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.
Wyszukiwarka
Podobne podstrony:
Chapt 02 Lect02
Chapt 04
Chapt 06 Lect02
Chapt 01 Lect02
Chapt 07 Lect02
Chapt 04 Lect01
Chapt 04 Lect03
Chapt 05 Lect02
Chapt 03 Lect02
Chapt 02 Lect02
Wykład 04
04 22 PAROTITE EPIDEMICA
04 Zabezpieczenia silnikówid 5252 ppt
Wyklad 04
Wyklad 04 2014 2015
04 WdK
04) Kod genetyczny i białka (wykład 4)
więcej podobnych podstron