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


Calculation method used for the reference solution
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* Without damping: the analytical solution of the one-element problem is:

:math:`{x}_{B}(t)=\frac{{F}_{x}}{m{\omega }_{0}^{2}}(1-\mathrm{cos}{\omega }_{0}t)`

:math:`m=\frac{1}{3}\rho \mathrm{SI}`, :math:`{\omega }_{0}^{2}=\frac{3E}{\rho {I}^{2}}`, :math:`{T}_{0}=\frac{2\pi }{{\omega }_{0}}`

where :math:`S` is the area of section :math:`(\pi {R}^{2})`.


* With amortization: the analytical solution of the one-element problem is:

:math:`{x}_{B}(t)=\frac{{F}_{x}}{m{\omega }_{0}^{2}}\left[1-\mathrm{exp}(-\frac{\mu +\lambda {\omega }_{0}^{2}}{2}\mathrm{.}t)\mathrm{.}(\frac{\mu +\lambda {\omega }_{0}^{2}}{2{\omega }_{1}}\mathrm{sin}({\omega }_{1}t)+\mathrm{cos}({\omega }_{1}t))\right]`

:math:`\lambda ,\mu` proportional damping coefficient :math:`C=\lambda K+\mu M`

:math:`{\omega }_{1}=\frac{\sqrt{(4-2\lambda \mu ){\omega }_{0}^{2}-{\mu }^{2}-{\lambda }^{2}{\omega }_{0}^{4}}}{2}`


Benchmark results
----------------------

Move :math:`{x}_{B}` to :math:`t=\frac{i{T}_{0}}{10}` :math:`i=\mathrm{1,}\mathrm{...},10`

with: :math:`{T}_{0}=\frac{2\pi }{{\omega }_{0}}`


Uncertainty about the solution
---------------------------

Analytical solution.

**Note:**

*The reference solution corresponds to the solution obtained with single-element discretization and keeping a full mass matrix. This allows the algorithm to be validated but it is not the step to solve the physical problem.*