2. Boundary conditions, loading and initial condition#
Only thermal boundary conditions leading to linear temperature equations are described here, which excludes radiation-type conditions.
2.1. 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).
\(T(r,t)={T}_{1}(r,t)\text{sur}{\Gamma }_{1}\)
where \({T}_{1}(r,t)\) is a function of the space and/or time variable.
2.2. 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
\(\sum _{i=1}^{n}{\alpha }_{i}{T}_{i}(r,t)=\beta (t)\)
between pairs of knots: with an equation of the form
\(\sum _{i=1}^{{n}_{1}}{\alpha }_{\mathrm{1i}}{T}_{i/{\Gamma }_{\text{12}}}(r,t)+\sum _{i=1}^{{n}_{2}}{\alpha }_{\mathrm{2i}}{T}_{i/{\Gamma }_{\text{21}}}(r,t)=\beta (t)\)
where \({\Gamma }_{\text{12}}\) and \({\Gamma }_{\text{21}}\) 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.
2.3. Normal flow imposed#
These are Neumann-type conditions, defining the flow entering the domain.
\(-q(r,t)\text{.}n=f(r,t)\text{sur}{\Gamma }_{2}\)
where \(f(r,t)\) is a function of the space and/or time variable and \(n\) refers to the normal at border \({\Gamma }_{2}\).
2.4. Exchange#
These are Neumann-type conditions modeling convective transfers at the edges of the domain.
\(-q(r,t)\text{.}n=h(r,t)({T}_{\text{ext}}(r,t)-T(r,t))\text{sur}{\Gamma }_{3}\)
where \({T}_{\text{ext}}(r,t)\) is a function of the space and/or time variable representing the temperature of the external environment, and \(h(r,t)\) is a function of the space and/or time variable representing the convective exchange coefficient on the border \({\Gamma }_{3}\).
2.5. 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.
\(\lambda \frac{\partial {T}_{1}}{\partial {n}_{1}}=h(r,t)({T}_{2}(r,t)-{T}_{1}(r,t))\text{sur}{\Gamma }_{\text{12}}\) |
|
\(\lambda \frac{\partial {T}_{2}}{\partial {n}_{2}}=h(r,t)({T}_{1}(r,t)-{T}_{2}(r,t))\text{sur}{\Gamma }_{\text{21}}\) |
(\({n}_{1}=-{n}_{2}\) in general) |
2.6. Volume source#
It is the term \(s(r,t)\) depending on the space and/or time variable.
2.7. Initial condition#
This is the expression of the temperature field at the initial instant \(t=0\):
\(T(r\mathrm{,0})={T}_{0}(r)\)
where \({T}_{0}(r)\) is a function of the space variable.