Rapid general Electromagnetic Analysis with computational conformal geometry via Conformal Energy Minimization

TL;DR

The paper introduces a fast framework for finite-frequency electromagnetic analysis on arbitrarily shaped metallic surfaces by combining computational conformal geometry with conformal energy minimization (CEM). By mapping complex surfaces to a disk and constructing orthonormal basis functions on the disk, the Green's function and EM fields are represented efficiently, with boundary conditions enforced through auxiliary boundary fields. The method yields a near-diagonal impedance matrix, rapid resonance extraction, and reveals low-energy, doubly degenerate modes that are challenging for conventional FEM approaches. Validation against Schwarz–Christoffel mappings and COMSOL demonstrates both accuracy and an orders-of-magnitude speedup, enabling real-time analysis and design of advanced EM devices.

Abstract

We recently found that the electromagnetic scattering problem can be very fast in an approach expressing the fields in terms of orthonormal basis functions. In this paper we apply computational conformal geometry with the conformal energy minimization (CEM) algorithm to make possible fast solution of finite-frequency electromagnetic problems involving arbitrarily shaped, simply-connected metallic surfaces. The CEM algorithm computes conformal maps with minimal angular distortion, enabling the transformation of arbitrary simply-connected surfaces into a disk, where orthogonal basis functions can be defined and electromagnetic analysis can be significantly simplified. We demonstrate the effectiveness and efficiency of our method by investigating the resonance characteristics of two metallic surfaces: a square plate and a four-petal plate. Compared to traditional finite element methods (e.g., COMSOL), our approach achieves a three-order-of-magnitude improvement in computational efficiency, requiring only seconds to extract resonant frequencies and fields. Moreover, it reveals low-energy, doubly degenerate resonance modes that are elusive to conventional methods. These findings not only provide a powerful tool for analyzing electromagnetic fields on complex geometries but also pave the way for the design of high-performance electromagnetic devices.

Figures (9)

• Figure 1: Discrete conformal maps computed using the CEM algorithm. Discrete conformal maps between figures on the left and those on the right: (a) the square plate to (b) the unit disk (N = 2,490 triangular elements); (c) the four-petal plate to (d) the unit disk (N = 2,488 triangular elements). Both plates have an edge length of 1.81.
• Figure 2: Comparison of numerical and analytical conformal mappings. Triangular mesh vertices obtained by numerical (CEM) and analytical (SC) methods for mapping the unit disk to the square.
• Figure 3: Fourier components $\bm{d(m)}$. Profiles $d(m)$ of the circular basis vectors projected onto the normal of the square plate (black dot-dashed line: SC transformation; red dot-dashed line: CEM method) and four-petal plate (blue dot-dashed line: CEM method).
• Figure 4: Circuit parameters of the square plate under two conformal mapping methods. (a) and (b) display $L_{X m, X m}\left(\kappa,\kappa^{\prime}\right)$ and $\tilde{C}_{N m, N m}^{-1}\left(\kappa,\kappa^{\prime}\right)$ for the V-type boundary condition; (c) and (d) illustrate $L_{X n, X m}\left(\kappa,\kappa\right)$ and $\tilde{C}_{N n, N m}^{-1}\left(\kappa,\kappa\right)$ for the D-type boundary condition. The conformal mapping methods applied in (a) and (c) are based on the SC transformation; those in (b) and (d), on the CEM. The units of the matrices $\mathbf{L}$ and $\mathbf{\tilde{C}}^{-1}$ are $2\pi\mu_{0}\epsilon^{-1}_{0}$ and $2\pi\epsilon^{-1}_{0}$ respectively.
• Figure 5: Circuit parameters for the four-petal plate with the CEM-based conformal mapping. (a) displays $L_{X m, X m}\left(\kappa,\kappa^{\prime}\right)$ and $\tilde{C}_{N m, N m}^{-1}\left(\kappa,\kappa^{\prime}\right)$ for the V-type boundary condition; (b) illustrates $L_{X n, X m}\left(\kappa,\kappa\right)$ and $\tilde{C}_{N n, N m}^{-1}\left(\kappa,\kappa\right)$ for the V-type boundary condition. The units of the matrices $\mathbf{L}$ and $\mathbf{\tilde{C}}^{-1}$ are $2\pi\mu_{0}\epsilon^{-1}_{0}$ and $2\pi\epsilon^{-1}_{0}$ respectively.