ALE Support for object having a large rigid body motion¶
ALE terms for rotating objects¶
This section present a set of bricks facilitating the use of an ALE formulation for rotating bodies having a rotational symmetry (typically a train wheel).
Theoretical background¶
This strategy consists in adopting an intermediary description between an Eulerian and a Lagrangian ones for a rotating body having a rotational symmetry. This intermediary description consist in a rotating axes with respect to the reference configuration. See for instance [Dr-La-Ek2014] and [Nackenhorst2004].
It is supposed that the considered body is submitted approximately to a rigid body motion
and may have additonal deformation (exptected smaller) with respect to this rigid motion, where
and

Note that the description is given for a three-dimensional body. For two-dimensional bodies, the third axes is neglected so that
We have then
If
With
Thanks to the rotation symmetry of the reference configuration

The denomination ALE of the method is justified by the fact that
the displacement with respect to this intermediate configuration, the advantage is that if this additional displacement with respect to the rigid body motion is small, it is possible to use a small deformation model (for instance linearized elasticity).
Due to the objectivity properties of standard constitutive laws, the expression of these laws in the intermediate configuration is most of the time identical to the expression in a standard reference configuration except for the expression of the time derivative which are modified because the change of coordinate is nonconstant in time :
Note that the term
This should not be forgotten that a correction has to be provided for each evolving variable for which the time derivative intervene in the considered model (think for instance to platic flow for plasticity). So that certain model bricks canot be used directly (plastic bricks for instance).
GetFEM bricks for structural mechanics are mainly considering the displacement as the amin unknown. The expression for the displacement is the following:
Weak formulation of the transient terms¶
Assuming
The third term in the right hand side can be integrated by part as follows:
Since
Thus globally
Note that two terms can deteriorate the coercivity of the problem and thus its well posedness and the stability of time integration schemes: the second one (convection term) and the fifth one. This may oblige to use additional stabilization techniques for large values of the angular velocity
The available bricks …¶
To be adapted
ind = getfem::brick_name(parmeters);
where parameters
are the parameters …
ALE terms for a uniformly translated part of an object¶
This section present a set of bricks facilitating the use of an ALE formulation for an object being potentially infinite in one direction and which whose part of interests (on which the computation is considered) is translated uniformly in that direction (typically a bar).
Theoretical background¶
Let us consider an object whose reference configuration
where
If
and
where

If we denote
the displacement on the intermediary configuration, then it is easy to check that
Weak formulation of the transient terms¶
Assuming
where