1. Introduction#

The objective of the option proposed here is to improve the treatment of singularities in the mesh adaptation strategies proposed in the Code_Aster. In fact, the presence of singularities (present in practice in any calculation of real structures using finite elements) implies two types of difficulties that will be described here as « theoretical » and « practical ».

The « theoretical » difficulties come from the fact that the contribution to the energy error of the elements affecting a singularity is of the form \(C{h}^{\alpha }\) (\(C\) a constant, \(h\) the size of the element and \(\alpha\) the order of the singularity) while the contribution of the elements outside the singularity is of the form \(C{h}^{q}\) (\(q\) depending only on the degree of interpolation of element shape functions). The adaptation of the mesh must take this difference into account in order to be as effective as possible. For example, to divide the error contribution by 4, it will be necessary to take elements 16 times smaller in the case of a crack (\(\alpha =1/2\)) and elements 2 times smaller in the case without a crack with quadratic elements (\(p=2\)).

The « practical » difficulties come from the fact that, in areas of singularities, the contributions to energy error are significant. If we aim to obtain a low energy error, we must therefore refine these areas very strongly. However, one can question the influence of these energy errors on physical quantities of interest to the engineer (displacement at such a point, maximum stress in such a sensitive zone, etc…). In other words, just because the areas of singularities cause significant errors in energy does not mean that they have a great influence on the calculation outside these areas. In practice, error estimators quickly designate the only areas of singularities as being to be refined: the areas of singularities mask the other errors, for example an area with a high gradient that we would like to refine.

The Mechanical Laboratory and CAO of Saint-Quentin has developed a method allowing, on the one hand, to detect singularities, and on the other hand, to determine, for a given global error, the size of the finite elements of the new mesh in the case of remeshing.

The use of these two pieces of information can be considered from two perspectives:

  • Finite elements considered to be « singular » by the method can be excluded from the cutting process,

  • the new size of the finite elements is given to a remailer so that this one can build the new mesh while respecting this new size map as best as possible. Currently, the HOMARD software cuts the element once (for example in 2D, a triangle is divided into 4 but no more). To continue with the division, you need to call HOMARD again. An evolution is therefore to be expected so that an element can be divided several times and thus best respect the size map of the new mesh. However, it is possible to use the free GMSH best, which takes a size card directly as input.

Note:

This document uses for the whole the note from a CRECO between LMCAO and department AMA whose reference is cited in the bibliography ([bib1]).