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 mufem wraps 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 need an averaging method to get an effective reluctivity $\nu$ for a sinusoidal flux density with peak value $B_p$ (T). $H(B)$ is the BH curve (A/m). The methods are: `NONE`: the table reluctivity at $B_p$. `SIMPLE_AVERAGE`: the energy-equivalent reluctivity $\nu = \tfrac{2}{B_p^2} \int_0^{B_p} H(B) \, dB$. `EFFECTIVE_HARMONIC`: the reluctivity that gives the fundamental harmonic of $H(t)$, $\nu = \tfrac{4}{\pi B_p} \int_0^{\pi/2} H(B_p \cos\theta) \cos\theta \, d\theta$. Its value is between the peak secant value $H(B_p)/B_p$ and the `SIMPLE_AVERAGE` value. All methods limit $\nu$ to the vacuum reluctivity $1/\mu_0$ (m/H) or less.
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 factor $\chi(T)$. A Temperature Model must be present. The point reluctivity at temperature $T$ is $$ \nu(B, T) = \frac{1}{\mu_0 + \chi(T)\,\bigl(\mu_s(B) - \mu_0\bigr)} , $$ and the effective reluctivity is its energy average over one cycle, $$ \nu_\text{eq}(B_p, T) = \frac{2}{B_p^2} \int_0^{B_p} B\,\nu(B, T)\, dB , $$ where: $\mu_s(B) = B / H(B)$ is the secant permeability of the reference BH curve (H/m), $\mu_0$ is the vacuum permeability (H/m), $\chi(T)$ is the susceptibility factor, linearly interpolated from the `temperature` (K) and `susceptibility` tables (dimensionless). Use $1$ at the reference temperature of the BH curve and $0$ above the Curie temperature. Values are limited to $[0, 1]$, $B_p$ is the peak magnetic flux density (T). mufem scales the BH curve before it calculates the average. This is the same energy average as `SIMPLE_AVERAGE`. Typical application: steel workpieces in induction heating that become non-magnetic above the Curie temperature.

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 $\sigma(T) = \sigma_\text{ref} + \alpha(T - T_\text{ref})$. A Temperature Model must 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. A Temperature Model must 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,
)