7. Implementation in Code_Aster#

7.1. Equations introduced#

The THER_LINEAIRE [U4.33.01] command makes it possible to treat the equation in the transient case as described above, but it also allows to solve the stationary problem which is reduced to the following equation:

\(-\text{div}(\lambda \nabla T)=s\text{dans}\Omega\)

and the following boundary conditions:

\(\{\begin{array}{ccc}T={T}_{1}(r,{t}_{s})& & \text{sur}{\Gamma }_{1}\\ \lambda \frac{\partial T}{\partial n}=q(r,{t}_{s})& & \text{sur}{\Gamma }_{2}\\ \lambda \frac{\partial T}{\partial n}=h(r,t)({T}_{\text{ext}}(r,{t}_{s})-T)& & \text{sur}{\Gamma }_{3}\end{array}\)

\({t}_{s}\) being the moment taken to assess the boundary conditions of the equation.

In the transitory case, it is necessary to provide an initial state; this initial state (temperature field) may be selected from among the following:

a field that can be uniform or any created by the CREA_CHAMP command,
a field result of a stationary problem described by the equations above, the calculation instant is taken at the first instant defined in the list of reals describing the time discretization defined by the user,
a field extracted from the result of a transitory problem.

The time discretization (value of \(\Delta t\)) must be provided in the form of one or more lists of moments. These lists are created by the user using the DEFI_LIST_REEL [U4.21.04] command.

A thermal transient can be calculated by making several calls to the command THER_LINEAIRE [U4.33.01] by enriching the same evol_ther concept by providing from the second call the initial moment for resuming the calculation (to obtain \({T}^{-}\)) and possibly the final instant.

The temperature fields resulting from a calculation contain both the value at the nodes of the mesh and at the Lagrange nodes. When the calculation is resumed, it is possible to vary the type of boundary conditions, the field used to initiate the new calculation internally is then reduced to only the nodes of the mesh. The evol_ther result concept will then contain fields at the nodes based on different numbers. The*Code_Aster* operators then only interpolate to mesh nodes when the numbering differs.

7.2. Main thermal options calculated in Code_Aster#

7.2.1. Boundary conditions and loads#

TEMP_IMPO	DDLI_R DDLI_F	\(\underset{{\Gamma }_{1}}{\int }{T}^{+}{\Phi }^{\ast }d{\Gamma }_{1}+\underset{{\Gamma }_{1}}{\int }{\Phi }^{\theta }{\mathrm{vdG}}_{1}\)
	DDLI_R DDLI_F	\(\underset{{\Gamma }_{1}}{\int }{\Phi }^{\ast }{T}_{1}d{\Gamma }_{1}\)
FLUX_REP	CHAR_THER_FLUN_R CHAR_THER_FLUN_F	\(\underset{{\Gamma }_{2}}{\int }{q}^{\theta }{\mathrm{vdG}}_{2}\)
ECHANGE	CHAR_THER_COEF_R CHAR_THER_COEF_F	\(\underset{{\Gamma }_{3}}{\int }\theta {h}^{+}{T}^{+}vd{\Gamma }_{3}\)
	CHAR_THER_TEXT_R CHAR_THER_TEXT_F	\(\underset{{\Gamma }_{3}}{\int }(({\text{hT}}_{\text{ext}}{)}^{\theta }-(1-\theta ){h}^{-}{T}^{-})vd{\Gamma }_{3}\)
ECHANGE_PAROI	RIGI_THER_PARO_R RIGI_THER_PARO_F	\(\underset{{\Gamma }_{\text{12}}}{\int }\theta {h}^{+}({T}_{/{\Gamma }_{\text{12}}}^{+}-{T}_{/{\Gamma }_{\text{21}}}^{+}){v}_{1}d{\Gamma }_{\text{12}}\)
	CHAR_THER_PARO_R CHAR_THER_PARO_F	\(\underset{{\Gamma }_{\text{12}}}{\int }(1-\theta ){h}^{-}({T}_{/{\Gamma }_{\text{21}}}^{-}-{T}_{/{\Gamma }_{\text{12}}}^{-}){v}_{1}d{\Gamma }_{\text{12}}\)
SOURCE	CHAR_THER_SOUR_R CHAR_THER_SOUR_F	\(\underset{\Omega }{\int }(\theta {s}^{+}+(1-\theta ){s}^{-})\mathrm{vd}\Omega\)

7.2.2. Calculation of elementary matrices and transitory term#

	RIGI_THER	\(\underset{\Omega }{\int }\theta \lambda \nabla {T}^{+}\text{.}\nabla \mathrm{vd}\Omega\)
	MASS_THER	\(\underset{\Omega }{\int }\frac{\rho {C}_{p}}{\Delta t}{T}^{+}\mathrm{vd}\Omega\)
	CHAR_THER_EVOL	\(\underset{\Omega }{\int }\frac{\rho {C}_{p}}{\Delta t}{T}^{-}\mathrm{vd}\Omega -\underset{\Omega }{\int }(1-\theta )\lambda \nabla {T}^{-}\text{.}\nabla \mathrm{vd}\Omega\)