
1
1
1
1
j
j
jjj
j
j
jjj
cd
bcc
ad
bdd
Put on upper triangular form:
Unknowns computed using
back-substitution:
1
1
j
jjj
j
N
NN
a
xdcx
d
cx
Nj ,.......3,2
1.........2,1 NNj
Jacobi
Easy but slow
bAx
N
j
ijij bxa
1
ii
ij
kjiji
ki
a
xab
x
1
In interation step k:
n
i
n
i
n
i
n
i SxTTT 21
1
11
1 2
2
2
111n
i
k
i
k
ik
i
SxTTT
T1,k+1 Ti,k+1Ti+1,k+1Ti-1,k+1 TN,k+1
T1,k Ti,kTi+1,kTi-1,k TN,k
Gauss-Seidel
bAx
N
j
ijij 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
n
i
n
i
n
i
n
i SxTTT 21
1
11
1 2
2
21
111n
i
k
i
k
ik
i
SxTTT
T1,k+1 Ti,k+1Ti+1,k+1Ti-1,k+1 TN,k+1
T1,k Ti,kTi+1,kTi-1,k TN,k
t
18
Successive Over-Relaxation (SOR)
• Accelerate convergence
• w > 1 overrelaxation
• w < 1 underrelaxation
(for stability)
k
T
Texact
T
k+1
Tk
Tk+1
k
T
Texact
wT
k+1
T*k
T*k+1
)( 11 k
i
k
i
k
i
k
i TTTT w
2
21
111n
i
k
i
k
ik
i
SxTTT
2
221
11
1
k
i
n
i
k
i
k
i
k
i
k
i
TSxTT
TT
w
ResidualsWhen should we stop
the iterations?
m
ji
m
jiRLu
,,
Relaxation techniquesPoint relaxation
y
xi i+1i-1
j
j+1
j-1
02
2
2
2
yx
0
222
1,1,
2
,1,1
yx
jiijjijiijji
2
1
1,1,
21
,1,11
12
k
ji
k
ji
k
ji
k
jik
ij
Example: Potential flow
Gauss-Seidel, point relaxation:
2
2
y
x
Relaxation techniquesLine relaxation
y
xi i+1i-1
j
j+1
j-1
2
1
1,1,
21
,1
1
,11
12
k
ji
k
ji
k
ji
k
jik
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 techniquesADI, alternating direction implicit
y
xi i+1i-1
j
j+1
j-1
2
2/1
1,1,
22/1
,1
2/1
,12/1
12
k
ji
k
ji
k
ji
k
jik
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
1
1,
1
1,
21
,1
2/1
,11
12
k
ji
k
ji
k
ji
k
jik
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 methodsMultigrid 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: jhxj 120 nj
nh
2
1
Discretisej
jjjf
h
uuu
2
112
Gauss-Seidelj
m
j
m
j
m
jfhuuu 21
112
nj 21
nj 21
01 uu
Multigrid methodsvon Neumann stability analysis
Use the numerical error*m
j
m
j
m
juu
1
112
m
j
m
j
m
jto rewrite the equation
ijn
mm
jec
12
0
Fourier modes of
the error:h
n
2
1lim 10
Gh
What does this tell us?
Amplification factor
Remember:
))sin()(cos( bibee aiba
Multigrid methods 1lim
10
Gh
Short wavelength (high frequency) errors damped 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 methodsExample of a linear problem, the Laplace equation
02
2
2
2
y
u
x
u
2
1,,1,
2
,1,,1
,
22
y
uuu
x
uuuLu
jijijijijiji
ji
On each grid, m, we solve:m
ji
m
jiRLu
,,
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
,M
jiLu
2. Calculate the residual and transfer it to the next
coarser grid, restriction: M
ji
M
M
M
jiRIR
,
11
,
Residual
Multigrid methodsExample 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,, m
ji
m
jiRuL
m
ji
m
ji
m
jiuuu
,,,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
m
ji
m
m
m
ji
m
jiuIuu
,
11
,
1
,ˆ
Multigrid methodsMultigrid cycles
V-cycle:
= relaxation
restrictionprolongation
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 help34
Multigrid methods