Arbitrary Lagrangian–Eulerian Formulation (ALE)
The partial differential equations of physics are usually formulated either in a spatial coordinate system, with coordinate axes fixed in space, or in a material coordinate system, fixed to the material in its reference configuration and following the material as it deforms. The former is often referred to as an Eulerian formulation, while the latter is a Lagrangian formulation.
Structural mechanics and other fields of physics dealing with a possibly anisotropic, solid material are most conveniently simulated using material coordinates. The Lagrangian formulation makes the anisotropic material properties independent of the current spatial orientation of the material.
However, if the focus is on simulating the physical state at fixed points in space, an Eulerian formulation is usually more convenient. In particular, when liquids and gases are involved, it is often unreasonable to track the state of individual material particles. Rather, the quantities of interest are pressure, temperature, concentration, and so forth, at fixed positions in space.
An inherent problem with the pure Eulerian formulation is that it cannot handle moving domain boundaries, since physical quantities are referred to fixed points in space, while the set of spatial points inside the domain boundaries changes with time. Therefore, to allow moving boundaries, the Eulerian equations must be rewritten so as to describe all physical quantities as functions of some coordinate system in which the domain boundaries are fixed. The finite element mesh offers one such system: the mesh coordinates.
In the mesh coordinate system, the domain is fixed, and there is a one-to-one map from the mesh coordinates to the current spatial configuration of the domain. Otherwise, the mesh coordinate system can be defined freely and separately from both the spatial and material systems. The natural choice is to let the mesh coordinate system, at least initially, coincide with the geometry coordinates. This follows immediately from the way meshes are created and means that points in the domain are identified by their position in the original geometry.
As the domain and mesh deform, the map from mesh coordinates to spatial coordinates can become increasingly ill-conditioned. Before the degradation of the mesh mapping goes too far, you can, using a remeshing operation, stop the simulation, create a new mesh in the current configuration of the domain, and map all quantities to the new mesh. When you restart the simulation, points in the domain are internally identified by their new mesh coordinates, which coincide with the spatial coordinates at the state where the simulation was stopped. Therefore, the geometry and mesh coordinates of a given point differ after remeshing the deformed geometry.
Rewriting physics equations in this way, on a freely moving mesh, results in an arbitrary Lagrangian–Eulerian (ALE) method. In the special case when the map from mesh coordinates to spatial coordinates follows the material deformation, a Lagrangian method is recovered. Similarly, when the map is an identity map, the ALE method becomes entirely Eulerian.
The ALE method is therefore an intermediate between the Lagrangian and Eulerian methods, and it combines the best features of both: it allows moving boundaries without the need for the mesh movement to follow the material.