General Material¶

Introduction¶

A material in the time-harmonic Maxwell model has a magnetic permeability $\mu$, electric permittivity $\varepsilon$, and electric conductivity $\sigma$. A general material with constant properties is created with the Constant factory:

my_material = TimeHarmonicMaxwellGeneralMaterial.Constant(
   name = "My Material",
   marker = my_material_marker,
   relative_magnetic_permeability = 1.0,
   relative_electric_permittivity = 2.25,
   electric_conductivity = 1e-15,
)

where

relative_magnetic_permeability is the relative magnetic permeability $\mu_r$ (default 1.0),
relative_electric_permittivity is the relative electric permittivity $\varepsilon_r$ (default 1.0),
electric_conductivity is the absolute electric conductivity $\sigma$ in S/m (default 0.0).

Each of the three properties accepts either a scalar float (isotropic material) or a $3\times3$ FixedMatrix (anisotropic material).

Material Properties¶

Magnetic Permeability¶

The magnetic permeability $\mu$ describes the response of a medium to an applied magnetic field. It is the proportionality factor between the magnetic field vectors $\tilde{\mathbf{B}}$ and $\tilde{\mathbf{H}}$: $$ \begin{align} \tilde{\mathbf{B}} = \mu \tilde{\mathbf{H}}. \end{align} $$

It is commonly expressed as $$ \begin{align} \mu = \mu_0 \mu_r, \end{align} $$

where $\mu_0$ denotes the magnetic permeability of free space and $\mu_r$ is the relative magnetic permeability of the medium.

In the most general case, the magnetic permeability $\mu$ may depend on the radiation frequency, spatial position, or propagation direction within the medium. But for most materials and practical applications described by the time-harmonic Maxwell model, $\mu$ is about equal to the vacuum permeability $\mu_0$. This corresponds to a relative magnetic permeability $\mu_r=1$.

The relative magnetic permeability $\mu_r$ is passed to the Constant factory as a scalar (isotropic) or as a $3\times3$ FixedMatrix (anisotropic).

Electric Permittivity¶

The electric permittivity $\varepsilon$ quantifies the response of a medium to an applied electric field and determines the relation between the electric displacement field $\tilde{\mathbf{D}}$ and the electric field $\tilde{\mathbf{E}}$: $$ \begin{align} \tilde{\mathbf{D}} = \varepsilon \tilde{\mathbf{E}}. \end{align} $$

It is conventionally written as $$ \begin{align} \varepsilon = \varepsilon_0 \varepsilon_r, \end{align} $$

where $\varepsilon_0$ is the permittivity of free space and $\varepsilon_r$ denotes the relative permittivity of the material.

In general, the permittivity $\varepsilon$ may vary with frequency, position, or direction within the medium. This reflects dispersive, inhomogeneous, or anisotropic behavior.

For many dielectric materials, the electric permittivity exceeds the vacuum value only moderately, with relative permittivity typically in the range $\varepsilon_r \approx 2 - 10$. Materials with stronger polarization response, such as polar dielectrics, may show larger values. Metals can be described by an effectively very large and frequency-dependent permittivity, which dominates their electromagnetic behavior.

The relative electric permittivity $\varepsilon_r$ is passed to the Constant factory as a scalar (isotropic) or as a $3\times3$ FixedMatrix (anisotropic).

Electric Conductivity¶

The electric conductivity $\sigma$ characterizes the ability of a medium to conduct electric current in response to an applied electric field. It relates the electric field $\tilde{\mathbf{E}}$ to the induced current density $\tilde{\mathbf{J}}$ through Ohm's law: $$ \begin{align} \tilde{\mathbf{J}} = \sigma \tilde{\mathbf{E}}. \end{align} $$

In general, the conductivity $\sigma$ may depend on frequency, spatial position, or direction within the material. This reflects dispersive, inhomogeneous, or anisotropic transport properties. In the time-harmonic electromagnetic models, $\sigma$ accounts for dissipative losses and contributes to the attenuation of electromagnetic waves in conductive media.

For many dielectric materials, the conductivity is negligibly small and can often be approximated as zero. But in conductors it attains large values and dominates the electromagnetic response.

The electric conductivity $\sigma$ (absolute, in S/m) is passed to the Constant factory as a scalar (isotropic) or as a $3\times3$ FixedMatrix (anisotropic).

Examples¶

Typical general materials for the time-harmonic Maxwell model are defined as follows:

from mufem.electromagnetics.timeharmonicmaxwell import (
    TimeHarmonicMaxwellGeneralMaterial,
)

# A typical quartz glass:
glass_material = TimeHarmonicMaxwellGeneralMaterial.Constant(
   name = "Glass",
   marker = glass_material_marker,
   relative_magnetic_permeability = 1.0,
   relative_electric_permittivity = 2.25,
   electric_conductivity = 1e-15,
)

# A typical metal:
metal_material = TimeHarmonicMaxwellGeneralMaterial.Constant(
   name = "Metal",
   marker = metal_material_marker,
   relative_magnetic_permeability = 1.0,
   relative_electric_permittivity = 100,
   electric_conductivity = 1e7,
)

An anisotropic material is defined by passing a $3\times3$ FixedMatrix for the respective property, e.g. via the Orthotropic factory:

from mufem import FixedMatrix

anisotropic_material = TimeHarmonicMaxwellGeneralMaterial.Constant(
   name = "Anisotropic Crystal",
   marker = crystal_marker,
   relative_electric_permittivity = FixedMatrix.Orthotropic(2.25, 2.25, 4.6),
)