Reinforcement learning for automatic quadrilateral mesh generation: a soft actor-critic approach

TL;DR

This work models 2D quadrilateral mesh generation as a Markov decision process and applies Soft Actor-Critic (SAC) to learn a fully automatic meshing policy, eliminating the need for human intervention and post-processing. It introduces a novel action formulation around a reference vertex, a compact partial-state representation of the boundary, and a carefully crafted reward that balances element quality, boundary integrity, and mesh density. Experimental results show SAC-based FreeMesh-RL can generalize from a single training domain to diverse geometries, scale to denser boundaries, and achieve fully quadrilateral meshes with competitive quality compared to established methods, while avoiding extensive preprocessing or cleanup. The study demonstrates the potential of combining rule-based state/action definitions with off-policy RL and highlights avenues for extending to 3D, transfer learning, and use as an RL testbed for mesh-generation research.

Abstract

This paper proposes, implements, and evaluates a reinforcement learning (RL)-based computational framework for automatic mesh generation. Mesh generation plays a fundamental role in numerical simulations in the area of computer aided design and engineering (CAD/E). It is identified as one of the critical issues in the NASA CFD Vision 2030 Study. Existing mesh generation methods suffer from high computational complexity, low mesh quality in complex geometries, and speed limitations. These methods and tools, including commercial software packages, are typically semiautomatic and they need inputs or help from human experts. By formulating the mesh generation as a Markov decision process (MDP) problem, we are able to use a state-of-the-art reinforcement learning (RL) algorithm called "soft actor-critic" to automatically learn from trials the policy of actions for mesh generation. The implementation of this RL algorithm for mesh generation allows us to build a fully automatic mesh generation system without human intervention and any extra clean-up operations, which fills the gap in the existing mesh generation tools. In the experiments to compare with two representative commercial software packages, our system demonstrates promising performance with respect to scalability, generalizability, and effectiveness.

Figures (17)

• Figure 1: Meshing problem. (a) The initial geometric boundary, $B$, is composed of piecewise linear segments, represented as a sequence of vertices $[V_1, V_2, ..., V_N]$; (b) The final mesh, $\Omega$, is composed of a set of quadrilateral elements $[Q_1, Q_2, ..., Q_M]$.
• Figure 2: A sequence of actions taken by the mesh generator to complete a mesh. At each time step $t_i$, an element (in red) is extracted from the current boundary (in blue). The boundary is then updated by cutting off the element and serves as the meshing boundary in the next time step $t_{i+1}$. This process continues until the updated boundary becomes an element.
• Figure 3: The RL-based computational framework for automatic mesh generation. The agent acts as the mesh generator by implementing various RL techniques. It generates an element after a state is perceived and improves the action by observing the rewards. The environment models the meshing boundary and updates the boundary by cutting off the element generated via the action.
• Figure 4: The angle calculation in reference vertex selection. For example, two surrounding vertices of $V_0$, $n_{rv} = 2$, are used to calculate the angles $\vartheta_1$ and $\vartheta_2$, where $\vartheta_1 = \angle V_{l,1}V_0 V_{r,1}$, $\vartheta_2 = \angle V_{l,2}V_0 V_{r,2}$. This calculation iterates along the boundary. The vertex with the least averaged angle by $\vartheta_1$ and $\vartheta_2$ is selected as the reference vertex.
• Figure 5: Action space for each rule type. Subfigures (a)-(c) correspond to three types of actions, respectively. The blue area is the space to select the candidate vertices with a radius $r$ and the origin vertex $V_0$ (reference vertex). The candidate vertices are $V_2$ (in type 1), and $V_2$ and $V_3$ (in type 2), respectively.