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Mathematical and numerical methods for modeling and computing the cyclic steady states in non-linear mechanics

Abstract : This work is focused on fast techniques for computing the steady cyclic states of evolution problems in non-linear mechanics with space and time periodicity conditions. This kind of problems can be faced, for instance, in the beating heart modeling. Another example concerns the rolling of a tyre with periodic sculptures, where the cyclic state satisfies "rolling" periodicity condition, including shifts both in time and space. More precisely, the state at any point is the same that at the corresponding point observed at the next sculpture one time period ago.Direct solvers for such problems are not very convenient, since they require inversion of very large matrices. In industrial applications, in order to avoid this, such a cyclic solution is usually computed as an asymptotic limit of the associated initial value problem with arbitrary initial data. However, when the relaxation time is high, convergence to the limit cycle can be very slow. In such cases nonetheless, one is not interested in the transient solution, but only in a fast access to the limit cycle. Thus, developing methods accelerating convergence to this limit is of high interest. This work is devoted to study and comparison of two techniques for fast calculation of the space-time periodic solution.The first is the well-known Newton-Krylov shooting method, looking for the initial state, which provides the space-time periodic solution. It considers the space-time periodicity condition as a non-linear equation on the unknown initial state, which is solved using Newton-Raphson technique. Since the associated Jacobian can not be expressed explicitly, the method uses one of the matrix-free Krylov iterative solvers. Using information stored while computing the residual to solve the linear system makes its calculation time negligible with respect to the residual calculation time. On the one hand, the algorithm is a shooting method, on the other side, it can be considered as an observer-controller method, correcting the transient solution after each cycle and accelerating convergence to the space-time periodic state.The second method, considered in this work, is an observer-controller type modification of the standard evolution to the limit cycle by introducing a feedback control term, based on the periodicity error. The time-delayed feedback control is a well-known powerful tool widely used for stabilization of unstable periodic orbits in deterministic chaotic systems. In this work the time-delayed feedback technique is applied to an a priori stable system in order to accelerate its convergence to the limit cycle. Moreover, given the space-time periodicity, along with the time-delay, the feedback term includes also a shift in space. One must then construct the gain operator, applied to the periodicity error in the control term. Our main result is to propose and to construct the optimal form of the gain operator for a very general class of linear evolution problems, providing the fastest convergence to the cyclic solution. The associated control term can be mechanically interpreted.Efficiency of the method increases with the problem's relaxation time. The method is presented in a simple predictor-corrector form, where correction is explicit and numerically cheap. In this later form, the feedback control has been also adapted and tested for a nonlinear problem.The discussed methods have been studied using academic applications and they also have been implemented into the Michelin industrial code, applied to a full 3D tyre model with periodic sculpture in presence of slip-stick frictional contact with the soil, and then compared to the standard asymptotic convergence.
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  • HAL Id : tel-03259609, version 1

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Ustim Khristenko. Mathematical and numerical methods for modeling and computing the cyclic steady states in non-linear mechanics. Solid mechanics [physics.class-ph]. Université Paris Saclay (COmUE), 2018. English. ⟨NNT : 2018SACLX001⟩. ⟨tel-03259609⟩

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