title: ÐÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸Ðµ Ð¼ÐµÑÐ¾Ð´Ð° Ð¿ÑÐ¾ÐµÐºÑÐ¸Ð¸ Ð³ÑÐ°Ð´Ð¸ÐµÐ½ÑÐ° Ð´Ð»Ñ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ Ð¿ÑÐ¾ÑÐµÑÑÐ° ÑÐ±Ð¾ÑÐºÐ¸ Ð´ÐµÑÐ¾ÑÐ¼Ð¸ÑÑÐµÐ¼ÑÑ ÐºÐ¾Ð½ÑÑÑÑÐºÑÐ¸Ð¹
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¼Ð°Ð³Ð¸ÑÑÑÐ°
titleType: Subtitle
title: Gradient projection method as applied to simulation of assembly process of compliant structures
Creators: name: ÐÐ°ÐºÐ»Ð°Ð½Ð¾Ð², Ð¡ÑÐ°Ð½Ð¸ÑÐ»Ð°Ð²
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2019
Subjects: ÐÑÐ¿ÑÐºÐ»Ð¾Ðµ Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ
ÐÐµÐ»Ð¸Ð½ÐµÐ¹Ð½Ð¾Ðµ Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ
ÐÑÐµÑÐ°ÑÐ¸Ñ
Ð¼ÐµÑÐ¾Ð´ Ð¿ÑÐ¾ÐµÐºÑÐ¸Ð¸ Ð³ÑÐ°Ð´Ð¸ÐµÐ½ÑÐ°
Ð·Ð°Ð´Ð°ÑÐ° ÑÐ±Ð¾ÑÐºÐ¸
Ð½ÐµÐ³Ð»Ð°Ð´ÐºÐ°Ñ ÑÐµÐ»ÐµÐ²Ð°Ñ ÑÑÐ½ÐºÑÐ¸Ñ
Ð´Ð²Ð¾Ð¹ÑÑÐ²ÐµÐ½Ð½Ð¾ÑÑÑ
ÑÑÐ±Ð³ÑÐ°Ð´Ð¸ÐµÐ½Ñ
ÑÐ°Ð·Ð»Ð¾Ð¶ÐµÐ½Ð¸Ðµ Ð¥Ð¾Ð»ÐµÑÐºÐ¾Ð³Ð¾
Ð¼ÐµÑÐ¾Ð´ ÑÐ¾Ð¿ÑÑÐ¶ÑÐ½Ð½ÑÑ Ð³ÑÐ°Ð´Ð¸ÐµÐ½ÑÐ¾Ð²
Ð¼ÐµÑÐ¾Ð´ Ð·Ð¾Ð»Ð¾ÑÐ¾Ð³Ð¾ ÑÐµÑÐµÐ½Ð¸Ñ
Ð¾ÑÐ¸Ð±ÐºÐ¸ Ð¾ÐºÑÑÐ³Ð»ÐµÐ½Ð¸Ñ

description: Ð Ð´Ð°Ð½Ð½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÐµ ÑÐ°ÑÑÐ¼Ð°ÑÑÐ¸Ð²Ð°ÐµÑÑÑ Ð¼ÐµÑÐ¾Ð´ Ð¿ÑÐ¾ÐµÐºÑÐ¸Ð¸ Ð³ÑÐ°Ð´Ð¸ÐµÐ½ÑÐ° Ð´Ð»Ñ ÑÐµÑÐµÐ½Ð¸Ñ Ð·Ð°Ð´Ð°ÑÐ¸ ÐºÐ²Ð°Ð´ÑÐ°ÑÐ¸ÑÐ½Ð¾Ð³Ð¾ Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ, Ð²Ð¾Ð·Ð½Ð¸ÐºÐ°ÑÑÐµÐ¹ Ð¿ÑÐ¸ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ð¸ Ð¿ÑÐ¾ÑÐµÑÑÐ° ÑÐ±Ð¾ÑÐºÐ¸. ÐÐ°Ð½Ð½Ð°Ñ Ð·Ð°Ð´Ð°ÑÐ° Ð¾Ð±Ð»Ð°Ð´Ð°ÐµÑ ÑÑÐ´Ð¾Ð¼ Ð¾ÑÐ¾Ð±ÐµÐ½Ð½Ð¾ÑÑÐµÐ¹: ÐÐµÑÑÐ¸Ð°Ð½ ÑÐµÐ»ÐµÐ²Ð¾Ð¹ ÑÑÐ½ÐºÑÐ¸Ð¸ Ð¿Ð»Ð¾ÑÐ¾ Ð¾Ð±ÑÑÐ»Ð¾Ð²Ð»ÐµÐ½, Ð¼Ð°ÑÑÐ¸ÑÐ° Ð¾Ð³ÑÐ°Ð½Ð¸ÑÐµÐ½Ð¸Ð¹ ÑÐ°Ð·ÑÐµÐ¶ÐµÐ½Ð°, ÑÐ°ÐºÐ¶Ðµ, ÐºÐ°Ðº Ð¿ÑÐ°Ð²Ð¸Ð»Ð¾, ÑÑÐµÐ±ÑÐµÑÑÑ Ð¿ÑÐ¾Ð¸Ð·Ð²Ð¾Ð´Ð¸ÑÑ Ð¼Ð°ÑÑÐ¾Ð²ÑÐµ ÑÐ°ÑÑÑÑÑ Ð´Ð»Ñ Ð¿Ð¾Ð´Ð¾Ð±Ð½ÑÑ Ð·Ð°Ð´Ð°Ñ ÐºÐ²Ð°Ð´ÑÐ°ÑÐ¸ÑÐ½Ð¾Ð³Ð¾ Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ. ÐÐ»Ñ ÑÐµÑÐµÐ½Ð¸Ñ Ð·Ð°Ð´Ð°ÑÐ¸ Ð¿ÑÐ¸Ð¼ÐµÐ½ÑÐµÑÑÑ Ð¿ÑÐ¾ÐµÐºÑÐ¸Ð¾Ð½Ð½ÑÐ¹ Ð¼ÐµÑÐ¾Ð´ Ñ Ð¼Ð¾Ð´Ð¸ÑÐ¸ÐºÐ°ÑÐ¸ÐµÐ¹ ÐÑÑÑÐ¾Ð½Ð° Ñ ÐºÐ²Ð°Ð´ÑÐ°ÑÐ¸ÑÐ½Ð¾Ð¹ ÑÐºÐ¾ÑÐ¾ÑÑÑÑ ÑÑÐ¾Ð´Ð¸Ð¼Ð¾ÑÑÐ¸, Ð¸ Ð¿ÑÐµÐ´Ð»Ð°Ð³Ð°ÑÑÑÑ Ð½ÐµÑÐºÐ¾Ð»ÑÐºÐ¾ Ð¼Ð¾Ð´Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¹ Ð´Ð»Ñ ÑÐ¼ÐµÐ½ÑÑÐµÐ½Ð¸Ñ Ð²ÑÐµÐ¼ÐµÐ½Ð¸ ÑÐµÑÐµÐ½Ð¸Ñ: Ð¼ÐµÑÐ¾Ð´ ÑÐµÑÐµÐ½Ð¸Ñ ÑÐ¸ÑÑÐµÐ¼Ñ Ð»Ð¸Ð½ÐµÐ¹Ð½ÑÑ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ð¹, Ð¼ÐµÑÐ¾Ð´ Ð·Ð¾Ð»Ð¾ÑÐ¾Ð³Ð¾ ÑÐµÑÐµÐ½Ð¸Ñ Ð´Ð»Ñ Ð²ÑÐ±Ð¾ÑÐ° ÑÐ°Ð·Ð¼ÐµÑÐ° ÑÐ°Ð³Ð° Ð¸ ÑÐ°Ðº Ð½Ð°Ð·ÑÐ²Ð°ÐµÐ¼ÑÐ¹ Ð¼ÐµÑÐ¾Ð´ Ð¿ÐµÑÐµÑÑÑÑÐ° Ð¾Ð³ÑÐ°Ð½Ð¸ÑÐµÐ½Ð¸Ð¹. Ð¢Ð°ÐºÐ¶Ðµ Ñ Ð¿Ð¾Ð¼Ð¾ÑÑÑ Ð¿ÑÐ¸Ð½ÑÐ¸Ð¿Ð° Ð´Ð²Ð¾Ð¹ÑÑÐ²ÐµÐ½Ð½Ð¾ÑÑÐ¸ ÑÐ¾ÑÐ¼ÑÐ»Ð¸ÑÑÑÑÑÑ Ð°Ð»ÑÑÐµÑÐ½Ð°ÑÐ¸Ð²Ð½ÑÐµ Ð·Ð°Ð´Ð°ÑÐ¸ ÐºÐ²Ð°Ð´ÑÐ°ÑÐ¸ÑÐ½Ð¾Ð³Ð¾ Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ, ÐºÐ¾ÑÐ¾ÑÑÐµ Ð¾Ð±ÑÑÐ½Ð¾ Ð¼Ð¾Ð³ÑÑ Ð±ÑÑÑ ÑÐµÑÐµÐ½Ñ Ð±ÑÑÑÑÐµÐµ. ÐÑÐ¸ Ð¿Ð¾ÑÑÑÐ¾ÐµÐ½Ð¸Ð¸ Ð´Ð°Ð½Ð½ÑÑ Ð·Ð°Ð´Ð°Ñ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·ÑÐµÑÑÑ Ð¼ÐµÑÐ¾Ð´, Ð¿Ð¾Ð·Ð²Ð¾Ð»ÑÑÑÐ¸Ð¹ ÑÐ¼ÐµÐ½ÑÑÐ¸ÑÑ Ð²ÑÑÐ¸ÑÐ»Ð¸ÑÐµÐ»ÑÐ½ÑÐµ Ð¾ÑÐ¸Ð±ÐºÐ¸. ÐÐ¿Ð¸ÑÑÐ²Ð°ÐµÑÑÑ Ð¼Ð¾Ð´Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ñ Ð¼ÐµÑÐ¾Ð´Ð° Ð¿ÑÐ¾ÐµÐºÑÐ¸Ð¸ Ð³ÑÐ°Ð´Ð¸ÐµÐ½ÑÐ° Ñ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð½Ð¸ÐµÐ¼ ÑÑÐ±Ð³ÑÐ°Ð´Ð¸ÐµÐ½ÑÐ° Ð´Ð»Ñ ÑÐµÑÐµÐ½Ð¸Ñ Ð²Ð¾Ð·Ð½Ð¸ÐºÐ°ÑÑÐµÐ¹ Ð² Ð¿ÑÐ¾ÑÐµÑÑÐµ ÑÐ±Ð¾ÑÐºÐ¸ Ð·Ð°Ð´Ð°ÑÐ¸ Ð½ÐµÐ»Ð¸Ð½ÐµÐ¹Ð½Ð¾Ð³Ð¾ Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ Ñ Ð½ÐµÐ³Ð»Ð°Ð´ÐºÐ¾Ð¹ ÑÐµÐ»ÐµÐ²Ð¾Ð¹ ÑÑÐ½ÐºÑÐ¸ÐµÐ¹ Ð¸ Ð´Ð¾ÐºÐ°Ð·ÑÐ²Ð°ÐµÑÑÑ ÑÑÐ¾Ð´Ð¸Ð¼Ð¾ÑÑÑ Ð¿ÑÐµÐ´ÑÑÐ°Ð²Ð»ÐµÐ½Ð½Ð¾Ð³Ð¾ Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼Ð°. ÐÐ°ÐºÐ¾Ð½ÐµÑ, Ð¼ÐµÑÐ¾Ð´ Ð¿ÑÐ¾ÐµÐºÑÐ¸Ð¸ Ð³ÑÐ°Ð´Ð¸ÐµÐ½ÑÐ° ÑÑÐ°Ð²Ð½Ð¸Ð²Ð°ÐµÑÑÑ Ñ Ð´ÑÑÐ³Ð¸Ð¼Ð¸ Ð¼ÐµÑÐ¾Ð´Ð°Ð¼Ð¸ ÐºÐ²Ð°Ð´ÑÐ°ÑÐ¸ÑÐ½Ð¾Ð³Ð¾ Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ.
description: In this work the gradient projection method for solving the quadratic programming problem which emerges in the assembly simulation of riveting process is considered. This particular class of problems has specific features such as ill-conditioned Hessian and a sparse matrix of constraints as well as a requirement for the massive solving of similar quadratic programming problems. A projected Newton method with the quadratic rage of convergence is used and some improvements to reduce the solving time are suggested: a method for solving the system of linear equations, a golden section method for a step size selection, and a so-called constraint recalculation method. Also the duality principle is used to formulate alternative forms of the minimization problem which, as a rule, can be solved faster. A method for the accurate calculation of data to prevent machine errors is presented. It is described how to solve the arising in assembly simulation nonlinear minimization problem with the nonsmooth objective function by modifying the gradient projection method employing subgradients. In addition, the convergence of the suggested algorithm is proved. Finally, the gradient projection method is compared with the other quadratic programming techniques on a number of assembly simulation problems.