$\texttt{CRLS}$: Convolutional Regularized Least Squares Framework for Reduced Order Modeling of Transonic Flows

TL;DR

The paper tackles the challenge of creating accurate reduced-order models for shock-dominated transonic flows, where standard POD struggles with discontinuities. It introduces CRLS, a two-stage pipeline that first smooths snapshots with a Gaussian convolution (using reflect padding) and builds POD bases on the smoothed data, then recovers sharp shocks through a regularized deconvolution, with hyperparameters automatically tuned via Bayesian optimization. The method combines POD coefficient interpolation with RBFs and a nearest-neighbor regularization in parameter space to yield accurate, data-efficient surrogates, demonstrated on inviscid transonic flow over the RAE2822 airfoil solved by the Euler equations, achieving improved shock location/strength, reduced surface-pressure and field errors, and a substantial reduction in required POD modes. The approach is nonintrusive and hardware-agnostic, with potential extensions to viscous/unsteady flows and further improvements in convolution/deconvolution strategies for broader aerodynamic design applications.

Abstract

We develop a convolutional regularized least squares ($\texttt{CRLS}$) framework for reduced-order modeling of transonic flows with shocks. Conventional proper orthogonal decomposition (POD) based reduced models are attractive because of their optimality and low online cost; however, but they perform poorly when snapshots contain parameter-dependent discontinuities, leading to smeared shocks, stair-stepping, or non-physical oscillations. In $\texttt{CRLS}$, we first map each full-order snapshot to a smoother representation by applying a one-dimensional Gaussian convolution with reflect padding along the flow field coordinates. The convolution hyperparameters (kernel width and support) are selected automatically by Bayesian optimization on a held-out set of snapshots. POD bases are then extracted from the smoothed data, and the parametric dependence of the POD coefficients is learned via radial basis function interpolation. To recover sharp shock structures, we introduce an efficient deconvolution step formulated as a regularized least squares problem, where the regularization centers the reconstruction around a nearest-neighbor reference snapshot in parameter space. The resulting $\texttt{CRLS}$ surrogate is evaluated on inviscid transonic flow over the RAE2822 airfoil, modeled by the steady compressible Euler equations solved with SU2 over a Latin hypercube sample of Mach number and angle of attack. Compared with standard POD and smoothed-POD baselines, $\texttt{CRLS}$ yields markedly improved shock location and strength, lower surface-pressure and field-level errors, and a $42$\% reduction in the number of POD modes required to capture a fixed fraction of snapshot energy. These results demonstrate that $\texttt{CRLS}$ provides an accurate, data-efficient, and largely automated route to shock-aware reduced order models for high-speed aerodynamic design.

Figures (13)

• Figure 1: FOM and prediciton of step function for different % energy at $\mu=14.5$ using POD
• Figure 2: FOM and prediciton of sigmoid function for different % energy at $\mu=14.5$ using POD
• Figure 3: Comparison of normalized singular values and and cumulative contribution for step functions (original) and sigmoid functions (smoothed)
• Figure 4: Convolution without padding
• Figure 5: Convolution with reflect padding