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:(1)

W_{I} (t + Δ t / 2) = \frac{1}{N} \sum_{J} W_{j} (t - Δ t / 2) + \frac{1}{N^{2}} \frac{a}{Δ t} \sum_{J} L_{I J} (t) \sum_{J} \frac{u_{J} (t) - u_{I} (t)}{L_{I J} (t)}

Where,

$1 - γ \leq \frac{w}{v} \leq 1 + γ$
$N$: Number of nodes connected to node I
$L_{I J}$: Distance between node I and node J
α, $γ$: adimensional factors given in input

Therefore, the grid displacement is given by:(2)

u (t + Δ t) = u (t) + w (t + Δ t / 2) Δ 2

Average Displacement Formulation (/DISP)

The average displacement formulation calculates average velocity to determine average displacement.(3)

u (t + Δ t) = \frac{1}{N} \sum_{J} w_{j} (t)

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:

$Δ T_{0}$: Typical time step (Must be greater than the time step of the current run.)
$0 < γ < 1$: Nonlinearity factor
$η$: 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.