# Automatic Grid Computation

There are three different grid velocity formulations that can be used in an ALE simulation. New keywords define the type of method used. The different formulations are:
• 0 - J. Donea Grid Formulation: use keyword /DONEA

(NWALE =0 for version < 4.1)

• 1 - Average Displacement Formulation: use keyword /DISP

(NWALE =1 for version < 4.1)

• 2 - Nonlinear Spring Formulation: use keyword /SPRING

(NWALE =2 for version < 4.1)

## J. Donea Grid Formulation (/DONEA)

This formulation 1 2 computes grid velocity using:

${W}_{I}\left(t+\text{Δ}t/2\right)=\frac{1}{N}\sum _{J}{W}_{j}\left(t-\text{Δ}t/2\right)+\frac{1}{{N}^{2}}\text{\hspace{0.17em}}\frac{a}{\text{Δ}t}\sum _{J}{L}_{IJ}\left(t\right)\sum _{J}\text{\hspace{0.17em}}\frac{{u}_{J}\left(t\right)-{u}_{I}\left(t\right)}{{L}_{IJ}\left(t\right)}$

Where,
$1-\gamma \le \frac{w}{v}\le 1+\gamma$
$N$
Number of nodes connected to node I
${L}_{IJ}$
Distance between node I and node J
$\alpha$ , $\gamma$

Therefore, the grid displacement is given by:

$u\left(t+\text{Δ}t\right)=u\left(t\right)+w\left(t+\text{Δ}t/2\right)\text{Δ}2$

## Average Displacement Formulation (/DISP)

The average displacement formulation calculates average velocity to determine average displacement.

$u\left(t+\text{Δ}t\right)=\frac{1}{N}\sum _{J}{w}_{j}\left(t\right)$

## Nonlinear Spring Formulation (/SPRING)

Each grid node is connected to neighboring grid nodes through a nonlinear viscous spring, similar to that shown in Figure 1.
The input parameters required are:
$\text{Δ}{T}_{0}$
Typical time step (Must be greater than the time step of the current run.)
$0<\gamma <1$
Nonlinearity factor
$\eta$
Damping coefficient
V
Shear factor (stiffness ratio between diagonal springs and springs along connectivities)

This formulation is the best of the three, but it is the most computationally expensive.

1 Donea J., An Arbitary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions, Computer methods in applied mechanics, 1982.
2 Brooks A.N. and Hughes T.J.R., Streamline Upwind /Petrov-Galerkin Formulations for Convection Dominated Flows with particular Emphasis on the Incompressible Navier-Stokes Equations, Computer Methods in Applied Mechanics and Engineering, Vol. 32, 1982.