Study of manifold interpolation techniques for acceleration of Stokes flow simulation

Abstract

This thesis investigates interpolation techniques for modeling the velocity field in the Lid-Driven Cavity with Stokes flow, inspired by the potential utility of these techniques in the aeronautical field to reduce computational costs. The aim is to compare and evaluate interpolation approaches for accurately predicting the velocity distribution during horizontal and vertical deformations. The study begins with one-parameter interpolation using right singular vectors. Singular value decomposition (SVD) is employed to decompose the data and approximate the velocity distribution for values of the horizontal elongation of the cavity which fall within the range of values considered during the study. This interpolation demonstrates to allow extrapolation within a limited scope. Building upon the insights gained, the research progresses to two-parameter interpolation using right singular vectors, considering both horizontal and vertical deformations. However, as deformations increase, accuracy significantly diminishes, raising concerns about result reliability. Interpolation using right singular vectors lacks compliance with the fundamental Navier-Stokes equations, so this thesis explores alternative approaches. Specifically, Model Order Reduction (MOR) is introduced as a means to reduce computational costs. It is important to note that MOR for the Navier-Stokes equations is not performed in this study. Instead, the focus is on investigating interpolation techniques that generate matrices of basis vectors, which play a key role in MOR. These matrices, if used in future MOR investigations, can facilitate estimating the resultant velocity field by interpolating parameter values. The interpolation technique using principal angles enables computation of the subspace matrix containing velocity field results, demonstrating sufficient accuracy through error analysis. However, the method has limitations in extrapolating beyond the studied deformation range and studying interpolation with two parameters. To overcome these challenges, the thesis explores the use of the Grassmann manifold for interpolation. The Grassmann manifold enables interpolation of the subspace matrix, providing a comprehensive understanding of velocity field behavior with low error and improved robustness compared to principal angles. However, it should be noted that this technique has limitations concerning the distribution of the studied parameter values, requiring them to be evenly spaced. Additionally, extrapolation is not recommended when using this type of interpolation. Comparing and evaluating these interpolation techniques yields valuable insights into their performance and limitations, highlighting the importance of accuracy, reliability, and the ability to handle different deformation scenarios when selecting the appropriate method. In conclusion, this thesis offers a comprehensive evaluation of interpolation techniques for modeling the velocity field in the Lid-Driven Cavity flow, driven by the potential utility of these techniques in the aeronautical field to reduce computational costs. It provides guidance for selecting suitable interpolation methods based on specific problem requirements and suggests avenues for future investigations, including exploring machine learning techniques.