Model order reduction of solidification problems

TL;DR

This paper explores the effectiveness of reduced order models in solidification problems, highlighting the importance of solution smoothness and the challenges posed by moving fronts in both 1D and 2D cases.

Contribution

It investigates the applicability of ROMs to solidification problems, focusing on the impact of solution smoothness and front movement on model accuracy.

Findings

Reduced spaces for moving 1D step functions are studied.

Comparison between 2D solidification and alloy mushy zone cases.

Solution smoothness significantly affects the decay of singular values.

Abstract

Advection driven problems are known to be difficult to model with a reduced basis because of a slow decay of the Kolmogorov $N$-width. This paper investigates how this challenge transfers to the context of solidification problems and tries to answer when and to what extend reduced order models (ROMs) work for solidification problems. In solidification problems, the challenge is not the advection per se, but rather a moving solidification front. This paper studies reduced spaces for 1D step functions that move in time, which can either be seen as advection of a quantity or as a moving solidification front. Furthermore, the reduced space of a 2D solidification test case is compared with the reduced space of an alloy solidification featuring a mushy zone. The results show that not only the PDE itself, but the smoothness of the solution is crucial for the decay of the singular values and thus the quality of a reduced space representation.