1. 1
FFinite Element Methodinite Element Method
FEM FOR 3D SOLIDS
for readers of all backgroundsfor readers of all backgrounds
G. R. Liu and S. S. Quek
CHAPTER 9:
2. 2Finite Element Method by G. R. Liu and S. S. Quek
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
3. 3Finite Element Method by G. R. Liu and S. S. Quek
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 3D solid element can have a tetrahedron and
hexahedron shape with flat or curved surfaces.
At any node there are three components in the x, y
and z directions for the displacement as well as
forces.
4. 4Finite Element Method by G. R. Liu and S. S. Quek
TETRAHEDRON ELEMENTTETRAHEDRON ELEMENT
3D solid meshed with tetrahedron elements
5. 5Finite Element Method by G. R. Liu and S. S. Quek
TETRAHEDRON ELEMENTTETRAHEDRON ELEMENT
z=Z
x=X
z=Z
y=Y
w4
v4
u4
w2
u2
u2
w1
u1
v1
w3
u3
v3
i
j
l
k1 =
4 =
2 =
3 =
fsy
fsz
fsx
Consider a four 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
6. 6Finite Element Method by G. R. Liu and S. S. Quek
Shape functionsShape functions
( , , ) ( , , )h
ex y z x y z=U 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
6447448 6447448 6447448 6447448
where
Use volume coordinates (Recall Area coordinates for
2D triangular element)
1234
234
1
V
V
L P
=
1=i
2=j
3=k
4=l
P
y
z
x
7. 7Finite Element Method by G. R. Liu and S. S. Quek
Shape functionsShape functions
Similarly,
1234
123
4
1234
124
3
1234
134
2 ,,
V
V
L
V
V
L
V
V
L PPP
===
Can also be viewed as ratio of distances
234 134 123124
1 2 3 4
1 234 1 234 1 234 1 234
, , ,P P PPd d dd
L L L L
d d d d
− − −−
− − − −
= = = =
1=i
2=j
3=k
4=l
P
y
z
x
14321 =+++ LLLL
since
1234123124134234 VVVVV PPPP =+++
(Partition of unity)
8. 8Finite Element Method by G. R. Liu and S. S. Quek
Shape functionsShape functions
=
jkl
i
Li
nodesremotetheat the0
nodehomeat the1
44332211
44332211
44332211
zLzLzLzLz
yLyLyLyLy
xLxLxLxLx
+++=
+++=
+++=
(Delta function property)
14321 =+++ LLLL
=
4
3
2
1
4321
4321
4321
11111
L
L
L
L
zzzz
yyyy
xxxx
z
y
x
9. 9Finite Element Method by G. R. Liu and S. S. Quek
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
10. 10Finite Element Method by G. R. Liu and S. S. Quek
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
det
6
1
(Volume of tetrahedron)
)(
6
1
zdycxba
V
LN iiiiii +++==Therefore,
11. 11Finite Element Method by G. R. Liu and S. S. Quek
Strain matrixStrain matrix
Since, ( , , ) ( , , )h
ex y z x y z=U N d
Therefore, ee BdLNdLU ===ε where NLNB
∂∂∂∂
∂∂∂∂
∂∂∂∂
∂∂
∂∂
∂∂
==
0
0
0
00
00
00
xy
xz
yz
z
y
x
=
44
44
44
4
4
4
33
33
33
3
3
3
22
22
22
2
2
2
11
11
11
1
1
1
0
0
0
00
00
00
0
0
0
00
00
00
0
0
0
00
00
00
0
0
0
00
00
00
2
1
bd
cd
bc
d
c
b
bd
cd
bc
d
c
b
bd
cd
bc
d
c
b
bd
cd
bc
d
c
b
V
B
(Constant strain element)
12. 12Finite Element Method by G. R. Liu and S. S. Quek
Element matricesElement matrices
e
T T
e 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 d
e e
T
e
V V
V Vρ ρ
= =
∫ ∫
N N N N
N N N N
m N N
N N N N
N N N N
where
=
ji
ji
ji
ij
NN
NN
NN
00
00
00
N
13. 13Finite Element Method by G. R. Liu and S. S. Quek
Element matricesElement matrices
1 2 3 4
! ! ! !
d 6
( 3)!e
m n p q
eV
m n p q
L 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
e
e
V
sy
ρ
=
m
14. 14Finite Element Method by G. R. Liu and S. S. Quek
Element matricesElement matrices
Alternative method for evaluating me: special natural
coordinate system
z
x
z=Z
y
i
j
l
k
1 =
4 =
2 =
3 =
ξ=0
ξ=1
ξ=1
ξ=constant
P
Q
15. 15Finite Element Method by G. R. Liu and S. S. Quek
Element matricesElement matrices
z
x
z=Z
y
i
j
l
k
1 =
4 =
2 =
3 =
η=0
η=0
η=1
η=constant
P
16. 16Finite Element Method by G. R. Liu and S. S. Quek
Element matricesElement matrices
z
x
z=Z
y
i
j
l
k
1 =
4 =
2 =
3 =
ζ=1
ζ=1
ζ=1
ζ=0
ζ=constant
P
QR
17. 17Finite Element Method by G. R. Liu and S. S. Quek
Element matricesElement matrices
z
x
z=Z
y
i
j
l
k
1 =
4 =
2 =
3 =
ξ=0
η=0
ζ=1
ξ=1
η=0
ζ=1
ξ=1
η=1
ζ=1
ζ=0
ζ=constant
P [xP=η(x3−x2)+x2, yP=η(y3−y2)+y2,0]
O
B
B [xB=ξ(xP −x1)+x1, yB=ξ[η(yP−y1)+y1],0]
O [x=(1−ζ)(x4−xB)+xB, y=(1−ζ)(y4−yB)+yB, z=(1−ζ)z4]
ξ=constant
ζ=constant
0
)(
)(
223
223
=
+−=
+−=
P
P
P
z
yyyy
xxxx
η
η
0
)()()(
)()()(
1122311
1122311
=
+−+−=+−=
+−+−=+−=
B
PB
PB
z
yyyyyyyyy
xxxxxxxxx
ξξηξ
ξξηξ
4
321214444
321214444
)1(
)()()()(
)()()()(
zz
yyyyyyyyyyy
xxxxxxxxxxx
B
B
ζ
ξζξζζζ
ξζξζζζ
−=
−−−+−−=−−=
−−−+−−=−−=
)1(
)1(
)1(
4
3
2
1
ζ
ηξζ
ξηζ
ζξ
−=
−=
=
−=
N
N
N
N
18. 18Finite Element Method by G. R. Liu and S. S. Quek
Element matricesElement matrices
Jacobian:
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
=
ζ
ζ
ζ
η
η
η
ξ
ξ
ξ
z
y
x
z
y
x
z
y
x
J
2
4
312141313121
312141313121
6
00
]det[ ξζξηξξζηζζ
ξηξξζηζζ
V
z
yyyyyy
xxxxxx
−=++−+
++−+
=J
1 1 1
0 0 0
d det d d d
e
T T
e
V
Vρ ρ ξ η ζ= =∫ ∫ ∫ ∫m N N N N [J]
11 12 13 14
1 1 1 21 22 23 242
0 0 0
31 32 33 34
41 42 43 44
6 d d de eV ρ ξζ ξ η ζ
= −
∫ ∫ ∫
N N N N
N N N N
m
N N N N
N N N N
19. 19Finite Element Method by G. R. Liu and S. S. Quek
Element matricesElement matrices
l
f
f
f
l
sz
sy
sx
e d][
43
T
∫
=
−
Nf
z=Z
x=X
z=Z
y=Y
w4
v4
u4
w2
u2
u2
w1
u1
v1
w3
u3
v3
i
j
l
k1 =
4 =
2 =
3 =
fsy
fsz
fsx
For uniformly distributed load:
{ }
{ }
{ }
{ }
{ }
{ }
=
×
×
×
×
×
×
−
13
13
13
13
13
13
43
2
1
0
0
0
0
0
0
f
sz
sy
sx
sz
sy
sx
e
f
f
f
f
f
f
l
20. 20Finite Element Method by G. R. Liu and S. S. Quek
HEXAHEDRON ELEMENTHEXAHEDRON ELEMENT
3D solid meshed with hexahedron elements
P P’
P’’P’’’
21. 21Finite Element Method by G. R. Liu and S. S. Quek
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
e
e
e
e
e
e
=
d
d
d
d
d
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 L
1
7
5 8
6 4
2
0
z
y
x
3
0
fsz
fsy
fsx
[ ]87654321 NNNNNNNNN =
)8,,2,1(
00
00
00
L=
= i
N
N
N
i
i
i
iN
22. 22Finite Element Method by G. R. Liu and S. S. Quek
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)
ξ
η
ζ
ii
i
ii
i
ii
i
zNz
yNy
xNx
),,(
),,(
),,(
8
1
8
1
8
1
ζηξ
ζηξ
ζηξ
∑
∑
∑
=
=
=
=
=
=
)1)(1)(1(
8
1
iiiiN ζζηηξξ +++=
(Tri-linear functions)
23. 23Finite Element Method by G. R. Liu and S. S. Quek
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 ===ε
24. 24Finite Element Method by G. R. Liu and S. S. Quek
Strain matrixStrain matrix
ζζζζ
ηηηη
ξξξξ
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
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
N
y
N
x
N
N
N
N
i
i
i
i
i
i
J
ζ
η
ξ
⇒ where
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
=
ζ
η
ξ
ζ
η
ξ
ζ
η
ξ
z
z
z
y
y
y
x
x
x
J
25. 25Finite Element Method by G. R. Liu and S. S. Quek
Strain matrixStrain matrix
8 8 8
1 1 1
( , , ) , ( , , ) , ( , , )i i i i i i
i i i
x N x y N y z N zξ η ζ ξ η ζ ξ η ζ
= = =
= = =∑ ∑ ∑Since,
or
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
=
∑
∑
∑
∑
∑
∑
∑
∑
∑
=
=
=
=
=
=
=
=
=
ζ
η
ξ
ζ
η
ξ
ζ
η
ξ
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
N
z
N
z
N
z
N
y
N
y
N
y
N
x
N
x
N
x
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 N
x y z
x y zN N N N NN N N
x y z
x y zN N N N N N N N
x y z
x y z
ξ ξ ξ ξ ξξ ξ ξ
η η η η η η η η
ζ ζ ζζ ζ ζ ζ ζ
∂ ∂ ∂ ∂ ∂∂ ∂ ∂
∂ ∂ ∂ ∂ ∂∂ ∂ ∂
∂ ∂ ∂ ∂ ∂∂ ∂ ∂ = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂∂ ∂ ∂ ∂ ∂
J
26. 26Finite Element Method by G. R. Liu and S. S. Quek
Strain matrixStrain matrix
∂
∂
∂
∂
∂
∂
=
∂
∂
∂
∂
∂
∂
−
ζ
η
ξ
i
i
i
i
i
i
N
N
N
z
N
y
N
x
N
1
J
∂∂∂∂
∂∂∂∂
∂∂∂∂
∂∂
∂∂
∂∂
==
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. ξ, η, ζ
27. 27Finite Element Method by G. R. Liu and S. S. Quek
Element matricesElement matrices
1 1 1
T T
1 1 1
d det[ ]d d d
e
e
V
A ξ η ζ
+ + +
− − −
= =∫ ∫ ∫ ∫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 ζηξηξηξ ∑∑∑∫ ∫ ∫ = = =
+
−
+
−
+
−
==
1 1 1
1 1 1
d det d d d
e
T T
e
V
Vρ ρ ξ η ζ
− − −
= =∫ ∫ ∫ ∫m N N N N [J]
28. 28Finite Element Method by G. R. Liu and S. S. Quek
Element matricesElement matrices
For rectangular hexahedron:
det eabc V= =[J]
=
88
7877
686766
58575655
4847464544
282726252433
28272625242322
1817161514131211
.
m
mm
mmm
mmmm
mmmmm
mmmmmm
mmmmmmm
mmmmmmmm
m
sy
e
29. 29Finite Element Method by G. R. Liu and S. S. Quek
Element matricesElement matrices
(Cont’d)
where
ζηξρ
ζηξρ
ζηξρ
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
30. 30Finite Element Method by G. R. Liu and S. S. Quek
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
3
1
3
1
3
1
1
1
1
1
1
1
1
1
1
1
jijiji
jijiji
jiij
hab
abc
NNabcm
ζζηηξξ
ρ
ζζζζζηηηηηξξξξξ
ρ
ζηξρ
+++=
++++++=
=
∫∫∫
∫ ∫
+
−
+
−
+
−
+
−
+
−
31. 31Finite Element Method by G. R. Liu and S. S. Quek
Element matricesElement matrices
(Cont’d)
E.g.
216
8)111)(111)(111(
8 3
1
3
1
3
1
33
abcabc
m
ρρ
×=××+××+××+=
216
1
216
2
216
4
216
8
46352817
184538276857473625162413
483726155814786756342312
8877665544332211
abc
mmmm
abc
mmmmmmmmmmmm
abc
mmmmmmmmmmmm
abc
mmmmmmmm
ρ
ρ
ρ
ρ
====
============
============
========
32. 32Finite Element Method by G. R. Liu and S. S. Quek
Element matricesElement matrices
(Cont’d)
=
8
48.
248
4248
42128
242148
1242248
21244248
216
sy
abc
ex
ρ
m
Note: For x direction only
(Rectangular hexahedron)
33. 33Finite Element Method by G. R. Liu and S. S. Quek
Element matricesElement matrices
l
f
f
f
l
sz
sy
sx
e d][
43
T
∫
=
−
Nf
1
7
5 8
6 4
2
0
z
y
x
3
0
fsz
fsy
fsx
{ }
{ }
{ }
{ }
{ }
{ }
=
×
×
×
×
×
×
−
13
13
13
13
13
13
43
2
1
0
0
0
0
0
0
f
sz
sy
sx
sz
sy
sx
e
f
f
f
f
f
f
l
For
uniformly
distributed
load:
34. 34Finite Element Method by G. R. Liu and S. S. Quek
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
8Hexahedron
made up of 5
tetrahedrons:
35. 35Finite Element Method by G. R. Liu and S. S. Quek
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
46
5 8
6 4
Break into three
Hexahedron
made up of six
tetrahedrons:
Element matrices can
be obtained by
assembly of
tetrahedron elements
36. 36Finite Element Method by G. R. Liu and S. S. Quek
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:
37. 37Finite Element Method by G. R. Liu and S. S. Quek
HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS
Tetrahedron elements (Cont’d)
20 nodes, cubic:
1
2
9 9
5 1 1 3 11 1 1 42 2
9 9
6 3 1 3 12 4 1 42 2
9 9
7 1 1 2 13 22 2
9
8 2 1 22
9
9 2 2 32
9
10 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
9
14 4 2 42
9
15 3 3 42
9
16 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
19
5
20
38. 38Finite Element Method by G. R. Liu and S. S. Quek
HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS
Brick elements
Lagrange type:
i(I,J,K)
(0,0,0)
η
ξ
ζ
(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 p
i I J K I J KN N N N l l lξ η ς= =
0 1 1 1
0 1 1 1
( )( ) ( )( ) ( )
( )
( )( ) ( )( ) ( )
n k k n
k
k k k k k k k n
l
ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ
ξ
ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ
− +
− +
− − − − −
=
− − − − −
L L
L L
where
39. 39Finite Element Method by G. R. Liu and S. S. Quek
HIGHER ORDERHIGHER 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)
ξ
η
ζ
9(1,0,-1)
10(0,1,-1)
11(-1,0,-1)
12(0-1,-1)
13
14
3
15
16
17 18
19
20
1
8
21
4
21
4
(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
ξ ξ η η ς ς ξ ξ η η ς ς
ξ η η ς ς
η ξ
= + + + + + −
=
= − + + =
= − +
L
21
4
)(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
ξ ς ς
ς ξ ξ η η
+ =
= − + + =
20 nodes, tri-quadratic:
40. 40Finite Element Method by G. R. Liu and S. S. Quek
HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS
Brick elements (Cont’d)
2 2 21
64
29
64
1
3
29
64
(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
ξ ξ η η ς ς ξ η ς
ξ ξ ξ η η ς ς
ξ η ς
η η
= + + + + + −
=
= − + + +
= ± = ± = ±
= − +
L
1
3
29
64
1
3
)(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
η ξ ξ ς ς
η ξ ς
ς ς ς ξ ξ η η
ς ξ η
+ +
= ± = ± = ±
= − + + +
= ± = ± = ±
32 nodes, tri-cubic:
ξ
η
ζ
41. 41Finite Element Method by G. R. Liu and S. S. Quek
ELEMENTS WITH CURVEDELEMENTS WITH CURVED
SURFACESSURFACES
1
4
98
7
10
2
5
6 3
7
18
16
12
15
14 11
13
5
17
19
20
6
109
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
ξ
η
ζ
42. 42Finite Element Method by G. R. Liu and S. S. Quek
CASE STUDYCASE STUDY
Stress and strain analysis of a quantum dot
heterostructure
Material E (Gpa) υ
GaAs 86.96 0.31
InAs 51.42 0.35
GaAs substrate
GaAs cap layer
InAs wetting
layer
InAs quantum dot
