title: ÐÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸Ðµ Ð¼ÐµÑÐ¾Ð´Ð° ÑÐ¸Ð½Ð³ÑÐ»ÑÑÐ½Ð¾Ð³Ð¾ ÑÐ°Ð·Ð»Ð¾Ð¶ÐµÐ½Ð¸Ñ Ð´Ð»Ñ Ð¿Ð¾ÑÑÑÐ¾ÐµÐ½Ð¸Ñ ROM-Ð¼Ð¾Ð´ÐµÐ»ÐµÐ¹
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð±Ð°ÐºÐ°Ð»Ð°Ð²ÑÐ°
titleType: Subtitle
title: Application of singular value decomposition for constructing rom models
Creators: name: Ð¥Ð¾ÑÐ¸ÐµÐ½ÐºÐ¾Ð², ÐÐ°Ð½Ð¸Ð»
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: 2024
Subjects: svd ÑÐ°Ð·Ð»Ð¾Ð¶ÐµÐ½Ð¸Ðµ
ÑÐ¸Ð½Ð³ÑÐ»ÑÑÐ½Ð¾Ðµ ÑÐ°Ð·Ð»Ð¾Ð¶ÐµÐ½Ð¸Ðµ Ð¼Ð°ÑÑÐ¸ÑÑ
rom-Ð¼Ð¾Ð´ÐµÐ»Ñ
Ð¼Ð°ÑÑÐ¸ÑÐ° ÑÐ¾ÑÑÐ¾ÑÐ½Ð¸Ñ
Ð¼Ð°ÑÑÐ¸ÑÐ° Ð»ÐµÐ²ÑÑ ÑÐ¸Ð½Ð³ÑÐ»ÑÑÐ½ÑÑ Ð²ÐµÐºÑÐ¾ÑÐ¾Ð²
Ð¼Ð°ÑÑÐ¸ÑÐ° Ð¿ÑÐ°Ð²ÑÑ ÑÐ¸Ð½Ð³ÑÐ»ÑÑÐ½ÑÑ Ð²ÐµÐºÑÐ¾ÑÐ¾Ð²
rbf Ð¸Ð½ÑÐµÑÐ¿Ð¾Ð»ÑÑÐ¸Ñ
Ð½ÐµÐ¹ÑÐ¾Ð½Ð½Ð°Ñ ÑÐµÑÑ
Ð¼Ð°ÑÐ¸Ð½Ð½Ð¾Ðµ Ð¾Ð±ÑÑÐµÐ½Ð¸Ðµ
svd decomposition
singular value decomposition
rom model
state matrix
matrix of left singular vectors
matrix of right singular vectors
rbf interpolation
neural network
machine learning

description: Ð Ð´Ð°Ð½Ð½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÐµ Ð¸Ð·Ð»Ð¾Ð¶ÐµÐ½ Ð¿Ð¾Ð´ÑÐ¾Ð´ ÑÐ½Ð¸Ð¶ÐµÐ½Ð¸Ñ ÑÐ°Ð·Ð¼ÐµÑÐ½Ð¾ÑÑÐ¸ Ð·Ð°Ð´Ð°ÑÐ¸ Ð¿Ð¾ ÑÑÐµÐ´ÑÑÐ²Ð°Ð¼ ÑÐ¸Ð½Ð³ÑÐ»ÑÑÐ½Ð¾Ð³Ð¾ ÑÐ°Ð·Ð»Ð¾Ð¶ÐµÐ½Ð¸Ñ Ð¼Ð°ÑÑÐ¸ÑÑ ÑÐ¾ÑÑÐ¾ÑÐ½Ð¸Ñ, Ð¿Ð¾Ð»ÑÑÐµÐ½Ð½Ð¾Ð¹ Ñ Ð¿Ð¾Ð¼Ð¾ÑÑÑ Ð¼ÐµÑÐ¾Ð´Ð° ÐºÐ¾Ð½ÐµÑÐ½ÑÑ ÑÐ»ÐµÐ¼ÐµÐ½ÑÐ¾Ð². Ð¢Ð°ÐºÐ¶Ðµ Ð±ÑÐ»Ð¾ Ð¿ÑÐ¾Ð²ÐµÐ´ÐµÐ½Ð¾ ÑÐ½Ð¸Ð¶ÐµÐ½Ð¸Ðµ ÑÐ°Ð·Ð¼ÐµÑÐ½Ð¾ÑÑÐ¸ Ð·Ð°Ð´Ð°ÑÐ¸ Ñ Ð¿Ð¾Ð¼Ð¾ÑÑÑ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½Ð¾Ð¹ ÑÐµÑÐ¸.Â  ÐÐ°Ð´Ð°ÑÐ¸, ÐºÐ¾ÑÐ¾ÑÑÐµ ÑÐµÑÐ°Ð»Ð¸ÑÑ Ð² ÑÐ¾Ð´Ðµ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ: 1. ÐÐ°Ð¿Ð¸ÑÐ°Ð½Ð¸Ðµ Ð¸ Ð¾Ð±ÑÑÐµÐ½Ð¸Ðµ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½Ð¾Ð¹ ÑÐµÑÐ¸, Ð¿Ð¾Ð·Ð²Ð¾Ð»ÑÑÑÐµÐ¹ Ð¿ÑÐµÐ´ÑÐºÐ°Ð·ÑÐ²Ð°ÑÑ ÑÐ°ÑÐ¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð¸Ðµ Ð¿Ð¾Ð»Ñ ÑÐµÐ¼Ð¿ÐµÑÐ°ÑÑÑ Ð´Ð»Ñ ÑÐ°Ð·Ð½ÑÑ Ð²ÑÐ¾Ð´Ð½ÑÑ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ¾Ð² ÑÐ¾ÑÑÐ¾ÑÐ½Ð¸Ñ, Ð° ÑÐ°ÐºÐ¶Ðµ ÑÐ¼ÐµÐ½ÑÑÐ°ÑÑ ÑÐ°Ð·Ð¼ÐµÑÐ½Ð¾ÑÑÑ Ð·Ð°Ð´Ð°ÑÐ¸. 2. Ð¡Ð¾ÑÑÐ°Ð²Ð»ÐµÐ½Ð¸Ðµ Ð¼Ð°ÑÑÐ¸ÑÑ ÑÐ¾ÑÑÐ¾ÑÐ½Ð¸Ñ Ð´Ð»Ñ Ð·Ð°Ð´Ð°ÑÐ¸ ÑÐµÐ¿Ð»Ð¾Ð¿ÑÐ¾Ð²Ð¾Ð´Ð½Ð¾ÑÑÐ¸ Ð¸ Ð¿Ð¾ÑÐ»ÐµÐ´ÑÑÑÐµÐµ ÑÐ¸Ð½Ð³ÑÐ»ÑÑÐ½Ð¾Ðµ ÑÐ°Ð·Ð»Ð¾Ð¶ÐµÐ½Ð¸Ðµ Ð¿Ð¾Ð»ÑÑÐµÐ½Ð½Ð¾Ð¹ Ð¼Ð°ÑÑÐ¸ÑÑ. 3. Ð¡Ð½Ð¸Ð¶ÐµÐ½Ð¸Ðµ ÑÐ°Ð·Ð¼ÐµÑÐ½Ð¾ÑÑÐ¸ Ð·Ð°Ð´Ð°ÑÐ¸ Ð¿Ð¾ ÑÑÐµÐ´ÑÑÐ²Ð°Ð¼ Ð¸Ð½ÑÐµÑÐ¿Ð¾Ð»ÑÑÐ¸Ð¸ ÑÐ°Ð´Ð¸Ð°Ð»ÑÐ½Ð¾Ð¹ Ð±Ð°Ð·Ð¸ÑÐ½Ð¾Ð¹ ÑÑÐ½ÐºÑÐ¸Ð¸. 4. ÐÑÐµÐ½ÐºÐ° Ð¿Ð¾Ð»ÑÑÐµÐ½Ð½ÑÑ Ð¿Ð¾Ð»ÐµÐ¹ ÑÐµÐ¼Ð¿ÐµÑÐ°ÑÑÑ Ð¿Ð¾ÑÐ»Ðµ ÑÐ½Ð¸Ð¶ÐµÐ½Ð¸Ñ ÑÐ°Ð·Ð¼ÐµÑÐ½Ð¾ÑÑÐ¸ Ð·Ð°Ð´Ð°ÑÐ¸. Ð Ð°Ð±Ð¾ÑÐ° Ð¿ÑÐ¾Ð²ÐµÐ´ÐµÐ½Ð° Ð½Ð° Ð±Ð°Ð·Ðµ Ð´Ð°Ð½Ð½ÑÑ, Ð¿Ð¾Ð»ÑÑÐµÐ½Ð½ÑÑ Ñ Ð¿Ð¾Ð¼Ð¾ÑÑÑ ÐºÐ¾Ð½ÐµÑÐ½Ð¾-ÑÐ»ÐµÐ¼ÐµÐ½ÑÐ½Ð¾Ð³Ð¾ ÑÐµÑÐµÐ½Ð¸Ñ Ð·Ð°Ð´Ð°ÑÐ¸ ÑÐµÐ¿Ð»Ð¾Ð¿ÑÐ¾Ð²Ð¾Ð´Ð½Ð¾ÑÑÐ¸ Ð² Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð½Ð¾Ð¼ ÐºÐ¾Ð¼Ð¿Ð»ÐµÐºÑÐµ ANSYS Workbench, ÐºÐ¾ÑÐ¾ÑÑÐµ Ð±ÑÐ»Ð¸ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð½Ñ Ð² ÐºÐ°ÑÐµÑÑÐ²Ðµ Ð¾Ð±ÑÑÐ°ÑÑÐµÐ¹ Ð²ÑÐ±Ð¾ÑÐºÐ¸ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½Ð¾Ð¹ ÑÐµÑÐ¸, Ð° ÑÐ°ÐºÐ¶Ðµ Ð´Ð»Ñ ÑÐ¾ÑÑÐ°Ð²Ð»ÐµÐ½Ð¸Ñ Ð¼Ð°ÑÑÐ¸ÑÑ ÑÐ¾ÑÑÐ¾ÑÐ½Ð¸Ñ Ð´Ð»Ñ SVD ÑÐ°Ð·Ð»Ð¾Ð¶ÐµÐ½Ð¸Ñ.
description: This work presents an approach to reducing the dimensionality of a problem using singular value decomposition of the state matrix obtained through the finite element method. Dimensionality reduction was also performed using a neural network. Tasks that were solved during the study: 1. Writing and training a neural network to predict the temperature field distribution for different input state parameters and to reduce the dimensionality of the problem. 2. Composing a state matrix for the heat conduction problem and subsequently performing singular value decomposition of the obtained matrix. 3. Reducing the dimensionality of the problem using radial basis function interpolation. 4. Evaluating the obtained temperature fields after dimensionality reduction. The work was conducted using data obtained from the finite element solution of the heat conduction problem in the ANSYS Workbench software suite. These data were used as the training set for the neural network and for composing the state matrix for SVD decomposition.