Reference solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------


* Upon heating, a uniform temperature rise from :math:`700` to :math:`900°C` in :math:`200s` is imposed.


* Analytical solution for thermal calculation (on cooling since :math:`900°C`).

:math:`T(x,y,t)=\theta (x,y,t)(T(x,y\mathrm{,0})-{T}_{\infty })+{T}_{\infty }`

where:

:math:`\theta (x,y,t)\mathrm{=}{\mathrm{\sum }_{i\mathrm{=}1}}^{\mathrm{\infty }}{A}_{i}{e}^{\mathrm{-}{\omega }_{i}^{2}\frac{\lambda }{\rho {C}_{P}}t}\mathrm{cos}{\omega }_{i}x\mathrm{\times }{\mathrm{\sum }_{A}}_{i}^{\mathrm{\infty }}{e}^{\mathrm{-}{\omega }_{i}^{2}\frac{\lambda }{\rho {C}_{P}}t}\mathrm{cos}{\omega }_{i}y`

with :math:`{\omega }_{i}` verifying:

:math:`{\omega }_{i}L\mathrm{tg}({\omega }_{i}L)=\frac{hL}{\lambda }=5.00`

and:

:math:`{A}_{i}=\frac{4\mathrm{sin}({\omega }_{i}L)}{2{\omega }_{i}L\mathrm{sin}({\omega }_{i}L)}`


* The reference values for metallurgical evolutions depend on the model and on the integration of behavioral relationships over time. Reference values are not available.


* Since the hardness of a material point depends on the metallurgical proportions of each phase, reference values are not available.


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

(Thermal calculation):


* temperature at points :math:`A`, :math:`B`, :math:`C` at time :math:`t=300s`,


* proportion of bainite at points :math:`A`, :math:`B`, :math:`C` at times :math:`t=\mathrm{410,}300` and :math:`300s` respectively,


* proportion of martensite at points :math:`A`, :math:`B`, :math:`C` at the moment :math:`t=\mathrm{410 }s`,


* proportion of austenite at point :math:`A` at times :math:`t=\mathrm{30 }s` and :math:`\mathrm{140 }s`.


* hardness at point :math:`O` at times :math:`t=\mathrm{30 }s,\mathrm{140 }s,300s` and :math:`410s`.


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

Less than 1% with 30 modes for each sum.


Bibliographical references
---------------------------


1. F.P. INCROPERA, D.P. DEWITT, J. WILEY. Fundamentals of heat and mass transfer. Third edition. 1990.