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

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  • FINITE ELEMENT METHOD IN FLUID DYNAMICSFLUID DYNAMICS

    P 2 Th fi i l h dPart 2: The finite element methodMarcelaB.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 (Bs verdes)

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  • Backward Facing Step Laminar flowLaminar flow

    Velocity = 0.00022 m/sReynolds = 440

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