Hydrosystemanalyse: Finite-Elemente-Methode (FEM)

V19: Finite-Elemente-Methode (FEM) 2-D 17.07.2013 Hydrosystemanalyse: Finite-Elemente-Methode (FEM) Prof. Dr.-Ing. habil. Olaf Kolditz 1 Helmholtz ...
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V19: Finite-Elemente-Methode (FEM) 2-D

17.07.2013

Hydrosystemanalyse: Finite-Elemente-Methode (FEM) Prof. Dr.-Ing. habil. Olaf Kolditz 1 Helmholtz

Centre for Environmental Research – UFZ, Leipzig

2 Technische

Universit¨ at Dresden – TUD, Dresden

Dresden, 17. Juli 2013

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Prof. Dr.-Ing. habil. Olaf Kolditz

Hydrosystemanalyse 2015

V19: Finite-Elemente-Methode (FEM) 2-D

17.07.2013

Vorlesungsplan SoSe 2015: Hydrosystemanalyse # 1 2 3 7 11 13 15 20 23 32 2/22

Datum Vorlesung 17.04.15 Einf¨ uhrung, Systemanalyse 17.04.15 Grundwasserhydraulik: Einzugsgebiet Bilanzierung, Vorlesung 24.04.15 Grundwasserhydraulik: Einzugsgebiet Bilan¨ zierung, Ubung 22.05.15 Grundwasserhydraulik: Finite-DifferenzenVerfahren, Rechteckaquifer 12.06.15 Grundwasserhydraulik: Finite-DifferenzenVerfahren, Selke 19.06.15 Grundwasserhydraulik: Finite-DifferenzenVerfahren, OOP, VTK, iFDM 26.06.15 Grundwasserhydraulik: implizites FiniteDifferenzen-Verfahren, Randbedingungen 03.07.15 Grundwasserhydraulik: Finite-ElementeVerfahren 10.07.15 UFZ-Exkursion: VISLAB 17.07.15 FEM 2-D, Klausurvorbereitung Prof. Dr.-Ing. habil. Olaf Kolditz

Hydrosystemanalyse 2015

¨ Ubung

Skript 1.1+2 1.3

BHYWI22-E3 BHYWI22-E4 BHYWI22-E15

1.4, 1.5 1.5, 1.6 1.5

V19: Finite-Elemente-Methode (FEM) 2-D

17.07.2013

Lecture Table of Contents

I

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FEM flexibility

Prof. Dr.-Ing. habil. Olaf Kolditz

Hydrosystemanalyse 2015

V19: Finite-Elemente-Methode (FEM) 2-D

17.07.2013

Finite-Elemente-Methode (FEM)

Abbildung: Modellierung eines Kluftsystems im Kristallin (Herbert Kunz, BGR) 4/22

Prof. Dr.-Ing. habil. Olaf Kolditz

Hydrosystemanalyse 2015

V19: Finite-Elemente-Methode (FEM) 2-D

17.07.2013

FEM: Element-Typen

Abbildung: M¨ ogliche Elementtypen f¨ ur die Finite Elemente Methode (FEM)

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Prof. Dr.-Ing. habil. Olaf Kolditz

Hydrosystemanalyse 2015

V19: Finite-Elemente-Methode (FEM) 2-D

17.07.2013

FEM: 2D Problem

Wir betrachten ein 2D station¨ares Grundwasserstr¨omungsproblem.     ∂ ∂h ∂ ∂h Kx + Ky =0 (1) ∂x ∂x ∂y ∂y

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Prof. Dr.-Ing. habil. Olaf Kolditz

Hydrosystemanalyse 2015

V19: Finite-Elemente-Methode (FEM) 2-D

17.07.2013

FEM: Residuum - Schwache Formulierung

Z N Ωe

∂ ∂x

∂ hˆ Kx ∂x

!

∂ + ∂y

∂ hˆ Ky ∂y

∂ 2 hˆ ∂ 2 hˆ N(x, y ) Kx 2 + Ky 2 ∂x ∂y Ωe

Z

!! dΩe = R e ≈ 0

! dΩe = R e ≈ 0

K: Warum k¨ onnen wir das tun?

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Prof. Dr.-Ing. habil. Olaf Kolditz

(2)

Hydrosystemanalyse 2015

(3)

V19: Finite-Elemente-Methode (FEM) 2-D

17.07.2013

FEM: Partielle Integration Dabei nutzen wir zwei mathematische ’Tricks’, die uns schon bekannt sind (Hydroinformatik II) 1. Kettenregel (integration by parts) ∇(NA) = N∇A + ∇NA N∇A = ∇(|{z} NA ) − ∇NA

(4)

B

2. Definition der Divergenz Z I ∇BdΩ = BdΓ Ω

(5)

∂Ω

3. Darcy’s law A = K∇h

(6)

4. Gradient ∇=( 8/22

∂ ∂ ∂ T , , ) ∂x ∂y ∂z

Prof. Dr.-Ing. habil. Olaf Kolditz

(7) Hydrosystemanalyse 2015

V19: Finite-Elemente-Methode (FEM) 2-D

17.07.2013

FEM:

∂ 2 hˆ ∂ 2 hˆ N(x, y ) Kx 2 + Ky 2 ∂x ∂y Ωe

Z

! dΩe = 0

(8)

Die Gleichung l¨asst sich f¨ ur 2D und 3D Problemstellungen erweitern Z

Z N∇(K∇h)dΩ = −



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I ∇NK∇hdΩ +



Prof. Dr.-Ing. habil. Olaf Kolditz

NK∇hdΓ = 0 ∂Ω

Hydrosystemanalyse 2015

(9)

V19: Finite-Elemente-Methode (FEM) 2-D

17.07.2013

FEM: N¨aherungsl¨osung Z

I ∇NK∇hdΩ =



NK∇hdΓ

(10)

∂Ω

3 X

Ni (x, y )hˆi (t)

(11)

ˆ h(x, y ) = N(x, y )h(t)

(12)

h(x, y , t) =

i=1

Z Ω

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ˆ dΩ = (∇NK∇N) h

I

ˆ dΓ (NK∇N)h

∂Ω

Prof. Dr.-Ing. habil. Olaf Kolditz

Hydrosystemanalyse 2015

(13)

V19: Finite-Elemente-Methode (FEM) 2-D

17.07.2013

FEM: Symbols

   N1  N2 N=   N3

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,

   h1  ˆ= h2 h   h3

Prof. Dr.-Ing. habil. Olaf Kolditz

,

  Kx 0 K=  0

Hydrosystemanalyse 2015

0 Ky 0

 0  0 (14)  Kz

V19: Finite-Elemente-Methode (FEM) 2-D

17.07.2013

FEM: Bildchens

Abbildung: Linear triangular element

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Prof. Dr.-Ing. habil. Olaf Kolditz

Hydrosystemanalyse 2015

V19: Finite-Elemente-Methode (FEM) 2-D

17.07.2013

FEM: Let’s talk about N Obviously, an arbitrary point in the triangle can be identified by use of the following local (area) coordinates (Fig. 3), N1 = A1 /A N2 = A2 /A

(15)

N3 = A3 /A where A is the area of triangular element x1 y1 1 1 x y 1 A = 2 2 2 x3 y3 1

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Prof. Dr.-Ing. habil. Olaf Kolditz

Hydrosystemanalyse 2015

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V19: Finite-Elemente-Methode (FEM) 2-D

17.07.2013

FEM:

From geometrical reasons we have 1 = N1 + N2 + N3

(17)

Furthermore we can write  1 : at node i Ni = 0 : at remaining nodes I

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Bildchens

Prof. Dr.-Ing. habil. Olaf Kolditz

Hydrosystemanalyse 2015

(18)

V19: Finite-Elemente-Methode (FEM) 2-D

17.07.2013

FEM: From this condition it can be concluded that 1 = N1 + N2 + N3 x

= N1 x1 + N2 x2 + N3 x3

y

= N1 y1 + N2 y2 + N3 y3

(19)

Now we write the above equations in following compact matrix form      1 1 1  N1   1  x N2 =  x1 x2 x3  (20)     y y1 y2 y3 N3

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Prof. Dr.-Ing. habil. Olaf Kolditz

Hydrosystemanalyse 2015

V19: Finite-Elemente-Methode (FEM) 2-D

17.07.2013

FEM:

Inversion gives      x2 y3 − x3 y2 y2 − y3 x3 − x2  1   N1  1  N2 x3 y1 − x1 y3 y3 − y1 x1 − x3  x = (21)   2A   N3 x1 y2 − x2 y1 y1 − y2 x2 − x1 y

x = AN N = A−1 x

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Prof. Dr.-Ing. habil. Olaf Kolditz

(22)

Hydrosystemanalyse 2015

V19: Finite-Elemente-Methode (FEM) 2-D

17.07.2013

FEM:

1 [(x2 y3 − x3 y2 ) + (y2 − y3 )x + (x3 − x2 )y ] 2A 1 N2 (x, y ) = [(x3 y1 − x1 y3 ) + (y3 − y1 )x + (x1 − x3 )y ] 2A 1 N3 (x, y ) = [(x1 y2 − x2 y1 ) + (y1 − y2 )x + (x2 − x1 )y ] 2A

N1 (x, y ) =

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Prof. Dr.-Ing. habil. Olaf Kolditz

Hydrosystemanalyse 2015

(23)

V19: Finite-Elemente-Methode (FEM) 2-D

17.07.2013

FEM:

Now the derivatives down.  ∂N1   =    ∂x  ∂N  ∂N2 = =  ∂x ∂x     ∂N3   = ∂x

of the shape functions can be easily written y2 − y3 2A y3 − y1 2A y1 − y2 2A

              

  ∂N1 x3 − x2     =    ∂y 2A       x1 − x3  ∂N2 ∂N = =  ∂y 2A  ∂y      ∂N x − x1    3 2     = ∂y 2A (24)

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Prof. Dr.-Ing. habil. Olaf Kolditz

Hydrosystemanalyse 2015

V19: Finite-Elemente-Methode (FEM) 2-D

17.07.2013

FEM: Stiffness Matrix #1

K

e

Z =

∇N K ∇NT dΩe

(25)

Ωe

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Prof. Dr.-Ing. habil. Olaf Kolditz

Hydrosystemanalyse 2015

V19: Finite-Elemente-Methode (FEM) 2-D

17.07.2013

FEM: Stiffness Matrix #2

Ke

Z =

∇N K ∇NT dΩe  (y2 − y3 )(y2 − y3 ) e Kxx  (y3 − y1 )(y2 − y3 ) 4A (y1 − y2 )(y2 − y3 )  (x3 − x2 )(x3 − x2 ) e Kyy  (x1 − x3 )(x3 − x2 ) 4A (x2 − x1 )(x3 − x2 ) Ωe

=

+

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Prof. Dr.-Ing. habil. Olaf Kolditz

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

Hydrosystemanalyse 2015

(26)  (y2 − y3 )(y1 − y2 ) (y3 − y1 )(y1 − y2 )  (y1 − y2 )(y1 − y2 )  (x3 − x2 )(x2 − x1 ) (x1 − x3 )(x2 − x1 )  (x2 − x1 )(x2 − x1 )

V19: Finite-Elemente-Methode (FEM) 2-D

17.07.2013

FEM: Stiffness Matrix #3

Kije

Z =

∂Ni ∂Nj dΩe (27) Kαβ ∂xα ∂xβ Z Z ∂Ni ∂Ni ∂Nj ∂Nj ∂Nj ∂Nj (Kx + Ky ) dΩe + (Kx + Ky ) dΩe ∂x ∂y ∂x ∂y Ωe ∂x Ωe ∂y Z Z ∂Ni ∂Nj ∂Ni ∂Nj dΩe + Ky dΩe Kx Ωe ∂x ∂x Ωe ∂y ∂y Z Z yj − yk yk − yi xk − xj xi − xk Kx dΩe + Ky dΩe 2A 2A 2A 2A Ωe Ωe Kx Ky (yj − yk )(yk − yi ) + (xk − xj )(xi − xk ) 4A 4A Ωe

= = = =

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Prof. Dr.-Ing. habil. Olaf Kolditz

Hydrosystemanalyse 2015

V19: Finite-Elemente-Methode (FEM) 2-D

17.07.2013

FEM: Exercise

tbd

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Prof. Dr.-Ing. habil. Olaf Kolditz

Hydrosystemanalyse 2015