V. Finite Element Method - .5.1 Introduction to Finite Element Method. ... Basic idea of Finite Element

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  • V. Finite Element Method

    5.1 Introduction to Finite Element Method

  • 5.1 Introduction to FEM

  • Boundary value problem Problem definition

    * 41

    ( )12

    x x x

    Variational principle

    Prob. 1

    Prob. 2

    1k Exact :

    22

    2, 0 1

    0 0, 1 0

    dx x

    dx

    212

    0

    1( 2 ( )

    2

    (0) 0, (1) 0

    dExtremize F x x dx

    dx

    subject to

    2 4

    1 2

    1( ) ( )

    12x x x x C x C

    Ritz method to differential equation

    0 0 , 1 0

    * x

    *F F x

    2

    1x x

    1x x

    2 1x x

    41

    12x x

  • ( 1)(2 1) 0y x x x 2 2( 1)Extremize y x x

    22

    2, 0 1, (0) 0, (1) 0

    d Tx x T T

    dx

    212

    0

    ( ) 2 ( )dT

    Extremize F T x T x dxdx

    (0) 0, (1) 0T T

    Trial function:

    1

    ( ) ( ), (0) 0, (1) 0n

    i ii

    T x C f x T T

    ( Prob. A)

    ( Prob. B)

    Ritz method to differential equation

    Extremization of a function and its related algebraic equation

    Extremization of a functional and its related differentiation equation

  • Ritz method to differential equation

    Necessary condition for to be extreme :

    Approximation

    21 2( ) 1 1x C x x C x x

    1 2

    2 3 4

    1 2 1 20

    1( ) 1 2 2 3 2 1 2 1

    2F C x C x x C x x C x x dx

    1 122 2 2 2

    1 200

    1 11 2 2 3

    2 2x dx C x x dx C

    1 1 1

    2 3 4

    1 2 1 20 0 0

    1 2 2 3 1 1x x x dx C C x x dx C x x dx C

    1 2

    0, 0F F

    C C

    2 2

    1 2 1 2 1 2 1 2

    1 1 1 1 1( , ) ( )

    6 15 6 20 30F C C F C C C C C C

    Linear equation: 1 1 22

    1 1 1 110 5 3 ,

    5 4 230 60 15 6

    CC C

    C

    2 31

    ( ) 2 3 - 530

    x x x x Approximate solution:

    Transformation: Function space Finite dimensional vector space

    * 41

    ( )12

    x x x Exactsolution

    21

    2

    0

    1( 2 ( )

    2

    (0) 0, (1) 0

    dExtremize F x x dx

    dx

    subject to

    Trial function

    1 2( , )F C C

  • 12

    0[ ( ) ( ) ( )] 0

    (0) 0, (1) 0

    ( )

    (0) 0 (1) 0

    where is arbitrary exc

    x x x x

    ept t

    d

    h

    x

    a

    an

    tx

    d

    22

    20, 0 1

    0 0, 1 0

    dx x

    dx

    0 0 , 1 0

    * x

    2

    1x x

    1x x

    2 1x x

    41

    12x x

    21

    2

    20( ) 0

    0 1 0

    dx x dx

    dx

    11 2

    0 0( ) ( ) ( ) ( ) ( ) 0

    (0) (1) 0

    Assume (0) (1) 0

    x x x x x x dx

    ( ) is arbitrary.x

    Weak form

    sin x

    tan x

    cos x

    logxe x

    2x

    0 0 , 1 0

    2

    1x x

    1x x

    2 1x x

    41

    12x x

    sin x

    tan x

    cos x

    logxe x

    2x

    b bb

    aa au v uv uv

    where is arbitrary except that

    Weighted residual approach to differential equation

    Prob. 1 Prob. 2

    Set of functions

  • Linear equations

    2 31

    ( ) 2 3 530

    x x x x Approx. solution :

    21 2 1 2( ) 1 1 ; and are unknownx C x x C x x C C

    21 2 1 2( ) 1 1 ; and are arbitraryx W x x W x x W W

    1 2 2 3 4

    1 2 1 2 1 20(1 2 ) (2 3 ) (1 2 ) (2 3 ) (1 ) (1 ) 0C x C x x W x W x x dx W x x W x x dx

    1 1 12 3

    1 1 20 0 0

    1 1 12 2 2 4

    2 1 20 0 0

    (1 2 )(1 2 ) (1 2 )(2 3 ) (1 )

    (2 3 )(1 2 ) (2 3 )(2 3 ) (1 ) 0

    W x x dx C x x x dx C x x dx

    W x x x dx C x x x x dx C x x dx

    2

    1 15 4

    0 0

    1 1, , etc.

    6 5x dx x dx

    1

    12

    2

    ( ) (1 )

    ( ) (1 )

    x x x

    x x x

    1 1 2 2 0W W

    W1 are W2 are arbitrary

    1x x

    2 1x x 4

    1

    12x x

    sin x

    logxe x

    0 0 , 1 0

    2

    1x x

    Galerkin approach

    11 2

    2

    1 1 1 110 5 3 ,

    5 4 230 60 15 6

    CC C

    C

    1

    2

    0

    0

    Trial function

    Basic function

    Approximate weighting function

    Set of approximate

    weighting functions

  • Characteristics of solution convergence

    Accuracy Accuracy

    Requirements on basic function

    Linearly independent

    or

    1

    1 1

    ( ( )) ( ( ))n n

    i i i ii i

    C x C x

    *1

    lim ( ) ( )n

    i ini

    x C x x

    i x

    pi x H 1( ) ( )pi x C

    Error 2.2%

    Error 20% *1 1

    0 , 0 , 12 15

    *1 7

    1 , 14 30

    * max max0.029165, 0.029799 x x

    Comparison between approximate and exact solutions

    Exact

    Approximate

    2 31

    ( ) 2 3 530

    x x x x

    * 41

    ( )12

    x x x

    maxx

    Accuracy of the approximate solution

  • Trial function : 1

    1

    1

    12 for 01 21 2

    12 2 (1 ) for 12

    C x xx C x

    C x x x

    7 7 1( )

    192 96 2x x Approximate solution :

    Eaxct

    FE solution

    11

    1 22

    x x

    1 x

    Superconvergence

    Basic idea of Finite Element Method (FEM)

  • Exact

    FE solutions

    FEM = Ritz or Galerkin method + FE discretization and interpolation (approximation)

    FE discretization and interpolation function: ( )iN x

    Finite element solutions

    3( )N x

    1 2 3 4

    2 ( )N x

    2 2 3 3( ) ( ) ( ) in Finite Element Methodx N x N x

    1 2 2( ) ( ) ( ) in Ritz or Galerkin methodx C x C x

    Technique of making basic functions

    FE solutions with different FEA models

    ( )iN x ( )iN x

    1 1

    ( )iN x

    1

    0 1 01

    ElementNode

  • Definition of a beam deflection problem

    4 2* 5( )

    12 2 12

    x xv x

    Solution:

    22

    21 , 0 1

    dx x

    dx

    (0) 0, (1) 0v v

    Variational principle:21

    2

    0

    ( ) 2( 1) ( )dv

    Extremize F v x v x dxdx

    (1) 0subject to v

    Ritz method

    It should satisfy essential boundary condition, (1)=0 v

    [Ex. 2.1]1( ) )v x C x = (1-

    [Ex. 2.2] 21 2( ) ) )v x C x C x = (1- (1-

    [Ex. 2-3] 2 21 2( ) ) )v x C x x C x= (1- (1-

    BVP based on beam theory:

    V

    ( )V x

    x

    0 1

    2

    x

    0 1

    1

    ( )bM x

    ( ) ( )bEIv x M x , (0) 0, ( ) 0v v L

    2( ) 1bM x x

    Ritz method to a beam deflection

    01, 2, 1EI w L

  • [Ex. 2.3] :2 2

    1 2( ) (1 ) (1 )v x C x x C x

    12 2 2 2 3 2

    1 2 1 20

    ( ) { (2 3 ) ( 2 )} 2( 1){ ( ) (1 )}F v C x x C x x C x x C x dx 2 2

    1 2 1 2 1 2 1 2

    5 4 1 1 16( , )

    12 3 3 10 15C C C C C C F C C

    1

    2

    4 1 1

    15 3 101 8 16

    3 3 15

    C

    C

    34 153 113

    27 270 270v x x x

    4 2* 5( )

    12 2 12

    x xv x Exact:

    Ritz method to a beam deflection

  • 5.2 FEM of Partial Differential

    Equations

  • Poissons equation

    Poissons equation

    TT T on S

    ( , ) 0

    k k f x yx x x y

    22

    2, 0 1

    0 0, 1 0

    dx x

    dx

    212

    0

    1( 2 ( )

    2

    (0) 0, (1) 0

    dExtremize F x x dx

    dx

    subject to

    221

    ( ) 2 ( , ) ( , )2

    Extremize F k k f x y x y dxdyx y

    subject toTT T on S

    1 1 2 2( , ) ( , ) ( , ) J JJ

    x y N x y N x y N ?, ? J J

    N

    2D

    1D

    = 3D

    Nodal value

  • Mesh

    Mesh system

    J

    Interpolation function ( , )JN x y

    1

    1

    -1

    -1

  • Intelligent remeshing of quadrilaterals

  • Intelligent remeshing of tetrahedrals

    Accurate description

    with minimum elements!

    As number of elements increases,that of remeshings does so much,which deteriorates solution accuracy.

  • 2D mesh systems with higher quality

    Initial

    Final

  • KSTP Fall 2016.

    3D mesh systems with higher quality

  • 4.5 Finite element formulation

    Equation of equilibrium

    Equation of heat conduction

    T

    ,ij ij

    g ij ijq , , T

  • Coordinate axis

    2

    3

    1

    , , 1, 2, 3 axis

    Mechanical quantities

    , 1, 2 12

    Unit vector

    , , , ,

    Kronecker delta

    0 1 =

    First and second order

    =

    = =

    11 12 1321 22 2331 32 33

    =

    Partial differentiation

    , =

    , , =

    2

    , , =

    , , =

    Summation

    =1

    3

    , + = 0 , + = 0

    : ( ) :

    Divergence theorem

    ,.., =

    ,..

    :

    Permutation symbol

    011

    = = =

    , , = 1,2,3 2,3,1 = 3,1,2

    , , = 1,3,2 2,1,3 = 3,2,1

    Examples

    = =

    .

    ,

    ,

    2 ,

    = =

  • Finite element analysis of 3D elastic problems of isotropic materials

    0ti

    ij ij i i i iV S V

    dV t dS f dV , 0 on

    ii i i uu u S

    Weak form

    S

    P

    S

    VV

    2 2 2ij ij xx xx yy yy zz zz xy xy yz yz zx zx

    i i

    Finite element equations

    T

    , , , , ,xx yy zz xy yz zx

    1, 1

    2, 2

    3, 3

    1, 2 2, 1

    2, 3 3, 2

    3, 1 1, 3

    2

    2

    2

    xx

    yy

    zz

    i

    xy

    yz

    zx

    i ij jD

    1, 1

    2, 2

    3, 3

    1, 2 2, 1

    2, 3 3, 2