MOAB: Mesh Oriented datABase  (version 5.3.1)
MBMesquite::TShape3DB2 Class Reference

3.3.8: 1/9 ^2(T) - 1, Kappa(T) = |T||T|^-1 = |T||adj(T)/ More...

#include <TShape3DB2.hpp>

Inheritance diagram for MBMesquite::TShape3DB2:
Collaboration diagram for MBMesquite::TShape3DB2:

## Public Member Functions

virtual MESQUITE_EXPORT ~TShape3DB2 ()
virtual MESQUITE_EXPORT std::string get_name () const
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 > &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.

## Detailed Description

3.3.8: 1/9 ^2(T) - 1, Kappa(T) = |T||T|^-1 = |T||adj(T)/

A target metric for volume elements that optimizes element shape

Definition at line 47 of file TShape3DB2.hpp.

## Constructor & Destructor Documentation

 MBMesquite::TShape3DB2::~TShape3DB2 ( )  [virtual]

Definition at line 41 of file TShape3DB2.cpp.

{}


## Member Function Documentation

 bool MBMesquite::TShape3DB2::evaluate ( const MsqMatrix< 3, 3 > & T, double & result, MsqError & err )  [virtual]

Evaluate $$\mu(T)$$.

Parameters:
 T 3x3 relative measure matrix (typically A W^-1) result Output: 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.

Definition at line 49 of file TShape3DB2.cpp.

{
double f = sqr_Frobenius( T );
double g = sqr_Frobenius( adj( T ) );
double d = det( T );
if( invalid_determinant( d ) )
{
MSQ_SETERR( err )( barrier_violated_msg, MsqError::BARRIER_VIOLATED );
return false;
}
result = ( f * g ) / ( 9 * d * d ) - 1;
return true;
}

 bool MBMesquite::TShape3DB2::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:
 T 3x3 relative measure matrix (typically A W^-1) result Output: value of function deriv_wrt_T Output: 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.

Definition at line 63 of file TShape3DB2.cpp.

{
double f = sqr_Frobenius( T );
double g = sqr_Frobenius( adj( T ) );
double d = det( T );
if( invalid_determinant( d ) )
{
MSQ_SETERR( err )( barrier_violated_msg, MsqError::BARRIER_VIOLATED );
return false;
}
result = ( f * g ) / ( 9 * d * d ) - 1;

wrt_T = T;
wrt_T *= ( g + f * f );
wrt_T -= f * ( T * transpose( T ) * T );
wrt_T -= f * g / d * transpose_adj( T );
wrt_T *= 2 / ( 9 * d * d );

return true;
}

 bool MBMesquite::TShape3DB2::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:
 T 3x3 relative measure matrix (typically A W^-1) result Output: value of function deriv_wrt_T Output: partial deriviatve of $$\mu$$ wrt each term of T, evaluated at passed T. second_wrt_T Output: 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.

Definition at line 85 of file TShape3DB2.cpp.

{
double f = sqr_Frobenius( T );
double g = sqr_Frobenius( adj( T ) );
double d = det( T );
if( invalid_determinant( d ) )
{
MSQ_SETERR( err )( barrier_violated_msg, MsqError::BARRIER_VIOLATED );
return false;
}
const double den = 1.0 / ( 9 * d * d );
result           = f * g * den - 1;

MsqMatrix< 3, 3 > dg   = 2 * ( f * T - T * transpose( T ) * T );
MsqMatrix< 3, 3 > df   = 2 * T;
MsqMatrix< 3, 3 > dtau = transpose_adj( T );

wrt_T = g * df + f * dg - 2 * f * g / d * transpose_adj( T );
wrt_T *= den;

set_scaled_2nd_deriv_norm_sqr_adj( second, den * f, T );
pluseq_scaled_I( second, 2 * den * g );
pluseq_scaled_sum_outer_product( second, den, dg, df );
pluseq_scaled_sum_outer_product( second, -2 * den * g / d, df, dtau );
pluseq_scaled_sum_outer_product( second, -2 * den * f / d, dg, dtau );
pluseq_scaled_outer_product( second, 6 * den * f * g / ( d * d ), dtau );
pluseq_scaled_2nd_deriv_of_det( second, -2 * den * f * g / d, T );

return true;
}

 std::string MBMesquite::TShape3DB2::get_name ( ) const [virtual]

Reimplemented from MBMesquite::TMetricBarrier.

Definition at line 43 of file TShape3DB2.cpp.

{
return "TShape3DB2";
}


List of all members.

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