Study on a Fast Solver for Combined Field Integral Equations of 3D Conducting Bodies Based on Graph Neural Networks

TL;DR

This work tackles the computational burden of solving combined-field integral equations (CFIEs) for 3D conducting bodies. It introduces GraphSolver, a graph neural network that represents RWG-based discretizations as graphs and directly predicts the real and imaginary surface current densities for each triangular element. The architecture combines an upsampling FCN, a Graph Convolutional Network, and six downsampling FCNs to produce per-node current components (x, y, z) for both their real and imaginary parts, enabling efficient solution across geometries of increasing complexity. Numerical results across basic shapes, missile-shaped, and airplane-shaped targets show close agreement with traditional MoM results and bistatic RCS curves, with faster training than physics-inspired baselines like PhiGRL. The approach holds promise for real-time EM simulations on unstructured meshes, with transfer learning aiding performance on more complex geometries.

Abstract

In this paper, we present a graph neural networks (GNNs)-based fast solver (GraphSolver) for solving combined field integral equations (CFIEs) of 3D conducting bodies. Rao-Wilton-Glisson (RWG) basis functions are employed to discretely and accurately represent the geometry of 3D conducting bodies. A concise and informative graph representation is then constructed by treating each RWG function as a node in the graph, enabling the flow of current between nodes. With the transformed graphs, GraphSolver is developed to directly predict real and imaginary parts of the x, y and z components of the surface current densities at each node (RWG function). Numerical results demonstrate the efficacy of GraphSolver in solving CFIEs for 3D conducting bodies with varying levels of geometric complexity, including basic 3D targets, missile-shaped targets, and airplane-shaped targets.

Figures (11)

• Figure 1: Relationship between RWG basis function and surface current density.
• Figure 2: Graph representation of RWG basis functions in a 3D conducting body. The 3D conducting body is first discretized into triangular meshes, with each triangular element represented as a node in the graph. Two nodes are connected to form an edge in the graph if their corresponding triangular elements share a common side.
• Figure 3: GraphSolver for solving CFIEs of 3D conducting bodies. It consists of one upsampling FCN, one GCN, and six downsampling FCNs. Their detailed architectures are illustrated. U-FCN is the upsampling FCN. D-FCN-x/y/z(r/i) denotes the downsampling FCN for the real (r) and imaginary (i) parts of the x, y and z components of the surface current density. Linear-$[a,b]$ denotes a linear layer with the input and output channels $a$ and $b$ respectively. GraphConv$k-[c,d,mean,w]$ denotes the $k$-th graph convolutional layer of which the input channel, output channel, aggregation function and kernel width are $c$, $d$, mean function and $w$.
• Figure 4: Schematic of basic 3D targets: spheroid, conical frustum, and hexahedron (from left to right). O, Ot and Ob denotes the body center, top center and base center.
• Figure 5: Convergence curve of GraphSolver for solving CFIEs of basic 3D targets.