# Numerical study of the parametric optimization of the forced oscillation frequencies of the shell of a minimal surface on a trapezoidal contour under thermal and power loading

## Authors

• Hryhoriy Ivanchenko Kyiv National University of Construction and Architecture, Ukraine
• Oleksandr Koshevyi Kyiv National University of Construction and Architecture, Ukraine
• Oleksandr Koshevyi Kyiv National University of Construction and Architecture, Ukraine
• Lyudmyla Grigoryеva Kyiv National University of Construction and Architecture, Ukraine

## Keywords:

optimization, parametric optimization, shape optimization, topological optimization, minimum surface hull, objective function, design variables, constraints, limit, forced frequency, hull weight

## Abstract

This research paper discusses various methods and approaches to the optimal design of structures. Methods for solving the optimization problem can be divided into two large groups. The first group includes methods that are based on the use of the necessary conditions for the extremes of the objective function. The second group consists of mathematical programming methods: linear, convex, dynamic programming, and random search. In mathematical terms, optimal design problems are optimization problems - the search for an extremum of the objective function and the values of the parameters at which the extremum is achieved. The choice of the optimality criterion is one of the main problems of optimal design. The most widely developed problems are those that have the optimization criterion of weight or volume of the structure while satisfying the conditions of strength, rigidity and stability.

Optimal design problemsare also divided into three large groups. The first group is parametric optimization problems, which involve the optimization of one or more parameters, called design variables, to minimize or maximize the objective function. The second group is topological optimization, in which unnecessary material is discarded, where the Mises stress is zero, thereby minimizing the objective function. The third group is optimization of the shape of the object under study, when the shape corresponds to internal forces, the shell with the smallest area is modeled on a given cone (shells of minimal surfaces), as well as methods of applied geometry, where the surface shape is modeled for a certain load.

To perform the parametric optimization of the forced vibrations of the shell of the minimum surface on a trapezoidal contour, the objective function is the weight of the spatial structure. The variables in the parametric optimization problem are the thickness of the finite elements from 1 to 100 mm. The structure constraint is imposed on the first forced oscillation frequency of 0.250 Hz. This type of problem is used to prevent resonance from process equipment that can affect the natural frequencies of the structure under external load.Subject of this study is an interesting applied problem for construction mechanics, as it is the first time to display the application of two types of optimization on one research object.

The results of a numerical study of the parametric optimization of the minimum surface shell on a trapezoidal cage under thermal power loading.  The parametric optimization helped to reduce the weight of the shell by 13.4%, which is 1810 kg of sheet steel. The first forced oscillation frequency meets the constraint of the optimization calculation. We constructed 10 forced vibration frequency shapes of the shell before and after optimization, and also presented the distribution of the shell thickness after the optimization calculation.

## Author Biographies

### Hryhoriy Ivanchenko, Kyiv National University of Construction and Architecture

Doctor of Technical Sciences, Professor, Dean of the Faculty of Civil Engineering

### Oleksandr Koshevyi, Kyiv National University of Construction and Architecture

PhD, associate professor of department of theoretical mechanics

### Oleksandr Koshevyi, Kyiv National University of Construction and Architecture

candidate of technical sciences, associate professor, head of Strength of Materials department

### Lyudmyla Grigoryеva, Kyiv National University of Construction and Architecture

candidate of physical and mathematical sciences, associate professor, associate professor of  Strength of Materials department

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2023-06-26

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