4. Implementation in Aster#

The residue estimator is implemented in 2D and 3D on all types of elements. It is calculated by the CALC_ERREUR command by activating the “ERME_ELEM” option.

This option calculates on each element:

  • the absolute error \(\eta (K)\) (see [éq 2-1]),

  • the \({\parallel {\sigma }_{h}\parallel }_{{L}^{2}(K)}\) stress tensor norm, which is used to normalize the absolute error,

  • the relative error \({\eta }_{\text{rel}}(K)=\text{100}\times \frac{\eta (K)}{\sqrt{\eta {(K)}^{2}+{\parallel {\sigma }_{h}\parallel }_{{L}^{2}(K)}^{2}}}\).

Note:

This definition of relative error implies that in areas where the constraints are very low, the relative error may be significant and not significant. It is then the absolute error that must be considered.

It also calculates at the global level:

  • the absolute error \(\eta (\Omega )={\left[\sum _{K\in T}\eta {(K)}^{2}\right]}^{1/2}\),

  • the global stress tensor standard \({\parallel {\sigma }_{h}\parallel }_{{L}^{2}(\Omega )}={\left[\sum _{K\in T}{\parallel {\sigma }_{h}\parallel }_{{L}^{2}(K)}^{2}\right]}^{1/2}\),

  • the relative error \({\eta }_{\text{rel}}(\Omega )=\text{100}\times \frac{\eta (\Omega )}{\sqrt{\eta {(\Omega )}^{2}+{\parallel {\sigma }_{h}\parallel }_{{L}^{2}(\Omega )}^{2}}}\).

From the expression [éq 2-1], we can see that to calculate the error indicator on the \(K\) mesh, we need to know:

  • the possible uploads \(f\) on \(K\) and \({g}_{N}\) on \(\partial K\cap {\Gamma }_{N}\) (or their discretization \({f}_{h}\) and \({g}_{\text{Nh}}\)),

  • the quantities \({h}_{K}\), \({h}_{F}\) and \(n\) linked to the geometry of the element,

  • the constraint field \({\sigma }_{h}\),

  • the list of neighbors of \(K\) to retrieve the constraints on these elements, necessary to calculate the 2nd term of [éq 2-1].

1 and 2 can be calculated or retrieved easily.

3 must have been calculated beforehand by one of the “SIGM_ELNO” or “SIEF_ELNO” options.

If not, a fatal error message is sent.

4 requires the calculation of a particular mesh-mesh connectivity, in addition to the standard mesh-node connectivity. This new object is stored in the mesh-type data structure.

For the details of the implementation in*Aster*, see [bib6].

For estimator validation, see [bib7].