Optimization of structures in the frequency domain

Abstract

The frequency-domain approach has been gaining great popularity over the last decades owing to the advantages it provides considering computational time and assessment of system dynamics, especially under random excitations. Nevertheless, the results achieved in the frequency domain might be difficult to apply to the physical world. On the other hand, the frequency-domain approach is getting attention in structural optimization studies. In this thesis study, structural optimization has been discussed with the frequency-domain approach, and solutions to structural optimization problems (e. g. numerical instabilities) mentioned in the literature were investigated. For this purpose, a novel constraint, which is adapted from the Nevanlinna-Pick (NP) interpolation theory, was imposed on the optimization problem. The NP interpolation theory states that the existence of a mapping function between two complex domains is contingent upon the positive definiteness of the Pick matrix. Particularly, the response of a system, which is a complex function, cannot have an independent amplitude from its derivative amplitude at certain frequency values. This mathematical theorem corresponds to a physical fact, the dissipativity of systems. In this context, Boundary Nevanlinna-Pick (B-NP) interpolation, which is a variant of NP interpolation, was set between the excitation frequency interval (or the frequency range of interest) and the complex values of the system response, which are computed through the transfer function. In other words, the transfer function was designed as a sort of mapping function (i.e., interpolant) in the NP interpolation theorem, while the frequency ranges of interest and the response values are the domains to be mapped. Hence, one cannot shape a physical system's frequency response arbitrarily at discrete frequencies. Consequently, the associated Pick matrix was formed, and its positive definiteness condition can be incorporated into the optimization problem as a non-linear constraint. In this way, a feasible design space was determined under the guidance of the B-NP constraint. When optimizing a physical system's frequency response, disregarding the NP interpolation theory may lead to numerical instability and impractical solutions. Even if the structural equations are linear, the constraints to be imposed due to natural frequencies, vibration amplitudes, fatigue damage, and stress limits transform the optimization problem into a nonlinear optimization problem in the frequency domain. Hence, achieving the optimal solutions becomes challenging, especially in the frequency domain. In this context, it is expected that the optimization capability in the frequency domain will be increased by determining the feasible regions where the results are achievable in the physical world with the help of the B-NP constraint. This goal will be achieved with the help of an interpolation method that has not been used before in any structural optimization study in the frequency domain. Furthermore, it is expected that employing constraints derived from B-NP interpolation theory leads to an efficient exploration of the frequency domain by affecting the direction and steps of the iterations during optimization. First of all, the numerical limitations of incorporating B-NP constraint into a structural optimization in the frequency domain were investigated.