MOAB: Mesh Oriented datABase  (version 5.4.1)
Public Member Functions | Private Member Functions | Private Attributes
MBMesquite::TUntangle1 Class Reference

Untangle metric. More...

#include <TUntangle1.hpp>

+ Inheritance diagram for MBMesquite::TUntangle1:
+ Collaboration diagram for MBMesquite::TUntangle1:

Public Member Functions

 TUntangle1 (double epsilon=1e-6, double tau_bar=1.0)
virtual MESQUITE_EXPORT ~TUntangle1 ()
virtual MESQUITE_EXPORT std::string get_name () const
virtual MESQUITE_EXPORT bool evaluate (const MsqMatrix< 2, 2 > &T, double &result, MsqError &err)
 Evaluate \(\mu(T)\).
virtual MESQUITE_EXPORT bool evaluate_with_grad (const MsqMatrix< 2, 2 > &T, double &result, MsqMatrix< 2, 2 > &deriv_wrt_T, MsqError &err)
 Gradient of \(\mu(T)\) with respect to components of T.
virtual MESQUITE_EXPORT bool evaluate_with_hess (const MsqMatrix< 2, 2 > &T, double &result, MsqMatrix< 2, 2 > &deriv_wrt_T, MsqMatrix< 2, 2 > second_wrt_T[3], MsqError &err)
 Hessian of \(\mu(T)\) with respect to components of T.
virtual MESQUITE_EXPORT bool evaluate (const MsqMatrix< 3, 3 > &T, double &result, MsqError &err)
 Evaluate \(\mu(T)\).
virtual MESQUITE_EXPORT bool evaluate_with_grad (const MsqMatrix< 3, 3 > &T, double &result, MsqMatrix< 3, 3 > &deriv_wrt_T, MsqError &err)
 Gradient of \(\mu(T)\) with respect to components of T.
virtual MESQUITE_EXPORT bool evaluate_with_hess (const MsqMatrix< 3, 3 > &T, double &result, MsqMatrix< 3, 3 > &deriv_wrt_T, MsqMatrix< 3, 3 > second_wrt_T[6], MsqError &err)
 Hessian of \(\mu(T)\) with respect to components of T.

Private Member Functions

template<unsigned D>
bool eval (const MsqMatrix< D, D > &T, double &result)
template<unsigned D>
bool grad (const MsqMatrix< D, D > &T, double &result, MsqMatrix< D, D > &first)
template<unsigned D>
bool hess (const MsqMatrix< D, D > &T, double &result, MsqMatrix< D, D > &first, MsqMatrix< D, D > *second)

Private Attributes

double mFactor

Detailed Description

Untangle metric.

\( \mu_n(T) = \frac{1}{2} \left( - \tau + \sqrt{\tau^2 + \left(\epsilon \, \bar{\tau} \right)^2} \right)\)

Section 3.2.8 of derivs.tex

Definition at line 48 of file TUntangle1.hpp.

Constructor & Destructor Documentation

MBMesquite::TUntangle1::TUntangle1 ( double  epsilon = 1e-6,
double  tau_bar = 1.0 
) [inline]

Definition at line 54 of file TUntangle1.hpp.

: mFactor( epsilon * epsilon * tau_bar * tau_bar ) {}
MBMesquite::TUntangle1::~TUntangle1 ( ) [virtual]

Definition at line 42 of file TUntangle1.cpp.

{}

Member Function Documentation

template<unsigned DIM>
bool MBMesquite::TUntangle1::eval ( const MsqMatrix< DIM, DIM > &  T,
double &  result 
) [inline, private]

Definition at line 50 of file TUntangle1.cpp.

References MBMesquite::det(), and mFactor.

{
    double tau = det( T );
    result     = 0.5 * ( sqrt( tau * tau + mFactor ) - tau );
    return true;
}
virtual MESQUITE_EXPORT bool MBMesquite::TUntangle1::evaluate ( const MsqMatrix< 2, 2 > &  T,
double &  result,
MsqError &  err 
) [virtual]

Evaluate \(\mu(T)\).

Parameters:
T2x2 relative measure matrix (typically A W^-1)
resultOutput: value of function
Returns:
false if function cannot be evaluated for given T (e.g. division by zero, etc.), true otherwise.

Reimplemented from MBMesquite::TMetric.

virtual MESQUITE_EXPORT bool MBMesquite::TUntangle1::evaluate ( const MsqMatrix< 3, 3 > &  T,
double &  result,
MsqError &  err 
) [virtual]

Evaluate \(\mu(T)\).

Parameters:
T3x3 relative measure matrix (typically A W^-1)
resultOutput: value of function
Returns:
false if function cannot be evaluated for given T (e.g. division by zero, etc.), true otherwise.

Reimplemented from MBMesquite::TMetric.

virtual MESQUITE_EXPORT bool MBMesquite::TUntangle1::evaluate_with_grad ( const MsqMatrix< 2, 2 > &  T,
double &  result,
MsqMatrix< 2, 2 > &  deriv_wrt_T,
MsqError &  err 
) [virtual]

Gradient of \(\mu(T)\) with respect to components of T.

Parameters:
T2x2 relative measure matrix (typically A W^-1)
resultOutput: value of function
deriv_wrt_TOutput: partial deriviatve of \(\mu\) wrt each term of T, evaluated at passed T.

\[\left[\begin{array}{cc} \frac{\partial\mu}{\partial T_{0,0}} & \frac{\partial\mu}{\partial T_{0,1}} \\ \frac{\partial\mu}{\partial T_{1,0}} & \frac{\partial\mu}{\partial T_{1,1}} \\ \end{array}\right]\]

Returns:
false if function cannot be evaluated for given T (e.g. division by zero, etc.), true otherwise.

Reimplemented from MBMesquite::TMetric.

virtual MESQUITE_EXPORT bool MBMesquite::TUntangle1::evaluate_with_grad ( const MsqMatrix< 3, 3 > &  T,
double &  result,
MsqMatrix< 3, 3 > &  deriv_wrt_T,
MsqError &  err 
) [virtual]

Gradient of \(\mu(T)\) with respect to components of T.

Parameters:
T3x3 relative measure matrix (typically A W^-1)
resultOutput: value of function
deriv_wrt_TOutput: partial deriviatve of \(\mu\) wrt each term of T, evaluated at passed T.

\[\left[\begin{array}{ccc} \frac{\partial\mu}{\partial T_{0,0}} & \frac{\partial\mu}{\partial T_{0,1}} & \frac{\partial\mu}{\partial T_{0,2}} \\ \frac{\partial\mu}{\partial T_{1,0}} & \frac{\partial\mu}{\partial T_{1,1}} & \frac{\partial\mu}{\partial T_{1,2}} \\ \frac{\partial\mu}{\partial T_{2,0}} & \frac{\partial\mu}{\partial T_{2,1}} & \frac{\partial\mu}{\partial T_{2,2}} \end{array}\right]\]

Returns:
false if function cannot be evaluated for given T (e.g. division by zero, etc.), true otherwise.

Reimplemented from MBMesquite::TMetric.

virtual MESQUITE_EXPORT bool MBMesquite::TUntangle1::evaluate_with_hess ( const MsqMatrix< 2, 2 > &  T,
double &  result,
MsqMatrix< 2, 2 > &  deriv_wrt_T,
MsqMatrix< 2, 2 >  second_wrt_T[3],
MsqError &  err 
) [virtual]

Hessian of \(\mu(T)\) with respect to components of T.

Parameters:
T3x3 relative measure matrix (typically A W^-1)
resultOutput: value of function
deriv_wrt_TOutput: partial deriviatve of \(\mu\) wrt each term of T, evaluated at passed T.
second_wrt_TOutput: 9x9 matrix of second partial deriviatve of \(\mu\) wrt each term of T, in row-major order. The symmetric matrix is decomposed into 3x3 blocks and only the upper diagonal blocks, in row-major order, are returned.

\[\left[\begin{array}{cc|cc} \frac{\partial^{2}\mu}{\partial T_{0,0}^2} & \frac{\partial^{2}\mu}{\partial T_{0,0}\partial A_{0,1}} & \frac{\partial^{2}\mu}{\partial T_{0,0}\partial A_{1,0}} & \frac{\partial^{2}\mu}{\partial T_{0,0}\partial A_{1,1}} \\ \frac{\partial^{2}\mu}{\partial T_{0,0}\partial A_{0,1}} & \frac{\partial^{2}\mu}{\partial T_{0,1}^2} & \frac{\partial^{2}\mu}{\partial T_{0,1}\partial A_{1,0}} & \frac{\partial^{2}\mu}{\partial T_{0,1}\partial A_{1,1}} \\ \hline & & \frac{\partial^{2}\mu}{\partial T_{1,0}^2} & \frac{\partial^{2}\mu}{\partial T_{1,0}\partial A_{1,1}} \\ & & \frac{\partial^{2}\mu}{\partial T_{1,0}\partial A_{1,1}} & \frac{\partial^{2}\mu}{\partial T_{1,1}^2} \\ \end{array}\right]\]

Returns:
false if function cannot be evaluated for given T (e.g. division by zero, etc.), true otherwise.

Reimplemented from MBMesquite::TMetric.

virtual MESQUITE_EXPORT bool MBMesquite::TUntangle1::evaluate_with_hess ( const MsqMatrix< 3, 3 > &  T,
double &  result,
MsqMatrix< 3, 3 > &  deriv_wrt_T,
MsqMatrix< 3, 3 >  second_wrt_T[6],
MsqError &  err 
) [virtual]

Hessian of \(\mu(T)\) with respect to components of T.

Parameters:
T3x3 relative measure matrix (typically A W^-1)
resultOutput: value of function
deriv_wrt_TOutput: partial deriviatve of \(\mu\) wrt each term of T, evaluated at passed T.
second_wrt_TOutput: 9x9 matrix of second partial deriviatve of \(\mu\) wrt each term of T, in row-major order. The symmetric matrix is decomposed into 3x3 blocks and only the upper diagonal blocks, in row-major order, are returned.

\[\left[\begin{array}{ccc|ccc|ccc} \frac{\partial^{2}\mu}{\partial T_{0,0}^2} & \frac{\partial^{2}\mu}{\partial T_{0,0}\partial T_{0,1}} & \frac{\partial^{2}\mu}{\partial T_{0,0}\partial T_{0,2}} & \frac{\partial^{2}\mu}{\partial T_{0,0}\partial T_{1,0}} & \frac{\partial^{2}\mu}{\partial T_{0,0}\partial T_{1,1}} & \frac{\partial^{2}\mu}{\partial T_{0,0}\partial T_{1,2}} & \frac{\partial^{2}\mu}{\partial T_{0,0}\partial T_{2,0}} & \frac{\partial^{2}\mu}{\partial T_{0,0}\partial T_{2,1}} & \frac{\partial^{2}\mu}{\partial T_{0,0}\partial T_{2,2}} \\ \frac{\partial^{2}\mu}{\partial T_{0,0}\partial T_{0,1}} & \frac{\partial^{2}\mu}{\partial T_{0,1}^2} & \frac{\partial^{2}\mu}{\partial T_{0,1}\partial T_{0,2}} & \frac{\partial^{2}\mu}{\partial T_{0,1}\partial T_{1,0}} & \frac{\partial^{2}\mu}{\partial T_{0,1}\partial T_{1,1}} & \frac{\partial^{2}\mu}{\partial T_{0,1}\partial T_{1,2}} & \frac{\partial^{2}\mu}{\partial T_{0,1}\partial T_{2,0}} & \frac{\partial^{2}\mu}{\partial T_{0,1}\partial T_{2,1}} & \frac{\partial^{2}\mu}{\partial T_{0,1}\partial T_{2,2}} \\ \frac{\partial^{2}\mu}{\partial T_{0,0}\partial T_{0,2}} & \frac{\partial^{2}\mu}{\partial T_{0,1}\partial T_{0,2}} & \frac{\partial^{2}\mu}{\partial T_{0,2}^2} & \frac{\partial^{2}\mu}{\partial T_{0,2}\partial T_{1,0}} & \frac{\partial^{2}\mu}{\partial T_{0,2}\partial T_{1,1}} & \frac{\partial^{2}\mu}{\partial T_{0,2}\partial T_{1,2}} & \frac{\partial^{2}\mu}{\partial T_{0,2}\partial T_{2,0}} & \frac{\partial^{2}\mu}{\partial T_{0,2}\partial T_{2,1}} & \frac{\partial^{2}\mu}{\partial T_{0,2}\partial T_{2,2}} \\ \hline & & & \frac{\partial^{2}\mu}{\partial T_{1,0}^2} & \frac{\partial^{2}\mu}{\partial T_{1,0}\partial T_{1,1}} & \frac{\partial^{2}\mu}{\partial T_{1,0}\partial T_{1,2}} & \frac{\partial^{2}\mu}{\partial T_{1,0}\partial T_{2,0}} & \frac{\partial^{2}\mu}{\partial T_{1,0}\partial T_{2,1}} & \frac{\partial^{2}\mu}{\partial T_{1,0}\partial T_{2,2}} \\ & & & \frac{\partial^{2}\mu}{\partial T_{1,0}\partial T_{1,1}} & \frac{\partial^{2}\mu}{\partial T_{1,1}^2} & \frac{\partial^{2}\mu}{\partial T_{1,1}\partial T_{1,2}} & \frac{\partial^{2}\mu}{\partial T_{1,1}\partial T_{2,0}} & \frac{\partial^{2}\mu}{\partial T_{1,1}\partial T_{2,1}} & \frac{\partial^{2}\mu}{\partial T_{1,1}\partial T_{2,2}} \\ & & & \frac{\partial^{2}\mu}{\partial T_{1,0}\partial T_{1,2}} & \frac{\partial^{2}\mu}{\partial T_{1,1}\partial T_{1,2}} & \frac{\partial^{2}\mu}{\partial T_{1,2}^2} & \frac{\partial^{2}\mu}{\partial T_{1,2}\partial T_{2,0}} & \frac{\partial^{2}\mu}{\partial T_{1,2}\partial T_{2,1}} & \frac{\partial^{2}\mu}{\partial T_{1,2}\partial T_{2,2}} \\ \hline & & & & & & \frac{\partial^{2}\mu}{\partial T_{2,0}^2} & \frac{\partial^{2}\mu}{\partial T_{2,0}\partial T_{2,1}} & \frac{\partial^{2}\mu}{\partial T_{2,0}\partial T_{2,2}} \\ & & & & & & \frac{\partial^{2}\mu}{\partial T_{2,0}\partial T_{2,1}} & \frac{\partial^{2}\mu}{\partial T_{2,1}^2} & \frac{\partial^{2}\mu}{\partial T_{2,1}\partial T_{2,2}} \\ & & & & & & \frac{\partial^{2}\mu}{\partial T_{2,0}\partial T_{2,2}} & \frac{\partial^{2}\mu}{\partial T_{2,1}\partial T_{2,2}} & \frac{\partial^{2}\mu}{\partial T_{2,2}^2} \\ \end{array}\right]\]

Returns:
false if function cannot be evaluated for given T (e.g. division by zero, etc.), true otherwise.

Reimplemented from MBMesquite::TMetric.

std::string MBMesquite::TUntangle1::get_name ( ) const [virtual]

Implements MBMesquite::TMetric.

Definition at line 44 of file TUntangle1.cpp.

{
    return "Untangle1";
}
template<unsigned DIM>
bool MBMesquite::TUntangle1::grad ( const MsqMatrix< DIM, DIM > &  T,
double &  result,
MsqMatrix< DIM, DIM > &  first 
) [inline, private]

Definition at line 58 of file TUntangle1.cpp.

References MBMesquite::det(), mFactor, and MBMesquite::transpose_adj().

{
    double tau = det( T );
    double g   = sqrt( tau * tau + mFactor );
    if( g > 1e-50 )
    {
        double f    = tau / g - 1;
        result      = 0.5 * ( g - tau );
        deriv_wrt_T = transpose_adj( T );
        deriv_wrt_T *= 0.5 * f;
        return true;
    }
    else
    {
        std::cout << "Warning: Division by zero avoided in TUntangle1::grad()" << std::endl;
        return false;
    }
}
template<unsigned DIM>
bool MBMesquite::TUntangle1::hess ( const MsqMatrix< DIM, DIM > &  T,
double &  result,
MsqMatrix< DIM, DIM > &  first,
MsqMatrix< DIM, DIM > *  second 
) [inline, private]

Definition at line 78 of file TUntangle1.cpp.

References MBMesquite::det(), mFactor, MBMesquite::pluseq_scaled_2nd_deriv_of_det(), MBMesquite::set_scaled_outer_product(), and MBMesquite::transpose_adj().

{
    const MsqMatrix< DIM, DIM > adjt = transpose_adj( T );
    double tau                       = det( T );
    double g                         = sqrt( tau * tau + mFactor );
    if( g == 0 ) return false;
    double f = 0.5 * ( tau / g - 1 );
    result   = 0.5 * ( g - tau );

    deriv_wrt_T = adjt;
    deriv_wrt_T *= f;

    set_scaled_outer_product( second_wrt_T, 0.5 * mFactor / ( g * g * g ), adjt );
    pluseq_scaled_2nd_deriv_of_det( second_wrt_T, f, T );

    return true;
}

Member Data Documentation

double MBMesquite::TUntangle1::mFactor [private]

Definition at line 51 of file TUntangle1.hpp.

Referenced by eval(), grad(), and hess().

List of all members.

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