9.4 Applying Topological Cell Optimization
349
be satisfied. The target function and the constraints are estimated with the help of
the thermoelastic finite element analysis. Information about object sensitivity and
constraints are estimated based on the algorithm of numerical optimization. The
latter process is repeated iterationally unless a convergence of the target function
is achieved. An important role in the mentioned process is associated with design
dependence on the heat loading. In contrary to the topological optimization regarding mechanical load which does not depend on the change of constraints topology,
the design heat dependence should be exhibited by sensitivity analysis. The final
mathematical modelling of the topological optimization takes the following form:
min f (ρ, U (ρ)),
(9.19)
under condition K (ρ) U = F (ρ), g i (ρ, U (ρ)) ≤ 0 , for the variables 0 ≤ x e ≤ 1 ,
for e = 1, 2, . . . , N .
The so far formulated problem is analogous of the general approach of the topological optimization (9.1) presented in the literature overview. However, the parameterization (9.19) is defined in the terms of physical variables associated with density
ρ e . The problem of optimization is still solved with regard to the projection variables x e . Recall that the physical densities are functions of the design variables as
ρ e (x) depending on a filter or projection, which undergo changes. The proposed
algorithm includes the elementary analysis and the complex analysis of sensitivity
in addition to the algorithms required for topological optimization (regularization,
filtration, optimizators, etc.).
Since a chosen method regarding density for the purpose of topological optimization is composed of a system of finite elements, it should be parametrized for
distribution of the design variables based on density which is taken into account in
the being designed finite elements. Here we employ the parametrization based on the
general thermoelastic space for design. The latter one denoted by is shown for the
2D case in Fig. 9.7. It consists of the boundary conditions, external loading located
on the surface parts and a given temperature distribution which can be homogenous
or area dependent. The space includes holes and the optimized space . Here we
employ parametrization based on the general thermoelastic design space. The design
space is discretized by N finite elements with N opt design and N nopt given elements.
Each of the finite elements is associated with a design variable 0 < x e ≤ 1 , for
e = 1, 2, . . . , N opt . All of the variables define a vector x. Design variables x are
coupled with physical densities ρ by the choice of a standard filter as it has been
described in Sect. 9.2.2.
System of equations for static space equilibrium is approximated by a choice of
finite elements with an account of both mechanical and temperature loads, and it has
the following form:
K (ρ) U (ρ) = F (ρ) ,
(9.20)
where K (ρ)—global matrix of stiffness, U (ρ)—vector of node displacements and
F (ρ) stands for a vector of nodal load. In a general case F (ρ) consists of a sum
349
be satisfied. The target function and the constraints are estimated with the help of
the thermoelastic finite element analysis. Information about object sensitivity and
constraints are estimated based on the algorithm of numerical optimization. The
latter process is repeated iterationally unless a convergence of the target function
is achieved. An important role in the mentioned process is associated with design
dependence on the heat loading. In contrary to the topological optimization regarding mechanical load which does not depend on the change of constraints topology,
the design heat dependence should be exhibited by sensitivity analysis. The final
mathematical modelling of the topological optimization takes the following form:
min f (ρ, U (ρ)),
(9.19)
under condition K (ρ) U = F (ρ), g i (ρ, U (ρ)) ≤ 0 , for the variables 0 ≤ x e ≤ 1 ,
for e = 1, 2, . . . , N .
The so far formulated problem is analogous of the general approach of the topological optimization (9.1) presented in the literature overview. However, the parameterization (9.19) is defined in the terms of physical variables associated with density
ρ e . The problem of optimization is still solved with regard to the projection variables x e . Recall that the physical densities are functions of the design variables as
ρ e (x) depending on a filter or projection, which undergo changes. The proposed
algorithm includes the elementary analysis and the complex analysis of sensitivity
in addition to the algorithms required for topological optimization (regularization,
filtration, optimizators, etc.).
Since a chosen method regarding density for the purpose of topological optimization is composed of a system of finite elements, it should be parametrized for
distribution of the design variables based on density which is taken into account in
the being designed finite elements. Here we employ the parametrization based on the
general thermoelastic space for design. The latter one denoted by is shown for the
2D case in Fig. 9.7. It consists of the boundary conditions, external loading located
on the surface parts and a given temperature distribution which can be homogenous
or area dependent. The space includes holes and the optimized space . Here we
employ parametrization based on the general thermoelastic design space. The design
space is discretized by N finite elements with N opt design and N nopt given elements.
Each of the finite elements is associated with a design variable 0 < x e ≤ 1 , for
e = 1, 2, . . . , N opt . All of the variables define a vector x. Design variables x are
coupled with physical densities ρ by the choice of a standard filter as it has been
described in Sect. 9.2.2.
System of equations for static space equilibrium is approximated by a choice of
finite elements with an account of both mechanical and temperature loads, and it has
the following form:
K (ρ) U (ρ) = F (ρ) ,
(9.20)
where K (ρ)—global matrix of stiffness, U (ρ)—vector of node displacements and
F (ρ) stands for a vector of nodal load. In a general case F (ρ) consists of a sum