### 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 $\Omega$ and Lagrangian reference configuration domain $\Omega-0$.

$x=\phi(\xi,t), \forall x\in \Omega,\xi\in \Omega_0$

Now assume a new reference region $\Omega_{\ast}$, one have another map $f$:

$x=f(c,t)=\phi(\xi,t),\forall c\in \Omega_{\ast}$

Here $f$ is identity mapping if $\Omega_{\ast}=\Omega$ while is $\phi$ if $\Omega_{\ast}=\Omega_0$. Similar to material velocity in spatial configuration:

$u(x,t)=u(\phi(\xi,t),t)=\phi_{,t}(\xi,t)|_{\xi}$

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

$u_{\ast}(x,t)=u_{\ast}(f(c,t),t)=f_{,t}(c,t)|_c$

With proper boundary condition $\phi(\xi,0)=\xi_0$ or $f(c,0)=c_0$ above two equations can be seen as PDE of $\phi$ or $f$ when corresponding velocities are given. Mesh motion $f$ is devised to reduce the mesh distortion during time evolution.

For a function $g(x),\forall x\in \Omega$, there is counterpart of $g$ in other reference configurations:

$g(x,t)=g_0(\xi,t)=g_{\ast}(c,t)$

Take material derivatives:

$\frac{Dg(x,t)}{Dt}|_{\xi}=g_{,t}(x,t)+g_{,i}(x,t)x_{i,t}|_{\xi}=g_{\ast,t}(c,t)+g_{\ast,i}(c,t)c_{i,t}|_{\xi}$

in which $x_{i,t}|_{\xi}\equiv u_i$ is the velocity of material particle $\xi$ in spatial configuration, and $c_{i,t}|_{\xi}\equiv w_i$ is the velocity of material particle $\xi$ in arbitrary reference configuration. On the other hand, fixing material particle marker while taking time derivative of both sides of

$\phi_j(\xi,t)=f_j(c,t)$

gives

$u_j=f_{j,t}|_c+f_{j,i}c_{i,t}|_{\xi}=u_{\ast j}+x_{j,i}c_{i,t}|_{\xi}\Longrightarrow \frac{\partial x_j}{\partial c_i}w_i=u_j-u_{\ast j}$

where $u_{\ast}$ is the velocity of fixed mesh point $c$ in spatial configuration.

Combining last two conclusion we have

$Dg(x,t)/Dt=\dot{g_0}=g_{\ast,t}|_c+g_{,j}(u_j-u_{\ast j})$

Define $b\equiv u-u_{\ast}$ 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 $g$ and $g_{\ast}$ is

$Dg/Dt|_{\xi}=g_{\ast,t}|_c+b\cdot \nabla_x g$

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