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.