Reduced order models for Lagrangian hydrodynamics

TL;DR

This paper develops advanced projection-based reduced order models for Lagrangian hydrodynamics, addressing challenges like moving meshes and shock fronts, and introduces hyper-reduction and error bounds for improved efficiency and accuracy.

Contribution

It introduces novel reduced basis techniques, time-windowing, and hyper-reduction methods tailored for Lagrangian hydrodynamics, enhancing computational efficiency and accuracy.

Findings

Reduced models achieve significant speed-up over full models.

Time-windowing improves accuracy in advection-dominated regimes.

Hyper-reduction techniques effectively handle nonlinearities.

Abstract

As a mathematical model of high-speed flow and shock wave propagation in a complex multimaterial setting, Lagrangian hydrodynamics is characterized by moving meshes, advection-dominated solutions, and moving shock fronts with sharp gradients. These challenges hinder the existing projection-based model reduction schemes from being practical. We develop several variations of projection-based reduced order model techniques for Lagrangian hydrodynamics by introducing three different reduced bases for position, velocity, and energy fields. A time-windowing approach is also developed to address the challenge imposed by the advection-dominated solutions. Lagrangian hydrodynamics is formulated as a nonlinear problem, which requires a proper hyper-reduction technique. Therefore, we apply the over-sampling DEIM and SNS approaches to reduce the complexity due to the nonlinear terms. Finally, we also present both a posteriori and a priori error bounds associated with our reduced order model. We compare the performance of the spatial and time-windowing reduced order modeling approaches in terms of accuracy and speed-up with respect to the corresponding full order model for several numerical examples, namely Sedov blast, Gresho vortices, Taylor-Green vortices, and triple-point problems.