Dimensional hyperreduction of nonlinear finite element models via empirical cubature with manifold-adaptive weights
Abstract
Nonlinear-manifold reduced-order models for parametrized finite element problems can achieve substantial compression both in the number of generalized (latent) coordinates and, through sampling-and-weighting hyperreduction, in the number of sampled elements/integration points. Yet current sampling-and-weighting approaches employ weights that remain fixed over the solution manifold. We contend that this restriction leaves hyperreduction potential untapped: allowing the weights to vary continuously and nonlinearly with the latent coordinates can further decrease the number of sampled spatial entities. To exploit this possibility, we propose the Manifold-Adaptive-Weight Empirical Cubature Method (MAW-ECM). Starting from a feasible fixed-weight ECM rule, a greedy pruning strategy removes sampled entities through convex quadratic weight-redistribution problems enforcing local conditions and positivity. The method is assessed on two nonlinear benchmarks: homogenization of a metamaterial unit cell exhibiting negative incremental stiffness, and a history-dependent continuum-damage problem. In both cases, the nonlinear manifold is constructed from an initial linear compression followed by an input-informed identification of the latent coordinates as general linear combinations of the retained modal amplitudes, incorporating graph information when relevant to seek the intrinsic dimensionality of the solution manifold. We show that combining the nonlinear-manifold representation with MAW-ECM reduces the number of sampled integration points by more than two orders of magnitude relative to the corresponding standard linear reduced model. Relative to the fixed-weight manifold models alone, the adaptive weights eliminate approximately 80% of the remaining points in the homogenization benchmark and more than 97% in the damage benchmark, while essentially preserving their accuracy.
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BibTeX
@misc{hernandez2026dimensional,
title = {{Dimensional hyperreduction of nonlinear finite element models via empirical cubature with manifold-adaptive weights}},
author = {Joaquín A. Hernández and S. Ares de Parga and Riccardo Rossi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03068},
eprint = {2609.03068},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03068},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dimensional hyperreduction of nonlinear finite element models via empirical cubature with manifold-adaptive weights
AU - Joaquín A. Hernández
AU - S. Ares de Parga
AU - Riccardo Rossi
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.03068
UR - https://arxiv.org/abs/2609.03068
AB - Nonlinear-manifold reduced-order models for parametrized finite element problems can achieve substantial compression both in the number of generalized (latent) coordinates and, through sampling-and-weighting hyperreduction, in the number of sampled elements/integration points. Yet current sampling-and-weighting approaches employ weights that remain fixed over the solution manifold. We contend that this restriction leaves hyperreduction potential untapped: allowing the weights to vary continuously and nonlinearly with the latent coordinates can further decrease the number of sampled spatial entities. To exploit this possibility, we propose the Manifold-Adaptive-Weight Empirical Cubature Method (MAW-ECM). Starting from a feasible fixed-weight ECM rule, a greedy pruning strategy removes sampled entities through convex quadratic weight-redistribution problems enforcing local conditions and positivity. The method is assessed on two nonlinear benchmarks: homogenization of a metamaterial unit cell exhibiting negative incremental stiffness, and a history-dependent continuum-damage problem. In both cases, the nonlinear manifold is constructed from an initial linear compression followed by an input-informed identification of the latent coordinates as general linear combinations of the retained modal amplitudes, incorporating graph information when relevant to seek the intrinsic dimensionality of the solution manifold. We show that combining the nonlinear-manifold representation with MAW-ECM reduces the number of sampled integration points by more than two orders of magnitude relative to the corresponding standard linear reduced model. Relative to the fixed-weight manifold models alone, the adaptive weights eliminate approximately 80% of the remaining points in the homogenization benchmark and more than 97% in the damage benchmark, while essentially preserving their accuracy.
ER -