Nonlinear Forced Vibration Analysis of Elastic Structures by Using Parallel Solvers for Large-Scale Systems

Geometrically nonlinear forced vibrations of three dimensional structures, due to harmonic excitations, are investigated in the frequency domain. Structures of elastic materials are considered and the discretized equation of motion is derived by the finite element method, using Elmer software. The shooting and the continuation methods are applied to the resulting large scale FEM system by using scalable parallel solvers. Periodic steady-state solutions are of interest and their computation is achieved by two techniques: shooting and continuation methods. The periodic solutions are obtained by shooting method, i.e. by solving a two-point boundary value problem defined by the periodicity condition. For that purpose, a time integration scheme, such as Newmark’s method is used and the correction of the initial guess is accomplished through a Newton-Raphson method. The next solution of the bifurcation diagram is obtained by the arc-length continuation method. A prediction for the new point from the bifurcation diagram is defined by using the previous solution and the new solution is obtained by correcting the prediction, i.e. by shooting method. The main objective of the current work is to investigate the potential of the proposed methods for the efficient computation of the bifurcation diagrams of large-scale dynamical systems, which result from the discretization in space of real-life structures, achieved by appropriate numerical techniques and parallel algorithms.