Lecture Notes: Introduction to Finite Element Method
Chapter 4. FE Modeling and Solution Techniques
© 1998 Yijun Liu, University of Cincinnati
107
II. Substructures (Superelements)
Substructuring is a process of analyzing a large structure as
a collection of (natural) components. The FE models for these
components are called substructures or superelements (SE).
Physical Meaning:
A finite element model of a portion of structure.
Mathematical Meaning:
Boundary matrices which are load and stiffness matrices
reduced (condensed) from the interior points to the exterior or
boundary points.
Lecture Notes: Introduction to Finite Element Method
Chapter 4. FE Modeling and Solution Techniques
© 1998 Yijun Liu, University of Cincinnati
108
Advantages of Using Substructures/Superelements:
•
Large problems (which will otherwise exceed your
computer capabilities)
•
Less CPU time per run once the superelements have
been processed (i.e., matrices have been saved)
•
Components may be modeled by different groups
•
Partial redesign requires only partial reanalysis (reduced
cost)
•
Efficient for problems with local nonlinearities (such as
confined plastic deformations) which can be placed in
one superelement (residual structure)
•
Exact for static stress analysis
Disadvantages:
•
Increased overhead for file management
•
Matrix condensation for dynamic problems introduce
new approximations
•
...
Lecture Notes: Introduction to Finite Element Method
Chapter 4. FE Modeling and Solution Techniques
© 1998 Yijun Liu, University of Cincinnati
109
III.
Equation Solving
Direct Methods (Gauss Elimination):
•
Solution time proportional to NB
2
(N is the dimension of
the matrix, B the bandwidth)
•
Suitable for small to medium problems, or slender
structures (small bandwidth)
•
Easy to handle multiple load cases
Iterative Methods:
•
Solution time is unknown beforehand
•
Reduced storage requirement
•
Suitable for large problems, or bulky structures (large
bandwidth, converge faster)
•
Need solving again for different load cases
Lecture Notes: Introduction to Finite Element Method
Chapter 4. FE Modeling and Solution Techniques
© 1998 Yijun Liu, University of Cincinnati
110
Gauss Elimination - Example:
−
=
−
−
−
−
3
1
2
3
3
0
3
4
2
0
2
8
3
2
1
x
x
x
or
b
Ax
=
.
Forward Elimination:
Form
−
−
−
−
−
3
1
2
3
3
0
3
4
2
0
2
8
)
3
(
)
2
(
)
1
(
;
(1) + 4 x (2)
⇒
(2):
−
−
−
−
3
2
2
3
3
0
12
14
0
0
2
8
)
3
(
)
2
(
)
1
(
;
(2) +
3
14
(3)
⇒
(3):
−
−
−
12
2
2
2
0
0
12
14
0
0
2
8
)
3
(
)
2
(
)
1
(
;
Back Substitution:
5
.
1
8
/
)
2
2
(
5
14
/
)
12
2
(
6
2
/
12
2
1
3
2
3
=
+
=
=
+
−
=
=
=
x
x
x
x
x
or
=
6
5
5
1.
x
.
Lecture Notes: Introduction to Finite Element Method
Chapter 4. FE Modeling and Solution Techniques
© 1998 Yijun Liu, University of Cincinnati
111
Iterative Method - Example:
The Gauss-Seidel Method
b
Ax
=
(A is symmetric)
or
.
...,
,
2
,
1
,
1
N
i
b
x
a
N
j
i
j
ij
=
=
∑
=
Start with an estimate
)
( 0
x
and then iterate using the following:
.
...,
,
2
,
1
for
,
1
1
1
1
)
(
)
1
(
)
1
(
N
i
x
a
x
a
b
a
x
i
j
N
i
j
k
j
ij
k
j
ij
i
ii
k
i
=
−
−
=
∑
∑
−
=
+
=
+
+
In vector form,
[
]
,
)
(
)
1
(
1
)
1
(
k
T
L
k
L
D
k
x
A
x
A
b
A
x
−
−
=
+
−
+
where
〉
〈
=
ii
D
a
A
is the diagonal matrix of A,
L
A
is the lower triangular matrix of A,
such that
.
T
L
L
D
A
A
A
A
+
+
=
Iterations continue until solution x converges, i.e.
,
)
(
)
(
)
1
(
ε
≤
−
+
k
k
k
x
x
x
where
ε
is the tolerance for convergence control.