Fourier Neural Operators for Two-Phase, 2D Mold-Filling Problems Related to Metal Casting

TL;DR

This paper addresses the computational bottleneck of modeling mold filling in metal casting by framing the problem as an operator learning task that maps geometry, initial and boundary data to time-resolved flow fields. It introduces a Fourier-Graph neural operator that encodes unstructured meshes, applies a Fourier-based spectral core for global coupling, and decodes onto a target mesh, with inlet-conditioned modulation and causal rollout training. The approach achieves mean relative errors around 5% across velocity, pressure, and liquid fraction, while delivering two to three orders of magnitude speedups over conventional CFD, enabling design-in-the-loop optimization of gating systems. The method generalizes across different gate configurations and remains data-efficient, though pressure fidelity remains more challenging; future work includes extending to 3D, coupling with thermal-solidification, and exploring active learning and multi-fidelity data strategies for even greater practicality in casting workflows.

Abstract

We formulate mold filling in metal casting as a 2D neural operator learning problem that maps geometry and boundary data on an unstructured mesh to time resolved flow quantities, replacing expensive transient CFD. In the proposed method, a graph based encoder aggregates local neighborhood information on the input mesh and encodes geometry and boundary data, a Fourier spectral core operates on a regular latent grid to capture global interactions across the domain, and a graph based decoder projects the latent fields to a target mesh. The model is trained to jointly predict velocity components, pressure, and liquid volume fraction over a fixed rollout horizon and generalizes across different ingate locations and process settings. On held out geometries and inlet conditions, it reproduces large scale advection and the fluid-air interface evolution with localized errors near steep gradients. The mean relative L2 error is about 5% across all fields, and inference is two to three orders of magnitude faster than conventional CFD, enabling design in the loop exploration. Ablation studies show monotonic accuracy degradation under stronger spatial subsampling of input vertices and a smoother decline under temporal subsampling. Halving the training set yields only a small increase in error. These results establish neural operators as accurate and data efficient surrogates for 2D mold filling and enable rapid optimization of gating systems in casting workflows.

Figures (7)

• Figure 1: Casting model. Figure \ref{['mold_exp']} shows an experimental concrete mold with sprue, cavity, and two risers. Figure \ref{['cavity']} depicts the parametric cavity model with filling colors representing numerical simulation outcomes. Figure \ref{['field_prediction']} illustrates an exemplarily velocity field prediction on a rectangular domain affinely normalized to $[0,1]^2$.
• Figure 2: Fourier–Graph neural solver. The model receives as input the unstructured mold geometry mesh, the inlet mask, and the initial fields at $_0=0$. A geometry-aware encoder lifts the inputs to a regular $2$D latent grid that is broadcasted to $3$D. The Fourier spectral core captures long-range couplings while modulating Fourier features through based on the inlet setup composed of inlet velocity , size , horizontal position , and angle . Finally, for each time step $\in$ , a decoder maps the $3$D latent grid to $(,,)$ fields at arbitrary $(,)$ spatial locations within the mold domain.
• Figure 3: Velocity field prediction. Velocity field prediction $$ (row $1$), target velocity field (row $2$), pointwise Euclidean error $\lVert-\rVert_2$ (row $3$), and relative $L_2$ error $_{\mathbf{u}}^{(k)}$(row $4$) at simulation steps $\in\{0,1,2,3,6,9\}$ with $\times0.06s$ of a mold filling simulation. Quiver overlays indicate flow direction (array orientation) and velocity magnitude (arrow length).
• Figure 4: Fluid-air interface tracking with iso-contour $=0.5$ as a proxy. Figs. \ref{['fig:pred']} and \ref{['fig:targ']} show the predicted and simulation target volume fraction fields, respectively. The background colormap and the continuous fluid-air interface overlay curves corresponds to step $=0$, while dashed curves denote the fluid-air interfaces at steps $\in \{3,5,7,9\}$ with $0.1s\times$ increments. Figure \ref{['fig:RMSE']} shows the relative $L_2$ error $_{\alpha}^{(k)}$ at each step , with filled circle markers corresponding to the errors between fluid-air interfaces depicted in Figs. \ref{['fig:pred']} and \ref{['fig:targ']}.
• Figure 5: Pressure field prediction. Pressure field prediction (row $1$), target pressure field (row $2$), pointwise Euclidean error $\lVert-\rVert_2$ (row $3$), and relative L$_2$ error $_{\alpha}^{(k)}$ at simulation steps $\in\{2,4,6,7,8,9\}$ with $\times0.06s$ of a mold filling simulation. Quiver overlays indicate flow direction (array orientation) and velocity magnitude (arrow length). Isocontour lines show distinct pressure levels.