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1.3 Boundary Integral Methods for Electromagnetics

by Jose Alves

Boundary integral methods, such as the Method of Moments (MoM) and the Boundary Element Method (BEM), are numerical computational techniques used to solve linear partial differential equations. These methods are formulated as integral equations that involve only the boundary of the considered domain. For applications like antenna radiation, magnetostatics, or eddy-current problems, volumetric methods such as the Finite Element Method (FEM) require the creation of an 'artificial box' to enclose the entire scenario. This entails discretizing the air region within the box and applying special boundary conditions on the box surfaces to manage the artificial truncation of the medium where radiation occurs, which may result in considerable modelling errors.

When employing boundary integral methods, only the surfaces of the objects under analysis need to be discretized. This approach eliminates the need for an air mesh, significantly reducing the complexity of modelling and meshing. Boundary integral methods are well-established for frequency domain problems, such as radiation issues in microwave and antenna engineering. They are also advantageous in simulations involving moving parts, as they avoid the need for remeshing and remapping. Furthermore, these methods are supported by a robust mathematical foundation that demonstrates their convergence.

Method of Moments

MoM belongs to the family of full-wave methods that discretely solve Maxwell's equations, which describe the physics of the problem. This method is commonly applied in scenarios such as antenna modelling and the evaluation of installed antenna performance on electrically small platforms. The MoM offers various boundary conditions depending on the material properties and the simulation's required accuracy, including:

• Electric Field Integral Equation (EFIE) for perfect conductors,
• Impedance Boundary Conditions (IBC) for lossy materials,
• Combined Field Integral Equation (CFIE) for either perfect or lossy materials if the surface is closed,
• Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) for bulk materials,
• Electric and Magnetic Current Combined Field Integral Equation (JMCFIE) for dielectric boundaries, typically suitable for low contrast dielectric boundaries.

Boundary Element Method

BEM, in many aspects similar to the MoM, are also used for addressing eddy-current, magnetostatic, and electrostatic problems. These methods are often coupled with FEM to model the generally nonlinear behaviour of electromagnetic components. A key advantage of this approach is the avoidance of complex air meshes and artificial truncation of the computational domain.

Special attention is required when solving the system of equations, particularly as results near 0 Hz in the frequency domain can lead to an ill-conditioned algebraic system. Thus, the use of preconditioners is essential to render the system of equations solvable. Identifying an effective yet computationally affordable preconditioner is crucial to minimise iteration numbers, although finding an appropriate preconditioner can be challenging.

Acceleration Techniques

When employing MoM, BEM, or other techniques from the Boundary-Integral family, the advantage of not needing an air-mesh is offset by certain challenges: singular integrals must be addressed, and the resulting system matrices are fully populated, in contrast to the data-sparse matrices typical of FEM. A straightforward implementation of these methods may prove ineffective for practical applications that require a large number of elements due to the computational complexity, which increases at least quadratically. This means that doubling the number of elements results in a fourfold increase in computational effort. To mitigate this, several acceleration techniques have been developed to reduce complexity to nearly linear levels, where doubling the number of elements only doubles the computational effort. These include Hierarchical Matrices, Adaptive Cross Approximation (ACA), methods based on Fast Fourier Transformations (FFT), and notably, the Fast Multipole Method (FMM).

All these acceleration techniques share a common approach: the computational domain is hierarchically divided into subdomains, with interactions between these clusters categorised into near- and far-field. By approximating far-field interactions, an almost linear computational complexity is achieved. Although not as straightforward as with FEM, these advanced methods can also be parallelized to scale effectively on HPC clusters.

The FMM significantly extends the applicability of MoM and BEM to electrically large structures, often up to hundreds of wavelengths. Typical uses include modelling of antennas and antenna arrays, reflectors, radome analysis (including those made of Frequency Selective Surfaces), installed antennas, and EMI/EMC risk mitigation, particularly where strong interactions occur between the antenna and surrounding structures, such as the bumper effect on automotive radars. Other applications include High-Intensity Radiated Fields risk assessment and Radar Cross Section analysis for individual components and complete structures. When applying acceleration techniques like the FMM to MoM/BEM, special care is needed to solve the larger systems of equations iteratively since a direct solution is generally unfeasible and unavailable with FMM. However, a meticulously implemented MoM or BEM, along with these described acceleration methods, equips electrical engineers with a powerful tool for achieving highly accurate results with significantly reduced modelling effort for many radiation-involved problems. In multiphysics contexts, no additional air-mesh is required, and the structural model alone is sufficient.

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