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Decoupled Divergence-Free Neural Networks Basis Method for Incompressible Fluid Problems

Jinbao Cheng, Jianguo Huang, Haoqin Wang, Tao Zhou

Abstract

We propose a decoupled divergence-free neural networks basis (Decoupled-DFNN) method for solving incompressible flow problems, including the Stokes and Navier-Stokes equations. To ensure the divergence free property exactly, the velocity field is represented as the curl of a stream function in two dimensions and as the curl of a vector potential in three dimensions. Beyond classical stream-function or velocity-vorticity formulations, we further utilize the properties of the curl operator to derive two specific decoupled subproblems for the velocity (through the stream function or vector potential) and the pressure, respectively. The proposed formulations enable a sequential solution strategy, in which the velocity and pressure are solved independently. To resolve the inherent nonlinearity of the Navier-Stokes equations, we employ a Gauss-Newton linearization strategy, transforming the nonlinear velocity subproblem into a sequence of linear subproblems. These decoupled subproblems for velocity and pressure are subsequently solved using the TransNet framework. Compared with existing methods, the proposed approach reduces computational cost while strictly preserving the incompressibility constraint.

Decoupled Divergence-Free Neural Networks Basis Method for Incompressible Fluid Problems

Abstract

We propose a decoupled divergence-free neural networks basis (Decoupled-DFNN) method for solving incompressible flow problems, including the Stokes and Navier-Stokes equations. To ensure the divergence free property exactly, the velocity field is represented as the curl of a stream function in two dimensions and as the curl of a vector potential in three dimensions. Beyond classical stream-function or velocity-vorticity formulations, we further utilize the properties of the curl operator to derive two specific decoupled subproblems for the velocity (through the stream function or vector potential) and the pressure, respectively. The proposed formulations enable a sequential solution strategy, in which the velocity and pressure are solved independently. To resolve the inherent nonlinearity of the Navier-Stokes equations, we employ a Gauss-Newton linearization strategy, transforming the nonlinear velocity subproblem into a sequence of linear subproblems. These decoupled subproblems for velocity and pressure are subsequently solved using the TransNet framework. Compared with existing methods, the proposed approach reduces computational cost while strictly preserving the incompressibility constraint.
Paper Structure (15 sections, 86 equations, 9 figures, 1 table)

This paper contains 15 sections, 86 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Decoupled divergence-free formulations for incompressible flows
  4. The 2D case: stream function representation
  5. The 3D case: vector potential representation
  6. The Neural networks basis method for incompressible flows
  7. The steady Stokes problem
  8. The steady Navier-Stokes problem
  9. Computational complexity
  10. Numerical experiments
  11. 2D Stokes problem
  12. 2D Navier-Stokes problem
  13. 3D Stokes problem
  14. 3D Navier-Stokes problem
  15. Conclusion

Figures (9)

  • Figure 5.1: Numerical results for the 2D Stokes problem with $\nu = 10^{-4}$ and varying numbers of basis functions.
  • Figure 5.2: Execution time for the 2D Stokes problem with $\nu = 10^{-4}$ across varying numbers of basis functions.
  • Figure 5.3: Numerical results for the 2D Stokes problem with $M=1000$ and varying kinematic viscosity.
  • Figure 5.4: Numerical results of Decoupled-DFNN and PINN for the 2D Stokes problem.
  • Figure 5.5: Numerical results for the 2D Navier-Stokes problem with $M=1000$ and varying $\nu$.
  • ...and 4 more figures

Theorems & Definitions (2)

  • proof
  • proof