A Gaussian process regression reduced order model for geometrically nonlinear structures

Abstract

Reduced order models, such as Hollkamp and Gordon’s Implicit Condensation and Expansion (ICE) model, are a highly efficient alternative to full-order finite element models (FEM) of geometrically nonlinear structures. However, a reduced order model (ROM) is typically only valid for one FEM. It does not capture how each ROM coefficient changes due to variations in the FEM (e.g., design parameters or uncertainties), so if the FEM is updated then the ROM needs to be re-computed with a new set of static load–displacement data. This study presents a data-driven reduced order modeling approach that creates a single ROM that incorporates design variations in FEM. The proposed method applies Gaussian Process Regression (GPR) to the ICE approach, making each coefficient in an ICE ROM a regression model with respect to a collection of FEMs with varying material properties or geometric parameters. Once the GPR ROM has been identified, one can immediately produce an ICE ROM for a set of FEM parameters without a need to solve any static load–displacement cases on the full FEM. This dramatically enhances the computational efficiency and could be helpful when model uncertainty needs to be considered or when seeking to update a model to correlate with measurements. Additionally, the coefficients of a ROM can often change considerably if the scale on the load–displacement data changes, so it can be difficult to know whether the scaling that was chosen has really identified an accurate ROM. The proposed GPR ROM estimates the mean ROM coefficients for a range of load scaling as well as the uncertainty on each ROM coefficient with respect to the load level. This can be used to gauge the success of the ROM identification and to eliminate ROM coefficients that are unimportant and hence highly variable. The proposed GPR ROM approach is evaluated by applying it to flat and curved beam structures, revealing that the advantages outlined above can be realized with a relatively modest increase in cost relative to a traditional ICE ROM.