Transcript EEG(MMN) Signal Processing

Gridless Method for Solving Moving Boundary Problems
Wang Hong
Department of Mathematical Information Technology
University of Jyväskyklä
28.05.2009
1/36
Content
Introduction
Principle of Gridless Method
Steady Simulation of Euler Equations and Applications
Unsteady Simulation of Euler Equations: Validation on Referenced
Airfoils
Conclusion and Future Research
2/36
Introduction
At present, CFD (Computational Fluid Dynamics) community has
many methods in solving moving boundary problems, for instance,
dynamic mesh method and fictitious domain method and so on.
The dynamic mesh method contains different techniques, for
example, mesh reconstruction methods and mesh deformation
methods.
3/36
Introduction
Mesh reconstruction methods have been proposed based on the
regeneration of mesh according to the moving boundaries. For
example, when we use unstructured Cartesian mesh to solve
moving boundary problems, we should fasten the mesh; the
moving boundaries cut the mesh elements. These kinds of methods
need to consider the cutting elements in every time step, and refine
and coarsen the grids to satisfy the distribution requirements.
4/36
Introduction
Mesh deformation methods are basically relied on the control models.
Spring approximation model was firstly introduced by Batina to
solve the vibrating airfoil flows. The basic idea is that treat every
edge of mesh elements as a spring, the coefficients of the spring are
related to the length of every edge of mesh elements; after the
movement of the boundaries, the new positions of the mesh points
are defined by solving the force equilibrium of the spring system.
5/36
Introduction
A fast dynamic cloud method based on Delaunay graph mapping
strategy is proposed in this presentation. A dynamic cloud method
makes use of algebraic mapping principles and therefore points can
be accurately redistributed in the flow field without any iteration. In
this way, the structure of the gridless clouds is not necessary
changed so that the clouds regeneration can be avoided
successfully.
6/36
Principle of Gridless Method
Gridless node definition
Spatial discretisation
7/36
Gridless Node Definition
Global and close-up views of typical structure for gridless clouds
8/36
Spatial Discretisation – Least Square Method
f k  fi  a1hk  a2lk  O(hk2 , lk2 )
If we keep the first order of fk , then we have the approximation
f k  fi  a1hk  a2lk
Define the total error in the cloud of points as
G    fk  fk 
M
k 1
2
f
f
hk  xk  xi
a2 
a1 
y i
x i
lk  yk  yi
in order to minimize the total error , let
G G

0
a1 a2
Ax＝b
a1  ik  fik  fi 
a2   ik  fik  fi 
9/36
Steady Simulation of Euler Equations and Applications
Governing Equations
Boundary Conditions
Spatial Discretisation
Time Discretisation
Steady Flow Simulation Results
NACA0012
RAE2822
10/36
Governing Equations
W E F


0
t x y
 W  [  ,  u,  v, e]T

2
T
 E  [  u,  u  p,  uv, (e  p )u ]
 F  [  v,  vu,  v 2  p, (e  p )v]T

Whereρis the density，u and v are the velocity components，p is the pressure,
e is the total energy per unit volume, for an idea gas, it can be written as
p
1
e
  (u 2  v 2 )
 1 2
11/36
Boundary Conditions – Solid Wall
pn
pn
V
V
Vn
Vn
V
V
V
pw
p
p
V
pw
h
h
pw
*
p
pw
h
h
*
p
V
(1) Direct method
V
V
*
V
*
(2) Mirror method
12/36
Boundary Conditions – Far Field
t
t
M<1
M<1
outer field
inner field
inner field
u
u+c
u-c
u
u+c
u
outer field
u-c
u
x
(1) Subsonic inflow
x
(2) Subsonic outflow
t
t
M>1
outer field
u+c
M>1
inner field
inner field
u
u u-c
u+c
x
(3) Supersonic inflow
outer field
u
u u-c
x
(4) Supersonic outflow
13/36
Spatial Discretisation
W  E F 


 0
t i  x y i
 E F 
Qi  



x

y

i
Qi    ik  Eik  Ei    ik  Fik  Fi 
   ik Eik  ik Fik    ik Ei  ik Fi 
   G ik  G i 
 U  u   v

G   E   F
 U

  uU   p 

G
  vU   p 


(
e

p
)
U


14/36
Spatial Discretisation
Qi    Gik  Gi 
i
L
R
k
ik
Gik  G  WL , WR 
Center node i and satellite node k.
(“+” denotes the right wave,
Roe Scheme
“－” denotes the left wave)
15/36
Time Discretisation
Win 1  Win
 Ri
t
 Wi(0)  Win
 (1)
(0)
(0)
W

W



t
R
i
i
1
i
i

 Wi(2)  Wi(0)   2 ti R i(1)
 (3)
(0)
(2)
 Wi  Wi   3ti R i
 Wi(4)  Wi(0)   4 ti R i(3)
 n 1
(4)
 Wi  Wi
CFL
ti 
max   A1  ,   A 2  , ,   A M 
 k (k  1, 2, , 4)
represents the stage coefficients, and
1  0.0833
  0.2069
 2

 3  0.4265
 4  1
 ( A)   u   v  c  2   2
16/36
Steady Flow Simulation Results – NACA0012
10
1
5
0.5
y
y
337 nodes on the airfoil and 5557 nodes in the flow field.
0
-5
-10
-10
0
-0.5
-5
0
x
5
10
-1
-1
-0.5
0
0.5
1
x
Global and close-up views of the computational domain for the NACA0012 airfoil.
17/36
Steady Flow Simulation Results – NACA0012
M  0.8,  0.0
-1.5
2
-1
1
Y
Cp
-0.5
0
0
0.5
-1
M = 0.8
 = 0.0o
1
1.5
0
0.2
0.4
M = 0.8
 = 0.0o
0.6
x/c
0.8
1
-2
-1
0
1
2
X
Flow field pressure coefficients and mach number distributions for NACA 0012 airfoil.
18/36
Steady Flow Simulation Results – NACA0012
M   0.8,  1.25
-1.5
2
-1
1
Y
Cp
-0.5
0
0
0.5
-1
M = 0.8o
 = 1.25
1
1.5
0
0.2
0.4
M = 0.8o
 = 1.25
0.6
x/c
0.8
1
-2
-1
0
1
2
X
Flow field pressure coefficients and mach number distributions for NACA 0012 airfoil.
19/36
Steady Flow Simulation Results – NACA0012
M   0.85,  1.0
-1.5
2
-1
1
Y
Cp
-0.5
0
0
0.5
-1
M = 0.85
 = 1.0o
1
1.5
0
0.2
0.4
0.6
x/c
M = 0.85
 = 1.0o
0.8
1
-2
-1
0
1
2
X
Flow field pressure coefficients and mach number distributions for NACA 0012 airfoil.
20/36
Steady Flow Simulation Results – NACA0012
M  1.2,  7.0
-1.5
2
-1
1
Y
Cp
-0.5
0
0
0.5
-1
M = 1.2
 = 7.0o
1
1.5
0
0.2
0.4
M = 1.2
 = 7.0o
0.6
x/c
0.8
1
-2
-1
0
1
2
X
Flow field pressure coefficients and mach number distributions for NACA 0012 airfoil.
21/36
Steady Flow Simulation Results – RAE2822
10
1
5
0.5
y
y
335 nodes on the airfoil and 5842 nodes in the flow field.
0
-5
-10
-10
0
-0.5
-5
0
x
5
10
-1
-1
-0.5
0
0.5
1
x
Global and close-up views of the computational domain for the RAE2822 airfoil.
22/36
Steady Flow Simulation Results – RAE2822
M  0.725,  2.55
-1.5
2
-1
1
0
Y
Cp
-0.5
0
0.5
M = 0.725
o
 = 2.55
1
1.5
0
0.2
0.4
0.6
x/c
-1
0.8
1
-2
-1
M = 0.725
o
 = 2.55
0
1
2
X
Flow field pressure coefficients and mach number distributions for RAE2822 airfoil.
23/36
Steady Flow Simulation Results – RAE2822
M   0.75,   3.0
2
-1.5
-1
1
Y
Cp
-0.5
0
0
0.5
-1
M = 0.75
 = 3.0o
1
1.5
0
0.2
0.4
0.6
x/c
M = 0.75
 = 3.0o
0.8
1
-2
-1
0
1
2
X
Flow field pressure coefficients and mach number distributions for RAE2822 airfoil.
24/36
Unsteady Simulation of Euler Equations and Validation
A Fast Dynamic Cloud Method
Unsteady Flow Simulation Results
NACA0012
NACA64A010
25/36
A Fast Dynamic Cloud Method
(a) Global view
(b) Close-up view
Back ground mesh for NACA0012 airfoil based on Delaunay triangulation
26/36
A Fast Dynamic Cloud Method
-0.355
-0.36
Y
-0.365
-0.37
-0.375
-0.38
-0.385
0.39
0.4
0.41
X
(a) Spring analogy strategy
(b) Delaunay graph mapping strategy
Moved gridless clouds of 30°pitching airfoil
27/36
Unsteady Flow Simulation Results – NACA0012
337 nodes on the airfoil and 5557 nodes in the flow field.
Close-up views of computational domain for the NACA0012 airfoil for 2.51° pitch.
28/36
Unsteady Flow Simulation Results – NACA0012
 (t )  m +0 sin(t ) Ma  0.755 m  0.016 0  2.51 k  0.0814
0.03
0.4
0.02
0.01
Cm
CL
0.2
0
0
-0.01
-0.2
-3
-2
-1
0
(o)
1
2
Experiment
Kirshman
Computation
-0.02
Experiment
Kirshman
Computation
-0.4
3
-0.03
-3
-2
-1
0
(o)
1
2
3
Comparisons of computed lift and moment coefficients with the experimental and
Kirshman’s data for prescribed oscillation of NACA0012 airfoil.
29/36
Unsteady Flow Simulation Results – NACA64A010
200 nodes on the airfoil and 4006 nodes in the flow field.
-0.5
0.5
0.5
0
0
y
0
y
y
0.5
-0.5
-0.5
0 computational
0.5
1 domain for the NACA64A010 airfoil for 1.01° pitch.
Close-up-0.5views of
0
0.5
1
-0.5
0
0.5
1 -0.5
x
x
x
30/36
Unsteady Flow Simulation Results – NACA64A010
 (t )  m +0 cos(t ) Ma  0.796 m  0.0 0  1.01 k  0.202
30
20
Experiment
Wanggang
Computation
20
Experiment
Wanggang
Computation
10
Im(Cp1)
Re(Cp1)
10
0
0
-10
-10
-20
-30
0
0.2
0.4
0.6
x/c
0.8
1
-20
0
0.2
0.4
0.6
0.8
1
x/c
Comparisons of the first Fourier mode component of surface pressure coefficients with
the experimental data for oscillating NACA64A010 airfoil.
31/36
 (t )  m +0 sin(t ) Ma  0.755 m  0.016 0  2.51 k  0.0814
 (t )  m +0 cos(t ) Ma  0.796 m  0.0 0  1.01 k  0.202
Mach number distribution for pitching airfoil (NACA0012 and NACA64A010)
Gridless Method with Dynamic Clouds of Points for Solving Unsteady CFD
Problems in Aerodynamics
– accepted by International Journal for Numerical Methods in Fluids
32/36
Conclusion and Future
Introduction
Principle of Gridless Method
Steady Simulation of Euler Equations
Unsteady Simulation of Euler Equations
Future Research
Inverse problem with one NACA 0012 airfoil using gridless method
Inverse problem with dual NACA 0012 airfoils using gridless method
Mesh/gridless hybridized algorithms to solve other boundary
moving problems
Our method is both flexible and efficient, therefore it is quite
suitable to solve optimization problems, such as:
Multi element airfoil lift optimization with Navier-Stokes flows in
Aerodynamics
33/36
Antennas optimal position in Telecommunications
Inverse problem of NACA 0012

Ma  0.5;   0.0
M
min f ( )   C p ( )  C p ( * )
i 1
-0.6
2
i
-0.4
-0.2
10
1
-2
0
10-3
0.2
10-4
0.6
f()
(o)
Cp
0.8
10
0.4
-5
0.4
10-6
0.6
0.2
10-7
0.0(target)
0.0006
0.8
1
1.5
2
2.5
generation
1
0
0.2
0.4
0.6
0.8
3
3.5
4
1
1.5
2
2.5
3
3.5
generation
1
x/c
Genetic Algorithms Using a Gridless Euler Solver for 2-D Unsteady Inverse
Problems in Aerodynamic Design
– will be submitted to EUROGEN 2009
34/36
4
Subsonic and Transonic Simulations of dual airfoils

Ma  0.5;   0.0

Ma  0.8;   1.25
Mach
0.70
0.67
0.64
0.61
0.58
0.55
0.52
0.49
0.46
0.43
0.40
0.37
0.34
0.31
0.28
0.25
0.22
0.19
0.16
0.13
0.10
0.07
0.04
1
y
0.5
0
-0.5
-1
-1
-0.5
0
0.5
1
1.5
2
Mach
1.5
1.50
1.44
1.37
1.31
1.25
1.18
1.12
1.05
0.99
0.93
0.86
0.80
0.74
0.67
0.61
0.55
0.48
0.42
0.35
0.29
0.23
0.16
0.10
1
0.5
y
1.5
0
-0.5
-1
-1
-0.5
0
x
0.5
1
1.5
2
x
Mach number distribution for dual aifoils
35/36
36/36