2. Benchmark solution

2.1. Calculation method used for the reference solution

• Elastostatic problem \(V\) electrical potential

In volume \(\Delta V=0.\)

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

electrical conductivity \(\sigma =1.\)

\(j\mathrm{.}n=-\sigma \nabla V\)

Axisymmetric solution

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

The boundary conditions on \(\mathrm{AD}\) and \(\mathrm{BC}\) require:

\({V}_{0}=10.\)

Note:

Knowledge of \(A\) is not required for thermal calculation.

• Thermal problem \(T\) temperature

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

Boundary conditions: \(T=0.\) on \(\mathrm{DA}\) and \(\mathrm{BC}\)

\(-\lambda \nabla T\mathrm{.}n=0\) out of \(\mathrm{DC}\) and \(\mathrm{AB}\)

Axisymmetric solution:

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

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

2.2. Benchmark results

\(T=588.9313°C\) (temperature at point \(M\)).

2.3. Uncertainty about the solution

Analytical solution.