# ALE formulation on moving mesh in CFD Toolbox for MATLAB

### ALE formulation – what is it?

ALE stands for the Arbitrary Lagrangian Eulerian formulation. In case of Computational Fluid Dynamics it comes down to a specific  transformation of Navier-Stokes equations (and/or advection-diffusion eqaution) to a form which enables simulation of flows in moving domains and deforming geometries. The key point that distinguishes the ALE formula from both the Lagrangian- or Eulerian-type is introduction of the computational domain, that can move freely and independently of the material. The reference domain’s motion is represented by a set of grid points that can be interpreted as a mesh of finite elements. Combining the advantages of both the Langrangian and Eulerial formulas, ALE is well-suited for handling distortions and entanglement of meshes and significant changes in the shape of the flow boundaries.

### ALE Applications

In particular, ALE is very useful for simulating flow in the domains that undergo significant deformations such as blood vessels present on moving organs (e.g heart) or flows in soft elements such as silicon or rubber pipes. Hence any kind of peristaltic or squeezing motion can be dealt with.

### ALE formulation in QuickerSim CFD Toolbox for MATLAB

We have implemented the ALE approach in QuickerSim CFD Toolbox for MATLAB and present the simulation of the peristaltic motion induced by the pipe walls’ deformation. This is quite similar to the case of an intestine where the material. Note that the rheology of the fluid can be of arbitrary complexity. Here we simulate the non-Newtonian fluid.

For more theory about ALE refer to a book J. Donea and A. Huerta “Finite Element Methods for Flow Problems”  for example.