Polarizablity of 2D and 3D conducting objects using method of moments

Abstract: Fundamental antenna limits of the gain-bandwidth product are derived from polarizability calculations. This electrostatic technique has significant value in many antenna evaluations. Polarizability is not available in closed form for most antenna shapes and no commercial electromagnetic packages have this facility. Numerical computation of the polarizability for arbitrary conducting bodies was undertaken using an unstructured triangular mesh over the surface of 2D and 3D objects. Numerical results compare favourably with analytical solutions and can be implemented efficiently for large structures of arbitrary shape.

• The paper introduces a MoM-based numerical method to compute polarizability tensors for arbitrary 2D and 3D conductors.
• It employs triangular mesh discretization and pulse basis functions to solve the electrostatic integral equation with errors on the order of 0.2–1%.
• The method is validated against canonical shapes and is applicable to antenna design, metamaterials, and nanoparticle analysis.

Polarizability Computation of 2D and 3D Conducting Objects via Method of Moments

Introduction and Motivation

Polarizability quantifies the static response of a conducting body under an external electric field, serving as a critical parameter in a variety of domains including electromagnetic scattering, antenna theory, metamaterials, and molecular physics. Notably, connections have been established between polarizability and fundamental antenna limits, particularly maximum gain–bandwidth products, as shown by Gustafsson et al. Consequently, precise numerical evaluation of polarizability for arbitrary geometries is indispensable for the design and analysis of general-purpose radiators and scatterers. However, polarizability lacks closed-form solutions except for basic canonical shapes, and prominent commercial EM solvers do not directly provide this metric.

This work presents a general numerical approach utilizing the Method of Moments (MoM) to compute the polarizability tensor for arbitrary 2D and 3D perfectly conducting objects. The authors implement a triangular mesh-based MoM formulation, validated against canonical analytical solutions, and achieve robust results for both thin (2D) and fully volumetric (3D) geometries.

Mathematical Formulation

The approach begins with the electrostatic integral form of Laplace's equation over the surface of a perfect electrical conductor. For a mesh discretization using $N$ triangular elements across the surface $S$, the integral equation for unknown surface charge density $\rho_j$ subject to unit field excitation along $\hat{x}_j$ is converted to a dense linear equation system:

$L \rho_j = g$

where the observation and source elements are accounted for via double surface integrals—diagonal matrix entries incorporate the necessary self-term singularity compensation, as detailed by Eibert and Hansen.

Pulse basis/testing functions over each mesh triangle enable efficient assembly of the impedance matrix $L$, where off-diagonal terms are approximated via centroid-based interaction, and diagonal terms are analytically evaluated using precise geometric relations between triangle vertices.

The right-hand excitation vector $g$ accounts for the external field and enforces charge conservation ($\int_S \rho_j = 0$). For offset or non-symmetric geometries, the proper determination of the corrective constant $C_j$ is necessary to enforce net zero surface charge—a closed-form expression for $C_j$ is derived as a function of the mesh and problem geometry.

The polarizability tensor component $\gamma_{ij}$ is ultimately computed as the surface moment of induced charge:

$\gamma_{ij} = \sum_{n=1}^N x_{i,n} (\rho_j)_n A_n$

which, when normalized by the cube of the minimum bounding sphere radius, yields a dimensionless and scalable metric for inter-geometric comparison.

Numerical Implementation

The methodology is implemented in MATLAB. Mesh generation is decoupled from the MoM solver, with FEKO used to create STL-format triangular meshes, enabling arbitrary geometry input. The MoM solver processes the mesh file, constructs and inverts the system matrix, computes the induced charge distribution, and outputs the polarizability tensor.

For efficiency, pulse basis functions are selected, but the authors note that higher-order bases (e.g., RWG functions) may improve accuracy, especially for finely tessellated or complex objects. Computational complexity is $O(N^3)$ due to matrix inversion, with runtimes scaling sharply with the number of surface elements.

Validation and Numerical Results

The solver's results are validated against closed-form solutions for a sphere, circular disk, and toroidal ring. The normalized polarizabilities computed numerically closely match analytical values (deviations on the order of 0.2–1%), demonstrating the credibility and accuracy of the numerical implementation.

Parametric studies on spheroids and rectangles with varying aspect ratios are conducted. For spheroids, both tangential and perpendicular polarizabilities track analytical predictions across a wide range of aspect ratios, capturing the asymmetric behavior as the object transitions from prolate to oblate forms. For rectangles, computed polarizabilities concur with literature results for 2D structures, validating the scheme's efficacy for thin conducting objects.

Implications and Future Directions

The presented approach allows precise and efficient computation of polarizability tensors for conducting bodies with arbitrary topology and aspect ratio, overcoming the limitations of analytic solutions and commercial simulation suites. Its direct applicability includes:

• Establishing fundamental bounds on small antenna performance and verifying gain–bandwidth tradeoffs for novel radiator designs.
• Quantifying maximal static and low-frequency scattering cross-sections, including for metamaterial inclusions.
• Evaluating the static electromagnetic response of nanoparticles and engineered scatterers.

The method is extendable to higher-order basis/testing schemes or hybrid integral–differential approaches for improved accuracy and reduced computational overhead. Incorporation with fast solvers or matrix compression techniques (e.g., FMM, H-matrices) could enable rapid evaluation for densely meshed or electrically large objects. The explicit calculation of $C_j$ for arbitrary offsets resolves a technical gap in robust large-scale modeling.

Conclusion

This work establishes a practical framework for computing electrostatic polarizability of arbitrary 2D/3D perfectly conducting objects via a triangular mesh-based MoM solver. The methodology is shown to be accurate, robust, and suitable for rapid theoretical analysis and engineering evaluation of electromagnetic scatterers and antennas, setting the stage for further advances in physical bounds and topology-driven EM design (1402.3681).