5. Conclusion#

An asymptotic analysis of the thermal equations in a thin structure when the thickness tends to zero results in a limit model characterized by an average temperature, the solution of a boundary problem, and a complementary parabolic term in thickness, defined locally.

From this we deduced the formulation of a model with \(3\) scalar fields defined on the mean surface of the shell, giving a parabolic representation of the temperature in the thickness. The differential operator obtained is of order \(2\); the thickness of the shell appears in its coefficients.

This model appears to be « optimal » for thin structures:

  • its limit when the thickness tends to zero is identical to the asymptotic limit model;

  • possible additional terms would tend to zero with thickness.

In a standard version, the curvature of the mean surface of the shell does not intervene directly. Test examples show a good adequacy of the temperature obtained with complete three-dimensional solutions.

This model therefore seems to be fully capable of:

  • be used in a finite element formulation to calculate the temperature in a thin shell of any shape; the solution obtained can easily be injected into a thermomechanical calculation of the shell; surface and linear elements are thus proposed for cases where a space variable does not intervene;

  • be introduced directly (or by coupling) into a method for solving the equations governing the thermohydraulic state of a pipe for example, in order to take into account the thermal restoration of the wall on the fluid;

  • be used as an integrated model in solving identification problems (inverse problem) based on experimental measurements (for example for stratified pipes);

  • search for analytical solutions in cases with simple geometry.

The model described here can also be used in thermal evolution problems, provided that thermal loads do not vary too quickly.

Finally, it remains to study the numerical methods to be used to avoid the blockage that could appear in a finite element calculation, when the thickness becomes small.