Two-phase flows simulation using the X-Mesh approach
(2026)
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- Abstract
- Two-phase flows with sharp interfaces arise in many natural and industrial processes, yet their numerical simulation remains challenging due to the complex dynamics of moving interfaces, topological changes, and discontinuities in material properties. Classical numerical approaches face intrinsic limitations: interface-capturing methods suffer from numerical diffusion and poor enforcement of interfacial conditions, while interface-tracking methods such as Arbitrary Lagrangian–Eulerian (ALE) formulations struggle with mesh distortion and remeshing, which compromises time continuity and conservation properties. This thesis use and develops the recently proposed X-Mesh approach, a novel finite element framework designed to overcome these limitations. The key idea of X-Mesh is to allow extreme mesh deformations, including the appearance of degenerate and zero-measure elements, while preserving a fixed mesh topology and a classical finite element approximation. This paradigm enables continuous interface tracking, sharp interface representation, and natural handling of topological changes without resorting to remeshing or enrichment techniques. A theoretical and numerical investigation of the impact of mesh degeneration on finite element convergence is conducted, leading to the formulation of the Tempered Finite Element Method (TFEM), which stabilizes computations on highly distorted meshes. Algorithms for interface representation and mesh deformation are developed, using level-set and front-tracking strategies within the X-Mesh framework. The proposed methodology is applied to incompressible two-phase flow problems governed by the Navier–Stokes equations, including surface tension effects and jump conditions at the interface. Extensive numerical experiments demonstrate that X-Mesh achieves accurate, stable, and high-order solutions while maintaining mesh quality and time continuity, outperforming classical ALE and interface-capturing approaches in challenging scenarios involving large interface deformations and topological transitions. Overall, this work establishes X-Mesh as a robust and versatile framework for sharp-interface multiphase flow simulations and opens new perspectives for the use of degenerate meshes in finite element methods.
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Quiriny, A. (2026). Two-phase flows simulation using the X-Mesh approach. https://hdl.handle.net/2078.5/273127