Boundary conditions, loading and initial condition
========================================================

Only thermal boundary conditions leading to linear temperature equations are described here, which excludes radiation-type conditions.




Imposed temperatures
---------------------

Dirichlet-type conditions are usually treated by dualization in *Code_Aster* (cf. [R3.03.01]), but they can also be eliminated in some cases (kinematic loads).

:math:`T(r,t)={T}_{1}(r,t)\text{sur}{\Gamma }_{1}`

where :math:`{T}_{1}(r,t)` is a function of the space and/or time variable.




Linear relationships
-------------------

These are Dirichlet-type conditions, making it possible to define a linear relationship between temperature values:


* between two or more nodes: with an equation of the form

:math:`\sum _{i=1}^{n}{\alpha }_{i}{T}_{i}(r,t)=\beta (t)`


* between pairs of knots: with an equation of the form

:math:`\sum _{i=1}^{{n}_{1}}{\alpha }_{\mathrm{1i}}{T}_{i/{\Gamma }_{\text{12}}}(r,t)+\sum _{i=1}^{{n}_{2}}{\alpha }_{\mathrm{2i}}{T}_{i/{\Gamma }_{\text{21}}}(r,t)=\beta (t)`

where :math:`{\Gamma }_{\text{12}}` and :math:`{\Gamma }_{\text{21}}` are two sub-parts of the border whose temperature values are linked two by two. This type of boundary condition makes it possible to define periodicity conditions.




Normal flow imposed
------------------

These are Neumann-type conditions, defining the flow entering the domain.

:math:`-q(r,t)\text{.}n=f(r,t)\text{sur}{\Gamma }_{2}`

where :math:`f(r,t)` is a function of the space and/or time variable and :math:`n` refers to the normal at border :math:`{\Gamma }_{2}`.




Exchange
-------

These are Neumann-type conditions modeling convective transfers at the edges of the domain.

:math:`-q(r,t)\text{.}n=h(r,t)({T}_{\text{ext}}(r,t)-T(r,t))\text{sur}{\Gamma }_{3}`

where :math:`{T}_{\text{ext}}(r,t)` is a function of the space and/or time variable representing the temperature of the external environment, and :math:`h(r,t)` is a function of the space and/or time variable representing the convective exchange coefficient on the border :math:`{\Gamma }_{3}`.




Wall exchange
-------------

These are Neumann-type conditions involving two sub-parts of the border opposite each other. This type of boundary condition models interface thermal resistance.


.. csv-table::

    ":math:`\lambda \frac{\partial {T}_{1}}{\partial {n}_{1}}=h(r,t)({T}_{2}(r,t)-{T}_{1}(r,t))\text{sur}{\Gamma }_{\text{12}}` "," :math:`{n}_{1}` normal outside to :math:`{\Gamma }_{\text{12}}`"
    ":math:`\lambda \frac{\partial {T}_{2}}{\partial {n}_{2}}=h(r,t)({T}_{1}(r,t)-{T}_{2}(r,t))\text{sur}{\Gamma }_{\text{21}}` "," :math:`{n}_{2}` normal outside to :math:`{\Gamma }_{\text{21}}`
    (:math:`{n}_{1}=-{n}_{2}` in general)"





Volume source
----------------

It is the term :math:`s(r,t)` depending on the space and/or time variable.




Initial condition
------------------

This is the expression of the temperature field at the initial instant :math:`t=0`:

:math:`T(r\mathrm{,0})={T}_{0}(r)`

where :math:`{T}_{0}(r)` is a function of the space variable.