336
9 Topologic Optimization of Vibrations of Size-Dependent Beams
the method of solid isotropic material with penalization (SIMP) [47], the method of
evolutionary structural optimization (ESO) [38] and the level set methods [48–51].
An important role is played in the matching of SIMP with other methods, and in
particular with the method of homogenization. The main idea of the latter approach
relies on the determination of the element parameters as the design variables, whereas
in the optimization process the appropriate parameters are estimated by establishing
links between the local parameters (for instance, density) and the global required
physical material properties (for instance, Young modulus and/or heat transfer coefficients). Various algorithms are then employed to solve the optimization problem
like the method of optimal criterion [52], the method of neighbourhood search based
on efficient sensitivity calculations [53] or the method of moving asymptotes [54].
It should be noticed that materials in the optimization domain are graded in a way
to achieve the required functional characteristics of the given constraints. Two main
directions in the FGM research can be distinguished: (i) problems dealing with constructions optimization [38, 44, 50, 51] and (ii) problems of achieving effective
functional material properties [55–60]. The design of composite structures requires
precise modelling of elastic, thermal and thermoelastic material properties. The associated widely popular elasticity criteria take into account only mechanical load [44],
whereas thermal criteria rely on the heat load [61, 62], and finally the thermoelastic
criterion includes both mechanical and heat loads [63, 64].
In the case of the problems of topological optimization, there are two main
approaches aimed on the inclusion of mechanical and thermal criteria. The first
class considers the given material thermoelastic properties but without a full coupling of heat and mechanic fields in the topological sense, i.e. usually the temperature
field is given and the thermal load is transformed to the mechanical load [63, 64].
In this statement, the given construction is controlled via mechanical characteristics.
The second class takes into account both heat and mechanical fields. The problem
is solved in the following way: first, the temperature field is calculated, and then it
is matched with the field of deformations. In the latter approach, the microstructure
material can be constructed with the help of the mechanical target function [63] or
with the help of the coupled mechanical and heat functions simultaneously [65].
The qualitative radical change in the design of composite materials was introduced by Sigmund [58, 60] who proposed the method of inverse homogenization.
He assumed that the composites are made from identical periodically distributed
cells. The topological optimization was employed for the appropriate localization of
either one-component [57] or multi-component [55, 59] materials in order to achieve
the required physical properties of the constructed composite. It is well known that a
direct homogenization allows consideration of the global composite properties based
on the structure of one elementary cell or on a basis of the representative cell volume
[66]. In contrary to this standard approach, the method of inverse homogenization
allows to estimate the appropriate/required material distribution in the introduced
model of an elementary cell [57]. A difference between the global homogenized
material properties and the required (target) values is controlled by finding a minimum with a help of the methods of topological optimization [55, 60], However, in the
latter approach, the elastic and heat material properties are usually treated separately.
9 Topologic Optimization of Vibrations of Size-Dependent Beams
the method of solid isotropic material with penalization (SIMP) [47], the method of
evolutionary structural optimization (ESO) [38] and the level set methods [48–51].
An important role is played in the matching of SIMP with other methods, and in
particular with the method of homogenization. The main idea of the latter approach
relies on the determination of the element parameters as the design variables, whereas
in the optimization process the appropriate parameters are estimated by establishing
links between the local parameters (for instance, density) and the global required
physical material properties (for instance, Young modulus and/or heat transfer coefficients). Various algorithms are then employed to solve the optimization problem
like the method of optimal criterion [52], the method of neighbourhood search based
on efficient sensitivity calculations [53] or the method of moving asymptotes [54].
It should be noticed that materials in the optimization domain are graded in a way
to achieve the required functional characteristics of the given constraints. Two main
directions in the FGM research can be distinguished: (i) problems dealing with constructions optimization [38, 44, 50, 51] and (ii) problems of achieving effective
functional material properties [55–60]. The design of composite structures requires
precise modelling of elastic, thermal and thermoelastic material properties. The associated widely popular elasticity criteria take into account only mechanical load [44],
whereas thermal criteria rely on the heat load [61, 62], and finally the thermoelastic
criterion includes both mechanical and heat loads [63, 64].
In the case of the problems of topological optimization, there are two main
approaches aimed on the inclusion of mechanical and thermal criteria. The first
class considers the given material thermoelastic properties but without a full coupling of heat and mechanic fields in the topological sense, i.e. usually the temperature
field is given and the thermal load is transformed to the mechanical load [63, 64].
In this statement, the given construction is controlled via mechanical characteristics.
The second class takes into account both heat and mechanical fields. The problem
is solved in the following way: first, the temperature field is calculated, and then it
is matched with the field of deformations. In the latter approach, the microstructure
material can be constructed with the help of the mechanical target function [63] or
with the help of the coupled mechanical and heat functions simultaneously [65].
The qualitative radical change in the design of composite materials was introduced by Sigmund [58, 60] who proposed the method of inverse homogenization.
He assumed that the composites are made from identical periodically distributed
cells. The topological optimization was employed for the appropriate localization of
either one-component [57] or multi-component [55, 59] materials in order to achieve
the required physical properties of the constructed composite. It is well known that a
direct homogenization allows consideration of the global composite properties based
on the structure of one elementary cell or on a basis of the representative cell volume
[66]. In contrary to this standard approach, the method of inverse homogenization
allows to estimate the appropriate/required material distribution in the introduced
model of an elementary cell [57]. A difference between the global homogenized
material properties and the required (target) values is controlled by finding a minimum with a help of the methods of topological optimization [55, 60], However, in the
latter approach, the elastic and heat material properties are usually treated separately.