Boundary element methods (BEM) are numerical approaches utilized in scientific and engineering fields to resolve partial differential equations. They have found extensive usage in the domain of electromagnetics to tackle problems like Maxwell’s equations.
BEM has a boundary-centric approach, meaning it only requires information on the boundaries of the solution domain, unlike finite element methods (FEM) which require solution of equations at each node within the interior of the domain.
The utilization of BEM in electromagnetics presents several advantages, one of which is its proficiency in handling complex geometries with a high level of precision. The solution domain is represented through its boundaries, which allows for an optimized representation of the problem, making BEM ideal for solving electromagnetic problems with complex geometries, for instance, problems with multiple objects of different shapes, sizes, and positions.
Additionally, BEM can handle both static and dynamic electromagnetic problems, solving time-invariant Maxwell’s equations for static problems, and time-dependent Maxwell’s equations for dynamic problems using time-stepping methods. BEM is also equipped to handle multiple physics, including coupled electro-magnetic-thermal problems, electromagnetic problems with material nonlinearities, and problems with time-varying magnetic permeability.
In terms of accuracy, BEM can provide high-resolution solutions for electromagnetic problems. The accuracy can be improved by refining the mesh, increasing the order of the interpolation functions, or through adaptive mesh refinement techniques.
However, there are some limitations to using BEM in electromagnetics. One of the major limitations is the computation time needed to solve the problem, as the solution necessitates the solution of a large system of equations. Furthermore, the solution may be vulnerable to the choice of mesh, interpolation functions, and boundary conditions, making it imperative to choose these carefully.
In conclusion, BEM is a widely utilized numerical method in the field of electromagnetics. Its capability to handle complex geometries with precision, handle multiple physics, and provide high-resolution solutions make it a potent tool for solving a wide range of electromagnetic problems. Despite its limitations, BEM continues to be a popular choice for solving electromagnetic problems due to its accuracy and efficiency.
BEM packages
BEM++ (Boundary Element Method++) is a free and open-source numerical library for solving boundary integral equations. Developed for the solution of partial differential equations (PDEs) in various fields such as electromagnetics, fluid mechanics, and acoustics, BEM++ provides a flexible and efficient framework for boundary element methods (BEMs), a class of numerical methods for solving PDEs on unbounded domains. The library is built on Python and offers a high-level interface for solving boundary integral equations.
BEM++ offers a comprehensive collection of solvers, including low- and high-frequency solvers and solvers for specific applications such as acoustic scattering and electromagnetic wave propagation. The solvers are based on advanced numerical techniques such as the Galerkin method and the collocation method. BEM++ also has parallel computing support, allowing users to run simulations on large and complex problems much faster than traditional serial algorithms. The library includes built-in support for parallel processing, including the use of multi-core processors and high-performance computing clusters.
Advanced visualization capabilities are also included in BEM++, allowing users to visualize the results of simulations including electric and magnetic fields, fluid flow, and pressure fields. This is crucial for verifying the accuracy of simulations and ensuring the results meet the requirements of the application. BEM++ also has a comprehensive API, making it easy to integrate into other software systems. The API includes a large collection of functions for manipulating boundary elements, including functions for meshing, evaluating integral operators, and assembling system matrices.
BEM++ is widely used in the fields of electromagnetics and fluid mechanics, and is well-suited for the simulation of complex physical processes like the interaction of electromagnetic fields with materials and structures and fluid flow in complex geometries. The library also utilizes Hierarchical Matrices (H-matrices) in its implementation of boundary element methods, providing improved performance and scalability, reduced memory requirements, and improved accuracy for long-range interactions. This makes it possible to handle large and complex problems with a high level of accuracy and reliability for simulations.