SYSTEM ANALYSIS AND COMPUTER SOFTWARE IN THE DESIGN OF OPTIMIZATION OF ENGINEERING STRUCTURES AND STRUCTURES

ABSTRACT

Optimization problems in engineering structures require diverse mathematical programming methods, from simplex algorithms to global random search techniques. These problems often exhibit unique characteristics, such as constraints defined by strength, rigidity, and stability, necessitating algorithm adaptations for improved efficiency. Direct structural calculations consume significantly more computational resources than objective function evaluations, highlighting the need for methods to reduce such calculations. Automation and formalization of model derivation, study, and algorithm selection are crucial in addressing these challenges. By leveraging these systems, researchers and engineers can enhance efficiency and minimize manual intervention, ultimately bridging theoretical development and practical application.

АННОТАЦИЯ

Задачи оптимизации инженерных конструкций требуют применения различных методов математического программирования, от симплекс-алгоритмов до глобальных алгоритмов случайного поиска. Эти задачи часто имеют уникальные характеристики, такие как ограничения, определяемые прочностью, жесткостью и устойчивостью, что требует адаптации алгоритмов для повышения эффективности. Прямые расчеты конструкций значительно более ресурсоемки по сравнению с оценкой целевых функций, что подчеркивает необходимость методов, уменьшающих такие расчеты. Автоматизация и формализация процесса вывода моделей, их изучения и выбора алгоритмов играют ключевую роль в решении этих задач. Использование этих систем позволяет исследователям и инженерам повысить эффективность и минимизировать ручное вмешательство, создавая мост между теоретической разработкой и практическим применением.

Keywords: Optimization problems, engineering structures, mathematical programming, algorithm adaptation, solid mechanics, continuum mechanics, structural mechanics, model formalization, algorithmic systems, computational efficiency.

Ключевые слова: Задачи оптимизации, инженерные конструкции, математическое программирование, адаптация алгоритмов, механика твердых тел, механика сплошных сред, механика конструкций, формализация моделей, алгоритмические системы, вычислительная эффективность.

Optimization problems, particularly those about engineering structures, necessitate the utilization of a diverse array of mathematical programming methodologies, spanning from the simplex algorithm to global random search algorithms. The formulation of optimization problems and inverse problems of calculating specific structures allows for the unification of the methods for their solution based on the application of the aforementioned methods. It is noteworthy that such problems exhibit several characteristics that differentiate them from abstract mathematical programming problems. This allows for the development of new algorithms or the modification of existing methods with accelerated convergence. The following features are highlighted: In the context of weight optimization of structures, it can be observed that the minimum of the objective function will invariably be situated at one of the intersections of the strength, rigidity, and stability constraints of the structures under consideration. This feature enables the parametric adaptation of search algorithms. Secondly, the direct calculation of a structure typically requires several orders of magnitude more computational time than calculating the objective function. It is therefore possible to adapt structural algorithms to minimize the number of direct calculations of structures. Thirdly, both direct calculations and inverse and optimization calculations for rather complex structures are performed using numerical methods. In this case, the ratio of the accuracy of the calculation model (which can be expressed in terms of the number of members of a series of coordinate functions, nodes of the difference grid, and finite elements) and the position of the search system in the search area is clearly beneficial.

Experimental. It is important to note that the aforementioned characteristics of solid mechanics problems merely indicate the potential for algorithm adaptation; they do not establish the necessity for the development of a single, universally optimal algorithm for all classes of problems. The inability to develop a universal algorithm is attributable to the heterogeneity of optimization problems, the technical specifications of the computer, and the time required to solve each task.

The plethora of available methods for solving optimisation problems gives rise to the challenge of identifying an optimal algorithm for a given problem. When one considers that each method corresponds to a multitude of algorithms with varying requirements and characteristics, it becomes evident that this problem can be highly intricate, even for an expert. For an unskilled user, however, it can prove intractable.

The set of requirements imposed by various algorithms on problems, such as convexity, differentiability a certain number of times, invertibility and sufficiently good conditioning of the matrix of second derivatives of the objective function and constraints, satisfaction of the Lipschitz condition of functions and their derivatives, the absence or presence of an initial point, etc., creates an extremely large number of situations in choosing a specific algorithm, which presents a significant challenge for researchers attempting to make an optimal decision. Conversely, the issue of selecting the most appropriate algorithm can and should be addressed through the automation of the requisite analysis, thereby enabling the computer to perform the necessary evaluation of potential options.

In order to make an informed choice regarding the classification of an algorithm, it is necessary to have access to more comprehensive information about the mathematical models of the problems in question than data on the convexity or non-convexity of the objective function and constraints. In order to obtain this information, it is necessary to study the models in question. This may be carried out analytically, with or without the use of a computer, or stochastically, with the definition of the distribution law of the objective function and constraints in the search area. Furthermore, the use of special algorithms that allow the identification of the contours of the area may also be employed. Furthermore, the study of models within a defined framework can be automated, thereby facilitating the more accurate selection of an algorithm or a sequence of algorithms to address a specific problem.

In order to overcome the difficulties that arise when setting, analysing and selecting algorithms to solve optimisation problems, it is necessary to adopt a systematic approach that allows for a comprehensive consideration and solution of the problems that emerge.

As can be observed from the analysis of optimisation methods, systems or packages designed to solve optimisation problems, which are parts of CAD, as well as individual systems, it can be seen that they are mainly invariant with respect to classes of objects. This situation presents two distinct aspects. The advantageous aspect of this invariance is that it permits the package to be linked to other systems (or to optimise other classes of objects when working in individual mode) essentially without modification.

One disadvantage of package invariance is that it does not consider the characteristics of objects that, in certain instances, can lead to a notable enhancement of algorithms designed to optimise a specific class of objects. The analysis further indicates that there is currently a dearth of attention devoted to the unification of problem statements, model research, and the utilisation of model research outcomes for the automatic selection and calibration of optimal algorithms.

When setting optimization problems in solid mechanics, there is no need to develop a separate mathematical model for each problem. This process can be successfully formalized and automated, which was convincingly demonstrated by the works of V.K. Kabulov and his students [1-9].

Based on the general conservation laws: mass, energy, etc. - with the help of a computer, systems of equations are derived that describe any mathematical model of solid mechanics.

Thus, to solve optimization and inverse problems (we will call them collectively extreme) of engineering structures, a chain is traced that is subject to formalization and automation: model derivation - model study - selection of an optimal algorithm that allows for maximum automation of the process of setting and solving optimization problems. This problem can be solved most fully on the basis of algorithmic methods developed by V.K. Kabulov [1-3]. The general scheme of algorithmization of research, setting and solving mechanics problems is presented in Fig. 1.

Figure 1. The general scheme of algorithmization of research, setting and solving mechanics problems

Experience is understood here in a broad, philosophical sense. This is the experience accumulated by mankind, reflected in monographs, and articles, obtained in laboratories and natural experiments. At the "experience" stage, it is assumed that information banks and extensive automation of experiments with the development of means for collecting, transmitting and processing data are created. At the "laws" stage, the accumulated experience is transformed into general laws of mechanics - conservation laws, etc. In algorithmization, known laws are encrypted and entered into the computer's memory. New laws are formulated based on the results of automated experiments.

Based on general laws, problems belonging to different classes are solved (for example, elasticity theory, deformation theory of plasticity, rheological problems, etc.). Classification of these problems and automatic recognition of classes are performed at the "problem" stage.

For each specific problem, according to known laws, at the "model" stage, systems of equations are automatically derived, which are mathematical models of the problems.

At the "algorithms" stage, a specific algorithm is selected to solve the obtained model, taking into account the issues of applicability and optimality.

After selecting the algorithm, it is necessary to move on to building programs that implement the calculation, which is done at the "mathematical support" stage.

At the last "calculation" stage, computing complexes are created associated with experimental setups, and the calculation is linked to experience.

Thus, the specified system allows for maximum automation of the researcher's creative work, starting with setting up an experiment and ending with processing the numerical results of the calculation, leaving him with the non-formalizable part of creativity.

Thus, the most promising assistance to researchers and engineers seems to us to be in the development of an algorithmic system for solving problems in continuum mechanics [1-9], as well as (taking into account data from modern computers) for narrower classes of problems in the mechanics of deformable solids and structural mechanics. It should be noted that it is also advisable to develop an optimizing system that is invariant concerning objects based on the specified principles of algorithmization, all stages of which are necessary for the functioning of the invariant system. It is necessary to slightly change the role of the "model" stage, since in this case, it is impossible to automate the derivation and formation of mathematical models, but the purpose of the "model" stage remains the study and identification of models.

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