Solid Temperature Material¶

SolidTemperatureMaterial defines the thermal properties on a marked region of the domain. It binds a Marker to the material laws used by the solid temperature model.

The material provides:

thermal conductivity material property \(\kappa\) [\(\rm{W}/(\rm{m} \cdot \rm{K})\)]
specific heat capacity material property \(c_p\) [\(\rm{J}/(\rm{kg} \cdot \rm{K})\)]
density method material property \(\rho\) [\(\rm{kg}/\rm{m}^3\)]

For steady problems, \(\kappa\) is required. For unsteady problems, the heat storage term uses the volumetric heat capacity \(\rho c_p\).

SolidTemperatureMaterial(
    name: str,
    marker: Marker,
    thermal_conductivity: ThermalConductivityPropertyMethodTrait,
    specific_heat_capacity: SpecificHeatCapacityPropertyMethodTrait,
    density: DensityPropertyMethodTrait,
)

Args:

name: Name of the material
marker: Marker of the volume region
thermal_conductivity: Thermal conductivity method
specific_heat_capacity: Specific heat capacity method
density: Density method

Material Properties¶

Thermal Conductivity¶

Thermal conductivity controls heat conduction via Fourier’s law \(\vec{q}=-\kappa\nabla T\).

Thermal Conductivity Methods
Name	Example	Description
Constant	`my_kappa = 132.0`	Specify a constant, isotropic thermal conductivity \( \kappa = \kappa_0 \) over the marked region.
Linear Temperature Coefficient	`my_kappa = mufem.methods.LinearTemperatureCoefficient( reference_value=25.0, temperature_coefficient=2.5, reference_temperature=300.0, )`	Specify an isotropic conductivity with a linear temperature dependence \( \kappa(T) = \kappa_0 + \alpha (T - T_{\text{ref}}) \), where: \( \kappa_0 \) is the reference thermal conductivity \( \alpha \) is the temperature coefficient \( T_{\text{ref}} \) is the reference temperature over the marked region.
Temperature Table	`my_kappa = mufem.methods.TemperatureTable( thermal_conductivity_table[:, 0], thermal_conductivity_table[:, 1], )`	Temperature-dependent conductivity \( \kappa(T) \) defined by a table (interpolated in \( T \)).

Specific Heat Capacity¶

Specific heat capacity controls transient heat storage via \(\rho c_p \,\partial T / \partial t\). It is provided as a scalar coefficient.

Specific Heat Capacity Methods
Name	Example	Description
Constant	`my_cp = 900.0`	Specify a constant heat capacity \( c_p = c_{p,0} \) over the marked region.
Linear Temperature Coefficient	`my_cp = mufem.methods.LinearTemperatureCoefficient( reference_value=900.0, temperature_coefficient=0.1, reference_temperature=300.0, )`	Specify a heat capacity with a linear temperature dependence \( c_p(T) = c_{p,0} + \alpha (T - T_{\text{ref}}) \), where: \( c_{p,0} \) is the reference heat capacity \( \alpha \) is the temperature coefficient \( T_{\text{ref}} \) is the reference temperature over the marked region.
Temperature Table	`my_cp = mufem.methods.TemperatureTable( specific_heat_capacity_table[:, 0], specific_heat_capacity_table[:, 1], )`	Temperature-dependent heat capacity \( c_p(T) \) defined by a table (interpolated in \( T \)).

Density¶

Density contributes to thermal storage via \( \rho c_p \) and is provided as a scalar coefficient.

Density Methods
Name	Example	Description
Constant	`my_rho = 2700.0`	Specify a constant density \( \rho = \rho_0 \) over the marked region.

Notes¶

The unsteady term uses the product \(\rho c_p\) (volumetric heat capacity). If either \(\rho\) or \(c_p\) depends on \(T\), the transient problem becomes nonlinear.

Example¶

silicon_material = SolidTemperatureMaterial(
    name="Silicon",
    marker="Thermal Domain" @ mufem.Vol,
    thermal_conductivity=mufem.methods.LinearTemperatureCoefficient(
        reference_value=25.0,
        temperature_coefficient=2.5,
        reference_temperature=300.0,
    ),
    specific_heat_capacity=129.0,
    density=2330,
)