# Stability Analysis and Numerical Study of Stefan Problems for Embedded Computation of Moving Internal Boundaries

Tiffanie Carlier, Brian T Helenbrook, Heloise Beaugendre, Mathieu Colin

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Tiffanie Carlier, Brian T Helenbrook, Heloise Beaugendre, Mathieu Colin. Stability Analysis and Numerical Study of Stefan Problems for Embedded Computation of Moving Internal Boundaries. CFC 2023 - 22nd Computational Fluids Conference, Apr 2023, Cannes, France. hal-04359856

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Submitted on 21 Dec 2023

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# Stability Analysis and Numerical Study of Stefan Problems for Embedded Computation of Moving Internal Boundaries

T. Carlier*, B. T. Helenbrook†, H. Beaugendre⁺ and M.Colin⁺

* Univ. Bordeaux, INRIA, CNRS, Bordeaux INP, IMB, UMR 5251, 33405, Talence, France
e-mail: tiffanie.carlier@inria.fr

† Department of Mechanical and Aerospace Engineering, Clarkson University, Potsdam,
NY13699-5725, USA

⁺ INRIA, Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, 200 Avenue de la Vieille Tour,
33405 Talence, France

## ABSTRACT

Classical finite element methods used to model problems with internal boundaries rely on body fitted
computational grids. However, those methods encounter computational challenges when the boundaries
are deformed or moved substantially. Embedded methods avoid boundary fitted grids in favor of im-
mersing the boundary in a pre-existing fixed grid. In this category of methods we are interested in the
shifted boundary method, where the physical boundary is replaced by a surrogate boundary, which is
updated after each displacement of the physical boundary [1, 2]. Here, we examine the application of
this technique to a Stefan problem. In the formulation, the moving boundary progresses at a speed de-
termined by the normal flux jump and is a source of instabilities which can impact the solution on the
whole domain. To understand this, a linear stability analysis of the numerical technique is performed
[3]. The stability analysis gives an understanding of the terms necessary to add in the weak formulation
for its stabilization. A comparison is made with and without these terms to highlight the improvement in
accuracy of the method.

## REFERENCES

[1] L. Nouveau, M. Ricchiuto and G. Scovazzi, *High-Order Gradients with the Shifted Boundary Method: An Embedded Enriched Mixed Formulation for Elliptic PDEs*. Journal of Computational Physics. 398, 2019.

[2] K. Li, N. Atallah, A. Main, and G. Scovazzi, *The Shifted Interface Method: A Flexible Approach to Embedded Interface Computations*. International Journal for Numerical Methods in Engineering. 121, 2019.

[3] B. Helenbrook and N. Barlowb, *Spatial-temporal stability analysis of faceted growth with application to horizontal ribbon growth*. Journal of Crystal Growth. Volume 454, 2016.