Chapter 9
Vibrations of Size-Dependent Beams
Under Topologic Optimization and
Temperature Field
9.1 Introduction
In this chapter, problems associated with multifunctional requirements with respect
to effective characteristics of composites consisting of two components as well as
composites with holes or technological inclusions have been studied. In the process
of investigation, a strong dependence of the optimal topology of the distribution of
materials in the microstructure of composites on the form of the target functions has
been detected. The study of transformations of the optimal topology of the composite
microstructure with a change in the weight coefficient from ω = 0 (maximization
of the heat transfer) up to ω = 1 (maximization of the mechanical moduli) has been
conducted. Moreover, a set of alternatives optimal in the Pareto sense has been
constructed. The considered examples clearly indicate the inability to achieve the
best/required properties simultaneously in both cases, which is caused by conflicting
criteria in the target function. In addition, nonlinear dynamics of the size-dependent
Euler-Bernoulli beams embedded into temperature field with topologically optimized
microstructure is studied. The following new results are presented:
(i) We have developed a mathematical model of the size-dependent nonlinear
beam, taking into account the topological optimization under the criterion
of maximum stiffness. The mathematical model is based on the BernoulliEuler, von Kármán and Duhamel-Newmann hypotheses. Also, an algorithm
and a computer program for numerical computations of the optimized beam
microstructure for the given boundary conditions, the form of external load
and the temperature have been developed (both static and dynamic problems
have been considered).
(ii) Reliability of the results was confirmed by investigating the convergence along
the spatial variable as well as by examining the solution to the Cauchy problems, and investigating Lyapunov exponents and time evolution of the frequency, obtained using wavelet analysis.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
J. Awrejcewicz et al., Mathematical Modelling and Numerical Analysis of Size-Dependent
Structural Members in Temperature Fields, Advanced Structured Materials 142,
https://doi.org/10.1007/978-3-030-55993-9_9
333
Vibrations of Size-Dependent Beams
Under Topologic Optimization and
Temperature Field
9.1 Introduction
In this chapter, problems associated with multifunctional requirements with respect
to effective characteristics of composites consisting of two components as well as
composites with holes or technological inclusions have been studied. In the process
of investigation, a strong dependence of the optimal topology of the distribution of
materials in the microstructure of composites on the form of the target functions has
been detected. The study of transformations of the optimal topology of the composite
microstructure with a change in the weight coefficient from ω = 0 (maximization
of the heat transfer) up to ω = 1 (maximization of the mechanical moduli) has been
conducted. Moreover, a set of alternatives optimal in the Pareto sense has been
constructed. The considered examples clearly indicate the inability to achieve the
best/required properties simultaneously in both cases, which is caused by conflicting
criteria in the target function. In addition, nonlinear dynamics of the size-dependent
Euler-Bernoulli beams embedded into temperature field with topologically optimized
microstructure is studied. The following new results are presented:
(i) We have developed a mathematical model of the size-dependent nonlinear
beam, taking into account the topological optimization under the criterion
of maximum stiffness. The mathematical model is based on the BernoulliEuler, von Kármán and Duhamel-Newmann hypotheses. Also, an algorithm
and a computer program for numerical computations of the optimized beam
microstructure for the given boundary conditions, the form of external load
and the temperature have been developed (both static and dynamic problems
have been considered).
(ii) Reliability of the results was confirmed by investigating the convergence along
the spatial variable as well as by examining the solution to the Cauchy problems, and investigating Lyapunov exponents and time evolution of the frequency, obtained using wavelet analysis.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
J. Awrejcewicz et al., Mathematical Modelling and Numerical Analysis of Size-Dependent
Structural Members in Temperature Fields, Advanced Structured Materials 142,
https://doi.org/10.1007/978-3-030-55993-9_9
333