2. Benchmark solution

2.1. Calculation method used for the reference solution

In this problem, the boundary conditions are adiabatic, the initial temperature is constant equal to \({T}_{0}\) and the load is reduced to the heat source as a function of the temperature \(r(T)\mathrm{=}{r}_{0}\mathrm{-}{r}_{1}T\) where \({r}_{1}\) is positive for reasons of thermal stability. These conditions ensure a homogeneous solution in space. The heat equation is reduced to:

\(\rho {C}_{p}\dot{T}\mathrm{=}{r}_{0}\mathrm{-}{r}_{1}T\); \(T(0)\mathrm{=}{T}_{0}\) [éq1]

By normalization, we can reduce ourselves without loss of generality to the following equation:

\(\dot{u}\mathrm{=}1\mathrm{-}\omega u\); \(u(0)\mathrm{=}0\) [éq2]

The solution to this first-order differential equation is then:

\(u(t)\mathrm{=}\frac{1}{\omega }(1\mathrm{-}{e}^{\mathrm{-}\omega t})\)

Rather than going back from \(u\) solution of [éq2] to \(T\) solution of [éq1], we prefer to adopt the following set of parameters, without paying attention to the units, which leads to \(T\mathrm{=}u\): \({T}_{0}\mathrm{=}0\), \({r}_{0}\mathrm{=}\rho C\) and \({r}_{1}\mathrm{=}\omega {r}_{0}\).

2.2. Benchmark results

The test case is conducted with \(\omega \mathrm{=}2\) and the temperature at \(t=1\) is examined at any node in the element. The data is as follows:

Thermal conductivity	LAMBDA
Volume heat capacity	RHO_CP
Initial temperature	\({T}_{0}\)
Heat source	\({r}_{0}\) \({r}_{1}\)	2. 4.

Size tested	\(T\) (\(t\mathrm{=}1\))
Reference value	\(0.432332\)