Expériences numériques

Ce document présente des expériences numériques en élasticité linéaire.

1. Cantilever

We consider a 2D Cantilever structure which is clamped on one side and free of constraints otherwise.

The dimensions are \(\Omega=[0,L]\times[0,1] \subset \RR^2\) with \(L=10\).

Table 1. Young Modulus and Poisson Ratio values

\(E\) (Pa)




The external forces are set to be the gravity, \(\mathbf{f}=-9.81 \vec\jmath\)

The boundary conditions read:

  • at \(x=0\), \(\disp{d} = \disp{0}\)

It means that on \(\partial \Omega \backslash x=0\), we have \(\stresst{\disp{d}} \normal = \disp{0}\).

The command line reads:

./feelpp_qs_elasticity_2d --config-file /feel/testcases/quickstart/elastivity/cantilever/cantilever.cfg

The results on Paraview

2. Cantilever with Pure Traction

We now consider a cantilever in pure traction. To ensure well-posedness, we add constraints on the translation and rotation of the cantilever, namely in 2D and 3D respectively

Constraints on translation and rotation in 2D
\[\int_\Omega \disp{d} = \disp{\tau}, \quad \int_\Omega \nabla \times \disp{d} = 2 \omega \disp{k}\]

where \(\disp{k}\) is the unit vector orthogonal to the 2D plane and \(\omega\) is the angular speed of \(d\).

Constraints in translation and rotation in 3D.
\[\int_\Omega \disp{d} = \disp{\tau}, \quad \int_\Omega \nabla \times \disp{d} = \disp{\omega}\]

We first display a case of pure rigid rotation, the translation is set to 0. The domain outlined in black is the reference cantilever domain, the arrows display the rigid rotation and the rotated cantilever is in grey.

feelpp_qs_elasticity_pure_traction_2d --config-file cantilever.cfg --functions.g="{0,0}" --functions.f="{0,0}"