#include <complex.h>
#include <stdlib.h>
#include <stdbool.h>

Data Structures
struct	Electrode

struct	Integration_data

struct	Field_integrand_data

Enumerations
enum	Integration_type { INTG_NONE = 1, INTG_DOUBLE = 2, INTG_SINGLE = 3, INTG_MHEM = 4 }

Functions
int	populate_electrode (Electrode *electrode, const double start_point[3], const double end_point[3], double radius)

bool	equal_electrodes (const Electrode sender, const Electrode receiver)

int	electrodes_file (const char file_name[], Electrode *electrodes, size_t num_electrodes)

int	nodes_file (const char file_name[], double *nodes, size_t num_nodes)

int	segment_electrode (Electrode electrodes, double nodes, size_t num_segments, const double start_point, const double end_point, double radius)

size_t	nodes_from_elecs (double nodes, Electrode electrodes, size_t num_electrodes)

int	integrand_double (unsigned ndim, const double t, void auxdata, unsigned fdim, double *fval)

int	integrand_single (unsigned ndim, const double t, void auxdata, unsigned fdim, double *fval)

int	logNf (unsigned ndim, const double t, void auxdata, unsigned fdim, double *fval)

_Complex double	self_integral (const Electrode *sender)

int	integral (const Electrode sender, const Electrode receiver, _Complex double gamma, size_t max_eval, double req_abs_error, double req_rel_error, int integration_type, double result[2], double error[2])

int	calculate_impedances (_Complex double zl, _Complex double zt, const Electrode *electrodes, size_t num_electrodes, _Complex double gamma, _Complex double s, double mur, _Complex double kappa, size_t max_eval, double req_abs_error, double req_rel_error, int integration_type)

int	impedances_images (_Complex double zl, _Complex double zt, const Electrode electrodes, const Electrode images, size_t num_electrodes, _Complex double gamma, _Complex double s, double mur, _Complex double kappa, _Complex double ref_l, _Complex double ref_t, size_t max_eval, double req_abs_error, double req_rel_error, int integration_type)

_Complex double	electric_potential (const double point, const Electrode electrodes, size_t num_electrodes, const _Complex double *it, _Complex double gamma, _Complex double kappa, size_t max_eval, double req_abs_error, double req_rel_error)

int	magnetic_potential (const double point, const Electrode electrodes, size_t num_electrodes, const _Complex double il, _Complex double gamma, double mur, size_t max_eval, double req_abs_error, double req_rel_error, _Complex double va)

int	elec_field_integrand (unsigned ndim, const double t, void auxdata, unsigned fdim, double *fval)

int	electric_field (const double point, const Electrode electrodes, size_t num_electrodes, const _Complex double il, const _Complex double it, _Complex double gamma, _Complex double s, double mur, _Complex double kappa, size_t max_eval, double req_abs_error, double req_rel_error, _Complex double *ve)

int	v_mag_pot_integrand (unsigned ndim, const double t, void auxdata, unsigned fdim, double *fval)

_Complex double	voltage (const double point1, const double point2, const Electrode electrodes, size_t num_electrodes, const _Complex double il, const _Complex double *it, _Complex double gamma, _Complex double s, double mur, _Complex double kappa, size_t max_eval, double req_abs_error, double req_rel_error)

Enumeration Type Documentation

◆ Integration_type

enum Integration_type

Type of integration and simplification thereof to be done.

Enumerator
INTG_NONE	No integration, returns the geometric distance between middle points
INTG_DOUBLE	Traditional integral along both electrodes $ \int_0^{L_i} \int_0^{L_k} \frac{e^{-\gamma r}}{r} d\ell_k \, d\ell_i $
INTG_SINGLE	Traditional integral along sender electrodes $ \int_0^{L_k} \frac{e^{-\gamma r}}{r} d\ell_k $
INTG_MHEM	Modified HEM integral $ \int_0^{L_k} \log_e \left( \frac{R_1 + R_2 + L_i}{R_1 + R_2 - L_i} \right) d\ell_k $

Definition at line 18 of file electrode.h.

Function Documentation

◆ calculate_impedances()

int calculate_impedances	(	_Complex double *	zl,
		_Complex double *	zt,
		const Electrode *	electrodes,
		size_t	num_electrodes,
		_Complex double	gamma,
		_Complex double	s,
		double	mur,
		_Complex double	kappa,
		size_t	max_eval,
		double	req_abs_error,
		double	req_rel_error,
		int	integration_type
	)

Calculates the impedance matrices $ Z_L, Z_T $. As they are symmetric, only their lower half is stored (set). If pointer zl = zt, then the resulting matrix will be filled with zt.
If integration_type == INTG_MHEM or integration_type == INTG_NONE, then parameters gamma, s, mur and kappa are ignored such that $ \frac{j\omega\mu}{4\pi} = \frac{1}{4\pi\,(\sigma + j\omega\varepsilon)} = 1 $.

Parameters

zl	longitudinal impedance matrix $ Z_L $ as a flat array of size $ m^2 $
zt	transversal impedance matrix $ Z_T $ as a flat array of size $ m^2 $
electrodes	array of electrodes
num_electrodes	number of electrodes $ m $
gamma	medium propagation constant $ \gamma = \sqrt{j\omega\mu(\sigma + j\omega\varepsilon)} $
s	complex angular frequency $ s = c + j\omega $ in [rad/s]
mur	relative magnetic permeability of the medium $ \mu_r $
kappa	medium complex conductivity $ \sigma + j\omega\varepsilon $ in [S/m]
max_eval	specifies a maximum number of function evaluations (0 for no limit)
req_abs_error	the absolute error requested (0 to ignore)
req_rel_error	the relative error requested (0 to ignore)
integration_type	type of integration to be done.

Returns: 0 on success

See also: integral; Integration_type; https://github.com/stevengj/cubature

◆ elec_field_integrand()

int elec_field_integrand	(	unsigned	ndim,
		const double *	t,
		void *	auxdata,
		unsigned	fdim,
		double *	fval
	)

Integrand to calculate the electric field caused by a differential transversal current $ dI_T $.

Parameters

ndim	should be 1
t	integration variable (electrode percentage 0 to 1)
auxdata	pointer to Field_integrand_data structure
fdim	should be 6 (Real and Imag.): $ (\Re(E_x), \Im(E_x), \Re(E_y), \Im(E_y), \Re(E_z), \Im(E_z)) $
fval	where the results are stored $ (\Re(E_x), \Im(E_x), \Re(E_y), \Im(E_y), \Re(E_z), \Im(E_z)) $ return 0 on success

See also: https://github.com/stevengj/cubature; electric_field

◆ electric_field()

int electric_field	(	const double *	point,
		const Electrode *	electrodes,
		size_t	num_electrodes,
		const _Complex double *	il,
		const _Complex double *	it,
		_Complex double	gamma,
		_Complex double	s,
		double	mur,
		_Complex double	kappa,
		size_t	max_eval,
		double	req_abs_error,
		double	req_rel_error,
		_Complex double *	ve
	)

Calculates the electric field at a point. $ \vec E = -\nabla u - s \vec A $

Parameters

point	array $ (x, y, z) $
electrodes	array of electrodes
num_electrodes	number of electrodes $ m $
il	longitudinal currents array $ I_L $
it	transversal currents array $ I_T $
gamma	medium propagation constant $ \gamma = \sqrt{j\omega\mu(\sigma + j\omega\varepsilon)} $
s	complex angular frequency $ s = c + j\omega $ in [rad/s]
mur	relative magnetic permeability of the medium $ \mu_r $
kappa	medium complex conductivity $ \sigma + j\omega\varepsilon $ in [S/m]
max_eval	specifies a maximum number of function evaluations (0 for no limit)
req_abs_error	the absolute error requested (0 to ignore)
req_rel_error	the relative error requested (0 to ignore)
ve	pointer to array of size 3 in which the electric field components $ (E_x, E_y, E_z) $ will be add to on return.

Returns: 0 on success

See also: elec_field_integrand

◆ electric_potential()

_Complex double electric_potential	(	const double *	point,
		const Electrode *	electrodes,
		size_t	num_electrodes,
		const _Complex double *	it,
		_Complex double	gamma,
		_Complex double	kappa,
		size_t	max_eval,
		double	req_abs_error,
		double	req_rel_error
	)

Calculates the scalar electric potential $ u $ to remote earth at a point.

Parameters

point	array $(x, y, z)$
electrodes	array of electrodes
num_electrodes	number of electrodes $ m $
it	transversal currents array $ I_T $
gamma	medium propagation constant $ \gamma = \sqrt{j\omega\mu(\sigma + j\omega\varepsilon)} $
kappa	medium complex conductivity $ \sigma + j\omega\varepsilon $ in [S/m]
max_eval	specifies a maximum number of function evaluations (0 for no limit)
req_abs_error	the absolute error requested (0 to ignore)
req_rel_error	the relative error requested (0 to ignore)

Returns: potential $ u $ to remote earth

See also: https://github.com/stevengj/cubature

◆ electrodes_file()

int electrodes_file	(	const char	file_name[],
		Electrode *	electrodes,
		size_t	num_electrodes
	)

Populates an Electrode array from a CSV file. Each Electrode must be defined in a single line with parameters: "x0, y0, z0, x1, y1, z1, radius".

Parameters

file_name	path to the file
electrodes	pointer to an array of Electrode to be filled
num_electrodes	number of electrodes to read (number of lines in file). Any line number greater than num_electrodes will be ignored.

Returns: error == 0 on success
error < 0: number of missing arguments from last line read (as a negative integer)
error > 0: bad input

◆ equal_electrodes()

bool equal_electrodes	(	const Electrode *	sender,
		const Electrode *	receiver
	)

Compares two electrodes for equality: have the same radius and coincide the starts and end points (independent of direction).

Parameters

sender	Electrode
receiver	Electrode

Returns: true or false

◆ impedances_images()

int impedances_images	(	_Complex double *	zl,
		_Complex double *	zt,
		const Electrode *	electrodes,
		const Electrode *	images,
		size_t	num_electrodes,
		_Complex double	gamma,
		_Complex double	s,
		double	mur,
		_Complex double	kappa,
		_Complex double	ref_l,
		_Complex double	ref_t,
		size_t	max_eval,
		double	req_abs_error,
		double	req_rel_error,
		int	integration_type
	)

Add the images' effect to the impedance matrices $ Z_L $ and $ Z_T $. As they are symmetric, only their lower half is stored (set). If pointer zl = zt, then the resulting matrix will be filled with zt.
If integration_type == INTG_MHEM or integration_type == INTG_NONE, then parameters gamma, s, mur, kappa, ref_l and ref_t are ignored such that $ \Gamma_L \frac{j\omega\mu}{4\pi} = \Gamma_T \frac{1}{4\pi\,(\sigma + j\omega\varepsilon)} = 1 $.

Parameters

zl	longitudinal impedance matrix $ Z_L $ as a flat array of size $ m^2 $
zt	transversal impedance matrix $ Z_T $ as a flat array of size $ m^2 $
electrodes	array of electrodes
images	array of electrodes
num_electrodes	number of electrodes $ m $
gamma	medium propagation constant $ \gamma = \sqrt{j\omega\mu(\sigma + j\omega\varepsilon)} $
s	complex angular frequency $ s = c + j\omega $ in [rad/s]
mur	relative magnetic permeability of the medium $ \mu_r $
kappa	medium complex conductivity $ \sigma + j\omega\varepsilon $ in [S/m]
ref_l	longitudinal current reflection coefficient $ \Gamma_L $
ref_t	transversal current reflection coefficient $ \Gamma_T $
max_eval	specifies a maximum number of function evaluations (0 for no limit)
req_abs_error	the absolute error requested (0 to ignore)
req_rel_error	the relative error requested (0 to ignore)
integration_type	type of integration to be done.

Returns: 0 on success

See also: integral; Integration_type; https://github.com/stevengj/cubature

◆ integral()

int integral	(	const Electrode *	sender,
		const Electrode *	receiver,
		_Complex double	gamma,
		size_t	max_eval,
		double	req_abs_error,
		double	req_rel_error,
		int	integration_type,
		double	result[2],
		double	error[2]
	)

Calculates the integral along the sender and receiver using Cubature. Note that the integration stops when either max_eval or req_abs_error or req_rel_error is reached.

Parameters

sender	Current carrying Electrode that generates the excitation.
receiver	Potential receiving Electrode that receives the excitation.
gamma	medium propagation constant $ \gamma = \sqrt{j\omega\mu(\sigma + j\omega\varepsilon)} $
max_eval	specifies a maximum number of function evaluations (0 for no limit)
req_abs_error	the absolute error requested (0 to ignore)
req_rel_error	the relative error requested (0 to ignore)
integration_type	type of integration to be done.
result	array[2] where to store the integral result (real and imaginary parts)
error	array[2] where to store the integral error (real and imaginary parts)

Returns: 0 on success
-10 if integration_type is unrecognized.

See also: Integration_type; https://github.com/stevengj/cubature

◆ integrand_double()

int integrand_double	(	unsigned	ndim,
		const double *	t,
		void *	auxdata,
		unsigned	fdim,
		double *	fval
	)

Calculates the integrand $ \frac{e^{-\gamma r}}{r} $ to be integrated using Cubature.

Parameters

ndim	must be = 2
t	array of size 2 of the electrodes' length as a percentage (0 to 1)
auxdata	Integration_data with electrodes and propagation constant
fdim	must be = 2
fval	pointer to where the result is stored

Returns: 0 on success

See also: Integration_type; Integration_data; integral; https://github.com/stevengj/cubature

◆ integrand_single()

int integrand_single	(	unsigned	ndim,
		const double *	t,
		void *	auxdata,
		unsigned	fdim,
		double *	fval
	)

Calculates the integrand $ \frac{e^{-\gamma r}}{r} $ to be integrated using Cubature; but the point on the receiver electrode is fixed at its middle.

Parameters

ndim	must be = 1
t	array of size 1 of the sender electrode's length as a percentage (0 to 1)
auxdata	Integration_data with electrodes and propagation constant
fdim	must be = 2
fval	pointer to where the result is stored

Returns: 0 on success

See also: Integration_type; Integration_data; integral; https://github.com/stevengj/cubature

◆ logNf()

int logNf	(	unsigned	ndim,
		const double *	t,
		void *	auxdata,
		unsigned	fdim,
		double *	fval
	)

Calculates the mHEM integrand $ \int_0^{L_k} \log_e \left( \frac{R_1 + R_2 + L_i}{R_1 + R_2 - L_i} \right) d\ell_k $ to be used in Cubature integration.

Parameters

ndim	must be = 1
t	array of size 1 with receiver's length percentage (0 to 1)
auxdata	Integration_data with electrodes and propagation constant
fdim	must be = 2
fval	pointer to where the result is stored

Returns: 0 on success

See also: Integration_type; Integration_data; integral; https://github.com/stevengj/cubature

◆ magnetic_potential()

int magnetic_potential	(	const double *	point,
		const Electrode *	electrodes,
		size_t	num_electrodes,
		const _Complex double *	il,
		_Complex double	gamma,
		double	mur,
		size_t	max_eval,
		double	req_abs_error,
		double	req_rel_error,
		_Complex double *	va
	)

Calculates the magnetic vector potential $ \vec A $.

Parameters

point	array $(x, y, z)$
electrodes	array of electrodes
num_electrodes	number of electrodes $ m $
il	longitudinal currents array $ I_L $
gamma	medium propagation constant $ \gamma = \sqrt{j\omega\mu(\sigma + j\omega\varepsilon)} $
mur	relative magnetic permeability of the medium $ \mu_r $
max_eval	specifies a maximum number of function evaluations (0 for no limit)
req_abs_error	the absolute error requested (0 to ignore)
req_rel_error	the relative error requested (0 to ignore)
va	pointer to array of size 3 in which the magnetic vector potential components $ (A_x,A_y,A_z) $ will be stored on return.

Returns: 0 on success

See also: https://github.com/stevengj/cubature

◆ nodes_file()

int nodes_file	(	const char	file_name[],
		double *	nodes,
		size_t	num_nodes
	)

Fill a nodes array from a CSV file. Each node must be defined in a single line with parameters: "x, y, z".

Parameters

file_name	path to the file
nodes	array of double[3] to be filled
num_nodes	number of nodes to read (number of lines in file). Any line number greater than num_nodes will be ignored.

Returns: error == 0 on success
error < 0: number of missing arguments from last line read (as a negative integer)
error > 0: bad input

◆ nodes_from_elecs()

size_t nodes_from_elecs	(	double *	nodes,
		Electrode *	electrodes,
		size_t	num_electrodes
	)

DON'T USE THIS FUNCTIONS, IT'S BUGGED.

Given an array of electrodes, populates the nodes array with all the unique nodes.

Parameters

nodes	flat array of size $n \ge 3(m + 1)$ to be filled with the unique nodes.
electrodes	array of electrodes
num_electrodes	number $m$ of electrodes

Returns: number of unique nodes $n_u$; can be used to realloc the nodes array

◆ populate_electrode()

int populate_electrode	(	Electrode *	electrode,
		const double	start_point[3],
		const double	end_point[3],
		double	radius
	)

Populates an Electrode structure.

Parameters

pointer	to an allocated memory
start_point	Array $(x,y,z)_0$ that defines the starting point of the electrode
start_point	Array $(x,y,z)_0$ that defines the ending point of the electrode
radius	electrode radius The Electrode radius

Returns: 0 on success

◆ segment_electrode()

int segment_electrode	(	Electrode *	electrodes,
		double *	nodes,
		size_t	num_segments,
		const double *	start_point,
		const double *	end_point,
		double	radius
	)

Segments an electrode populating a passed array of electrodes and an array of nodes.

Parameters

electrodes	pointer to an array of Electrode to be filled
nodes	flat array $3 (n + 1)$ to be filled by the new nodes that are created
num_segments	number $ n $ of segments to create
start_point	electrode's start point
end_point	electrode's end point
radius	electrode's radius

Returns: 0 on success

◆ self_integral()

_Complex double self_integral ( const Electrode * sender )

Calculates the mHEM closed form solution of the integral when the sender and receiver segments are the same (self impedance). $ 2 L_k \left[ \log_e \left( \frac{\sqrt{1 + (b/L_k)^2} + 1}{b/L_k} \right) - \sqrt{1 + \left(\frac{b}{L_k}\right)^2} + \frac{b}{L_k} \right] $

◆ v_mag_pot_integrand()

int v_mag_pot_integrand	(	unsigned	ndim,
		const double *	t,
		void *	auxdata,
		unsigned	fdim,
		double *	fval
	)

Interface to pass magnetic_potential as an integrand to Cubature when calculating the voltage along a path.

Parameters

ndim	should be 1
t	integration variable (electrode percentage 0 to 1)
auxdata	pointer to Field_integrand_data structure
fdim	should be 2: $ \Re(\vec A \cdot d\vec\ell) + \Im(\vec A \cdot d\vec\ell) $
fval	where the results are stored (array of size 2) return 0 on success

See also: https://github.com/stevengj/cubature

◆ voltage()

_Complex double voltage	(	const double *	point1,
		const double *	point2,
		const Electrode *	electrodes,
		size_t	num_electrodes,
		const _Complex double *	il,
		const _Complex double *	it,
		_Complex double	gamma,
		_Complex double	s,
		double	mur,
		_Complex double	kappa,
		size_t	max_eval,
		double	req_abs_error,
		double	req_rel_error
	)

Calculates the voltage $ \Delta U_{12} $ along a straight line from $ \vec p_1 $ to $ \vec p_2 $.
$ \Delta U_{12} = \int_{\vec p_1}^{\vec p_2} \left( -\nabla u - s \vec A \right) d\vec\ell $

Parameters

point1	Line integral start point $ \vec p_1 = (x,y,z)_1 $
point2	Line integral start point $ \vec p_2 = (x,y,z)_2 $
electrodes	array of electrodes
num_electrodes	number of electrodes $ m $
il	longitudinal currents array $ I_L $
gamma	medium propagation constant $ \gamma = \sqrt{j\omega\mu(\sigma + j\omega\varepsilon)} $
s	complex angular frequency $ s = c + j\omega $ in [rad/s]
mur	relative magnetic permeability of the medium $ \mu_r $
kappa	medium complex conductivity $ \sigma + j\omega\varepsilon $ in [S/m]
max_eval	specifies a maximum number of function evaluations (0 for no limit)
req_abs_error	the absolute error requested (0 to ignore)
req_rel_error	the relative error requested (0 to ignore)

Returns: the voltage along the line

See also: https://github.com/stevengj/cubature

zl	longitudinal impedance matrix \( Z_L \) as a flat array of size \( m^2 \)
zt	transversal impedance matrix \( Z_T \) as a flat array of size \( m^2 \)
electrodes	array of electrodes
num_electrodes	number of electrodes \( m \)
gamma	medium propagation constant \( \gamma = \sqrt{j\omega\mu(\sigma + j\omega\varepsilon)} \)
s	complex angular frequency \( s = c + j\omega \) in [rad/s]
mur	relative magnetic permeability of the medium \( \mu_r \)
kappa	medium complex conductivity \( \sigma + j\omega\varepsilon \) in [S/m]
max_eval	specifies a maximum number of function evaluations (0 for no limit)
req_abs_error	the absolute error requested (0 to ignore)
req_rel_error	the relative error requested (0 to ignore)
integration_type	type of integration to be done.

point	array \( (x, y, z) \)
electrodes	array of electrodes
num_electrodes	number of electrodes \( m \)
il	longitudinal currents array \( I_L \)
it	transversal currents array \( I_T \)
gamma	medium propagation constant \( \gamma = \sqrt{j\omega\mu(\sigma + j\omega\varepsilon)} \)
s	complex angular frequency \( s = c + j\omega \) in [rad/s]
mur	relative magnetic permeability of the medium \( \mu_r \)
kappa	medium complex conductivity \( \sigma + j\omega\varepsilon \) in [S/m]
max_eval	specifies a maximum number of function evaluations (0 for no limit)
req_abs_error	the absolute error requested (0 to ignore)
req_rel_error	the relative error requested (0 to ignore)
ve	pointer to array of size 3 in which the electric field components \( (E_x, E_y, E_z) \) will be add to on return.

point	array \((x, y, z)\)
electrodes	array of electrodes
num_electrodes	number of electrodes \( m \)
it	transversal currents array \( I_T \)
gamma	medium propagation constant \( \gamma = \sqrt{j\omega\mu(\sigma + j\omega\varepsilon)} \)
kappa	medium complex conductivity \( \sigma + j\omega\varepsilon \) in [S/m]
max_eval	specifies a maximum number of function evaluations (0 for no limit)
req_abs_error	the absolute error requested (0 to ignore)
req_rel_error	the relative error requested (0 to ignore)

nodes	flat array of size \(n \ge 3(m + 1)\) to be filled with the unique nodes.
electrodes	array of electrodes
num_electrodes	number $m$ of electrodes

point1	Line integral start point \( \vec p_1 = (x,y,z)_1 \)
point2	Line integral start point \( \vec p_2 = (x,y,z)_2 \)
electrodes	array of electrodes
num_electrodes	number of electrodes \( m \)
il	longitudinal currents array \( I_L \)
gamma	medium propagation constant \( \gamma = \sqrt{j\omega\mu(\sigma + j\omega\varepsilon)} \)
s	complex angular frequency \( s = c + j\omega \) in [rad/s]
mur	relative magnetic permeability of the medium \( \mu_r \)
kappa	medium complex conductivity \( \sigma + j\omega\varepsilon \) in [S/m]
max_eval	specifies a maximum number of function evaluations (0 for no limit)
req_abs_error	the absolute error requested (0 to ignore)
req_rel_error	the relative error requested (0 to ignore)

Data Structures

Enumerations

Functions

Enumeration Type Documentation

◆ Integration_type

Function Documentation

◆ calculate_impedances()

◆ elec_field_integrand()

◆ electric_field()

◆ electric_potential()

◆ electrodes_file()

◆ equal_electrodes()

◆ impedances_images()

◆ integral()

◆ integrand_double()

◆ integrand_single()

◆ logNf()

◆ magnetic_potential()

◆ nodes_file()

◆ nodes_from_elecs()

◆ populate_electrode()

◆ segment_electrode()

◆ self_integral()

◆ v_mag_pot_integrand()

◆ voltage()