Machine Learning (ML) based Reduced Order Modeling (ROM) for linear and non-linear solid and structural mechanics

TL;DR

The paper introduces a lightly intrusive reduced order model (ROM) that couples Proper Orthogonal Decomposition (POD) with machine learning to predict the inverse of the reduced stiffness matrix, enabling real-time simulations for affine and non-affine parametric linear and geometrically nonlinear solid mechanics. By performing a two-stage POD and using ML (Random Forest) to estimate reduced coefficients, it reconstructs reduced matrices from full-order model (FOM) snapshots without requiring expert intrusion into commercial solvers. The approach is validated on linear elastostatic plates, nonlinear geometrically nonlinear plates, and a complex airfoil geometry, achieving stiffness prediction errors under ~2% and displacement errors typically within a few percent of full FEM or POD-ROM baselines. This work facilitates robust, efficient parametric studies and digital-twin workflows by delivering accurate online predictions while leveraging existing FE software and data. The method shows promise for real-time design optimization, health monitoring, and multi-physics co-simulation in engineering contexts.

Abstract

Multiple model reduction techniques have been proposed to tackle linear and non linear problems. Intrusive model order reduction techniques exhibit high accuracy levels, however, they are rarely used as a standalone industrial tool, because of the required high level knowledge involved in the construction and usage of these techniques. Moreover, the computation time benefit is compromised for highly nonlinear problems. On the other hand, non-intrusive methods often struggle with accuracy in nonlinear cases, typically requiring a large design of experiment and a large number of snapshots achieve a reliable performance. However, generating the stiffness matrix in a non-intrusive approach presents an optimal way to align accuracy with efficiency, allying the advantages of both intrusive and non-intrusive methods.This work introduces a lightly intrusive model order reduction technique that employs machine learning within a Proper Orthogonal Decomposition framework to achieve this alliance. By leveraging outputs from commercial full-order models, this method constructs a reduced-order model that operates effectively without requiring expert user intervention. The proposed technique has the possibility to approximate linear non affine as well as non linear terms. It is showcased for linear and nonlinear structural mechanics problems.

Figures (17)

• Figure 1: Linear plate problem: Static point load applied at the centre of a thin metal plate.
• Figure 2: Linear plate problem: Predicted versus reference $\bm\theta_1$ coefficients on the training data set.
• Figure 3: Linear plate problem: Evaluating precision through a comparison of the Frobenius norm between the reference FEM stiffness matrix and the matrix predicted using machine Learning (ML)- based Reduced Order Modeling (ROM).
• Figure 4: Linear plate problem: Assessing the precision of the method on displacement and rotation values for all degrees of freedom associated with different training and validation datasets.
• Figure 5: Linear plate problem: Comparison of the plate deformation, scaled by a factor of $1000$ for sake of better visibility. In blue the results obtained with the Machine Learning (ML) based Reduced Order Model (ROM) and in red the corresponding finite element based Full Order Model (FOM) results.