Методы оптимизации с применением поверхностей отклика, адаптированные к решению задач анализа и синтеза конструктивных параметров тонкостенных машиностроительных конструкций
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Дата
2016
ORCID
DOI
Науковий ступінь
Рівень дисертації
Шифр та назва спеціальності
Рада захисту
Установа захисту
Науковий керівник
Члени комітету
Видавець
НТУ "ХПИ"
Анотація
Учитывая особенности формы поверхностей отклика характеристик напряженного состояния тонкостенных машиностроительных конструкций в случае их нелинейного поведения, разработан подход к решению задачи оптимизации. Он состоит в глобальной аппроксимации рассматриваемой функции квадратичной формой,
последующем уменьшении приращений независимых переменных и движении от минимума квадратичной формы, построенной на предыдущем шаге, к минимуму вновь построенной аппроксимации. Недостатком предложенного подхода является то, что найденный оптимум может оказаться локальным экстремумом, если минимизируемая функция имеет полихолмистый характер. Тем не менее, он может быть рекомендован для решения практических задач. Разработанный алгоритм был опробован на ряде тестовых задач и продемонстрировал хорошую сходимость. Ключевые слова: машиностроительная конструкция; нелинейное поведение; оптимизация; поверхность отклика; глобальная аппроксимация; квадратичная форма
Response surfaces of being evaluated stress state characteristics of thin-walled structures in the case of nonlinear behavior in many cases are characterized by smooth, gentle slope and presence of a plurality of faintly expressed local extrema. This constitute the main difficulty during optimization. An approach to solving the optimization problem that takes into account these peculiarities of these functions shape is developed. It is in the global approximation of the function by quadratic form founding the Hessian matrix using finite difference method for large increments of the independent variables, equal to the intervals of their variation, the subsequent reducing of the independent variables increments and the movement from the minimum of the quadratic form constructed at the previous step, to the minimum of newly constructed approximation. The disadvantage of this approach is that the founded optimum can be a local extremum in the case when minimized function has poli-hilly character. Nevertheless, it can be recommended for solving practical problems because it may be irrational to seek global optimum of function if the first founded optimum is little different of it. Also, it is not worthwhile to seek a global optimum in the case of improving existing structure, but not the design of new one due to the fact that in this case, the optimum located near the nominal values of variable design parameters is of more interest than the remote optimum. The algorithm was tested on a series of classical test problems and it showed good convergence. By improving this approach, it was proposed to "freeze" the Hessian matrix at several steps of the iterative process. For some of test functions considered this correction has allowed to reduce the number of computational iterations, but with a reduction in accuracy, and for some functions it showed its inapplicability. Thus, in practice the proposed algorithm with using the Hessian matrix update correction should take into account the specificity of the response function in each case.
Response surfaces of being evaluated stress state characteristics of thin-walled structures in the case of nonlinear behavior in many cases are characterized by smooth, gentle slope and presence of a plurality of faintly expressed local extrema. This constitute the main difficulty during optimization. An approach to solving the optimization problem that takes into account these peculiarities of these functions shape is developed. It is in the global approximation of the function by quadratic form founding the Hessian matrix using finite difference method for large increments of the independent variables, equal to the intervals of their variation, the subsequent reducing of the independent variables increments and the movement from the minimum of the quadratic form constructed at the previous step, to the minimum of newly constructed approximation. The disadvantage of this approach is that the founded optimum can be a local extremum in the case when minimized function has poli-hilly character. Nevertheless, it can be recommended for solving practical problems because it may be irrational to seek global optimum of function if the first founded optimum is little different of it. Also, it is not worthwhile to seek a global optimum in the case of improving existing structure, but not the design of new one due to the fact that in this case, the optimum located near the nominal values of variable design parameters is of more interest than the remote optimum. The algorithm was tested on a series of classical test problems and it showed good convergence. By improving this approach, it was proposed to "freeze" the Hessian matrix at several steps of the iterative process. For some of test functions considered this correction has allowed to reduce the number of computational iterations, but with a reduction in accuracy, and for some functions it showed its inapplicability. Thus, in practice the proposed algorithm with using the Hessian matrix update correction should take into account the specificity of the response function in each case.
Опис
Ключові слова
формы поверхностей отклика, нелинейное поведение, глобальная аппроксимация, квадратичная форма, machine engineering design, nonlinear behavior, optimization, response surface, global approximation
Бібліографічний опис
Бондаренко, М. А. Методы оптимизации с применением поверхностей отклика, адаптированные к решению задач анализа и синтеза конструктивных параметров тонкостенных машиностроительных конструкций / М. А. Бондаренко // Вісник Нац. техн. ун-ту "ХПІ" : зб. наук. пр. Сер. : Нові рішення в сучасних технологіях = Bulletin of National Technical University "KhPI" : coll. of sci. papers. Ser. : New solutions in modern technologies. – Харків : НТУ "ХПІ", 2016. – № 42 (1214). – С. 22-28.