2. Formulation of the residue estimator

Let \(\Omega\) be an open of \({R}^{n}\), \(n=2\), or \(3\), of border \(\Gamma\), and \(T\) a regular triangulation of \(\Omega\).

In linear elasticity, the continuous problem is written as:

find \((u,\sigma )\) such as:

\(\{\begin{array}{cc}\text{div}\sigma +f=0& \text{dans}\Omega \\ u={u}_{D}& \text{sur}{\Gamma }_{D}\\ \sigma \text{.}n={g}_{N}& \text{sur}{\Gamma }_{N}\end{array}\)

\({\Gamma }_{D}\) is the Dirichlet border of the mesh

\({u}_{D}\) is the displacement imposed on this border

\({\Gamma }_{N}\) is the Neumann border

\(n\) the unit normal to \({\Gamma }_{N}\)

\({g}_{N}\) is the load applied to this border; it can be continuous or discretized.

\(f\) is a force of the volume type (gravity, rotation); it can be continuous or discretized.

\({\sigma }_{h}\) is the constraint obtained by solving the discrete problem:

\(\{\begin{array}{cc}\text{div}{\sigma }_{h}+f=0& \text{dans}\Omega \\ {u}_{h}={u}_{D}& \text{sur}{\Gamma }_{D}\\ {\sigma }_{h}\text{.}n={g}_{N}& \text{sur}{\Gamma }_{N}\end{array}\)

with relationship \({\sigma }_{h}=\text{DB}{u}_{h}\) where:

\(D\) is the Hooke matrix

\(B\) is the linearized deformation operator

If \(K\) designates a current element of the mesh, the error estimator (noted \(\eta (\Omega )\)) is defined as being the root mean square of the local error indicators, noted \(\eta (K)\):

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

The local residue indicator

The indicator is composed of three terms; the first represents the residue of the equilibrium equation on each mesh, the second term the jump in normal stresses on the interfaces, the third term the difference between the normal stresses and the load imposed on \({\Gamma }_{N}\) if the element intersects \({\Gamma }_{N}\).

Figure 2-a: Internal elements of a mesh

• the first term of the estimator is the norm \({L}^{2}\) of the residue of the equilibrium equation on cell \(K\), multiplied by \({h}_{K}\) which is, either the diameter of the circumscribed circle for a triangular finite element, or the maximum diagonal for a quadrangle,

• the second term is the integral, on \(S(K)\) defined [Figure 2-a], of the normal stress jumps integrated on each edge \(F\) of the element that has a neighbor, and multiplied by the root of \({h}_{F}\), which is the length of the edge \(F\),

Figure 2-b: Elements located on the border of a mesh

• the third term is the integral, at the intersection of each edge \(F\) of the edges \(\partial K\) of the current element \(K\) with the Neumann border \({\Gamma }_{N}\), jumps between the normal stresses of the element and the Neumann force \({g}_{N}\), multiplied by the root of \({h}_{F}\), length of the edge \(F\).

So we have the following formula for the residue estimator:

\(\eta (K)={h}_{K}{\parallel f+\text{div}{\sigma }_{h}\parallel }_{{L}^{2}(K)}+\frac{1}{2}\sum _{F\in S(K)}{h}_{F}^{1/2}{\parallel \left[{\sigma }_{h}\text{.}n\right]\parallel }_{{L}^{2}(F)}+\sum _{F\subset \partial K\subset {\Gamma }_{N}}{h}_{F}^{1/2}{\parallel {g}_{N}-{\sigma }_{h}\text{.}n\parallel }_{{L}^{2}(F)}\) eq 2-1

For the choice of the various terms in [éq 2-1], refer to [bib1].