### ALE formulation

There is a old post for Lagrangian configuration, which is popular in solid mechanics but not in CFD. ALE is dedicated for combining both Eularian and Lagrangian configuration. In Lagrangian, one has the map between Eularian configuration region and Lagrangian reference configuration domain .

Now assume a new reference region , one have another map :

Here is identity mapping if while is if . Similar to material velocity in spatial configuration:

new reference configuration also has a velocity, called “mesh velocity” in spatial configuration:

With proper boundary condition or above two equations can be seen as PDE of or when corresponding velocities are given. Mesh motion is devised to reduce the mesh distortion during time evolution.

For a function , there is counterpart of in other reference configurations:

Take material derivatives:

in which is the velocity of material particle in spatial configuration, and is the velocity of material particle in arbitrary reference configuration. On the other hand, fixing material particle marker while taking time derivative of both sides of

gives

where is the velocity of fixed mesh point in spatial configuration.

Combining last two conclusion we have

Define as the difference between material velocity and mesh velocity in spatial configuration, when both material particle marker and mesh point marker are at the same spatial position at that moment. Finally, the transformation of and is

All the conservation laws in ALE formulation can be derived from expression above.