Simultaneous Regression and Selection in Nonlinear Modal Model Identification

Christopher Van Damme,Joseph Hollkamp,Alecio Madrid,Matthew Allen

doi:10.3390/vibration4010016

Abstract

High fidelity finite element (FE) models are widely used to simulate the dynamic responses of geometrically nonlinear structures. The high computational cost of running long time duration analyses, however, has made nonlinear reduced order models (ROMs) attractive alternatives. While there are a variety of reduced order modeling techniques, in general, their shared goal is to project the nonlinear response of the system onto a smaller number of degrees of freedom. Implicit Condensation (IC), a popular and non-intrusive technique, identifies the ROM parameters by fitting a polynomial model to static force-displacement data from FE model simulations. A notable drawback of these models, however, is that the number of polynomial coefficients increases cubically with the number of modes included within the basis set of the ROM. As a result, model correlation, updating and validation become increasingly more expensive as the size of the ROM increases. This work presents simultaneous regression and selection as a method for filtering the polynomial coefficients of a ROM based on their contributions to the nonlinear response. In particular, this work utilizes the method of least absolute shrinkage and selection (LASSO) to identify a sparse set of ROM coefficients during the IC regression step. Cross-validation is used to demonstrate accuracy of the sparse models over a range of loading conditions.

Highlights

Finite element (FE) modeling is a common approach for simulating nonlinear dynamical systems
The nonlinear normal modes (NNMs) were computed using the Multi-Harmonic Balance (MHB) method as in [25] with three harmonics included within the solution
This work has demonstrated that simultaneous regression and selection via least absolute shrinkage and selection (LASSO) is an effective procedure to reduce the number nonlinear stiffness terms in a reduced order models (ROMs) while maintaining accurate response predictions

Summary

Introduction

Finite element (FE) modeling is a common approach for simulating nonlinear dynamical systems. As the nonlinear equations of motion are not generally known in closed form, most simulation techniques require numerical integration in the time domain. This is computationally expensive for large models and is often unfeasible for large systems; implicit integration and its many forms, see [1], are desirable for linear structural dynamics applications because they are able to take large time steps, yet for geometrically nonlinear analysis they have the drawback that at each iteration of each time step the stiffness matrix must be reassembled and re-factored [1]. Reduced order modeling techniques project a nonlinear system onto a much smaller number of basis vectors. For an introduction to intrusive methods, the reader is refereed to [2]

Results

Discussion

Conclusion