Proposing an intelligent mesh smoothing method with graph neural networks

TL;DR

CFD mesh smoothing is essential but traditionally expensive when using optimization-based methods. GMSNet introduces a lightweight graph neural network that uses one-ring neighbor features to predict smoothed node positions, guided by a MetricLoss that eliminates the need for labeled high-quality meshes and a shift-truncation mechanism to prevent negative-volume elements. The approach yields up to 13.56× acceleration over optimization-based smoothing while maintaining competitive smoothing quality and generalizing to unseen meshes, with far fewer parameters than prior NN-based smoothing. These results demonstrate a practical, robust smoothing workflow for unstructured 2D triangular meshes and point to extensions to other element types and integration with broader CFD mesh-improvement pipelines.

Abstract

In CFD, mesh smoothing methods are commonly utilized to refine the mesh quality to achieve high-precision numerical simulations. Specifically, optimization-based smoothing is used for high-quality mesh smoothing, but it incurs significant computational overhead. Pioneer works improve its smoothing efficiency by adopting supervised learning to learn smoothing methods from high-quality meshes. However, they pose difficulty in smoothing the mesh nodes with varying degrees and also need data augmentation to address the node input sequence problem. Additionally, the required labeled high-quality meshes further limit the applicability of the proposed method. In this paper, we present GMSNet, a lightweight neural network model for intelligent mesh smoothing. GMSNet adopts graph neural networks to extract features of the node's neighbors and output the optimal node position. During smoothing, we also introduce a fault-tolerance mechanism to prevent GMSNet from generating negative volume elements. With a lightweight model, GMSNet can effectively smoothing mesh nodes with varying degrees and remain unaffected by the order of input data. A novel loss function, MetricLoss, is also developed to eliminate the need for high-quality meshes, which provides a stable and rapid convergence during training. We compare GMSNet with commonly used mesh smoothing methods on two-dimensional triangle meshes. The experimental results show that GMSNet achieves outstanding mesh smoothing performances with 5% model parameters of the previous model, and attains 13.56 times faster than optimization-based smoothing.

Figures (12)

• Figure 1: a) Laplacian smoothing. The StarPolygon of point $\boldsymbol{x}$ is formed by $\mathbf{S}(\boldsymbol{x}) = \{\boldsymbol{x}_1, \boldsymbol{x}_2, \dots, \boldsymbol{x}_6\}$ as a hexagon. The optimized mesh nodes are then located at the arithmetic average of the coordinates of the nodes in the StarPolygon. b) Angle-Based smoothing. The figure shows the optimal position based on $\boldsymbol{x}_1$. This is achieved by rotating $\boldsymbol{x}$ to the angle bisector of the angle where $\boldsymbol{x}_1$ is located, resulting in $\boldsymbol{x}_1^*$. This process is repeated for other points in the mesh, and the final optimized points are obtained through arithmetic averaging.
• Figure 2: The architecture of GMSNet. GMSNet smoothes the mesh by calculating the optimized position for each mesh node. The process begins with the normalization of the input StarPolygon. Next, feature transformation is performed on the normalized features, and information from the StarPolygon nodes is integrated using graph convolution. Finally, the model predicts the optimized node positions through fully connected (FC) layers. In the figure, GNorm represents the GraphNorm operation, and IS-Norm represents the InstanceNorm operation. The model iterates $N$ times over the entire mesh to perform the smoothing operation. After training, GMSNet can be applied to previously unseen meshes, such as the pipe's mesh.
• Figure 3: Data normalization. The StarPolygon to be processed is normalized to the center of the coordinate axis. In the figure, $d$ represents the maximum size of the StarPolygon, $x$ and $y$ represent the coordinates to be scaled, and $c_x$ and $c_y$ are the normalized coordinates of the target node after the scaling.
• Figure 4: MetricLoss. We use the mean transformed aspect ratio as the loss function for model optimization.
• Figure 5: a) For non-convex StarPolygons, Laplace smoothing may move nodes outside the StarPolygon. b) Computing the circumcenter of triangles is necessary for CVT smoothing. For highly distorted mesh element, it circumcenter is far away from the StarPolygon, resulting in negative volume elements. b) Shift truncation. In the training and inference phases of the model, we truncate the shift to avoid generating negative volume elements.