354
9 Topologic Optimization of Vibrations of Size-Dependent Beams
which defined sensitivity for the whole vector of reactive load. The computational
requirements can be decreased if the sensitivity is computed only for a separate
reaction
d R i
d ρ j
=
d K i
d ρ j
U + V
T
i
d F
th
d ρ j
−
d K
d ρ j
U
−
d F i
th
d ρ j
,
(9.43)
where the complex vector V i is yielded as a solution to the conjugated system having
the same form as Eq. (9.37):
KV i = K i .
(9.44)
Here K i is a column of the global stiffness matrix, and F i
th stands for the global
component of the heat load corresponding to the ith degree of freedom.
9.5.2 Formulating Problems of Topological Optimization
We consider three methods of topological optimization. They can be employed to
problems with other geometry, load conditions and target functions.
9.5.2.1 Achieving Minimum Flexibility Under Thermal Load
The first formulation is based on getting minimum flexibility (maximum of stiffness)
with constraints put on the volume. As it has been already mentioned, application of
the target function in the form of flexibility for topological optimization with heat
loads are questionable. However, in what follows we consider this problem and we
show why it cannot yield reliable results. The mathematical statement of the problem
for the case of topological optimization aimed on achieving flexibility minimum is
governed by the following conditions:
min c (ρ) = F
T
(ρ) U (ρ) = U
T
(ρ) K (ρ) U (ρ) ,
under condition K (ρ) U (ρ) = F
th
(ρ),
g(ρ) =
N opt
e=1
ρ e v e ≤ ρ 0 V f ,
(9.45)
and the variables 0 < x min ≤ x e ≤ 1 for e = 1, 2, . . . , N opt .
Here the following notation is used: c—flexibility, ρ e —physical density of element e (coupled with projection variable x e via filter), v e —volume of element e,
V f —allowed volume of fraction, N opt —general number of optimized elements, and
x min —minimum allowed value of the design variables.
9 Topologic Optimization of Vibrations of Size-Dependent Beams
which defined sensitivity for the whole vector of reactive load. The computational
requirements can be decreased if the sensitivity is computed only for a separate
reaction
d R i
d ρ j
=
d K i
d ρ j
U + V
T
i
d F
th
d ρ j
−
d K
d ρ j
U
−
d F i
th
d ρ j
,
(9.43)
where the complex vector V i is yielded as a solution to the conjugated system having
the same form as Eq. (9.37):
KV i = K i .
(9.44)
Here K i is a column of the global stiffness matrix, and F i
th stands for the global
component of the heat load corresponding to the ith degree of freedom.
9.5.2 Formulating Problems of Topological Optimization
We consider three methods of topological optimization. They can be employed to
problems with other geometry, load conditions and target functions.
9.5.2.1 Achieving Minimum Flexibility Under Thermal Load
The first formulation is based on getting minimum flexibility (maximum of stiffness)
with constraints put on the volume. As it has been already mentioned, application of
the target function in the form of flexibility for topological optimization with heat
loads are questionable. However, in what follows we consider this problem and we
show why it cannot yield reliable results. The mathematical statement of the problem
for the case of topological optimization aimed on achieving flexibility minimum is
governed by the following conditions:
min c (ρ) = F
T
(ρ) U (ρ) = U
T
(ρ) K (ρ) U (ρ) ,
under condition K (ρ) U (ρ) = F
th
(ρ),
g(ρ) =
N opt
e=1
ρ e v e ≤ ρ 0 V f ,
(9.45)
and the variables 0 < x min ≤ x e ≤ 1 for e = 1, 2, . . . , N opt .
Here the following notation is used: c—flexibility, ρ e —physical density of element e (coupled with projection variable x e via filter), v e —volume of element e,
V f —allowed volume of fraction, N opt —general number of optimized elements, and
x min —minimum allowed value of the design variables.