The optimal design in applied mechanics is used to improve the efficiency of minimal surface shells, this process is carried out until the design can no longer be better given the input data, using a certain optimization method. It has several advantages while building structures are designing. One of the passes is the optimization process, which provides a systematization, a logical procedure for the design of shells. With the correct use of the optimization algorithm, it is possible to reduce the probability of a designer's mistake. Modern optimization methods can be applied to problems that have more than a million design variables and constraints. Optimization algorithms work effectively when there is some regularity in the objective function, such as a convexity or a depression, so when choosing an optimization algorithm, it is necessary to consider their advantages and disadvantages. While minimizing the objective function fmin, the main task is to find the point of the global minimum, the value of fmin (X) will be minimal, taking into account the restrictions. Determining the global minimum is quite a difficult task. Much more often, a point has a local minimum, and the task of the designer is to investigate where the point of the local and global minimum is. The optimization algorithm for single-criteria parametric optimization is performed as follows: the objective function is the weight of the shell of the minimum surface on the square contour, the design variables are the shell thickness from 1 to 100 mm, the constraints are presented - the first forced oscillation frequency is 0.10 Hz. The results of changing the objective function are reduction in the weight of the shell, which is in the percentage equivalent of 9.6% without losing the strength and stability of the minimum surface shell on the square contour. The first forced oscillation frequency after the optimization calculation from the thermoforce load is 0.10187155 Hz, which is actually represented by the limitation. Using the author's methodology and software, it is possible to perform an effective optimization calculation of the minimum surface shell on the square contour.

The article considers the numerical study of the multicriteria parametric optimization of the minimum surface shell on the square contour under thermoforce loading. The mathematical algorithm for solving the problem of multicriteria parametric optimization of minimal surfaces shellsis described: the necessary output data are given, which target functions can be used, design variables and constraints are described. The influence of initial data on the numerical study of multicriteria parametric optimization was shown. The calculation of necessary components and matrices for multicriteria parametric optimization of minimal surfaces shells, as well as methods of its simplification, depending on the initial data was made. Was shown in which cases the algorithm is executed successfully, and where it was necessary to return to change the initial parameters. The sensitivity analysis of multi-criteria parametric optimization of minimal surfaces shells is presented. In article reveals the inner essence of the structure's work while the optimization calculation was done. Work with internal forces and stresses, the connection with the finite element method and its stiffness matrix were made. The process and methodology of setting ofthermoforce loadare revealed.All initial parameters, as well as how the load works with multicriteria parametric optimization are shown. The numerical experiment made it possible to reduce the weight of minimum surface shell on the square contour by 5%, and the movement along the coordinate axes by 38%, which are the target functions of multicriteria parametric optimization. Redistribution of the thickness of the minimum surface shell on the square contour from 1 mm to 31 mm was realized. The interaction of the objective functions is shown - they conflict, as well as the graphs of changes in Mises stresses, shear stresses and strain energy, which makes it possible to reveal the internal processes of multi-criteria parametric optimization of the minimum surface shell on a square contour. 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.

Practical using of curvilinear shape shells is related with significant problems during their production especially for metal structures. Therefore during such shells production curvilinear shape is replaced by faceted. Realization of this method when designing needs additional investigations performing of faceted shells bearing capacity on the basis of appropriate numerical calculation method using. Problems of solving such tasks are practically not displayed in the literature. Break-in of the middle surface affect significantly to the shell stress-strain state. Accounting of temperature fields’ influence in the problems of their stability complicates their behavior research even more. In this paper the research results comparing analysis of static problems about smooth and faceted shells nonlinear deformation and stability under mechanical loads is presented. The problem is solving with using of software that are based on the finite element method: by method that realized the moment finite-element scheme and using software package LIRA. The solving method that used the moment finite-element scheme is based on the geometrically nonlinear equations of the 3D theory of thermoelasticity without application of theory of shells simplifying hypothesis and on the applications of the universal three-dimensional solid finite element.

Providing all parameters according to two groups of limit states at the design stage is a multi–criteria task. Optimal design helps to perform such tasks. When the phenomenon of resonance occurs, it is necessary to change the frequency of forced oscillations of the load–bearing structures of various tanks and coatings in order to ensure sufficient strength and stability of the structure and the building in general. For this purpose, there is a separate type of problem in optimal design that allows us to change the forced frequencies of oscillations using the example of the shell of the minimum surface on a rectangular contour. In mathematical point of view, optimal design tasks are optimization tasks, finding the extremum of the target function (maximum or minimum). Methods of solving optimal design problems can be conditionally divided into two groups. One of them will include methods that are based on using necessary conditions for the extremum of the objective function. The second group includes methods: linear, convex and dynamic programming, random search methods. The application of modern methods of mathematical optimization requires powerful PCs with a large RAM, therefore, when choosing optimization, it is necessary to take into account the capabilities of computer equipment. The optimization algorithm for single–criteria parametric optimization is performed as follows: the objective function is the weight of the shell of the minimum surface on a rectangular contour, the design variables are the shell thickness from 1 to 50 mm, the constraints are presented with first forced oscillation frequency is 0,15 Hz. Using software complex Femap with Nastran and our own software, a single–criterion parametric optimization was made. The results of changing the objective function are reducing the weight of the shell by 2300 kg of C240 steel, which is equivalent to 10,3% without losing the strength and stability of the minimum surface shell on a rectangular contour. Using author's methodology and own software, it is possible to perform an effective optimization calculation for the minimum surface shell on a rectangular contour.

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.