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

• View
302

9

Embed Size (px)

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

• 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 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

Recommended ##### Finite Element Method - Massachusetts Institute of .Finite Element Method January 12, 2004 ... -
Documents ##### Non-linear finite element method The finite element .The finite element method is first developed
Documents ##### FOUNDMENTALS FOR FINITE ELEMENT METHOD CHAPTER1: The Finite Element Method A Practical Course The Finite Element Method A Practical Course
Documents ##### Introduction to Finite Element Method - .Introduction to Finite Element Method By ... The finite
Documents ##### การวิเคราะห และ ... foundation on Ground ... Discrete element method - Finite difference method - Finite element method (FEM) - Finite grid method (FGM)
Documents ##### 1 09 - finite element method 09 - finite element method - density growth - implementation
Documents ##### AN INTRODUCTION TO THE FINITE ELEMENT METHOD FOR آ  C. Finite Element Method The finite element method
Documents ##### Finite Element Method AERSP 301 Finite Element Method Jose Palacios July 2008
Documents ##### Finite Element Method Applied in Electromagnetic .Finite Element Method Applied in Electromagnetic
Documents ##### Finite element methods in scientific Finite element method (FEM) Finite volume method (FVM) Finite difference
Documents Documents