Delayed Feedback Control Method For Calculating Space-Time Periodic Solutions Of Viscoelastic Problems
Résumé
We are interested in fast techniques for calculating a periodic solution to viscoelastic
evolution problems with a space-time periodic condition of ”rolling” type. Such a solution is
usually computed as an asymptotic limit of the initial value problem with arbitrary initial data.
We want to invent a control method, accelerating the convergence. The main idea is to modify
our problem by introducing a feedback control term, based on a periodicity error, in order to
accelerate the convergence to the desired periodic solution of the problem.
First, an abstract evolution problem has been studied. From the analytic solution of the
modified (controlled) problem, an efficient control has been found, optimizing the spectrum of
the problem. The proposed control term can be mechanically interpreted, and its efficiency
increases with the relaxation time.
In order to confirm numerically the theoretical results, a finite element simulation has been
carried out on a full 2D model for a steady rolling of a viscoelastic tyre with periodic sculpture.
It has demonstrated that the controlled solution converges indeed faster than the non-controlled
one, and that the efficiency of the method increases with the problem’s relaxation time, that is
when the memory of the underlying problem is large.