Optimization of 2D Finite Element Meshes using Valence-Aware Neural Smoothing

The quality of computational meshes is a critical factor in ensuring the accuracy, stability, and efficiency of numerical simulations, particularly within the framework of finite element analysis (FEA). Classical mesh smoothing techniques, such as Laplacian smoothing and optimization-based methods, are commonly employed to enhance element shape and distribution. However, these approaches are constrained either by the inability to explicitly account for mesh quality metrics, in the case of Laplacian smoothing, or by high computational cost, in the case of optimization-based methods.Recent advances in machine learning, and neural networks in particular, provide a promising alternative to address these limitations. Yet, existing approaches typically require separate models for each mesh topology or rely on computationally intensive architectures, limiting their scalability in practice.To overcome these challenges, we propose a Valence-Aware Smoothing Optimization Network (VASONet), a dual-branch neural architecture capable of handling multiple mesh configurations within a single model. By explicitly encoding the valence degree of local nodal neighborhoods into the learning process, VASONet achieves both flexibility across topologies and efficiency in computation, eliminating the need for multiple specialized models.Numerical experiments demonstrate that VASONet provides a fast and robust alternative to conventional optimization-based methods, generating high-quality meshes with improved geometric and numerical properties while reducing the computational time by 34\((\times)\), in average. These results highlight the potential of VASONet as a scalable and generalizable framework for mesh smoothing and optimization in computational mechanics.