finite element analysis of boundary value problem

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Finite Element

Analysis of Boundary

Value Problem


FE Analysis of 1D Bars

00 ), ()0(

0)(

Qdx

duEAuu

qdx

duEA

dx

d

Lx

The DE is in the form of

q is the distributed load and

Q0 is the axial force.

Q0

q(x)

x L

Physical Model

FE Model 1 2 3 4 5 6


In FE analysis, we seek an approximation solution over each element.

x x

A B

1 2 Axxdx

duEAQ

1

Bxxdx

duEAQ

2

0)()(

BBAA

x

x

QxwQxwdxwqdx

du

dx

dwEA

B

A

BBAA

x

x

x

x

QxwQxwdxwqwl

dxdx

du

dx

dwEAuwB

B

A

B

A

)()()(

),(

)(),( wluwB

Weak form



Approximation of the solution

1- The approximation solution should be continuous and differentiable

as required by the weak form. (nonzero coefficient matrix)

2- It should be a complete polynomial (capture all possible States, e.g.

constant, linear, ….)

3- It should be an interpolant of variables at the nodes (satisfy EBCs)

2211 )(,)(, uxUuxUbxaU

/, /1 21

2211

xNxN

uNuNU

First order

332211

2 )(,)(,)(, uxUuxUuxUcxbxaU

( / )( / ), / ( / ), / ( / )1 2 3N 1 x 1 2x N 4x 1 x N x 1 2x Second Order

1 2

1 2

3 1 1 2 2 3 3U N u N u N u



FE Model

j

n

j

jj

Nw

NuUu

1

0)()(

BBAA

x

x

QxwQxwdxwqdx

du

dx

dwEA

B

A

and

0)(

0)(

0)(

11

1

22

1

2

1

11

1

1

n

j

jjn

x

x

n

n

j

j

jn

n

j

jj

x

x

n

j

j

j

n

j

jj

x

x

n

j

j

j

QxNdxqNdx

dNu

dx

dNEA

QxNdxqNdx

dNu

dx

dNEA

QxNdxqNdx

dNu

dx

dNEA

B

A

B

A

B

A

If n > 2 then the above integral should modify to

include interior nodal forces

ni

QfuKn

j

iijij

,,2,1

01

Stiffness matrix Force vector

Primary nodal

DOF

Secondary nodal

DOF



FE Model

where

n

j

ijij

i

x

x

ii

ji

x

x

jiij

QQxN

NldxqNf

NNBdxdx

dN

dx

dNEAK

B

A

B

A

1

)(

)(

),(

Note

Note that the problem has 2n unknowns for each element, i.e. ui

and Qi, so it cannot be solved without having another n

conditions. Some of these will be provided by BCs and the

remainder by balance of the secondary variables (forces) at node

common to several element. (assembling process)



FE Model (Linear Element)

/, /1 21

2211

xNxN

uNuNU

/)/1)(/1(

/)/1)(/1(

/)/1)(/1(

0

22

0

12

0

11

AEdxEAK

AEdxEAK

AEdxEAK

qdxxqf

qdxxqf

2/1)/(

2/1)/1(

0

2

0

1

1

1

2}; {

11

11][

qf

AEK eEventually for

Linear shape function



FE Model (Quadratic Element)

)/21(/), /1(/4), /21)(/1( 321

332211

xxNxxNxxN

uNuNuNU

3/8)/8/4)(/4/3(

3/7)/4/3)(/4/3(

0

2

2112

0

22

11

AEdxxxEAKK

AEdxxxEAK

qdxxxqf

qdxxxqf

6/4)/1()/4(

6/1))/(2/31(

0

2

0

2

1



FE Model (Quadratic Element)

For quadratic Shape function

1

4

1

6}; {

781

8168

187

3][

qf

AEK e



Assembly (or connectivity) of elements

In driving the element equation

-Isolate the element from mesh

-Formulate weak form (variational form)

-Developed its finite element model

To solve the total problem

-put the element in its original position

-Impose continuity of PVs at nodal points

-Balance of SVs at connecting nodes

1

1

ee

n uu

.

e e 1

n 1

0 0

0 if no external point source is appliedQ Q

Q if an external point source of Q is applied

1 2 1 2

e 1e

eQ1 eQ2

1

1

eQ 1

2

eQ


(*)


Assembly (or connectivity) of elements (For linear element n=2)

The interelement continuity of the primary variables is imposed by

renaming the two variable une and u1

e+1 at x=xN as one and same,

namely the value of u at the global node N: e e 1

n 1 Nu u U


( )N n 1 e 1 where

For a mesh of E linear finite elements (n=2): 1

1 1

1 2

2 1 2

2 3

2 1 3

E 1 E

2 1 E

E

2 E 1

u U

u u U

u u U

u u U

u U



To enforce balance of secondary variables Qie , eq. (*), we must

add nth equation of the element e to the first equation of the

element e+1 :


n

e e e e

nj j n n

j 1

K u f Q

and

n

e 1 e 1 e 1 e 1

1 j j 1 1

j 1

K u f Q

to give

( ) ( )n

e e e 1 e 1 e e 1 e e 1

nj j 1 j j n 1 n 1

j 1

e e 1

n 1 0

K u K u f f Q Q

f f Q

This process reduces the number of equations from 2E to E+1.



The first equation of the first element and the last equation of the

last element will remain unchanged, except for renaming of the

primary variables. The left-hand of the equation can be written in

terms of the global nodal values as


( )

( ) ( )

( )

( )

( )

e e e e e e e 1 e 1 e 1 e 1 e 1 e 1

n1 1 n2 2 nn n 11 1 12 2 1n n

e e e

n1 N n2 N 1 nn N n 1

e 1 e 1 e 1

11 N n 1 12 N n 1n N 2n 2

e e e

n1 N n2 N 1 n n 1 N n 2

e e 1

nn 11 N

K u K u K u K u K u K u

K U K U K U

K U K U K U

K U K U K U

K K U

e 1 e 1

n 1 12 N n 1n N 2n 2K U K U

where ( )N n 1 e 1




( )

( )

( )

( )

(

1 1 1 1

11 1 12 2 1 1

1 1 2 2 1 2 1 2

21 1 22 11 2 12 3 2 1 2 1

2 2 3 3 2 3 2 3

21 2 22 11 3 12 4 2 1 2 1

E 1 E 1 E E E 1 E E 1 E

21 E 1 22 11 E 12 E 1 2 1 2 1

E E E E

21 E 22 E 1 2 2

K U K U f Q unchanged

K U K K U K U f f Q Q



K U K U f Q un

)changed

For a mesh of E linear finite elements (n=2):




0

.................................................................................

0

1 1

11 12 1

1 1 2 2

12 22 11 12 2

2 2 3321 22 11

E 1 E EE22 11 12

EE 1

22

1

1

2

K K U

K K K K U

UK K K

UK K K

UK

f

f

... ...

1

1

1 2 1 2

1 2 1

2 3 2 3

2 1 2 1

E 1 E E 1 E

2 1 2 1

E E

2 2

Q

f Q Q

f f Q Q

f f Q Q

f Q

In matrix form


FE Analysis of BEAM

Lxxfdx

wdb

dx

d 0) ()(

2

2

2

2


f(x)

L

M0

F0


FE Analysis of BEAM

Weak form

0)(1

2

2

2

2

dxfdx

wdb

dx

dv

e

e

x

x

0)()( 431212

2

2

2

1

1

e

x

e

e

e

x

e

e

x

x

Qdx

dvQxvQ

dx

dvQxvdxvf

dx

wd

dx

vdb

ee

e

e

or

e

x

e

e

e

x

e

e

x

x

x

x

Qdx

dvQxvQ

dx

dvQxvdxvfvl

dxdx

wd

dx

vdbwvB

ee

e

e

e

e

43121

2

2

2

2

1

1

1

)()()(

),(

where

11

2

2

42

2

3

2

2

22

2

1

;

;

ee

ee

x

e

x

e

x

e

x

e

dx

wdbQ

dx

wdb

dx

dQ

dx

wdbQ

dx

wdb

dx

dQ

BCs


FE Analysis of BEAM

Approximation of the solution

1- The approximation solution should be continuous and differentiable


2- It should be a complete polynomial (capture all possible States, e.g.


3- It should be an interpolant of variables at the nodes (satisfy EBCs)

211211

3

4

2

321

)(,)(,)(,)(

eeee xxwxwwxw

xcxcxccw

First order

1 2

1

41321

3

4

2

321

),(;),(

ee x

e

e

e

x

e

e

e

dx

dwuxwu

dx

dwuxwu

xcxcxccw

or


FE Analysis of BEAM

Shape Functions

4

1

)(j

j

e

j

e Nuxw

Calculating Ci and substituting in the equation for w

The interpolation functions in term of local coordinates are

h

x

h

xxN

h

x

h

xN

h

xxN

h

x

h

xN

2

4

32

3

2

2

32

1

;23

1;231


FE Analysis of BEAM

Hermite cubic interpolation function


FE Analysis of BEAM

FE Model

0)(114

12

2

2

2

e

i

x

x

ij

j

x

x

ji QfdxNudxdx

Nd

dx

Ndb

e

e

e

e

or 04

1

ij

j

ij FuK

For b=EI constant and also a constant f over the element.

4

3

2

1

22

22

3 6

6

12}; {

233

3636

323

3636

2][

Q

Q

Q

Q

h

hfhF

hhhh

hh

hhhh

hh

h

EIK


FE Analysis of 1D FIN

Model Boundary Value Problem

00 , )0(

)(

Qx

TkAQTT

TPAqTPdx

dTkA

dx

d


k is thermal conductivity

is convection heat transfer coefficient

is the ambient temperature

q is the heat energy generated per unit volume

Physical Model

FE Model 1 2 3 4 5 6

A, cross section

P, Perimeter

T



Weak form

BA

B

A

B

A

x

e

x

e

x

x

ii

x

x

ji

jiij

dx

dTkAQ

dx

dTkAQ

dxTPqANf

dxNNPdx

dN

dx

dNkAK

21 ;

)(

Assume the lateral surfaces of the bar are isolated and the BCs

21 )(, )0(

)(

TLTTT

Aqdx

dTkA

dx

d



Approximation of the solution 1- the approximation solution should be continuous and differentiable


2- it should be a complete polynomial (capture all possible States, e.g.


3- it should be an interpolant of variables at the nodes (satisfy EBCs)

2211 )(,)(, TxTTxTbxaT

/, /1 21

2211

xNxN

TNTNT

First order

332211

2 )(,)(,)(, TxTTxTTxTcxbxaT

)/21(/), /1(/4), /21)(/1( 321

332211

xxNxxNxxN

TNTNTNT

Second Order

1 2

1 2

3



FE Model

Evaluating the integral using linear shape function

e

e

e

e

eeee

Q

QAq

T

TkA

QfTK

2

1

2

1

1

1

211

11

}{}{}]{[

For a uniform mesh and after assembling NL /

N

N Q

QQ

Q

Aq

T

T

T

kA

1

2

1

1

2

1

1

1

2

1

1

2

1

2

1100

121

011



FE Model

11

11

NN TT

TT

NeforQQ ee ,,3,2 01

1

2

Heat balance at global nodes 2,3,….,N

Boundary conditions at nodes 1 and N+1

N

n Q

Q

Aq

T

T

T

kA

1

1

1

1

2

1

0

1

2

1

2

1100

121

011

After applying the above conditions:


Virtual work as the „weak form‟ of equilibrium

equations for analysis of solids

In a general three-dimensional continuum the equilibrium

equations of an elementary volume can be written in terms of the

components of the symmetric cartesian stress tensor as

1

2

3

0

0 ,

0

xyx xz

x

xy y yz

y

yzxz z

z

bx y z

L

L bx y z

L

bx y z

u x 0L

T

x x xb b bb The body forces acting per unit volume

T

u v wu The displacement vector




The weighting function vector defined as

2 3( ) v ( ) ( ) v

0

xyT x xz

xV Vd u b v L w L d

x y z

u uL

T

u v w u

Integrating each term by parts and rearranging we can write this as

We can now write the integral statement of equilibrium equations as

v

+ ( ) (..) (..)

x xy x y zV

x x xy y xz z

u u vub vb wb d

x y x

u n n n v w d

(*)




where

are tractions acting per unit area

of external boundary surface

of the solid

x x x xy y xz z

y xy x y y yz z

xz x yz y z zz

t n n n

t n n n

n n nt

t

In the first set of bracketed terms in eq. (*) we can recognize

immediately the small strain operators acting on δu, which can be

termed a virtual displacement.

We can therefore introduce a virtual strain defined as

, , , , ,

T

Tu v w u v v w v w

x y z y x z y z y

D u

Arranging the six stress components in a vector σ in an order

corresponding to that used for δε, we can write Eq. (*) simply as




we see from the above that the virtual work statement is precisely

the weak form of equilibrium equations and is valid for non-linear

as well as linear stress–strain (or stress–strain rate) relations.

v vT T T

V Vd d d

u b u t


Appendix

From “Energy Principles and Variational Methods in Applied Mechanics”, by: J. N. Reddy,

2nd ed., John Wiley (2002). (pp. 441--447)


Appendix


Galerkin‟s Method in Elasticity

FE Model

Governing equations

Interpolated Displ Field Interpolated Weighting Function

ii uzyxNu ,,

jj uzyxNv ,,

kk uzyxNw ,,

ixix zyxN ,,

jyjy zyxN ,,

kzkz zyxN ,,



Integrate by part…

0

dVfzyx

fzyx

fzyx

zzzzyxz

yy

yzyxy

V

xxxzxyx

0~ V

dVPuL



Galerkin’s Method in Elasticity Virtual Work

Compare to Total Potential Energy

Virtual Total Potential Energy

i

i

T

iS

T

V

T

V

T dSdVdV PuTufuεσ2

1



0

0

0 ,

0

on .

ˆ on

xyx xz

x

xy y yz

y

yzxz z

z

u

bx y z

bx y z

bx y z

A

A

E u x 0

u u 0B u x 0

σ n t 0

finite element analysis of boundary value problem

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