Introduction to the Finite Element Method in Electromagnetics

The Finite Element Method (FEM) is a powerful numerical tool for solving electromagnetic problems, particularly well-suited for complex geometries and boundary conditions. For beginners, mastering its fundamental principles and implementation steps is crucial.

In electromagnetics, FEM is typically used to solve Maxwell's equations by discretizing continuous space, transforming complex partial differential equations into linear algebraic problems. The core implementation steps include:

Mesh Generation: Dividing the solution domain into finite elements (such as triangles or quadrilaterals) to create discrete node and element information. Code implementation often involves algorithms like Delaunay triangulation or advancing front method for 2D/3D mesh creation.

Basis Function Selection: Approximating field quantities (like electric or magnetic fields) within each element using basis functions, typically employing linear or higher-order polynomials. Implementation requires defining shape functions that satisfy continuity conditions across element boundaries.

Stiffness Matrix Assembly: Transforming differential equations into matrix equations through variational principles or weighted residual methods. This involves numerical integration over each element and careful indexing for global matrix construction.

Boundary Condition Application: Handling Dirichlet or Neumann boundary conditions to ensure solution uniqueness and physical合理性. Code implementation typically modifies the global matrix and right-hand side vector accordingly.

Linear System Solution: Solving the final system of linear equations using direct methods (like LU decomposition) or iterative methods (such as conjugate gradient). The choice depends on problem size and matrix sparsity patterns.

The accompanying source code provides 1D and 2D implementation examples covering typical problems like electrostatic fields and waveguide propagation. Through these examples, beginners can progressively understand the complete workflow from mesh generation and matrix assembly to post-processing analysis, laying the foundation for more complex 3D problems.