Parametric model order reduction for dynamic non-linear thermoelastic problems in functionally graded materials

Abstract

Functionally graded materials subjected to thermoelastic loading are increasingly utilized in a wide range of industrial applications. The coupled temperature–displacement analysis of such complex structures is typically performed using finite element analysis. However, high-fidelity finite element models often result in significant computational costs. Furthermore, during the design phase, it is desirable to explore variations in material gradation to optimize performance, which further amplifies the computational demand. To address this, a parametric model order reduction framework is proposed in this study to accelerate the dynamic simulation of functionally graded materials under thermoelastic loading. In many applications, mechanical responses remain linear due to small deformations, while thermal non-linearity dominates due to high temperature. Exploiting this structure, a hybrid reduced-order model is introduced, which employs Krylov-based reduction for the mechanical model while retaining the thermal model at full-scale. This hybrid reduced order model is further extended to incorporate parametric dependencies inherent in functionally graded materials through various parametric model order reduction techniques. The spatial variation of material properties is captured using the generalized isoparametric formulation. Material gradation is modeled using either a power-law or exponential-law distribution, with the corresponding exponents treated as parameters of interest. Parametric variations are managed through interpolation of local bases and a locally reduced order model. Four distinct parametric reduced order models are developed based on different combinations of these interpolation strategies. The effectiveness and accuracy of the proposed models are validated using a 2D planar benchmark problem featuring spatially varying material properties. It is observed that, for the mechanical part, reduced order models employing interpolation of local bases achieve higher speed-ups than those based on interpolation of reduced system matrices. In the thermal part, all models utilize local basis interpolation with hyper-reduction via either the discrete empirical interpolation method or the energy conserving sampling and weighting method; among these, energy conserving sampling and weighting-based approaches offer better accuracy. The developed framework demonstrates speed-ups of up to 50 compared to full-scale simulations.