Arbitrary Lagrangian–Eulerian (ALE) Formulation: Theory, Numerical Methods, and Applications

Report, 2026

This report presents a systematic formulation of the Arbitrary Lagrangian–Eulerian (ALE) framework for continuum mechanics problems involving time- dependent domains. Such problems arise naturally in fluid–structure interaction (FSI), free-surface flows, and other applications where the computational domain evolves as part of the solution. The central challenge is the consistent formulation of conservation laws when the domain Ω(t) is itself a function of time. To address this, the ALE description introduces a kinematic decoupling be- tween material motion and mesh motion, enabling a unified treatment that bridges classical Lagrangian and Eulerian viewpoints. Starting from first princi- ples, conservation laws are reformulated on moving domains, leading to the ALE Reynolds Transport Theorem and the corresponding Navier–Stokes equations expressed in terms of the relative (convective) velocity.
Particular emphasis is placed on the Geometric Conservation Law (GCL), which provides a necessary consistency condition between mesh motion and vol- ume evolution to ensure numerical stability and accuracy. Both continuous and discrete forms of the GCL are discussed, highlighting their role in preventing spurious sources in moving-mesh simulations. The report further interprets the ALE framework in the context of fluid– structure interaction, where boundary-fitted meshes enable high-fidelity resolu- tion of interface physics. Finally, the advantages and limitations of ALE are critically assessed, with comparisons to alternative approaches such as fixed-grid and immersed methods, providing guidance on the appropriate use of ALE in practical simulations.