Shape Functions generation, requitirements, etc. · 2011-11-21 · Displacement Approximation ( ) ~...

Ruhr Universität BochumFakultät für Bauingenieurwesen

Computational EngineeringFinite Elements in Structural Mechanics

Shape Functions generation, i t trequirements, etc.

Student presentationStudent presentation

S d Silj k EStudent: Siljak EnesJanuary, 2009

Displacement Approximation

(1))()(~)(

bygiven is ntsdisplacemeofion approximatelement finiteA eN ii uxNuxuxu ≈ ∑

p pp

li hi ddfbhhdntsdisplaceme nodal are functions, or element are where

(1) )()()(

N

N

iii

u

uxNuxuxu

ioninterpolat shape

==≈ ∑

. called is element.an with associatednodesofnumber over theranges sumtheand

matrix function shapeN

(2) )()()(~)(

:bygiven are sexpression thein ely,Alternativ

N

form ricisoparamete

i

ii uNuuu ξξξξ ==≈ ∑~ )()()( N

i

iii

xNxx ξξξ ==∑

ion interpolatthat theinsuresThis others.allat zeroandnodeat the one valuethe

must takeit that states function shapeelement on the Thethi

Ncondition ioninterpolat i

nodes at thecorrect isp

Requirements for Shape Functionsq p

Requirements for shape functions are motivated by convergence: as the mesh is fi d th FEM l ti h ld h th l ti l l ti f threfined the FEM solution should approach the analytical solution of the

mathematical model.

1 The requirement for compatibility: The interpolation has to be such that field1. The requirement for compatibility: The interpolation has to be such that field of displacements is :1. continual and derivable inside the element2 continual across the element border2. continual across the element border

The finite elements that satisfy this property are called conforming, or compatible. (The use of elements that violate this property, nonconforming or incompatible elements is however common)

2. The requirement for completeness: The interpolation has to be able to represent:1 th i id b d di l t1. the rigid body displacement2. constant strain state

Requirements for Shape FunctionsRequirement for Compatibility:The shape functions should provide displacement continuity between elements.

q p

Physically this insure that no material gaps appear as the elements deform. As the mesh is refined, such gaps would multiply and may absorb or release spurious energy.

Figure 1. Compatibility violation by using different types of elements.Figure 1. Compatibility violation by using different types of elements.a) Discretization and load; b) Deformed shape (left gap, right overlapping)

Requirement for Completeness: The interpolation has to be able to represent:

1. The rigid body displacement2. Constant strain state

Rigid body translation Rigid body rotation Deformation

Figure 2. a) Deformation of cantilever beam. b) Rigid body displacement and deformation of hatched element

Requirements for Shape Functionsi f hi h d i i i i din terms of highest derivative in integrand

If the stiffness integrands involve derivatives of order m, then requirements for shape functions can be formulated as follows:

( 1)1. The requirement for compatibility: The shape functions must be C(m-1)

continuous between elements, and Cm piecewise differentiable inside each element.

2. The requirement for completeness: The element shape functions must represent exactly all polynomial terms of order ≤ m in the Cartesian coordinates. A set of shape functions that satisfies this condition is called m completeshape functions that satisfies this condition is called m-complete.

Highest derivative in integrand ‐ reminder

[ ] [ ] ~~~ ~:bygiven isproblemldimensiona-threegeneralfor theexpressionwork Virtual

∑ ∫∑ ∫∑ ∫ Γ−Ω=Ω tTT

uT

u ddd tubuuDuD δδδ εε C

Ph i lTruss (N=1)

T di i l blEuler‐Bernoulli beam

∑ ∫∑ ∫∑ ∫Γ

ΩΩ e et

e ee e

Physical or mechanical problem

Heat conductionTwo‐dimensional problem

(N=2)Three‐dimensional continuum (N=3)

(N=1)Thin plate (N=2)(Kirchhoff plate)

Thick plates (N=2)(Reissner‐Mindlin plate)

Independent primary variables

Temperature1

DisplacementsN

Transverse displacement1

Transverse displacement 1Rotation N

⎤⎡ ∂∂ 0/ ⎤⎡ 22 ⎥⎤

⎢⎡

∂∂∂∂

/000/0

yx

OperatorDεu

(for N=2)⎥⎦

⎤⎢⎣

⎡∂∂∂∂yx

//

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∂∂∂∂

∂∂

∂∂

xy

y

x

//0

/0/

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∂∂∂∂∂∂∂

yxyx

/2//

2

22

22

⎥⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢⎢

⎣ ∂∂∂∂

∂∂∂∂∂∂

10/01///0/00

yx

xyy

Required continuity C0 C0 C1 C0

⎥⎦⎢⎣ ∂∂ 10/ y

Table 1. Differential operator Dεu for different types of physical or mechanical problems

Generation of shape functionsOne‐dimensional elementsOne‐dimensional elements

Physical and natural coordinates

In the case of a truss element, the position of a point relative to the longitudinal axis is measured in terms of the physical coordinate X1 or the natural coordinate ξ1

⎥⎦

⎤⎢⎣

⎡−∈⎥⎦

⎤⎢⎣⎡−∈ 1 ,1

2,

2 11 ξLLX⎦⎣⎦⎣

The relationship between the coordinates X1 and ξ1 is described by the tiequation:

11 2ξLX = 11 2

Local polynomial Approximation of One‐dimensional functionsOne dimensional functions

1

11111111 )()()(~)( ==≈ ∑

+

=

p

i

eieiNuuu uN ξξξξ

e1

:wayfollowingin thedefinedare u theand )N( thewhere vectorntdisplacemeelementfunctionsshapeofmatrix ξ

[ ] [ ]nodes.)element ofnumber thesymbolizes 1(

12

11

1112

11

1 ... )( ... )( )()(

:wayfollowing in the definedare

+=

==pNN

TeNNeeeNN uuuNNN uN ξξξξ

−=∏

+

N

pp k

i )(

:productby theformedbecan degreeofpolynomialion interpolat Lagrangian1

11 ξξξ−

=∏≠=

N

ikk

iki )(1 11 ξξ

ξ

⎧ = kii

Nk

k

ik

for1 node the toassociatedfunction shape the

and node theofposition thezingcharacteri with 1ξ

⎩⎨⎧

≠=

=kiki

N ki

for 0for 1

)( 1ξ

Linear, Quadratic and Cubic Shape functionsHere the Lagrange interpolation polynomials used for one-dimensional finite elements

Linear p = 1 Quadratic p = 2 Cubic p = 3

must be defined for p = 1, p = 2 and p = 3.

)(

)(

12

11

ξ

ξ

N

N2

1

1121

1

)1(

ξ

ξξ

−

−

)1(

)1(

121

121

ξ

ξ

+

−

))(1(

))(1(

31

12

11627

912

11169

−−

−−

ξξ

ξξ

)(

)(

14

13

ξ

ξ

N

N 1121 )1( ξξ+

12

))(1(

))(1(

912

11169

31

12

11627

316

−+

+−

ξξ

ξξ

Shape Functions for Beam ElementAdmissible displacements must be C1 continuous.The nodal displacements (3) are used to define uniquely the variation of the transverse displacement

112 and : :1 node:endson theConditions

L dxdwwwx θ==−=Figure 3: The two node Beam element with four DOFs.

2432

222

ld lfl l i32/

and: :2 node L

xxdxdw

dxdwwwx

ααα

θ

++=

===[ ]= 2211 (3) θθ wweu

( ) ( )( )2

324

2232211

:valuesnodalfor gcalculatinLLLw αααα −+−=

+++ 32)(

be ion willinterpolatfreedom of degreesfour with

αααα xxxxw ( )( ) ( )324

2232212

2242321 32

LLL

LL

w αααα

αααθ

+++=

+−=

⎥⎥⎤

⎢⎢⎡

+++=

2

1

32

4321

]1[)(

)(

αααααα xxxxw

( )2242322 32 LL αααθ ++=⎥⎥⎥

⎦⎢⎢⎢

⎣

=

4

3

232 ]1[)(

αα

xxxxw

Shape Functions for Beam Element

⎤⎡ ⎞⎛⎞⎛

:formmatrix in Written 32 LLL

⎥⎤

⎢⎡

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

⎟⎞

⎜⎛−

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛−

⎥⎤

⎢⎡ 2

3210

2221

11 LL

LLL

w αθ

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠

⎜⎝=

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

2221

23

2210

4

3

232

2

2

1

LLLwααα

θ

θ

⎦⎣

⎥⎥⎥

⎦⎢⎢⎢

⎣⎟⎠⎞

⎜⎝⎛

⎠⎝⎠⎝⎦⎣2

23

2210

42

LL

⎥⎤

⎢⎡

⎥⎤

⎢⎡ −

⎥⎤

⎢⎡ 22

:scoeficient for Solving

14343

1 waaaaα

α

⎟⎠⎞

⎜⎝⎛ =

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

−−−−

=

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

2 :with

1100

3341

2

122

3232

33

2 Lawaa

aaaaa

θ

θ

ααα

⎦⎣⎦⎣ −⎦⎣ 11 24 aa θα


⎥⎥⎤

⎢⎢⎡

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

−−−

−

1

1

32 41

43

41

43

421

421

]1[)(θw

aa

aa

⎥⎥⎥

⎦⎢⎢⎢

⎣⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢

⎣−

−=

2

2

2323

32

41

41

41

41

410

410

4444]1[)(

θw

aaaa

aa

aaxxxxw

[ ] ⎥⎥⎤

⎢⎢⎡

⎦⎣

1

1

4444

θw

aaaa

[ ]⎥⎥⎥

⎦⎢⎢⎢

⎣

=

2

2

12211)(

θ

θθθ w

NNNNxw ww

⎞⎛ 33

⎟⎟⎞

⎜⎜⎛

+−−=+−−=

⎟⎟⎠

⎞⎜⎜⎝

⎛+−=+−=

1)(

2341

443

21)(

2332

1

3

3

3

3

1

xxxaxxxaxN

ax

ax

ax

axxNw

θ

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−=−+=

⎟⎟⎠

⎜⎜⎝

+=+=

2341

443

21)(

144444

)(

3

3

3

3

2

2321

ax

ax

ax

axxN

aaaaaxN

w

θ

⎟⎟⎠

⎞⎜⎜⎝

⎛−−+=++−−=

⎠⎝

144444

)( 2

2

3

3

2

32

2 ax

ax

axa

ax

axxaxNθ


( )1

:2

with

3

⎟⎠⎞

⎜⎝⎛ ==

Laaxξ

( )

( )14

)(

2341)(

231

31

+−−=

+−=

ξξξξ

ξξξ

θaN

Nw

( )

( )1)(

2341)(

4

23

32 −−−=

ξξξξ

ξξξ

aN

Nw

( )14

)( 232 −−+= ξξξξθ

aN

f ihcalledarefunctions These

b functionsshape cubicHermitian

:curvatureget totwiceatingDifferenti

Figure 4. Hermitian⎤⎡

=====4422

2

22

2

ξξξξ

κdd

Ldwd

Ldxwd eee u'N'BuuN

cubic shape functions⎥⎦⎤

⎢⎣⎡ +−−= 1361361 ξξξξ

LLLB

Shape Functions of Plane Elements

Classification of shape functions according to:

• the element form:– triangular elements,

rectangular elements– rectangular elements.

• polynomial degree of the shape functions:– linearlinear – quadratic – cubic– ……

• type of the shape functions– Lagrange shape functionsg g p– serendipity shape functions

Rectangular elements – Lagrange family

:direction onein polynomialion interpolat Lagrange

)(0 11

111

n

kii

ik

inkl −

−=∏

≠= ξξ

ξξξ

generating of method systematic andeasy An

ki≠

:scoordinatetwo in the spolynomial Lagrange of products simpleby

achievedbecan noworder any of functionsshape

:where)()(

:scoordinatetwo

21mJ

nIIJa llNN == ξξ

)(2 )(221

e

e

e

e

byy

axx −

=−

= ξξ

element theof dimensions are ,elementtheofcenter theofscoordinate are,

ee

ee

bayx Figure 5. Generation of a typical shape

function for a Lagrangian element


Figure 6: The Quadrilateral Lagrangian elements: a) bilinear, b) biquadratic c) bicubic

Figure 7: The Quadrilateral Lagrangian elements: a) quadratic-linear, b) linear-cubic c) quadratic-cubic, d) quartic-quadratic


Figure 8: Complete two-dimensional Lagrange ansatz polynomials in the Pascal triangle

The Four‐Node Bilinear Quadrilateral

Figure 9: The four-node bilinear quadrilateral: (a) element geometry; (b) perspective view of shape function

11 31

(b) perspective view of shape function

)1)(1(41)( )1)(1(

41)(

)1)(1(41)( )1)(1(

41)(

214

212

213

211

ξξξξ

ξξξξ

+−=−+=

++=−−=

ξξ

ξξ

NN

NN

44

The Four‐Node Bilinear QuadrilateralCheck of compatibilityec o co pa b y

Figure 10: Assemblage of four bilinear quadrilateral elements

Change of N5 along the edge is linear and it is uniquely defined by two nodes

Derivative inside element exists, and on the boundary has finite discontinuity

Figure 11: Partial derivatives with respect to x and y of the shape functions N5

Check of completeness

:assuchmotionsnt displacemelinear any exactly represent can they ifelement continuumafor completeisfunctions shapeofset A

(1) ,:assuch

210210 ++=++= yx yxuyxu βββααα

(2) ,:arefieldnt displaceme thistoingcorrespondntsdisplacemepoint nodalThe

210210 ++=++= iiyiiixi yxuyxu βββααα

(2).by given are ntsdisplacemepoint nodalelement en theelement wh the obtained be tohave (1) ntsdisplaceme The within

(3) ,

:ioninterpolatnt displacemethehaven weformulatioricisoparametIn the

11∑∑==

==n

i

eiyiy

n

i

eixix NuuNuu

:nt displaceme for then Computatio

11 ==

x

ii

u

( ) (4) 1

21

11

01

210 ∑∑∑∑====

++=++=n

i

eii

n

i

eii

n

i

ei

e

i

eiiix NyNxNNyxu αααααα

edinterpolatarescoordinaten theformulatioricisoparamet in theSince

,

:usecan wents,displaceme theas way same in the

∑∑ ==n

eii

neii NyyNxx

(5)

:obtain to11

∑

∑∑==

ne

ii

N

id d(1)iihh(5)id fi ddi lTh

(5) 211

0∑=

++=i

eix yxNu ααα

:element in thepoint any for that provided(1),in given thoseassametheare(5)in definedntsdisplacemeThe

(6) 1 0∑=

=n

iiN

satisfiedbetotsrequiremensscompletene for the functionsion interpolat on thecondition theis (6)relation The

satisfiedbetotsrequiremen sscompletene

The Nine‐Node Biquadratic Quadrilateral

2222111141 )1()1( ξξξξξξξξ iiiiiN ++=

0)1)(1(

0 , )1)(1(21

21111222

1

=+=

=+−=iiii

iiii

N

N

ξξξξξξ

ξξξξξξ

Figure 12: The nine-node

0 ,)1)(1( 1222212 =+−=N ξξξξξξ

)1)(1( 22

21

9 ξξ −−=Nbiquadratic quadrilateral

222

15 )1()1(1)( ξξξ −−=ξN )1)(1()( 2

22

19 ξξ −−=ξN

21211 )1)(1(1)( ξξξξ −−=ξN 221 )1()1(

2)( ξξξξN

Figure 13: Perspective view of shape functions for nodes 1, 5 and 9 of the nine-node biquadratic quadrilateral

2121 )1)(1(4

)( ξξξξξN

The 16‐Node Bicubic Quadrilateral

))()(1)(1( 229

1219

12211256

81 ξξξξξξ −−++= iiiN

)1)(3)()(1( 221131

912

22

1256243 ξξξξξξ iiiN ++−−=

Figure 14: The 16-node bicubic quadrilateral

)1)(3)()(1( 112231

912

122256

243 ξξξξξξ iiiN ++−−=

)3)(3)(1)(1( 2231

11312

22

1256729 ξξξξξξ iiiN ++−−=bicubic quadrilateral )3)(3)()(( 22311321256 ξξξξξξN

Figure 15: Perspective view of shape functions for nodes 1, 5 and 13 of the 16-node bicubic quadrilateral

Serendipity elements

Serendipity elements are constructed with nodes only on the element b dboundary

Figure 17 Two dimensional serendipity polynomials

Figure 16. Serendipity quadrilateral elements: a) bilinear , b) biquadratique, c) bicubic

Figure 17. Two dimensional serendipity polynomialsof quadrilateral elements in Pascal triangle

Serendipity Biquadratic Shape functionsFor mid-side nodes a lagrangian interpolation of quadratic x linear type suffices to determine Ni at nodes 5 to 8. For corner nodes start with bilinear lagragian family (step 1), and successive subtraction (step 2, step 3) ensures zero value at nodes 5, 8

)1)(1( 22

1215 ξξ −−=N )1)(1( 2

21218 ξξ −−=N

)1)(1(ˆ 11 ξξ −−=N

)1)(1)(1()( 2121411 ξξξξN −−−−−=ξ

)1)(1)(1()( 2211221141 ξξξξN iiiii ξξξξ ++−++=ξ

)1)(1( 214 ξξ=N

5211ˆ NN − 2

815111 ˆ NNNN −−= )1)(1()( 215 ξξN =ξ22 NNNN = )1)(1()( 212 ξξN −−=ξ

0for , )1)(1()(

0for , )1)(1()(

222112

1

122212

1

=−+=

=+−=iii

iii

ξξN

ξξN

ξξ

ξξ

ξ

ξ

[ ])1()1(1)1)(1(

)1)(1()1)(1()1)(1(

212141

2212

121

22

121

21

21411

ξξξξ

ξξξξξξ

−−−−−−

=−−⋅−−−⋅−−−=N

)1)(1)(1( 2121411 ξξξξ −−−−−=N

Figure 19. Biquadratic serendipity element shape functionsFigure 18. Construction of serendipity element shape functions

Serendipity Shape functions

In general serendipity shape functions can be obtained with the following expression:

)(1,N,-1)(NN 2i

1ii )1()1(),( 12

122

121 ξξξξξξ ++−=

(1,-1)N(-1,-1)N

)(-1,N,1)(N

)(1,N, 1)(NN

ii

2i

1i

21

)1)(1()1)(1(

)1()1(

)1()1(),(

2141

2141

121

221

122221

ξξξξ

ξξξξ

ξξξξξξ

−+−−−−

−+++

++

(-1,1)N(1,1)N ii )1)(1()1)(1( 2141

2141 ξξξξ +−−++−

boundary ngcorespondi thealong ionsinterpolat lagrangian are ),1( ),1,( ),,1( ),1,( functions where 2121 −− iiii NNNN ξξξξ

corrnerson ion interpolat of valuesrepresent and1or 0 valueshave )1 ,1( ),1 ,1( ),1,1( ),1,1( valuesand −−−− iiii NNNN

Serendipity Shape functionsAs an example shape function of the node N3 of the element on Fig. 20 will be generated .

Figure 20. Construction of N3

)1(3 ξN

of N3

),1( 23 ξ−N

)1,( 1ξN

)1)(1(),( 21141

213 ξξξξξ ++=N

)1,( 13 −ξN

),1( 23 ξN

Figure 21. View of shape function N3

1)1)(1()1()1()1()1(

)1)(1()1()1(),(

2141

1121

221

221

121

3214

1322

1312

121

3

ξξξξξξξ

ξξξξξξξξ

=⋅++−+⋅+++⋅+

=++−+++=

(1,1)N ,1)(N)(1,NN 12

)1)(1( 21141 ξξξ ++

Quadrilateral element with variable number of nodes (4 to 16 nodes)a ab e u be o odes ( o 6 odes)

)1)(1(250)1)(1(250

4to1nodesoffunctionsShape

213

211 ξξξξ ++=−−= NN

)1)(1(25.0 )1)(1(25.0

)1)(1(25.0 )1)(1(25.0

214

212

2121

ξξξξ

ξξξξ

+−=−+=

++

NN

NN


absent is 6 node if 0 present; is 6 node if )1)(1(5.0

absent is 5 node if 0 present; is 5 node if )1)(1(5.0

p

727

6221

6

52

21

5

NN

NN

=−+=

=−−=

ξξ

ξξ

:nodescorner offunctionsshapeofsCorrectionabsent is 8 node if 0 present; is 8 node if )1)(1(5.0

absentis7nodeif 0 present;is7 nodeif )1)(1(5.082

218

72

21

7

NN

NN

=−−=

=+−=

ξξ

ξξ

)( 5.0 )( 5.0

)( 5.0 )( 5.0

p

87446522

76335811

NNNNNNNN

NNNNNNNN

+−←+−←

+−←+−←

:sidestheofmiddlein theandnodescorneroffunctionsshapeofCorrection0not if ;lagrangian cbiquadrati iselement if )1)(1(

9nodeinternaloffunction Shape92

22

19 =−−= NN ξξ

)8 ,7 ,6 ,5( 5.0 )4 ,3 ,2 ,1( 25.0

:sidestheofmiddlein theandnodescorner offunctionsshapeofCorrection99 =−==+= iNNNiNNN iiii


borderon the 12 to9nodesoffunctionsShape

absent is 10 node if 0 present; is 10 node if )31)(1)(1(28125.0


11211

102

221

10

912

21

9

=+−+=

=+−−=

NN

NN

ξξξ

ξξξ


absentis11nodeif 0 present; is11nodeif )31)(1)(1(28125.012

2221

12

1112

21

11

=−−−=

=−+−=

NN

NN

ξξξ

ξξξ

3/)(25.0125.0

3/)(25.0125.0

nodescorner of functionsshapeofsCorrection

1096522

9125811

NNNNNN

NNNNNN

−−−+←

−−−+←

3/)(25.0125.0

3/)(25.0125.0

3/)(25.0125.0

12118744

11107633

NNNNNN

NNNNNN

NNNNNN

−−−+←

−−−+←

+←

12881066

1177955

125.1125.1

125.1 125.1

borderon thenodes offunctionsshapeofsCorrection

NNNNNN

NNNNNN

−←−←

−←−←

125.1 125.1 NNNNNN ←←


absentis13nodeif0present;is13nodeif256/)31)(1)(31)(1(81

16to13nodes internaloffunctionsShape132213 NN ξξξξ

absent is 15 node if 0 present; is 15 node if 256/)31)(1)(31)(1(81

absent is 14 node if 0 present; is 14 node if 256/)31)(1)(31)(1(81

absentis13nodeif 0 present;is 13 nodeif 256/)31)(1)(31)(1(81

152

221

21

15

142

221

21

142211

=+−+−=

=−−+−=

=−−−−=

NN

NN

NN

ξξξξ

ξξξξ

ξξξξ

nodescorner of functionsshapeofsCorrection1615141311

absent is 16 node if 0 present; is 16 node if 256/)31)(1)(31)(1(81 162

221

21

16 =+−−−= NN ξξξξ

9/)5.0 5.0 250(4

9/)25.05.0 50 (4

9/)5.0 25.05.0 (4

1615141333

1615141322

1615141311

NNNN.NN

NNNN.NN

NNNNNN

++++←

++++←

++++←

9/) 5.0 25.050 (4

)(1615141344 NNNN.NN

NNNNNN

++++←

borderon thenodestheoffunctionsshapeofsCorrection

3/)2(3/)2(

3/)2( 3/)2(

3/)2( 3/)2(

13161111141577

16151010131466

151499161355

NNNNNNNN

NNNNNNNN

NNNNNNNN

+←+←

+−←+−←

+−←+−←

3/)2( 3/)2(

3/)2( 3/)2(14131212151688 NNNNNNNN

NNNNNNNN

+−←+−←

+−←+−←

Triangular elementsThree‐Node Triangle Elementg

Triangular elementsThree‐Node Triangle Elementg

31

221

1 1

ξ

ξ

ξξ

=

−−=

N

N

N

2ξ=N

1 1 ξξN 12 ξ=N 2

3 ξ=N

Figure 24. Isometric view of shape functions,

211 ξξ −−=N 1ξN 2ξN

d tdbf tiSh

Six‐Node Quadratic Triangle

...

:functionslinear ofproductsasexpressedbecan functionsShape

21= LLLcN nii

vanisheson which

lines ofequation shomogeneou theare

,...,1 , 0 where ==

N

njL

i

j

tcoefficienion normalizat a is and

vanisheson which

1

cN

i

)12(

)12(

)221)(1(

223

112

21211

ξξ

ξξ

ξξξξ

−=

−=

−−−−=

N

N

N

)1(

)12(

4211

4

22

ξξξ

ξξ

−−= cN

N

i

4

)1(4

4 1)1()0,(

215

2114

21

21

214

ξξ

ξξξ

=

−−=

=⇒=−=

N

N

ccN ii

)221)(1( 21211 ξξξξ −−−−=N )1(4 211

4 ξξξ −−=N

)1(4

4

2126

21

ξξξ

ξξ

−−=

=

N

NFigure 25. Shape functions N1 and N 4

for the quadratic triangle

Ten‐Node Qubic Triangle

:anishesfunction v shapeon which edgesofproducts asgCalculatin

)1)(332)(331(

12)0,0(

)1)(332)(331(

212121211

21

111

21212111

ξξξξξξ

ξξξξξξ

−−−−−−=

=⇒==

−−−−−−=

N

ccN

cN

))((

))((

1)0,1( ))((

))()((

219332

131

11292

29

292

22

32

131

1122

2121212

ξξξ

ξξξ

ξξξ

ξξξξξξ

−−=

=⇒==−−=

N

N

ccNcN

)13(

)1)(332(

))((

121295

21211294

32

231

22293

ξξξ

ξξξξξ

ξξξ

−=

−−−−=

−−=

N

N

N

)13(

)1)(13(

)1)(13(

221298

2111297

2122296

ξξξ

ξξξξ

ξξξξ

−=

−−−=

−−−=

N

N

N

)1(27

)1)(332(

)(

212110

21212299

2212

ξξξξ

ξξξξξ

ξξξ

−−=

−−−−=

N

N

Figure 26. Shape functions N1, N4 and N10 for the Qubic triangle

Shape functions of Three‐dimensional elements

Figure 27. Examples of Solid finite elements:

a) Hexahedral b) Prismatic

c) Tetrahedral element

Rectangular PrismsLagrange family

Shape functions for this element will be generated by a direct product of three Lagrange polynomials:

)()()( 321pK

mJ

nIIJKa lllNN == ξξξ

)(2)(2)(2:and sideeach along nssubdivisio , , ,for

zzyyxx

pmn

−=−=−= ξξξ )( ,)( , )( 321 eee zzc

yyb

xxa

−=−=−= ξξξ

The eight Node Hexahedron

The shape functions are:

)1)(1)(1( )1)(1)(1(

)1)(1)(1( )1)(1)(1(

321816

321812

321815

321811

ξξξξξξ

ξξξξξξ

+−+=−−+=

+−−=−−−=

NN

NNThe shape functions are:

)1)(1)(1( )1)(1)(1(

)1)(1)(1( )1)(1)(1(

321818

321814

321817

321813

ξξξξξξ

ξξξξξξ

++−=−+−=

+++=−++=

NN

NN

These eight formulas can be summarized in a single expression:)1)(1)(1( 3322118

1 ξξξξξξ iiiiN +++=

where denote the coordinates of the ith node.iii31 ,, ξξξ 2

The 20‐Node (Serendipity) Hexahedron

The 20-node hexahedron is equivalent to the 8-node “serendipity” quadrilateral.p y q

The shape functions are:The shape functions are:

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

:8...,,2,1nodescorner For the

33221133221181 ξξξξξξξξξξξξ iiiiiiiN

i

−+++++=

=

)1)(1)(1(

:15 ,13 ,11 ,9 nodes midside For the

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

33222

141

3322113322118

ξξξξξ

ξξξξξξξξξξξξ

iiiN

i

N

++−=

=

+++++

:20,19,18,17nodesmidside For the)1)(1)(1(

:16,14,12,10nodesmidside For the

3311224

1 ξξξξξ iii

iN

i

=

++−=

=

)1)(1)(1(

,,,

2211234

1 ξξξξξ iiiN ++−=

Shape functions of Hexahedron with variable number of nodes (8‐27 nodes)

)1)(1)(1(1250)1)(1)(1(1250

)1)(1)(1(125.0 )1)(1)(1(125.0


62321

5321

1

ξξξξξξ

ξξξξξξ

+−+=−−+=

+−−=−−−=

NN

NN

)1)(1)(1(125.0 )1)(1)(1(125.0

)1)(1)(1(125.0 )1)(1)(1(125.0

)1)(1)(1(125.0 )1)(1)(1(125.0

3218

3214

3217

3213

321321

ξξξξξξ

ξξξξξξ

ξξξξξξ

++−=−+−=

+++=−++=

+−+=−−+=

NN

NN

NN

present;not if 0 present;is10nodeif )1)(1)(1(25.0

present;not if 0 present; is 9 node if )1)(1)(1(25.0

borderthealong20to9nodes of functionsShape

103

221

10

932

21

9

=−−+=

=−−−=

NN

NN

ξξξ

ξξξ

present;notif0present;is13nodeif)1)(1)(1(250



p ;p ;))()((

1332

21

13

123

221

12

1132

21

11321

=+−−=

=−−−=

=−+−=

NN

NN

NN

ξξξ

ξξξ

ξξξ

ξξξ

ttif0ti16dif)1)(1)(1(250




16216

1532

21

15

143

221

14321

=++−=

=+−+=

=+=

NN

NN

NN

NN

ξξξ

ξξξ

ξξξ

ξξξ




182321

18

172321

17

163

221

16

=−−+=

=−−−=

=+−−=

NN

NN

NN

ξξξ

ξξξ

ξξξ



32120

192321

19

=−+−=

=−++=

NN

NN

ξξξ

ξξξ


)(50)(50

)(5.0 )(5.0

:nodescorner offunctionsshapeofCorrection

181413661810922

171316551791211

NNNNNNNNNN

NNNNNNNNNN ++−←++−←

)(5.0 )(5.0

)(5.0 )(5.0

)(5.0 )(5.0

2016158820121144

1915147719111033

181413661810922

NNNNNNNNNN

NNNNNNNNNN

NNNNNNNNNN

++−←++−←

++−←++−←

++−←++−←

absentis22nodeif0present;is22nodeif)1)(1)(1(50


:26to21nodesface-midoffunctionsShape

222222

213

22

21

21

=+=

=−−−=

NN

NN

ξξξ

ξξξ

bi2dif0i2dif)1)(1)(1(0



absentis22nodeif 0 present;is22nodeif )1)(1)(1(5.0

252225

2423

221

24

23232

21

23321

=−−+=

=−−−=

=+−−=

NN

NN

NN

ξξξ

ξξξ

ξξξ

ξξξ


absentis25nodeif 0 present;is25nodeif )1)(1)(1(5.0262

3221

26

25232

21

25

=−−−=

=−+−=

NN

NN

ξξξ

ξξξ


)(25.0 )(25.0

)(25.0 )(25.0

)(25.0 )(25.0

2524227725242133

2423226624232122

2326225523262111

NNNNNNNNNN

NNNNNNNNNN

NNNNNNNNNN

+++←+++←

+++←+++←

+++←+++←

)(25.0 )(25.0

)()(2625228826252144 NNNNNNNNNN +++←+++←


)(50)(50

)(5.0 )(5.0

:borderon the functionsshapeof Correction

2622161624211010

25221515232199

NNNNNNNN

NNNNNNNN +−←+−←

)(5.0 )(5.0

)(5.0 )(5.0

)(5.0 )(5.0

2423181826211212

2326171725211111

2622161624211010

NNNNNNNN

NNNNNNNN

NNNNNNNN

+−←+−←

+−←+−←

+−←+−←

)(5.0 )(5.0

)(5.0 )(5.02625202024221414

2524191923221313

NNNNNNNN

NNNNNNNN

+−←+−←

+−←+−←

absent is 27 node if 0 present; is 27 node if )1)(1)(1(

:27node internaltheoffunction Shape272

322

21

27 =−−−= NN ξξξ

:26to1nodesoffunctionsshapeofCorrection

)20 ,...,10 ,9( 25.0

)8 ,...,2 ,1( 125.0

:26to1nodes offunctionsshapeofCorrection

27

27

27

=+←

=−←

iNNN

iNNN

ii

ii

ii

)26,...,22 ,21( 5.0 27 =−← iNNN ii

Shape functions of Prismatic element with variable number of nodes (6‐18 nodes)

)1(50)1(50

)1)(1(5.0 )1)(1(5.0


52321

4321

1

ξξξξ

ξξξξξξ

+=−=

+−−=−−−=

NN

NN

)1( 5.0 )1( 5.0

)1( 5.0 )1( 5.0

326

323

3131

ξξξξ

ξξξξ

+=−=

+=−=

NN

NN

edgethealong15to7nodesof functionsShape

present;not if 0 present; is 9 node if )1)(1(2

present;not if 0 present; is 8 node if )1( 2


93212

9

8321

8

73211

7

=−−−=

=−=

=−−−=

NN

NN

NN

ξξξξ

ξξξ

ξξξξ

present;not if 0 present;is12node if )1)(1(2

present;not if 0 present; is 11 node if )1( 2


123212

12

11321

11

103211

10

=+−−=

=+=

=+−−=

NN

NN

NN

ξξξξ

ξξξ

ξξξξ

present;notif0present;is15nodeif)1(

present;not if 0 present; is 14 node if )1(

present;not if 0 present; is 13 node if )1)(1(

p ;p ;))((

15215

14231

14

132321

133212

==

=−=

=−−−=

NN

NN

NN

ξξ

ξξ

ξξξ

ξξξξ

present;not if 0 present;is15nodeif )1( 32 =−= NN ξξ

)()(

)(5.0 )(5.0


14111055148722

13101244137911 NNNNNNNNNN ++−←++−←

)(5.0 )(5.0

)(5.0 )(5.015121166159833

14111055148722

NNNNNNNNNN

NNNNNNNNNN

++−←++−←

++−←++−←

Shape functions of Prismatic Element with variable number of nodes (6‐18 nodes)

if0i1dif)1(4


faceon the 18 to16nodesoffunctionsShape

17217

1623211

16 =−−−= NN

ξξξ

ξξξξ


present;not if 0 present;is17nodeif )1( 4182

321218

172321

17

=−−−=

=−=

NN

NN

ξξξξ

ξξξ

:nodescorneroffunctionsshapeofCorrection

)(25.0 )(25.0

)(25.0 )(25.0

:nodescorner of functionsshapeofCorrection

181766181733

171655171622

161844161811

NNNNNNNN

NNNNNNNN

++←++←

++←++←

)(25.0 )(25.0 181766181733 NNNNNNNN ++←++←

505050

:edgeon thecorner offunctionsshapeofCorrection189917881677 NNNNNNNNN −←−←−←

)(50

)(5.0 )(5.0

5.0 5.0 5.0

5.0 5.0 5.0

18171515

1716141416181313

181212171111161010

NNNN

NNNNNNNN

NNNNNNNNN

NNNNNNNNN

+←

+−←+−←

−←−←−←

←←←

)(5.0 NNNN +−←

Shape functions of Tetrahedron Elementwith variable number of nodes (4‐10 nodes)

422

3321

1 1

:4to1nodescorner offunctionsShape

ξξξξ =−−−= NN

34

12 ξξ == NN

:borderon the10to5nodesof functionsShape

absent is 7 node if 0 present; is 7 node if)1(4absent is 6 node if 0 present; is 6 node if4absent is 5 node if 0 present; is 5 node if)1(4

73212

7

621

6

53211

5

=−−−====−−−=

NNNNNN

ξξξξξξ

ξξξξ

absent is 10 node if 0 present; is 10 node if4absent is 9 node if 0 present; is 9 node if4absent is 8 node if 0 present; is 8 node if)1(4

1032

10

931

9

83213

8

=====−−−=

NNNNNN

ξξξξ

ξξξξ

)(0)(0

)(5.0 )(5.0

:nodescorner offunctionsshape of sCorrection

10984496522

10763387511 NNNNNNNNNN ++−←++−←

)(5.0 )(5.0 10984496522 NNNNNNNNNN ++−←++−←

Literature

1. Bathe K.J. : Finite Element Procedures, Prentice Hall of India, New Delhi 20072 Felippa C : Introduction to Finite Element Methods2. Felippa C. : Introduction to Finite Element Methods,

http://www.colorado.edu/engineering/CAS/courses.d/IFEM.d/

3. Gmür T. : Méthode des éléments finis, Presses polytechniques et universitaires romandes, 2000

4. Knothe K. , Wessels H. : Finite Elemente. Eine Einfürung für Ingenieure. Springer Verlag Berlin 2007

5. Kuhl D. , Meschke G. : Lecture Notes, Finite Element Methods in Linear Structural Mechanics, 6. Edition, Institute for Structural Mechanics, Ruhr Universität Bochum, 2008

6. Zienkiewicz O.C. , Taylor R.L. , Zhu J.Z. : The Finite Element Method: Its Basis d d l 6 di i l i h H i 200and Fundamentals, 6. Edition, Elsevier Butterworth‐Heinemann, 2005

Shape Functions generation, requitirements, etc. · 2011-11-21 · Displacement Approximation ( ) ~...

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