  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.
