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.