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Deep learning accelerated solutions of incompressible Navier-Stokes equations on non-uniform Cartesian grids

Heming Bai, Dong Zhang, Shengze Cai, Xin Bian

Abstract

The pressure Poisson equation (PPE) represents the primary computational bottleneck in fractional step methods for incompressible flow simulations, requiring iterative solutions of large-scale linear systems. We previously introduced HyDEA, a hybrid approach to accelerate the PPE solution process. However, its current implementation is limited to uniform Cartesian grids. Accurately resolving complex flow dynamics near solid boundaries requires local grid refinement, yet extending the original HyDEA to non-uniform Cartesian grids is fundamentally challenging, as its standard convolution operators are inherently ill-suited for processing data with spatially varying resolutions. To address this limitation, we adopt the Mesh-Conv (MConv) operator, which explicitly incorporates grid spacing information into convolution operations. Specifically, MConv operator replaces a subset of the standard convolution operators within the U-Net-based branch network of the deep operator network, with the necessary grid spacing information computed via a novel multi-level distance vector map construction strategy. Building upon this enhanced architecture, the framework seamlessly extends to simulate flows interacting with solid structures using a decoupled immersed boundary projection method. Furthermore, by training exclusively on fabricated linear systems rather than conventional flow-dependent datasets, the model generalizes effortlessly across diverse immersed obstacle geometries with fixed neural network weights. Benchmark results demonstrate that the MConv-based HyDEA significantly outperforms both standalone preconditioned conjugate gradient methods and the standard convolution-based HyDEA in convergence performance on strongly non-uniform Cartesian grids. The robustness and generalizability of the MConv-based HyDEA underscore its potential for real-world computational fluid dynamics applications.

Deep learning accelerated solutions of incompressible Navier-Stokes equations on non-uniform Cartesian grids

Abstract

The pressure Poisson equation (PPE) represents the primary computational bottleneck in fractional step methods for incompressible flow simulations, requiring iterative solutions of large-scale linear systems. We previously introduced HyDEA, a hybrid approach to accelerate the PPE solution process. However, its current implementation is limited to uniform Cartesian grids. Accurately resolving complex flow dynamics near solid boundaries requires local grid refinement, yet extending the original HyDEA to non-uniform Cartesian grids is fundamentally challenging, as its standard convolution operators are inherently ill-suited for processing data with spatially varying resolutions. To address this limitation, we adopt the Mesh-Conv (MConv) operator, which explicitly incorporates grid spacing information into convolution operations. Specifically, MConv operator replaces a subset of the standard convolution operators within the U-Net-based branch network of the deep operator network, with the necessary grid spacing information computed via a novel multi-level distance vector map construction strategy. Building upon this enhanced architecture, the framework seamlessly extends to simulate flows interacting with solid structures using a decoupled immersed boundary projection method. Furthermore, by training exclusively on fabricated linear systems rather than conventional flow-dependent datasets, the model generalizes effortlessly across diverse immersed obstacle geometries with fixed neural network weights. Benchmark results demonstrate that the MConv-based HyDEA significantly outperforms both standalone preconditioned conjugate gradient methods and the standard convolution-based HyDEA in convergence performance on strongly non-uniform Cartesian grids. The robustness and generalizability of the MConv-based HyDEA underscore its potential for real-world computational fluid dynamics applications.

Paper Structure

This paper contains 17 sections, 12 equations, 30 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Decoupled immersed boundary projection method
  4. Hybrid deep learning line-search directions and iterative methods for accelerated solutions
  5. Dataset construction
  6. Neural Network architecture and model training
  7. Technical details
  8. Results and discussions
  9. Case 1: 2D lid-driven cavity flow with an embedded stationary circular cylinder
  10. Case 2: 2D flows past an obstacle
  11. 2D flow past a circular cylinder at $Re=100$
  12. 2D flow past an elliptical cylinder
  13. 2D flow past the DARPA SUBOFF profile
  14. 2D flow past an inline oscillating cylinder
  15. Case 3: Flapping elliptical wing
  16. ...and 2 more sections

Figures (30)

  • Figure 1: The workflow of HyDEA.
  • Figure 2: Schematic illustration of the distribution of target and source nodes within the non-uniform local Cartesian grid.
  • Figure 3: The overall computational workflow of MConv operator based on the simplified local weight. The tensor shape $[bs,c,h,w]$ represents the feature map dimensions, corresponding to batch size, the number of channel, height, and width.
  • Figure 4: Schematic illustration of the multi-level distance vector map construction strategy.
  • Figure 5: Architecture of deep operator network.
  • ...and 25 more figures