6. Conclusion

The models AXIS_DIAG, PLAN_DIAG and 3D_ DIAG are proposed in order to solve the problems of exceeding the maximum with oscillation of the solution in space and time that appear during certain transient thermal calculations with sudden variation in load.

At the discrete level, the analysis leads to a sufficient condition of non-oscillation on the time discretization step which must belong to an interval:

\(\Delta {t}_{\text{min}}\le \Delta t\le \Delta {t}_{\text{max}}\)

where the values of \(\Delta {t}_{\text{min}}\) and \(\Delta {t}_{\text{max}}\) depend on the coefficients of thermal mass and stiffness matrices as well as on the time discretization parameter \(\theta\).

In practice, if the oscillations come from a time step that is too big \((\Delta t>\Delta {t}_{\text{max}})\), we suggest the choice of an implicit pattern in time \((\theta =1)\). If the time steps are too small, diagonalizing the mass matrix can make it possible to eliminate the oscillations.

For linear elements, it is shown that diagonalization effectively makes it possible to avoid oscillations in the solution. For quadratic elements, a direct diagonalization is not enough to avoid oscillations, so this possibility is forbidden.