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Mul�disciplinary design op�miza�on in computa�onal mechanics Applica�on case – 2D wing Piotr Breitkopf 26.11.2014
2
Li�
Angle of a�ack Air velocity
Drag Aerodynamic center
Chord Reference line
3
Hypothesis
Simplifying assump�ons – incompressible, non-‐viscous (non-‐rota�onal) flow
@v @u @v + = 0, @x @y @x
@u =0 @y
– velocity field determined by stream func�on
Wortmain airfoil FX60.126 - computing domain
2
1.5
1
0.5
0
@ ,v = u= @y
@ @x
– pressure given by Bernoulli's principle
-0.5
-1
-1.5
-2
⇢ 2 P (x, y) = P1 + (Vwind V 2 (x, y)) 2 far field pressure P 1 , flow velocity Vwind
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
4
3
Streamlines
Curves that are instantaneously tangent to the velocity vector of the flow
0 1
n
2
5
Integral (weak formula�on)
differen�al formula�on
∆ (x, y) = 0, (x, y) 2 ⌦
residual weighted by test func�on
W =
Z
δ
∆ (x, y)δ d⌦ = 0 ⌦
a�er integra�ng by parts and using Green’s theorem
W =
Z
⌦
(rδ , r )d⌦ −
I
@⌦
δ (r , n)d(@⌦) = W⌦ − W@⌦ = 0 6
Finite element discre�za�on
Integral terms
W⌦ = W@⌦ =
I
Z
⌦
ne Z X
(rδ , r )d⌦ =
δ (r , n)d(@⌦) = @⌦
e=1 ⌦e ne I X e=1
(rδ , r )d⌦e δ (r , n)d(@⌦e )
@⌦e
are computed over surface and boundary mesh 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2
7 -2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Finite element approxima�on (x1 , y1 )
l1 l2
(x2 , y2 )
A3 (x, y)
(x, y)
(x1 , y1 )
A1
(x2 , y2 ) Ni (l) = li /L 2 X (x, y) = Ni (l) (xi , yi ) r (x, y) =
i=1 2 X i=1
rNi (l) (xi , yi )
A2
(x3 , y3 )
Ni (x, y) = Ai /A, i = 1...3 3 X (x, y) = Ni (x, y) (xi , yi ) r (x, y) =
i=1 3 X i=1
rNi (x, y) (xi , yi ) 8
Explicit form of shape func�ons and deriva�ves for a triangle
1 A = ((x2 x1 )(y3 y1 ) (x3 x1 )(y2 y1 )) 2 1 ((x2 x)(y3 y) (x3 x)(y2 y)) N1 = 2A 1 ((x x1 )(y3 y1 ) (x3 x1 )(y y1 )) N2 = 2A 1 ((x2 x1 )(y y1 ) (x x1 )(y2 y1 )) N3 = 2A
1 @N1 = (y3 @x 2A 1 @N2 = (y3 @x 2A 1 @N3 = (y1 @x 2A
y2 )
1 @N1 = (x3 @y 2A
x2 )
y1 )
1 @N2 = (x1 @y 2A
x3 )
y2 )
1 @N3 = (x2 @y 2A
x1 ) 9
Surface term
δ
r (x, y) =
3 X i=1
e
⇥
= N1 (x, y)
rNi (x, y)
i
N2 (x, y)
L y2 − y3 = 6 x3 − x2
1 y2 Be = 2A x3
W⌦e = δ
y3 x2
y3 x1
2
⇤ δ N3 (x, y) 4δ δ
y3 − y1 x1 − x3
y1 x3
T e Ke e , Ke
y1 x2
1 2 3
3 5 �
2
y1 − y2 4 x2 − x1
y2 x1
�
1 2 3
3
5 = Be
= AB T B
10
e
Boundary term
⇥
⇤
(r , n) = N1 (s) N2 (s) (ny u¯n − nx v¯n ) � ⇥ ⇤ δ 1 δ e = N1 (s) N2 (s) δ 2 W@⌦e = δ
T ¯n e Me (ny u
nx v¯), u ¯n =
�
✓
u ¯1 u ¯2
◆
✓ ◆ v¯ , v¯n = 1 v¯2
L 2 1 Me = 6 1 2
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Finite element linear system W =
e X
We =
T
e=1
(K − F ) = 0 )K
=F
boundary condi�ons – Neumann at the external boundary – intergrated in the RHS – Dirichlet
K11 T K12 1
K12 K22
�
= K111 (F
1
¯2
�
=
F1 0
�
K12 ¯2 )
12
@ , ( @y
@ ) = (¯ u, v¯) @x
non-‐rota�onal flow
=0
-2
-1
0
1
@ , ( @y 2
@ ) = (¯ u, v¯) 13 @x 3
=1
Circular flow
=0 =1
=1
=1
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Ku�a (Joukowski) condi�on
Ku�a condi�on: no circula�on around the trailing edge resultant velocity follows reference line y
0.25
y
0.25
0.2
0.2
0.2
0.15
0.15
0.15
0.1
0.1
0.1
0.05
0.05
0.05
0
0
0
-0.05
-0.05
-0.05
-0.1
-0.1
-0.1
-0.15
-0.15
-0.15
-0.2
-0.2
-0.2
-0.25 0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.2
-0.25 0.75
0.8
0.85
0.9
V1
0.95
1
1.05
p
0.25
1.1
1.15
1.2
1.25
-0.25
0.8
V2
0.9
1
g
1.1
1.2
V = V1 + ↵V2
weighted sum of uniform and circular flows Li� and li� coefficient
L = ⇢V ,
I
L = (rφ, n), cL = 1 2 @⌦ 2 ⇢V l 15
Summed flows p
2
g
1.5
1
0.5
0
-0.5
-1
-1.5
-2
16 -2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Finite element 2D wing example summary
generate domain mesh – surface elements – linear elements on the boundary solve two problems – uniform flow – circular flow assemble flows – compute Ku�a coefficient – sum up stream func�ons post-‐process quan��es of interest – veloci�es – pressure – li�, drag, pitching moment – li�/drag coefficients – ... detailed course materials at :
pressure
#10 4
0.8 10.5
0.6
10
0.4
0.2 9.5
0
-0.2
9
-0.4
8.5
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
– h�p://www.utc.fr/~mecagom4/MECAWEB/EXEMPLE/EX07/SAAA1.htm
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