math.NAFeb 27, 2026

A Boundary Integral-based Neural Operator for Mesh Deformation

Authors: Zhengyu Wu, Jun Liu, Wei Wang

Organizations: School of Electronics and Information Engineering, Hangzhou Dianzi University, Hangzhou, China

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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.

Figures & tables

Figure 1: Deformation of Flexible beam : (a) initial mesh (1980 triangles). (b) deformed mesh (mesh topology and the number of triangles remain the same).
Figure 2: Element quality metrics of flexible beam distorted from the top and bottom layers of beam: (a) the deformed mesh generated by FEM. (b) the deformed mesh generated by the proposed surrogate-model.
Figure 3: The proposed surrogate model for the solution of Eq. ( 1 ) in this case : (a) the solution in XX direction. (b) the solution in YY direction. Both evaluate the approximation accuracy by the point-wise squared error.
Figure 4: Deformation of NACA 0012 airfoil : (a) initial mesh (2603 triangles). (b) deformed mesh.
Figure 5: Element quality metrics of NACA 0012 airfoil under combined translational and rotational rigid-body motion: (a) deformed mesh generated by FEM. (b) deformed mesh generated by the proposed surrogate model.
Figure 6: The proposed surrogate model for the solution of Eq. ( 1 ) in this case : (a) the solution in XX direction. (b) the solution in YY direction. Both evaluate the approximation accuracy by the point-wise squared error.
Figure 7: The verification of the scale invariance of the NACA 0012 airfoil example: (a) The deformed mesh used as the scale reference. (b) The quantitative relationship between the scale scaling factor and output response.
Figure 8: The verification of the scale invariance of the flexible beam example: (a) The deformed mesh used as the scale reference. (b) The quantitative relationship between the scale scaling factor and output response.
Scale kkRLEScale kkRLE
0.13.1335×10−73.1335\times 10^{-7}1.23.3118×10−73.3118\times 10^{-7}
0.50.0000×1000.0000\times 10^{0}1.53.2918×10−63.2918\times 10^{-6}
0.83.1335×10−73.1335\times 10^{-7}2.00.0000×1000.0000\times 10^{0}
1.00.0000×1000.0000\times 10^{0}2.53.3552×10−73.3552\times 10^{-7}
Table 1: Relative Linearization Error (RLE) for Different Scale Parameter Combinations in the NACA 0012 airfoil
Scale kkRLEScale kkRLE
0.11.5924×10−71.5924\times 10^{-7}1.21.4415×10−71.4415\times 10^{-7}
0.50.0000×1000.0000\times 10^{0}1.51.6150×10−61.6150\times 10^{-6}
0.81.5924×10−71.5924\times 10^{-7}2.00.0000×1000.0000\times 10^{0}
1.00.0000×1000.0000\times 10^{0}2.51.6491×10−71.6491\times 10^{-7}
Table 2: Relative Linearization Error (RLE) for Different Scale Parameter Combinations in the flexible beam
Figure 9: The verification of the superposition principle of the surrogate model: (a) is the deformed mesh generated by the NACA 0012 airfoil after rotation and translation superposition under the standard scale (a=1, b=1). (b) is the superposition error under the standard scale.
CombinationParameter aaParameter bbRSE
11.01.03.7663×10−63.7663\times 10^{-6}
22.00.54.4212×10−74.4212\times 10^{-7}
3-1.01.03.0857×10−73.0857\times 10^{-7}
40.5-1.53.2926×10−73.2926\times 10^{-7}
50.12.03.6614×10−73.6614\times 10^{-7}
62.52.53.7587×10−63.7587\times 10^{-6}
Table 3: Relative Superposition Error (RSE) for Different Superposition Parameter Combinations
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