geom::SimplexModCondNum< N, T > Class Template Reference

Implements the modified condition number quality metric. More...

#include <SimplexModCondNum.h>

Inheritance diagram for geom::SimplexModCondNum< N, T >:


Public Types
typedef T	Number
	The number type.
typedef Base::Vertex	Vertex
	The class for a vertex.
typedef Base::Simplex	Simplex
	The simplex type.
typedef Base::Matrix	Matrix
	An NxN matrix.
Public Member Functions
Constructors etc.
	SimplexModCondNum ()
	Default constructor. Un-initialized memory.
	SimplexModCondNum (const SimplexModCondNum &other)
	Copy constructor.
	SimplexModCondNum (const Simplex &s)
	Construct from a simplex.
SimplexModCondNum &	operator= (const SimplexModCondNum &other)
	Assignment operator.
	~SimplexModCondNum ()
	Trivial destructor.
Mathematical functions
Number	operator() () const
	Return the modified condition number (kappa) quality metric.
Number	operator() (const Simplex &simplex) const
	Return the modified condition number (kappa) quality metric.
Number	operator() (Number minDeterminant) const
	Return the modified condition number (kappa) quality metric.
void	computeGradient (Vertex *gradient) const
	Calculate the gradient of the modified condition number (kappa) quality metric.
void	computeGradient (Number minDeterminant, Vertex *gradient) const
	Calculate the gradient of the modified condition number (kappa) quality metric.
Protected Member Functions
Number	computeFunction (const Number snj, const Number sna) const
	Return the modified quality metric given and $\|\Sigma\|^2$ .
Number	computeFunction (Number minDeterminant, Number snj, Number sna) const
	Return the modified quality metric given and $\|\Sigma\|^2$ .

Detailed Description

template<int N, typename T = double>
class geom::SimplexModCondNum< N, T >

Implements the modified condition number quality metric.

Parameters:

	N	is the dimension.
	T	is the number type. By default it is double.

This class implements the modified condition number metric.

operator()() returns the modified condition number metric.
operator()(const Simplex& simplex) returns the modified condition number metric as a function of the simplex.
operator()(const Number minDeterminant) returns the modified metric.
computeGradient(Vertex& grad) calculates the gradient of the metric. If the determinant is positive, it is the gradient of the unmodified metric. Otherwise it is the modified metric.
computeGradient(const Number minDeterminant,Vertex& gradient) is useful when this simplex is part of a complex if simplices.

Before evaluating the metric, you must set the Jacobian matrix with setFunction() or set(). Before evaluating the gradient of the metric, you must set the Jacobian matrix and its gradient with set().

Member Function Documentation

template<int N, typename T = double>

void geom::SimplexModCondNum< N, T >::computeGradient	(	Number	minDeterminant,
		Vertex *	gradient
	)			const

Calculate the gradient of the modified condition number (kappa) quality metric.

min_determinant is the minimum determinant of the simplices currently being considered. If the quality of a single simplex is being computed, then min_determinant should be the Jacobian determinant of this simplex. If the quality of the simplices adjacent to a vertex is being considered, then min_determinant should be the minimum determinant among these simplices.

Precondition:: The Jacobian determinant need not be positive.

Let

be the Jacobian matrix, $\Sigma$ be its scaled inverse, $\sigma$ be the Jacobian determinant, $\sigma_m$ be the minimum Jacobian determinant, $\epsilon$ be the value of epsilon() and $| \cdot |$ be the Frobenius norm. If $\sigma_m \geq \epsilon$ then the kappa quality metric is

$\frac{ |S| |\Sigma| }{ N \sigma }.$

Otherwise the modified kappa quality metric is

$\frac{ |S| |\Sigma| }{ N h(\sigma_m) }.$

template<int N, typename T = double>

void geom::SimplexModCondNum< N, T >::computeGradient ( Vertex * gradient ) const [inline]

Calculate the gradient of the modified condition number (kappa) quality metric.

Precondition:: The Jacobian determinant need not be positive.

Let

be the Jacobian matrix, $\Sigma$ be its scaled inverse, $\sigma$ be the Jacobian determinant and $| \cdot |$ be the Frobenius norm. The modified kappa function is

$\frac{ |S| |\Sigma| }{ N h(\sigma) }.$

Reimplemented from geom::SimplexCondNum< N, T >.

template<int N, typename T = double>

Number geom::SimplexModCondNum< N, T >::operator() ( Number minDeterminant ) const

Return the modified condition number (kappa) quality metric.

Parameters:

minDeterminant is the minimum determinant of the simplices currently being considered. If the quality of a single simplex is being computed, then min_determinant should be the Jacobian determinant of this simplex. If the quality of the simplices adjacent to a vertex is being considered, then min_determinant should be the minimum determinant among these simplices.