358
9 Topologic Optimization of Vibrations of Size-Dependent Beams
parameters as design variables and derive a relationship between local (for instance,
density) and global physical properties of the material (for instance, Young modulus
or thermal conductivity) in the computational process aimed at finding parameters in
the optimization process. To design microstructures composed of different materials,
the solid isotropic material with penalization (SIMP) method [47], the evolutionary
structural optimization (ESO) [38] and the level set method (LS) [49–51, 101] are
used. In particular, the use of homogenization with SIMP is becoming increasingly
popular and useful.
By virtue of numerous algorithms aimed at finding an optimum, such as the method
of optimal criterion (OC) [52], successive linear programming (SLP) or the method
of moving asymptotes (MMA) [54], the optimized material localization allows us
to achieve the required functional material characteristics. The current research has
focused on solving problems aimed at either optimizing the constructions [38, 44, 51]
or getting functional properties of materials [50, 55–60]. When designing composite
structures, either elastic or thermal/thermoelastic properties are usually taken into
account. In the case of the elasticity criterion, only the mechanical load is considered
[44, 103], whereas thermal criteria take into account the thermal load [61, 62], and
thermoelastic criteria cover both mechanical and thermal loads [63, 64].
It should be emphasized that there are two approaches that include both thermal
and mechanical criteria in topological optimization problems. The first one considers
the material as thermoelastic without the coupling between the thermal and mechanical fields in the topological sense (the thermal load is converted into mechanical
load) [63, 64]. In this case, the structure is controlled by the mechanical characteristics. On the other hand, the second approach takes into account both the heat and
mechanical fields. Namely, the temperature field is considered first, and then it is
matched with the field of deformations. In the mentioned cases, the design can be
guided either by the mechanical target function [63] or by coupled mechanical and
heat functions [65].
The qualitative seminal results in the problem of design of materials were obtained
by Sigmund [58, 60] who introduced the method of inverse optimization. It is
assumed that the composites are built from identical periodic base cells. The methodology of the topological optimization was employed in the case of the proper distribution of one-component [57] or multi-phase materials [55, 59] in order to achieve
the required physical properties. It is known that direct homogenization makes it
possible to consider global properties based on the structure of the elementary cell
or the representative volume [66]. On the other hand, the inverse homogenization is
aimed at searching the optimized material distributions within the elementary cell
[57]. Differences between the global homogenized material properties and their target values are minimized with the help of topological optimization methods [55, 57,
60]. In this context, the elastic and heat material properties are considered separately.
Optimal microstructures of materials have been developed in the problems of
maximizing bulk and shear moduli [55, 67]. Using this approach, novel composites
with negative Poisson’s coefficient [57] and materials with extreme or negative coefficients of thermal expansion [60] have been designed, which demonstrated wide
possibilities of this method in the design of elastic and thermoelastic composites.
9 Topologic Optimization of Vibrations of Size-Dependent Beams
parameters as design variables and derive a relationship between local (for instance,
density) and global physical properties of the material (for instance, Young modulus
or thermal conductivity) in the computational process aimed at finding parameters in
the optimization process. To design microstructures composed of different materials,
the solid isotropic material with penalization (SIMP) method [47], the evolutionary
structural optimization (ESO) [38] and the level set method (LS) [49–51, 101] are
used. In particular, the use of homogenization with SIMP is becoming increasingly
popular and useful.
By virtue of numerous algorithms aimed at finding an optimum, such as the method
of optimal criterion (OC) [52], successive linear programming (SLP) or the method
of moving asymptotes (MMA) [54], the optimized material localization allows us
to achieve the required functional material characteristics. The current research has
focused on solving problems aimed at either optimizing the constructions [38, 44, 51]
or getting functional properties of materials [50, 55–60]. When designing composite
structures, either elastic or thermal/thermoelastic properties are usually taken into
account. In the case of the elasticity criterion, only the mechanical load is considered
[44, 103], whereas thermal criteria take into account the thermal load [61, 62], and
thermoelastic criteria cover both mechanical and thermal loads [63, 64].
It should be emphasized that there are two approaches that include both thermal
and mechanical criteria in topological optimization problems. The first one considers
the material as thermoelastic without the coupling between the thermal and mechanical fields in the topological sense (the thermal load is converted into mechanical
load) [63, 64]. In this case, the structure is controlled by the mechanical characteristics. On the other hand, the second approach takes into account both the heat and
mechanical fields. Namely, the temperature field is considered first, and then it is
matched with the field of deformations. In the mentioned cases, the design can be
guided either by the mechanical target function [63] or by coupled mechanical and
heat functions [65].
The qualitative seminal results in the problem of design of materials were obtained
by Sigmund [58, 60] who introduced the method of inverse optimization. It is
assumed that the composites are built from identical periodic base cells. The methodology of the topological optimization was employed in the case of the proper distribution of one-component [57] or multi-phase materials [55, 59] in order to achieve
the required physical properties. It is known that direct homogenization makes it
possible to consider global properties based on the structure of the elementary cell
or the representative volume [66]. On the other hand, the inverse homogenization is
aimed at searching the optimized material distributions within the elementary cell
[57]. Differences between the global homogenized material properties and their target values are minimized with the help of topological optimization methods [55, 57,
60]. In this context, the elastic and heat material properties are considered separately.
Optimal microstructures of materials have been developed in the problems of
maximizing bulk and shear moduli [55, 67]. Using this approach, novel composites
with negative Poisson’s coefficient [57] and materials with extreme or negative coefficients of thermal expansion [60] have been designed, which demonstrated wide
possibilities of this method in the design of elastic and thermoelastic composites.