Accurate numerical solution of Euler
equations by optimal control of dissipation
SM Deshpande
EMU, JNCASR, Bangalore
smd@jncasr.ac.in
N Anil
Omesh Reshi
Dept. of Aerospace Engg.
IISc, Bangalore
L & T, Mumbai
French-Indian workshop on “Numerical Simulation, Control and
Design for Aeronautical and Space Applications”. INRIA, Sophia
Antipolis, France, Nov 29-Dec 01, 2006.
OUTLINE
• Introduction to KFVS
• Basic theory of m-KFVS
• Numerical Results
• Optimal control of dissipation
• A numerical test case
• Conclusions
INTRODUCTION TO KFVS
Consider the 1D Euler equations
U G

0
t
x
which are moments of the 1D Boltzmann equation
in the Euler limit
F
F
v
0
t
x
where F - Maxwellian velocity distribution function
F

I0

2

exp    v  u   I / I 0 



INTRODUCTION TO KFVS
The Euler equations can be written as
U G
F
F

 ,
v
0
t
x
t
x
 is the moment function vector
  1 v I  v / 2 
2
T
The inner product is defined as
, f 

R R
f  v dvdI
INTRODUCTION TO KFVS
Using CIR splitting, 1D Boltzmann equation gives
F v  | v | F v  | v | F


0
t
2 x
2 x
Taking moments, we get KFVS Euler equations
U G  G 


0
t
x
x
The split fluxes are given by
v | v |
G   
FdvdI
2
R R

INTRODUCTION TO KFVS
Consider the CIR split Boltzmann equation
F v  | v | F v  | v | F


0
t
2 x
2 x
Using suitable finite difference approximations
 F  v  | v | Fj  Fj 1 v  | v | Fj 1  Fj

0

 
2
x
2
x
 t  j
The mpde analysis gives
F
F x
2
v

| v | Fxx  O  x 
t
x
2
KFVS – first order accurate and dissipative !
INTRODUCTION TO KFVS
• Numerical dissipation can be reduced by using
higher order upwind schemes
• In the case of 1D, second order scheme requires
5 points in the stencil
• Is it possible to develop a scheme which requires
only 3 point stencil but still gives results better
than first order and close to second order ?
BASIC THEORY OF MKFVS
MCIR Splitting
The modified CIR splitting is defined as
v | v |  v | v | 
v

2
2
The mpde based on MCIR splitting is given by
F
F x
2
v

| v |  Fxx  O  x 
t
x
2
BASIC THEORY OF MKFVS …
Taking moments we get mpde at Euler level
U G x
2



|
v
|

F
dvdI

O

x




xx
t x
2 RR
The modified split fluxes are given by
Gm   ,
v | v | 
F
2
•

  1  Gm  Gkfvs
•
  0  Central differencing
•
By tuning  we can control dissipation and
order of accuracy
BASIC THEORY OF MKFVS …
Mathematical models for  :
Consider the mpde
U G x
2


 | v |  F  xx dvdI  O  x 

t x
2 R  R
When   1
 
2
  u   2
t x
x
 x

|
v
|



I0
 2 R R
2

 num  
x 2


2
exp     v  u   I / I 0  dvdI 




BASIC THEORY OF MKFVS …
 num = Numerical kinematic viscosity
 num
2
 x 

   v u 

|
v
|

e
dv

R
2



When   1
• Particles with c   v  u   0 contribute maximum to
• Particles with | c |  contribute very little to
 num
 num
BASIC THEORY OF MKFVS …
The choice   e


|v |
• Low velocity particles for which
| v |     0 - Contribute very little to  num
• High velocity particles for which
| v |     1 - Contribute max. to  num
BASIC THEORY OF MKFVS …
 |v|
Another choice   e
• Low velocity particles for which
 | v | 1    1 - Contribute max. to  num
• High velocity particles for which
 | v | 1    0 - Contribute very little to  num
BASIC THEORY OF MKFVS …
Consider the steady 1D Boltzmann equation with a
BGK model for the collision term
f
v
 A F  f 
x
Using the IF  e Ax / v the solution is given by
f  x, v   f  x0 , v  e

A
 x  x0 
v

x
x0
A
  x x0 
A
F  x  e v
dx, v  0
x
Assuming F  x  to be constant over x0  x  x
f  x, v   f  x0 , v  e

A
 x  x0 
v
A
  x  x0  

 F 1  e v



BASIC THEORY OF MKFVS …
• The low velocity molecules are almost always lost
• While high velocity molecules are lost negligibly,
i.e., they travel over x without any collision
• The choice
 e


|v |
is consistent with the earlier argument
but doesn’t have closed form expressions for the split fluxes
• The second choice
for split fluxes
  e |v| leads to much simpler formulae
BASIC THEORY OF MKFVS …
For
  e |v| the modified split fluxes are given by
   2  S  
G 1

4
 
Gm   e
G u 

2 2
2 



  e

2

 S 

 4



G u 

2 



 
 
• G - full flux, G  - kfvs fluxes,  non - dimensional parameter
• mkfvs fluxes are simpler and similar to kfvs fluxes
• Takes almost the same amount of time as kfvs fluxes
• Smooth for all Mach number range
• Continuous and differentible functions
BASIC THEORY OF MKFVS …
Dissipation Analysis :
The mpde for 1D Euler equations based on mkfvs is given by
U
U   x 
2

 U 
A
 
Gm  Gm 

O

x



t
x x  2 U
x 
The dissipation matrix is given by
x 
Gm   Gm  

2 U
x

Am   Am  

2
D
BASIC THEORY OF MKFVS …
Dissipation Analysis :
The numerical kinematic viscosity is given by
 num 
x
max   D 
2
• Here   D  are the eigenvalues of the matrix D
• For large values of
 the eigenvalues   D   0
BASIC THEORY OF MKFVS …
Max  of
the matrix D
for increase
in 
NUMERICAL RESULTS
1D inviscid test cases:
• Convergent Divergent nozzle problem
• Shock tube problem
2D inviscid test cases:
• Subsonic flow past Williams airfoil
• Transonic flow past NACA 0012 airfoil
• Supersonic flow past NACA 0012 airfoil
NUMERICAL RESULTS …
Convergent – divergent nozzle
NUMERICAL RESULTS …
Convergent – divergent nozzle test case:
• Governing equations are quasi-one-dimensional
• Steady flow
• Number of grid points : 60
• Computations are performed with
  e |v|
NUMERICAL RESULTS …
Non-dimensional pressure along
the nozzle using
  e |v|
Non-dimensional value    / 
along the nozzle using
NUMERICAL RESULTS …
Variation of density along the shock tube. No. of points = 500
NUMERICAL RESULTS …
Non-dimensional value    /  along the shock tube
NUMERICAL RESULTS …
Variation of density for higher values of 
NUMERICAL RESULTS …
Role of
 in numerical simulations:
• Increase of
•
 enhances accuracy
 has been used to suppress wiggles
NUMERICAL RESULTS …
Features of 2D cell centred FVM solver based
on m-KFVS
• m-KFVS fluxes for flux evaluation
• modified kinetic wall and outer boundary conditions
• Local time stepping
• LUSGS for convergence acceleration
NUMERICAL RESULTS …
Subsonic flow past Williams airfoil
M = 0.15
aoa = 0.0
Grid points = 50000
NUMERICAL RESULTS …
KFVS
Entropy contours
of m-KFVS method
are compared with
KFVS and q-KFVS
q-KFVS
m-KFVS
NUMERICAL RESULTS …
KFVS
Pressure contours
of m-KFVS method
are compared with
KFVS and q-KFVS
q-KFVS
m-KFVS
NUMERICAL RESULTS …
Cp – distribution: kfvs, m-kfvs and exact
Main airfoil
Flap
NUMERICAL RESULTS …
Transonic flow past NACA 0012 airfoil
M = 0.85
aoa = 1.0
320x120 structured grid
NUMERICAL RESULTS …
KFVS
Pressure contours
of m-KFVS method
are compared with
KFVS and q-KFVS
q-KFVS
m-KFVS
NUMERICAL RESULTS …
Cp-Plot of m-KFVS
are compared with
KFVS and q-KFVS
NUMERICAL RESULTS …
Pressure distribution
on unstructured grid
Cp-Plot of m-KFVS
are compared with
KFVS and q-KFVS
NUMERICAL RESULTS …
Supersonic flow past NACA 0012 airfoil
M = 1.2, aoa = 0.0
320x120 structured grid
NUMERICAL RESULTS …
KFVS
Pressure contours
of m-KFVS method
are compared with
KFVS and q-KFVS
q-KFVS
m-KFVS
NUMERICAL RESULTS …
Comments on m-KFVS:
• It gives results as accurate as second
order accurate q-KFVS
• It does not have spurious wiggles near
shock, which are present in q-KFVS
OPTIMAL CONTROL OF DISSIPATION
• Consider the 2D subsonic flow past NACA 0012 airfoil
• Accurate capture of suction peak depends on numerical
entropy generated by the scheme
• The low dissipative m-KFVS method captures the suction
peak more accurately compared to KFVS
• Further, by optimal control of dissipation parameter  ,
we can get even better accuracy
Optimisation Problem :


Cost function = I U    Sum of squares of entropy at cells
Min I U  

OPTIMAL CONTROL OF DISSIPATION
Discrete adjoint approach
Minimise: The cost function I
The discrete form of the cost function is given by
N
I U       si 
2
i 1
  p /   
i

where si  change in entropy = ln 

  p /   
U - Vector of conserved variables
 - Dissipation control variables vector
N - Number of cells in the computational domain
OPTIMAL CONTROL OF DISSIPATION
Subject to: The governing state equations
G1 G 2
R U ,   

0
x
y
Where G1 and G2 are the fluxes along coordinate directions
Few observations:
• The cost function doesn’t explicitly depend on 
• It depends on  through its dependence on solution U
• The number of design variables = N = Number of cells
OPTIMAL CONTROL OF DISSIPATION
The first variation in the cost function is given by
I
I
U 

U

N  4

I
I
l 
  
Ui 
 i 
l 
 i
i 1  l 1 U i

I 
 4 I
l  
  
Ui 
l 
i 1  l 1 U i

N
The first variation in the state equation is given by
R 
R
R
U 
  0
U

OPTIMAL CONTROL OF DISSIPATION
At any cell i the first variation is given by
 Ri   R U i ,U j ,  i ,  j   0, j  nbhd  i 
nbhd  i 
 Ri

Ri
Ri
Ri

Ui 
 i   
U j 
 j   0
U i
 i
 j
j 1 
 U j

The above equation can be written in expanded notation
l 
 Ri

Ri 
 m  Ri

Ui 
 i
 m
 i
m 1 U i
4

nbhd  i 

j 1
l
l
l 
 4 R  l 


R
m


i
i
U j 
 j   0, l  1 to 4 and

 m
 j
 m 1 U j

i  1 to N
OPTIMAL CONTROL OF DISSIPATION
Introducing the vector of adjoint variables, V and using
the state equation as a constraint
N
 I   I  ViT  Ri
i 1
N 4
 4 I
l  
l 
l 
   

U

V

R

i   i
i
l 

U
i 1  l 1
i
 i 1 l 1
N
Collecting all the terms which are coefficients of  U i and
equating them to zero we get the adjoint euqtion for cell i
R 
* T
i
nbhd  i 
R j

R
I
T
T
i

 Vi
  Vj
 0, i  1...N
U i
U i
U i
j 1
where  R* i is the adjoint residual for the cell i
T
OPTIMAL CONTROL OF DISSIPATION
Further expanding
R 
*
l 
i
 m
 R j
 I 
 Ri
 m
  l     
Vi    
l 
l

U

U

U
j 1 m 1 
i 
i
 i  m1 

i  1 to N and
T
4
 m  T
nbhd  i  4
T
  m
 Vj  0

l  1 to 4
Introducing a pseudo time stepping
Vi  
* l 
Ai
  R  , i  1 to N and l  1 to 4
i
t
l
We have used TAPENADE to compute the gradients
I
 R 
and 
 V
U
 U 
T
OPTIMAL CONTROL OF DISSIPATION
The adjoint variables are then substituted in the equation
for cost function variation
nbhd  i 



R
Ri
T
i
 I   Vi 
 i  
 j 
i 1
j 1  j
  i

nbhd  i 
l 
l 
N
4



R

R
l 
i
i
  Vi 
 i  
 j 
i 1 l 1
j 1  j
  i

 GT 
N
Again TAPENADE is used to compute the gradient
R

OPTIMAL CONTROL OF DISSIPATION
One of the early and simplest methods is steepest descent
method in which
   G,   0
Using the above method the cost function will decrease
since  I   G G   G  0 for   0
T
2
Now better methods are available. We have here used the
above steepest descent method as a first step
The values of  can be updated using
i n1  i n   Gi ,  >0 and i  1...N
A NUMERICAL TEST CASE
Comments on the m-KFVS-AD solver
• Non-linear m-KFVS solver written in C++
• m-KFVS-AD solver written in FORTRAN 77
• TAPENADE is used to get the subroutines,
which are used to compute the gradients
in the AD solver
A NUMERICAL TEST CASE
Subsonic flow past NACA 0012 Airfoil
• M = 0.63, aoa = 2.0o
• Unstructured grid with 4733 points
• Non-linear solver – 2D cell centred FVM
based on m-KFVS
A NUMERICAL TEST CASE
Entropy contours
of adjoint based
m-KFVS solver is
compared with KFVS
Scheme
Max. entropy
KFVS
0.03129
m-KFVS-AD
0.00306
A NUMERICAL TEST CASE
Surface pressure distribution
A NUMERICAL TEST CASE
Cost function variation w.r.to optimisation cycles
A NUMERICAL TEST CASE
Alpha contours after 35 design cycles
A NUMERICAL TEST CASE
Alpha contours near the leading edge
A NUMERICAL TEST CASE
Alpha contours near the suction peak
A NUMERICAL TEST CASE
Alpha contours near the trailing edge
CONCLUSIONS
• The m-KFVS method has dissipation control function
 which can be suitably chosen to reduce numerical
dissipation or to reduce numerical entropy generated
• The eigenvalues of dissipation matrix D can be
made vanishingly small by increasing 
• Near the discontinuities

plays the role of a limiter
• m-KFVS code can be combined with adjoint method to
obtain optimal control vector  for minimal entropy
and hence maximum accuracy of numerical solution
Thank You