General-Purpose Finite Element-Based Path Following of Nonlinear Dynamical Systems

This paper summarizes ongoing efforts toward an advanced integration of algorithmsfrom the finite-element and bifurcation domains, thus enabling an accurate and effective unfoldingof the bifurcation and post-bifurcation scenarios of nonautonomous PDEs typical of nonlinearstructural dynamics including multi-physics applications. The developed tools rely on a nonlinearfinite-element (FE) formulation coupled with a MATLAB-based platform for continuation of periodicorbits called COCO.The paper demonstrates the versatility of the approach through a comparison between the predictionsof the frequency-response curves for a hinged-hinged nonlinear beam subject to a primaryresonanceharmonic transverse load using an in-house FE discretization of the approximate Mettlermodel and using a FE discretization of the special Cosserat model of rods implemented in COMSOLMultiphysics. As the latter accounts for the full nonlinearities and extensional/flexural/shearing deformations,it is expected to yield physically more realistic results. As demonstrated in this paper, italso allows for a straightforward analysis of different sets of boundary conditions without the needfor further simplifications or model reductions.