Title: Finite Element Method
1Finite Element Method
for readers of all backgrounds
G. R. Liu and S. S. Quek
CHAPTER 9
2CONTENTS
- 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
3INTRODUCTION
- 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.
4TETRAHEDRON ELEMENT
- 3D solid meshed with tetrahedron elements
5TETRAHEDRON ELEMENT
Consider a four node tetrahedron element
6Shape functions
where
Use volume coordinates (Recall Area coordinates
for 2D triangular element)
7Shape functions
Similarly,
Can also be viewed as ratio of distances
(Partition of unity)
since
8Shape functions
(Delta function property)
9Shape functions
(Adjoint matrix)
i 1,2
Therefore,
i
l 4,1
j
l
j 2,3
k
(Cofactors)
k 3,4
where
10Shape functions
(Volume of tetrahedron)
Therefore,
11Strain matrix
Since,
Therefore,
where
(Constant strain element)
12Element matrices
where
13Element matrices
Eisenberg and Malvern 1973
14Element matrices
Alternative method for evaluating me special
natural coordinate system
15Element matrices
16Element matrices
17Element matrices
18Element matrices
Jacobian
19Element matrices
For uniformly distributed load
20HEXAHEDRON ELEMENT
- 3D solid meshed with hexahedron elements
21Shape functions
22Shape functions
(Tri-linear functions)
23Strain matrix
whereby
Note Shape functions are expressed in natural
coordinates chain rule of differentiation
24Strain matrix
Chain rule of differentiation
?
where
25Strain matrix
Since,
or
26Strain matrix
Used to replace derivatives w.r.t. x, y, z with
derivatives w.r.t. ?, ?, ?
27Element matrices
Gauss integration
28Element matrices
For rectangular hexahedron
29Element matrices
(Contd)
where
30Element matrices
(Contd)
or
where
31Element matrices
(Contd)
E.g.
32Element matrices
(Contd)
Note For x direction only
(Rectangular hexahedron)
33Element matrices
For uniformly distributed load
34Using tetrahedrons to form hexahedrons
- Hexahedrons can be made up of several tetrahedrons
Hexahedron made up of 5 tetrahedrons
35Using tetrahedrons to form hexahedrons
- Element matrices can be obtained by assembly of
tetrahedron elements
Hexahedron made up of six tetrahedrons
36HIGHER ORDER ELEMENTS
10 nodes, quadratic
37HIGHER ORDER ELEMENTS
- Tetrahedron elements (Contd)
20 nodes, cubic
38HIGHER ORDER ELEMENTS
(nd(n1)(m1)(p1) nodes)
Lagrange type
where
39HIGHER ORDER ELEMENTS
Serendipity type elements
20 nodes, tri-quadratic
40HIGHER ORDER ELEMENTS
32 nodes, tri-cubic
41ELEMENTS WITH CURVED SURFACES
42CASE STUDY
- Stress and strain analysis of a quantum dot
heterostructure
GaAs cap layer
InAs wetting layer
InAs quantum dot
GaAs substrate
43CASE STUDY
44CASE STUDY
45CASE STUDY
46CASE STUDY