3. Modeling A#

3.1. Characteristics of modeling#

Modeling: C_ PLAN. It is used to validate the “VERSION3” keyword used for ALGO_LAGR in DEFI_CONTACT. The structure is a healthy square, in which an arc-shaped interface is introduced.

Contact/friction is treated with quadratic X-FEM elements \(\text{P2}\) (displacement) and \(\text{P1}\) (pressure), i.e. bearing the degrees of freedom of movement on all the nodes and the Lagranges of contact/friction on the vertex nodes.

3.2. Characteristics of the mesh#

Number of knots: 4961

Number of meshes and types: 1600 QUAD8 for the plate and 160 SEG3 for the edges.

_images/100000000000032500000321C31A1D2561C3901A.png

Figure 3.2‑a: Quadrangle 2D mesh

3.3. Tested sizes and results#

The values only appear at the interface nodes from the new mesh.

Identification

Reference type

Reference value

Precision

\(\text{LAGS\_C}\) at points \(A\) and \(C\) (MIN)

“ANALYTIQUE”

-0.1

0.1%

\(\text{LAGS\_C}\) at point \(B\) (MAX)

“ANALYTIQUE”

-0.06110

0.1%

\(\text{LAGS\_F1}\) at point \(B\) (MIN)

“ANALYTIQUE”

0

0.1%

\(\text{LAGS\_F1}\) at point \(A\) (MAX)

“ANALYTIQUE”

0.39894

0.5%

3.4. Comments#

This test validates:

  • the calculation of the stiffness matrix (the right offset when filling the matrix because the nodes do not have the same number of degrees of freedom),

  • the calculation of contact matrices (integration on a SE3 at Gauss points),

  • subdivision (curved interface configurations and elements with straight edges),

  • the X- FEM post-processing of the \(\text{P2P1}\) elements,

  • the version 3 algorithm of equality relationships on cut edges for quadratic elements in small slips, in order to satisfy LBB.

Solver MUMPS detects a singularity in the matrix. This problem has already occurred on numerous other X- FEM test cases in contact. The readjustment of the normal level set makes it possible to improve the conditioning of the matrix but at the expense of an excessive discretization error of the level set. The option of deactivating the solver’s singularity detection is therefore retained before proposing a more satisfactory solution.