9.4 Applying Topological Cell Optimization
347
gradually decreased and in a limit of this process the initial form (input) problem of
the introduced constraints form (9.13) is obtained [90]. The so far described approach
has been followed by other researchers [93–98]. Le et al. [99] proposed qualitatively
different approach to remove the problem of singularity by introduction of a simple
penalty based on the stresses estimation with the help of the interpolating function.
Let the function interpolating stresses be η σ (ρ), then the constrained of stresses can
be presented by the following weak inequality
g(ρ) = η σ (ρ)σ (ρ) ≤ σ r .
(9.15)
It has been shown that using interpolation η = ρ
p and an appropriate choice of
the penalty parameter for interpolation of stresses, stiffness and volume, the singular
subspaces of design space can be achieved in an effective way. Le et al. [99] proposed
ρ = 1/2, 3 for stresses, stiffness and volume, respectively.
A high-order nonlinear character of stress constraints is implied by a high sensitivity of the stresses in a finite element generated by a neighbourhood changes
in topology. The authors [99] recommended the use of Heaviside filters to remove
the occurred problem.
The local character of stress constraints plays a crucial role in the majority of investigations within the topological optimization. In engineering problems of design, the
occurred stresses should be bounded in each point of a construction. It can be easily
achieved by optimizing the sizes and forms of a construction, using as engineering
intuition in order to satisfy the required constraints in critical places (critical elements) on the cases of the structure geometry and employed loads. However, the
process of a topological optimization does not allow to predict the elements with
critical stresses, since their geometric configuration is a priori not known. Therefore,
the possible solutions should satisfy the introduced stress constraints in all finite
elements. On the other hand, a decrease of computational time requires a decrease
of the constraints’ number, and therefore it seems to be an attractive introduction of
one constraint of the maximum stresses as follows
g(ρ) = max
e=1..n
(σ e (ρ)) ≤ ¯
σ .
(9.16)
Unfortunately, the maximum operator (9.16) is not differentiable, which does not
allow to describe formulas for sensitivity in an analytical way. It motivated many
researchers to introduce continuous aggregating functions matching local stresses
into one formula [72, 95]. The latter methods are named global ones and they usually
use variation of the Kresselmeier-Steinhauser (KS) functions in the following form:
σ K S =
1
p
ln
N es
e=1
exp
F(σ e )
¯
σ
(9.17)
or its equivalent approximated p norm
347
gradually decreased and in a limit of this process the initial form (input) problem of
the introduced constraints form (9.13) is obtained [90]. The so far described approach
has been followed by other researchers [93–98]. Le et al. [99] proposed qualitatively
different approach to remove the problem of singularity by introduction of a simple
penalty based on the stresses estimation with the help of the interpolating function.
Let the function interpolating stresses be η σ (ρ), then the constrained of stresses can
be presented by the following weak inequality
g(ρ) = η σ (ρ)σ (ρ) ≤ σ r .
(9.15)
It has been shown that using interpolation η = ρ
p and an appropriate choice of
the penalty parameter for interpolation of stresses, stiffness and volume, the singular
subspaces of design space can be achieved in an effective way. Le et al. [99] proposed
ρ = 1/2, 3 for stresses, stiffness and volume, respectively.
A high-order nonlinear character of stress constraints is implied by a high sensitivity of the stresses in a finite element generated by a neighbourhood changes
in topology. The authors [99] recommended the use of Heaviside filters to remove
the occurred problem.
The local character of stress constraints plays a crucial role in the majority of investigations within the topological optimization. In engineering problems of design, the
occurred stresses should be bounded in each point of a construction. It can be easily
achieved by optimizing the sizes and forms of a construction, using as engineering
intuition in order to satisfy the required constraints in critical places (critical elements) on the cases of the structure geometry and employed loads. However, the
process of a topological optimization does not allow to predict the elements with
critical stresses, since their geometric configuration is a priori not known. Therefore,
the possible solutions should satisfy the introduced stress constraints in all finite
elements. On the other hand, a decrease of computational time requires a decrease
of the constraints’ number, and therefore it seems to be an attractive introduction of
one constraint of the maximum stresses as follows
g(ρ) = max
e=1..n
(σ e (ρ)) ≤ ¯
σ .
(9.16)
Unfortunately, the maximum operator (9.16) is not differentiable, which does not
allow to describe formulas for sensitivity in an analytical way. It motivated many
researchers to introduce continuous aggregating functions matching local stresses
into one formula [72, 95]. The latter methods are named global ones and they usually
use variation of the Kresselmeier-Steinhauser (KS) functions in the following form:
σ K S =
1
p
ln
N es
e=1
exp
F(σ e )
¯
σ
(9.17)
or its equivalent approximated p norm