2. Benchmark solution

2.1. Calculation method

When the exact solution of the problem studied has singularities, the order of convergence of the finite elements solution is modified. For example, consider a discretized plane elasticity problem with triangular elements of degree \(p\).

If the exact solution \({U}_{\mathrm{ex}}\) is regular, we know that ([bib1]):

\({\parallel u-{u}_{h}\parallel }_{\Omega }={\parallel e\parallel }_{\Omega }\le C{h}^{p}\) eq 2.1-1

With \(C\) a constant, \(h\) the size of the elements.

Where \({\parallel e\parallel }_{\Omega }\le C{h}^{p}\) is the contribution to the energy error, which is:

\({\parallel e\parallel }_{\Omega }\le \frac{1}{2}\underset{\Omega }{\int }\varepsilon ({e}_{h})K\varepsilon ({e}_{h})d\Omega\) eq 2.1-2

On the other hand, if the exact solution has a singularity, for example if, locally in the vicinity of a point \({M}_{0}\), the field of displacement is of the form (with \(r\) and \(\theta\) polar coordinates in the vicinity of point \({M}_{0}\)):

\({U}_{\mathrm{ex}}={r}^{\alpha }V(\theta )+W\) with \(0<\alpha <1\) eq 2.1-3

With \(V\) a function of \(\theta\) and \(W\) a constant.

So, we show that [bib1]:

\({\parallel {e}_{h}\parallel }_{\Omega }\le C{h}^{\alpha }\) eq 2.1-4

As a result, the convergence rate of the global energy error becomes independent of the degree \(p\) of the finite elements used and the same is true of that of the measurement of the error (for example, if \(p=1\) or \(p=2\) then \(\alpha =1/2\) for a crack).

Thus, at the crack point, the order of the singularity will be equal to 0.5, and far from the singularity (where the finite element solution is regular, the order of the singularity is equal to \(p\) (1 for linear elements, 2 for quadratic elements).

2.2. Reference quantities and results

We will test the value of the singularity at the crack point (analytical solution), in its vicinity (non-regression) and far from the crack (analytical solution).

We also test the size ratio to be applied to the mesh for a target error (non-regression) and the new size of the elements (non-regression).

It should be noted that for modeling A, the target error is the error (in quantity of interest) on average displacement on the structure. For the other models, the error used is the error in the energy norm.

2.3. Uncertainty about the solution

Analytical and non-regression solution

2.4. Bibliographical references

1. STRANG & FIX: An analysis of the finite element method, Prentice Hall, 1976.