variational approach in fem part ii

CONTINUUM & FINITE ELEMENT METHOD

Variational Approach in FEM Part II

Prof. Seong Jin ParkMechanical Engineering, POSTECH

Find to minimize .)(x )(x

One wants to obtain an approximate solution to minimize a functional . One of the

historically famous approximate methods for this kind of problem is Rayleigh-Ritz Method, and

the other modern method is the Finite Element Method.

Finite Element Method vs. Rayleigh-Ritz Method

)(x

: approximate solution satisfying the essential B.C.

: trial functions (defined over the whole domain)where

i) Rayleigh-Ritz Method:

n

i

iic1

)()(~

xx

)(xi

Then, ),,,()~

( 21 nccc to be minimized w.r.t. ci.

Therefore,ni

ci

,,1,0)

~(

: n equations for n unknown ci’s.

This method is very simple and easy to understand. However, it is not easy to find a family of trial functions forthe entrie domain satisfying the essential boundary conditions when geometry is complicated. The solution tothis troublesome point can be found in the Finite Element Method.


Then,

iiN )()(~

xx

),,,()~

( 21 n

nii

,,1,0

to be minimized w.r.t. i

: n equations for n unknown i’s.

functions shape :

values nodal :

)(xi

i

N


ii) Finite Element Method

In this case, the shape functions can be found more easily than the trial functions without having to worryabout satisfying the essential boundary conditions, which makes FEM much more useful than Ralyleigh-RitzMethod. In this regard, the Finite Element Method is as modernized approximation method suitable forcomputer environment.

Example: Tight string problem via two methods

minimized. be to dxwydx

dyTxy

l

0

2

2

1)(

* Special case: w(x)=w (constant)

w(x)

y(x)

x

020

02

120

2

2

2

2

1

1

lTA

A

lTA

lw-

A

0 ,4

23

2

1

AT

wlA

l

x

T

wly

sin

4~3

2

T

wly

T

wly

22

125.02

1129.0

2

1~

vs. (Note: )

i) Rayleigh-Ritz Method

l

xA

l

xAy

2sinsin~

21 (Note: Trial functions satisfy essential B.C.)


dxwydx

dyTxy

l

0

2

2

1)( to be minimized is equivalent to

.00

δydxywdx

dy

dx

dyT

l

any for

Introduce elements to the system as depicted below.

1 ① 2 ② 3 ③ 4 x

y e

y1 ey2lx

el

①,② ,③ : element number1, 2, 3, 4 : node number

: local coordinate lx

ii) Finite Element Method


Introduce the approximate solution via interpolation functions (or shape functions) for each element.

)()( e

ii

e yxNxy

Then,

e

ii

e

ii

e

ii

e

ydx

dNy

dx

d

dx

dy

yNxy

ydx

dN

dx

dy

)(

and the variation of functional over each element is summed to result in the variation of thewhole system, i.e.,

)3()2()1( e

e


Let us consider forany element

el

lee dxyw

dx

dy

dx

dyT

0 (henceforth xl → x for convenience)

form) matrix a (in

form) indicial an (in

e

e

e

ee

T

e

e

e

i

e

j

e

ij

e

i

le

i

e

j

leije

i

lee

ii

e

i

ie

j

je

f

f

y

yK

y

y

fyKy

dxwNydxdx

dN

dx

dNTy

dxywNydx

dNy

dx

dNT

2

1

2

1

2

1

00

0


where

force nodal equivalent- work:

matrix stiffnesselement :

0

0

le

i

e

i

leije

ij

dxwNf

dxdx

dN

dx

dNTK

summation

iijiji

e

e

yFyKy

any for 0

FyK


Linear element for simplicity

eee yxNyxNxy 2211 )()()(

)(

1)(

2

1

N

N)(

e

l

l

x

d

dN

ldx

d

d

dN

dx

dN i

e

ii 1

ee ldx

dN

ldx

dN 1 ,

1 21

)(1 N )(2 N

1 1

0 1


11

11

e

e

l

TK

1

1

202

1 ele wl

dxN

Nwf

e

2/1

1

1

2/1

3

1100

1210

0121

0011

3

4

3

2

1

wl

y

y

y

y

l

T

Global matrix equation

B.C. y1 = y4 =0T

wlyy

2

329

1



Notes:

1. 1st and 4th equations are not to be used (or obtained more precisely) since and are notarbitrary, but zero. As a matter of fact, however, introduction of boundary conditions replacesthose equations. The reaction force and can be obtained from the 1st and 4th equations,respectively.

2. Advantages of variational approach over the direct one: 1) Use of scalar quantity (energy) versusvectors, 2) Ease in treatment of distributed load

3. Treatment of concentrated loads:

1y 4y

1F 4F

cF

1

y cy

FEM for Second-Order Elliptic Partial Differential Equation

* Steady state heat conduction, flow though porous media, torsion, etc.

2S

1S

x

y

z

n̂

in ),,( zyxf

zk

zyk

yxk

xzyx

B.C.

2

1

on 0),,(),,(

on ),,(

Szyxhzyxgnz

kny

knx

k

Szyx

zzyyxx

.any for 0Ji.e.


Minimize

The above partial differential equation with boundary conditions is equivalent to the following variational principle:

With many elements introduced, one can sum the contributions of each element to the functionals as described below:

e

e

e

e

JJ

JJ

e

ii

e

e

ii

e

N

N

Introduce the approximate solution in terms of shape functions and nodal values of over each element:

e

e

2S


e

j

S

ij

e

i

S

i

e

i

i

e

i

e

jij

zij

yij

x

e

ie

ee

e

e

dSNhNdSN

dVfN

dVz

N

z

Nk

y

N

y

Nk

x

N

x

NkJ

22

g

e

Si

e

fi

e

j

e

Sij

e

j

e

Cij

e

ie RRKKJ


where

e

dVz

N

z

Nk

y

N

y

Nk

x

N

x

NkK

jiz

jiy

jix

e

Cij

2S

ij

e

Sij

e

dSNhNK

e

dVfNR i

e

fi

2

gS

i

e

Si

e

dSNR

(stiffness matrix due to conduction)

(forcing matrix due to convection)

(forcing matrix due to distributed heat sink)

(forcing matrix due to distributed heat outflux)


After the assembly procedure, one can obtain

1

0 for any arbitray , with 0 for nodes on .

ee

i Cij j Sij j fi Si

i

J J

K K R R

S

0 SfSC RRKK

FK

SC KKK

Sf RRF forcing matrix due to heat source and heat flux


For a given statically admissible stress field , consider any kinematically admissible virtual displacement and external work due to the virtual displacement.

ij

iu

A. Principle of Virtual Displacement (Work)

Variational Principle for Deformation

jiji nt

0, ijij f

ntdisplaceme virtual: iu

state mequilibriu

dV

dVufx

dVx

u

dVufdSun

dVufdSutW

ijij

ii

j

ij

j

iij

iiijij

iiiiext

where

0 ,

2

1 ,

2

1

ijijijij

j

i

i

j

j

i

ij

i

j

j

i

ij

x

u

x

u

x

u

x

u

x

u


dVdVufdSutW ijijiiiiext

0

dVufdSutdV iiiiijij

Physical meaning

iuFind Any virtual displacement iiij ft ,,if are in equilibrium

Principle of virtual velocity (power)

dVt

dVt

ufdS

t

ut

t

W ij

ij

i

i

i

i

ext

Principle of virtual displacement


( can be replaced with respectively.)

Notes:

1. Kinematically admissible virtual displacement where prescribed.

2. Statically admissible stress field satisfies not only the equilibrium equation but also theprescribed traction boundary condition (i.e., natural boundary condition), i.e.

3. Principle of virtual velocity (power)

4. If inertial term is included in the body force term, the principle of virtual displacement canbe extended to fictitious equilibrium state.


0 ii uu ii uu

ijiji tnt

dVt

dVt

ufdS

t

ut

t

W ij

ij

i

i

i

i

ext

ijii δdvv

iu

dVdVuvfdSut ijijiiiii ijiu ,iji dv ,

B. Principle of Minimum Potential Energy

Principle of virtual displacement

dVdVufdSutW ijijiiiiext

For an elastic body, there exists strain energy density, Uo, such thatij

oij

U

then UdVUdVUdVU

dV ooij

ij

oijij


ii ρft and

where

dVUU o: strain energy

UWext

Define the potential energy V as

dVufdSutV iiii with fixed

VWext Then

Therefore 0VU


Defining the total potential energy asp

VUp 0 p

With and ij

o

ij

U

ijjiij uu ,,

2

1


yields

In summary, the deformation of any elastic body (linear or nonlinear) is governed by minimizing the functional, the total potential energy,

dVufdSutdVUu iiiioip

Note: Principle of Virtual Work is valid for any material, whereas Principle of Minimum Potential Energy is valid only for elastic materials.

C. Principle of Complementary Virtual Work

)( on

in

Stn

f

ijij

ijij

0,

dV

dVufdVu

dVfudSnu

dVfudStuW

ijij

iijijijji

iijiji

iiii

,,

*

d

d

*W

extW


Consider a variation of statically admissible stress field and external forces while keeping kinematically admissible displacement .

ijii tf ,

iu

Define a complementary virtual work as*W

ij

o

ij

U

*

(a constitutive equation)


Now as a counterpart to the Principle of Minimum Potential Energy for an elastic material, consider the case for an elastic material for which a complementary strain energy density exists as below:

Then we can rewrite the right hand side of the expression for as*W

***

*

* UdVUdVUdVU

dVW ooij

ij

o

ijij

and the left hand side can be rewritten in terms of the complementary potential energy as follows

dVfudStuV iiii

*

dVfudStuV iiii

*with iu(keeping fixed)

*** ,, VUft iiij

0*

Therefore, one can have

<Total Complementary potential energy>

For any statically admissible stress, force system


Note:

ip u

iiij ft ,,*

: leads to displacement-based FEM yielding a stiffness matrix

: leads to equilibrium-based FEM yielding a flexibility matrix

Principle of Virtual Displacement (Work): 0

dVufdSutdV iiiiijijp

fixed) (with iiijiiiiijijp ftdVufdSutdV ,,

Let us apply the principle of virtual displacement (or principle of minimum potential energy) to the two-dimensional elastic deformation problem.

Principle of Minimum Potential Energy:

dVufdSutdVUu iiiioip

ij

o

ij

U

ijjiij uu ,,

2

1

0)()(

k

i

k

k

iiiiiiijijp uFdVuufdSutdV

where

Variational expression:

Displacement-based FEM for Elasticity

0)()(

k

i

k

k

iiiiiiijijp uFdVuufdSutdV


In this section, let us include the inertia force and concentrated forces as the most generalFEM formulation for elasticity. So, consider the following variational expression as thestarting form.

VNyxv

yxuu

),(

),(

3

3

2

2

1

1

321

321

000

000

v

u

v

u

v

u

NNN

NNN

v

uu

VNv

uu

Displacement approximation via shape functions

Acceleration

1

2

3


VN

xy

y

x

x

N

y

N

x

N

y

N

x

N

y

N

y

N

y

N

y

Nx

N

x

N

x

N

N

332211

321

321

000

000

xy

y

x

xy

y

x

,

C

Strain matrix:

e.g.

Stress-Strain relation(constitutive law):


stress plane for

2

100

01

01

1 2

EC

strain plane for

2

2100

01

01

)21)(1(

EC

dVCUT

2

1

)()(

2

1 k

i

k

k

iiiiii

T

p uFdVuufdSutdVC

0)()(

k

i

k

k

iiiiii

T

p uFdVuufdSutdVC

and the total potential energy becomes


Note: With this notation, the strain energy can be represented as

y

x

t

tt

y

x

f

ff

)(

)(

)(

k

y

k

xk

F

FF

Define the force matrices as follows:

: traction force

: body force

: concentrated force applied at k-th position


Note: In case of initial strain , is to be replaced with . In case of initial stress,

init init

initC

VNyxv

yxuu

),(

),(

VN

xy

yy

xx

k

kTT

e

e

p

k

k

T

k

k

e

e

pp

FNV

Fv

u

k

)(

)(

)(

)(

)(

x

eeee

eee

dVuudVfudStudVC

dVuufdSutdVC

TT

S

TT

iii

S

ii

Te

p

2

2

Virtual displacement and corresponding strain:

Variation of total potential energy:

For an element:


eee

b

e

d

eeTe

TT

TT

S

TTTTe

p

VmFFVKV

VdVNNV

dVfNVdStNVVdVNCNV

e

eee

2

VVMFFFVKV cbd

T

p any for 0

Introducing the matrix notations for approximated displacement field andconstitutive law and so on into the above equation gives:

After assembly, one finally obtains:


dVNNmTe

e

dVNCNKTe

e

2S

Te

d

e

dStNF

dVfNFTe

b

e

ek

kTe

C FNFc

)(

x

FFFFVKVM cbd

where : element mass matrix

: element stiffness matrix

: work equivalent nodal force

: work equivalent nodal force

: concentrated force


is the functional to be minimized. results in .)

FVVKVTT

p 2

1

0 p FVK

variational approach in fem part ii

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