[Paper Review] Hybrid Neural Network Reduced Order Modelling for Turbulent Flows with Geometric Parameters

Matteo Zancanaro|arXiv (Cornell University)|Jul 20, 2021

Model Reduction and Neural NetworksPhysics and Astronomy39 references21 citations

TL;DR

This paper proposes a hybrid reduced order model (ROM) for geometrically parametrized turbulent incompressible flows by combining POD-Galerkin projection for momentum and continuity equations with a data-driven neural network to reconstruct the eddy viscosity field. The method achieves high accuracy across diverse geometries—demonstrated on a backstep and Ahmed body—while maintaining computational efficiency and flexibility across turbulence models, with mean drag coefficient error of 2.4 drag counts over 10 test cases.

ABSTRACT

Geometrically parametrized partial differential equations are currently widely used in many different fields, such as shape optimization processes or patient-specific surgery studies. The focus of this work is some advances on this topic, capable of increasing the accuracy with respect to previous approaches while relying on a high cost–benefit ratio performance. The main scope of this paper is the introduction of a new technique combining a classical Galerkin-projection approach together with a data-driven method to obtain a versatile and accurate algorithm for the resolution of geometrically parametrized incompressible turbulent Navier–Stokes problems. The effectiveness of this procedure is demonstrated on two different test cases: a classical academic back step problem and a shape deformation Ahmed body application. The results provide insight into details about the properties of the architecture we developed while exposing possible future perspectives for this work.

Motivation & Objective

• . To develop a reduced order model (ROM) that maintains physical consistency for geometrically parametrized turbulent flows.
• . To address the challenge of non-affine and nonlinear turbulence models in ROMs by decoupling physical projection from data-driven eddy viscosity reconstruction.
• . To ensure high accuracy and efficiency in online simulations for industrial applications such as shape optimization.
• . To enable application across multiple eddy viscosity models without retraining the ROM, enhancing versatility.

Proposed method

• . Uses Proper Orthogonal Decomposition (POD) to extract a low-dimensional basis from high-fidelity finite volume solutions.
• . Applies Galerkin projection of the steady-state incompressible Navier-Stokes equations onto the POD subspace to preserve momentum and continuity.
• . Employs a radial basis function (RBF) interpolation for mesh deformation due to geometric parameters, ensuring mesh quality.
• . Uses a feedforward neural network to learn the mapping from geometric parameters to the eddy viscosity field, replacing traditional model-specific ROMs.
• . Implements a reduced SIMPLE algorithm to enforce pressure-velocity coupling in the online phase.
• . Trains the neural network on full-order model (FOM) solutions, including both converged and intermediate iterations, to improve generalization and convergence.

Experimental results

Research questions

• RQ1 . Can a hybrid ROM combining projection-based and data-driven methods achieve high accuracy for geometrically parametrized turbulent flows?
• RQ2 . How does the proposed method perform across different turbulence models without retraining?
• RQ3 . What is the impact of neural network-based eddy viscosity reconstruction on pressure and drag coefficient prediction accuracy?
• RQ4 . How do errors in velocity and pressure fields correlate with drag coefficient errors in the ROM?
• RQ5 . Can the method maintain accuracy and convergence across varying geometric parameters, such as slant angles in the Ahmed body case?

Key findings

• . The hybrid ROM achieves a mean drag coefficient error of 2.4 drag counts across 10 test samples of the Ahmed body with varying slant angles.
• . The highest error in drag coefficient occurs at the 15.4° slant angle, with a relative L2 error of 16.5% for pressure and 15.4% for velocity.
• . Pressure field prediction is the main source of error, with relative L2 errors one magnitude higher than for velocity, indicating room for improvement.
• . The velocity field prediction error is close to the projection error lower bound, indicating good approximation quality.
• . The pressure field error remains significantly above the projection error, suggesting that the neural network reconstruction of eddy viscosity limits accuracy.
• . Qualitative comparisons show good agreement between FOM and ROM for both velocity and pressure fields, with highest errors localized in the wake and underbody regions for the 15.4° case.