1. Reference problem

1.1. Geometry

We consider a one-dimensional structure (a bar whose lateral faces are subject to adiabatic conditions) of length \(\mathrm{2L}\) occupying the domain \([-L;L]\).

Since the temperature is homogeneous in the directions normal to the bar, the calculation can be considered as 1D.

1.2. Material properties

\(\lambda \mathrm{=}2\)	thermal conductivity
\(\rho C\mathrm{=}2\)	volume heat

1.3. Boundary conditions and loads

Non-linear volume source loading, temperature dependent:

\(s(T)\mathrm{=}2\mathrm{-}2\mathrm{\times }w\mathrm{\times }T\) with \(w\mathrm{=}2\)

The boundary conditions are adiabatic on the lateral faces and of the imposed zero temperature type at the end of the bar; a symmetry condition is implemented on the symmetry plane (which is equivalent to an adiabatic boundary condition).

The time range \(\mathrm{[}0.;1.\mathrm{]}\) is discretized into 100 time steps (duration of each time step equal to \(0.01\)).

1.4. Initial conditions

An initial analytical state is provided. See the developments in the paragraph detailing the reference solution.