346
9 Topologic Optimization of Vibrations of Size-Dependent Beams
9.4.3 Topological Optimization of Stress State
While conducting topological optimization the following question arises: how essentially can we decrease thermal stresses while designing thermoelastic structures?
Dusynix and Bendose [90] belong to the first few people who attack the problem of
reaching minimum stress while carrying out the topological optimization of structures. After that three fundamental problems related to the use of criteria of stresses
estimation in the topological optimization have been raised, i.e. a so-called occurrence of singularity, strongly nonlinear behaviour of stresses with possible changes
of topology and local character of stresses [72]. The earlier carried out investigations
based on stresses estimation have been abandoned and more correct and reliable
criteria based on flexibility have been employed. In parallel to that recent trend,
the transitional structural optimization has been based on achieving a minimum of
material volume in conditions of constraints employed on stresses. The problem of
singularity occurs due to the generation of singular subspaces of the n-dimensional
allowed design space, i.e. some of the n projection variables are removed from the
studied problem [91]. Besides, it has been found that often the global optimal construction lies in some of these singular subspaces. The latter problem has been solved
by Duysinix and Bendose [90] with a help of β—relaxation modified by an analogous
problem dealing with form optimization where a stress element becomes singular
since a transversal cross section of the optimized space tends to be zero [92]. In a
general case, the constraints employed on stresses in topological optimization can
be formulated in the following way:
g(ρ) = σ (ρ) ≤ ¯
σ , ρ > 0,
(9.12)
where σ —elementary stress (for instance, von Mises stress), ¯
σ —allowed stress, and
inequality ρ > 0 points out existence of an element. In order to remove the condition
ρ > 0 from constraints, one may use the following modified formula
g(ρ) = ρ
σ (ρ)
¯
σ
− 1
≤ 0.
(9.13)
Besides, in order to avoid singularity based on employment of ε-relaxation, the
following counterpart constraints can be used
g(ρ) = ρ
σ (ρ)
¯
σ
− 1
≤ ε,
(9.14)
where ε stands for a given parameter. For ε = 0, one deals with the input problem
with stress constraints. However, for arbitrary ε > 0, the problem is characterized
by a non-singular space, i.e. the optimal solutions are distributed in subspaces of
design space with non-zero ρ, and now they can be achieved by using traditional
algorithms of structural optimization. In the process of optimization, the value of ε is
9 Topologic Optimization of Vibrations of Size-Dependent Beams
9.4.3 Topological Optimization of Stress State
While conducting topological optimization the following question arises: how essentially can we decrease thermal stresses while designing thermoelastic structures?
Dusynix and Bendose [90] belong to the first few people who attack the problem of
reaching minimum stress while carrying out the topological optimization of structures. After that three fundamental problems related to the use of criteria of stresses
estimation in the topological optimization have been raised, i.e. a so-called occurrence of singularity, strongly nonlinear behaviour of stresses with possible changes
of topology and local character of stresses [72]. The earlier carried out investigations
based on stresses estimation have been abandoned and more correct and reliable
criteria based on flexibility have been employed. In parallel to that recent trend,
the transitional structural optimization has been based on achieving a minimum of
material volume in conditions of constraints employed on stresses. The problem of
singularity occurs due to the generation of singular subspaces of the n-dimensional
allowed design space, i.e. some of the n projection variables are removed from the
studied problem [91]. Besides, it has been found that often the global optimal construction lies in some of these singular subspaces. The latter problem has been solved
by Duysinix and Bendose [90] with a help of β—relaxation modified by an analogous
problem dealing with form optimization where a stress element becomes singular
since a transversal cross section of the optimized space tends to be zero [92]. In a
general case, the constraints employed on stresses in topological optimization can
be formulated in the following way:
g(ρ) = σ (ρ) ≤ ¯
σ , ρ > 0,
(9.12)
where σ —elementary stress (for instance, von Mises stress), ¯
σ —allowed stress, and
inequality ρ > 0 points out existence of an element. In order to remove the condition
ρ > 0 from constraints, one may use the following modified formula
g(ρ) = ρ
σ (ρ)
¯
σ
− 1
≤ 0.
(9.13)
Besides, in order to avoid singularity based on employment of ε-relaxation, the
following counterpart constraints can be used
g(ρ) = ρ
σ (ρ)
¯
σ
− 1
≤ ε,
(9.14)
where ε stands for a given parameter. For ε = 0, one deals with the input problem
with stress constraints. However, for arbitrary ε > 0, the problem is characterized
by a non-singular space, i.e. the optimal solutions are distributed in subspaces of
design space with non-zero ρ, and now they can be achieved by using traditional
algorithms of structural optimization. In the process of optimization, the value of ε is