# 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:

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

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 $
- adimensional factors given in input

Therefore, the grid displacement is given by:

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

## Average Displacement Formulation (/DISP)

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

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

## 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:

- $\text{\Delta}{T}_{0}$
- 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.