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

TL;DR

This work addresses the computational burden of high-fidelity SPH simulations in multi-query contexts by introducing an intrusive meshless PMOR built on modal reference spaces. The method maps SPH data onto a fixed reference space, applies POD to extract low-dimensional modal dynamics, and back-projects to the meshless domain for time integration using Galerkin POD or Adjoint Petrov–Galerkin projections. Key findings show dramatic compression (up to $9\times10^{4}$) with velocity fields reconstructed within $\sim$10% error across three benchmark flows, while pressure accuracy is more sensitive to projection errors and benefits from APG stabilization. This framework paves the way for significant speedups in SPH-based MQ tasks, with future work targeting hyper-reduction, error bounds, and robust density-pressure handling.

Abstract

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.

Figures (36)

• Figure 1: Illustration of the SPH numerical discretization of a periodic field function, $\bm{f}(\bm{x})$, over domain, $\Omega$, with boundary, $\partial \Omega$. The discretization of the field function is projected onto a two-dimensional plane, where each blue particle represents a subdomain, $\Omega_i$, with diameter, $\Delta x$, that has a kernel support, $\Omega_s$.
• Figure 1: Illustration of the SPH reference space on a periodic domain. Field quantities from the Lagragian space (i.e., the SPH data) are mapped onto the reference space.
• Figure 1: TGV at $Re=100$; FOM snapshot at $t=0.5$ s.
• Figure 2: Illustration of a cell containing particles of interest (red) and particles inside neighboring cells (black particles). Local polyharmonic splines are constructed for each local neighborhood belonging to each cell of interest.
• Figure 2: Singular values derived from Lagrangian and reference space snapshot matrices. Vertical colored lines are meant to highlight the differences in singular value decay and cumulative energy between Lagrangian and reference space at $M=2$ (yellow), $M=5$ (purple), and $M=10$ (green).