\( |T|^3 - 3 \sqrt{3} \tau + \gamma (\tau - 1)^2 \) More...

#include <TShapeSize3DNB1.hpp>

Inheritance diagram for MBMesquite::TShapeSize3DNB1:

Collaboration diagram for MBMesquite::TShapeSize3DNB1:

Public Member Functions
	TShapeSize3DNB1 (double gamma=2.0)
virtual MESQUITE_EXPORT	~TShapeSize3DNB1 ()
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 > &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 Attributes
double	mGamma

Detailed Description

\( |T|^3 - 3 \sqrt{3} \tau + \gamma (\tau - 1)^2 \)

A target metric for volume elements that optimizes element shape and size

Definition at line 46 of file TShapeSize3DNB1.hpp.

Constructor & Destructor Documentation

MBMesquite::TShapeSize3DNB1::TShapeSize3DNB1 ( double gamma = 2.0 ) [inline]

Definition at line 52 of file TShapeSize3DNB1.hpp.

: mGamma( gamma ) {}

MBMesquite::TShapeSize3DNB1::~TShapeSize3DNB1 ( ) [virtual]

Definition at line 45 of file TShapeSize3DNB1.cpp.

{}

Member Function Documentation

bool MBMesquite::TShapeSize3DNB1::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 47 of file TShapeSize3DNB1.cpp.

References MBMesquite::det(), MBMesquite::Frobenius(), mGamma, and MBMesquite::MSQ_SQRT_THREE.

{
    const double nT   = Frobenius( T );
    const double tau  = det( T );
    const double tau1 = tau - 1;
    result            = nT * nT * nT - 3 * MSQ_SQRT_THREE * tau + mGamma * tau1 * tau1;
    return true;
}

bool MBMesquite::TShapeSize3DNB1::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 56 of file TShapeSize3DNB1.cpp.

References MBMesquite::det(), MBMesquite::Frobenius(), mGamma, MBMesquite::MSQ_SQRT_THREE, T, and MBMesquite::transpose_adj().

{
    const double nT   = Frobenius( T );
    const double tau  = det( T );
    const double tau1 = tau - 1;
    result            = nT * nT * nT - 3 * MSQ_SQRT_THREE * tau + mGamma * tau1 * tau1;

    wrt_T = T;
    wrt_T *= 3 * nT;
    wrt_T -= ( 3 * MSQ_SQRT_THREE - 2 * mGamma * tau1 ) * transpose_adj( T );

    return true;
}

bool MBMesquite::TShapeSize3DNB1::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 73 of file TShapeSize3DNB1.cpp.

References MBMesquite::det(), MBMesquite::Frobenius(), mGamma, MBMesquite::MSQ_SQRT_THREE, MBMesquite::pluseq_scaled_2nd_deriv_of_det(), MBMesquite::pluseq_scaled_I(), MBMesquite::pluseq_scaled_outer_product(), MBMesquite::set_scaled_outer_product(), T, and MBMesquite::transpose_adj().

{
    const double nT   = Frobenius( T );
    const double tau  = det( T );
    const double tau1 = tau - 1;
    result            = nT * nT * nT - 3 * MSQ_SQRT_THREE * tau + mGamma * tau1 * tau1;

    const double f               = ( 3 * MSQ_SQRT_THREE - 2 * mGamma * tau1 );
    const MsqMatrix< 3, 3 > adjt = transpose_adj( T );
    wrt_T                        = T;
    wrt_T *= 3 * nT;
    wrt_T -= f * adjt;

    set_scaled_outer_product( second, 2 * mGamma, adjt );
    pluseq_scaled_2nd_deriv_of_det( second, -f, T );
    pluseq_scaled_I( second, 3 * nT );
    // Could perturb T a bit if the norm is zero, but that would just
    // result in the coefficent of the outer product being practically
    // zero, so just skip the outer product in that case.
    // Anyway nT approaches zero as T does, so the limit of this term
    // as nT approaches zero is zero.
    if( nT > 1e-100 )  // NOTE: nT is always positive
        pluseq_scaled_outer_product( second, 3 / nT, T );

    return true;
}

std::string MBMesquite::TShapeSize3DNB1::get_name ( ) const [virtual]

Implements MBMesquite::TMetric.

Definition at line 40 of file TShapeSize3DNB1.cpp.

{
    return "TShapeSize3DNB1";
}

Member Data Documentation

double MBMesquite::TShapeSize3DNB1::mGamma [private]

Definition at line 49 of file TShapeSize3DNB1.hpp.

Referenced by evaluate(), evaluate_with_grad(), and evaluate_with_hess().

List of all members.

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

Public Member Functions

Private Attributes

Detailed Description

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation