Electromagnetics

Electromagnetic problems are commonly described by Maxwell’s equations, which govern the behavior of electric and magnetic fields. The Finite Element Method (FEM) provides a powerful numerical approach for solving these equations in complex geometries. This section provides a concise overview of the mathematical formulation for electromagnetic problems using finite elements.

Maxwell’s Equations

The time-harmonic Maxwell’s equations in free space are given by:

\[\begin{split} \begin{align} \nabla \times \mathbf{E} &= -j\omega \mu \mathbf{H}, \label{eq:maxwell1} \\ \nabla \times \mathbf{H} &= j\omega \varepsilon \mathbf{E}, \label{eq:maxwell2} \\ \nabla \cdot \mathbf{B} &= 0, \label{eq:maxwell3} \\ \nabla \cdot \mathbf{D} &= 0, \label{eq:maxwell4} \end{align} \end{split}\]

where \(\mathbf{E}\) is the electric field, \(\mathbf{H}\) is the magnetic field, \(\mathbf{B}\) is the magnetic flux density, \(\mathbf{D}\) is the electric displacement field, \(\omega\) is the angular frequency, \(\mu\) is the permeability, and \(\varepsilon\) is the permittivity.

Weak Formulation

The weak form of Maxwell’s equations is obtained by multiplying each equation with suitable test functions and integrating over the domain \(\Omega\). For the electric field \(\mathbf{E}\), the weak form is given by:

\[ \begin{equation} \int_{\Omega} \nabla \times \mathbf{E} \cdot \nabla \times \mathbf{v} \, d\Omega + \omega^2 \mu \varepsilon \mathbf{E} \cdot \mathbf{v} \, d\Omega = 0, \end{equation} \]

where \(\mathbf{v}\) is a test function belonging to the function space \(H(\text{curl}; \Omega)\).

Similarly, for the magnetic field \(\mathbf{H}\), the weak form is given by:

\[ \begin{equation} \int_{\Omega} \nabla \times \mathbf{H} \cdot \nabla \times \mathbf{u} \, d\Omega - \omega^2 \mu \varepsilon \mathbf{H} \cdot \mathbf{u} \, d\Omega = 0, \end{equation} \]

where \(\mathbf{u}\) is a test function belonging to the function space \(H(\text{curl}; \Omega)\).

Finite Element Discretization

To apply the finite element method, the domain \(\Omega\) is discretized into elements, and the electromagnetic fields are approximated using piecewise basis functions. The discretized weak form leads to a system of linear equations, which can be solved numerically to obtain the finite element solution.

References

The following references provide in-depth coverage of the mathematical background for electromagnetism using finite elements: [] [] []