Transcript Tentative Outline

Outline
• Introduction to Finite Element Formulations
• Direct Stiffness Method for Linear Elasticity
Problems
• Element Shape Functions, Tetrahedron T4
Implementation
• Generalized Finite Element Formulations (Weighted
Residuals Methods and Variational Methods)
• Tetrahedron T10 Implementation
• “Hierarchical” Shape Functions, Tetrahedron TP2
Implementation
Mathematics
Engineering
Structural
analog
substitution
Direct
continuum
elements
Trial functions for
differential equations
Variational
methods
Present-Day
Finite Element Method
Weighted
residuals
u: unknown displ.
in element e
N: prescribed element
shape functions
ae: unknown nodal
displ. for element e
e
 ai 
 
u  uˆ  [N i ,N j ,...]a j   Na e
 .
 
For a particular location (x, y) in element e
y
i
e
uˆ ( x, y )
u  uˆ  

v
(
x
,
y
)
ˆ


 ( a x )i 


uˆ ( x, y )  [ N i ( x, y ) N j ( x, y ) N k ( x, y )](a x ) j 
(a ) N i
 x k
k
x
y
1.0
i
N i ( xi , yi )  1.0
??
Ni ( x j , y j )  Ni ( xk , yk )  0.0
??
j
x
u  Na e
  u
  Bae
where
B  N
: strains
 : suitable differential operator
In 2D linear elasticity:
 x  
u / x
0 
  /  x
u 
  
 

   y   
v / y
 / y  
 0
 v 
   u /  y   v /  x   /  y  /  x 
 

 xy  

  D
: stresses
: strains
D: constitutive matrix containing
proper material properties
In 2D linear elasticity (plane stress):
 x 
0   x 
1 v
E 
 
 

v 1
0
   y  
y 
2

  1  v 0 0 (1  v) / 2  

  xy 
 xy 
D
Boundary Conditions
q ie 
e  e
q  q j 
q e 
 k
qe: equivalent nodal forces
(qy)i
i
y
i
e
k
e
j
x
(qx)i
k
(qy)j
j
(qx)j
e

a
Impose an arbitrary (virtual) nodal displacement
Internal Virtual Work = External Virtual Work
T

  dV
(ae )T (qe )
Ve
(a e )T (  BT DB dV )(a e )
Ve
(ae )T (  (N)T D(N) dV )(ae )
Ve
K ea e  q e
For the whole domain
n
n
e1
e1
( K e )a  ( q e )
Linear algebra problem
Tetrahedron T4 Implementation
(Linear Shape Function)
t
2
• N0 = 1 when r=1
• N0 = 0 when r=0
3
1
0
r
N0 = r
s
Tetrahedron T4 Implementation
*T4Shape::shape_functions
N0 = r
N1 = s
N2 = t
N3 = 1.0-r-s-t = u
shape->at(0) = r
shape->at(1) = s
shape->at(2) = t
shape->at(3) = u
*T4Shape::shape_derivatives
dN0/dr= 1.0
dN1/dr= 0.0
dN2/dr= 0.0
dN3/dr= -1.0
etc.
deriv->at(0,0) = 1.0
deriv->at(0,1) = 0.0
deriv->at(0,2) = 0.0
deriv->at(0,3) = -1.0
etc.
Direct Stiffness Method
Generalization of FEM Concept
Differential equations
u  uˆ  Na
  u  Na
  D


T
  (N) D(N) dV (a)  q


V

Ka  q
 A(u)  0 on 

B(u)  0 on 


Approximate u  uˆ  Na
Cast in an integral form (weak form) by:
• Weighted Residual Methods
• Variational Methods
Leads to Ka=q if A and B are linear
differential operators
Differential equations
 A(u)  0 on 

B(u)  0 on 
u  uˆ
Residuals
Weighted Residual Methods
Elasticity Problems


 A(uˆ )  R A on 

B(uˆ )  R B on 
Find weighting functions w so:
  w T R A d  0


T
w
R B d  0


If w=N, we have Galerkin Method

t
 ij,j  0 on 

ij n j  t i on 

Galerkin Method
T
N
 ()dV  0
V
Integration by parts
  (N)T ()dV   (N)T (n)dS  0
V
S


 (N)T (D)(N)dV a  (N)T (t )dS


V

S


Ka  q
Variational Methods
Differential equations
 A(u)  0 on 

B(u)  0 on 
Elasticity Problems

S

Define a functional 
   F(u)d   G(u) d

t

Solution is a function u that makes
 stationary w.r.t. small
changes  u
  0
(Principal of Min. Potential Energy)
V
   (T )dV   (uT t )dS
V
S
  0
T
T
(

)

dV

(

u)
tdS


V
S


 (N)T (D)(N)dV a  (N)T (t )dS


V

S


Ka  q
Tetrahedron T10 Implementation
(Quadratic Shape Function)
t
End node 0
• N0 = 1 when r=1
• N0 = 0 at other nodal locations
2
7
3
6
9
0
r
5
8
4
N0 = r(2r-1)
1
s
Midside node 4
• N4 = 1 when r=1/2 and s=1/2
• N4 = 0 at other nodal locations
N4 = 4rs
Tetrahedron TP2 Implementation
(Hierarchical Quadratic Shape Function)
t
End node 0
• N0 = 1 when r=1
• N0 = 0 at other nodal locations
2
7
3
6
9
0
r
4
5
8
N0 = r
1
s
Edge 4
• Any quadratic polynomial of r and s
that yields 0 at node 0 (r=1, s=0)
and node 1 (s=1, r=0).
N4 = 4rs
Hierarchical Shape Function (Polynomial Order = 2)
N0 = r
N1 = s
N2 = t
P=1
N3 = u
N4 = 4rs
N5 = 4st
N6 = 4rt
N7 = 4tu
N8 = 4su
N9 = 4ru
P=2
What to study next?
Finite Element Books
• Zienkiewicz, O.C. and Taylor, R.L.: The Finite Element Method,
Volume 1 1989 and Volume 2, 1991
• Bathe, K.J.: Finite Element Procedures, 1996
• Hughes, T.J.R.: The Finite Element Method (Linear Static and
Dynamic Finite Element Analysis), 1987
• Cook, R.D. et al.: Concepts and Applications of Finite Element
Analysis, 1989
• Szabo and Babuska: Finite Element Analysis, 1991
Finite Element Programs
•
•
•
•
Software developed at CFG
Commercial codes: ABAQUS, ANSYS, NASTRAN, DIANA
Proprietary codes: WARP3D, STAGS
Internet resource: www.engr.usask.ca/~macphed/finite/fe_resources