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.

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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.

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Constructor & Destructor Documentation

MBMesquite::TShape3DB2::~TShape3DB2

(

)

[virtual]

Definition at line 41 of file TShape3DB2.cpp.

{}

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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.

References MBMesquite::adj(), MBMesquite::MsqError::BARRIER_VIOLATED, MBMesquite::barrier_violated_msg, MBMesquite::det(), MBMesquite::TMetricBarrier::invalid_determinant(), MSQ_SETERR, and MBMesquite::sqr_Frobenius().

{
    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.

References MBMesquite::adj(), MBMesquite::MsqError::BARRIER_VIOLATED, MBMesquite::barrier_violated_msg, MBMesquite::det(), MBMesquite::TMetricBarrier::invalid_determinant(), MSQ_SETERR, MBMesquite::sqr_Frobenius(), T, MBMesquite::transpose(), and MBMesquite::transpose_adj().

{
    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 87 of file TShape3DB2.cpp.

References MBMesquite::adj(), MBMesquite::MsqError::BARRIER_VIOLATED, MBMesquite::barrier_violated_msg, MBMesquite::det(), MBMesquite::TMetricBarrier::invalid_determinant(), MSQ_SETERR, MBMesquite::pluseq_scaled_2nd_deriv_of_det(), MBMesquite::pluseq_scaled_I(), MBMesquite::pluseq_scaled_outer_product(), MBMesquite::pluseq_scaled_sum_outer_product(), MBMesquite::set_scaled_2nd_deriv_norm_sqr_adj(), MBMesquite::sqr_Frobenius(), T, MBMesquite::transpose(), and MBMesquite::transpose_adj().

{
    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.

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

• TShape3DB2.hpp
• TShape3DB2.cpp