IFOSMONDI Co-simulation Algorithm with Jacobian-Free Methods in PETSc

IFOSMONDI iterative algorithm for implicit co-simulation of coupled physical systems (introduced by the authors in july 2019 during the Simultech conference, p.176-186) enables us to solve the nonlinear coupling function while keeping the smoothness of interfaces without introducing a delay. Moreover, it automatically adapts the size of the steps between data exchanges among the systems according to the difficulty of the solving of the coupling constraint. The latter was solved by a fixed-point algorithm in the original implementation whereas this paper introduces the JFM version (standing for Jacobian-Free Methods). Most implementations of Newton-like methods require a jacobian matrix which can be difficult to compute in the co-simulation context, except in the case where the interfaces are represented by a Zero-Order-Hold (ZOH). As far as IFOSMONDI coupling algorithm uses Hermite interpolation for smoothness enhancement (up to Third-Order-Hold), we propose hereafter a new formulation of the non-linear coupling function including both the values and the time-derivatives of the coupling variables. This formulation is well designed for solving the coupling through jacobian-free Newton type methods. Consequently, successive function evaluations consist in multiple simulations of the systems on a co-simulation time step using rollback. The orchestrator-workers structure of the algorithm enables us to combine the PETSc framework on the orchestrator side for the non-linear Newton-type solvers with the parallel integrations of the systems on the workers side thanks to MPI processes. Different nonlinear methods will be compared to one another and to the original fixed-point implementation on a newly proposed 2-systems academic test-case (mass-spring-damper type) with direct feedthrough on both sides.