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FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

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Page 1: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15
Page 2: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

FINITE ELEMENT METHOD IN FLUID DYNAMICSFLUID DYNAMICS

P 2 Th fi i l h dPart 2: The finite element methodMarcela B. Goldschmit

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Page 3: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

The finite element method2D / 3D Problems2D / 3D Problems

Elements and nodes

The interpolation functions ~ VhVNNOD

∑The interpolation functions inside an element

i t

,,,,,""var:1

f til tih

TpvvvexamplesknodeatiableV

VhV

zyxk

kk= ∑

""0""1

int:

knodeathknodeath

functionerpolationh

k

k

k

≠==

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k

Page 4: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

The finite element method2D example2D example

s 1

Natural coordinate systeminside each element (r, s)

r=1s=1

2

r=‐14

s=‐1

3

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Page 5: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

The finite element method2D four – nodes element2D four nodes element

s

12

h1+1

r

1

3 4

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Page 6: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

The finite element method2D four – nodes element2D four nodes element

s

12

h2+1

r

1

3 4

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Page 7: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

The finite element method2D four – nodes element2D four nodes element

s

12

h3

r

3

+1

3 4

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Page 8: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

The finite element method2D four – nodes element2D four nodes element

s

12

h4

r

1

+1

3 4

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Page 9: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

The finite element method

to be able to represent constant temperature situationsp

Are these meshes acceptable?

YESYES

NO YES YES

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Page 10: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

The finite element methodGood MeshGood Mesh

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Page 11: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

The finite element methodGood MeshGood Mesh

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Page 12: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

The finite element methodGood Mesh

However!!!

Minimize the element distortions to have good predictive capability

Target for each element: det(J)=const

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Page 13: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

The finite element methodIsoparametric elementsIsoparametric elements

kk TtsrhtsrT ),,(),,(~ =

kk xtsrhtsrx ),,(),,( =

kk

kk

ztsrhtsrzytsrhtsry

),,(),,(),,(),,(

==

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Page 14: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

The finite element method

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From Bathe, Finite Element Procedures

Page 15: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

The finite element method

From Bathe, Finite ElementProcedures

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Page 16: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

FEM heat transfer

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Page 17: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

FEM heat transfer

SUPG: Streamline Upwind Petrov Galerkin Method :The weighted functions are different of the approximation are different of the approximation functions

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Page 18: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

FEM heat transferSUPG: Streamline Upwind Petrov Galerkin MethodSUPG: Streamline Upwind Petrov Galerkin Method

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Page 19: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

Laminar Flow

Boundary conditions inin

in

in

Initial conditions

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in

Page 20: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

Laminar FlowNon-linearityNon linearity

Picard Method

N t R h M th dNewton-Raphson Method

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Page 21: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

Laminar FlowGalerkin Method – Newton Raphson MethodGalerkin Method Newton Raphson Method

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Page 22: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

Laminar FlowSUPG MethodSUPG MethodVP formulation

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Page 23: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

Laminar Flow

SUPG MethodPenalization formulation

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Page 24: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

ChannelGeometryGeometry

Boundary conditions at the walls: non-slip

Boundary conditions at the entrance: velocities

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Page 25: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

ChannelLaminar flowLaminar flow

Velocity = 10e-4 m/sReynolds = 10

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y

Page 26: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

ChannelLaminar flowLaminar flow

Velocities in the central line

Velocity = 10e-4 m/s

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Reynolds = 10

Page 27: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

ChannelLaminar flowLaminar flow

Velocity = 10e-3 m/sReynolds = 100

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Page 28: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

ChannelLaminar flowLaminar flow

Velocity = 0.01 m/sReynolds = 1000

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Page 29: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

Cavity with sliding wallGeometryGeometry

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Page 30: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

Cavity with sliding wallBoundary conditionsBoundary conditions

Velocities at the upper wall No slip at the other walls

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Page 31: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

Laminar flow

Cavity with sliding wallLaminar flow

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Page 32: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

Backward Facing Step GeometryGeometry

Boundary conditions at the walls:Boundary conditions at the walls:No slip (B´s verdes)

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Page 33: FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element Procedures. The finite element method From Bathe, Finite Element Procedures 15

Backward Facing Step Laminar flowLaminar flow

Velocity = 0.00022 m/sReynolds = 440

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