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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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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
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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The finite element method2D four – nodes element2D four nodes element
s
12
h1+1
r
1
3 4
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The finite element method2D four – nodes element2D four nodes element
s
12
h2+1
r
1
3 4
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The finite element method2D four – nodes element2D four nodes element
s
12
h3
r
3
+1
3 4
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The finite element method2D four – nodes element2D four nodes element
s
12
h4
r
1
+1
3 4
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The finite element method
to be able to represent constant temperature situationsp
Are these meshes acceptable?
YESYES
NO YES YES
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The finite element methodGood MeshGood Mesh
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The finite element methodGood MeshGood Mesh
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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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The finite element methodIsoparametric elementsIsoparametric elements
kk TtsrhtsrT ),,(),,(~ =
kk xtsrhtsrx ),,(),,( =
kk
kk
ztsrhtsrzytsrhtsry
),,(),,(),,(),,(
==
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The finite element method
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From Bathe, Finite Element Procedures
The finite element method
From Bathe, Finite ElementProcedures
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FEM heat transfer
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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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FEM heat transferSUPG: Streamline Upwind Petrov Galerkin MethodSUPG: Streamline Upwind Petrov Galerkin Method
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Laminar Flow
Boundary conditions inin
in
in
Initial conditions
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in
Laminar FlowNon-linearityNon linearity
Picard Method
N t R h M th dNewton-Raphson Method
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Laminar FlowGalerkin Method – Newton Raphson MethodGalerkin Method Newton Raphson Method
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Laminar FlowSUPG MethodSUPG MethodVP formulation
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Laminar Flow
SUPG MethodPenalization formulation
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ChannelGeometryGeometry
Boundary conditions at the walls: non-slip
Boundary conditions at the entrance: velocities
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ChannelLaminar flowLaminar flow
Velocity = 10e-4 m/sReynolds = 10
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y
ChannelLaminar flowLaminar flow
Velocities in the central line
Velocity = 10e-4 m/s
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Reynolds = 10
ChannelLaminar flowLaminar flow
Velocity = 10e-3 m/sReynolds = 100
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ChannelLaminar flowLaminar flow
Velocity = 0.01 m/sReynolds = 1000
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Cavity with sliding wallGeometryGeometry
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Cavity with sliding wallBoundary conditionsBoundary conditions
Velocities at the upper wall No slip at the other walls
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Laminar flow
Cavity with sliding wallLaminar flow
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Backward Facing Step GeometryGeometry
Boundary conditions at the walls:Boundary conditions at the walls:No slip (B´s verdes)
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Backward Facing Step Laminar flowLaminar flow
Velocity = 0.00022 m/sReynolds = 440
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