Solving the Discretised Multiphase Flow Equations with Interface Capturing on Structured Grids Using Machine Learning Libraries

TL;DR

The paper tackles solving discretised multiphase flow equations by reinterpreting standard numerical discretisations as neural network operations (NN4PDEs) within machine‑learning libraries. It extends this framework to three‑dimensional incompressible Navier–Stokes flow with surface tension on structured grids, introducing a compressive algebraic VoF interface capturing and a four‑stage segregated solver that uses a U‑Net multigrid for pressure corrections. Validation on 2D and 3D collapsing water column problems and a rising bubble demonstrates good agreement with experiments and literature, with higher‑order ConvFEM offering a trade‑off between accuracy and speed when combined with diffusion‑controlled diffusion for the volume fraction. The approach yields CPU/GPU portability and aligns CFD discretisations with AI workflows, enabling easy integration with surrogate modelling, data assimilation, and digital twins while achieving substantial performance on modern accelerators.

Abstract

This paper solves the discretised multiphase flow equations using tools and methods from machine-learning libraries. The idea comes from the observation that convolutional layers can be used to express a discretisation as a neural network whose weights are determined by the numerical method, rather than by training, and hence, we refer to this approach as Neural Networks for PDEs (NN4PDEs). To solve the discretised multiphase flow equations, a multigrid solver is implemented through a convolutional neural network with a U-Net architecture. Immiscible two-phase flow is modelled by the 3D incompressible Navier-Stokes equations with surface tension and advection of a volume fraction field, which describes the interface between the fluids. A new compressive algebraic volume-of-fluids method is introduced, based on a residual formulation using Petrov-Galerkin for accuracy and designed with NN4PDEs in mind. High-order finite-element based schemes are chosen to model a collapsing water column and a rising bubble. Results compare well with experimental data and other numerical results from the literature, demonstrating that, for the first time, finite element discretisations of multiphase flows can be solved using an approach based on (untrained) convolutional neural networks. A benefit of expressing numerical discretisations as neural networks is that the code can run, without modification, on CPUs, GPUs or the latest accelerators designed especially to run AI codes.

Figures (18)

• Figure 1: On the left (in orange) is a 5 by 5 image with pixel values $C_{ij}$ ($i,\,j\in\{1,\,2,\,\ldots,\,5\}$). A filter (in blue) of a convolutional layer, with weights as shown, is applied to the 3 by 3 part of the image centred around pixel $C_{33}$. The sum of the products of the weights and the pixel values determines the central value of the resulting feature map $Y_{33}$ (in orange). Equivalently, on the left (in orange) is a 5 by 5 solution field $C_{ij}$ ($i,\,j\in\{1,\,2,\,\ldots,\,5\}$) to which is applied the stencil of the diffusion operator discretised by finite differences (in blue). This results in the value $Y_{33}$ (in orange). (The finite difference nodes are located at the centre of the cells.) Forming a halo around the inner domain are the white cells or ghost cells, through which boundary conditions can be applied.
• Figure 2: Diagrams showing how boundary conditions are applied through halo values when using the quadratic ConvFEM. Linear ConvFEM can be realised simply by ignoring the second layer of halo values. The indices 1, 2, 3 and 4 indicate the cell or node number within the solution domain. These same indices are used in the halo nodes to indicate where the information for forming the halo values is obtained. The basic idea behind the boundary conditions is to specify the value of the halo values such that when one interpolates these with the values inside the domain then one obtains the desired boundary conditions along the boundaries (indicated here by thick blue lines).
• Figure 3: Schematic of the multiphase flow solver as a neural network containing the four stages of the overall solution method. See Figure \ref{['fig:NN-NS2']} for a schematic of the velocity and the non-hydrostatic pressure correction.
• Figure 4: Schematic of the velocity and the non-hydrostatic pressure correction necessary to satisfy the continuity equation. Details of a single iteration of the pressure and velocity correction are shown in top left and top right, respectively. Three of these iterations are combined to form an overall pressure and velocity correction to satisfy the continuity equation, the schematic of which is shown bottom right. (See Figure \ref{['fig:NN-NS']} for a key of the colours).
• Figure 5: Saw-tooth multigrid method, based on a U-Net architecture, used to solve for the hydrostatic and surface-tension pressure and the non-hydrostatic non-surface-tension pressure correction.