Meshless data-driven decompositions with RBF-based inner products

TL;DR

This paper introduces a meshless, data-driven method for modal decompositions using RBFs, enabling analysis of scattered data from experiments or simulations without requiring uniform sampling.

Contribution

It extends previous RBF-based approaches to compute inner products directly from scattered data, reducing computational cost and improving structure recovery in various flow datasets.

Findings

RBF-based decompositions outperform classical binning methods.

The approach effectively captures relevant flow structures across different data densities.

It reduces computational cost by using time-constant basis functions.

Abstract

Data-driven modal decompositions are useful tools for compressing data or identifying dominant structures. Popular ones like the dynamic mode decomposition (DMD) and the proper orthogonal decomposition (POD) are defined with continuous inner products. These are usually approximated with samples of data uniform in space and time. However, not every dataset fulfills this requirement. Numerical simulations with smoothed particle hydrodynamics or experiments with Lagrangian particle tracking velocimetry produce scattered data varying in time and space, rendering sample-based inner products impossible. In this work, we extend a previous approach that computes the modal decompositions with meshfree radial basis functions (RBFs). We regress the data and use the continuous representation of the RBFs to compute the required inner products. We choose our basis to be constant in time, greatly reducing the computational cost since the inner product of the data reduces to the inner product of the basis functions. We use this approach in the most popular decompositions, namely the POD, DMD, multi-scale POD, and the two versions of the spectral POD. For all decompositions, the RBFs give a mesh-free representation of the spatial structures. Two test cases are considered: particle image velocimetry measurements of an impinging jet and large eddy simulations of the flow past a transitional airfoil. In both cases, the RBF-based approach outperforms classical binning and better recovers relevant structures across all data densities.