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• 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
e1
e1
( 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