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Scaled-up prediction of steady Navier-Stokes equation with component reduced order modeling

Seung Whan Chung, Youngsoo Choi, Pratanu Roy, Thomas Roy, Tiras Y. Lin, Du T. Nguyen, Christopher Hahn, Eric B. Duoss, Sarah E. Baker

Abstract

Scaling up new scientific technologies from laboratory to industry often involves demonstrating performance on a larger scale. Computer simulations can accelerate design and predictions in the deployment process, though traditional numerical methods are computationally intractable even for intermediate pilot plant scales. Recently, component reduced order modeling method is developed to tackle this challenge by combining projection reduced order modeling and discontinuous Galerkin domain decomposition. However, while many scientific or engineering applications involve nonlinear physics, this method has been only demonstrated for various linear systems. In this work, the component reduced order modeling method is extended to steady Navier-Stokes flow, with application to general nonlinear physics in view. Large-scale, global domain is decomposed into combination of small-scale unit component. Linear subspaces for flow velocity and pressure are identified via proper orthogonal decomposition over sample snapshots collected at small scale unit component. Velocity bases are augmented with pressure supremizer, in order to satisfy inf-sup condition for stable pressure prediction. Two different nonlinear reduced order modeling methods are employed and compared for efficient evaluation of nonlinear advection: 3rd-order tensor projection operator and empirical quadrature procedure. The proposed method is demonstrated on flow over arrays of five different unit objects, achieving $23$ times faster prediction with less than $4\%$ relative error up to $256$ times larger scale domain than unit components. Furthermore, a numerical experiment with pressure supremizer strongly indicates the need of supremizer for stable pressure prediction. A comparison between tensorial approach and empirical quadrature procedure is performed, which suggests a slight advantage for empirical quadrature procedure.

Scaled-up prediction of steady Navier-Stokes equation with component reduced order modeling

Abstract

Scaling up new scientific technologies from laboratory to industry often involves demonstrating performance on a larger scale. Computer simulations can accelerate design and predictions in the deployment process, though traditional numerical methods are computationally intractable even for intermediate pilot plant scales. Recently, component reduced order modeling method is developed to tackle this challenge by combining projection reduced order modeling and discontinuous Galerkin domain decomposition. However, while many scientific or engineering applications involve nonlinear physics, this method has been only demonstrated for various linear systems. In this work, the component reduced order modeling method is extended to steady Navier-Stokes flow, with application to general nonlinear physics in view. Large-scale, global domain is decomposed into combination of small-scale unit component. Linear subspaces for flow velocity and pressure are identified via proper orthogonal decomposition over sample snapshots collected at small scale unit component. Velocity bases are augmented with pressure supremizer, in order to satisfy inf-sup condition for stable pressure prediction. Two different nonlinear reduced order modeling methods are employed and compared for efficient evaluation of nonlinear advection: 3rd-order tensor projection operator and empirical quadrature procedure. The proposed method is demonstrated on flow over arrays of five different unit objects, achieving times faster prediction with less than relative error up to times larger scale domain than unit components. Furthermore, a numerical experiment with pressure supremizer strongly indicates the need of supremizer for stable pressure prediction. A comparison between tensorial approach and empirical quadrature procedure is performed, which suggests a slight advantage for empirical quadrature procedure.

Paper Structure

This paper contains 19 sections, 82 equations, 7 figures.

Table of Contents

  1. Introduction
  2. General framework
  3. Governing physics equations
  4. Domain decomposition
  5. Full order model with Discontinuous Galerkin (DG)
  6. Linear subspace approximation
  7. Proper orthogonal decomposition
  8. Supremizer enrichment
  9. POD-Galerkin ROM
  10. Overview and linear operators
  11. Nonlinear advection operator: (1) tensorial approach
  12. Nonlinear advection operator: (2) empirical quadrature procedure
  13. Global ROM system
  14. Results
  15. Sample generation and POD bases
  16. ...and 4 more sections

Figures (7)

  • Figure 1: Domain decomposition showing the global domain, subdomains and reference components.
  • Figure 2: Singular value spectrum of the component POD bases: (a) velocity; and (b) pressure.
  • Figure 3: Energy missing ratio (\ref{['eq:pod-eps']}) of the component POD bases: (a) velocity; and (b) pressure.
  • Figure 4: Scaled-up ROM prediction of a $16\times16$ array at $\mathrm{Re}=25$ with basis sizes $R_{u,r}=R_{p,r}=Z_r=40$.
  • Figure 5: Prediction performance between FOM, ROM with tensorial approach (\ref{['eq:rom-adv-tensor']}) and EQP (\ref{['eq:eqp']}) in (a) assembly time, (b) computation time and (c) accuracy. Error bars indicate $95\%$ confidence interval over 100 test cases.
  • ...and 2 more figures