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 between material motion and mesh motion, enabling a unified treatment that bridges classical Lagrangian and Eulerian viewpoints. Starting from first principles, the full set of conservation laws (mass, momentum, and energy) is reformulated on moving domains, leading to the ALE Reynolds Transport Theorem and the corresponding governing equations expressed in terms of the relative (convective) velocity c = v − w, where v is the material velocity and w is the mesh velocity. Particular emphasis is placed on the Geometric Conservation Law (GCL), which provides a necessary consistency condition between mesh motion and volume 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 covers both major spatial discretization strategies for moving-domain problems. For finite element methods, the variational ALE form is derived, and Streamline-Upwind Petrov–Galerkin (SUPG) and Pressure-Stabilizing Petrov–Galerkin (PSPG) stabilizations are formulated in terms of the relative velocity. For finite volume methods, the ALE flux construction and donor-cell reconstruction strategies are presented alongside the DGCL-consistent time integration algorithm. Mesh motion strategies are surveyed, including spring analogy, Laplacian smoothing, and elastic analogy methods, together with mesh quality metrics that guide the choice of remeshing strategy. The report further interprets the ALE framework in the context of fluid–structure interaction, where boundary-fitted meshes enable high-fidelity resolution 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.