ME424 524 Presentation 08

7/25/2019 ME424 524 Presentation 08

1/23

Isoparametric Elements

Element Stifness Matrices

Structural Mechanics

Displacement-basedFormulations


2/23

General Approach Specic

Example

We ill loo! at manipulation o" the mechanics#uantities $displacement% strain% stress& usin' shape"unctions

(he approach is #uite 'eneral% and is used to

"ormulate a number o" diferent elements We ill use a specic example to ma!e the

de)elopment more concrete $*+&

We ill start "rom the nodal displacement

representation% or! toard strain and stress% andnall, element stifness

(here is a lot 'oin' on here% pa, attention to boththe o)erall themes and the detailed steps


3/23

Master Element Mappin'

.ote/ e ill use a"or and b"or because I can0tremember% pronounce% or le'ibl, rite 1xi2 and 1eta2

master elementactual element


4/23

3ilinear *uadrilateral $*+&

Interpolation in)ol)es the summation o" nodal )aluesmultiplied b, correspondin' shapes "unctions

x

y

=

Nix iNiy i

= N[ ] c{ }

u

v

=

NiuiNiv i

= N[ ] d{ }

c{ } = x1 y1 x2 y2 x3 y3 x4 y4[ ]T

d{ } = u1 v1 u2 v2 u3 v3 u4 v4[ ]T

N[ ] =N1 0 N2 0 N3 0 N4 0

0 N1 0 N2 0 N3 0 N4

- here -'eometr, interpolation eld )ariable interpolation

nodal coordinates

nodal displacements

shape "unctions


5/23

Start ith the element displacement eld

We ha)e to 'i)e it some "unctional "orm in order to or! ith it

4et it be dened o)er the element b, our interpolation scheme

{u} 5 displacements continuousl, dened $all components& o)er

an element[N] 5 the element shape "unctions in master element coordinates

{d} 5 the nodal $discrete& displacement )alues

*+ - Displacements

u{ } = N[ ] d{ }

N1 =

1

4 1 a( ) 1b( )

N2 =1

41+a( ) 1b( )

N3 =

1

4 1+a( ) 1+ b( )

N4 =1

41 a( ) 1+ b( )


6/23

.o calculate element strains "rom the displacementeld

(his is 6ust the usual strain-displacement relationshipritten in compact "orm ith an operator matrix

Strain "rom {u}

{ } = [ ] u{ }

[ ] =

x

0

0

y

y

x

x

y

xy

=

x

0

0

y

y

x

u

v


7/23

4et0s no or! toard an expression "or element strain

We ha)e a bit o" a di7cult, here ith direct substitution

(he shape "unctions $N1, N2, N3, N4& are dened in terms o" themaster element coordinates $a,b&

3ut e need to diferentiate in terms o" the 'lobal coordinates$x,y&

*+ Strain "rom {d}

{ } = [ ] N[ ] d{ }

x

y

xy

=

x0

0

y

y

x

N1 0 N2 0 N3 0 N4 0

0 N1 0 N2 0 N3 0 N4

u1

v1

u2

v2

u3

v3

u4

v4

{ } = B[ ] d{ }

- or -

this operationcannot be donedirectl,


8/23

8oordinate (rans"ormation

Gi)en an, "unction o" the master element coordinates $a,b&/

We can nd deri)ati)es ith respect to 'lobal $x,y& b, usin'

the chain rule/

We can combine and rearran'e these relationships to 'etour deri)ati)es/

f = f a,b( )

f

a=f

x

x

a+f

y

y

a

f

b=f

x

x

b+f

y

y

b

f,a

f,b

= J[ ]

f,x

f,y

f,x

f,y

= J[ ]

1 f,a

f,b


9/23

(he 9acobian

(he 9acobian matrix is an important part o" element"ormulation/

For the *+ element this becomes/

J[ ] =x,a y,a

x,b y,b

=

Ni,ax i Ni,ay iNi,bx i Ni,by i

J[ ] =1

4

1b( ) 1b( ) 1+ b( ) 1+ b( )

1 a( ) 1+a( ) 1+a( ) 1 a( )

x1 y

1

x2 y

2

x3 y

3

x4 y

4

local coordinatederi)ati)es o" the shape

"unctions'lobal coordinatelocations o" theelement nodes

note the9acobian matrixis a "unction o"

location ithinthe masterelement


10/23

9acobian Interpretation

(he 9acobian contains in"ormation about element si:e andshape

(he 9acobian determinant $j& is a scalin' "actor that relatesthe diferential area o" the actual element to the diferentialarea o" the master element

(he 9acobian in)erse $& relates 'lobal coordinate s,stem$x,y& "unction deri)ati)es to master element coordinates,stem $a,b& "unction deri)ati)es

J[ ] =J11 J12

J21

J22

j = det J[ ] =J11J22J21J22

[ ] = J[ ]1=1

j

J22 J12

J21 J11

=11 12

21

22


11/23

Intra-Element 9acobian;ariation


12/23

9acobian $determinant&=atio

(his is one measure o" element #ualit, $hich afects element accurac,&

=atio o" the hi'hest to loest #uadrature point 9acobian determinant

It is >?@ "or an, s#uare or rectan'ular element $samejthrou'hout element&

It increases as element distortion increases


13/23

StrainDisplacement "or *+

Start ith the usual strain-displacement relationship in asli'htl, diferent "orm/

.o add the 9acobian approach to master'lobal coordinatederi)ati)e trans"ormation/

{ } =

x

y

xy

=

1 0 0 0

0 0 0 1

0 1 1 0

u,x

u,y

v,x

v,y

u,x

u,y

v,x

v,y

=

11

120 0

21

220 0

0 0 11

12

0 0 21

22

?

?

?????

?

?

?????

u,a

u,b

v,a

v,b

?

?

??

?

??

?

?

??

?

??


14/23

StrainDisplacement cont?

.o represent the displacement eld master elementderi)ati)es in terms o" the shape "unctions/

u,a

u,b

v,a

v,b

?

?

??

?

??

?

?

??

?

??

=

N1,a 0 N2,a 0 N3,a 0 N4,a 0

N1,b 0 N2,b 0 N3,b 0 N4,b 0

0 N1,a 0 N2,a 0 N3,a 0 N4,a

0 N1,b 0 N2,b 0 N3,b 0 N4,b

?

?

????

?

?

?

????

?

u1

v1

u2

v2

u3

v3

u4

v4

?

?

?

?????

?

??

????

?

?

?

?????

?

??

????


15/23

All (o'ether .o

x

y

xy

=

1 0 0 0

0 0 0 1

0 1 1 0

11

120 0

21

220 0

0 0 11 120 0 21 22

N1,a 0 N2,a 0 N3,a 0 N4,a 0

N1,b

0 N2,b

0 N3,b

0 N4,b

0

0 N1,a 0 N2,a 0 N3,a 0 N4,a

0 N1,b 0 N2,b 0 N3,b 0 N4,b

u1

v1

u2

v2

u3

v3

u4

v4

{ } = B[ ] d{ }

- or -nodal

displacements% 'lobalcoordinates

shape

"unctionderi)ati)es%master

coordinates

9acobianin)erseterms% master

to 'lobalcoordinate

trans"ormation

or'ani:ation


16/23

Stress

I" e ha)e strain% e can 'et to stress b, brin'in' inmaterial properties

We ha)e to be a little care"ul here% this simple expressionassumes/ .o initial $residual% assembl,& stresses present

4inear elastic beha)ior

(he 'eneral "orm abo)e does accommodate anisotropic beha)ior

I" e "urther limit oursel)es to BD% isotropic% plane stress%e can rite/

{ } = E[ ] { } = E[ ] B[ ] d{ }

E[ ] = E

12

1 0

1 0

0 0 1( ) 2

E[ ]1=

1E

E 0

E

1E

0

0 0 1G

G =

E

2 1+( )


17/23

Element Stifness Matrix

=ecall here the element stifness matrix ts into the nite element

"ormulation/

(a!e it as a 'i)en "or the present that the element stifness matrixCk is/

An inte'ral o)er the element area in 'lobal coordinates $t5thic!ness&

Wh, is inte'ration re#uired

(hin! about hat Ck does For displacements applied to the element nodes% it determines the re#uired "orce

I" one element is lar'er than another% the "orce re#uired ou'ht to be 'reater "orthe same nodal displacements

I" an element has a rotated orientation% a coordinate axis displacement canproduce "orces ith multiple coordinate components

k[ ] = B[ ]T

E[ ]B[ ]t dA

k[ ] d{ } = r{ }


18/23

Inte'ration in Master8oordinates

It is not eas, to inte'rate "or the terms in Ck usin' the'lobal coordinate s,stem $elements are 'enerall, distortedand not ali'ned ith 'lobal axes&

3ut e can do this instead $matrix dimensions "or a *+element&/

Inte'rate o)er the master element It is undistorted and ali'ned ith the coordinate s,stem

Ad6ust "or the chan'e in coordinates b, brin'in' in the9acobian determinantj

k[ ]8x8symm

= B[ ]8x3

T

1

1

1

1

E[ ]3x3

B[ ]3x8

t j da db


19/23

*uadrature

=ead 1#uadrature2 as 1numerical inte'ration2

Wh, do e ant to numericall, inte'rate to establish Ck

(o inte'rate directl, is still computationall, expensi)e% e)en

ith the chan'e to local coordinates

*uadrature in)ol)es samplin' at discrete points% multipl,in'b, a ei'htin' "actor% and summin' to 'et an estimate o"the inte'ral

k[ ]8x8

symm

= B[ ]8x3

T

1

1

1

1

E[ ]3x3

B[ ]3x8

t j da db

this )aries point-b,-point too

these contain9acobian in)erseterms hich )ar,

point-b,-pointithin the element


20/23

Gauss oints

Gauss #uadrature is a method o" numerical inte'ration that has optimalcharacteristics hen the underl,in' "unctions ha)e pol,nomial "orm

(he 'ure shos Gauss points "or Bndorder and rdorder #uadrature For $a&% all "our points ha)e a ei'ht o" >?@ $total 5 +?@&

For $b&/ >%%H% ei'ht 5 ?@JKL B%+%K%J ei'ht 5 ?+JL ei'ht 5 ?H@> $total 5 +?@&

.ote/ the #uadrature rule is independent o" element order $*+% *J% *&


21/23

Element Distortion

Nne o" the reasons a distorted element is lessthan ideal/ (he inte'ral is estimated b, discrete samplin' at specic

locations ithin the element

I" the element is not distorted% the sampled points are hi'hl,representati)e o" the un-sampled near b, re'ions o" theelement

I" the element is hi'hl, distorted% the sampled points are notrepresentati)e o" the un-sampled re'ions o" the element


22/23

Element .ormal ;ectors

I" ,ou 'et 1inside out element2 errors

;eri",-Element-.ormals as a "rin'e or )ector plot $rotate themodel to see the )ector orientation&


23/23

Element .ormal ;ectors

Ose Modi",-Element-=e)erse to 'et them all 'oin' inthe same $positi)e P% I thin!% chec! this& direction

Documents

ME424 524 Presentation 08