Meshless data-driven decompositions with RBF-based inner products

TL;DR

This work introduces a meshless, RBF-based framework to compute data-driven modal decompositions directly from scattered data, enabling POD, DMD, SPOD, and mPOD without grids. By representing the velocity field with a regression-based RBF expansion and using an RBF-induced inner product, the authors construct spatial and temporal correlation matrices that preserve essential dynamics even at strong data downsampling. The approach yields improved spectral fidelity and spatial structures compared with traditional binning, demonstrated on PIV and LES datasets. The framework offers a scalable, grid-free route to high-resolution, physically consistent modal analyses applicable to Lagrangian and scattered-data measurements and simulations.

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.

Figures (11)

• Figure 1: Test case 1. Mean subtracted velocity magnitude for the impinging jet. Example PIV snapshot (top) and the time step interpolated onto scattered points with downsampling factor $k=1$ (middle) and $k=16$ (bottom).
• Figure 2: Test case 2. Mean subtracted velocity magnitude for the airfoil flow. Example LES snapshot (top) and the time step interpolated onto scattered points with downsampling factor $k=2$ (middle) and $k=32$ (bottom). The aspect ratio is slightly increased for better visualization.
• Figure 3: Normalized $L_2$ error in the temporal correlation matrix $\mathbf{K}$\ref{['eq:k_err']} versus the downsampling factor for binning and RBFs for the jet (left) and the airfoil (right). The dashed circles correspond to the downsampling factors analyzed in detail in the remainder of the article.
• Figure 4: Temporal correlation matrix $\mathbf{K}$ of the jet data (left) and airfoil data (right). The upper left panel on each side shows the gridded (ground truth) data while the second and third row respectively show the binning and RBF results. The columns on each side show different downsampling factors, and the colorscale is shared across all subpanels. For better visualization, only the first 200 time steps of each dataset are displayed.
• Figure 5: Diagonal of the power spectral density of the temporal correlation matrix $\mathbf{K}$ over the Strouhal number for the jet (top) and the airfoil (bottom). The power spectral density is computed using Welch's method with Hanning windows, 75% overlap and 500.0 (Jet) and 4000.0 (Airfoil) samples per segment. The columns correspond to increasing downsampling factors according to the title of each subpanel. The legend is shared across all subpanels and in regions of strong overlap, the curves match almost completely.