I want to solve a thermal analysis using finite elements. The governing equation is $$C \frac{dT}{dt}+K T = Q$$. When using backward differencing for time, the resulting equation is quite straight forward.
When I now try to solve this equation for a number of time steps, I can see that the results can oscillate a lot and even yield negative temperatures in unit Kelvin which is physically not correct.
I have seen that commercial codes like MSC Marc perform recycles inside a time increment.
My question now is, what is being done when a recycle is performed? What is changed in the matrices and right hand side etc?

  • 4
    $\begingroup$ We don't know what the term "recycle" means, and what exactly MSC Marc is or does. You'll have to provide more details. $\endgroup$ Commented Dec 17, 2019 at 19:44
  • $\begingroup$ The problem for the heat transfer analysis is, that even with an implicit time discretization, the method is not unconditionally stable. How the time step size should be chosen is usually given in the manual. $\endgroup$
    – P. G.
    Commented Dec 21, 2019 at 21:26

1 Answer 1


The numerical oscillations that you're seeing are most likely due to a too coarse time-step. For instance for the explicit Euler scheme there is a maximum time-step to avoid oscillations which is given by $Fo \leq 1/4$ in 2D problems and $Fo \leq 1/6$ in 3D problems where $Fo$ is the Fourier number of the smallest element. The derivation is given in chapter 4 of the book "Finite Element Simulation of Heat Transfer" by Bergheau and Fortunier 1. I've never used MSC Marc but my first guess would be that those "recycle" mean using a smaller time-step.


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