Why the Mathematical Analysis is Important
In this chapter we illustrate by using simple model examples which types of problems are solved in sizing and shape optimization (SSO). Further, we present some difficulties one may meet in their practical realization. Finally we try to convince the reader of the helpfulness of a thorough mathematic...
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| Published in | Introduction to Shape Optimization pp. 2 - 11 |
|---|---|
| Main Authors | , |
| Format | Book Chapter |
| Language | English |
| Published |
Society for Industrial and Applied Mathematics (SIAM)
2003
Society for Industrial and Applied Mathematics |
| Series | Advances in Design and Control |
| Subjects | |
| Online Access | Get full text |
| ISBN | 0898715369 9780898715361 |
| DOI | 10.1137/1.9780898718690.ch1 |
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| Summary: | In this chapter we illustrate by using simple model examples which types of problems are solved in sizing and shape optimization (SSO). Further, we present some difficulties one may meet in their practical realization. Finally we try to convince the reader of the helpfulness of a thorough mathematical analysis of the problems to be solved.
We start this chapter with a simple sizing optimization problem whose exact solution can be found by hand. Let us consider a simply supported beam of variable thickness e represented by the interval I = [0, 1]. The beam is under a uniform vertical load f
0
. One wants to find a thickness distribution to maximize the stiffness of the beam. The deflection u≔ u (e) of the beam solves the following fourth order boundary value problem:
(
β
e
3
u
″
(
x
)
)
″
=
ƒ
0
in
[
0
,
1
]
,
u
(
0
)
=
u
(
1
)
=
(
β
e
3
u
″
)
(
0
)
=
(
β
e
3
u
″
)
(
1
)
=
0
,
(
P
′
(
e
)
)
where β is a given positive constant. The stiffness of the beam is characterized by the compliance functional J defined by
J
(
u
(
e
)
)
=
∫
0
1
ƒ
0
u
(
e
)
d
x
,
where u (e) solves
(
P
′
(
e
)
)
.
The stiffer the construction is, the lower the value J attains. Therefore the stiffness maximization is equivalent to the compliance minimization. We formulate the following sizing optimization problem:
Find
e
*
∈
U
a
d
such that
J
(
u
(
e
*
)
)
≤
J
(
u
(
e
)
)
∀
e
∈
U
a
d
,
(
ℙ
ℙ
1
)
where Uad is the set of admissible thicknesses defined as follows:
U
a
d
=
{
e
:
[
0
,
1
]
→
ℝ
+
|
∫
0
1
e
(
x
)
d
x
=
γ
}
,
γ
>
0
given
.
The integral constraint appearing in the definition of Uad says that the volume of the beam is preserved. Next we show how to find a solution to (ℙ
1
). |
|---|---|
| ISBN: | 0898715369 9780898715361 |
| DOI: | 10.1137/1.9780898718690.ch1 |