Тонкостінні конструкції: аналіз напружено-деформованого стану та обґрунтування параметрів

The approach is developed to substantiate technical solutions for thin-walled machine building structures. It implies that the problem is considered in the space of generalized parameters. These parameters combine design and technological factors, as well as operating conditions. In addition, we introduce criterial and constraint dependences to a given space. In the generated uniform parametric space an approximated response surface is constructed, which stretches over a discrete set of solutions to analysis problems. For example, based on the results of examining the stresses-strained state, maximum stresses or displacements, mass or other controlled magnitudes are determined. They are unambiguously computed (a point in a common parametric space) for a specific set of variable generalized parameters. Having a cloud of such points, it is possible to construct an approximated response surface. Approximation constraints are also built on it. Next, by using the methods of nonlinear programming, we search on the set of permissible values for the minimum (or maximum) of quality function of the examined structure.Specifically, for the thin-walled structures, important parameters are the shape and dimensions in a plan, as well as thickness of individual elements. Using a number of structures as examples, authors of present work performed analysis of influence of these parameters on the strength of designed structures.Разработан подход к обоснованию технических решений для тонкостенных машиностроительных конструкций. Задача рассматривается в пространстве обобщенных параметров, которые объединяют проектные и технологические факторы и условия эксплуатации. В сформированном параметрическом пространстве строится аппроксимированная поверхность отклика. В дополнение вводятся критериальные и ограничительные зависимости. После этого проводится поиск оптимума функции качества исследуемой конструкцииРозроблено підхід до обґрунтування технічних рішень для тонкостінних машинобудівних конструкцій. Задача розглядається у просторі узагальнених параметрів, які об’єднують проектні й технологічні чинники та умови експлуатації. У сформованому параметричному просторі будується апроксимована поверхня відгуку. На додаток вводяться критеріальні та обмежувальні залежності. Після цього проводиться пошук оптимуму функції якості досліджуваної конструкці

A Primal-Dual Augmented Lagrangian Penalty-Interior-Point Algorithm for Nonlinear Programming

This thesis treats a new numerical solution method for large-scale nonlinear optimization problems. Nonlinear programs occur in a wide range of engineering and academic applications like discretized optimal control processes and parameter identification of physical systems. The most efficient and robust solution approaches for this problem class have been shown to be sequential quadratic programming and primal-dual interior-point methods. The proposed algorithm combines a variant of the latter with a special penalty function to increase its robustness due to an automatic regularization of the nonlinear constraints caused by the penalty term. In detail, a modified barrier function and a primal-dual augmented Lagrangian approach with an exact l2-penalty is used. Both share the property that for certain Lagrangian multiplier estimates the barrier and penalty parameter do not have to converge to zero or diverge, respectively. This improves the conditioning of the internal linear equation systems near the optimal solution, handles rank-deficiency of the constraint derivatives for all non-feasible iterates and helps with identifying infeasible problem formulations. Although the resulting merit function is non-smooth, a certain step direction is a guaranteed descent. The algorithm includes an adaptive update strategy for the barrier and penalty parameters as well as the Lagrangian multiplier estimates based on a sensitivity analysis. Global convergence is proven to yield a first-order optimal solution, a certificate of infeasibility or a Fritz-John point and is maintained by combining the merit function with a filter or piecewise linear penalty function. Unlike the majority of filter methods, no separate feasibility restoration phase is required. For a fixed barrier parameter the method has a quadratic order of convergence. Furthermore, a sensitivity based iterative refinement strategy is developed to approximate the optimal solution of a parameter dependent nonlinear program under parameter changes. It exploits special sensitivity derivative approximations and converges locally with a linear convergence order to a feasible point that further satisfies the perturbed complementarity condition of the modified barrier method. Thereby, active-set changes from active to inactive can be handled. Due to a certain update of the Lagrangian multiplier estimate, the refinement is suitable in the context of warmstarting the penalty-interior-point approach. A special focus of the thesis is the development of an algorithm with excellent performance in practice. Details on an implementation of the proposed primal-dual penalty-interior-point algorithm in the nonlinear programming solver WORHP and a numerical study based on the CUTEst test collection is provided. The efficiency and robustness of the algorithm is further compared to state-of-the-art nonlinear programming solvers, in particular the interior-point solvers IPOPT and KNITRO as well as the sequential quadratic programming solvers SNOPT and WORHP