1. Introduction

We are interested in transient thermal calculations where sudden changes in loads occur - for example, thermal shocks. In some cases, it is noted that the temperature oscillates spatially and temporally. In addition, if a temperature profile is observed at a given moment of transition, the temperature at some nodes may exceed the min. and max. limits imposed by the initial conditions and the boundary conditions. This physically unacceptable result violates what is called the « principle of the maximum. »

Diagonalizing the mass matrix can solve these problems of exceeding the maximum. This is detailed in the note [bib1]. We are content here with recalling the main results.

We recall the principle of the maximum in the continuous case, then we express sufficient conditions that make it possible to verify it for discrete equations. In particular, it is shown that the diagonalization of the thermal mass matrix is one of these sufficient conditions and various methods for diagonalizing \(\mathrm{M}\) are presented.

Another sufficient condition depends on the thermal stiffness matrix (conduction). More particularly, from this point of view, the finite thermal elements used in Code_Aster are studied.

As a result, in the case of linear elements, all the conditions sufficient to verify the principle of the maximum are met. In particular, the diagonalization of the mass effectively makes it possible to obtain a regular solution. On the other hand, for quadratic elements, oscillations cannot be prevented.

We therefore describe the solution proposed in Code_Aster: the models developed in 2D (AXIS_DIAG, PLAN_DIAG) and 3D work with linear elements only.

A numerical study of a thermal shock on a cylinder makes it possible to illustrate these results.