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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
Shape Functions for Beam Element
⎥⎥⎤
⎢⎢⎡
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
−−−
−
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
Shape Functions for Beam Element
( )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
Rectangular elements – Lagrange family
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
Rectangular elements – Lagrange family
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
8to5nodesoffunctionsShape
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
Quadrilateral element with variable number of nodes (4 to 16 nodes)a ab e u be o odes ( o 6 odes)
borderon the 12 to9nodesoffunctionsShape
absent is 10 node if 0 present; is 10 node if )31)(1)(1(28125.0
absent is 9 node if 0 present; is 9 node if )31)(1)(1(28125.0
11211
102
221
10
912
21
9
=+−+=
=+−−=
NN
NN
ξξξ
ξξξ
absent is 12 node if 0 present; is 12 node if )31)(1)(1(28125.0
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 ←←
Quadrilateral element with variable number of nodes (4 to 16 nodes)a ab e u be o odes ( o 6 odes)
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
8to1nodesoffunctionsShape
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
present;not if 0 present; is 12 node if )1)(1)(1(25.0
present;not if 0 present; is 11 node if )1)(1)(1(25.0
p ;p ;))()((
1332
21
13
123
221
12
1132
21
11321
=+−−=
=−−−=
=−+−=
NN
NN
NN
ξξξ
ξξξ
ξξξ
ξξξ
ttif0ti16dif)1)(1)(1(250
present;not if 0 present; is 15 node if )1)(1)(1(25.0
present;not if 0 present; is 14 node if )1)(1)(1(25.0
present;not if 0 present;is13nodeif )1)(1)(1(25.0
16216
1532
21
15
143
221
14321
=++−=
=+−+=
=+=
NN
NN
NN
NN
ξξξ
ξξξ
ξξξ
ξξξ
present;not if 0 present; is 18 node if )1)(1)(1(25.0
present;not if 0 present; is 17 node if )1)(1)(1(25.0
present;not if 0 present;is16nodeif )1)(1)(1(25.0
182321
18
172321
17
163
221
16
=−−+=
=−−−=
=+−−=
NN
NN
NN
ξξξ
ξξξ
ξξξ
present;not if 0 present; is 20 node if )1)(1)(1(25.0
present;not if 0 present; is 19 node if )1)(1)(1(25.0202
32120
192321
19
=−+−=
=−++=
NN
NN
ξξξ
ξξξ
Shape functions of Hexahedron with variable number of nodes (8‐27 nodes)
)(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
absent is 21 node if 0 present; is 21 node if )1)(1)(1(5.0
:26to21nodesface-midoffunctionsShape
222222
213
22
21
21
=+=
=−−−=
NN
NN
ξξξ
ξξξ
bi2dif0i2dif)1)(1)(1(0
absent is 24 node if 0 present; is 24 node if )1)(1)(1(5.0
absent is 23 node if 0 present; is 23 node if )1)(1)(1(5.0
absentis22nodeif 0 present;is22nodeif )1)(1)(1(5.0
252225
2423
221
24
23232
21
23321
=−−+=
=−−−=
=+−−=
NN
NN
NN
ξξξ
ξξξ
ξξξ
ξξξ
absent is 26 node if 0 present; is 26 node if )1)(1)(1(5.0
absentis25nodeif 0 present;is25nodeif )1)(1)(1(5.0262
3221
26
25232
21
25
=−−−=
=−+−=
NN
NN
ξξξ
ξξξ
:nodescorner offunctionsshapeofCorrection
)(25.0 )(25.0
)(25.0 )(25.0
)(25.0 )(25.0
2524227725242133
2423226624232122
2326225523262111
NNNNNNNNNN
NNNNNNNNNN
NNNNNNNNNN
+++←+++←
+++←+++←
+++←+++←
)(25.0 )(25.0
)()(2625228826252144 NNNNNNNNNN +++←+++←
Shape functions of Hexahedron with variable number of nodes (8‐27 nodes)
)(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
6to1nodesoffunctionsShape
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
present;not if 0 present; is 7 node if )1)(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
present;not if 0 present; is 10 node if )1)(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
:nodescorner offunctionsshapeofCorrection
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
present;not if 0 present; is 16 node if )1)(1(4
faceon the 18 to16nodesoffunctionsShape
17217
1623211
16 =−−−= NN
ξξξ
ξξξξ
present;not if 0 present; is 18 node if )1)(1(4
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