A Krylov accelerated Newton–Raphson scheme for efficient pseudo-arclength pathfollowing. (October 2022) - Formica, Giovanni; Milicchio, Franco; Lacarbonara, Walter
Abstract: The computational efficiency of an enhanced version of a pseudo-arclength pathfollowing scheme tailored for general multi-degree-of-freedom (multi-dof) nonlinear dynamical systems is discussed. The pathfollowing approach is based on the numerical computation of the Poincaré map and its Jacobian in order to tackle nonautonomous systems with discontinuous vector fields. The scheme is applied to obtain frequency response curves of multi-dof hysteretic systems with a state vector size up to 120, as well as various reduced-order models of single and multiple cantilever beams on a shuttle mass. The proposed approach is shown to drastically increase the speed of convergence in the modified Newton–Raphson scheme thanks to a Krylov sub-space iteration which makes use of the LU decomposition of a frozen Jacobian matrix, which, upon convergence, becomes the monodromy matrix. Several numerical tests performed on mechanical systems with material or geometric nonlinearities corroborate the efficiency of the numerical strategy. The C++ code implementing the proposed methodology is freely available at https://doi.org/10.5281/zenodo.6616482 . Highlights: Efficient pseudo-arclength pathfollowing for large nondifferentiable nonlinear ODEs Modified Newton-Raphson scheme with Krylov subspace-based acceleration procedure Tested nonlinear and nonsmooth dynamic systems proving efficiency and robustness Approach implemented in C++ saving over 90% CPU time with respect to standard schemes Free and open source C++ algorithms with ready-to-use examples and an iOS frontend