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Parametric vector flows for registration fields in bounded domains with applications to nonlinear interpolation of shock-dominated flows

Jon Labatut, Jean-Baptiste Chapelier, Angelo Iollo, Tommaso Taddei

Abstract

We present a registration procedure for parametric model order reduction (MOR) in two- and three-dimensional bounded domains. In the MOR framework, registration methods exploit solution snapshots to identify a parametric coordinate transformation that improves the approximation of the solution set through linear subspaces. For each training parameter, optimization-based (or variational) registration methods minimize a target function that measures the alignment of the coherent structures of interest (e.g., shocks, shear layers, cracks) for different parameter values, over a family of bijections of the computational domain $Ω$. We consider diffeomorphisms $Φ$ that are vector flows of given velocity fields $v$ with vanishing normal component on $\partial Ω$; we rely on a sensor to extract appropriate point clouds from the solution snapshots and we develop an expectation-maximization procedure to simultaneously solve the point cloud matching problem and to determine the velocity $v$ (and thus the bijection $Φ$); finally, we combine our registration method with the nonlinear interpolation technique of [Iollo, Taddei, J. Comput. Phys., 2022] to perform accurate interpolations of fluid dynamic fields in the presence of shocks. Numerical results for a two-dimensional inviscid transonic flow past a NACA airfoil and a three-dimensional viscous transonic flow past an ONERA M6 wing illustrate the many elements of the methodology and demonstrate the effectiveness of nonlinear interpolation for shock-dominated fields.

Parametric vector flows for registration fields in bounded domains with applications to nonlinear interpolation of shock-dominated flows

Abstract

We present a registration procedure for parametric model order reduction (MOR) in two- and three-dimensional bounded domains. In the MOR framework, registration methods exploit solution snapshots to identify a parametric coordinate transformation that improves the approximation of the solution set through linear subspaces. For each training parameter, optimization-based (or variational) registration methods minimize a target function that measures the alignment of the coherent structures of interest (e.g., shocks, shear layers, cracks) for different parameter values, over a family of bijections of the computational domain . We consider diffeomorphisms that are vector flows of given velocity fields with vanishing normal component on ; we rely on a sensor to extract appropriate point clouds from the solution snapshots and we develop an expectation-maximization procedure to simultaneously solve the point cloud matching problem and to determine the velocity (and thus the bijection ); finally, we combine our registration method with the nonlinear interpolation technique of [Iollo, Taddei, J. Comput. Phys., 2022] to perform accurate interpolations of fluid dynamic fields in the presence of shocks. Numerical results for a two-dimensional inviscid transonic flow past a NACA airfoil and a three-dimensional viscous transonic flow past an ONERA M6 wing illustrate the many elements of the methodology and demonstrate the effectiveness of nonlinear interpolation for shock-dominated fields.
Paper Structure (28 sections, 1 theorem, 54 equations, 18 figures, 3 algorithms)

This paper contains 28 sections, 1 theorem, 54 equations, 18 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Formulation
  3. Vector flows
  4. Target function
  5. Convex displacement interpolation
  6. Methodology
  7. Finite element discretization
  8. Choice of the sensor
  9. Evaluation of the target function
  10. Choice of the penalty term
  11. Expectation-maximization procedure
  12. Abstract procedure
  13. Probabilistic model
  14. EM procedure for point-set registration
  15. Solution to the optimization problem
  16. ...and 13 more sections

Key Result

Proposition 2.1

Let $\Omega\subset \mathbb{R}^d$ be a Lipschitz domain; we denote by $\mathcal{W}_h:= {\rm span} \{\phi_i \}_{i=1}^{N_h} \subset \mathcal{V}_0(\Omega)$ the approximation space associated to eq:VB_maps. Then, the following hold.

Figures (18)

  • Figure 1: transonic flow past a NACA airfoil. (a) geometric configuration. (b) computational mesh. (c)-(d) density profiles for two values of the free-stream Mach number and angle of attack $0.4^o$.
  • Figure 2: transonic flow past a NACA airfoil; selected points for the two snapshots, $U_i=U(\mu_i)$, $i=0,1$, with the sensor \ref{['eq:tecplot_sensor']} ($\texttt{tol}=0.7$).
  • Figure 3: transonic flow past a NACA airfoil; evolution of the residuals $R_1,R_2,R_3$ in \ref{['eq:termination_condition']} with respect to the EM iteration count, for different FE meshes with $N_e$. (a)-(b)-(c) $\Delta t=0.2$, linear elements ($p=1)$. (d)-(e)-(f) $\Delta t=0.05$, quadratic elements ($p=2)$.
  • Figure 4: transonic flow past a NACA airfoil; evolution of the geometric error \ref{['eq:geometric_error_naca']} with respect the FE mesh size for linear ($p=1$) and quadratic ($p=2$) polynomials.
  • Figure 5: transonic flow past a NACA airfoil; HF estimates of the pressure field for three parameter values.
  • ...and 13 more figures

Theorems & Definitions (3)

  • Definition 2.1
  • Proposition 2.1
  • Remark 3.1