Validation of Nonlinear Reduced Order Models with Time Integration Targeted at Nonlinear Normal Modes

Abstract

Recently, nonlinear reduced order models (ROMs) of large scale finite element models have been used to approximate the nonlinear normal modes (NNMs) of detailed structures with geometric nonlinearity distributed throughout all of its elements. The ROMs provide a low order representation of the full model, and are readily used with numerical continuation algorithms to compute the NNMs of the system. In this work, the NNMs computed from the reduced equations serve as candidate periodic solutions for the full order model. A subset of these are used to define a set of initial conditions and integration periods for the full order model and then the full model is integrated to check the quality of the NNM estimated from the ROM. If the resulting solution is not periodic, then the initial conditions can be iteratively adjusted using a shooting algorithm and a Newton–Raphson approach. These converged solutions give the true NNM of the finite element model, as they satisfy the full order equations, and they can be compared to the ROM predictions to validate the ROM at selected points along the NNM branch. This gives a load-independent metric that may provide confidence in the accuracy of the ROM while avoiding the excessive cost of computing the complete NNM of the full order model. This approach is demonstrated on two models with geometric nonlinearity: a beam with clamped-clamped boundary conditions, and a cantilevered plate used to study fatigue and crack propagation.