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


Calculation method used for the reference solution
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* Elastostatic problem :math:`V` electrical potential

In volume :math:`\Delta V=0.`

Boundary conditions NEUMANN :math:`\{\begin{array}{cc}\mathrm{j.n}=0.& \text{sur}\mathrm{CD}\text{et}\mathrm{AB}\\ \mathrm{j.n}=-10.& \text{sur}\mathrm{AD}\\ \mathrm{j.n}=3.6787944& \text{sur}\mathrm{BC}\end{array}`

electrical conductivity :math:`\sigma =1.`

:math:`j\mathrm{.}n=-\sigma \nabla V`

Axisymmetric solution

:math:`\frac{1}{r}\frac{\partial }{\partial r}(r\frac{\partial V}{\partial r})=0\Rightarrow V={V}_{0}\mathrm{log}\frac{r}{A}`

The boundary conditions on :math:`\mathrm{AD}` and :math:`\mathrm{BC}` require:

:math:`{V}_{0}=10.`

Note:

*Knowledge of* :math:`A` *is not required for thermal calculation.*


* Thermal problem :math:`T` temperature

:math:`-\lambda \Delta T=s` with a :math:`s=\sigma {(\nabla V)}^{2}` volume source

Boundary conditions: :math:`T=0.` on :math:`\mathrm{DA}` and :math:`\mathrm{BC}`

                     :math:`-\lambda \nabla T\mathrm{.}n=0` out of :math:`\mathrm{DC}` and :math:`\mathrm{AB}`

Axisymmetric solution:

:math:`\frac{1}{r}\frac{\partial }{\partial r}(r\frac{\partial T}{\partial r})=-\frac{\sigma }{1}\frac{{v}_{0}^{2}}{{r}^{2}}` :math:`\Rightarrow` considering boundary conditions

:math:`T(r)\mathrm{=}\mathrm{-}\frac{1}{2}\sigma \frac{{v}_{0}^{2}}{\lambda }\mathrm{log}(\frac{r}{{R}_{0}})\mathrm{log}(\frac{r}{{R}_{1}})`


Benchmark results
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:math:`T=588.9313°C` (temperature at point :math:`M`).


Uncertainty about the solution
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Analytical solution.