Download - Chpt06-FEM for 3D Solidsnew
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FFinite Element Methodinite Element Method
FEM FOR 3D SOLIDS
A Practical CourseA Practical Course
CHAPTER 6:
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CONTENTSCONTENTS INTRODUCTION TETRAHEDRON ELEMENT
– Shape functions– Strain matrix– Element matrices
HEXAHEDRON ELEMENT– Shape functions– Strain matrix– Element matrices– Using tetrahedrons to form hexahedrons
HIGHER ORDER ELEMENTS ELEMENTS WITH CURVED SURFACES CASE STUDY
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INTRODUCTIONINTRODUCTION
For 3D solids, all the field variables are dependent of x, y and z coordinates – most general element.
The element is often known as a 3D solid element or simply a solid element.
A 3-D solid element can have a tetrahedron and hexahedron shape with flat or curved surfaces.
At any node there are three components in x, y and z directions for the displacement as well as forces.
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TETRAHEDRON ELEMENTTETRAHEDRON ELEMENT
3D solid meshed with tetrahedron elements
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TETRAHEDRON ELEMENTTETRAHEDRON ELEMENT
z=Z
x=Xz = Z
y=Y
w4
v4
u4
w2
u2
u2
w1
u1
v1
w3
u3
v3 i
j
l
k 1 =
4 =
2 =
3 =
fsy
fsz
fsx
Consider a 4 node tetrahedron element
1
1
1
2
2
2
3
3
3
4
4
4
node 1
node 2
node 3
node 4
e
u
v
w
u
v
w
u
v
w
u
v
w
d
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Shape functionsShape functions
( , , ) ( , , )hex y z x y zU N d
1 2 3 4
1 2 3 4
1 2 3 4
node 1 node 2 node 3 node 4
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
N N N N
N N N N
N N N N
N
where
Use volume coordinates (Recall Area coordinates for 2D triangular element)
1234
2341 V
VL P
1=i
2=j
3=k
4=l
P
y
z
x
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Shape functionsShape functions
Similarly,1234
1234
1234
1243
1234
1342 , ,
V
VL
V
VL
V
VL PPP
Can also be viewed as ratio of distances
234 134 1231241 2 3 4
1 234 1 234 1 234 1 234
, , , P P PPd d ddL L L L
d d d d
1=i
2=j
3=k
4=l
P
y
z
x
1 4321 LLLL
since
1234123124134234 VVVVV PPPP
(Partition of unity)
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Shape functionsShape functions
jkl
iLi nodes remote theat the 0
node home at the 1
44332211
44332211
44332211
zLzLzLzLz
yLyLyLyLy
xLxLxLxLx
(Delta function property)
1 4321 LLLL
4
3
2
1
4321
4321
4321
1 1 1 11
L
L
L
L
zzzz
yyyy
xxxx
z
y
x
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Shape functionsShape functions
Therefore,
where
z
y
x
dcba
dcba
dcba
dcba
V
L
L
L
L 1
6
1
4444
3333
2222
1111
4
3
2
1
1
det , det 1
1
1 1
det 1 , det 1
1 1
j j j j j
i k k k i k k
l l l l l
j j j j
i k k i k k
l l l l
x y z y z
a x y z b y z
x y z y z
y z y z
c y z d y z
y z y z
(Adjoint matrix)
(Cofactors)
i
j
k
l
i= 1,2
j = 2,3
k = 3,4
l = 4,1
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Shape functionsShape functions
l
k
j
i
l
k
j
i
l
k
j
i
z
z
z
z
y
y
y
y
x
x
x
x
V
1
1
1
1
det6
1(Volume of tetrahedron)
)(6
1zdycxba
VLN iiiiii Therefore,
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Strain matrixStrain matrix
Since, ( , , ) ( , , )hex y z x y zU N d
Therefore, ee BdLNdLU where NLNB
0
0
0
00
00
00
xy
xz
yz
z
y
x
(Constant strain element)
31 2 4
31 2 4
31 2 4
3 31 1 2 2 4 4
3 31 1 2 2 3 4
3 31 1 2 2 4 4
0 00 0 0 0 0 0
0 00 0 0 0 0 0
0 00 0 0 0 0 0100 0 02
00 0 0
00 0 0
bb b b
cc c c
dd d d
d cd c d c d cV
d bd b d b d b
c bc b c b c b
B
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Element matricesElement matrices
e
T Te eV
dV V k B cB B cB
11 12 13 14
21 22 23 24
31 32 33 34
41 42 43 44
d de e
Te
V V
V V
N N N N
N N N Nm N N
N N N N
N N N N
where
ji
ji
ji
ij
NN
NN
NN
00
00
00
N
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Element matricesElement matrices
1 2 3 4
! ! ! !d 6
( 3)!e
m n p qeV
m n p qL L L L V V
m n p q
Eisenberg and Malvern, 1973 :
2 0 0 1 0 0 1 0 0 1 0 0
2 0 0 1 0 0 1 0 0 1 0
2 0 0 1 0 0 1 0 0 1
2 0 0 1 0 0 1 0 0
2 0 0 1 0 0 1 0
2 0 0 1 0 0 1
2 0 0 1 0 020
2 0 0 1 0
2 0 0 1
. 2 0 0
2 0
2
ee
V
sy
m
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Element matricesElement matrices
Alternative method for evaluating me: special natural coordinate system
z
x
y
i
j
l
k
1 =
4 =
2 =
3 =
=0
=1
=1
=constant
P
Q
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Element matricesElement matrices
z
x
y
i
j
l
k
1 =
4 =
2 =
3 =
=0
=0
=1
=constant
P
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Element matricesElement matrices
z
x
y
i
j
l
k
1 =
4 =
2 =
3 =
=1
=1
=1
=0
=constant
P
Q R
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Element matricesElement matrices3 2 2
3 2 2
3 2 2
( )
( )
( )
P
P
P
x x x x
y y y y
z z z z
1 1 3 2 2 1 1
1 1 3 2 2 1 1
1 1 3 2 2 1 1
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
B P
B P
B P
x x x x x x x x x
y y y y y y y y y
z z z z z z z z z
4 4 4 4 1 2 1 2 3
4 4 4 4 1 2 1 2 3
4 4 4 4 1 2 1 2 3
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
B
B
B
x x x x x x x x x x x
y y y y y y y y y y y
z z z z z z z z z z z
1
2
3
4
(1 )
(1 )
(1 )
N
N
N
N
z
x z=Z
y
i
j
l
k
1 =
4 =
2 =
3 =
=0 =0 =1
=1 =0 =1
=1 =1 =1
=0
= constant
P [xP(x3 x2)+x2, yP(y3 y2)+y2,zP(z3 z2)+z2]
O
B
B [xB(xP x1)+x1, yB(yP y1)y1, zB(zP y1)z1]
O [x=(1 )(x4 xB)xB, y=(1 )(y4 yB)yB, z=(1 )(z4 zB)zB]
=constant
= constant
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Element matricesElement matrices
Jacobian:
z
y
x
z
y
x
z
y
x
J
1 1 1
0 0 0d det d d d
e
T Te
V
V m N N N N [J]
11 12 13 14
1 1 1 21 22 23 242
0 0 031 32 33 34
41 42 43 44
6 d d de eV
N N N N
N N N Nm
N N N N
N N N N
21 31 31 41 21 312
21 31 31 41 21 31
21 31 31 41 21 31
det[ ] 6
x x x x x x
y y y y y y V
z z z z z z
J
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Element matricesElement matrices
T
2 3[ ] d
sx
e syl
sz
f
f l
f
f N
z=Z
x=Xz=Z
y=Y
w 4
v4
u4
w2
u2
u2
w 1
u1
v1
w3
u3
v3 i
j
l
k 1 =
4 =
2 =
3 =
fsy
fsz
fsx
For uniformly distributed load:
3 1
2 3
3 1
1
2
sx
sy
sz
e
sx
sy
sz
f
f
fl
f
f
f
0
f
0
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HEXAHEDRON ELEMENTHEXAHEDRON ELEMENT
3D solid meshed with hexahedron elements
P P’
P’’ P’’’
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Shape functionsShape functions
eNdU
1
2
3
4
5
6
7
8
displacement components at node 1
displacement components at node 2
displacement components at node 3
displacement components at node 4
displacement co
e
e
e
ee
e
e
e
e
d
d
d
dd
d
d
d
d
mponents at node 5
displacement components at node 6
displacement components at node 7
displacement components at node 8
1
1
1
( 1, 2, ,8) ei
u
v i
w
d
17
5 8
6 4
2
0
z
y
x
3
0
fsz
fsyfsx
87654321 NNNNNNNNN
)8,,2,1(
00
00
00
i
N
N
N
i
i
i
iN
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Shape functionsShape functions
4(-1, 1, -1)
(1, -1, 1)6
(1, -1, -1)2
1 7
5 8
6 4
2 0
z
y
x
3
0
fsz
fsy fsx
8(-1, 1, 1)
7 (1, 1, 1)
(-1, -1, 1)5
(-1, -1, -1)1
3(1, 1, -1)
iii
iii
iii
zNz
yNy
xNx
),,(
),,(
),,(
8
1
8
1
8
1
)1)(1)(1(
8
1iiiiN
(Tri-linear functions)
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Strain matrixStrain matrix
87654321 BBBBBBBBB
whereby
0
0
0
00
00
00
xNyN
xNzN
yNzN
zN
yN
xN
ii
ii
ii
i
i
i
ii LNB
Note: Shape functions are expressed in natural coordinates – chain rule of differentiation
ee BdLNdLU
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Strain matrixStrain matrix
z
z
Ny
y
Nx
x
NN
z
z
Ny
y
Nx
x
NN
z
z
Ny
y
Nx
x
NN
iiii
iiii
iiii
Chain rule of differentiation
z
Ny
Nx
N
N
N
N
i
i
i
i
i
i
J
where
z
z
z
y
y
y
x
x
x
J
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Strain matrixStrain matrix8 8 8
1 1 1
( , , ) , ( , , ) , ( , , )i i i i i ii i i
x N x y N y z N z
Since,
or
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
i
ii
i
ii
i
ii
i
ii
i
ii
i
ii
i
ii
i
ii
i
ii
Nz
Nz
Nz
Ny
Ny
Ny
Nx
Nx
Nx
J
1 1 1
2 2 23 5 6 7 81 2 4
3 3 3
4 4 43 5 6 7 81 2 4
5 5 5
6 6 61 2 3 4 5 6 7 8
7 7 7
8 8 8
x y z
x y zN N N N NN N Nx y z
x y zN N N N NN N Nx y z
x y zN N N N N N N N
x y z
x y z
J
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Strain matrixStrain matrix
i
i
i
i
i
i
N
N
N
z
Ny
Nx
N
1J
0
0
0
00
00
00
xNyN
xNzN
yNzN
zN
yN
xN
ii
ii
ii
i
i
i
ii LNB
Used to replace derivatives w.r.t. x, y, z with derivatives w.r.t. , ,
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Element matricesElement matrices
1 1 1T T
1 1 1d det[ ]d d d
e
e
V
V
k B cB B cB J
Gauss integration: ),,(d)d,(1 1 1
1
1
1
1
1
1 jjikji
n
i
m
j
l
k
fwwwfI
1 1 1
1 1 1d det d d d
e
T Te
V
V
m N N N N [J]
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Element matricesElement matrices
For rectangular hexahedron:
det / 8eabc V [J]
11 12 13 14 15 16 17 18
22 23 24 25 26 27 28
33 34 35 36 37 38
44 45 46 47 48
55 56 57 58
66 67 68
77 78
88
.
e
sy
m m m m m m m m
m m m m m m m
m m m m m m
m m m m mm
m m m m
m m m
m m
m
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Element matricesElement matrices
(Cont’d)
where
ddd
00
00
00
ddd
00
00
00
00
00
00
ddd
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
ji
ji
ji
j
j
j
i
i
i
jiij
NN
NN
NN
abc
N
N
N
N
N
N
abc
abc NNm
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Element matricesElement matrices
(Cont’d)
or
ij
ij
ij
ij
m
m
m
00
00
00
m
where
)1)(1)(1(8
d)1)(1(d)1)(1(d)1)(1(64
ddd
31
31
31
1
1
1
1
1
1
1
1
1
1
jijiji
jijiji
jiij
hab
abc
NNabcm
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Element matricesElement matrices
(Cont’d)
E.g.216
8)111)(111)(111(8 3
131
31
33
abcabcm
216
1216
2216
4
216
8
46352817
184538276857473625162413
483726155814786756342312
8877665544332211
abcmmmm
abcmmmmmmmmmmmm
abcmmmmmmmmmmmm
abc
mmmmmmmm
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Element matricesElement matrices
(Cont’d)
8
48.
248
4248
42128
242148
1242248
21244248
216
sy
abcex
m
Note: For x direction only
(Rectangular hexahedron)
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Element matricesElement matrices
l
f
f
f
l
sz
sy
sx
e d ][43
T
Nf
17
5 8
6 4
2
0
z
y
x
3
0
fsz
fsyfsx
13
13
13
13
13
13
432
1
0
0
0
0
0
0
f
sz
sy
sx
sz
sy
sx
e
f
f
ff
f
f
l
For uniformly distributed load:
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Using tetrahedrons to form hexahedronsUsing tetrahedrons to form hexahedrons
Hexahedrons can be made up of several tetrahedrons
1
5
6
8 1 4
3
8
1
2 3
4
5
7
8
3
1 6
8
6
3
2
1
6
3
6 7
8 Hexahedron made up of 5 tetrahedrons:
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Using tetrahedrons to form hexahedronsUsing tetrahedrons to form hexahedrons
1
2 3
4
5
7
8
6
1
2
4
5 8
6
2 3
7
8
6 4
1 4
5
6
1
2
4 6
5 8
6 4
Break into three
Hexahedron made up of 6 tetrahedrons:
Element matrices can be obtained by assembly of tetrahedron elements
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HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS
Tetrahedron elements
1
9
8
7 10
2
5
6
3
4
5 2 3
6 1 3
7 1 2
8 1 4
9 2 4
10 3 4
(2 -1) for corner nodes 1,2,3,4
4
4
4 for mid-edge nodes
4
4
4
i i iN L L i
N L L
N L L
N L L
N L L
N L L
N L L
10 nodes, quadratic:
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HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS
Tetrahedron elements (Cont’d)20 nodes, cubic:
12
9 95 1 1 3 11 1 1 42 2
9 96 3 1 3 12 4 1 42 2
9 97 1 1 2 13 22 2
98 2 1 22
99 2 2 32
910 3 2 32
(3 1)(3 2) for corner nodes 1,2,3,4
(3 1) (3 1)
(3 1) (3 1)
(3 1) (3 1)
(3 1)
(3 1)
(3 1)
i i i iN L L L i
N L L L N L L L
N L L L N L L L
N L L L N L L
N L L L
N L L L
N L L L
2 4
914 4 2 42
915 3 3 42
916 4 3 42
17 2 3 4
18 1 2 3
19 1 3 4
20 1 2 4
for edge nodes(3 1)
(3 1)
(3 1)
27
27 for center surface nodes
27
27
L
N L L L
N L L L
N L L L
N L L L
N L L L
N L L L
N L L L
1
13 12
7
15
2
9
6 3
4
5
8
10
11
14
16
17
18
195
20
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HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS
Brick elements
Lagrange type:
i(I,J,K)
(0,0,0)
(n,m,p)
(n,0,0)
(n,m,0)
nd=(n+1)(m+1)(p+1) nodes
1 1 1 ( ) ( ) ( )D D D n m pi I J K I J KN N N N l l l
0 1 1 1
0 1 1 1
( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )n k k nk
k k k k k k k n
l
where
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HIGHER ORDER HIGHER ORDER ELEMENTSELEMENTS
Brick elements (Cont’d)
Serendipity type elements:
4(-1, 1, -1)
(1, -1, 1)6
(1, -1, -1)2
8(-1, 1, 1)
7 (1, 1, 1)
(-1, -1, 1)5
(-1,-1,-1)1
3(1, 1, -1)
9(1,0,-1)
10(0,1,-1)
11(-1,0,-1) 12(0-1,-1)
13 143
15
16
17 18
19 20
18
214
214
(1 )(1 )(1 )( 2)
for corner nodes 1, , 8
(1 )(1 )(1 ) for mid-side nodes 10,12,14,16
(1 )(1
j j j j j j i
j j j
j
N
j
N j
N
214
)(1 ) for mid-side nodes 9,11,13,15
(1 )(1 )(1 ) for mid-side nodes 17,18,19,20
j j
j j j
j
N j
20 nodes, tri-quadratic:
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HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS
Brick elements (Cont’d)
2 2 2164
2964
13
2964
(1 )(1 )(1 )(9 9 9 19)
for corner nodes 1, , 8
(1 )(1 9 )(1 )(1 )
for side nodes with , 1 and 1
(1 )(1 9
j j j j
j j j j
j j j
j
N
j
N
N
13
2964
13
)(1 )(1 )
for side nodes with , 1 and 1
(1 )(1 9 )(1 )(1 )
for side nodes with , 1 and 1
j j j
j j j
j j j j
j j j
N
32 nodes, tri-cubic:
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ELEMENTS WITH CURVED ELEMENTS WITH CURVED SURFACESSURFACES
1
4
9 8
7 10
2 5
6 3
7 18
16
12 15
14 11
13
5 17 19
20
6
10 9
8
2
1
4 3
9 8
7 10
2
5
6 3
1
4
13 7 18 16
12 15
14 11
5 17 19
20
6
10
9
8
2
1 4
3
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CASE STUDYCASE STUDY
Stress and strain analysis of a quantum dot heterostructure
Material E (Gpa)
GaAs 86.96 0.31
InAs 51.42 0.35
GaAs substrate
GaAs cap layer
InAs wetting layer
InAs quantum dot
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CASE STUDYCASE STUDY
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CASE STUDYCASE STUDY30 nm
30 nm
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CASE STUDYCASE STUDY
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CASE STUDYCASE STUDY