# ALE Grid Calculation

In the ALE formulation, the freedom of moving the mesh is very appealing as it helps to combine the respective advantages of Lagrangian and Eulerian formulations. However, it is not easy to specify a grid velocity well-suited to the particular problem under consideration. As a consequence, the practical implementation of the ALE description requires that an automatic mesh-displacement prescription algorithm be supplied.

In Radioss, the following automatic grid computations exist.

## /ALE/GRID/DONEA

This is the standard method applicable to the most of problems. It is based upon a combination of the material and grid velocities of the neighboring nodes:(1)
${W}_{I}\left(t+\text{Δ}t}{2}\right)=\frac{1}{N}\sum _{J}{W}_{J}\left(t-\text{Δ}t}{2}\right)+\frac{1}{{N}^{2}}\frac{\alpha }{\text{Δ}t}\sum _{J}{L}_{IJ}\left(t\right)\sum _{J}\frac{{u}_{J}\left(t\right)-{u}_{I}\left(t\right)}{{L}_{IJ}\left(t\right)}$
Where,(2)
$1-\gamma \le \frac{W}{v}\le 1+\gamma$
$N$
Number of nodes connected to node
${L}_{IJ}$
Distance between node $I$ and node $J$
α and $\gamma$
The adimensional factors given in input
Therefore, the grid displacement is given as:(3)
$u\left(t+\text{Δ}t\right)=u\left(t\right)+w\left(\frac{t+\text{Δ}t}{2}\right)\text{Δ}t$

## /ALE/GRID/DISP

The average displacement formulation calculates the average velocity to determine the average displacement.(4)
$w\left(t+\text{Δ}t\right)=\frac{1}{N}\sum _{J}{W}_{J}\left(t\right)$

## /ALE/GRID/SPRING

Each grid node is connected to neighboring grid nodes through a nonlinear viscous spring, similar to that shown in Figure 1.

The stiffness of each spring is given by $M\text{Δ}{t}_{02}$ (where, $M$ is the mass of the node and $\text{Δ}{t}_{02}$ is a user input typical time step), a viscosity and a ratio between shear spring stiffness and traction-compression stiffness of the springs can be defined.

It must be noted that those springs only affect the grid node velocity; they have no influence on the material velocity.

This method is very accurate and robust, but highly expensive in terms of CPU time.

## /ALE/GRID/ZERO

No automatic grid calculation is performed for the grid. The grid velocity is either constant (0 if no initial grid velocity is specified, the formulation is therefore Eulerian) or imposed by Property TYPE15 for parts with a rigid body movement.