Finite Element 2D Wing

Mul�disciplinary  design  op�miza�on    in  computa�onal  mechanics  

 Applica�on  case  –  2D  wing  

Piotr  Breitkopf  26.11.2014  

2  

Li�  

Drag  

Angle  of  a�ack  

Reference  line  Aerodynamic  center    

Chord  Air  velocity  

3  

Hypothesis  

4  

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Wortmain airfoil FX60.126 - computing domain

  Simplifying  assump�ons  –  incompressible,  non-­‐viscous  (non-­‐rota�onal)  flow  

–  velocity  field  determined  by  stream  func�on  

–  pressure  given  by  Bernoulli's  principle  

  far  field  pressure                          ,  flow  velocity  P1 Vwind

@u

@x+

@v

@y= 0,

@v

@x @u

@y= 0

u =@

@y, v = @

@x

P (x, y) = P1 +⇢

2(V 2

wind V 2(x, y))

Streamlines  

5  

  Curves  that  are  instantaneously  tangent  to  the  velocity  vector  of  the  flow  

n

1

2

0

Integral  (weak  formula�on)  

6  

W =

Z

⌦

(rδ ,r )d⌦−I

@⌦

δ (r , n)d(@⌦) = W⌦ −W@⌦ = 0

  differen�al  formula�on  

  residual  weighted  by  test  func�on  

     a�er  integra�ng  by  parts  and  using  Green’s  theorem  

δ

W =

Z

⌦

∆ (x, y)δ d⌦ = 0

∆ (x, y) = 0, (x, y) 2 ⌦

Finite  element  discre�za�on  

7  

  Integral  terms  

  are  computed  over  surface  and  boundary  mesh  

W⌦ =

Z

⌦

(rδ ,r )d⌦ =

neX

e=1

Z

⌦e

(rδ ,r )d⌦e

W@⌦ =

I

@⌦

δ (r , n)d(@⌦) =neX

e=1

I

@⌦e

δ (r , n)d(@⌦e)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Finite  element  approxima�on  

8  

A1

A2A3

(x, y)

(x1, y1)

(x2, y2)(x3, y3)

(x1, y1)

(x2, y2)

(x, y)

l1l2

(x, y) =3X

i=1

Ni(x, y) (xi, yi)

Ni(l) = li/L Ni(x, y) = Ai/A, i = 1...3

(x, y) =2X

i=1

Ni(l) (xi, yi)

r (x, y) =2X

i=1

rNi(l) (xi, yi) r (x, y) =3X

i=1

rNi(x, y) (xi, yi)

Explicit  form  of  shape  func�ons  and  deriva�ves  for  a  triangle  

9  

A =1

2((x2 x1)(y3 y1) (x3 x1)(y2 y1))

@N3

@x=

1

2A(y1 y2)

@N3

@y=

1

2A(x2 x1)

N1 =1

2A((x2 x)(y3 y) (x3 x)(y2 y))

N2 =1

2A((x x1)(y3 y1) (x3 x1)(y y1))

@N1

@y=

1

2A(x3 x2)

@N2

@x=

1

2A(y3 y1)

@N2

@y=

1

2A(x1 x3)

N3 =1

2A((x2 x1)(y y1) (x x1)(y2 y1))

@N1

@x=

1

2A(y3 y2)

Surface  term  

10  

Be =1

2A

y2 y3 y3 y1 y1 y2x3 x2 x1 x3 x2 x1

�

W⌦e= δ T

e Ke e,Ke = ABTB

r (x, y) =3X

i=1

rNi(x, y) i =L

6

y2 − y3 y3 − y1 y1 − y2x3 − x2 x1 − x3 x2 − x1

�2

4 1

2

3

3

5 = Be e

δ e =⇥N1(x, y) N2(x, y) N3(x, y)

⇤2

4δ 1

δ 2

δ 3

3

5

Boundary  term  

11  

Me =L

6

2 11 2

�

W@⌦e= δ T

e Me(nyun nxv), un =

✓u1

u2

◆, vn =

✓v1v2

◆

(r , n) =⇥N1(s) N2(s)

⇤(nyun − nxvn)

δ e =⇥N1(s) N2(s)

⇤ δ 1

δ 2

�

Finite  element  linear  system  

12  

W =

eX

e=1

We = T (K − F ) = 0

) K = F

  boundary  condi�ons  –  Neumann  at  the  external  boundary  –  intergrated  in  the  RHS  –  Dirichlet    

1 = K111 (F K12 2)

K11 K12

KT12 K22

� 1

2

�=

F1

0

�

non-­‐rota�onal  flow  

13  -2 -1 0 1 2 3

= 0

(@

@y,@

@x) = (u, v)

(@

@y,@

@x) = (u, v)

Circular  flow  

14  

= 0

= 1 = 1

= 1

= 1

Ku�a  (Joukowski)  condi�on  

15  

0.8 0.9 1 1.1 1.2-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25 p g

0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25 y

0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.2-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25 y

V = V1 + ↵V2V1 V2

  Ku�a  condi�on:  no  circula�on  around  the  trailing  edge      resultant  velocity  follows  reference  line  

   weighted  sum  of  uniform  and  circular  flows    Li�  and  li�  coefficient  

L = ⇢V , =

I

@⌦

(rφ, n), cL =L

12⇢V

2l

Summed  flows  

16  -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2 p g

Finite  element  2D  wing  example  summary  

17  

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

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-0.4

-0.2

0

0.2

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0.8

pressure #104

8.5

9

9.5

10

10.5

  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  :  –  h�p://www.utc.fr/~mecagom4/MECAWEB/EXEMPLE/EX07/SAAA1.htm