A Boundary Integral-based Neural Operator for Mesh Deformation

TL;DR

This paper introduces a boundary integral-based neural operator for mesh deformation that efficiently models linear elasticity problems by focusing on boundary displacements, enabling accurate and computationally efficient shape manipulation.

Contribution

It proposes a novel neural operator framework that leverages boundary integral formulations to handle mesh deformation with improved efficiency and generalization capabilities.

Findings

High accuracy in large deformation simulations

Robust generalization across diverse boundary conditions

Efficient mesh quality preservation during deformation

Abstract

This paper presents an efficient mesh deformation method based on boundary integration and neural operators, formulating the problem as a linear elasticity boundary value problem (BVP). To overcome the high computational cost of traditional finite element methods and the limitations of existing neural operators in handling Dirichlet boundary conditions for vector fields, we introduce a direct boundary integral representation using a Dirichlet-type Green's tensor. This formulation expresses the internal displacement field solely as a function of boundary displacements, eliminating the need to solve for unknown tractions. Building on this, we design a Boundary-Integral-based Neural Operator (BINO) that learns the geometry- and material-aware Green's traction kernel. A key technical advantage of our framework is the mathematical decoupling of the physical integration process from the geometric representation via geometric descriptors. While this study primarily demonstrates robust generalization across diverse boundary conditions, the architecture inherently possesses potential for cross-geometry adaptation. Numerical experiments, including large deformations of flexible beams and rigid-body motions of NACA airfoils, confirm the model's high accuracy and strict adherence to the principles of linearity and superposition. The results demonstrate that the proposed framework ensures mesh quality and computational efficiency, providing a reliable new paradigm for parametric mesh generation and shape optimization in engineering.