Ruhr Universität Bochum Fakultät für Bauingenieurwesen Computational Engineering Finite Elements in Structural Mechanics Shape Functions generation, i t t requirements, et c. Student presentation Student presentation S d Silj k E Student: SiljakEnes January, 2009
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Shape Functions generation, requitirements, etc. · 2011-11-21 · Displacement Approximation ( ) ~ ( ) ( ) (1) A finite element approximation of displacements is given by u x ≈u
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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.
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
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
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
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,
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