General Material¶

Introduction¶

A time-domain magnetic general material is defined by its magnetic permeability $\mu$, electric conductivity $\sigma$, and remanent magnetic flux density $\mathbf{B}_r$ — plus a flag indicating whether eddy currents are computed in this material.

A material is created by specifying a property method for each of the three quantities:

my_material = TimeDomainMagneticGeneralMaterial(
    name="My Material",
    marker=my_material_marker,
    magnetic_permeability=my_mu,
    electric_conductivity=my_sigma,
    remanent_flux_density=my_br,
    has_eddy_currents=True,
)

Each property accepts either an explicit method object (see tables below) or a convenient shorthand (a scalar, a list, a LinearTemperatureCoefficient, …) that is wrapped automatically. Passing None (the default) selects the trivial method (non-magnetic / insulating / zero remanence).

Typical applications¶

The general material covers most regions encountered in low-frequency electromagnetic devices:

Air, plastics, encapsulants — non-magnetic, insulating (defaults).
Copper / aluminium windings — non-magnetic, finite conductivity; eddy currents disabled for stranded coils, enabled for solid conductors.
Soft-magnetic iron (motor / transformer cores) — nonlinear BH curve, optional eddy currents.
Permanent magnets (NdFeB, SmCo, ferrite) — near-unity permeability with a constant remanent flux density; magnetization may vary in space for Halbach-style arrays.
Conductive shells and shielding — high conductivity with eddy currents to capture induced losses.

For superconducting materials, see the Superconductor Module; for stacks of thin ferromagnetic sheets, see the Lamination Module. A hysteresis module is planned.

Material Properties¶

Magnetic Permeability¶

The magnetic permeability links the magnetic field $\mathbf{H}$ to the magnetic flux density $\mathbf{B}$ through the constitutive law $\mathbf{B} = \mu \mathbf{H}$. In general, $\mu$ is a symmetric rank-2 tensor

\[ \begin{align} \mu &= \left( \begin{array}{ccc} \mu_{xx} & \mu_{xy} & \mu_{xz} \\ & \mu_{yy} & \mu_{yz} \\ & & \mu_{zz} \end{array} \right), \end{align} \]

whose components may further depend on the magnetic flux density (nonlinear materials) or on other physical quantities.

The available methods are summarized in Table 1.

Table 1: Magnetic Permeability Methods
Name	Example	Description
Non-magnetic (default)	`my_mu = None` Equivalently: `my_mu = MagneticPermeabilityMethodNonMagnetic()`	Vacuum permeability $ \mu = \mu_0 $.
Linear Isotropic	`my_mu = 132.0` Equivalently: `my_mu = MagneticPermeabilityMethodLinearIsotropic( relative_permeability_value=132.0, )`	Constant, isotropic relative permeability $\mu_r$, so that $ \mu = \mu_0 \mu_r $.
BH Curve	`my_mu = ([0., 120., 330., 500., 3100.], [0.00, 0.10, 1.00, 1.30, 1.65])` Equivalently: `my_mu = MagneticPermeabilityMethodBHCurve( magnetic_field_strength=[0, 120, 330, 500, 3100], magnetic_flux_density=[0.00, 0.10, 1.00, 1.30, 1.65], )`	Nonlinear isotropic permeability obtained from a tabulated $\|\mathbf{B}\|$–$\|\mathbf{H}\|$ curve via $$ \mu(\|\mathbf{H}\|) = \frac{\|\mathbf{B}(\|\mathbf{H}\|)\|}{\|\mathbf{H}\|}. $$ The table must be strictly monotonic.
Linear Anisotropic	`my_mu = MagneticPermeabilityMethodLinearAnisotropic( permeability_tensor=FixedMatrix([ [mu_xx, mu_xy, mu_xz], [mu_xy, mu_yy, mu_yz], [mu_xz, mu_yz, mu_zz], ]), )`	Constant, fully anisotropic permeability tensor.

Electric Conductivity¶

The electric conductivity $\sigma$ links the electric field $\mathbf{E}$ to the electric current density $\mathbf{J}$ via Ohm's law $\mathbf{J} = \sigma \mathbf{E}$. It is expressed in siemens per meter (S/m). In general, $\sigma$ is a symmetric rank-2 tensor

\[ \begin{align} \sigma &= \left( \begin{array}{ccc} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ & \sigma_{yy} & \sigma_{yz} \\ & & \sigma_{zz} \end{array} \right), \end{align} \]

whose components may depend on temperature, current density, etc. In many practical cases the material is assumed isotropic.

Table 2: Electric Conductivity Methods
Name	Example	Description
Insulating (default)	`my_sigma = None` Equivalently: `my_sigma = ElectricConductivityMethodInsulating()`	Zero conductivity: $ \sigma = 0 $.
Linear Isotropic	`my_sigma = 5.8e7` Equivalently: `my_sigma = ElectricConductivityMethodLinearIsotropic( conductivity_value=5.8e7, )`	Constant, isotropic conductivity.
Linear Anisotropic	`my_sigma = ElectricConductivityMethodLinearAnisotropic( conductivity_tensor=FixedMatrix([ [5.8e7, 1.0e6, 0.0], [1.0e6, 4.2e7, 0.0], [0.0, 0.0, 2.0e7], ]), )`	Constant, fully anisotropic conductivity tensor.
Linear Temperature Coefficient	`my_sigma = mufem.methods.LinearTemperatureCoefficient( reference_value=5.8e7, temperature_coefficient=-3.9e5, reference_temperature=293.15, )`	Isotropic conductivity with linear temperature dependence $$ \sigma(T) = \sigma_{\text{ref}} + \alpha \left( T - T_{\text{ref}} \right). $$ Requires a Temperature Model to be present.
Temperature Table	`my_sigma = mufem.methods.TemperatureTable( temperature=[293.15, 400.0, 600.0, 800.0], values=[5.8e7, 4.9e7, 3.8e7, 3.0e7], )`	Isotropic conductivity tabulated against temperature; linearly interpolated. Requires a Temperature Model to be present.

Remanent Flux Density¶

The remanent flux density $\mathbf{B}_r$ accounts for permanent magnetization. With the convention $\mathbf{B} = \mu \mathbf{H} + \mathbf{B}_r$, it is the residual magnetic flux density retained after the external magnetizing field is removed. It is generally a vector

\[ \begin{align} \mathbf{B}_r = B_r \mathbf{d}, \end{align} \]

where $B_r$ is the magnitude and $\mathbf{d}$ the unit vector of the magnetization direction. The available methods are summarized in Table 3.

Table 3: Remanent Flux Density Methods
Name	Example	Description
Zero (default)	`my_br = None` Equivalently: `my_br = MagnetizationMethodNoMagnetization()`	$ \mathbf{B}_r = 0 $.
Constant Direction	`my_br = [0.0, 0.0, 1.2]` Equivalently: `my_br = MagnetizationMethodDirection( x=0.0, y=0.0, z=1.2, )`	Constant remanent flux density vector.
Variable	`halbach_components = mufem.CffConstantVector((0, 1, 0)) halbach_remanence = mufem.CffCylindricalCoordinate( vec=halbach_components, ) my_br = MagnetizationMethodVariable( magnetization=halbach_remanence, )`	Spatially varying remanent flux density driven by a user-provided vector coefficient. Useful for Halbach arrays, poloidal patterns, and any magnetization built from the existing coefficient building blocks (cylindrical / spherical coordinates, expressions, etc.).

Example¶

A linear permanent-magnet material can be created in two equivalent ways.

Using the shorthand form:

from mufem.electromagnetics.timedomainmagnetic import TimeDomainMagneticGeneralMaterial

magnet_material = TimeDomainMagneticGeneralMaterial(
    name="Magnet",
    marker="Magnet" @ Vol,
    magnetic_permeability=1.12,
    remanent_flux_density=[1.02, 0.0, 0.0],
    has_eddy_currents=False,
)

Using explicit method objects:

from mufem.electromagnetics.timedomainmagnetic import (
    ElectricConductivityMethodInsulating,
    MagneticPermeabilityMethodLinearIsotropic,
    MagnetizationMethodDirection,
    TimeDomainMagneticGeneralMaterial,
)

magnet_material = TimeDomainMagneticGeneralMaterial(
    name="Magnet",
    marker="Magnet" @ Vol,
    magnetic_permeability=MagneticPermeabilityMethodLinearIsotropic(1.12),
    electric_conductivity=ElectricConductivityMethodInsulating(),
    remanent_flux_density=MagnetizationMethodDirection(1.02, 0.0, 0.0),
    has_eddy_currents=False,
)

Name	Example	Description
Non-magnetic (default)	`my_mu = None` Equivalently: `my_mu = MagneticPermeabilityMethodNonMagnetic()`	Vacuum permeability \( \mu = \mu_0 \).
Linear Isotropic	`my_mu = 132.0` Equivalently: `my_mu = MagneticPermeabilityMethodLinearIsotropic( relative_permeability_value=132.0, )`	Constant, isotropic relative permeability $\mu_r$, so that \( \mu = \mu_0 \mu_r \).
BH Curve	`my_mu = ([0., 120., 330., 500., 3100.], [0.00, 0.10, 1.00, 1.30, 1.65])` Equivalently: `my_mu = MagneticPermeabilityMethodBHCurve( magnetic_field_strength=[0, 120, 330, 500, 3100], magnetic_flux_density=[0.00, 0.10, 1.00, 1.30, 1.65], )`	Nonlinear isotropic permeability obtained from a tabulated $\|\mathbf{B}\|$–$\|\mathbf{H}\|$ curve via $$ \mu(\|\mathbf{H}\|) = \frac{\|\mathbf{B}(\|\mathbf{H}\|)\|}{\|\mathbf{H}\|}. $$ The table must be strictly monotonic.
Linear Anisotropic	`my_mu = MagneticPermeabilityMethodLinearAnisotropic( permeability_tensor=FixedMatrix([ [mu_xx, mu_xy, mu_xz], [mu_xy, mu_yy, mu_yz], [mu_xz, mu_yz, mu_zz], ]), )`	Constant, fully anisotropic permeability tensor.

Name	Example	Description
Insulating (default)	`my_sigma = None` Equivalently: `my_sigma = ElectricConductivityMethodInsulating()`	Zero conductivity: \( \sigma = 0 \).
Linear Isotropic	`my_sigma = 5.8e7` Equivalently: `my_sigma = ElectricConductivityMethodLinearIsotropic( conductivity_value=5.8e7, )`	Constant, isotropic conductivity.
Linear Anisotropic	`my_sigma = ElectricConductivityMethodLinearAnisotropic( conductivity_tensor=FixedMatrix([ [5.8e7, 1.0e6, 0.0], [1.0e6, 4.2e7, 0.0], [0.0, 0.0, 2.0e7], ]), )`	Constant, fully anisotropic conductivity tensor.
Linear Temperature Coefficient	`my_sigma = mufem.methods.LinearTemperatureCoefficient( reference_value=5.8e7, temperature_coefficient=-3.9e5, reference_temperature=293.15, )`	Isotropic conductivity with linear temperature dependence $$ \sigma(T) = \sigma_{\text{ref}} + \alpha \left( T - T_{\text{ref}} \right). $$ Requires a Temperature Model to be present.
Temperature Table	`my_sigma = mufem.methods.TemperatureTable( temperature=[293.15, 400.0, 600.0, 800.0], values=[5.8e7, 4.9e7, 3.8e7, 3.0e7], )`	Isotropic conductivity tabulated against temperature; linearly interpolated. Requires a Temperature Model to be present.

Name	Example	Description
Zero (default)	`my_br = None` Equivalently: `my_br = MagnetizationMethodNoMagnetization()`	\( \mathbf{B}_r = 0 \).
Constant Direction	`my_br = [0.0, 0.0, 1.2]` Equivalently: `my_br = MagnetizationMethodDirection( x=0.0, y=0.0, z=1.2, )`	Constant remanent flux density vector.
Variable	`halbach_components = mufem.CffConstantVector((0, 1, 0)) halbach_remanence = mufem.CffCylindricalCoordinate( vec=halbach_components, ) my_br = MagnetizationMethodVariable( magnetization=halbach_remanence, )`	Spatially varying remanent flux density driven by a user-provided vector coefficient. Useful for Halbach arrays, poloidal patterns, and any magnetization built from the existing coefficient building blocks (cylindrical / spherical coordinates, expressions, etc.).