Calibration-Based ALE Model Order Reduction for Hyperbolic Problems with Self-Similar Travelling Discontinuities

TL;DR

The paper tackles model order reduction for hyperbolic PDEs with multiple moving shocks by introducing a calibration-based map $T[\bm{w}(\bm{\mu})]$ built from control points $\bm{w}(\bm{\mu})$ and learned via an ANN to predict reduced coefficients. It handles (quasi) self-similar solutions through two pathways: direct self-similar calibration of $\bm{\theta}$ and quasi-self-similar calibration via a POD-projected residual onto $V_{POD}$. Numerical results on 1D Sod, 2D double Mach reflection, and a triple-point problem show faster POD eigenvalue decay, reduced online errors with a small basis, and a fully non-intrusive online reconstruction, avoiding explicit shock detectors. The approach extends MOR to transport-dominated hyperbolic problems with multiple moving discontinuities and demonstrates robustness to reference choices, albeit within rectangular domains; future work includes non-rectangular geometries and hyper-reduced schemes for improved scalability.

Abstract

We propose a novel Model Order Reduction framework that is able to handle solutions of hyperbolic problems characterized by multiple travelling discontinuities. By means of an optimization based approach, we introduce suitable calibration maps that allow us to transform the original solution manifold into a lower dimensional one. The novelty of the methodology is represented by the fact that the optimization process does not require the knowledge of the discontinuities location. The optimization can be carried out simply by choosing some reference control points, thus avoiding the use of some implicit shock tracking techniques, which would translate into an increased computational effort during the offline phase. In the online phase, we rely on a non-intrusive approach, where the coefficients of the projection of the reduced order solution onto the reduced space are recovered by means of an Artificial Neural Network. To validate the methodology, we present numerical results for the 1D Sod shock tube problem, for the 2D double Mach reflection problem, also in the parametric case, and for the triple point problem.

Figures (24)

• Figure 1: The density $\rho$, solution of the Sod shock tube problem at different timesteps, and the corresponding POD modes (right).
• Figure 1: Example of PCHIP calibration: solution on physical domain with calibration control points (left), solution on reference domain with reference control points (center) and PCHIP transformation with all control points (right)
• Figure 1: Example of the architecture of an ANN
• Figure 1: FOM simulation of Sod 1D problem non parameteric at times 0.04 (green), 0.1 (orange) and 0.16 (blue), on the original domain (top) and calibrated on the reference domain (bottom)
• Figure 2: Example of a control point grid in 2D with $M_1=3$ and $M_2=4$ on the reference domain (left) and on the physical domain (right). Note that the coordinates of the reference control points are the tensor product of unidirectional control points. We highlight in red the free coordinates of these control points

Theorems & Definitions (1)

• Definition 1.1