Benchmark solution
=====================


"Rectangular" crack
------------------------------

The aim of this test is to validate the definition of this crack by calculating the associated level sets. The test actually only looks at the value of :math:`\mathrm{lst}` at points :math:`\mathrm{P4}` and :math:`\mathrm{P5}`. A quick calculation gives :math:`\mathrm{lst}(\mathrm{P4})=\sqrt{({(\frac{1}{2}-(a-r))}^{2}+{(1-(b-r))}^{2})}-r` and :math:`\mathit{lst}(\mathit{P5})\mathrm{=}\mathrm{0,2}`.


"Cylinder" crack
---------------------------

The aim of this test is to validate the definition of this crack by calculating the associated level sets. The test focuses on the values of :math:`\mathit{lsn}` and :math:`\mathrm{lst}` at points :math:`\mathrm{P4}` and :math:`\mathrm{P5}`. A quick calculation gives:

:math:`\mathit{lsn}(\mathit{P4})\mathrm{=}\sqrt{2}\mathrm{-}r` and :math:`\mathit{lst}(\mathit{P5})\mathrm{=}1\mathrm{-}r`,

:math:`\mathit{lst}(\mathit{P4})\mathrm{=}\mathrm{-}\mathrm{0,5}` and :math:`\mathit{lst}(\mathit{P5})\mathrm{=}\mathrm{-}\mathrm{0,5}`.