Delayed feedback control method for computing the cyclic steady states of evolution problems

Khristenko, Ustim; Tallec, Patrick Le

doi:10.1016/j.cma.2018.04.015

Cited by 2 publications

(3 citation statements)

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“…One of the issues is to accelerate the convergence. 14 An alternative method was presented in the work of Qi et al, 15 but it is mainly restricted to small time periods. Another strategy is to use shooting-type techniques, which solve iteratively an initial value problem over one time period, and successively correct the initial state to cancel the periodicity error observed at the end of the period.…”

Section: Motivation and State Of The Artmentioning

confidence: 99%

“…Time discretization of evolution problem (14) consists in introducing a sequence of calculation times t 0 < t i < t m = t 0 + T with time steps Δt i = t i + 1 − t i , and in replacing the implicit time evolution (14) by the induction 44…”

Section: Time-discrete Problemmentioning

confidence: 99%

“…Note that, during each Newton iteration n, computing the periodicity error n per requires to perform all the evolution from t = 0 to t = T, ie, to solve the initial value problem (17) with initial state u n 0 , to obtain Φ 0 (u n 0 ). Therefore, the residual computation for the "global" Newton loop, solving (16) or its time approximation (21), includes solving the nonlinear evolution problem (14), or respectively, (20), with possible "local" Newton loops at each time step during one period, if one uses a time implicit scheme.…”

Section: Newton Shooting Techniquementioning

confidence: 99%

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Newton‐Krylov method for computing the cyclic steady states of evolution problems in nonlinear mechanics

Khristenko

Tallec

2018

Numerical Meth Engineering

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This work is focused on the Newton-Krylov technique for computing the steady cyclic states of evolution problems in nonlinear mechanics with space-time periodicity conditions. This kind of problems can be faced, for instance, in the modeling of a rolling tire with a periodic tread pattern, where the cyclic state satisfies "rolling" periodicity condition, including shifts both in time and space. The Newton-Krylov method is a combination of a Newton nonlinear solver with a Krylov linear solver, looking for the initial state, which provides the space-time periodic solution. The convergence of the Krylov iterations is proved to hold in presence of an adequate preconditioner. After preconditioning, the Newton-Krylov method can be also considered as an observer-controller method, correcting the transient solution of the initial value problem after each period. Using information stored while computing the residual, the Krylov solver computation time becomes negligible with respect to the residual computation time. The method has been analyzed and tested on academic applications and compared with the standard evolution (fixed point) method. Finally, it has been implemented into the Michelin industrial code, applied to a full 3D rolling tire model.

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Section: Motivation and State Of The Artmentioning

confidence: 99%

Section: Time-discrete Problemmentioning

confidence: 99%