General Material¶

Introduction¶

A time-harmonic magnetic general material is defined by its magnetic permeability $\mu$ and electric conductivity $\sigma$ — plus a flag indicating whether eddy currents are computed in this material.

A material is created by specifying a property method for each quantity:

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

Each property accepts either an explicit method object (see tables below) or a convenient shorthand (a scalar, a LinearTemperatureCoefficient, …) that is wrapped automatically. Passing None (the default) selects the trivial method ($\mu_r = 1$ / insulating).

Typical applications¶

The general material covers the regions most often encountered in frequency-domain eddy-current studies:

Air, plastics, encapsulants — non-magnetic, insulating (defaults).
Stranded coils (litz wire, multi-strand windings) — finite $\sigma$ but has_eddy_currents=False: the source current is imposed externally and no induced current is solved for in the conductor.
Solid conductors (bus-bars, end-windings, squirrel-cage rotors) — finite $\sigma$ with has_eddy_currents=True to resolve the skin- and proximity-effect current distribution.
Soft-magnetic cores (transformer laminations, motor stators) — nonlinear BH curve with harmonic averaging to model saturation under sinusoidal excitation.
Induction-heating workpieces (steel billets, tubes, aluminium pans) — temperature-dependent $\sigma$ and BH curve; couples naturally to a Temperature Model for full thermo-electromagnetic analysis.

Permanent magnets have no remanence support in the time-harmonic model — use the time-domain model instead. For stacks of thin ferromagnetic sheets, use the Lamination Module, which provides a dedicated time-harmonic material with anisotropic homogenisation and in-sheet eddy-current correction.

Material Properties¶

Magnetic Permeability¶

The magnetic permeability $\mu$ links the magnetic field $\tilde{\mathbf{H}}$ to the magnetic flux density $\tilde{\mathbf{B}}$ through $\tilde{\mathbf{B}} = \mu \tilde{\mathbf{H}}$. In the time-harmonic model the permeability is treated as a real scalar that may depend on field magnitude (nonlinear materials) or temperature. The available methods are summarized in Table 1.

Table 1: Magnetic Permeability Methods
Name	Example	Description
Linear Isotropic (default $\mu_r = 1$)	`my_mu = None` Equivalently: `my_mu = MagneticPermeabilityPropertyMethodConstant( relative_permeability_value=1.0, )`	Constant, isotropic relative permeability $\mu_r$ so that $\mu = \mu_0 \mu_r$. Defaults to vacuum permeability.
Linear Isotropic	`my_mu = 132.0` Equivalently: `my_mu = MagneticPermeabilityPropertyMethodConstant( relative_permeability_value=132.0, )`	Constant, isotropic relative permeability.
Nonlinear Isotropic (BH curve)	`my_mu = MagneticPermeabilityMethodNonlinearIsotropic( magnetic_field_strength=[0, 120, 330, 500, 3100], magnetic_flux_density=[0.00, 0.10, 1.00, 1.30, 1.65], averaging_method=( MagneticPermeabilityMethodNonlinearIsotropic .AveragingMethod.SIMPLE_AVERAGE ), )`	Nonlinear isotropic material from a tabulated $(H, B)$ curve. Time-harmonic problems require an averaging procedure to obtain an effective permeability for a sinusoidal excitation; supported methods are `NONE` and `SIMPLE_AVERAGE` (the latter uses $B_\text{eff}(H) = \tfrac{2}{H} \int_0^H B(H') \, dH'$).
Temperature-Dependent Susceptibility	`my_mu = MagneticPermeabilityMethodTemperatureTableSusceptibility( magnetic_field_strength=[0, 120, 330, 500, 3100], magnetic_flux_density=[0.00, 0.10, 1.00, 1.30, 1.65], temperature=[293.15, 400.0, 600.0, 800.0], susceptibility=[1.00, 0.80, 0.60, 0.20], )`	Temperature-dependent magnetic response: a reference BH curve scaled by a temperature-dependent susceptibility $\chi(T)$. Requires a Temperature Model to be present.

Electric Conductivity¶

The electric conductivity $\sigma$ relates the electric field $\tilde{\mathbf{E}}$ to the conduction current density via Ohm's law $\tilde{\mathbf{J}} = \sigma \tilde{\mathbf{E}}$. Available methods are summarized in Table 2.

Table 2: Electric Conductivity Methods
Name	Example	Description
Insulating (default)	`my_sigma = None` Equivalently: `my_sigma = ElectricConductivityPropertyMethodConstant( conductivity=0.0, )`	Zero conductivity: $\sigma = 0$.
Linear Isotropic	`my_sigma = 5.8e7` Equivalently: `my_sigma = ElectricConductivityPropertyMethodConstant( conductivity=5.8e7, )`	Constant, isotropic conductivity.
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 $\sigma(T) = \sigma_\text{ref} + \alpha(T - T_\text{ref})$. 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], )` Equivalently: `my_sigma = ElectricConductivityMethodTemperatureTable( temperature_table=mufem.methods.TemperatureTable(...), )`	Isotropic conductivity tabulated against temperature; linearly interpolated. Requires a Temperature Model to be present.

Example¶

A solid iron material with a nonlinear BH curve and a temperature-dependent electric conductivity, suitable for induction heating:

from mufem.electromagnetics.timeharmonicmagnetic import (
    ElectricConductivityMethodTemperatureTable,
    MagneticPermeabilityMethodNonlinearIsotropic,
    TimeHarmonicMagneticGeneralMaterial,
)

iron_permeability = MagneticPermeabilityMethodNonlinearIsotropic(
    magnetic_field_strength=bh_curve[:, 0],
    magnetic_flux_density=bh_curve[:, 1],
    averaging_method=(
        MagneticPermeabilityMethodNonlinearIsotropic
        .AveragingMethod.SIMPLE_AVERAGE
    ),
)

iron_conductivity = ElectricConductivityMethodTemperatureTable(
    mufem.methods.TemperatureTable(
        temperature=sigma_T[:, 0],
        values=sigma_T[:, 1],
    ),
)

iron_material = TimeHarmonicMagneticGeneralMaterial(
    name="Iron",
    marker="Iron" @ Vol,
    magnetic_permeability=iron_permeability,
    electric_conductivity=iron_conductivity,
    has_eddy_currents=True,
)