CGCoordinates::XYZ Class Reference

Defines the position of the CGPoint, in the non-averaged directions, i.e. all directions on which spatial coarse-graining is applied (all directions for XYZ); all other directions are averaged over homogeneously. More...

#include <XYZ.h>

Inheritance diagram for CGCoordinates::XYZ:

Public Member Functions
void	write (std::ostream &os) const
	Writes the coordinates in human-readable form to an ostream. More...
Mdouble	getDistanceSquared (const Vec3D &p) const
	Returns the square of the distance between the particle p and the current CGPoint, in the non-averaged directions. More...
void	setXYZ (Vec3D p)
	Returns the position of the current CGPoint, in the non-averaged directions. More...
void	setHalfExtend (Vec3D halfExtend)
	Sets the extend of the current grid cell. More...
Mdouble	getINormal (const BaseInteraction &c, const Vec3D &normal) const
	For the Interaction between particles/walls P and I, this function returns the dot product between the normal vector of the interaction and the branch vector from the current CGPoint towards I. More...
Mdouble	getPNormal (const BaseInteraction &c, const Vec3D &normal) const
	For the Interaction between particles/walls P and I, this function returns the dot product between the normal vector of the interaction and the branch vector from the current CGPoint towards P. More...
Mdouble	getCNormal (const BaseInteraction &c, const Vec3D &normal) const
	For the Interaction between particles/walls P and I, this function returns the dot product between the normal vector of the interaction and the branch vector from the current CGPoint towards the contact point. More...
Mdouble	getTangentialSquared (const BaseInteraction &c, Mdouble pNormal) const
	For the Interaction between particles/walls P and I, this function returns the square of the minimum distance between the the current CGPoint and the branch vector between P and I. More...
Public Member Functions inherited from CGCoordinates::BaseCoordinates
virtual Mdouble	getWeight ()

Static Public Member Functions
static void	writeNames (std::ostream &os)
	Writes the coordinate names in human-readable form to an ostream. More...
static Mdouble	getVolumeOfAveragedDimensions (const Vec3D &min, const Vec3D &max)
	returns the factor the CGFunction has to be divided by, due to integrating the variables over the averaged dimensions, 1.0 for XYZ. More...
static Mdouble	getLength (const Vec3D &p)
	Returns the length of the input vector in the non-averaged directions. More...
static Mdouble	getGaussPrefactor (Mdouble width, Mdouble cutoff)
	Computes the prefactor of the Gauss CGFunction, which is dependent on the number of non-averaged dimensions. More...
static Mdouble	getGaussIntegralPrefactor (Mdouble distance, Mdouble width, Mdouble cutoff)
	Computes the prefactor of the Gauss line integral, which is dependent on the number of non-averaged dimensions. More...
static void	normalisePolynomialCoefficients (std::vector< Mdouble > &coefficients, Mdouble cutoff)
	Normalises the coefficients of Polynomial CGFunction such that the integral over all non-averaged dimensions is unity. More...
static const unsigned	countVariables ()
static bool	isResolvedIn (unsigned i)
static std::string	getName ()
Static Public Member Functions inherited from CGCoordinates::BaseCoordinates
static Mdouble	getDomainVolume (const Vec3D &min, const Vec3D &max)

Protected Attributes
Vec3D	p_
Vec3D	pointHalfExtend_

Detailed Description

Defines the position of the CGPoint, in the non-averaged directions, i.e. all directions on which spatial coarse-graining is applied (all directions for XYZ); all other directions are averaged over homogeneously.

In addition to defining the spatial variable, the classes in CGCoordinates contain all functions that only depend on how many coordinate directions are locally resolved (i.e. not averaged over); e.g. getDistanceSquared and getVolumeOfAveragedDimensions.

The CGCoordinates class should be chosen such that the system in homogeneous (symmetric) in the averaged directions. E.g., periodic chutes are homogeneous in x, y and t, but varying in z, so Z should be used.

See member CGCoordinates for more details.

Member Function Documentation

countVariables()

const unsigned XYZ::countVariables ( )

static

returns the number of variables (in this case three)

{
    return 3;
}

getCNormal()

Mdouble XYZ::getCNormal	(	const BaseInteraction &	c,
		const Vec3D &	normal
	)		const

For the Interaction between particles/walls P and I, this function returns the dot product between the normal vector of the interaction and the branch vector from the current CGPoint towards the contact point.

Parameters

[in]

c

the Interaction object from which iNormal is computed

Returns

cNormal, one of the three distances needed to calculate the line integral \(\psi\) which defines the stress distribution (see image).

Illustration of the line integral

{
    return Vec3D::dot(c.getContactPoint() - p_, c.getNormal());
}

getDistanceSquared()

Mdouble XYZ::getDistanceSquared

(

const Vec3D &

p

)

const

Returns the square of the distance between the particle p and the current CGPoint, in the non-averaged directions.

This function is needed to evaluate the CGFunction, as this function has the distance between the CGPoint and the Particle as an argument. To properly account for the averaging, the distance is only computed in the non-averaged directions.

Parameters

[in]

p

the position of a particle for which the distance is computed.

{
    return Vec3D::getLengthSquared(p_ - p);
}

References Vec3D::getLengthSquared(), p, and p_.

getGaussIntegralPrefactor()

Mdouble XYZ::getGaussIntegralPrefactor ( Mdouble  distance, Mdouble  width, Mdouble  cutoff  )

static

Computes the prefactor of the Gauss line integral, which is dependent on the number of non-averaged dimensions.

The prefactor of the Gauss line integral is set such that the integral over the non-averaged dimensions is unity.

Parameters

[in]	distance	length of the branch vector along which the line integral is evaluated.
[in]	width	width (equals the standard deviation in 1D) of the Gauss CGFunction.
[in]	cutoff	cutoff of the Gauss CGFunction

Returns

the prefactor of the Gauss CGFunction.

{
    Mdouble widthSqrt2 = width * constants::sqrt_2;
    Mdouble a = -cutoff;
    Mdouble b = cutoff + distance;
    //full 2D prefactor
    Mdouble prefactor = 1.0 / (constants::sqrt_2 * constants::sqrt_pi * width);
    prefactor = mathsFunc::square(prefactor) / (1.0 - exp(-0.5 * mathsFunc::square(cutoff / width)));
    return prefactor * 0.5 / (
            +erf(b / widthSqrt2) * b
            + widthSqrt2 / constants::sqrt_pi * exp(-mathsFunc::square(b / widthSqrt2))
            - erf(a / widthSqrt2) * a
            - widthSqrt2 / constants::sqrt_pi * exp(-mathsFunc::square(a / widthSqrt2))
    );
}

References a, b, Eigen::bfloat16_impl::exp(), constants::sqrt_2, constants::sqrt_pi, and mathsFunc::square().

getGaussPrefactor()

Mdouble XYZ::getGaussPrefactor ( Mdouble  width, Mdouble  cutoff  )

static

Computes the prefactor of the Gauss CGFunction, which is dependent on the number of non-averaged dimensions.

The prefactor of the Gauss CGFunction is set such that the integral over the non-averaged dimensions is unity.

Parameters

[in]	width	width (equals the standard deviation in 1D) of the Gauss CGFunction.
[in]	cutoff	cutoff of the Gauss CGFunction

Returns

the prefactor of the Gauss CGFunction.

{
    //Wolfram alpha: erf(c/(sqrt(2) w))-(sqrt(2/pi) c e^(-c^2/(2 w^2)))/w
    Mdouble prefactor = 1.0 / (constants::sqrt_2 * constants::sqrt_pi * width);
    Mdouble cw = cutoff / width;
    return mathsFunc::cubic(prefactor) / (
            erf(cw / constants::sqrt_2)
            - constants::sqrt_2 / constants::sqrt_pi * cw * exp(-0.5 * mathsFunc::square(cw))
    );
}

References mathsFunc::cubic(), Eigen::bfloat16_impl::exp(), constants::sqrt_2, constants::sqrt_pi, and mathsFunc::square().

getINormal()

Mdouble XYZ::getINormal	(	const BaseInteraction &	c,
		const Vec3D &	normal
	)		const

For the Interaction between particles/walls P and I, this function returns the dot product between the normal vector of the interaction and the branch vector from the current CGPoint towards I.

Parameters

[in]

c

the Interaction object from which iNormal is computed

Returns

iNormal, one of the three distances needed to calculate the line integral \(\psi\) which defines the stress distribution (see image).

Illustration of the line integral

{
    return Vec3D::dot(c.getI()->getPosition() - p_, c.getNormal());
}

References calibrate::c, Vec3D::dot(), and p_.

getLength()

Mdouble XYZ::getLength ( const Vec3D &  p )

static

Returns the length of the input vector in the non-averaged directions.

Parameters

[in]

p

vector whose length should be determined

Returns

length of the vector in the non-averaged directions

Todo:

{
    return sqrt(p.X * p.X + p.Y * p.Y + p.Z * p.Z);
}

References p, and sqrt().

getName()

std::string XYZ::getName ( )

static

{
    return "XYZ";
}

getPNormal()

Mdouble XYZ::getPNormal	(	const BaseInteraction &	c,
		const Vec3D &	normal
	)		const

For the Interaction between particles/walls P and I, this function returns the dot product between the normal vector of the interaction and the branch vector from the current CGPoint towards P.

Parameters

[in]

c

the Interaction object from which iNormal is computed

Returns

pNormal, one of the three distances needed to calculate the line integral \(\psi\) which defines the stress distribution (see image).

Illustration of the line integral

{
    return Vec3D::dot(c.getP()->getPosition() - p_, c.getNormal());
}

getTangentialSquared()

Mdouble XYZ::getTangentialSquared	(	const BaseInteraction &	c,
		Mdouble	pNormal
	)		const

For the Interaction between particles/walls P and I, this function returns the square of the minimum distance between the the current CGPoint and the branch vector between P and I.

Parameters

[in]	c	the Interaction object from which iNormal is computed
[in]	pNormal	the output of getPNormal needed for the computation.

Returns

iNormal, one of the three distances needed to calculate the line integral \(\psi\) which defines the stress distribution (see image).

Illustration of the line integral

{
    return Vec3D::getLengthSquared(c.getP()->getPosition() - p_) - mathsFunc::square(pNormal);
}

References calibrate::c, Vec3D::getLengthSquared(), p_, and mathsFunc::square().

getVolumeOfAveragedDimensions()

Mdouble XYZ::getVolumeOfAveragedDimensions ( const Vec3D &  min, const Vec3D &  max  )

static

returns the factor the CGFunction has to be divided by, due to integrating the variables over the averaged dimensions, 1.0 for XYZ.

If averaged dimensions are present (i.e. for all Coordinates except XYZ), the CGFunction has to be divided by a factor (here called volume) due to integrating the variables over the averaged dimensions.

Parameters

[in]	min	the lower limits of the mesh domain (xMin, yMin, zMin)
[in]	max	the upper limits of the mesh domain (xMax, yMax, zMax)

Returns

the volume factor

{
    return 1.0;
}

isResolvedIn()

static bool CGCoordinates::XYZ::isResolvedIn ( unsigned  i )

inlinestatic

{return true;}

normalisePolynomialCoefficients()

void XYZ::normalisePolynomialCoefficients ( std::vector< Mdouble > &  coefficients, Mdouble  cutoff  )

static

Normalises the coefficients of Polynomial CGFunction such that the integral over all non-averaged dimensions is unity.

The volume is computed as

\[volume= \int_0^1\sum_{i=1}^n c_i r/c^i 4 pi r^2 dr = 4 pi \sum_{i=1}^n c_i/(i+3) \]

with 4 pi r^2 the surface area of a sphere.

Parameters

[in,out]	coefficients	the coefficients of Polynomial CGFunctions.
[in]	cutoff	cutoff of the Gauss CGFunction

{
    Mdouble volume = 0.0;
    for (std::size_t i = 0; i < coefficients.size(); i++)
        volume += coefficients[i] / static_cast<Mdouble>(i + 3);
    volume *= 4.0 * constants::pi * mathsFunc::cubic(cutoff);
    for (double& coefficient : coefficients)
        coefficient /= volume;
}

References mathsFunc::cubic(), i, and constants::pi.

setHalfExtend()

void CGCoordinates::XYZ::setHalfExtend ( Vec3D  halfExtend )

inline

Sets the extend of the current grid cell.

{ pointHalfExtend_ = halfExtend; }

References pointHalfExtend_.

setXYZ()

void XYZ::setXYZ

(

Vec3D

p

)

Returns the position of the current CGPoint, in the non-averaged directions.

Parameters

[in]

p

the position that is to be set.

{
    p_ = p;
}

References p, and p_.

Referenced by VolumeCoupling::getProjectionMatrix().

write()

void XYZ::write

(

std::ostream &

os

)

const

Writes the coordinates in human-readable form to an ostream.

Parameters

[out]

os

the ostream file to which the position is written

{
    os << p_ << ' ';
}

References p_.

writeNames()

void XYZ::writeNames ( std::ostream &  os )

static

Writes the coordinate names in human-readable form to an ostream.

{
    os << "x y z ";
}

Member Data Documentation

p_

Vec3D CGCoordinates::XYZ::p_

protected

The position of the current CGPoint.

Referenced by getDistanceSquared(), getINormal(), getTangentialSquared(), setXYZ(), and write().

pointHalfExtend_

Vec3D CGCoordinates::XYZ::pointHalfExtend_

protected

Referenced by setHalfExtend().

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

• XYZ.h
• XYZ.cc