FINITE ELEMENT METHOD IN - Sim&Tec · FINITE ELEMENT METHOD IN ... From Bathe, Finite Element...

FINITE ELEMENT METHOD IN FLUID DYNAMICSFLUID DYNAMICS

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

2www.simytec.com

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

≠==

3www.simytec.com

k

The finite element method2D example2D example

s 1

Natural coordinate systeminside each element (r, s)

r=1s=1

2

r=‐14

s=‐1

3

4www.simytec.com

The finite element method2D four – nodes element2D four nodes element

s

12

h1+1

r

1

3 4

5www.simytec.com

The finite element method2D four – nodes element2D four nodes element

s

12

h2+1

r

1

3 4

6www.simytec.com

The finite element method2D four – nodes element2D four nodes element

s

12

h3

r

3

+1

3 4

7www.simytec.com

The finite element method2D four – nodes element2D four nodes element

s

12

h4

r

1

+1

3 4

8www.simytec.com

The finite element method

to be able to represent constant temperature situationsp

Are these meshes acceptable?

YESYES

NO YES YES

9www.simytec.com

The finite element methodGood MeshGood Mesh

10www.simytec.com

The finite element methodGood MeshGood Mesh

11www.simytec.com

The finite element methodGood Mesh

However!!!

Minimize the element distortions to have good predictive capability

Target for each element: det(J)=const

12www.simytec.com

The finite element methodIsoparametric elementsIsoparametric elements

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

kk xtsrhtsrx ),,(),,( =

kk

kk

ztsrhtsrzytsrhtsry

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

==

13www.simytec.com

The finite element method

14www.simytec.com

From Bathe, Finite Element Procedures

The finite element method

From Bathe, Finite ElementProcedures

15www.simytec.com

FEM heat transfer

16www.simytec.com

FEM heat transfer

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

17www.simytec.com

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

18www.simytec.com

Laminar Flow

Boundary conditions inin

in

in

Initial conditions

19www.simytec.com

in

Laminar FlowNon-linearityNon linearity

Picard Method

N t R h M th dNewton-Raphson Method

20www.simytec.com

Laminar FlowGalerkin Method – Newton Raphson MethodGalerkin Method Newton Raphson Method

21www.simytec.com

Laminar FlowSUPG MethodSUPG MethodVP formulation

22www.simytec.com

Laminar Flow

SUPG MethodPenalization formulation

23www.simytec.com

ChannelGeometryGeometry

Boundary conditions at the walls: non-slip

Boundary conditions at the entrance: velocities

24www.simytec.com

ChannelLaminar flowLaminar flow

Velocity = 10e-4 m/sReynolds = 10

25www.simytec.com

y

ChannelLaminar flowLaminar flow

Velocities in the central line

Velocity = 10e-4 m/s

26www.simytec.com

Reynolds = 10

ChannelLaminar flowLaminar flow

Velocity = 10e-3 m/sReynolds = 100

27www.simytec.com

ChannelLaminar flowLaminar flow

Velocity = 0.01 m/sReynolds = 1000

28www.simytec.com

Cavity with sliding wallGeometryGeometry

29www.simytec.com

Cavity with sliding wallBoundary conditionsBoundary conditions

Velocities at the upper wall No slip at the other walls

30www.simytec.com

Laminar flow

Cavity with sliding wallLaminar flow

31www.simytec.com

Backward Facing Step GeometryGeometry

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

32www.simytec.com

Backward Facing Step Laminar flowLaminar flow

Velocity = 0.00022 m/sReynolds = 440

33www.simytec.com