Author : Thomas Nguyen (Graduate student)
Publisher :
Page : 0 pages
File Size : 10,76 MB
Release : 2022
Category :
ISBN :
Different methods have been developed to solve the Boltzmann equation during the past decades: the direct simulation Monte Carlo method, the lattice Boltzmann method, and the direct deterministic methods for computing the Boltzmann equation. However, computational costs of the existing methods are still prohibitive for simulating complex flows in three dimensions and flows of multi-component gases with real gas effects. Methods of increased efficiency need to be proposed in order to continue advancement in these areas. In this thesis, we explore use of neural networks for solving the Boltzmann equation for a class of problems of spatially homogeneous relaxation of sums of two Maxwellian streams. The data set for training the neural networks is generated by solving the Boltzmann equation using classical methods. We consider applications of deep autoencoder to learn a compressed representation of the solution dataset and to filtering of truncation errors in numerical solutions. The Boltzmann collision operator is approximated using deep convolutional neural networks (CNNs). Accuracy of the trained autoencoders and CNNs was investigated. We use the trained CNNs and Euler method to numerically solve the spatially homogeneous Boltzmann equation. The results are compared to solutions obtained by deterministic solvers. The solutions obtained by CNNs showed good agreement with the results obtained by classical methods while providing at least three orders of magnitude acceleration. The computer memory requirements were found to be comparable to requirements of the classical methods. Small violations of conservation of mass and energy are observed as solution are reaching the steady state. Additionally, the solutions appear to be not stable on an infinite time interval. However, both issues can be corrected using established numerical methods for kinetic equations.