arXiv subjects

Liam K. Magargal

Publications and source records attributed to Liam K. Magargal.

3 recordsLinked to original sources

arXiv2026

Hyper-reduction-free reduced-order Newton solvers for projection-based model-order reduction of nonlinear dynamical systems

This study proposes an intrusive projection-based model-order reduction framework for nonlinear problems with a polynomial structure, solved iteratively using a Newton solver in the reduced space. It is demonstrated that for the targeted class of polynomial nonlinearities, all operators appearing in the projected approximate residual and Jacobian can be precomputed in the offline phase, eliminating the need for hyper-reduction. Additionally, the evaluation of both the projected approximate residual and its Jacobian scales only with the dimension of the reduced space, and does not depend on the dimension of the full-order model, enabling effective offline-online decomposition. The proposed hyper-reduction-free (HRF) framework is applied to both Galerkin (HRF-G) and least-squares Petrov-Galerkin (HRF-LSPG) projection schemes. The accuracy and computational efficiency of the proposed HRF schemes are evaluated in two numerical experiments and compared with a commonly used hyper-reduction scheme, namely the energy-conserving sampling and weighting method, for both the Galerkin and LSPG schemes. In the first numerical example, a parametric Burgers' equation is used to assess the predictive capabilities of the considered model reduction approaches on parameter sets not seen in the training snapshots. In the second example, a parametric heat equation with a cubic reaction term is studied, for which a lifting transformation is employed to expose the desired structure. The efficacy of the HRF methods in accurately reducing the dimensionality of the lifted formulation is investigated. For the studied problems, the results show that HRF-G and HRF-LSPG achieve two and one order of magnitude speedup, respectively, with respect to the full-order model while resulting in state prediction errors below O(10^-2).

cs.CE↗

arXiv2025

Meshless projection model-order reduction via reference spaces for smoothed-particle hydrodynamics

A model-order reduction framework for the meshless smoothed-particle hydrodynamics (SPH) method is presented. The proposed framework introduces the concept of modal reference spaces to overcome the challenges of discovering low-dimensional subspaces from unstructured, dynamic, and mixing numerical topology that occurs in SPH simulations. These reference spaces enable a low-dimensional representation of the field equations while maintaining the inherent meshless qualities of SPH. Modal reference spaces are constructed by projecting snapshot data onto a reference space where low-dimensionality of field quantities can be discovered via traditional modal decomposition techniques (e.g., the proper orthogonal decomposition (POD)). Modal quantities are mapped back to the meshless SPH space via scattered data interpolation during the online predictive stage. The proposed model-order reduction framework is cast into the meshless Galerkin POD and the Adjoint Petrov-Galerkin projection model-order reduction (PMOR) formulation. The PMORs are tested on three numerical experiments: 1) the Taylor--Green vortex; 2) the lid-driven cavity; and 3) the flow past an open cavity. Results show good agreement in reconstructed and predictive velocity fields, which showcase the ability of this framework to evolve the field equations in a low-dimensional subspace on an unstructured, dynamic, and mixing numerical topology. Results also show that the pressure field is sensitive to the projection error due to the stiff weakly-compressible assumption made in the current SPH framework, but this sensitivity can be alleviated through nonlinear approximations, such as the APG approach. The proposed meshless model-order reduction framework reports up to 90,000x dimensional compression within 10% error in quantities of interest, marking a step toward drastic cost reduction in SPH simulations.

cs.CE↗

arXiv2025

Projection-based model-order reduction via graph autoencoders suited for unstructured meshes

This paper presents the development of a graph autoencoder architecture capable of performing projection-based model-order reduction (PMOR) using a nonlinear manifold least-squares Petrov-Galerkin (LSPG) projection scheme. The architecture is particularly useful for advection-dominated flows modeled by unstructured meshes, as it provides a robust nonlinear mapping that can be leveraged in a PMOR setting. The presented graph autoencoder is constructed with a two-part process that consists of (1) generating a hierarchy of reduced graphs to emulate the compressive abilities of convolutional neural networks (CNNs) and (2) training a message passing operation at each step in the hierarchy of reduced graphs to emulate the filtering process of a CNN. The resulting framework provides improved flexibility over traditional CNN-based autoencoders because it is readily extendable to unstructured meshes. We provide an analysis of the interpretability of the graph autoencoder's latent state variables, where we find that the Jacobian of the decoder for the proposed graph autoencoder provides interpretable mode shapes akin to traditional proper orthogonal decomposition modes. To highlight the capabilities of the proposed framework, which is named geometric deep least-squares Petrov-Galerkin (GD-LSPG), we benchmark the method on a one-dimensional Burgers' model with a structured mesh and demonstrate the flexibility of GD-LSPG by deploying it on two test cases for two-dimensional Euler equations that use an unstructured mesh. The proposed framework is more flexible than using a traditional CNN-based autoencoder and provides considerable improvement in accuracy for very low-dimensional latent spaces in comparison with traditional affine projections.

cs.CE↗