1. Linear modeling options#

In this document, only the linear modeling of the physical phenomenon of temperature evolution in a continuous medium is considered. All the coefficients involved in the heat equation will be constants or functions that may depend on time or space. The boundary conditions can only be linear functions of temperature.

By default the material is assumed to be isotropic, Fourier’s law relating the heat flow to the temperature gradient involves a scalar coefficient \(\lambda\) the thermal conductivity:

\(q\mathrm{=}\mathrm{-}\lambda \mathrm{\nabla }T\)

In the general case, in any medium, this relationship is expressed with a thermal conductivity tensor. Since the associated matrix is positive, it is always possible to refer to a diagonal matrix in the coordinate system associated with the proper directions. The treatment of thermal anisotropy (see [bib 1]) is therefore carried out in Code_Aster by providing the thermal conductivity values for each main direction and the proper coordinate system. The evaluation of the elementary terms is then carried out by recovering the various coefficients and by changing the frame of reference. Two types of anisotropy are treated in Code_Aster, these are:

  • Cartesian anisotropy where the preferred directions remain fixed in a Cartesian coordinate system, the data of the three nautical angles \(\alpha\), \(\beta\) and \(\gamma\) makes it possible to pass from the global coordinate system to the main anisotropy coordinate system,

  • cylindrical anisotropy where the preferred directions remain fixed in a cylindrical coordinate system, the data of the two nautical angles \(\alpha\) and \(\beta\) defining the direction of the axis and of the three coordinates of a point on this axis makes it possible to pass from the global coordinate system to the main anisotropy coordinate system.

The variational formulation of the linear heat equation (cf. [R5.02.01]) leads to the evaluation of a certain number of expressions in the form of integrals that ultimately constitute a matrix system. The matrix and the second member are built from different bricks: calculation options that combine one or more integrals. The options described here are common to all isoparametric finite elements. Their evaluation depends on the type of element: degree of form functions, number and family of integration points used.

Reference may be made to documents [U3.23.01], [U3.23.02] and [U3.24.01] concerning the various models (type of mesh supporting finite elements).