338
9 Topologic Optimization of Vibrations of Size-Dependent Beams
Besides, it should be mentioned, that our research fulfils a gap in the existing
research, since there are no reports of the results of chaotic vibrations of the sizedependent FGM in the available literature.
9.3 Topological Optimization Problem
The topological optimization stands for a process of determination of coupling and
possible forms of localization of inclusions inside a given space occupied by an elastic material. It can be understood as the determination of the best distribution of the
materials inside of the given space [72]. It guarantees a large freedom for designers in comparison to the process of optimization of the size and space forms. The
latter takes into account the variables like fitness and surface of the cross sections
of the being constructed elements (sizes) and the geometric peculiarities (forms) of
the given design elements. Therefore, optimization of a topology plays a crucial role
in both preliminary design studies where changes exhibit essential influence on the
final effectiveness of the phase components. Owing to the mentioned possibility of
finding the optimal material microstructure, the topological optimization belongs to
the perspective tool for the thermoelastic design. There exist two types of topological optimizations: discrete and continuous [73, 74]. In what follows, we consider
only continuous optimization. There are available a few different methods based on
finite element for the topological optimization of a continuum. The general problem
consists of determination of either existence or a lack of material in a given subspace
of the space undergoing the design process. In what follows, we briefly describe the
nowadays employed methods of topological optimization.
9.3.1 Methods Based on Density
Nowadays the most widely employed methods of topological optimization of structures are based on the explicit parameterization also known as methods based on
density. Those methods operate on the space divided into finite elements. However,
instead of characterization of elastic properties of microstructure, each of the finite
elements contains only one design variable. The latter one is often understood as
the material element density ρ e . Properties of each material element, for instance,
the Young moduli and the heat transfer coefficient are the functions of the variable
density through the given interpolating functions. The most known are widely used
in the solid isotropic material with penalization method (SIMP). The fundamental
mathematical formulations of the problem of the topological optimization based on
the density consist of target functions, a set of constraints (they often include constraints because of the material use) and the finite-dimensional representation of the
physical system, for instance, on a basis of the method of finite element (FEM). The
9 Topologic Optimization of Vibrations of Size-Dependent Beams
Besides, it should be mentioned, that our research fulfils a gap in the existing
research, since there are no reports of the results of chaotic vibrations of the sizedependent FGM in the available literature.
9.3 Topological Optimization Problem
The topological optimization stands for a process of determination of coupling and
possible forms of localization of inclusions inside a given space occupied by an elastic material. It can be understood as the determination of the best distribution of the
materials inside of the given space [72]. It guarantees a large freedom for designers in comparison to the process of optimization of the size and space forms. The
latter takes into account the variables like fitness and surface of the cross sections
of the being constructed elements (sizes) and the geometric peculiarities (forms) of
the given design elements. Therefore, optimization of a topology plays a crucial role
in both preliminary design studies where changes exhibit essential influence on the
final effectiveness of the phase components. Owing to the mentioned possibility of
finding the optimal material microstructure, the topological optimization belongs to
the perspective tool for the thermoelastic design. There exist two types of topological optimizations: discrete and continuous [73, 74]. In what follows, we consider
only continuous optimization. There are available a few different methods based on
finite element for the topological optimization of a continuum. The general problem
consists of determination of either existence or a lack of material in a given subspace
of the space undergoing the design process. In what follows, we briefly describe the
nowadays employed methods of topological optimization.
9.3.1 Methods Based on Density
Nowadays the most widely employed methods of topological optimization of structures are based on the explicit parameterization also known as methods based on
density. Those methods operate on the space divided into finite elements. However,
instead of characterization of elastic properties of microstructure, each of the finite
elements contains only one design variable. The latter one is often understood as
the material element density ρ e . Properties of each material element, for instance,
the Young moduli and the heat transfer coefficient are the functions of the variable
density through the given interpolating functions. The most known are widely used
in the solid isotropic material with penalization method (SIMP). The fundamental
mathematical formulations of the problem of the topological optimization based on
the density consist of target functions, a set of constraints (they often include constraints because of the material use) and the finite-dimensional representation of the
physical system, for instance, on a basis of the method of finite element (FEM). The