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Transcript of 1 Finite Element Method FEM FOR 2D SOLIDS for readers of all backgrounds G. R. Liu and S. S. Quek...
1
FFinite Element Methodinite Element Method
FEM FOR 2D SOLIDS
for readers of all backgroundsfor readers of all backgrounds
G. R. Liu and S. S. Quek
CHAPTER 7:
2Finite Element Method by G. R. Liu and S. S. Quek
CONTENTSCONTENTS INTRODUCTION LINEAR TRIANGULAR ELEMENTS
– Field variable interpolation– Shape functions construction– Using area coordinates– Strain matrix– Element matrices
LINEAR RECTANGULAR ELEMENTS– Shape functions construction– Strain matrix– Element matrices– Gauss integration– Evaluation of me
3Finite Element Method by G. R. Liu and S. S. Quek
CONTENTSCONTENTS
LINEAR QUADRILATERAL ELEMENTS– Coordinate mapping– Strain matrix– Element matrices– Remarks
HIGHER ORDER ELEMENTS COMMENTS (GAUSS INTEGRATION)
4Finite Element Method by G. R. Liu and S. S. Quek
INTRODUCTIONINTRODUCTION
2D solid elements are applicable for the analysis of plane strain and plane stress problems.
A 2D solid element can have a triangular, rectangular or quadrilateral shape with straight or curved edges.
A 2D solid element can deform only in the plane of the 2D solid.
At any point, there are two components in the x and y directions for the displacement as well as forces.
5Finite Element Method by G. R. Liu and S. S. Quek
INTRODUCTIONINTRODUCTION
For plane strain problems, the thickness of the element is unit, but for plane stress problems, the actual thickness must be used.
In this course, it is assumed that the element has a uniform thickness h.
Formulating 2D elements with a given variation of thickness is also straightforward, as the procedure is the same as that for a uniform element.
6Finite Element Method by G. R. Liu and S. S. Quek
2D solids – plane stress and plane strain2D solids – plane stress and plane strain
fx
fy
x
y
x
y fx
z
Plane stress Plane strain
7Finite Element Method by G. R. Liu and S. S. Quek
LINEAR TRIANGULAR LINEAR TRIANGULAR ELEMENTSELEMENTS
Less accurate than quadrilateral elementsUsed by most mesh generators for complex
geometryA linear triangular element:
x, u
y, v
1 (x1, y1) (u1, v1)
2 (x2, y2) (u2, v2)
3 (x3, y3) (u3, v3)
A fsx
fsy
8Finite Element Method by G. R. Liu and S. S. Quek
Field variable interpolationField variable interpolation
( , ) ( , )hex y x yU N d
3 nodeat ntsdisplaceme
2 nodeat ntsdisplaceme
1 nodeat ntsdisplaceme
3
3
2
2
1
1
v
u
v
u
v
u
ed
31 2
31 2
Node 2Node 1 Node 3
00 0
00 0
NN N
NN N
N
where
x, u
y, v
1 (x1, y1) (u1, v1)
2 (x2, y2) (u2, v2)
3 (x3, y3) (u3, v3)
A fsx
fsy
(Shape functions)
9Finite Element Method by G. R. Liu and S. S. Quek
Shape functions constructionShape functions construction
1 1 1 1N a b x c y
2 2 2 2N a b x c y
3 3 3 3N a b x c y
i i i iN a b x c y
Assume,
i= 1, 2, 3
1 T
T
i
i i
i
a
N x y b
c
p
p
or
10Finite Element Method by G. R. Liu and S. S. Quek
Shape functions constructionShape functions construction
Delta function property:
1 for ( , )
0 for i j j
i jN x y
i j
1 1 1
1 2 2
1 3 3
( , ) 1
( , ) 0
( , ) 0
N x y
N x y
N x y
Therefore, 1 1 1 1 1 1 1 1
1 2 2 1 1 2 1 2
1 3 3 1 1 3 1 3
( , ) 1
( , ) 0
( , ) 0
N x y a b x c y
N x y a b x c y
N x y a b x c y
Solving, 2 3 3 2 2 3 3 21 1 1, ,
2 2 2e e e
x y x y y y x xa b c
A A A
11Finite Element Method by G. R. Liu and S. S. Quek
Shape functions constructionShape functions construction
1 1
2 2 2 3 3 2 2 3 1 3 2 1
3 3
11 1 1
1 [( ) ( ) ( ) ]2 2 2
1e
x y
A x y x y x y y y x x x y
x y
P
Area of triangle Moment matrix
Substitute a1, b1 and c1 back into N1 = a1 + b1x + c1y:
1 2 3 2 3 2 2
1[( )( ) ( )( )]
2 e
N y y x x x x y yA
12Finite Element Method by G. R. Liu and S. S. Quek
Shape functions constructionShape functions construction
Similarly,
2 1 1
2 2 2
2 3 3
( , ) 0
( , ) 1
( , ) 0
N x y
N x y
N x y
2 3 1 1 3 3 1 1 3
3 1 3 1 3 3
1[( ) ( ) ( ) ]
2
1[( )( ) ( )( )]
2
e
e
N x y x y y y x x x yA
y y x x x x y yA
3 1 1
3 2 2
3 3 3
( , ) 0
( , ) 0
( , ) 1
N x y
N x y
N x y
3 1 2 1 1 1 2 2 1
1 2 1 2 1 1
1[( ) ( ) ( ) ]
2
1[( )( ) ( )( )]
2
e
e
N x y x y y y x x x yA
y y x x x x y yA
13Finite Element Method by G. R. Liu and S. S. Quek
Shape functions constructionShape functions construction
i i i iN a b x c y
1( )
2
1( )
2
1( )
2
i j k k je
i j ke
i k je
a x y x yA
b y yA
c x xA
where
i
jk
i= 1, 2, 3
J, k determined from cyclic permutation
i = 1, 2
j = 2, 3k = 3, 1
14Finite Element Method by G. R. Liu and S. S. Quek
Using area coordinatesUsing area coordinates
Alternative method of constructing shape functions
i, 1 j, 2
k, 3
x
y
P
A1
1 2 2 2 3 3 2 2 3 3 2
3 3
11 1
1 [( ) ( ) ( ) ]2 2
1
x y
A x y x y x y y y x x x y
x y
11
e
AL
A
2-3-P:
Similarly, 3-1-P A2
1-2-P A3
22
e
AL
A
33
e
AL
A
15Finite Element Method by G. R. Liu and S. S. Quek
Using area coordinatesUsing area coordinates
1 2 3 1L L L Partitions of unity:
3 2 2 32 21 2 3 1
e e e e
A A A AA AL L L
A A A A
Delta function property: e.g. L1 = 0 at if P at nodes 2 or 3
Therefore,1 1 2 2 3 3, , N L N L N L
( , ) ( , )hex y x yU N d
16Finite Element Method by G. R. Liu and S. S. Quek
Strain matrixStrain matrix
xx
yy
xy
u
xv
y
u v
y x
LU where
0
0
x
y
y x
L
ee BdLNdLU
0
0
x
y
y x
B LN N1 2 3
1 2 3
1 1 2 2 3 3
0 0 0
0 0 0
a a a
b b b
b a b a b a
B
(constant strain element)
17Finite Element Method by G. R. Liu and S. S. Quek
Element matricesElement matrices
0d ( d ) d d
e e e
hT T Te
V A A
V z A h A k B cB B cB B cB
Constant matrix T
e ehAk B cB
0d d d d
e e e
hT T Te
V A A
V x A h A m N N N N N N
18Finite Element Method by G. R. Liu and S. S. Quek
Element matricesElement matrices
1 1 1 2 1 3
1 1 1 2 1 3
2 1 2 2 2 3
2 1 2 2 2 3
3 1 3 2 3 3
3 1 3 2 3 3
0 0 0
0 0 0
0 0 0d
0 0 0
0 0 0
0 0 0
e
e
A
N N N N N N
N N N N N N
N N N N N Nh A
N N N N N N
N N N N N N
N N N N N N
m
For elements with uniform density and thickness,
Apnm
pnmALLL pn
A
m 2)!2(
!!!d321
Eisenberg and Malvern (1973):
19Finite Element Method by G. R. Liu and S. S. Quek
Element matricesElement matrices
2
02.
102
0102
10102
010102
12
sy
hAe
m
x, u
y, v
1 (x1, y1) (u1, v1)
2 (x2, y2) (u2, v2)
3 (x3, y3) (u3, v3)
A fsx
fsy
lf
fl
sy
sxe d ][
32
T
Nf
y
x
y
xe
f
f
f
fl
0
0
2
132fUniform distributed load:
20Finite Element Method by G. R. Liu and S. S. Quek
LINEAR RECTANGULAR LINEAR RECTANGULAR ELEMENTSELEMENTS
Non-constant strain matrixMore accurate representation of stress and
strainRegular shape makes formulation easy
21Finite Element Method by G. R. Liu and S. S. Quek
Shape functions constructionShape functions construction
x, u
y, v
1 (x1, y1) (u1, v1)
2 (x2, y2) (u2, v2)
3 (x3, y3) (u3, v3)
2a
fsy fsx
4 (x4, y4) (u4, v4)
2b
Consider a rectangular element
1
1
2
2
3
3
4
4
displacements at node 1
displacements at node 2
displacements at node 3
displacements at node 4
e
u
v
u
u
u
u
u
u
d
22Finite Element Method by G. R. Liu and S. S. Quek
Shape functions constructionShape functions construction
x, u
y, v
1 (x1, y1) (u1, v1)
2 (x2, y2) (u2, v2)
3 (x3, y3) (u3, v3)
2a
fsy fsx
4 (x4, y4) (u4, v4)
2b
1 ( 1, 1) (u1, v1)
2 (1, 1) (u2, v2)
3 (1, +1) (u3, v3)
2a
4 ( 1, +1) (u4, v4)
2b
, byax
( , ) ( , )hex y x yU N d 31 2 4
31 2 4
Node 2 Node 3Node 1 Node 4
00 0 0
00 0 0
NN N N
NN N N
N
where
(Interpolation)
23Finite Element Method by G. R. Liu and S. S. Quek
Shape functions constructionShape functions construction
)1)(1(
)1)(1(
)1)(1(
)1)(1(
41
4
41
3
41
2
41
1
N
N
N
N
113 4at node 1 1
113 4at node 2 1
113 4at node 3 1
113 4at node 4 1
(1 )(1 ) 0
(1 )(1 ) 0
(1 )(1 ) 1
(1 )(1 ) 0
N
N
N
N
Delta function property
4
1 2 3 41
14
14
[(1 )(1 ) (1 )(1 ) (1 )(1 ) (1 )(1 )]
[2(1 ) 2(1 )] 1
ii
N N N N N
Partition of unity
)1)(1(41 jjjN
1 ( 1, 1) (u1, v1)
2 (1, 1) (u2, v2)
3 (1, +1) (u3, v3)
2a
4 ( 1, +1) (u4, v4)
2b
24Finite Element Method by G. R. Liu and S. S. Quek
Strain matrixStrain matrix
abababab
bbbb
aaaa
11111111
1111
1111
0000
0000
LNB
Note: No longer a constant matrix!
25Finite Element Method by G. R. Liu and S. S. Quek
Element matricesElement matrices
, byax dxdy = ab dd
Therefore,
ddd T1
1
1
1
T cBBcBBk habAhA
e
dddddd1
1
1
10NNNNNNNNm TT
A
T
A
hT
V
e abhAhAxV
26Finite Element Method by G. R. Liu and S. S. Quek
Element matricesElement matrices
lf
fl
sy
sxe d ][
32
T
Nf
x, u
y, v
1 (x1, y1) (u1, v1)
2 (x2, y2) (u2, v2)
3 (x3, y3) (u3, v3)
2a
fsy fsx
4 (x4, y4) (u4, v4)
2b
For uniformly distributed load,
0
0
0
0
y
x
y
x
e
ff
f
f
bf
27Finite Element Method by G. R. Liu and S. S. Quek
Gauss integrationGauss integration
For evaluation of integrals in ke and me (in practice)
In 1 direction: )()d(1
1
1 jj
m
j
fwfI
m gauss points gives exact solution of polynomial integrand of n = 2m - 1
1 1
1 11 1
( , )d d ( , )yx
nn
i j i ji j
I f w w f
In 2 directions:
28Finite Element Method by G. R. Liu and S. S. Quek
Gauss integrationGauss integrationm j wj Accuracy n
1 0 2 1
2 -1/3, 1/3 1, 1 3
3 -0.6, 0, 0.6 5/9, 8/9, 5/9 5
4 -0.861136, -0.339981, 0.339981, 0.861136
0.347855, 0.652145, 0.652145, 0.347855
7
5 -0.906180, -0.538469, 0,
0.538469, 0.906180
0.236927, 0.478629, 0.568889, 0.478629, 0.236927
9
6 -0.932470, -0.661209,
-0.238619, 0.238619, 0.661209, 0.932470
0.171324, 0.360762, 0.467914, 0.467914, 0.360762, 0.171324
11
29Finite Element Method by G. R. Liu and S. S. Quek
Evaluation of Evaluation of mmee
4
04.
204
0204
10204
010204
2010204
02010204
9
sy
habe
m
30Finite Element Method by G. R. Liu and S. S. Quek
Evaluation of Evaluation of mmee
E.g.
)1)(1(4
)1)(1()1)(1(16
31
31
1
1
1
1
1
1
1
1
jiji
jiji
jiij
hab
ddhab
ddNNhabm
94)111)(111(
4 31
31
33
habhabm
Note: In practice, Gauss integration is often used
31Finite Element Method by G. R. Liu and S. S. Quek
LINEAR QUADRILATERAL LINEAR QUADRILATERAL ELEMENTSELEMENTS
Rectangular elements have limited applicationQuadrilateral elements with unparallel edges are
more useful Irregular shape requires coordinate mapping
before using Gauss integration
32Finite Element Method by G. R. Liu and S. S. Quek
Coordinate mappingCoordinate mapping
2 (x2, y2)
y
x 1 (1, 1) 2 (1, 1)
3 (1, +1) 4 (1, +1)
3 (x3, y3) 4 (x4, y4)
1 (x1, y1)
Physical coordinates Natural coordinates
( , ) ( , )he U N d (Interpolation of displacements)
( , ) ( , ) e X N x (Interpolation of coordinates)
33Finite Element Method by G. R. Liu and S. S. Quek
Coordinate mappingCoordinate mapping
( , ) ( , ) e X N x
wherex
y
X ,
1
1
2
2
3
3
4
4
coordinate at node 1
coordinate at node 2
coordinate at node 3
coordinate at node 4
e
x
y
x
y
x
y
x
y
x
)1)(1(
)1)(1(
)1)(1(
)1)(1(
41
4
41
3
41
2
41
1
N
N
N
N
iii
xNx ),(4
1
iii
yNy ),(4
1
34Finite Element Method by G. R. Liu and S. S. Quek
Coordinate mappingCoordinate mapping
Substitute 1 into iii
xNx ),(4
1
2 (x2, y2)
y
x 1 ( 1, 1) 2 (1, 1)
3 (1, +1) 4 ( 1, +1)
3 (x3, y3) 4 (x4, y4)
1 (x1, y1)
321
221
321
221
)1()1(
)1()1(
yyy
xxx
or
)()(
)()(
2321
3221
2321
3221
yyyyy
xxxxx
Eliminating , )()}({)(
)(322
1322
1
23
23 yyxxxyy
xxy
35Finite Element Method by G. R. Liu and S. S. Quek
Strain matrixStrain matrix
y
y
Nx
x
NN
y
y
Nx
x
NN
iii
iii i i
ii
N N
xNNy
Jor
x y
x y
Jwhere (Jacobian matrix)
1 131 2 4
2 2
3 331 2 4
4 4
x yNN N N
x y
x yNN N N
x y
JSince ( , ) ( , ) e X N x ,
36Finite Element Method by G. R. Liu and S. S. Quek
Strain matrixStrain matrix
1
ii
i i
NN
xN Ny
JTherefore,
NLNB
xy
y
x
0
0
Replace differentials of Ni w.r.t. x and y with differentials of Ni w.r.t. and
(Relationship between differentials of shape functions w.r.t. physical coordinates and differentials w.r.t. natural coordinates)
37Finite Element Method by G. R. Liu and S. S. Quek
Element matricesElement matrices
Murnaghan (1951) : dA=det |J | dd
1 1 T
1 1det d de h
k B cB J
dddet
dddd
1
1
1
1
0
JNN
NNNNNNm
T
T
A
T
A
hT
V
e
h
AhAxV
38Finite Element Method by G. R. Liu and S. S. Quek
RemarksRemarks
Shape functions used for interpolating the coordinates are the same as the shape functions used for interpolation of the displacement field. Therefore, the element is called an isoparametric element.
Note that the shape functions for coordinate interpolation and displacement interpolation do not have to be the same.
Using the different shape functions for coordinate interpolation and displacement interpolation, respectively, will lead to the development of so-called subparametric or superparametric elements.
39Finite Element Method by G. R. Liu and S. S. Quek
HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS
Higher order triangular elements
i (I,J,K)
(p,0,0) (0,p,0)
(0,0,p)
(p 1,1,0)
L1
L3
L2
(0,p 1,1)
(0,1,p 1) (1,0,p 1)
(2,0,p 2)
nd = (p+1)(p+2)/2
I J K p Node i,
Argyris, 1968 :
1 2 3( ) ( ) ( )I J Ki I J KN l L l L l L
0 1 ( 1)
0 1 ( 1)
( )( ) ( )( )
( )( ) ( )I I I
L L L L L Ll L
L L L L L L
40Finite Element Method by G. R. Liu and S. S. Quek
HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS
Higher order triangular elements (Cont’d)
x , u
y , v
1
2
3
4
5 6
1 2 2 1 1( 2 1 )N N N L L
4 5 6 1 24N N N L L
x , u
y , v
1
2
3
4 5
6
7 8
9 1 0
1 2 3 1 1 1
1( 3 1 ) ( 3 2 )
2N N N L L L
4 9 1 2 1
9( 3 1 )
2N N L L L
1 0 1 2 32 7N L L L
Cubic element
Quadratic element
41Finite Element Method by G. R. Liu and S. S. Quek
HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS
Higher order rectangular elements
(0,0)
0
(n,0)
(0,m) (n,m)
i(n,m)
Lagrange type:
1 1 ( ) ( )D D n mi I J I JN N N l l
0 1 1 1
0 1 1 1
( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )n k k nk
k k k k k k k n
l
[Zienkiewicz et al., 2000]
42Finite Element Method by G. R. Liu and S. S. Quek
HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS
Higher order rectangular elements (Cont’d)
1 2
3 4
5
6
7
8 9
1 11 1 1
1 12 2 1
1 13 2 2
1 14 1 2
1( ) ( ) (1 ) (1 )
41
( ) ( ) (1 ) (1 )4
1( ) ( ) (1 )(1 )
41
( ) ( ) (1 )(1 )4
D D
D D
D D
D D
N N N
N N N
N N N
N N N
1 15 3 1
1 16 2 3
1 17 3 2
1 18 1 1
1 1 2 29 3 3
1( ) ( ) (1 )(1 )(1 )
21
( ) ( ) (1 )(1 )(1 )21
( ) ( ) (1 )(1 )(1 )2
1( ) ( ) (1 )(1 )
2
( ) ( ) (1 )(1 )
D D
D D
D D
D D
D D
N N N
N N N
N N N
N N N
N N N
(nine node quadratic element)
43Finite Element Method by G. R. Liu and S. S. Quek
HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS
Higher order rectangular elements (Cont’d)
Serendipity type:
1 2
3 4
5
6
7
8 0
=1
= 1
14
212
212
(1 )(1 )( 1) 1, 2, 3, 4
(1 )(1 ) 5, 7
(1 )(1 ) 6,8
j j j j j
j j
j j
N j
N j
N j
(eight node quadratic element)
44Finite Element Method by G. R. Liu and S. S. Quek
HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS
Higher order rectangular elements (Cont’d)
1 2
3 4
5 6
7
8
9 10
11
12
2 2132
2932
13
2932
(1 )(1 )(9 9 10)
for corner nodes 1, 2, 3, 4
(1 )(1 )(1 9 )
for side nodes 7, 8, 11, 12 where 1 and
(1 )(1 )(1
j j j
j j j
j j
j j
N
j
N
j
N
13
9 )
for side nodes 5, 6, 9, 10 where and 1
j
j jj
(twelve node cubic element)
45Finite Element Method by G. R. Liu and S. S. Quek
ELEMENT WITH CURVED ELEMENT WITH CURVED EDGESEDGES
4
2
3
8
1 5
7
6
1 4 2
5
3
6
1
2
3
4
5 6
1 2
3 4
5
6
7
8
46Finite Element Method by G. R. Liu and S. S. Quek
COMMENTS (GAUSS COMMENTS (GAUSS INTEGRATION)INTEGRATION)
When the Gauss integration scheme is used, one has to decide how many Gauss points should be used.
Theoretically, for a one-dimensional integral, using m points can give the exact solution for the integral of a polynomial integrand of up to an order of (2m1).
As a general rule of thumb, more points should be used for
a higher order of elements.
47Finite Element Method by G. R. Liu and S. S. Quek
COMMENTS (GAUSS COMMENTS (GAUSS INTEGRATION)INTEGRATION)
Using a smaller number of Gauss points tends to counteract the over-stiff behaviour associated with the displacement-based method.
Displacement in an element is assumed using shape functions. This implies that the deformation of the element is somehow prescribed in a fashion of the shape function. This prescription gives a constraint to the element. The so-constrained element behaves stiffer than it should. It is often observed that higher order elements are usually softer than lower order ones. This is because using higher order elements gives fewer constraint to the elements.
48Finite Element Method by G. R. Liu and S. S. Quek
COMMENTS ON GAUSS COMMENTS ON GAUSS INTEGRATIONINTEGRATION
Two Gauss points for linear elements, and two or three points for quadratic elements in each direction should be sufficient for most cases.
Most of the explicit FEM codes based on explicit formulation tend to use one-point integration to achieve the best performance in saving CPU time.
49Finite Element Method by G. R. Liu and S. S. Quek
CASE STUDYCASE STUDY
Side drive micro-motor
50Finite Element Method by G. R. Liu and S. S. Quek
CASE STUDYCASE STUDY
Elastic Properties of Polysilicon
Young’s Modulus, E 169GPa
Poisson’s ratio, 0.262
Density, 2300kgm-3
10N/m
10N/m
10N/m
51Finite Element Method by G. R. Liu and S. S. Quek
CASE STUDYCASE STUDY
Analysis no. 1: Von Mises stress distribution using 24 bilinear
quadrilateral elements (41 nodes)
52Finite Element Method by G. R. Liu and S. S. Quek
CASE STUDYCASE STUDY
Analysis no. 2: Von Mises stress distribution using 96 bilinear
quadrilateral elements (129 nodes)
53Finite Element Method by G. R. Liu and S. S. Quek
CASE STUDYCASE STUDY
Analysis no. 3: Von Mises stress distribution using 144 bilinear
quadrilateral elements (185 nodes)
54Finite Element Method by G. R. Liu and S. S. Quek
CASE STUDYCASE STUDY
Analysis no. 4: Von Mises stress distribution using 24 eight-nodal, quadratic elements (105 nodes)
55Finite Element Method by G. R. Liu and S. S. Quek
CASE STUDYCASE STUDY
Analysis no. 5: Von Mises stress distribution using 192 three-nodal,
triangular elements (129 nodes)
56Finite Element Method by G. R. Liu and S. S. Quek
CASE STUDYCASE STUDYAnalysis
no.Number / type of
elements
Total number of nodes in
model
Maximum Von Mises
Stress (GPa)
124 bilinear,
quadrilateral 41 0.0139
296 bilinear, quadrilateral
129 0.0180
3144 bilinear, quadrilateral
185 0.0197
424 quadratic, quadrilateral
105 0.0191
5 192 linear, triangular
129 0.0167