belfem::Material Class Reference

Base class for all materials in BELFEM. More...

#include <cl_Material.hpp>

Inheritance diagram for belfem::Material:

Collaboration diagram for belfem::Material:

Public Member Functions
	Material (const MaterialType aType, const bool aIsIsotropic=true)
	Constructor.
virtual	~Material ()
	Destructor - deletes owned B-H curves and Jc/n functions.
	Material (const Material &)=delete
Material &	operator= (const Material &)=delete
	Material (Material &&)=delete
Material &	operator= (Material &&)=delete
bool	have (const MaterialProperty aProperty) const
	Check if a material property is available.
bool	have_defect () const
	Check if the material has a defect.
bool	have_heating () const
	Check if the material has a heating function.
bool	use_piecewise () const
	Check if we use piecewise instead of power-law.
bool	depends (const MaterialProperty aProperty, const MaterialDependency aDependency) const
	Check if a property depends on a specific variable.
const MaterialDependencyBitset *	dependencies (const MaterialProperty aProperty) const
	Get all dependencies for a property.
bool	is_isotropic () const
	Check if material is isotropic.
MaterialType	type () const
	Get the material type.
const string &	label () const
	Get material label.
const string &	number () const
	Get material number/identifier.
void	load_bh_curve (const material::BhCurve *aCurve)
	Activate a B-H curve as the permeability source (ferromagnets).
void	flag (const uint8_t aIndex=0)
	Set flag (multi-purpose flag used by Kernel).
void	unflag (const uint8_t aIndex=0)
	Clear flag.
bool	is_flagged (const uint8_t aIndex=0) const
	Check if material is flagged.
bool	is_constant (const MaterialProperty aProperty) const
	Check if a property is constant (temperature-independent).
virtual void	set_RRR (const real RRR)
	Set the residual resistivity ratio (for noble metals).
virtual real	density (const real T=gTroom) const
	Density as a function of temperature.
real	ref_density () const
	Reference density at reference temperature.
real	M () const
	Molar mass.
virtual real	cp (const real T=gTroom) const
	Specific heat capacity.
virtual real	dcpdT (const real T=gTroom) const
virtual real	d2cpdT2 (const real T=gTroom) const
virtual real	lambda (const real T=gTroom) const
	Thermal conductivity (isotropic).
virtual real	dlambdadT (const real T=gTroom) const
virtual real	lambda (const real T, const real B, const real beta) const
	Thermal conductivity for noble metals with magnetoresistance.
virtual real	dlambdadT (const real T, const real B, const real beta) const
virtual real	dlambdadB (const real T, const real B, const real beta) const
virtual real	dlambdadbeta (const real T, const real B, const real beta) const
virtual real	lambda (const real T, const real B_par, const real B_perp, const real J) const
	Thermal conductivity for HTS materials.
virtual real	dlambdadT (const real T, const real B_par, const real B_perp, const real J) const
virtual real	rho (const real T) const
	Electrical resistivity (isotropic).
virtual real	drhodT (const real T) const
virtual real	rho (const real T, const real B, const real beta) const
	Electrical resistivity for noble metals with magnetoresistance.
virtual real	drhodT (const real T, const real B, const real beta) const
virtual real	drhodB (const real T, const real B, const real beta) const
virtual real	drhodbeta (const real T, const real B, const real beta) const
real	rho_powerlaw (const real normJ) const
	Power law resistivity for HTS (constant jc and n).
real	rho_powerlaw (const real normJ, const real x, const real y, const real z, const real t) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Constant \(J_c\) / \(n\) with spatially-dependent defect modulation \(J_c \rightarrow J_c \cdot d(x,y,z,t)\).
real	rho_powerlaw (const real normJ, real normB, const real angleNxB) const
	Power law resistivity for HTS (field-dependent jc, constant n).
real	rho_powerlaw (const real normJ, real normB, const real angleNxB, const real x, const real y, const real z, const real t) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Field-dependent \(J_c(|B|, \angle)\) with spatial defect modulation.
real	rho_powerlaw (const real normJ, const real T, real normB, const real angleNxB) const
	Power law resistivity for HTS (full temperature and field dependence).
real	rho_powerlaw (const real normJ, const real T, real normB, const real angleNxB, const real x, const real y, const real z, const real t) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Fully \((|B|, \angle, T)\)-dependent variant with spatial defect modulation applied to \(J_c\).
real	rho_powerlaw (const real normJ, const real T) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Temperature-only variant.
real	rho_powerlaw (const real normJ, const real T, const real x, const real y, const real z, const real t) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Temperature-only custom-callback variant with spatial defect modulation.
real	rho_piecewise (const real normJ) const
	Piecewise resistivity for HTS (constant jc and n).
real	rho_piecewise (const real normJ, const real x, const real y, const real z, const real t) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Constant \(J_c\) / \(n\) with spatially-dependent defect modulation \(J_c \rightarrow J_c \cdot d(x,y,z,t)\).
real	rho_piecewise (const real normJ, real normB, const real angleNxB) const
	Piecewise resistivity for HTS (field-dependent jc, constant n).
real	rho_piecewise (const real normJ, real normB, const real angleNxB, const real x, const real y, const real z, const real t) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Field-dependent \(J_c(|B|, \angle)\) with spatial defect modulation.
real	rho_piecewise (const real normJ, const real T, real normB, const real angleNxB) const
	Piecewise resistivity for HTS (full temperature and field dependence).
real	rho_piecewise (const real normJ, const real T, real normB, const real angleNxB, const real x, const real y, const real z, const real t) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Fully \((|B|, \angle, T)\)-dependent variant with spatial defect modulation applied to \(J_c\).
real	rho_piecewise (const real normJ, const real T) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Temperature-only variant: \(J_c(T)\) and \(n(T)\) from the user-supplied callbacks Material::jc_custom() / Material::n_custom().
real	rho_piecewise (const real normJ, const real T, const real x, const real y, const real z, const real t) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Temperature-only custom-callback variant with spatial defect modulation.
real	rho_riva (const real normJ, const real T, const real normB, const real angleNxB) const
	Riva-law resistivity: the superconducting power-law channel in parallel with the normal-state channel.
real	rho_riva (const real normJ, const real T, const real normB, const real angleNxB, const real x, const real y, const real z, const real t) const
	defect overload: jc_eff = D(x,y,z,t)·jc throughout
real	drho_riva_dJ (const real normJ, const real T, const real normB, const real angleNxB) const
	dρ/d|J| of rho_riva: w²·dρ_PL/dJ with w = ρ_n/(ρ_PL+ρ_n)
real	drho_riva_dJ (const real normJ, const real T, const real normB, const real angleNxB, const real x, const real y, const real z, const real t) const
real	drho_riva_dB (const real normJ, const real T, const real normB, const real angleNxB) const
	dρ/d|B| of rho_riva: w²·dρ_PL/dB ( dρ_n/dB = 0 on this path )
real	drho_riva_dB (const real normJ, const real T, const real normB, const real angleNxB, const real x, const real y, const real z, const real t) const
real	drho_riva_dT (const real normJ, const real T, const real normB, const real angleNxB) const
	dρ/dT of rho_riva: w²·dρ_PL/dT + (1−w)²·dρ_n/dT
real	drho_riva_dT (const real normJ, const real T, const real normB, const real angleNxB, const real x, const real y, const real z, const real t) const
real	n (const real normB, const real angleNxB, const real T) const
	Direct evaluation of the n-value as a function of \(|B|\), \(\angle(n,B)\) and \(T\).
real	n (const real normB, const real angleNxB) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Field-dependent \(n(|B|, \angle(n,B))\) without temperature dependence.
real	jc (const real normB, const real angleNxB, const real T) const
	Direct evaluation of the critical current density \(J_c(|B|, \angle(n,B), T)\).
real	jc (const real normB, const real angleNxB) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Field-dependent \(J_c(|B|, \angle(n,B))\) without temperature dependence.
real	jc (const real normB, const real angleNxB, const real T, const real x, const real y, const real z, const real t) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Variant with spatial defect modulation \(J_c \rightarrow J_c \cdot d(x,y,z,t)\).
real	jc (const real normB, const real angleNxB, const real x, const real y, const real z, const real t) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Field-dependent \(J_c(|B|, \angle(n,B))\) with spatial defect modulation \(J_c \rightarrow J_c \cdot d(x,y,z,t)\).
real	drho_powerlaw_dJ (const real normJ, const real T, const real normB, const real angleNxB) const
	Derivative of power-law resistivity with respect to current density magnitude.
real	drho_powerlaw_dJ (const real normJ, const real T, const real normB, const real angleNxB, const real x, const real y, const real z, const real t) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Fully \((|B|, \angle, T)\)-dependent variant with spatial defect modulation applied to \(J_c\).
real	drho_powerlaw_dJ (const real normJ) const
	Analytic Jacobian \(d\rho_{eff}/dJ\) of the power-law model.
real	drho_powerlaw_dJ (const real normJ, const real x, const real y, const real z, const real t) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Constant \(J_c\) / \(n\) with spatial defect modulation applied to \(J_c\).
real	drho_powerlaw_dJ (const real normJ, const real T) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Temperature-only variant: \(J_c(T)\) and \(n(T)\) from Material::jc_custom() / Material::n_custom().
real	drho_powerlaw_dJ (const real normJ, const real T, const real x, const real y, const real z, const real t) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Temperature-only custom-callback variant with spatial defect modulation.
real	drho_powerlaw_dJ (const real normJ, const real normB, const real angleNxB) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Field-dependent \(J_c(|B|, \angle)\) via mJcFunction; \(n\) remains constant.
real	drho_powerlaw_dJ (const real normJ, const real normB, const real angleNxB, const real x, const real y, const real z, const real t) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Field-dependent \(J_c(|B|, \angle)\) with spatial defect modulation; \(n\) remains constant.
real	drho_piecewise_dJ (const real normJ, const real T, const real normB, const real angleNxB) const
	Derivative of piecewise resistivity with respect to current density magnitude.
real	drho_piecewise_dJ (const real normJ, const real T, const real normB, const real angleNxB, const real x, const real y, const real z, const real t) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Fully \((|B|, \angle, T)\)-dependent variant with spatial defect modulation applied to \(J_c\).
real	drho_powerlaw_dB (const real normJ, const real T, const real normB, const real angleNxB) const
	Derivative of power-law resistivity with respect to |B| at fixed J, T, θ ( jc = jc(T,|B|,θ), n = n(T,|B|,θ) ).
real	drho_powerlaw_dB (const real normJ, const real T, const real normB, const real angleNxB, const real x, const real y, const real z, const real t) const
	defect overload: jc_eff = D(x)·jc throughout, djc_eff = D·djc
real	drho_powerlaw_dT (const real normJ, const real T, const real normB, const real angleNxB) const
	Derivative of power-law resistivity with respect to T at fixed J, |B|, θ ( T-leg: jc(T), n(T) AND ρ_n(T) all move — the quench-feedback tangent ).
real	drho_powerlaw_dT (const real normJ, const real T, const real normB, const real angleNxB, const real x, const real y, const real z, const real t) const
	defect overload: jc_eff = D(x)·jc throughout, djc_eff = D·djc
real	drho_piecewise_dB (const real normJ, const real T, const real normB, const real angleNxB) const
	Derivative of piecewise resistivity with respect to |B| at fixed J, T, θ
real	drho_piecewise_dB (const real normJ, const real T, const real normB, const real angleNxB, const real x, const real y, const real z, const real t) const
	defect overload: jc_eff = D(x)·jc throughout, djc_eff = D·djc
real	drho_piecewise_dT (const real normJ, const real T, const real normB, const real angleNxB) const
	Derivative of piecewise resistivity with respect to T at fixed J, |B|, θ ( T-leg ).
real	drho_piecewise_dT (const real normJ, const real T, const real normB, const real angleNxB, const real x, const real y, const real z, const real t) const
	defect overload: jc_eff = D(x)·jc throughout, djc_eff = D·djc
real	drho_piecewise_dJ (const real normJ) const
	Branch-aware Jacobian \(d\rho/dJ\) of the three-regime piecewise model.
real	drho_piecewise_dJ (const real normJ, const real x, const real y, const real z, const real t) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Constant \(J_c\) / \(n\) with spatial defect modulation applied to \(J_c\).
real	drho_piecewise_dJ (const real normJ, const real T) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Temperature-only variant: \(J_c(T)\) and \(n(T)\) from Material::jc_custom() / Material::n_custom().
real	drho_piecewise_dJ (const real normJ, const real T, const real x, const real y, const real z, const real t) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Temperature-only custom-callback variant with spatial defect modulation.
real	drho_piecewise_dJ (const real normJ, const real normB, const real angleNxB) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Field-dependent \(J_c(|B|, \angle)\) via mJcFunction; \(n\) remains constant.
real	drho_piecewise_dJ (const real normJ, const real normB, const real angleNxB, const real x, const real y, const real z, const real t) const
	This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Field-dependent \(J_c(|B|, \angle)\) with spatial defect modulation; \(n\) remains constant.
virtual real	rho_i (const real T) const
	Intrinsic (phonon) electrical resistivity of a metal, ρ = ρ_i(T) + ρ_0.
virtual real	H (const real B) const
	Magnetic field strength from flux density.
virtual real	mu (const real H, const real T=BELFEM_QUIET_NAN) const
	Magnetic permeability.
virtual void	dmudH (const real H, real &mu, real &dmudH) const
	Magnetic permeability and its derivative.
virtual real	E (real T=gTroom) const
	Young's modulus.
virtual real	nu (real T=gTroom) const
	Poisson's ratio.
virtual real	G (real T=gTroom) const
	Shear modulus.
virtual real	K (real T=gTroom) const
	Bulk modulus.
virtual real	alpha (real T=gTroom) const
	Thermal expansion coefficient.
virtual real	l (real T) const
	Relative length after thermal expansion.
virtual real	Rp02 (real T=gTroom) const
	Yield stress (0.2% offset).
virtual real	debye (const real T) const
	Debye temperature.
virtual void	set_bh_curve (const material::BhCurve *aCurve)
	Assign a B-H curve to this material.
void	set_jc_function (const material::JcFunction *aFunction)
	Assign a critical current density function.
void	set_n_function (const material::JcFunction *aFunction)
	Assign a power law exponent function.
void	set_piecewise (const bool aUsePiecewise)
void	set_resistivity_law (const ResistivityLaw aLaw)
ResistivityLaw	resistivity_law () const
bool	use_riva () const
virtual void	set_user_defined_function (const MaterialProperty Property, const MaterialDependency Dependency, MatFunc1 *Function)
	Set a user-defined function with one dependency (for UserDefinedMaterial).
virtual void	set_user_defined_function (const MaterialProperty Property, const MaterialDependency Dependency1, const MaterialDependency Dependency2, MatFunc2 *Function)
	Set a user-defined function with two dependencies (for UserDefinedMaterial).
virtual void	set_user_defined_function (const MaterialProperty Property, const MaterialDependency Dependency1, const MaterialDependency Dependency2, const MaterialDependency Dependency3, MatFunc3 *Function)
	Set a user-defined function with three dependencies (for UserDefinedMaterial).
void	set_user_defined_defect (DefectFunc *Function)
	Set a user-defined defect function with x, y, z, t dependencies.
void	read_defect (const string &aLibraryPath, const string &aLabel)
	Read a defect from the library.
void	set_user_defined_heating (HeatFunc *Function)
	Set a user-defined volumetric heat load [ W/m³ ] as a function of x, y, z [ m ] and t [ s ].
void	read_heating (const string &aLibraryPath, const string &aLabel)
	Read a heating function from a plugin library.
virtual real	evaluate_polynomial (const MaterialProperty Property, const real T) const
	Evaluate a polynomial for a given property (for UserDefinedMaterial).
virtual real	evaluate_derivative_of_polynomial (const MaterialProperty Property, const real T) const
virtual void	set_user_defined_polynomial (const MaterialProperty Property, const std::vector< real > &Coefficients)
	Set a polynomial function for a property (for UserDefinedMaterial).
virtual void	set_user_defined_polynomial (const MaterialProperty Property, const Cell< real > &Coefficients)
void	set_label (const string &aLabel)
	PROTECTED INTERFACE FOR DERIVED MATERIALS.
void	set_number (const string &aNumber)
	Set material number/identifier (for derived classes).
void	set_have (const MaterialProperty aProperty, const bool aHave=true)
	Mark a property as available.
void	reset_dependencies (const MaterialProperty aProperty)
	Reset all dependencies for a property.
void	set_dependency (const MaterialProperty aProperty, const MaterialDependency aDependency)
	Add a dependency to a property.
void	set_constant (const MaterialProperty aProperty, const real aValue)
	Define a property as constant.
void	set_custom (const MaterialProperty aProperty)
	Mark property as using custom evaluation function.
real	constant_property (const MaterialProperty aProperty) const
	Get constant property value.
virtual real	spline_property (const MaterialProperty aProperty, const real aX) const
	Evaluate property using spline interpolation.
virtual real	dspline_property (const MaterialProperty aProperty, const real aX) const
virtual real	ddspline_property (const MaterialProperty aProperty, const real aX) const
virtual void	set_table_flags (const bool aFlag)
virtual real	jc_custom (const real T) const
virtual real	n_custom (const real T) const
real	volumetric_heatload (const real x, const real y, const real z, const real time) const
	Artificial volumetric heat load [ W/m³ ] from the heating plugin.

Protected Member Functions
real	jc_eval (const real T, const real normB, const real angleNxB) const
	O1 "full-signature policy" helpers: the assembly path always passes the full ( T, normB, angleNxB ) set; the material consumes what its Jc / n functions depend on and ignores the rest.
real	djc_eval_dB (const real T, const real normB, const real angleNxB) const
real	dn_eval_dB (const real T, const real normB, const real angleNxB) const
	d(n)/d|B|, same routing as djc_eval_dB
real	djc_eval_dT (const real T, const real normB, const real angleNxB) const
real	dn_eval_dT (const real T, const real normB, const real angleNxB) const
	d(n)/dT, same routing as djc_eval_dT
real	n_eval (const real T, const real normB, const real angleNxB) const
void	check_riva_n_source () const
real	n_eval_raw (const real T, const real normB, const real angleNxB) const
bool	riva_rho_pl (const real normJ, const real jc, const real n, const real ec, real &rhoPL) const
void	create_spline (const MaterialProperty aProperty, const real adYdX0=BELFEM_QUIET_NAN, const real adXdX1=BELFEM_QUIET_NAN)
	PROPERTY EVALUATION FUNCTIONS.
real	alpha_switch_temperature () const
	Temperature below which alpha is taken from the Grueneisen branch.
real	density_const (const real T) const
virtual real	density_custom (const real T) const
real	return_zero (const real T) const
	Shared binding for derivative channels that are identically zero.
real	cp_const (const real T) const
real	dcpdT_finite_difference (const real T) const
real	d2cpdT2_finite_difference (const real T) const
real	cp_spline (const real T) const
real	dcpdT_spline (const real T) const
real	d2cpdT2_spline (const real T) const
virtual real	cp_custom (const real T) const
virtual real	dcpdT_custom (const real T) const
virtual real	d2cpdT2_custom (const real T) const
real	lambda_const (const real T) const
real	dlambdadT_finite_difference (const real T) const
real	lambda_spline (const real T) const
real	dlambdadT_spline (const real T) const
virtual real	lambda_custom (const real T) const
virtual real	dlambdadT_custom (const real T) const
real	drhodT_finite_difference (const real T) const
real	rho_const (const real T) const
virtual real	rho_custom (const real T) const
virtual real	drhodT_custom (const real T) const
virtual real	rho_spline (const real T) const
virtual real	drhodT_spline (const real T) const
virtual real	rho_kohler (const real T, const real normB, const real angleJxB) const
virtual real	drhodT_kohler (const real T, const real normB, const real angleJxB) const
virtual real	drhodB_kohler (const real T, const real normB, const real angleJxB) const
virtual real	drhodbeta_kohler (const real T, const real normB, const real angleJxB) const
virtual real	rho_table (const real T, const real normB, const real angleJxB) const
virtual real	drhodT_table (const real T, const real normB, const real angleJxB) const
virtual real	drhodB_table (const real T, const real normB, const real angleJxB) const
virtual real	drhodbeta_table (const real T, const real normB, const real angleJxB) const
virtual real	lambda_custom (const real T, const real normB, const real angle) const
virtual real	lambda_table (const real T, const real normB, const real angle) const
real	H_const (const real B) const
virtual real	H_bhcurve (const real B) const
real	mu_const (const real H, const real T) const
virtual real	mu_bhcurve (const real H, const real T) const
virtual real	mu_custom (const real H, const real T) const
void	dmudH_const (const real H, real &mu, real &dmudH) const
virtual void	dmudH_bhcurve (const real H, real &mu, real &dmudH) const
real	E_const (const real T) const
real	E_spline (const real T) const
virtual real	E_custom (const real T) const
virtual real	dEdT_custom (const real T) const
real	nu_const (const real T) const
real	nu_spline (const real T) const
virtual real	nu_custom (const real T) const
real	alpha_const (const real T) const
real	alpha_spline (const real T) const
virtual real	alpha_custom (const real T) const
real	Rp02_const (const real T) const
real	Rp02_spline (const real T) const
virtual real	Rp02_custom (const real T) const
real	debye_const (const real T) const
real	debye_spline (const real T) const
virtual real	debye_custom (const real T) const
virtual real	rho_i_custom (const real T) const
real	rho_i_spline (const real T) const
virtual void	reset_spline (const MaterialProperty aProperty)
virtual void	create_spline (real(Material::*aFunction)(const real aT) const, const MaterialProperty aProperty, const uint aStartBC, const uint aEndBC, const real adYdX0=BELFEM_QUIET_NAN, const real adYdX1=BELFEM_QUIET_NAN)

Protected Attributes
real(Material::*)	mFunctionRhoKohler (const real T, const real normB, const real angle) const = nullptr
real(Material::*)	mFunctiondRhoKohlerdT (const real T, const real normB, const real angle) const = nullptr
real(Material::*)	mFunctiondRhoKohlerdB (const real T, const real normB, const real angle) const = nullptr
real(Material::*)	mFunctiondRhoKohlerdbeta (const real T, const real normB, const real angle) const = nullptr
real(Material::*)	mFunctionH (const real B) const = nullptr
real(Material::*)	mFunctionMu (const real H, const real T) const = nullptr
void(Material::*)	mFunctionDMuDH (double, double &, double &) const = nullptr
const material::JcFunction *	mJcFunction = nullptr
const material::JcFunction *	mNFunction = nullptr

Detailed Description

Base class for all materials in BELFEM.

The Material class provides a unified interface for accessing physical properties of materials. Properties can be constant, temperature-dependent, or depend on multiple variables (field, current, etc.) depending on the material type.

Key features:

• Automatic property function dispatching based on dependencies
• Support for both constant and variable properties
• Spline-based interpolation for tabulated data
• Custom functions for complex material behavior
• Ownership management for B-H curves and Jc functions

Usage:

MaterialFactory factory;
Material* cu = factory.create_material("copper");
cu->set_RRR(100);
real rho = cu->rho(77.0);  // Resistivity at 77K

IMPORTANT NOTES:

• For HTS materials, use rho_powerlaw() instead of rho()
• For thermal M-matrix, use ref_density() or density(gTroom), not density(T)
• Material owns and deletes assigned B-H curves and Jc/n functions
• Variable names in material functions are exempt from BELFEM naming rules

See also

BELFEM Materials Module - Usage Guide

Constructor & Destructor Documentation

Material() [1/3]

belfem::Material::Material	(	const MaterialType	aType,
		const bool	aIsIsotropic = true )

Constructor.

Parameters

aType	Material type (Ferro, HTS, PureMetal, etc.)
aIsIsotropic	True if material properties are isotropic (default)

~Material()

belfem::Material::~Material ( )

virtual

Destructor - deletes owned B-H curves and Jc/n functions.

Material() [2/3]

belfem::Material::Material ( const Material & )

delete

Material() [3/3]

belfem::Material::Material ( Material && )

delete

Member Function Documentation

alpha()

real belfem::Material::alpha ( real T = gTroom ) const

inlinevirtual

Thermal expansion coefficient.

Parameters

T	Temperature [K] (default: room temperature)

Returns

Thermal expansion coefficient [1/K]

Defined as α = (1/l)·∂l/∂T, NOT (1/l)·Δl/ΔT!

alpha_const()

real belfem::Material::alpha_const ( const real T ) const

inlineprotected

alpha_custom()

real belfem::Material::alpha_custom ( const real T ) const

protectedvirtual

Reimplemented in belfem::material::Aluminum, belfem::material::Chromium, belfem::material::Copper, belfem::material::HastelloyC276, belfem::material::Indium, belfem::material::Iron, belfem::material::Lead, belfem::material::Magnesia, belfem::material::Nickel, belfem::material::Silver, belfem::material::UserDefinedMaterial, belfem::material::WhiteTin, and belfem::material::YBCO.

alpha_spline()

real belfem::Material::alpha_spline ( const real T ) const

inlineprotected

alpha_switch_temperature()

real belfem::Material::alpha_switch_temperature ( ) const

inlineprotected

Temperature below which alpha is taken from the Grueneisen branch.

min( 0.618 * theta_Debye, gTAlphaSwitchMax ). The Debye factor is a convention, not a derivation - it sits comfortably above the temperature at which dln(C)/dT changes sign, which is where the fitted expansion curve stops being consistent with the heat capacity.

check_riva_n_source()

void belfem::Material::check_riva_n_source ( ) const

protected

setup gate for the riva law: refuses a PROVABLY bad n source ( table bound or constant at or below 1 ). Called from set_resistivity_law and set_n_function, so the outcome does not depend on which of the two is called first.

constant_property()

real belfem::Material::constant_property ( const MaterialProperty aProperty ) const

inline

Get constant property value.

cp()

real belfem::Material::cp ( const real T = gTroom ) const

inlinevirtual

Specific heat capacity.

Parameters

T	Temperature [K] (default: room temperature)

Returns

Specific heat capacity [J/(kg·K)]

cp_const()

real belfem::Material::cp_const ( const real T ) const

inlineprotected

cp_custom()

real belfem::Material::cp_custom ( const real T ) const

protectedvirtual

Reimplemented in belfem::material::HastelloyC276, belfem::material::Magnesia, belfem::material::Metal, belfem::material::UserDefinedMaterial, and belfem::material::YBCO.

cp_spline()

real belfem::Material::cp_spline ( const real T ) const

inlineprotected

create_spline() [1/2]

void belfem::Material::create_spline ( const MaterialProperty aProperty, const real adYdX0 = BELFEM_QUIET_NAN, const real adXdX1 = BELFEM_QUIET_NAN )

protected

PROPERTY EVALUATION FUNCTIONS.

These functions are called via function pointers for property evaluation. Each property has up to four variants:

• *_const: Returns constant value
• *_spline: Interpolates from spline
• *_custom: Calls derived class's custom implementation
• *_table: Uses lookup table (for field-dependent properties)

The appropriate function is selected during material initialization based on how the property is defined.

Create spline for a property with optional boundary derivatives

create_spline() [2/2]

void belfem::Material::create_spline ( real(Material::* aFunction )(const real aT) const, const MaterialProperty aProperty, const uint aStartBC, const uint aEndBC, const real adYdX0 = BELFEM_QUIET_NAN, const real adYdX1 = BELFEM_QUIET_NAN )

protectedvirtual

Reimplemented in belfem::material::SplineLookupTable.

d2cpdT2()

real belfem::Material::d2cpdT2 ( const real T = gTroom ) const

inlinevirtual

d2cpdT2_custom()

real belfem::Material::d2cpdT2_custom ( const real T ) const

protectedvirtual

Reimplemented in belfem::material::Metal.

d2cpdT2_finite_difference()

real belfem::Material::d2cpdT2_finite_difference ( const real T ) const

inlineprotected

d2cpdT2_spline()

real belfem::Material::d2cpdT2_spline ( const real T ) const

inlineprotected

dcpdT()

real belfem::Material::dcpdT ( const real T = gTroom ) const

inlinevirtual

dcpdT_custom()

real belfem::Material::dcpdT_custom ( const real T ) const

protectedvirtual

Reimplemented in belfem::material::Metal.

dcpdT_finite_difference()

real belfem::Material::dcpdT_finite_difference ( const real T ) const

inlineprotected

dcpdT_spline()

real belfem::Material::dcpdT_spline ( const real T ) const

inlineprotected

ddspline_property()

real belfem::Material::ddspline_property ( const MaterialProperty aProperty, const real aX ) const

virtual

Reimplemented in belfem::material::SplineLookupTable.

debye()

real belfem::Material::debye ( const real T ) const

inlinevirtual

Debye temperature.

Parameters

T	Temperature [K]

Returns

Debye temperature [K]

debye_const()

real belfem::Material::debye_const ( const real T ) const

inlineprotected

debye_custom()

real belfem::Material::debye_custom ( const real T ) const

protectedvirtual

Reimplemented in belfem::material::Aluminum, belfem::material::Chromium, belfem::material::Copper, belfem::material::Indium, belfem::material::Iron, belfem::material::Lead, belfem::material::Nickel, belfem::material::Silver, and belfem::material::WhiteTin.

debye_spline()

real belfem::Material::debye_spline ( const real T ) const

inlineprotected

dEdT_custom()

real belfem::Material::dEdT_custom ( const real T ) const

protectedvirtual

Reimplemented in belfem::material::Metal, and belfem::material::Nickel.

density()

real belfem::Material::density ( const real T = gTroom ) const

inlinevirtual

Density as a function of temperature.

Parameters

T	Temperature [K] (default: room temperature)

Returns

Density [kg/m³]

IMPORTANT: For thermal M-matrix, use ref_density() or density(gTroom), not density(T), because BELFEM computes on the undeformed mesh.

density_const()

real belfem::Material::density_const ( const real T ) const

inlineprotected

density_custom()

real belfem::Material::density_custom ( const real T ) const

inlineprotectedvirtual

dependencies()

const MaterialDependencyBitset * belfem::Material::dependencies ( const MaterialProperty aProperty ) const

inline

Get all dependencies for a property.

Parameters

aProperty

The material property

Returns

Pointer to bitset containing all dependencies

depends()

bool belfem::Material::depends ( const MaterialProperty aProperty, const MaterialDependency aDependency ) const

inline

Check if a property depends on a specific variable.

Parameters

aProperty	The material property
aDependency	The dependency to check (T, B, angle, etc.)

Returns

True if the property depends on the specified variable

djc_eval_dB()

real belfem::Material::djc_eval_dB ( const real T, const real normB, const real angleNxB ) const

inlineprotected

d(jc)/d|B| routed like jc_eval: JcFunction::deval_dB when a function is attached, 0 for the constants fallback ( the exact derivative of a constant ) — so constant-jc decks are bit-identical

djc_eval_dT()

real belfem::Material::djc_eval_dT ( const real T, const real normB, const real angleNxB ) const

inlineprotected

d(jc)/dT, T-leg twin of djc_eval_dB: JcFunction::deval_dT when a function is attached, 0 for the constants fallback. For a T-dependent UserDefined without a deval_dT override the base-class 0 is a conservative fallback, not an exact derivative

dlambdadB()

real belfem::Material::dlambdadB ( const real T, const real B, const real beta ) const

inlinevirtual

Reimplemented in belfem::material::Metal.

dlambdadbeta()

real belfem::Material::dlambdadbeta ( const real T, const real B, const real beta ) const

inlinevirtual

Reimplemented in belfem::material::Metal.

dlambdadT() [1/3]

real belfem::Material::dlambdadT ( const real T, const real B, const real beta ) const

inlinevirtual

Reimplemented in belfem::material::Alloy, and belfem::material::Metal.

dlambdadT() [2/3]

real belfem::Material::dlambdadT ( const real T, const real B_par, const real B_perp, const real J ) const

inlinevirtual

Reimplemented in belfem::material::Alloy, and belfem::material::Metal.

dlambdadT() [3/3]

real belfem::Material::dlambdadT ( const real T = gTroom ) const

inlinevirtual

Reimplemented in belfem::material::Alloy, and belfem::material::Metal.

dlambdadT_custom()

real belfem::Material::dlambdadT_custom ( const real T ) const

protectedvirtual

dlambdadT_finite_difference()

real belfem::Material::dlambdadT_finite_difference ( const real T ) const

inlineprotected

dlambdadT_spline()

real belfem::Material::dlambdadT_spline ( const real T ) const

inlineprotected

dmudH()

void belfem::Material::dmudH ( const real H, real & mu, real & dmudH ) const

inlinevirtual

Magnetic permeability and its derivative.

Parameters

H	Magnetic field strength [A/m]
mu	Output: magnetic permeability [H/m]
dmudH	Output: derivative dμ/dH [H/m²]

dmudH_bhcurve()

void belfem::Material::dmudH_bhcurve ( const real H, real & mu, real & dmudH ) const

protectedvirtual

Reimplemented in belfem::material::Metal.

dmudH_const()

void belfem::Material::dmudH_const ( const real H, real & mu, real & dmudH ) const

inlineprotected

dn_eval_dB()

real belfem::Material::dn_eval_dB ( const real T, const real normB, const real angleNxB ) const

inlineprotected

d(n)/d|B|, same routing as djc_eval_dB

dn_eval_dT()

real belfem::Material::dn_eval_dT ( const real T, const real normB, const real angleNxB ) const

inlineprotected

d(n)/dT, same routing as djc_eval_dT

drho_piecewise_dB() [1/2]

real belfem::Material::drho_piecewise_dB ( const real normJ, const real T, const real normB, const real angleNxB ) const

inline

Derivative of piecewise resistivity with respect to |B| at fixed J, T, θ

Returns

dρ/d|B| [Ω·m/T]

Exact per regime where the regime's law permits, staged where it does not ( 2026-08-13 audit, C5 accepted by both voices ):

• power-law regime ( J ≤ j1 ): dp/d|B| WITHOUT the parallel factor, because rho_piecewise returns the RAW power law there — consistency is with the residual, not with drho_powerlaw_dB.
• normal regime ( J > j3 ) and T > T_crit: exactly 0 ( ρ = ρ_n is independent of jc, n ).
• Bézier blend ( j1 < J ≤ j3 ): 0 for now — the boundaries j1..j3 and the control points also move with jc, and that derivative is deferred; a zero here is today's global behaviour, locally. The tangent therefore has a jump at j1.

drho_piecewise_dB() [2/2]

real belfem::Material::drho_piecewise_dB ( const real normJ, const real T, const real normB, const real angleNxB, const real x, const real y, const real z, const real t ) const

inline

defect overload: jc_eff = D(x)·jc throughout, djc_eff = D·djc

drho_piecewise_dJ() [1/8]

real belfem::Material::drho_piecewise_dJ ( const real normJ ) const

inline

Branch-aware Jacobian \(d\rho/dJ\) of the three-regime piecewise model.

Branches mirror those of rho_piecewise():

• Power-law regime \((J \le J_1)\): the parallel-combined chain rule of drho_powerlaw_dJ() is applied here. NOTE: rho_piecewise() returns the RAW power law in this regime (no parallel combination), so unlike drho_piecewise_dB/dT this tangent is not consistent with the residual there.
• Normal regime \((J > J_3)\): \(\rho\) is constant, so the derivative is exactly zero.
• Flux-flow transition \((J_1 < J \le J_3)\): differentiate the Bezier blend in log–log space. With \(L_\rho \equiv \log_{10}\rho\),

  \[ \frac{d\rho}{dJ} \;=\; \rho\,\ln 10\, \frac{dL_\rho}{dt}\, \frac{dt}{dJ}, \]

  where the parametric derivatives follow from the implicit relation \(a t^2 - 2bt - c(J) = 0\) defined in rho_piecewise(). Above \(T_{crit}\) the derivative is taken as zero (only \(\rho_n\) is returned by rho_piecewise()), and for \(|J| < \)BELFEM_EPSILON the derivative is zero to avoid the singular \(\log J\) term.

Required by the nonlinear iteration in the HTS solver (Messe et al. 2023, Sec. 2.7) so that Newton-Raphson can drive the residual through the curved transition without losing quadratic convergence.

This overload assumes constant \(J_c\) and \(n\).

Parameters

normJ

current-density magnitude \(|J|\) [A/m²]

Returns

\(d\rho/dJ\) [Ω·m / (A/m²)]

drho_piecewise_dJ() [2/8]

real belfem::Material::drho_piecewise_dJ ( const real normJ, const real normB, const real angleNxB ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Field-dependent \(J_c(|B|, \angle)\) via mJcFunction; \(n\) remains constant.

drho_piecewise_dJ() [3/8]

real belfem::Material::drho_piecewise_dJ ( const real normJ, const real normB, const real angleNxB, const real x, const real y, const real z, const real t ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Field-dependent \(J_c(|B|, \angle)\) with spatial defect modulation; \(n\) remains constant.

drho_piecewise_dJ() [4/8]

real belfem::Material::drho_piecewise_dJ ( const real normJ, const real T ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Temperature-only variant: \(J_c(T)\) and \(n(T)\) from Material::jc_custom() / Material::n_custom().

drho_piecewise_dJ() [5/8]

real belfem::Material::drho_piecewise_dJ ( const real normJ, const real T, const real normB, const real angleNxB ) const

inline

Derivative of piecewise resistivity with respect to current density magnitude.

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Fully \((|B|, \angle, T)\)-dependent variant: both \(J_c\) and \(n\) come from JcFunction objects.

Parameters

normJ	Current density magnitude [A/m²]
normB	Magnetic field magnitude [T]
angleNxB	Angle between surface normal and B [rad]
T	Temperature [K]

Returns

Derivative dρ/d||J|| [Ω·m³/A]

For Newton-Raphson: computes dρ/d||J|| for the piecewise model

drho_piecewise_dJ() [6/8]

real belfem::Material::drho_piecewise_dJ ( const real normJ, const real T, const real normB, const real angleNxB, const real x, const real y, const real z, const real t ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Fully \((|B|, \angle, T)\)-dependent variant with spatial defect modulation applied to \(J_c\).

drho_piecewise_dJ() [7/8]

real belfem::Material::drho_piecewise_dJ ( const real normJ, const real T, const real x, const real y, const real z, const real t ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Temperature-only custom-callback variant with spatial defect modulation.

drho_piecewise_dJ() [8/8]

real belfem::Material::drho_piecewise_dJ ( const real normJ, const real x, const real y, const real z, const real t ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Constant \(J_c\) / \(n\) with spatial defect modulation applied to \(J_c\).

drho_piecewise_dT() [1/2]

real belfem::Material::drho_piecewise_dT ( const real normJ, const real T, const real normB, const real angleNxB ) const

inline

Derivative of piecewise resistivity with respect to T at fixed J, |B|, θ ( T-leg ).

Returns

dρ/dT [Ω·m/K]

Follows rho_piecewise's OWN residual branch by branch:

• T > T_crit and normal regime ( J > j3 ): exactly dρ_n/dT.
• power-law regime ( J ≤ j1 ): raw-law derivative, NO parallel factor and NO ρ_n term — consistency is with the residual.
• Bézier blend ( j1 < J ≤ j3 ): BOTH parts — the frozen-knot partial and the knot motion dtParam/dT through j1(T)..j3(T); the knot motion dominates. Degrades to the frozen part alone at the degenerate n−1 == mNff transition. Jumps at j1 and T_crit are inherited from the residual itself.

drho_piecewise_dT() [2/2]

real belfem::Material::drho_piecewise_dT ( const real normJ, const real T, const real normB, const real angleNxB, const real x, const real y, const real z, const real t ) const

inline

defect overload: jc_eff = D(x)·jc throughout, djc_eff = D·djc

drho_powerlaw_dB() [1/2]

real belfem::Material::drho_powerlaw_dB ( const real normJ, const real T, const real normB, const real angleNxB ) const

inline

Derivative of power-law resistivity with respect to |B| at fixed J, T, θ ( jc = jc(T,|B|,θ), n = n(T,|B|,θ) ).

Returns

dρ/d|B| [Ω·m/T]

Chain: dρ/d|B| = (r/p)² · [ ∂p/∂jc·djc/d|B| + ∂p/∂n·dn/d|B| ] with p the power-law resistivity, r the parallel combination with ρ_n, ∂p/∂jc = −n·p/jc and ∂p/∂n = p·ln(J/jc). Parallel factor and floor convention mirror drho_powerlaw_dJ exactly ( floored p in the factor, unfloored law differentiated ).

drho_powerlaw_dB() [2/2]

real belfem::Material::drho_powerlaw_dB ( const real normJ, const real T, const real normB, const real angleNxB, const real x, const real y, const real z, const real t ) const

inline

defect overload: jc_eff = D(x)·jc throughout, djc_eff = D·djc

drho_powerlaw_dJ() [1/8]

real belfem::Material::drho_powerlaw_dJ ( const real normJ ) const

inline

Analytic Jacobian \(d\rho_{eff}/dJ\) of the power-law model.

For the parallel-combined model \(\rho_{eff} = (1/\rho_n + 1/\rho_{PL})^{-1}\) the chain rule gives

\[ \frac{d\rho_{eff}}{dJ} \;=\; \frac{d\rho_{PL}/dJ}{(1 + \rho_{PL}/\rho_n)^2}, \qquad \frac{d\rho_{PL}}{dJ} \;=\; \frac{E_c}{J_c^{2}}\,(n-1)\,\left(\frac{|J|}{J_c}\right)^{n-2}. \]

Required by the nonlinear (Newton-Raphson) iteration in the HTS solver to drive the residual below \(10^{-11}\) as recommended in Messe et al. 2023, Sec. 2.7. For \(|J| < \)BELFEM_EPSILON the derivative is taken as zero to avoid the \(0^{n-2}\) singularity when \(n < 2\).

This overload assumes constant \(J_c\) and \(n\).

Parameters

normJ

current-density magnitude \(|J|\) [A/m²]

Returns

\(d\rho_{eff}/dJ\) [Ω·m / (A/m²)]

drho_powerlaw_dJ() [2/8]

real belfem::Material::drho_powerlaw_dJ ( const real normJ, const real normB, const real angleNxB ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Field-dependent \(J_c(|B|, \angle)\) via mJcFunction; \(n\) remains constant.

drho_powerlaw_dJ() [3/8]

real belfem::Material::drho_powerlaw_dJ ( const real normJ, const real normB, const real angleNxB, const real x, const real y, const real z, const real t ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Field-dependent \(J_c(|B|, \angle)\) with spatial defect modulation; \(n\) remains constant.

drho_powerlaw_dJ() [4/8]

real belfem::Material::drho_powerlaw_dJ ( const real normJ, const real T ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Temperature-only variant: \(J_c(T)\) and \(n(T)\) from Material::jc_custom() / Material::n_custom().

drho_powerlaw_dJ() [5/8]

real belfem::Material::drho_powerlaw_dJ ( const real normJ, const real T, const real normB, const real angleNxB ) const

inline

Derivative of power-law resistivity with respect to current density magnitude.

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Fully \((|B|, \angle, T)\)-dependent variant: both \(J_c\) and \(n\) come from JcFunction objects.

Parameters

normJ	Current density magnitude [A/m²]
normB	Magnetic field magnitude [T]
angleNxB	Angle between surface normal and B [rad]
T	Temperature [K]

Returns

Derivative dρ/d||J|| [Ω·m³/A]

For Newton-Raphson: computes dρ/d||J|| where ρ = ρ(||J||, B, θ, T)

drho_powerlaw_dJ() [6/8]

real belfem::Material::drho_powerlaw_dJ ( const real normJ, const real T, const real normB, const real angleNxB, const real x, const real y, const real z, const real t ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Fully \((|B|, \angle, T)\)-dependent variant with spatial defect modulation applied to \(J_c\).

drho_powerlaw_dJ() [7/8]

real belfem::Material::drho_powerlaw_dJ ( const real normJ, const real T, const real x, const real y, const real z, const real t ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Temperature-only custom-callback variant with spatial defect modulation.

drho_powerlaw_dJ() [8/8]

real belfem::Material::drho_powerlaw_dJ ( const real normJ, const real x, const real y, const real z, const real t ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Constant \(J_c\) / \(n\) with spatial defect modulation applied to \(J_c\).

drho_powerlaw_dT() [1/2]

real belfem::Material::drho_powerlaw_dT ( const real normJ, const real T, const real normB, const real angleNxB ) const

inline

Derivative of power-law resistivity with respect to T at fixed J, |B|, θ ( T-leg: jc(T), n(T) AND ρ_n(T) all move — the quench-feedback tangent ).

Returns

dρ/dT [Ω·m/K]

Chain: dρ/dT = dp₀/dT / (1+ρ_PL/ρ_n)² + dρ_n/dT · (ρ_PL/(ρ_n+ρ_PL))² with dp₀/dT = p₀·( −(n/jc)·djc/dT + ln(J/jc)·dn/dT ). Floor convention mirrors drho_powerlaw_dB ( floored ρ_PL in the parallel factors, unfloored law differentiated ). No (djc==0 && dn==0) early-out: the ρ_n term is the only correct T-dependence of a constant-jc material and must survive.

drho_powerlaw_dT() [2/2]

real belfem::Material::drho_powerlaw_dT ( const real normJ, const real T, const real normB, const real angleNxB, const real x, const real y, const real z, const real t ) const

inline

defect overload: jc_eff = D(x)·jc throughout, djc_eff = D·djc

drho_riva_dB() [1/2]

real belfem::Material::drho_riva_dB ( const real normJ, const real T, const real normB, const real angleNxB ) const

inline

dρ/d|B| of rho_riva: w²·dρ_PL/dB ( dρ_n/dB = 0 on this path )

drho_riva_dB() [2/2]

real belfem::Material::drho_riva_dB ( const real normJ, const real T, const real normB, const real angleNxB, const real x, const real y, const real z, const real t ) const

inline

drho_riva_dJ() [1/2]

real belfem::Material::drho_riva_dJ ( const real normJ, const real T, const real normB, const real angleNxB ) const

inline

dρ/d|J| of rho_riva: w²·dρ_PL/dJ with w = ρ_n/(ρ_PL+ρ_n)

drho_riva_dJ() [2/2]

real belfem::Material::drho_riva_dJ ( const real normJ, const real T, const real normB, const real angleNxB, const real x, const real y, const real z, const real t ) const

inline

drho_riva_dT() [1/2]

real belfem::Material::drho_riva_dT ( const real normJ, const real T, const real normB, const real angleNxB ) const

inline

dρ/dT of rho_riva: w²·dρ_PL/dT + (1−w)²·dρ_n/dT

drho_riva_dT() [2/2]

real belfem::Material::drho_riva_dT ( const real normJ, const real T, const real normB, const real angleNxB, const real x, const real y, const real z, const real t ) const

inline

drhodB()

real belfem::Material::drhodB ( const real T, const real B, const real beta ) const

inlinevirtual

Reimplemented in belfem::material::Alloy, and belfem::material::Metal.

drhodB_kohler()

real belfem::Material::drhodB_kohler ( const real T, const real normB, const real angleJxB ) const

protectedvirtual

Reimplemented in belfem::material::Metal.

drhodB_table()

real belfem::Material::drhodB_table ( const real T, const real normB, const real angleJxB ) const

protectedvirtual

Reimplemented in belfem::material::Metal.

drhodbeta()

real belfem::Material::drhodbeta ( const real T, const real B, const real beta ) const

inlinevirtual

Reimplemented in belfem::material::Alloy, and belfem::material::Metal.

drhodbeta_kohler()

real belfem::Material::drhodbeta_kohler ( const real T, const real normB, const real angleJxB ) const

protectedvirtual

Reimplemented in belfem::material::Metal.

drhodbeta_table()

real belfem::Material::drhodbeta_table ( const real T, const real normB, const real angleJxB ) const

protectedvirtual

Reimplemented in belfem::material::Metal.

drhodT() [1/2]

real belfem::Material::drhodT ( const real T ) const

inlinevirtual

Reimplemented in belfem::material::Alloy, and belfem::material::Metal.

drhodT() [2/2]

real belfem::Material::drhodT ( const real T, const real B, const real beta ) const

inlinevirtual

Reimplemented in belfem::material::Alloy, and belfem::material::Metal.

drhodT_custom()

real belfem::Material::drhodT_custom ( const real T ) const

inlineprotectedvirtual

drhodT_finite_difference()

real belfem::Material::drhodT_finite_difference ( const real T ) const

inlineprotected

drhodT_kohler()

real belfem::Material::drhodT_kohler ( const real T, const real normB, const real angleJxB ) const

protectedvirtual

Reimplemented in belfem::material::Metal.

drhodT_spline()

real belfem::Material::drhodT_spline ( const real T ) const

inlineprotectedvirtual

drhodT_table()

real belfem::Material::drhodT_table ( const real T, const real normB, const real angleJxB ) const

protectedvirtual

Reimplemented in belfem::material::Metal.

dspline_property()

real belfem::Material::dspline_property ( const MaterialProperty aProperty, const real aX ) const

virtual

Reimplemented in belfem::material::SplineLookupTable.

E()

real belfem::Material::E ( real T = gTroom ) const

inlinevirtual

Young's modulus.

Parameters

T	Temperature [K] (default: room temperature)

Returns

Young's modulus [Pa]

E_const()

real belfem::Material::E_const ( const real T ) const

inlineprotected

E_custom()

real belfem::Material::E_custom ( const real T ) const

protectedvirtual

Reimplemented in belfem::material::HastelloyC276, belfem::material::Magnesia, belfem::material::Metal, belfem::material::Nickel, belfem::material::UserDefinedMaterial, and belfem::material::YBCO.

E_spline()

real belfem::Material::E_spline ( const real T ) const

inlineprotected

evaluate_derivative_of_polynomial()

real belfem::Material::evaluate_derivative_of_polynomial ( const MaterialProperty Property, const real T ) const

virtual

Reimplemented in belfem::material::UserDefinedMaterial.

evaluate_polynomial()

real belfem::Material::evaluate_polynomial ( const MaterialProperty Property, const real T ) const

virtual

Evaluate a polynomial for a given property (for UserDefinedMaterial).

Internal method used by UserDefinedMaterial to evaluate polynomial-based property functions. Uses Horner's method for efficient evaluation.

Parameters

Property	The material property
T	Temperature [K]

Returns

Property value at temperature T

Note

This is a virtual method overridden by UserDefinedMaterial

Reimplemented in belfem::material::UserDefinedMaterial.

flag()

void belfem::Material::flag ( const uint8_t aIndex = 0 )

inline

Set flag (multi-purpose flag used by Kernel).

G()

real belfem::Material::G ( real T = gTroom ) const

inlinevirtual

Shear modulus.

Parameters

T	Temperature [K] (default: room temperature)

Returns

Shear modulus G = E/(2(1+ν)) [Pa]

H()

real belfem::Material::H ( const real B ) const

inlinevirtual

Magnetic field strength from flux density.

Parameters

B	Magnetic flux density [T]

Returns

Magnetic field strength H [A/m]

H_bhcurve()

real belfem::Material::H_bhcurve ( const real B ) const

protectedvirtual

Reimplemented in belfem::material::Metal.

H_const()

real belfem::Material::H_const ( const real B ) const

inlineprotected

have()

bool belfem::Material::have ( const MaterialProperty aProperty ) const

inline

Check if a material property is available.

Parameters

aProperty

The property to check

Returns

True if the property is implemented for this material

have_defect()

bool belfem::Material::have_defect ( ) const

inline

Check if the material has a defect.

Returns

True if a defect function was defined

have_heating()

bool belfem::Material::have_heating ( ) const

inline

Check if the material has a heating function.

Returns

True if a heating function was defined

is_constant()

bool belfem::Material::is_constant ( const MaterialProperty aProperty ) const

inline

Check if a property is constant (temperature-independent).

Parameters

aProperty

The property to check

Returns

True if the property is constant

is_flagged()

bool belfem::Material::is_flagged ( const uint8_t aIndex = 0 ) const

inline

Check if material is flagged.

Returns

True if flag is set

is_isotropic()

bool belfem::Material::is_isotropic ( ) const

inline

Check if material is isotropic.

Returns

True if all properties are isotropic

jc() [1/4]

real belfem::Material::jc ( const real normB, const real angleNxB ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Field-dependent \(J_c(|B|, \angle(n,B))\) without temperature dependence.

jc() [2/4]

real belfem::Material::jc ( const real B, const real angleNxB, const real T ) const

inline

Direct evaluation of the critical current density \(J_c(|B|, \angle(n,B), T)\).

Used by post-processing and by overloads of rho_powerlaw() / rho_piecewise() that take a JcFunction-supplied \(J_c(B,\angle,T)\). For materials with a constant \(J_c\), use Material::constant_property(MaterialProperty::jc) instead.

Precondition

mJcFunction must have been installed (checked in debug mode). Note: Material::have(jc) is true even when \(J_c\) was set as a plain constant, so it is not sufficient to guard the function pointer — the explicit nullptr check is required.

jc() [3/4]

real belfem::Material::jc ( const real normB, const real angleNxB, const real T, const real x, const real y, const real z, const real t ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Variant with spatial defect modulation \(J_c \rightarrow J_c \cdot d(x,y,z,t)\).

jc() [4/4]

real belfem::Material::jc ( const real normB, const real angleNxB, const real x, const real y, const real z, const real t ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Field-dependent \(J_c(|B|, \angle(n,B))\) with spatial defect modulation \(J_c \rightarrow J_c \cdot d(x,y,z,t)\).

jc_custom()

real belfem::Material::jc_custom ( const real T ) const

inlinevirtual

Reimplemented in belfem::material::UserDefinedMaterial.

jc_eval()

real belfem::Material::jc_eval ( const real T, const real normB, const real angleNxB ) const

inlineprotected

O1 "full-signature policy" helpers: the assembly path always passes the full ( T, normB, angleNxB ) set; the material consumes what its Jc / n functions depend on and ignores the rest.

JcFunction subclasses override either eval( B, angle ) or eval( B, angle, T ) — the respective other default throws — so we route on the declared T-dependence. Falls back to the plain constants when no function is attached.

K()

real belfem::Material::K ( real T = gTroom ) const

inlinevirtual

Bulk modulus.

Parameters

T	Temperature [K] (default: room temperature)

Returns

Bulk modulus K = E/(3(1-2ν)) [Pa]

l()

real belfem::Material::l ( real T ) const

virtual

Relative length after thermal expansion.

Parameters

T	Temperature [K]

Returns

Relative length l/l₀ [-]

Reimplemented in belfem::material::Metal, and belfem::material::SplineLookupTable.

label()

const string & belfem::Material::label ( ) const

inline

Get material label.

Returns

Material label string (e.g., "Copper", "YBCO")

lambda() [1/3]

real belfem::Material::lambda ( const real T, const real B, const real beta ) const

inlinevirtual

Thermal conductivity for noble metals with magnetoresistance.

Parameters

T	Temperature [K]
B	Magnetic field magnitude [T]
beta	Angle between field and current [rad]

Returns

Thermal conductivity [W/(m·K)]

NOTE: metals evaluate Kohler's rule directly; when a metal was constructed with tables enabled, set_RRR() builds <label>_RRR<n>.hdf5 and the field-dependent evaluation reads it instead. Alloys require the table.

Reimplemented in belfem::material::Alloy, and belfem::material::Metal.

lambda() [2/3]

real belfem::Material::lambda ( const real T, const real B_par, const real B_perp, const real J ) const

inlinevirtual

Thermal conductivity for HTS materials.

Parameters

T	Temperature [K]
B_par	Parallel magnetic field component [T]
B_perp	Perpendicular magnetic field component [T]
J	Current density magnitude [A/m²]

Returns

Thermal conductivity [W/(m·K)]

Reimplemented in belfem::material::Alloy, and belfem::material::Metal.

lambda() [3/3]

real belfem::Material::lambda ( const real T = gTroom ) const

inlinevirtual

Thermal conductivity (isotropic).

Parameters

T	Temperature [K] (default: room temperature)

Returns

Thermal conductivity [W/(m·K)]

Reimplemented in belfem::material::Alloy, and belfem::material::Metal.

lambda_const()

real belfem::Material::lambda_const ( const real T ) const

inlineprotected

lambda_custom() [1/2]

real belfem::Material::lambda_custom ( const real T ) const

protectedvirtual

Reimplemented in belfem::material::HastelloyC276, belfem::material::Magnesia, belfem::material::Metal, belfem::material::UserDefinedMaterial, and belfem::material::YBCO.

lambda_custom() [2/2]

real belfem::Material::lambda_custom ( const real T, const real normB, const real angle ) const

protectedvirtual

Reimplemented in belfem::material::UserDefinedMaterial.

lambda_spline()

real belfem::Material::lambda_spline ( const real T ) const

inlineprotected

lambda_table()

real belfem::Material::lambda_table ( const real T, const real normB, const real angle ) const

protectedvirtual

load_bh_curve()

void belfem::Material::load_bh_curve

(

const material::BhCurve *

aCurve

)

Activate a B-H curve as the permeability source (ferromagnets).

Parameters

aCurve

Concrete B-H curve; the material takes ownership.

Kept free of the concrete curve type: the caller (MaterialFactory) builds the linalg-backed BhSplineCurve, so base Material never depends on Spline/linalg. Stores the curve via set_bh_curve() and routes mu/H/dmudH through it.

M()

real belfem::Material::M ( ) const

inline

Molar mass.

Returns

Molar mass [kg/mol]

mu()

real belfem::Material::mu ( const real H, const real T = BELFEM_QUIET_NAN ) const

inlinevirtual

Magnetic permeability.

Parameters

H	Magnetic field strength [A/m]
T	Temperature [K] (used by user-defined / polynomial mu(H,T); ignored by the constant and B-H-curve bindings)

Returns

Magnetic permeability [H/m]

mu_bhcurve()

real belfem::Material::mu_bhcurve ( const real H, const real T ) const

protectedvirtual

Reimplemented in belfem::material::Metal.

mu_const()

real belfem::Material::mu_const ( const real H, const real T ) const

inlineprotected

mu_custom()

real belfem::Material::mu_custom ( const real H, const real T ) const

protectedvirtual

Reimplemented in belfem::material::HastelloyC276, and belfem::material::UserDefinedMaterial.

n() [1/2]

real belfem::Material::n ( const real normB, const real angleNxB ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Field-dependent \(n(|B|, \angle(n,B))\) without temperature dependence.

n() [2/2]

real belfem::Material::n ( const real normB, const real angleNxB, const real T ) const

inline

Direct evaluation of the n-value as a function of \(|B|\), \(\angle(n,B)\) and \(T\).

Used by post-processing and by overloads of rho_powerlaw() / rho_piecewise() that take a JcFunction-supplied \(n(B,\angle,T)\). For materials with a constant \(n\), use Material::constant_property(MaterialProperty::n) instead.

Precondition

mNFunction must have been installed (checked in debug mode).

n_custom()

real belfem::Material::n_custom ( const real T ) const

inlinevirtual

Reimplemented in belfem::material::UserDefinedMaterial.

n_eval()

real belfem::Material::n_eval ( const real T, const real normB, const real angleNxB ) const

inlineprotected

n_eval_raw()

real belfem::Material::n_eval_raw ( const real T, const real normB, const real angleNxB ) const

inlineprotected

unfloored n as routed from table / custom / constant — n_eval clamps this at 1, and the dn_eval_* derivatives return 0 while the clamp binds

nu()

real belfem::Material::nu ( real T = gTroom ) const

inlinevirtual

Poisson's ratio.

Parameters

T	Temperature [K] (default: room temperature)

Returns

Poisson's ratio [-]

nu_const()

real belfem::Material::nu_const ( const real T ) const

inlineprotected

nu_custom()

real belfem::Material::nu_custom ( const real T ) const

protectedvirtual

Reimplemented in belfem::material::HastelloyC276, belfem::material::Magnesia, belfem::material::UserDefinedMaterial, and belfem::material::YBCO.

nu_spline()

real belfem::Material::nu_spline ( const real T ) const

inlineprotected

number()

const string & belfem::Material::number ( ) const

inline

Get material number/identifier.

Returns

Material number if it exists, empty string otherwise

operator=() [1/2]

Material & belfem::Material::operator= ( const Material & )

delete

operator=() [2/2]

Material & belfem::Material::operator= ( Material && )

delete

read_defect()

void belfem::Material::read_defect	(	const string &	aLibraryPath,
		const string &	aLabel )

Read a defect from the library.

Parameters

aLibraryPath	path to the shared library holding the defect, resolved through material::data_file(): the run directory first, then $BELFEM_DATA/material, and otherwise handed to dlopen unchanged
aLabel	name of the defect to read

read_heating()

void belfem::Material::read_heating	(	const string &	aLibraryPath,
		const string &	aLabel )

Read a heating function from a plugin library.

Parameters

aLibraryPath	path to the shared library holding the heating function, resolved through material::data_file(): the run directory first, then $BELFEM_DATA/material, and otherwise handed to dlopen unchanged
aLabel	name of the heating function; the library must export <label>_init( Material * )

ref_density()

real belfem::Material::ref_density ( ) const

inline

Reference density at reference temperature.

Returns

Reference density [kg/m³]

Use this for thermal M-matrix calculations.

reset_dependencies()

void belfem::Material::reset_dependencies

(

const MaterialProperty

aProperty

)

Reset all dependencies for a property.

reset_spline()

void belfem::Material::reset_spline ( const MaterialProperty aProperty )

protectedvirtual

Reimplemented in belfem::material::SplineLookupTable.

resistivity_law()

ResistivityLaw belfem::Material::resistivity_law ( ) const

inline

return_zero()

real belfem::Material::return_zero ( const real T ) const

inlineprotected

Shared binding for derivative channels that are identically zero.

Bound wherever a property is held constant, so that its first and higher derivatives vanish. One function rather than one clone per property and per derivative order.

rho() [1/2]

real belfem::Material::rho ( const real T ) const

inlinevirtual

Electrical resistivity (isotropic).

Parameters

T	Temperature [K]

Returns

Electrical resistivity [Ω·m]

For HTS materials, use rho_powerlaw() instead.

Reimplemented in belfem::material::Alloy, and belfem::material::Metal.

rho() [2/2]

real belfem::Material::rho ( const real T, const real B, const real beta ) const

inlinevirtual

Electrical resistivity for noble metals with magnetoresistance.

Parameters

T	Temperature [K]
B	Magnetic field magnitude [T]
beta	Angle between field and current [rad]

Returns

Electrical resistivity [Ω·m]

NOTE: metals evaluate Kohler's rule directly; when a metal was constructed with tables enabled, set_RRR() builds <label>_RRR<n>.hdf5 and the field-dependent evaluation reads it instead. Alloys require the table.

Reimplemented in belfem::material::Alloy, and belfem::material::Metal.

rho_const()

real belfem::Material::rho_const ( const real T ) const

inlineprotected

rho_custom()

real belfem::Material::rho_custom ( const real T ) const

inlineprotectedvirtual

Reimplemented in belfem::material::Ferromagnetic, belfem::material::HastelloyC276, belfem::material::Metal, and belfem::material::UserDefinedMaterial.

rho_i()

real belfem::Material::rho_i ( const real T ) const

inlinevirtual

Intrinsic (phonon) electrical resistivity of a metal, ρ = ρ_i(T) + ρ_0.

Parameters

T	Temperature [K]

Returns

Intrinsic electrical resistivity [Ω·m]

rho_i_custom()

real belfem::Material::rho_i_custom ( const real T ) const

protectedvirtual

Reimplemented in belfem::material::Metal.

rho_i_spline()

real belfem::Material::rho_i_spline ( const real T ) const

inlineprotected

rho_kohler()

real belfem::Material::rho_kohler ( const real T, const real normB, const real angleJxB ) const

protectedvirtual

Reimplemented in belfem::material::Metal, and belfem::material::UserDefinedMaterial.

rho_piecewise() [1/8]

real belfem::Material::rho_piecewise ( const real normJ ) const

inline

Piecewise resistivity for HTS (constant jc and n).

Three-regime resistivity (power law / flux-flow / normal) with a smooth Bezier transition in log–log space.

Parameters and return value are documented on the definition in powerlaws.hpp, together with the derivation and its references.

Use this for HTS materials instead of rho(T).

The pure power law diverges as \(J \rightarrow \infty\), which is unphysical above the critical regime: an HTS conductor saturates at the resistivity of its normal-state matrix once the superconductor is fully driven. This model splits the \(J\)-axis into three regimes connected smoothly in \((\log_{10} J,\,\log_{10}\rho)\) space:

• Power-law regime \((J \le J_1)\): standard \(\rho_{PL}(J) = (E_c/J_c)\,(J/J_c)^{n-1}\).
• Flux-flow transition \((J_1 < J \le J_3)\): a quadratic Bezier curve in \((\log_{10} J,\,\log_{10}\rho)\) connecting \((\log J_1, \log \rho_1)\) to \((\log J_3, \log \rho_n)\) with the middle control point at \((\log J_2, \log \rho_n)\).
• Normal regime \((J > J_3)\): \(\rho = \rho_n(T)\).

The transition currents are

\[ J_1 \;=\; J_c \cdot 10^{D/n}, \qquad J_2 \;=\; J_1 \cdot \left(\rho_n / \rho_1\right)^{1/(n-1)}, \qquad J_3 \;=\; J_1 \cdot \left(\rho_n / \rho_1\right)^{1/N_{ff}}, \]

where the knee is placed a fixed \(D = 2.5\) decades above \(J_c\) ( hardcoded; the member mD is not consulted ) and mNff (default 3) controls the slope of the flux-flow regime. Above the assumed critical temperature \(T_{crit}\), only \(\rho_n\) is returned.

The Bezier parameter \(t \in [0,1]\) for a queried \(J\) is the positive root of \(a\,t^2 - 2\,b\,t - c = 0\) with \(a = \log J_1 - 2\log J_2 + \log J_3\), \(b = \log J_1 - \log J_2\), and \(c = \log J - \log J_1\). The actual resistivity is then

\[ \log_{10}\rho_{FF}(J) = (1-t)^2 \log\rho_1 + 2(1-t)\,t \log\rho_n + t^2 \log\rho_n . \]

This three-regime extension is BELFEM-specific; it complements the pure Rhyner power law (Messe et al. 2023, Sec. 2.6) so that the same material can be evaluated above the critical regime in quench simulations without numerical blow-up.

Parameters

normJ

current-density magnitude \(|J|\) [A/m²]

Returns

effective electric resistivity [Ω·m]

Precondition

\(J_c > 0\) and \(n > 1\) (checked in debug mode).

The Bezier coefficient \(|a|\) must be non-degenerate (checked in debug mode for the flux-flow branch).

rho_piecewise() [2/8]

real belfem::Material::rho_piecewise ( const real normJ, const real T ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Temperature-only variant: \(J_c(T)\) and \(n(T)\) from the user-supplied callbacks Material::jc_custom() / Material::n_custom().

rho_piecewise() [3/8]

real belfem::Material::rho_piecewise ( const real normJ, const real T, const real x, const real y, const real z, const real t ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Temperature-only custom-callback variant with spatial defect modulation.

rho_piecewise() [4/8]

real belfem::Material::rho_piecewise ( const real normJ, const real T, real normB, const real angleNxB ) const

inline

Piecewise resistivity for HTS (full temperature and field dependence).

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Fully \((|B|, \angle, T)\)-dependent variant: both \(J_c\) and \(n\) come from JcFunction objects, and \(\rho_n\) is evaluated at the local temperature T.

Parameters

normJ	Current density magnitude [A/m²]
normB	Magnetic field magnitude [T]
angleNxB	Angle between surface normal and B [rad]
T	Temperature [K]

Returns

Electrical resistivity [Ω·m]

rho_piecewise() [5/8]

real belfem::Material::rho_piecewise ( const real normJ, const real T, real normB, const real angleNxB, const real x, const real y, const real z, const real t ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Fully \((|B|, \angle, T)\)-dependent variant with spatial defect modulation applied to \(J_c\).

rho_piecewise() [6/8]

real belfem::Material::rho_piecewise ( const real normJ, const real x, const real y, const real z, const real t ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Constant \(J_c\) / \(n\) with spatially-dependent defect modulation \(J_c \rightarrow J_c \cdot d(x,y,z,t)\).

rho_piecewise() [7/8]

real belfem::Material::rho_piecewise ( const real normJ, real normB, const real angleNxB ) const

inline

Piecewise resistivity for HTS (field-dependent jc, constant n).

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Field-dependent \(J_c(|B|, \angle)\) via mJcFunction; \(n\) remains constant.

Parameters

normJ	Current density magnitude [A/m²]
normB	Magnetic field magnitude [T]
angleNxB	Angle between surface normal and B [rad]

Returns

Electrical resistivity [Ω·m]

rho_piecewise() [8/8]

real belfem::Material::rho_piecewise ( const real normJ, real normB, const real angleNxB, const real x, const real y, const real z, const real t ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Field-dependent \(J_c(|B|, \angle)\) with spatial defect modulation.

rho_powerlaw() [1/8]

real belfem::Material::rho_powerlaw ( const real normJ ) const

inline

Power law resistivity for HTS (constant jc and n).

Effective resistivity from the E–J power law (Rhyner 1993), parallel-combined with the normal-state resistivity.

Parameters and return value are documented on the definition in powerlaws.hpp, together with the derivation and its references.

Computes ρ = ( 1/ρ_n + 1/ρ_PL )^-1 with ρ_PL = (Ec/jc)·(J/jc)^(n-1) for constant jc and n ( parallel combination with the normal-state resistivity ρ_n, see powerlaws.hpp ). Use this for HTS materials instead of rho(T).

The intrinsic HTS power law is

\[ \rho_{PL}(J) \;=\; \frac{E_c}{J_c}\left(\frac{|J|}{J_c}\right)^{n-1}, \]

where \(E_c\) is the critical-field criterion (typically \(10^{-4}\) V/m), \(J_c\) is the critical current density and \(n\) is the resistive-transition exponent ( \(n \rightarrow 1\) → linear, \(n \rightarrow \infty\) → ideal critical-state).

To bound \(\rho\) from above as \(J \gg J_c\) and from below as \(J \rightarrow 0\), the effective resistivity is the parallel combination of \(\rho_{PL}\) with the normal-state resistivity \(\rho_n(T)\):

\[ \rho_{eff}(J) \;=\; \left(\frac{1}{\rho_n} + \frac{1}{\rho_{PL}}\right)^{-1}. \]

\(\rho_{PL}\) is floored at mRhoMin, which is ZERO by default (2026-08-10): a positive floor desynchronizes the value from drho_powerlaw_dJ, which differentiates the unfloored law — with the old 1e-16 the Newton tangent was wrong for all \(J \lesssim 0.87\,J_c\) on typical tape constants. Anyone raising mRhoMin again must also zero the derivative while the floor binds.

In this overload \(J_c\) and \(n\) are read as plain constants (configured via Material::set_constant) and \(\rho_n\) is evaluated at the global bulk temperature gTbulk.

Parameters

normJ

current-density magnitude \(|J|\) [A/m²]

Returns

effective electric resistivity [Ω·m]

See also

Rhyner, Physica C 212 (1993) 292–300.

Messe et al., Supercond. Sci. Technol. 36 (2023) 114001, Sec. 2.6.

rho_powerlaw() [2/8]

real belfem::Material::rho_powerlaw ( const real normJ, const real T ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Temperature-only variant.

\(J_c(T)\) and \(n(T)\) are obtained from the user-supplied callbacks Material::jc_custom() and Material::n_custom(), allowing materials with bespoke analytic \(T\)-laws (e.g. Kim-style fits) without going through a JcFunction.

rho_powerlaw() [3/8]

real belfem::Material::rho_powerlaw ( const real normJ, const real T, const real x, const real y, const real z, const real t ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Temperature-only custom-callback variant with spatial defect modulation.

rho_powerlaw() [4/8]

real belfem::Material::rho_powerlaw ( const real normJ, const real T, real normB, const real angleNxB ) const

inline

Power law resistivity for HTS (full temperature and field dependence).

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Fully field- and temperature-dependent variant: \(J_c = J_c(|B|, \angle, T)\) and \(n = n(|B|, \angle, T)\), both supplied via mJcFunction / mNFunction.

Parameters

normJ	Current density magnitude [A/m²]
normB	Magnetic field magnitude [T]
angleNxB	Angle between surface normal and B [rad]
T	Temperature [K]

Returns

Electrical resistivity [Ω·m]

Computes ρ = ( 1/ρ_n + 1/ρ_PL )^-1 with ρ_PL = (Ec/jc)·(J/jc)^(n-1) where jc = jc(B, angle, T) and n = n(B, angle, T).

\(\rho_n\) is evaluated at the local temperature T (not gTbulk).

rho_powerlaw() [5/8]

real belfem::Material::rho_powerlaw ( const real normJ, const real T, real normB, const real angleNxB, const real x, const real y, const real z, const real t ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Fully \((|B|, \angle, T)\)-dependent variant with spatial defect modulation applied to \(J_c\).

rho_powerlaw() [6/8]

real belfem::Material::rho_powerlaw ( const real normJ, const real x, const real y, const real z, const real t ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Constant \(J_c\) / \(n\) with spatially-dependent defect modulation \(J_c \rightarrow J_c \cdot d(x,y,z,t)\).

rho_powerlaw() [7/8]

real belfem::Material::rho_powerlaw ( const real normJ, real normB, const real angleNxB ) const

inline

Power law resistivity for HTS (field-dependent jc, constant n).

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Field-dependent critical current: \(J_c = J_c(|B|, \angle(n,B))\) supplied via mJcFunction.

Parameters

normJ	Current density magnitude [A/m²]
normB	Magnetic field magnitude [T]
angleNxB	Angle between surface normal and B [rad]

Returns

Electrical resistivity [Ω·m]

Computes ρ = ( 1/ρ_n + 1/ρ_PL )^-1 with ρ_PL = (Ec/jc)·(J/jc)^(n-1) where jc = jc(B, angle).

\(n\) remains a constant.

rho_powerlaw() [8/8]

real belfem::Material::rho_powerlaw ( const real normJ, real normB, const real angleNxB, const real x, const real y, const real z, const real t ) const

inline

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts. Field-dependent \(J_c(|B|, \angle)\) with spatial defect modulation.

rho_riva() [1/2]

real belfem::Material::rho_riva ( const real normJ, const real T, const real normB, const real angleNxB ) const

inline

Riva-law resistivity: the superconducting power-law channel in parallel with the normal-state channel.

Returns

ρ = ρ_PL·ρ_n / ( ρ_PL + ρ_n ) [Ω·m]

The parallel combination of Duron et al. 2004, used by Riva 2021 (EPFL thesis 8754, Eq. 5.4). Same model core as rho_powerlaw, but total over the full table range: jc_eff ≤ 0 or nonfinite (dead defect, underflow) falls back to the fully normal branch, the power-law channel is evaluated in log10 space with an overflow early-out, and n arrives pre-floored at 1 from n_eval (ohmic limit ρ_PL = ec/jc, J-independent). Only the assembly-path signatures (J,T,|B|,θ) ± defect exist; jc and n arrive through the usual dependency routing ( table or constants ).

rho_riva() [2/2]

real belfem::Material::rho_riva ( const real normJ, const real T, const real normB, const real angleNxB, const real x, const real y, const real z, const real t ) const

inline

defect overload: jc_eff = D(x,y,z,t)·jc throughout

rho_spline()

real belfem::Material::rho_spline ( const real T ) const

inlineprotectedvirtual

rho_table()

real belfem::Material::rho_table ( const real T, const real normB, const real angleJxB ) const

protectedvirtual

Reimplemented in belfem::material::Metal.

riva_rho_pl()

bool belfem::Material::riva_rho_pl ( const real normJ, const real jc, const real n, const real ec, real & rhoPL ) const

inlineprotected

shared riva kernel: power-law channel resistivity in log10 space. Returns false when the channel has overflowed past the normal state ( caller takes the fully-normal branch: ρ = ρn )

Rp02()

real belfem::Material::Rp02 ( real T = gTroom ) const

inlinevirtual

Yield stress (0.2% offset).

Parameters

T	Temperature [K] (default: room temperature)

Returns

Yield stress Rp0.2 [Pa]

Rp02_const()

real belfem::Material::Rp02_const ( const real T ) const

inlineprotected

Rp02_custom()

real belfem::Material::Rp02_custom ( const real T ) const

protectedvirtual

Reimplemented in belfem::material::UserDefinedMaterial.

Rp02_spline()

real belfem::Material::Rp02_spline ( const real T ) const

inlineprotected

set_bh_curve()

void belfem::Material::set_bh_curve ( const material::BhCurve * aCurve )

virtual

Assign a B-H curve to this material.

Parameters

aCurve

Pointer to B-H curve (material takes ownership)

IMPORTANT: Material takes ownership and will delete the curve.

Virtual so that load_bh_curve() (defined on the base) dispatches to the B-H-capable override in Metal; the base implementation errors.

Reimplemented in belfem::material::Metal.

set_constant()

void belfem::Material::set_constant	(	const MaterialProperty	aProperty,
		const real	aValue )

Define a property as constant.

set_custom()

void belfem::Material::set_custom

(

const MaterialProperty

aProperty

)

Mark property as using custom evaluation function.

set_dependency()

void belfem::Material::set_dependency	(	const MaterialProperty	aProperty,
		const MaterialDependency	aDependency )

Add a dependency to a property.

set_have()

void belfem::Material::set_have	(	const MaterialProperty	aProperty,
		const bool	aHave = true )

Mark a property as available.

set_jc_function()

void belfem::Material::set_jc_function

(

const material::JcFunction *

aFunction

)

Assign a critical current density function.

Parameters

aFunction

Pointer to Jc function (material takes ownership)

IMPORTANT: Material takes ownership and will delete the function.

set_label()

void belfem::Material::set_label

(

const string &

aLabel

)

PROTECTED INTERFACE FOR DERIVED MATERIALS.

This section contains utility functions for derived material classes to set up their properties. The Material class uses a strategy pattern where properties can be:

1. Constant - stored directly in mConstantProperties
2. Spline-interpolated - using tabulated data
3. Custom - using specialized functions in derived classes

When a property is set up, the appropriate function pointer is assigned (e.g., mFunctionRho points to rho_const, rho_spline, or rho_custom).

Set material label (for derived classes)

set_n_function()

void belfem::Material::set_n_function

(

const material::JcFunction *

aFunction

)

Assign a power law exponent function.

Parameters

aFunction

Pointer to n function (material takes ownership)

IMPORTANT: Material takes ownership and will delete the function.

set_number()

void belfem::Material::set_number

(

const string &

aNumber

)

Set material number/identifier (for derived classes).

set_piecewise()

void belfem::Material::set_piecewise

(

const bool

aUsePiecewise

)

set_resistivity_law()

void belfem::Material::set_resistivity_law

(

const ResistivityLaw

aLaw

)

set_RRR()

void belfem::Material::set_RRR ( const real RRR )

virtual

Set the residual resistivity ratio (for noble metals).

Parameters

RRR	Residual resistivity ratio (ρ(273K)/ρ(0K))

Reimplemented in belfem::material::Metal.

set_table_flags()

void belfem::Material::set_table_flags ( const bool aFlag )

virtual

Reimplemented in belfem::material::Metal.

set_user_defined_defect()

void belfem::Material::set_user_defined_defect

(

DefectFunc *

Function

)

Set a user-defined defect function with x, y, z, t dependencies.

Parameters

Function

Pointer to user function with signature: real(real, real, real, real)

set_user_defined_function() [1/3]

void belfem::Material::set_user_defined_function ( const MaterialProperty Property, const MaterialDependency Dependency, MatFunc1 * Function )

virtual

Set a user-defined function with one dependency (for UserDefinedMaterial).

This method is primarily used by UserDefinedMaterial to assign custom property evaluation functions. Most users should not call this directly; instead, use MaterialFactory::create_material() with a user library.

Parameters

Property	The material property to define
Dependency	The dependency type (typically MaterialDependency::T)
Function	Pointer to user function with signature: real(const Material*, real)

Note

This is a virtual method overridden by UserDefinedMaterial

Reimplemented in belfem::material::UserDefinedMaterial.

set_user_defined_function() [2/3]

void belfem::Material::set_user_defined_function ( const MaterialProperty Property, const MaterialDependency Dependency1, const MaterialDependency Dependency2, const MaterialDependency Dependency3, MatFunc3 * Function )

virtual

Set a user-defined function with three dependencies (for UserDefinedMaterial).

This method is primarily used by UserDefinedMaterial to assign custom property evaluation functions for three-parameter properties like rho(T,B,angle).

Parameters

Property	The material property to define
Dependency1	First dependency
Dependency2	Second dependency
Dependency3	Third dependency
Function	Pointer to user function with signature: real(const Material*, real, real, real)

Note

This is a virtual method overridden by UserDefinedMaterial

Reimplemented in belfem::material::UserDefinedMaterial.

set_user_defined_function() [3/3]

void belfem::Material::set_user_defined_function ( const MaterialProperty Property, const MaterialDependency Dependency1, const MaterialDependency Dependency2, MatFunc2 * Function )

virtual

Set a user-defined function with two dependencies (for UserDefinedMaterial).

This method is primarily used by UserDefinedMaterial to assign custom property evaluation functions for two-parameter properties like mu(H,T) or jc(B,angle).

Parameters

Property	The material property to define
Dependency1	First dependency
Dependency2	Second dependency
Function	Pointer to user function with signature: real(const Material*, real, real)

Note

This is a virtual method overridden by UserDefinedMaterial

Reimplemented in belfem::material::UserDefinedMaterial.

set_user_defined_heating()

void belfem::Material::set_user_defined_heating

(

HeatFunc *

Function

)

Set a user-defined volumetric heat load [ W/m³ ] as a function of x, y, z [ m ] and t [ s ].

Parameters

Function

Pointer to user function with signature: real(real, real, real, real)

set_user_defined_polynomial() [1/2]

void belfem::Material::set_user_defined_polynomial ( const MaterialProperty Property, const Cell< real > & Coefficients )

virtual

Reimplemented in belfem::material::UserDefinedMaterial.

set_user_defined_polynomial() [2/2]

void belfem::Material::set_user_defined_polynomial ( const MaterialProperty Property, const std::vector< real > & Coefficients )

virtual

Set a polynomial function for a property (for UserDefinedMaterial).

Defines a property as a polynomial in temperature: f(T) = c₀T^n + c₁T^(n-1) + ... + c_n This is a convenience function for user-defined materials.

IMPORTANT: Coefficients are in DESCENDING order (MATLAB style), highest degree first.

Parameters

Property	The material property to define
Coefficients	Polynomial coefficients [c₀, c₁, c₂, ...] in DESCENDING order (highest degree first)

Note

This is a virtual method overridden by UserDefinedMaterial

Reimplemented in belfem::material::UserDefinedMaterial.

spline_property()

real belfem::Material::spline_property ( const MaterialProperty aProperty, const real aX ) const

virtual

Evaluate property using spline interpolation.

Reimplemented in belfem::material::Metal, and belfem::material::SplineLookupTable.

type()

MaterialType belfem::Material::type ( ) const

inline

Get the material type.

Returns

Material type (Ferro, HTS, PureMetal, etc.)

unflag()

void belfem::Material::unflag ( const uint8_t aIndex = 0 )

inline

Clear flag.

use_piecewise()

bool belfem::Material::use_piecewise ( ) const

inline

Check if we use piecewise instead of power-law.

Returns

True if a we use piecewise resistivity

use_riva()

bool belfem::Material::use_riva ( ) const

inline

volumetric_heatload()

real belfem::Material::volumetric_heatload ( const real x, const real y, const real z, const real time ) const

inline

Artificial volumetric heat load [ W/m³ ] from the heating plugin.

Member Data Documentation

mFunctionDMuDH

void(Material::*) belfem::Material::mFunctionDMuDH(double, double &, double &) const = nullptr

protected

mFunctiondRhoKohlerdB

real(Material::*) belfem::Material::mFunctiondRhoKohlerdB(const real T, const real normB, const real angle) const = nullptr

protected

mFunctiondRhoKohlerdbeta

real(Material::*) belfem::Material::mFunctiondRhoKohlerdbeta(const real T, const real normB, const real angle) const = nullptr

protected

mFunctiondRhoKohlerdT

real(Material::*) belfem::Material::mFunctiondRhoKohlerdT(const real T, const real normB, const real angle) const = nullptr

protected

mFunctionH

real(Material::*) belfem::Material::mFunctionH(const real B) const = nullptr

protected

mFunctionMu

real(Material::*) belfem::Material::mFunctionMu(const real H, const real T) const = nullptr

protected

mFunctionRhoKohler

real(Material::*) belfem::Material::mFunctionRhoKohler(const real T, const real normB, const real angle) const = nullptr

protected

mJcFunction

const material::JcFunction* belfem::Material::mJcFunction = nullptr

protected

mNFunction

const material::JcFunction* belfem::Material::mNFunction = nullptr

protected

---

The documentation for this class was generated from the following files:

• src/physics/materials/cl_Material.hpp
• src/physics/materials/cl_Material.cpp
• src/physics/materials/powerlaws.hpp