Electromagnetic waves

Electric field

Strong formulation

We start from Maxwell’s equations along with the material relations

\begin{align} \boldsymbol{B} &= \boldsymbol{\mu} \boldsymbol{H} + \boldsymbol{B_r} \\[5pt] \boldsymbol{D} &= \boldsymbol{\varepsilon} \boldsymbol{E} + \boldsymbol{D_r} \\[5pt] \boldsymbol{J} &= \boldsymbol{\sigma} \boldsymbol{E} + \boldsymbol{J_r}\ . \end{align}

Where $\boldsymbol{\varepsilon}$ is the electric permittivity tensor, $\boldsymbol{\mu}$ is the magnetic permeability tensor and $\boldsymbol{\sigma}$ is the electric conductivity tensor. Quantities $\boldsymbol{D_r}$ , $\boldsymbol{B_r}$ and $\boldsymbol{J_r}$ are typically associated with remanent effects. We assume that quantities $\boldsymbol{\epsilon}$ , $\boldsymbol{\mu}$ , $\boldsymbol{\sigma}$ , $\boldsymbol{D_r}$ , $\boldsymbol{B_r}$ and $\boldsymbol{J_r}$ change slowly over time compared to the electromagnetic waves frequency. This allows to neglect their time derivatives.

Using this assumptions we can rewrite Faraday’s law as

\begin{align} \nabla \times \boldsymbol{E} &= -\boldsymbol{\mu} \frac{\partial \boldsymbol{H}}{\partial t} \\[10pt] \boldsymbol{\mu}^{-1} (\nabla \times \boldsymbol{E}) &= -\frac{\partial \boldsymbol{H}}{\partial t}. \end{align}

We apply the curl operator to both sides and assume the curl and time derivative can be interchanged using the time derivative of a curl identity

\begin{align} \nabla \times (\boldsymbol{\mu}^{-1} (\nabla \times \boldsymbol{E})) &= -\frac{\partial}{\partial t} (\nabla \times \boldsymbol{H}). \end{align}

Substituting in the Ampère-Maxwell, material relations (2) and (3) and by rearranging we obtain

\begin{align} \nabla \times (\boldsymbol{\mu}^{-1} (\nabla \times \boldsymbol{E})) + \boldsymbol{\sigma} \frac{\partial \boldsymbol{E}}{\partial t} + \boldsymbol{\varepsilon} \frac{\partial^2 \boldsymbol{E}}{\partial t^2} &= 0. \end{align}

If we assume isotropic material, all tensor quantities become scalars, the equation results in

\begin{align} \nabla \times (\frac{1}{\mu} (\nabla \times \boldsymbol{E})) + \sigma \frac{\partial \boldsymbol{E}}{\partial t} + \varepsilon \frac{\partial^2 \boldsymbol{E}}{\partial t^2} &= 0. \end{align}

Weak formulation

To obtain the weak formulation of (7) we multiply the equation by a test function $\boldsymbol{E}'$ and integrating over the electromagnetic waves physics domain $\Omega$ :

\begin{align} \int_\Omega\ \nabla \times (\boldsymbol{\mu}^{-1} \, \nabla \times \boldsymbol{E}) \cdot \boldsymbol{E}' \, d\Omega + \int_\Omega\ \boldsymbol{\sigma} \frac{\partial \, \boldsymbol{E}}{\partial t} \cdot \boldsymbol{E}' \, d\Omega + \int_\Omega\ \boldsymbol{\epsilon} \frac{\partial^2 \, \boldsymbol{E}}{\partial t^2} \cdot \boldsymbol{E}' \, d\Omega &= 0. \end{align}

We can rewrite the first term using the divergence of cross product identity

\begin{align} \int_{\Omega}\ \nabla \cdot ((\boldsymbol{\mu}^{-1} \nabla \times \boldsymbol{E}) \times \boldsymbol{E}^{\prime})\ d\Omega + \int_{\Omega}\ (\boldsymbol{\mu}^{-1} \nabla \times \boldsymbol{E}) \cdot \ (\nabla \times \boldsymbol{E}^{\prime}), \end{align}

where the divergence theorem can rewritten using the divergence theorem

\begin{align} \int_{\Gamma}\ ((\boldsymbol{\mu}^{-1} \nabla \times \boldsymbol{E}) \times \boldsymbol{E}^{\prime}) \cdot \boldsymbol{n}\ d\Gamma + \int_{\Omega}\ (\boldsymbol{\mu}^{-1} \nabla \times \boldsymbol{E}) \cdot \ (\nabla \times \boldsymbol{E}^{\prime})\ d\Omega. \end{align}

Substituting in the Faraday’s law and using the material relation (1) we obtain

\begin{align} -\int_{\Gamma}\ (\frac{\partial \boldsymbol{H}}{\partial t} \times \boldsymbol{E}^{\prime}) \cdot \boldsymbol{n}\ d\Gamma + \int_{\Omega}\ (\boldsymbol{\mu}^{-1} \nabla \times \boldsymbol{E}) \cdot \ (\nabla \times \boldsymbol{E}^{\prime})\ d\Omega. \end{align}

Finally, by using the scalar triple product identity we derive the weak formulation

\begin{align} -\int_{\Gamma}\ (\boldsymbol{n} \times \frac{\partial \boldsymbol{H}}{\partial t}) \cdot \boldsymbol{E}^{\prime}\ d\Gamma + \int_{\Omega}\ (\boldsymbol{\mu}^{-1} \nabla \times \boldsymbol{E}) \cdot \ (\nabla \times \boldsymbol{E}^{\prime})\ d\Omega \\[10pt] + \int_{\Omega}\ \boldsymbol{\sigma} \frac{\partial \boldsymbol{E}}{\partial t} \cdot \boldsymbol{E}^{\prime}\ d\Omega + \int_{\Omega}\ \boldsymbol{\epsilon} \frac{\partial^2 \boldsymbol{E}}{\partial t^2} \cdot \boldsymbol{E}^{\prime}\ d\Omega &= 0. \end{align}

Available Interactions

Constraint

Fixes the electromagnetic field value at the defined region, enforcing idealized behavior such as perfect reflection.

Perfectly matched layer

Implements an absorbing boundary region to simulate infinite space and prevent reflections from truncated boundaries. Placed at the outer boundary of the air domain, it absorbs outgoing waves without reflection.

Key properties:

Used to simulate open boundaries for radiating structures.

Two types available:

AML type: Suitable for smooth boundaries.
Box PML: Suitable for rectangular or cornered boundaries.
The number of PML layers can be tuned in the Shared PML settings.

Rectangular port

A special case of the Eigenmode port, used when the port geometry and material are well-defined and homogeneous. Provides analytical solutions for the field distribution under certain geometric assumptions.

Assumptions:

The port surface must be rectangular.
The dielectric across the port must be uniform.
All four port edges must be bounded by perfect conductors.

Key properties:

Perfectly absorbing (impedance matched, no reflection).
Provides Electric field (E) and Magnetic field (H) access, but not Voltage (V) or Current (I).

Perfect conductor

Imposes a Dirichlet boundary condition on the electric field. Sets E = 0 on the target boundary or region.

Key properties:

Models highly conductive materials as ideal conductors.
Reduces the number of unknowns and computational cost.

Eigenmode port

Defines a port that perfectly absorbs the excited mode with matched impedance (no reflection). This port type is typically used when the cut-off frequency or mode shapes are unknown. It supports both low and high-frequency signals.

Key properties:

No Voltage (V) or Current (I) output — only Electric field (E) and Magnetic field (H) are available.

Dielectric Loss

Models energy absorption in dielectric materials as the electromagnetic wave propagates through them. The absorbed energy is dissipated as heat.

Key properties:

Target region: dielectric domains.
Characterized by loss tangent, defined as $tan \ \delta = \frac{\epsilon^{\prime\prime}}{\epsilon^\prime}$ where $\epsilon^{\prime}$ is the real part and $\epsilon^{\prime\prime}$ is the imaginary part of the complex permittivity $\epsilon = \epsilon^{\prime} - j\epsilon^{\prime\prime}$ .
The real part is taken from the material’s permittivity; the imaginary part is computed as $\epsilon^{\prime\prime} = - tan \ \delta * \epsilon^{\prime}$
The predefinedemwave function in Allsolve takes as arguments, among many, the real part and imaginary part of the complex permittivity.

Lump V/I

Defines a lumped voltage or current source for feeding compact structures where the port dimensions are much smaller than the wavelength. This type of port assumes that the electric potential v is physically meaningful and that the local electric field is derived from $E = –\nabla v$ .

Key properties:

Provides direct access to Voltage (V) and Current (I) at the port.
Allows connecting external circuit elements (RLC components).
Suitable for compact feed structures such as microstrip, CPW, or GCPW.

Periodicity

Imposes periodic boundary conditions for the electromagnetic fields. Reduces computational domain size for symmetric problems.

Absorbing boundary

Boundary impedance

Use cases

Electromagnetic waves: Patch antenna