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


Calculation method used for the reference solution 
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Given the symmetry of the boundary conditions, the solution is independent of :math:`y`. For the mechanical part, the mechanical balance of the skeleton is written as a projection on the :math:`(\mathrm{0x})` axis:

:math:`({\lambda }_{1}^{M}+2{\lambda }_{2}^{M})\frac{{\partial }^{2}{u}_{x}}{\partial {x}^{2}}-b\frac{\partial p}{\partial x}-\mathrm{rg}=0`

where :math:`{\lambda }_{1}^{M}` and :math:`{\lambda }_{2}^{M}` designate the Lamé coefficients of the material. For the hydraulic part, the conservation of the mass of water is written

:math:`\frac{\partial {M}_{x}}{\partial x}=0`

with

:math:`\frac{{M}_{x}}{\rho }={\lambda }^{H}(\frac{\partial p}{\partial x}-\rho g)`

where :math:`{\lambda }^{H}` refers to hydraulic conductivity. The pressure is then given by the formula

:math:`p={P}_{0}+\rho g(L-x)`

Horizontal displacements :math:`{u}_{x}` are given by

:math:`{u}_{x}=\frac{1}{2}\frac{r-b\rho }{{\lambda }_{1}^{M}+2{\lambda }_{2}^{M}}\mathrm{gx}(x-2L)+\frac{b{P}_{0}}{{\lambda }_{1}^{M}+2{\lambda }_{2}^{M}}x`


Benchmark results
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We recall the formulas giving the Lamé coefficients as a function of the Young's modulus and the Poisson's ratio.

:math:`{\lambda }_{1}^{M}=\frac{E\nu }{(1+\nu )(1-2\nu )}` and :math:`{\lambda }_{2}^{M}=\frac{E\nu }{2(1+\nu )}`

The reference result is the value of the movements and the pressure at point :math:`P`.


Uncertainty about the solution
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Analytical solution.