A templated (3 x 3) symmetric tensor of objects of <T>, effectively containing 6 elements, derived from VectorSpace. More...

Inheritance diagram for SymmTensor< Cmpt >:

[legend]

Collaboration diagram for SymmTensor< Cmpt >:

[legend]

Public Types
enum	components { XX, XY, XZ, YY, YZ, ZZ }
	Component labeling enumeration. More...

typedef SymmTensor< label >	labelType
	Equivalent type of labels used for valid component indexing. More...

Public Types inherited from VectorSpace< SymmTensor< Cmpt >, Cmpt, 6 >
typedef VectorSpace< SymmTensor< Cmpt >, Cmpt, Ncmpts >	vsType
	VectorSpace type. More...

typedef Cmpt	cmptType
	Component type. More...

typedef Cmpt	magType
	Magnitude type. More...

typedef Cmpt *	iterator
	Random access iterator for traversing VectorSpace. More...

typedef const Cmpt *	const_iterator
	Random access iterator for traversing VectorSpace. More...

Public Member Functions
	SymmTensor ()=default
	Default construct. More...

	SymmTensor (const SymmTensor &)=default
	Copy construct. More...

SymmTensor &	operator= (const SymmTensor &)=default
	Copy assignment. More...

	SymmTensor (const Foam::zero)
	Construct initialized to zero. More...

template<class Cmpt2 >
	SymmTensor (const VectorSpace< SymmTensor< Cmpt2 >, Cmpt2, 6 > &)
	Construct given VectorSpace of the same rank. More...

	SymmTensor (const SphericalTensor< Cmpt > &)
	Construct given SphericalTensor. More...

	SymmTensor (const Vector< Cmpt > &x, const Vector< Cmpt > &y, const Vector< Cmpt > &z, const bool transposed=false)
	Construct given the three row (or column) vectors. More...

	SymmTensor (const Cmpt txx, const Cmpt txy, const Cmpt txz, const Cmpt tyy, const Cmpt tyz, const Cmpt tzz)
	Construct given the six components. More...

	SymmTensor (Istream &is)
	Construct from Istream. More...

const Cmpt &	xx () const noexcept

const Cmpt &	xy () const noexcept

const Cmpt &	xz () const noexcept

const Cmpt &	yx () const noexcept

const Cmpt &	yy () const noexcept

const Cmpt &	yz () const noexcept

const Cmpt &	zx () const noexcept

const Cmpt &	zy () const noexcept

const Cmpt &	zz () const noexcept

Cmpt &	xx () noexcept

Cmpt &	xy () noexcept

Cmpt &	xz () noexcept

Cmpt &	yx () noexcept

Cmpt &	yy () noexcept

Cmpt &	yz () noexcept

Cmpt &	zx () noexcept

Cmpt &	zy () noexcept

Cmpt &	zz () noexcept

Vector< Cmpt >	cx () const
	Extract vector for column 0. More...

Vector< Cmpt >	cy () const
	Extract vector for column 1. More...

Vector< Cmpt >	cz () const
	Extract vector for column 2. More...

template<direction Idx>
Vector< Cmpt >	col () const
	Extract vector for given column: compile-time check of index. More...

Vector< Cmpt >	col (const direction c) const
	Extract vector for given column (0,1,2): runtime check of index. More...

void	col (const direction c, const Vector< Cmpt > &v)
	Set values of given column (0,1,2): runtime check of index. More...

void	cols (const Vector< Cmpt > &x, const Vector< Cmpt > &y, const Vector< Cmpt > &z)
	Set column values. More...

Vector< Cmpt >	x () const
	Extract vector for row 0. More...

Vector< Cmpt >	y () const
	Extract vector for row 1. More...

Vector< Cmpt >	z () const
	Extract vector for row 2. More...

template<direction Row>
Vector< Cmpt >	row () const
	Extract vector for given row: compile-time check of index. More...

Vector< Cmpt >	row (const direction r) const
	Extract vector for given row (0,1,2): runtime check of index. More...

template<direction Idx>
void	row (const Vector< Cmpt > &v)
	Set values of given row: compile-time check of index. More...

void	row (const direction r, const Vector< Cmpt > &v)
	Set values of given row (0,1,2): runtime check of row. More...

void	rows (const Vector< Cmpt > &x, const Vector< Cmpt > &y, const Vector< Cmpt > &z)
	Set row values. More...

Vector< Cmpt >	diag () const
	Extract the diagonal as a vector. More...

void	diag (const Vector< Cmpt > &v)
	Set values of the diagonal. More...

void	addDiag (const Vector< Cmpt > &v)
	Add to the diagonal. More...

void	subtractDiag (const Vector< Cmpt > &v)
	Subtract from the diagonal. More...

scalar	diagSqr () const
	The L2-norm squared of the diagonal. More...

const SymmTensor< Cmpt > &	T () const noexcept
	Return non-Hermitian transpose. More...

Cmpt	det () const
	The determinate. More...

Cmpt	det2D (const direction excludeCmpt) const
	The 2D determinant by excluding given direction. More...

SymmTensor< Cmpt >	adjunct () const
	Return adjunct matrix (transpose of cofactor matrix) More...

SymmTensor< Cmpt >	cof () const
	Return cofactor matrix (transpose of adjunct matrix) More...

SymmTensor< Cmpt >	adjunct2D (const direction excludeCmpt) const
	Return 2D adjunct matrix by excluding given direction. More...

SymmTensor< Cmpt >	inv () const
	Return inverse. More...

SymmTensor< Cmpt >	safeInv () const
	Return inverse, with (ad hoc) failsafe handling of 2D tensors. More...

SymmTensor< Cmpt >	inv2D (const direction excludeCmpt) const
	Return inverse of 2D tensor (by excluding given direction) More...

void	operator= (const SphericalTensor< Cmpt > &)
	Assign to given SphericalTensor. More...

template<Foam::direction Idx>
Foam::Vector< Cmpt >	row () const

Public Member Functions inherited from VectorSpace< SymmTensor< Cmpt >, Cmpt, 6 >
	VectorSpace ()=default
	Default construct. More...

	VectorSpace (const Foam::zero)
	Construct initialized to zero. More...

	VectorSpace (const VectorSpace< SymmTensor< Cmpt >, Cmpt, Ncmpts > &vs)
	Copy construct. More...

	VectorSpace (const VectorSpace< Form2, Cmpt2, Ncmpts > &)
	Copy construct of a VectorSpace with the same size. More...

	VectorSpace (Istream &is)
	Construct from Istream. More...

const Cmpt &	component (const direction) const

Cmpt &	component (const direction)

void	component (Cmpt &, const direction) const

void	replace (const direction, const Cmpt &)

const Cmpt *	cdata () const noexcept
	Return const pointer to the first data element. More...

Cmpt *	data () noexcept
	Return pointer to the first data element. More...

const ConstBlock< SubVector, BStart >	block () const

const Foam::VectorSpace< SymmTensor< Cmpt >, Cmpt, Ncmpts >::template ConstBlock< SubVector, BStart >	block () const

const Cmpt &	operator[] (const direction) const

Cmpt &	operator[] (const direction)

void	operator= (const VectorSpace< SymmTensor< Cmpt >, Cmpt, Ncmpts > &)

void	operator= (const Foam::zero)

void	operator+= (const VectorSpace< SymmTensor< Cmpt >, Cmpt, Ncmpts > &)

void	operator-= (const VectorSpace< SymmTensor< Cmpt >, Cmpt, Ncmpts > &)

void	operator*= (const scalar)

void	operator/= (const scalar)

iterator	begin () noexcept
	Return an iterator to begin of VectorSpace. More...

const_iterator	begin () const noexcept
	Return const_iterator to begin of VectorSpace. More...

iterator	end () noexcept
	Return an iterator to end of VectorSpace. More...

const_iterator	end () const noexcept
	Return const_iterator to end of VectorSpace. More...

const_iterator	cbegin () const noexcept
	Return const_iterator to begin of VectorSpace. More...

const_iterator	cend () const noexcept
	Return const_iterator to end of VectorSpace. More...

Static Public Attributes
static constexpr direction	rank = 2
	Rank of SymmTensor is 2. More...

static const SymmTensor	I

Static Public Attributes inherited from VectorSpace< SymmTensor< Cmpt >, Cmpt, 6 >
static constexpr direction	dim
	Dimensionality of space. More...

static constexpr direction	nComponents
	Number of components in this vector space. More...

static constexpr direction	mRows

static constexpr direction	nCols

static const char *const	typeName

static const char *const	componentNames []

static const SymmTensor< Cmpt >	zero

static const SymmTensor< Cmpt >	one

static const SymmTensor< Cmpt >	max

static const SymmTensor< Cmpt >	min

static const SymmTensor< Cmpt >	rootMax

static const SymmTensor< Cmpt >	rootMin

Additional Inherited Members
Static Public Member Functions inherited from VectorSpace< SymmTensor< Cmpt >, Cmpt, 6 >
static constexpr direction	size () noexcept
	The number of elements in the VectorSpace = Ncmpts. More...

static SymmTensor< Cmpt >	uniform (const Cmpt &s)
	Return a VectorSpace with all elements = s. More...

Public Attributes inherited from VectorSpace< SymmTensor< Cmpt >, Cmpt, 6 >
Cmpt	v_ [Ncmpts]
	The components of this vector space. More...

Detailed Description

template<class Cmpt>
class Foam::SymmTensor< Cmpt >

A templated (3 x 3) symmetric tensor of objects of <T>, effectively containing 6 elements, derived from VectorSpace.

Source files

Definition at line 50 of file SymmTensor.H.

Member Typedef Documentation

◆ labelType

typedef SymmTensor<label> labelType

Equivalent type of labels used for valid component indexing.

Definition at line 61 of file SymmTensor.H.

Member Enumeration Documentation

◆ components

enum components

Component labeling enumeration.

Enumerator
XX
XY
XZ
YY
YZ
ZZ

Definition at line 80 of file SymmTensor.H.

Constructor & Destructor Documentation

◆ SymmTensor() [1/8]

SymmTensor ( )

default

Default construct.

◆ SymmTensor() [2/8]

SymmTensor ( const SymmTensor< Cmpt > & )

default

Copy construct.

◆ SymmTensor() [3/8]

SymmTensor ( const Foam::zero )

inline

Construct initialized to zero.

Definition at line 27 of file SymmTensorI.H.

◆ SymmTensor() [4/8]

SymmTensor ( const VectorSpace< SymmTensor< Cmpt2 >, Cmpt2, 6 > & vs )

inline

Construct given VectorSpace of the same rank.

Definition at line 36 of file SymmTensorI.H.

◆ SymmTensor() [5/8]

SymmTensor ( const SphericalTensor< Cmpt > & st )

inline

Construct given SphericalTensor.

Definition at line 45 of file SymmTensorI.H.

◆ SymmTensor() [6/8]

SymmTensor	(	const Vector< Cmpt > &	x,
		const Vector< Cmpt > &	y,
		const Vector< Cmpt > &	z,
		const bool	transposed = `false`
	)

inline

Construct given the three row (or column) vectors.

Definition at line 55 of file SymmTensorI.H.

◆ SymmTensor() [7/8]

SymmTensor	(	const Cmpt	txx,
		const Cmpt	txy,
		const Cmpt	txz,
		const Cmpt	tyy,
		const Cmpt	tyz,
		const Cmpt	tzz
	)

inline

Construct given the six components.

Definition at line 68 of file SymmTensorI.H.

◆ SymmTensor() [8/8]

SymmTensor ( Istream & is )

inlineexplicit

Construct from Istream.

Definition at line 81 of file SymmTensorI.H.

Member Function Documentation

◆ operator=() [1/2]

SymmTensor& operator= ( const SymmTensor< Cmpt > & )

default

Copy assignment.

◆ xx() [1/2]

const Cmpt& xx ( ) const

inlinenoexcept

◆ xy() [1/2]

const Cmpt& xy ( ) const

inlinenoexcept

Definition at line 151 of file SymmTensor.H.

Referenced by ReynoldsStress< RASModel< BasicTurbulenceModel > >::checkRealizabilityConditions(), turbulentDFSEMInletFvPatchVectorField::checkStresses(), Foam::innerSqr(), Foam::invariantII(), Foam::invTransform(), Foam::lerp(), Foam::magSqr(), Foam::operator &(), Foam::operator &&(), Foam::operator*(), Foam::operator+(), Foam::operator-(), Foam::operator/(), Tensor< scalar >::operator=(), Foam::pinv(), Tensor< scalar >::Tensor(), and Foam::transform().

Here is the caller graph for this function:

◆ xz() [1/2]

const Cmpt& xz ( ) const

inlinenoexcept

Definition at line 152 of file SymmTensor.H.

Referenced by Foam::innerSqr(), Foam::invariantII(), Foam::invTransform(), Foam::lerp(), Foam::magSqr(), Foam::operator &(), Foam::operator &&(), Foam::operator*(), Foam::operator+(), Foam::operator-(), Foam::operator/(), Tensor< scalar >::operator=(), Foam::pinv(), Tensor< scalar >::Tensor(), and Foam::transform().

Here is the caller graph for this function:

◆ yx() [1/2]

const Cmpt& yx ( ) const

inlinenoexcept

Definition at line 153 of file SymmTensor.H.

Referenced by Foam::pinv().

Here is the caller graph for this function:

◆ yy() [1/2]

const Cmpt& yy ( ) const

inlinenoexcept

◆ yz() [1/2]

const Cmpt& yz ( ) const

inlinenoexcept

Definition at line 155 of file SymmTensor.H.

Referenced by Foam::innerSqr(), Foam::invariantII(), Foam::invTransform(), Foam::lerp(), Foam::magSqr(), Foam::operator &(), Foam::operator &&(), Foam::operator*(), Foam::operator+(), Foam::operator-(), Foam::operator/(), Tensor< scalar >::operator=(), Foam::pinv(), Tensor< scalar >::Tensor(), and Foam::transform().

Here is the caller graph for this function:

◆ zx() [1/2]

const Cmpt& zx ( ) const

inlinenoexcept

Definition at line 156 of file SymmTensor.H.

Referenced by Foam::pinv().

Here is the caller graph for this function:

◆ zy() [1/2]

const Cmpt& zy ( ) const

inlinenoexcept

Definition at line 157 of file SymmTensor.H.

Referenced by Foam::pinv().

Here is the caller graph for this function:

◆ zz() [1/2]

const Cmpt& zz ( ) const

inlinenoexcept

Definition at line 158 of file SymmTensor.H.

Referenced by primitiveMeshTools::cellDeterminant(), molecule::constantProperties::constantProperties(), Foam::diag(), Foam::innerSqr(), Foam::invariantII(), Foam::invTransform(), Foam::lerp(), Foam::magSqr(), Foam::operator &(), Foam::operator &&(), Foam::operator+(), Foam::operator-(), Foam::operator/(), Tensor< scalar >::operator=(), Foam::pinv(), SymmTensor< scalar >::safeInv(), Tensor< scalar >::Tensor(), Foam::tr(), and Foam::transform().

Here is the caller graph for this function:

◆ xx() [2/2]

Cmpt& xx ( )

inlinenoexcept

Definition at line 160 of file SymmTensor.H.

◆ xy() [2/2]

Cmpt& xy ( )

inlinenoexcept

Definition at line 161 of file SymmTensor.H.

◆ xz() [2/2]

Cmpt& xz ( )

inlinenoexcept

Definition at line 162 of file SymmTensor.H.

◆ yx() [2/2]

Cmpt& yx ( )

inlinenoexcept

Definition at line 163 of file SymmTensor.H.

◆ yy() [2/2]

Cmpt& yy ( )

inlinenoexcept

Definition at line 164 of file SymmTensor.H.

◆ yz() [2/2]

Cmpt& yz ( )

inlinenoexcept

Definition at line 165 of file SymmTensor.H.

◆ zx() [2/2]

Cmpt& zx ( )

inlinenoexcept

Definition at line 166 of file SymmTensor.H.

◆ zy() [2/2]

Cmpt& zy ( )

inlinenoexcept

Definition at line 167 of file SymmTensor.H.

◆ zz() [2/2]

Cmpt& zz ( )

inlinenoexcept

Definition at line 168 of file SymmTensor.H.

◆ cx()

Vector<Cmpt> cx ( ) const

inline

Extract vector for column 0.

Definition at line 176 of file SymmTensor.H.

◆ cy()

Vector<Cmpt> cy ( ) const

inline

Extract vector for column 1.

Definition at line 181 of file SymmTensor.H.

◆ cz()

Vector<Cmpt> cz ( ) const

inline

Extract vector for column 2.

Definition at line 186 of file SymmTensor.H.

◆ col() [1/3]

Vector<Cmpt> col ( ) const

inline

Extract vector for given column: compile-time check of index.

Definition at line 192 of file SymmTensor.H.

◆ col() [2/3]

Vector<Cmpt> col ( const direction c ) const

inline

Extract vector for given column (0,1,2): runtime check of index.

Definition at line 197 of file SymmTensor.H.

◆ col() [3/3]

void col	(	const direction	c,
		const Vector< Cmpt > &	v
	)

inline

Set values of given column (0,1,2): runtime check of index.

Definition at line 202 of file SymmTensor.H.

◆ cols()

void cols	(	const Vector< Cmpt > &	x,
		const Vector< Cmpt > &	y,
		const Vector< Cmpt > &	z
	)

inline

Set column values.

Definition at line 208 of file SymmTensor.H.

◆ x()

Foam::Vector< Cmpt > x ( ) const

inline

Extract vector for row 0.

Definition at line 90 of file SymmTensorI.H.

Referenced by SymmTensor< scalar >::cx().

Here is the caller graph for this function:

◆ y()

Foam::Vector< Cmpt > y ( ) const

inline

Extract vector for row 1.

Definition at line 97 of file SymmTensorI.H.

Referenced by SymmTensor< scalar >::cols(), and SymmTensor< scalar >::cy().

Here is the caller graph for this function:

◆ z()

Foam::Vector< Cmpt > z ( ) const

inline

Extract vector for row 2.

Definition at line 104 of file SymmTensorI.H.

Referenced by SymmTensor< scalar >::cols(), and SymmTensor< scalar >::cz().

Here is the caller graph for this function:

◆ row() [1/5]

Vector<Cmpt> row ( ) const

inline

Extract vector for given row: compile-time check of index.

Referenced by SymmTensor< scalar >::col().

Here is the caller graph for this function:

◆ row() [2/5]

Foam::Vector< Cmpt > row ( const direction r ) const

inline

Extract vector for given row (0,1,2): runtime check of index.

Definition at line 124 of file SymmTensorI.H.

◆ row() [3/5]

void row ( const Vector< Cmpt > & v )

inline

Set values of given row: compile-time check of index.

Definition at line 142 of file SymmTensorI.H.

◆ row() [4/5]

void row	(	const direction	r,
		const Vector< Cmpt > &	v
	)

inline

Set values of given row (0,1,2): runtime check of row.

Definition at line 177 of file SymmTensorI.H.

◆ rows()

void rows	(	const Vector< Cmpt > &	x,
		const Vector< Cmpt > &	y,
		const Vector< Cmpt > &	z
	)

inline

Set row values.

Definition at line 163 of file SymmTensorI.H.

Referenced by SymmTensor< scalar >::cols().

Here is the caller graph for this function:

◆ diag() [1/2]

Foam::Vector< Cmpt > diag ( ) const

inline

Extract the diagonal as a vector.

Definition at line 195 of file SymmTensorI.H.

◆ diag() [2/2]

void diag ( const Vector< Cmpt > & v )

inline

Set values of the diagonal.

Definition at line 202 of file SymmTensorI.H.

◆ addDiag()

void addDiag ( const Vector< Cmpt > & v )

inline

Add to the diagonal.

Definition at line 209 of file SymmTensorI.H.

◆ subtractDiag()

void subtractDiag ( const Vector< Cmpt > & v )

inline

Subtract from the diagonal.

Definition at line 216 of file SymmTensorI.H.

◆ diagSqr()

Foam::scalar diagSqr ( ) const

inline

The L2-norm squared of the diagonal.

Definition at line 223 of file SymmTensorI.H.

◆ T()

const SymmTensor<Cmpt>& T ( ) const

inlinenoexcept

Return non-Hermitian transpose.

Definition at line 301 of file SymmTensor.H.

◆ det()

Cmpt det ( ) const

inline

The determinate.

Definition at line 235 of file SymmTensorI.H.

Referenced by Foam::det(), Foam::pinv(), and SymmTensor< scalar >::safeInv().

Here is the caller graph for this function:

◆ det2D()

Cmpt det2D ( const direction excludeCmpt ) const

inline

The 2D determinant by excluding given direction.

Definition at line 247 of file SymmTensorI.H.

◆ adjunct()

Foam::SymmTensor< Cmpt > adjunct ( ) const

inline

Return adjunct matrix (transpose of cofactor matrix)

Definition at line 268 of file SymmTensorI.H.

Referenced by Foam::inv(), and SymmTensor< scalar >::safeInv().

Here is the caller graph for this function:

◆ cof()

Foam::SymmTensor< Cmpt > cof ( ) const

inline

Return cofactor matrix (transpose of adjunct matrix)

Definition at line 281 of file SymmTensorI.H.

Referenced by Foam::cof().

Here is the caller graph for this function:

◆ adjunct2D()

Foam::SymmTensor< Cmpt > adjunct2D ( const direction excludeCmpt ) const

inline

Return 2D adjunct matrix by excluding given direction.

Definition at line 290 of file SymmTensorI.H.

◆ inv()

Foam::SymmTensor< Cmpt > inv ( ) const

inline

Return inverse.

Definition at line 341 of file SymmTensorI.H.

Referenced by Foam::inv().

Here is the caller graph for this function:

◆ safeInv()

Foam::SymmTensor< Cmpt > safeInv ( ) const

inline

Return inverse, with (ad hoc) failsafe handling of 2D tensors.

Definition at line 361 of file SymmTensorI.H.

◆ inv2D()

Foam::SymmTensor< Cmpt > inv2D ( const direction excludeCmpt ) const

inline

Return inverse of 2D tensor (by excluding given direction)

Definition at line 329 of file SymmTensorI.H.

◆ operator=() [2/2]

void operator= ( const SphericalTensor< Cmpt > & st )

inline

Assign to given SphericalTensor.

Definition at line 419 of file SymmTensorI.H.

◆ row() [5/5]

Foam::Vector<Cmpt> row ( ) const

inline

Definition at line 112 of file SymmTensorI.H.

Member Data Documentation

◆ rank

constexpr direction rank = 2

static

Rank of SymmTensor is 2.

Definition at line 69 of file SymmTensor.H.

◆ I

const Foam::labelSymmTensor I

static

Definition at line 74 of file SymmTensor.H.

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

src/OpenFOAM/primitives/SymmTensor/SymmTensor.H
src/OpenFOAM/primitives/SymmTensor/labelSymmTensor/labelSymmTensor.C
src/OpenFOAM/primitives/SymmTensor/SymmTensorI.H

Public Types

Public Member Functions

Static Public Attributes

Additional Inherited Members

Detailed Description

template<class Cmpt> class Foam::SymmTensor< Cmpt >

Member Typedef Documentation

◆ labelType

Member Enumeration Documentation

◆ components

Constructor & Destructor Documentation

◆ SymmTensor() [1/8]

◆ SymmTensor() [2/8]

◆ SymmTensor() [3/8]

◆ SymmTensor() [4/8]

◆ SymmTensor() [5/8]

◆ SymmTensor() [6/8]

◆ SymmTensor() [7/8]

◆ SymmTensor() [8/8]

Member Function Documentation

◆ operator=() [1/2]

◆ xx() [1/2]

◆ xy() [1/2]

◆ xz() [1/2]

◆ yx() [1/2]

◆ yy() [1/2]

◆ yz() [1/2]

◆ zx() [1/2]

◆ zy() [1/2]

◆ zz() [1/2]

◆ xx() [2/2]

◆ xy() [2/2]

◆ xz() [2/2]

◆ yx() [2/2]

◆ yy() [2/2]

◆ yz() [2/2]

◆ zx() [2/2]

◆ zy() [2/2]

◆ zz() [2/2]

◆ cx()

◆ cy()

◆ cz()

◆ col() [1/3]

◆ col() [2/3]

◆ col() [3/3]

◆ cols()

◆ x()

◆ y()

◆ z()

◆ row() [1/5]

◆ row() [2/5]

◆ row() [3/5]

◆ row() [4/5]

◆ rows()

◆ diag() [1/2]

◆ diag() [2/2]

◆ addDiag()

◆ subtractDiag()

◆ diagSqr()

◆ T()

◆ det()

◆ det2D()

◆ adjunct()

◆ cof()

◆ adjunct2D()

◆ inv()

◆ safeInv()

◆ inv2D()

◆ operator=() [2/2]

◆ row() [5/5]

Member Data Documentation

◆ rank

◆ I

template<class Cmpt>
class Foam::SymmTensor< Cmpt >