1. Reference problem#

1.1. Geometry#

1.1.1. Meshing for validation on linear elements#

The mesh consists of three plates oriented in the 3 directions of space:

_images/10000000000001EF00000215B5D3F67028B774B0.png

Group COQ_X

 _images/10000000000001CF00000213559EA4321FBEF8C6.png

Group COQ_Y

 _images/10000000000001F90000023A02A50D03ED18B0DB.png

Group COQ_Z

These three plates make it possible to simply validate the dependence of the pre-deformation functions on the space variables \(X\), \(Y\) and \(Z\). They are all rectangular with dimensions 3 meters by 1 meter.

For each mesh plate, several groups of meshes and nodes are defined:

  • ENCX, ENCY and ENCZ: Node groups made up of the two points at the corners \(X=0\), \(Y=0\), and \(Z=0\).

  • FLX, FLY and FLZ: Nodes located in \(X=\mathrm{3,}Y=0\), \(Y=\mathrm{3,}X=0\) and \(Z=\mathrm{3,}Y=0\) respectively.

  • G1X, G1Y and G1Z: Groups of three cells located between 0 and 1 according to the coordinate of the length of the plate.

  • G2X, G2Y and G2Z: Groups of three cells located between 1 and 2 according to the coordinate of the length of the plate.

  • G3X, G3Y and G3Z: Groups of three cells located between 2 and 3 according to the coordinate of the length of the plate.

1.1.2. Meshing for validation on quadratic elements#

It is a mesh composed of 27 3D elements. The volume has a length of 3 meters, a width of one meter and a thickness of 0.5 meters.

The face located in \(X=0\) constitutes the group ENCAST, the surface meshes of the upper face are duplicated to form the group GRILLE. In the same way as in the previous mesh, we have the groups G1X, G1Y and G1Z. Finally, we define node FLX located in \((\mathrm{3,0,0})\).

_images/100000000000026D00000221B506055E6A33481D.png

1.2. Material properties#

The material is linear elastic.

Young’s module \(E=1E9\mathit{Pa}\)

Poisson’s ratio: \(\mathrm{\nu }=0.3\)

1.3. Boundary conditions and loads#

1.3.1. Linear mesh#

Embedded in ENCX, ENCY and ENCZ: \(\mathit{DX}=\mathit{DY}=\mathit{DZ}=\mathit{DRX}=\mathit{DRY}=\mathit{DRZ}=0\).

Pre-deformation loads will be detailed in each model. As for the loads by functions, they will depend on the following function each time:

_images/100002000000017E0000017888BB21BBC454B6B0.png

« Staircase » function

1.3.2. Quadratic mesh#

Embedding in ENCAST: \(\mathit{DX}=\mathit{DY}=\mathit{DZ}=0\).

Pre-deformation loads will be detailed in each model. As for the loads by functions, they will depend each time on the « staircase » function.