Various mathematical utilitiy functions. More...

#include <math_utils.h>

Collaboration diagram for Kratos::MathUtils< TDataType >:

Type Definitions
using	MatrixType = Matrix
	The matrix type. More...

using	VectorType = Vector
	The vector type. More...

using	SizeType = std::size_t
	The size type. More...

using	IndexType = std::size_t
	The index type. More...

using	IndirectArrayType = boost::numeric::ublas::indirect_array< DenseVector< std::size_t > >
	The indirect array type. More...

static constexpr double	ZeroTolerance = std::numeric_limits<double>::epsilon()
	The machine precision. More...

Operations
	TDim

TDim &	rA

TDim BoundedMatrix< double, TDim, TDim > &	rEigenVectorsMatrix

TDim BoundedMatrix< double, TDim, TDim > BoundedMatrix< double, TDim, TDim > &	rEigenValuesMatrix

TDim BoundedMatrix< double, TDim, TDim > BoundedMatrix< double, TDim, TDim > const double	Tolerance = 1.0e-18

TDim BoundedMatrix< double, TDim, TDim > BoundedMatrix< double, TDim, TDim > const double const SizeType	MaxIterations

static double	GetZeroTolerance ()
	This function returns the machine precision. More...

template<bool TCheck>
static double	Heron (double a, double b, double c)
	In geometry, Heron's formula (sometimes called Hero's formula), named after Hero of Alexandria, gives the area of a triangle by requiring no arbitrary choice of side as base or vertex as origin, contrary to other formulas for the area of a triangle, such as half the base times the height or half the norm of a cross product of two sides. More...

template<class TMatrixType >
static double	Cofactor (const TMatrixType &rMat, IndexType i, IndexType j)
	Calculates the cofactor. More...

template<class TMatrixType >
static MatrixType	CofactorMatrix (const TMatrixType &rMat)
	Calculates the cofactor matrix. More...

template<SizeType TDim>
static BoundedMatrix< double, TDim, TDim >	InvertMatrix (const BoundedMatrix< double, TDim, TDim > &rInputMatrix, double &rInputMatrixDet, const double Tolerance=ZeroTolerance)
	Calculates the inverse of a 2x2, 3x3 and 4x4 matrices (using bounded matrix for performance) More...

template<class TMatrix1 , class TMatrix2 >
static bool	CheckConditionNumber (const TMatrix1 &rInputMatrix, TMatrix2 &rInvertedMatrix, const double Tolerance=std::numeric_limits< double >::epsilon(), const bool ThrowError=true)
	This method checks the condition number of amtrix. More...

template<class TMatrix1 , class TMatrix2 >
static void	GeneralizedInvertMatrix (const TMatrix1 &rInputMatrix, TMatrix2 &rInvertedMatrix, double &rInputMatrixDet, const double Tolerance=ZeroTolerance)
	It inverts non square matrices (https://en.wikipedia.org/wiki/Inverse_element#Matrices) More...

static void	Solve (MatrixType A, VectorType &rX, const VectorType &rB)
	This function is designed to be called when a dense linear system is needed to be solved. More...

template<class TMatrix1 , class TMatrix2 >
static void	InvertMatrix (const TMatrix1 &rInputMatrix, TMatrix2 &rInvertedMatrix, double &rInputMatrixDet, const double Tolerance=ZeroTolerance)
	It inverts matrices of order 2, 3 and 4. More...

template<class TMatrix1 , class TMatrix2 >
static void	InvertMatrix2 (const TMatrix1 &rInputMatrix, TMatrix2 &rInvertedMatrix, double &rInputMatrixDet)
	It inverts matrices of order 2. More...

template<class TMatrix1 , class TMatrix2 >
static void	InvertMatrix3 (const TMatrix1 &rInputMatrix, TMatrix2 &rInvertedMatrix, double &rInputMatrixDet)
	It inverts matrices of order 3. More...

template<class TMatrix1 , class TMatrix2 >
static void	InvertMatrix4 (const TMatrix1 &rInputMatrix, TMatrix2 &rInvertedMatrix, double &rInputMatrixDet)
	It inverts matrices of order 4. More...

template<class TMatrixType >
static double	Det2 (const TMatrixType &rA)
	Calculates the determinant of a matrix of dimension 2x2 (no check performed on matrix size) More...

template<class TMatrixType >
static double	Det3 (const TMatrixType &rA)
	Calculates the determinant of a matrix of dimension 3*3 (no check performed on matrix size) More...

template<class TMatrixType >
static double	Det4 (const TMatrixType &rA)
	Calculates the determinant of a matrix of dimension 4*4 (no check performed on matrix size) More...

template<class TMatrixType >
static double	Det (const TMatrixType &rA)
	Calculates the determinant of a matrix of a square matrix of any size (no check performed on release mode) More...

template<class TMatrixType >
static double	GeneralizedDet (const TMatrixType &rA)
	Calculates the determinant of any matrix (no check performed on matrix size) More...

static double	Dot3 (const Vector &a, const Vector &b)
	Performs the dot product of two vectors of dimension 3. More...

static double	Dot (const Vector &rFirstVector, const Vector &rSecondVector)
	Performs the dot product of two vectors of arbitrary size. More...

template<class TVectorType >
static double	Norm3 (const TVectorType &a)
	Calculates the norm of vector "a" which is assumed to be of size 3. More...

static double	Norm (const Vector &a)
	Calculates the norm of vector "a". More...

static double	StableNorm (const Vector &a)
	Calculates the norm of vector "a" while avoiding underflow and overflow. More...

template<class T >
static T	CrossProduct (const T &a, const T &b)
	Performs the vector product of the two input vectors a,b. More...

template<class T1 , class T2 >
static std::enable_if< std::is_same< T1, T2 >::value, bool >::type	CheckIsAlias (T1 &value1, T2 &value2)
	Checks there is aliasing. More...

template<class T1 , class T2 >
static std::enable_if<!std::is_same< T1, T2 >::value, bool >::type	CheckIsAlias (T1 &value1, T2 &value2)
	Checks there is aliasing. More...

template<class T1 , class T2 , class T3 >
static void	CrossProduct (T1 &c, const T2 &a, const T3 &b)
	Performs the cross product of the two input vectors a,b. More...

template<class T1 , class T2 , class T3 >
static void	UnitCrossProduct (T1 &c, const T2 &a, const T3 &b)
	Performs the unitary cross product of the two input vectors a,b. More...

template<class T1 , class T2 , class T3 >
static void	OrthonormalBasis (const T1 &c, T2 &a, T3 &b, const IndexType Type=0)
	This computes a orthonormal basis from a given vector (Frisvad method) More...

template<class T1 , class T2 , class T3 >
static void	OrthonormalBasisHughesMoeller (const T1 &c, T2 &a, T3 &b)
	This computes a orthonormal basis from a given vector (Hughes Moeller method) More...

template<class T1 , class T2 , class T3 >
static void	OrthonormalBasisFrisvad (const T1 &c, T2 &a, T3 &b)
	This computes a orthonormal basis from a given vector (Frisvad method) More...

template<class T1 , class T2 , class T3 >
static void	OrthonormalBasisNaive (const T1 &c, T2 &a, T3 &b)
	This computes a orthonormal basis from a given vector (Naive method) More...

template<class T1 , class T2 >
static double	VectorsAngle (const T1 &rV1, const T2 &rV2)
	Computes the angle between two vectors in 3D. More...

static MatrixType	TensorProduct3 (const Vector &a, const Vector &b)
	Returns a matrix : A = a.tensorproduct.b. More...

template<class TMatrixType1 , class TMatrixType2 >
static void	AddMatrix (TMatrixType1 &rDestination, const TMatrixType2 &rInputMatrix, const IndexType InitialRow, const IndexType InitialCol)
	"rInputMatrix" is ADDED to "Destination" matrix starting from InitialRow and InitialCol of the destination matrix More...

template<class TVectorType1 , class TVectorType2 >
static void	AddVector (TVectorType1 &rDestination, const TVectorType2 &rInputVector, const IndexType InitialIndex)
	"rInputVector" is ADDED to "Destination" vector starting from InitialIndex of the destination matrix More...

static void	SubtractMatrix (MatrixType &rDestination, const MatrixType &rInputMatrix, const IndexType InitialRow, const IndexType InitialCol)
	"rInputMatrix" is SUBTRACTED to "rDestination" matrix starting from InitialRow and InitialCol of the destination matrix More...

static void	WriteMatrix (MatrixType &rDestination, const MatrixType &rInputMatrix, const IndexType InitialRow, const IndexType InitialCol)
	"rInputMatrix" is WRITTEN on "Destination" matrix starting from InitialRow and InitialCol of the destination matrix More...

static void	ExpandReducedMatrix (MatrixType &rDestination, const MatrixType &rReducedMatrix, const SizeType Dimension)
	Performs the Kroneker product of the Reduced Matrix with the identity matrix of size "dimension". More...

static void	ExpandAndAddReducedMatrix (MatrixType &rDestination, const MatrixType &rReducedMatrix, const SizeType Dimension)
	Performs the Kroneker product of the Reduced Matrix with the identity matrix of size "dimension" ADDING to the destination matrix. More...

static void	VecAdd (Vector &rX, const double coeff, Vector &rY)
	Performs rX += coeff*rY. no check on bounds is performed. More...

template<class TVector , class TMatrixType = MatrixType>
static TMatrixType	StressVectorToTensor (const TVector &rStressVector)
	Transforms a stess vector into a matrix. Stresses are assumed to be stored in the following way: \( [ s11, s22, s33, s12, s23, s13 ] \) for 3D case and \( [ s11, s22, s33, s12 ] \) for 2D case. \( [ s11, s22, s12 ] \) for 2D case. More...

template<class TVector , class TMatrixType = MatrixType>
static TMatrixType	VectorToSymmetricTensor (const TVector &rVector)
	Transforms a vector into a symmetric matrix. More...

static int	Sign (const double &ThisDataType)
	Sign function. More...

template<class TVector , class TMatrixType = MatrixType>
static TMatrixType	StrainVectorToTensor (const TVector &rStrainVector)
	Transforms a strain vector into a matrix. Strains are assumed to be stored in the following way: \( [ e11, e22, e33, 2e12, 2e23, 2e13 ] \) for 3D case and \( [ e11, e22, e33, 2e12 ] \) for 2D case. \( [ e11, e22, 2*e12 ] \) for 2D case. More...

template<class TMatrixType , class TVector = Vector>
static Vector	StrainTensorToVector (const TMatrixType &rStrainTensor, SizeType rSize=0)
	Transforms a given symmetric Strain Tensor to Voigt Notation: More...

template<class TMatrixType , class TVector = Vector>
static TVector	StressTensorToVector (const TMatrixType &rStressTensor, SizeType rSize=0)
	Transforms a given symmetric Stress Tensor to Voigt Notation: More...

template<class TMatrixType , class TVector = Vector>
static TVector	SymmetricTensorToVector (const TMatrixType &rTensor, SizeType rSize=0)
	Transforms a given symmetric Tensor to Voigt Notation: More...

template<class TMatrixType1 , class TMatrixType2 , class TMatrixType3 >
static void	BtDBProductOperation (TMatrixType1 &rA, const TMatrixType2 &rD, const TMatrixType3 &rB)
	Calculates the product operation B'DB. More...

template<class TMatrixType1 , class TMatrixType2 , class TMatrixType3 >
static void	BDBtProductOperation (TMatrixType1 &rA, const TMatrixType2 &rD, const TMatrixType3 &rB)
	Calculates the product operation BDB'. More...

template<class TMatrixType1 , class TMatrixType2 >
static bool	GaussSeidelEigenSystem (const TMatrixType1 &rA, TMatrixType2 &rEigenVectorsMatrix, TMatrixType2 &rEigenValuesMatrix, const double Tolerance=1.0e-18, const SizeType MaxIterations=20)
	Calculates the eigenvectors and eigenvalues of given symmetric matrix. More...

template<class TIntegerType >
static TIntegerType	Factorial (const TIntegerType Number)
	Calculates the Factorial of a number k, Factorial = k! More...

template<class TMatrixType >
static void	CalculateExponentialOfMatrix (const TMatrixType &rMatrix, TMatrixType &rExponentialMatrix, const double Tolerance=1000.0 *ZeroTolerance, const SizeType MaxTerms=200)
	Calculates the exponential of a matrix. More...

static double	DegreesToRadians (double AngleInDegrees)

template<SizeType TDim>
	KRATOS_DEPRECATED_MESSAGE ("Please use GaussSeidelEigenSystem() instead. Note the resulting EigenVectors matrix is transposed respect GaussSeidelEigenSystem()") static inline bool EigenSystem(const BoundedMatrix< double
	Calculates the eigenvectors and eigenvalues of given symmetric TDimxTDim matrix. More...

TDim BoundedMatrix< double, TDim, TDim > BoundedMatrix< double, TDim, TDim > const double const SizeType class TMatrixType2 static inline bool	MatrixSquareRoot (const TMatrixType1 &rA, TMatrixType2 &rMatrixSquareRoot, const double Tolerance=1.0e-18, const SizeType MaxIterations=20)

Detailed Description

template<class TDataType = double>
class Kratos::MathUtils< TDataType >

Various mathematical utilitiy functions.

Various mathematical utilitiy functions. Defines several utility functions.

Author: Riccardo Rossi; Pooyan Dadvand

Member Typedef Documentation

◆ IndexType

template<class TDataType = double>

using Kratos::MathUtils< TDataType >::IndexType = std::size_t

The index type.

◆ IndirectArrayType

template<class TDataType = double>

using Kratos::MathUtils< TDataType >::IndirectArrayType = boost::numeric::ublas::indirect_array<DenseVector<std::size_t> >

The indirect array type.

◆ MatrixType

template<class TDataType = double>

using Kratos::MathUtils< TDataType >::MatrixType = Matrix

The matrix type.

◆ SizeType

template<class TDataType = double>

using Kratos::MathUtils< TDataType >::SizeType = std::size_t

The size type.

◆ VectorType

template<class TDataType = double>

using Kratos::MathUtils< TDataType >::VectorType = Vector

The vector type.

Member Function Documentation

◆ AddMatrix()

template<class TDataType = double>

template<class TMatrixType1 , class TMatrixType2 >

static void Kratos::MathUtils< TDataType >::AddMatrix	(	TMatrixType1 &	rDestination,
		const TMatrixType2 &	rInputMatrix,
		const IndexType	InitialRow,
		const IndexType	InitialCol
	)

inlinestatic

"rInputMatrix" is ADDED to "Destination" matrix starting from InitialRow and InitialCol of the destination matrix

"Destination" is assumed to be able to contain the "input matrix" (no check is performed on the bounds)

Parameters

rDestination	The matrix destination
rInputMatrix	The input matrix to be added
InitialRow	The initial row
InitialCol	The initial column

◆ AddVector()

template<class TDataType = double>

template<class TVectorType1 , class TVectorType2 >

static void Kratos::MathUtils< TDataType >::AddVector	(	TVectorType1 &	rDestination,
		const TVectorType2 &	rInputVector,
		const IndexType	InitialIndex
	)

inlinestatic

"rInputVector" is ADDED to "Destination" vector starting from InitialIndex of the destination matrix

"Destination" is assumed to be able to contain the "input vector" (no check is performed on the bounds)

Parameters

rDestination	The vector destination
rInputVector	The input vector to be added
InitialIndex	The initial index

◆ BDBtProductOperation()

template<class TDataType = double>

template<class TMatrixType1 , class TMatrixType2 , class TMatrixType3 >

static void Kratos::MathUtils< TDataType >::BDBtProductOperation	(	TMatrixType1 &	rA,
		const TMatrixType2 &	rD,
		const TMatrixType3 &	rB
	)

inlinestatic

Calculates the product operation BDB'.

Parameters

rA	The resulting matrix
rD	The "center" matrix
rB	The matrices to be transposed

Template Parameters

TMatrixType1	The type of matrix considered (1)
TMatrixType2	The type of matrix considered (2)
TMatrixType3	The type of matrix considered (3)

◆ BtDBProductOperation()

template<class TDataType = double>

template<class TMatrixType1 , class TMatrixType2 , class TMatrixType3 >

static void Kratos::MathUtils< TDataType >::BtDBProductOperation	(	TMatrixType1 &	rA,
		const TMatrixType2 &	rD,
		const TMatrixType3 &	rB
	)

inlinestatic

Calculates the product operation B'DB.

Parameters

rA	The resulting matrix
rD	The "center" matrix
rB	The matrices to be transposed

Template Parameters

TMatrixType1	The type of matrix considered (1)
TMatrixType2	The type of matrix considered (2)
TMatrixType3	The type of matrix considered (3)

◆ CalculateExponentialOfMatrix()

template<class TDataType = double>

template<class TMatrixType >

static void Kratos::MathUtils< TDataType >::CalculateExponentialOfMatrix	(	const TMatrixType &	rMatrix,
		TMatrixType &	rExponentialMatrix,
		const double	Tolerance = `1000.0*ZeroTolerance`,
		const SizeType	MaxTerms = `200`
	)

inlinestatic

Calculates the exponential of a matrix.

see https://mathworld.wolfram.com/MatrixExponential.html

Template Parameters

rMatrix	the matrix A of which exp is calculated
rExponentialMatrix	exp(A)

◆ CheckConditionNumber()

template<class TDataType = double>

template<class TMatrix1 , class TMatrix2 >

static bool Kratos::MathUtils< TDataType >::CheckConditionNumber	(	const TMatrix1 &	rInputMatrix,
		TMatrix2 &	rInvertedMatrix,
		const double	Tolerance = `std::numeric_limits<double>::epsilon()`,
		const bool	ThrowError = `true`
	)

inlinestatic

This method checks the condition number of amtrix.

Parameters

rInputMatrix	Is the input matrix (unchanged at output)
rInvertedMatrix	Is the inverse of the input matrix
Tolerance	The maximum tolerance considered

◆ CheckIsAlias() [1/2]

template<class TDataType = double>

template<class T1 , class T2 >

static std::enable_if<std::is_same<T1, T2>::value, bool>::type Kratos::MathUtils< TDataType >::CheckIsAlias	(	T1 &	value1,
		T2 &	value2
	)

inlinestatic

Checks there is aliasing.

Parameters

value1	The first value
value2	The second value

◆ CheckIsAlias() [2/2]

template<class TDataType = double>

template<class T1 , class T2 >

static std::enable_if<!std::is_same<T1, T2>::value, bool>::type Kratos::MathUtils< TDataType >::CheckIsAlias	(	T1 &	value1,
		T2 &	value2
	)

inlinestatic

Checks there is aliasing.

Parameters

value1	The first value
value2	The second value

◆ Cofactor()

template<class TDataType = double>

template<class TMatrixType >

static double Kratos::MathUtils< TDataType >::Cofactor	(	const TMatrixType &	rMat,
		IndexType	i,
		IndexType	j
	)

inlinestatic

Calculates the cofactor.

Parameters

rMat	The matrix to calculate
i	The index i
j	The index j

Returns: The cofactor of the matrix

◆ CofactorMatrix()

template<class TDataType = double>

template<class TMatrixType >

static MatrixType Kratos::MathUtils< TDataType >::CofactorMatrix ( const TMatrixType & rMat )

inlinestatic

Calculates the cofactor matrix.

Parameters

rMat	The matrix to calculate

Returns: The cofactor matrix

◆ CrossProduct() [1/2]

template<class TDataType = double>

template<class T >

static T Kratos::MathUtils< TDataType >::CrossProduct	(	const T &	a,
		const T &	b
	)

inlinestatic

Performs the vector product of the two input vectors a,b.

a,b are assumed to be of size 3 (no check is performed on vector sizes)

Parameters

a	First input vector
b	Second input vector

Returns: The resulting vector

◆ CrossProduct() [2/2]

template<class TDataType = double>

template<class T1 , class T2 , class T3 >

static void Kratos::MathUtils< TDataType >::CrossProduct	(	T1 &	c,
		const T2 &	a,
		const T3 &	b
	)

inlinestatic

Performs the cross product of the two input vectors a,b.

a,b are assumed to be of size 3 (check is only performed on vector sizes in debug mode)

Parameters

a	First input vector
b	Second input vector
c	The resulting vector

◆ DegreesToRadians()

template<class TDataType = double>

static double Kratos::MathUtils< TDataType >::DegreesToRadians ( double AngleInDegrees )

inlinestatic

◆ Det()

template<class TDataType = double>

template<class TMatrixType >

static double Kratos::MathUtils< TDataType >::Det ( const TMatrixType & rA )

inlinestatic

Calculates the determinant of a matrix of a square matrix of any size (no check performed on release mode)

Parameters

rA	Is the input matrix

Returns: The determinant of any size matrix

◆ Det2()

template<class TDataType = double>

template<class TMatrixType >

static double Kratos::MathUtils< TDataType >::Det2 ( const TMatrixType & rA )

inlinestatic

Calculates the determinant of a matrix of dimension 2x2 (no check performed on matrix size)

Parameters

rA	Is the input matrix

Returns: The determinant of the 2x2 matrix

◆ Det3()

template<class TDataType = double>

template<class TMatrixType >

static double Kratos::MathUtils< TDataType >::Det3 ( const TMatrixType & rA )

inlinestatic

Calculates the determinant of a matrix of dimension 3*3 (no check performed on matrix size)

Parameters

rA	Is the input matrix

Returns: The determinant of the 3x3 matrix

◆ Det4()

template<class TDataType = double>

template<class TMatrixType >

static double Kratos::MathUtils< TDataType >::Det4 ( const TMatrixType & rA )

inlinestatic

Calculates the determinant of a matrix of dimension 4*4 (no check performed on matrix size)

Parameters

rA	Is the input matrix

Returns: The determinant of the 4x4 matrix

◆ Dot()

template<class TDataType = double>

static double Kratos::MathUtils< TDataType >::Dot	(	const Vector &	rFirstVector,
		const Vector &	rSecondVector
	)

inlinestatic

Performs the dot product of two vectors of arbitrary size.

No check performed on vector sizes

Parameters

rFirstVector	First input vector
rSecondVector	Second input vector

Returns: The resulting norm

◆ Dot3()

template<class TDataType = double>

static double Kratos::MathUtils< TDataType >::Dot3	(	const Vector &	a,
		const Vector &	b
	)

inlinestatic

Performs the dot product of two vectors of dimension 3.

No check performed on vector sizes

Parameters

a	First input vector
b	Second input vector

Returns: The resulting norm

◆ ExpandAndAddReducedMatrix()

template<class TDataType = double>

static void Kratos::MathUtils< TDataType >::ExpandAndAddReducedMatrix	(	MatrixType &	rDestination,
		const MatrixType &	rReducedMatrix,
		const SizeType	Dimension
	)

inlinestatic

Performs the Kroneker product of the Reduced Matrix with the identity matrix of size "dimension" ADDING to the destination matrix.

Parameters

rDestination	The matric destination
rReducedMatrix	The reduced matrix to be added
Dimension	The dimension where we work

◆ ExpandReducedMatrix()

template<class TDataType = double>

static void Kratos::MathUtils< TDataType >::ExpandReducedMatrix	(	MatrixType &	rDestination,
		const MatrixType &	rReducedMatrix,
		const SizeType	Dimension
	)

inlinestatic

Performs the Kroneker product of the Reduced Matrix with the identity matrix of size "dimension".

Parameters

rDestination	The matric destination
rReducedMatrix	The reduced matrix to be computed
Dimension	The dimension where we work

◆ Factorial()

template<class TDataType = double>

template<class TIntegerType >

static TIntegerType Kratos::MathUtils< TDataType >::Factorial ( const TIntegerType Number )

inlinestatic

Calculates the Factorial of a number k, Factorial = k!

Template Parameters

Number The number of which the Factorial is computed

◆ GaussSeidelEigenSystem()

template<class TDataType = double>

template<class TMatrixType1 , class TMatrixType2 >

static bool Kratos::MathUtils< TDataType >::GaussSeidelEigenSystem	(	const TMatrixType1 &	rA,
		TMatrixType2 &	rEigenVectorsMatrix,
		TMatrixType2 &	rEigenValuesMatrix,
		const double	Tolerance = `1.0e-18`,
		const SizeType	MaxIterations = `20`
	)

inlinestatic

Calculates the eigenvectors and eigenvalues of given symmetric matrix.

The eigenvectors and eigenvalues are calculated using the iterative Gauss-Seidel-method. The resulting decomposition is LDL'

Note: See https://en.wikipedia.org/wiki/Gauss%E2%80%93Seidel_method

Parameters

rA	The given symmetric matrix the eigenvectors are to be calculated.
rEigenVectorsMatrix	The result matrix (will be overwritten with the eigenvectors)
rEigenValuesMatrix	The result diagonal matrix with the eigenvalues
Tolerance	The largest value considered to be zero
MaxIterations	Maximum number of iterations

Template Parameters

TMatrixType1	The type of matrix considered (1)
TMatrixType2	The type of matrix considered (2)

◆ GeneralizedDet()

template<class TDataType = double>

template<class TMatrixType >

static double Kratos::MathUtils< TDataType >::GeneralizedDet ( const TMatrixType & rA )

inlinestatic

Calculates the determinant of any matrix (no check performed on matrix size)

Parameters

rA	Is the input matrix

Returns: The determinant of the 2x2 matrix

◆ GeneralizedInvertMatrix()

template<class TDataType = double>

template<class TMatrix1 , class TMatrix2 >

static void Kratos::MathUtils< TDataType >::GeneralizedInvertMatrix	(	const TMatrix1 &	rInputMatrix,
		TMatrix2 &	rInvertedMatrix,
		double &	rInputMatrixDet,
		const double	Tolerance = `ZeroTolerance`
	)

inlinestatic

It inverts non square matrices (https://en.wikipedia.org/wiki/Inverse_element#Matrices)

Parameters

rInputMatrix	Is the input matrix (unchanged at output)
rInvertedMatrix	Is the inverse of the input matrix
rInputMatrixDet	Is the determinant of the input matrix
Tolerance	The maximum tolerance considered

Template Parameters

TMatrix1	The type of the input matrix
TMatrix2	Is the type of the output matrix

◆ GetZeroTolerance()

template<class TDataType = double>

static double Kratos::MathUtils< TDataType >::GetZeroTolerance ( )

inlinestatic

This function returns the machine precision.

Returns: The corresponding epsilon for the double

◆ Heron()

template<class TDataType = double>

template<bool TCheck>

static double Kratos::MathUtils< TDataType >::Heron	(	double	a,
		double	b,
		double	c
	)

inlinestatic

In geometry, Heron's formula (sometimes called Hero's formula), named after Hero of Alexandria, gives the area of a triangle by requiring no arbitrary choice of side as base or vertex as origin, contrary to other formulas for the area of a triangle, such as half the base times the height or half the norm of a cross product of two sides.

Parameters

a	First length
b	Second length
c	Third length

Returns: Heron solution: Heron's formula states that the area of a triangle whose sides have lengths a, b, and c

◆ InvertMatrix() [1/2]

template<class TDataType = double>

template<SizeType TDim>

static BoundedMatrix<double, TDim, TDim> Kratos::MathUtils< TDataType >::InvertMatrix	(	const BoundedMatrix< double, TDim, TDim > &	rInputMatrix,
		double &	rInputMatrixDet,
		const double	Tolerance = `ZeroTolerance`
	)

inlinestatic

Calculates the inverse of a 2x2, 3x3 and 4x4 matrices (using bounded matrix for performance)

Parameters

rInputMatrix	The matrix to invert
rInputMatrixDet	The determinant of the matrix
Tolerance	The maximum tolerance considered

Returns: InvertMatrix: The inverted matrix

◆ InvertMatrix() [2/2]

template<class TDataType = double>

template<class TMatrix1 , class TMatrix2 >

static void Kratos::MathUtils< TDataType >::InvertMatrix	(	const TMatrix1 &	rInputMatrix,
		TMatrix2 &	rInvertedMatrix,
		double &	rInputMatrixDet,
		const double	Tolerance = `ZeroTolerance`
	)

inlinestatic

It inverts matrices of order 2, 3 and 4.

Parameters

rInputMatrix	Is the input matrix (unchanged at output)
rInvertedMatrix	Is the inverse of the input matrix
rInputMatrixDet	Is the determinant of the input matrix

Template Parameters

TMatrix1	The type of the input matrix
TMatrix2	Is the type of the output matrix

◆ InvertMatrix2()

template<class TDataType = double>

template<class TMatrix1 , class TMatrix2 >

static void Kratos::MathUtils< TDataType >::InvertMatrix2	(	const TMatrix1 &	rInputMatrix,
		TMatrix2 &	rInvertedMatrix,
		double &	rInputMatrixDet
	)

inlinestatic

It inverts matrices of order 2.

Parameters

rInputMatrix	Is the input matrix (unchanged at output)
rInvertedMatrix	Is the inverse of the input matrix
rInputMatrixDet	Is the determinant of the input matrix

◆ InvertMatrix3()

template<class TDataType = double>

template<class TMatrix1 , class TMatrix2 >

static void Kratos::MathUtils< TDataType >::InvertMatrix3	(	const TMatrix1 &	rInputMatrix,
		TMatrix2 &	rInvertedMatrix,
		double &	rInputMatrixDet
	)

inlinestatic

It inverts matrices of order 3.

Parameters

rInputMatrix	Is the input matrix (unchanged at output)
rInvertedMatrix	Is the inverse of the input matrix
rInputMatrixDet	Is the determinant of the input matrix

◆ InvertMatrix4()

template<class TDataType = double>

template<class TMatrix1 , class TMatrix2 >

static void Kratos::MathUtils< TDataType >::InvertMatrix4	(	const TMatrix1 &	rInputMatrix,
		TMatrix2 &	rInvertedMatrix,
		double &	rInputMatrixDet
	)

inlinestatic

It inverts matrices of order 4.

Parameters

rInputMatrix	Is the input matrix (unchanged at output)
rInvertedMatrix	Is the inverse of the input matrix
rInputMatrixDet	Is the determinant of the input matrix

◆ KRATOS_DEPRECATED_MESSAGE()

template<class TDataType = double>

template<SizeType TDim>

Kratos::MathUtils< TDataType >::KRATOS_DEPRECATED_MESSAGE ( "Please use GaussSeidelEigenSystem() instead. Note the resulting EigenVectors matrix is transposed respect GaussSeidelEigenSystem()" ) const

Calculates the eigenvectors and eigenvalues of given symmetric TDimxTDim matrix.

The eigenvectors and eigenvalues are calculated using the iterative Gauss-Seidel-method. The resulting decomposition is L'DL

Note: See https://en.wikipedia.org/wiki/Gauss%E2%80%93Seidel_method

Parameters

A	The given symmetric matrix the eigenvectors are to be calculated.
rEigenVectorsMatrix	The result matrix (will be overwritten with the eigenvectors)
rEigenValuesMatrix	The result diagonal matrix with the eigenvalues
Tolerance	The largest value considered to be zero
MaxIterations	Maximum number of iterations

Template Parameters

TDim	The size of the bounded matrix

Warning: This method is deprecated. Will be removed soon

◆ MatrixSquareRoot()

template<class TDataType = double>

TDim BoundedMatrix<double, TDim, TDim> BoundedMatrix<double, TDim, TDim> const double const SizeType class TMatrixType2 static inline bool Kratos::MathUtils< TDataType >::MatrixSquareRoot	(	const TMatrixType1 &	rA,
		TMatrixType2 &	rMatrixSquareRoot,
		const double	Tolerance = `1.0e-18`,
		const SizeType	MaxIterations = `20`
	)

inline

◆ Norm()

template<class TDataType = double>

static double Kratos::MathUtils< TDataType >::Norm ( const Vector & a )

inlinestatic

Calculates the norm of vector "a".

Parameters

a	Input vector

Returns: The resulting norm

◆ Norm3()

template<class TDataType = double>

template<class TVectorType >

static double Kratos::MathUtils< TDataType >::Norm3 ( const TVectorType & a )

inlinestatic

Calculates the norm of vector "a" which is assumed to be of size 3.

No check is performed on the vector's size

Parameters

a	Input vector

Returns: The resulting norm

◆ OrthonormalBasis()

template<class TDataType = double>

template<class T1 , class T2 , class T3 >

static void Kratos::MathUtils< TDataType >::OrthonormalBasis	(	const T1 &	c,
		T2 &	a,
		T3 &	b,
		const IndexType	Type = `0`
	)

inlinestatic

This computes a orthonormal basis from a given vector (Frisvad method)

Parameters

c	The input vector
a	First resulting vector
b	Second resulting vector
Type	The type of method employed, 0 is HughesMoeller, 1 is Frisvad and otherwise Naive

◆ OrthonormalBasisFrisvad()

template<class TDataType = double>

template<class T1 , class T2 , class T3 >

static void Kratos::MathUtils< TDataType >::OrthonormalBasisFrisvad	(	const T1 &	c,
		T2 &	a,
		T3 &	b
	)

inlinestatic

This computes a orthonormal basis from a given vector (Frisvad method)

Parameters

c	The input vector
a	First resulting vector
b	Second resulting vector

Note: Orthonormal basis taken from: http://orbit.dtu.dk/files/126824972/onb_frisvad_jgt2012_v2.pdf

◆ OrthonormalBasisHughesMoeller()

template<class TDataType = double>

template<class T1 , class T2 , class T3 >

static void Kratos::MathUtils< TDataType >::OrthonormalBasisHughesMoeller	(	const T1 &	c,
		T2 &	a,
		T3 &	b
	)

inlinestatic

This computes a orthonormal basis from a given vector (Hughes Moeller method)

Parameters

c	The input vector
a	First resulting vector
b	Second resulting vector

Note: Orthonormal basis taken from: http://orbit.dtu.dk/files/126824972/onb_frisvad_jgt2012_v2.pdf

◆ OrthonormalBasisNaive()

template<class TDataType = double>

template<class T1 , class T2 , class T3 >

static void Kratos::MathUtils< TDataType >::OrthonormalBasisNaive	(	const T1 &	c,
		T2 &	a,
		T3 &	b
	)

inlinestatic

This computes a orthonormal basis from a given vector (Naive method)

Parameters

c	The input vector
a	First resulting vector
b	Second resulting vector

Note: Orthonormal basis taken from: http://orbit.dtu.dk/files/126824972/onb_frisvad_jgt2012_v2.pdf

◆ Sign()

template<class TDataType = double>

static int Kratos::MathUtils< TDataType >::Sign ( const double & ThisDataType )

inlinestatic

Sign function.

Parameters

ThisDataType The value to extract the sign

Returns: The sign of the value

◆ Solve()

template<class TDataType >

void Kratos::MathUtils< TDataType >::Solve	(	MatrixType	A,
		VectorType &	rX,
		const VectorType &	rB
	)

static

This function is designed to be called when a dense linear system is needed to be solved.

Parameters

A	System matrix
rX	Solution vector. it's also the initial guess for iterative linear solvers.
rB	Right hand side vector.

◆ StableNorm()

template<class TDataType = double>

static double Kratos::MathUtils< TDataType >::StableNorm ( const Vector & a )

inlinestatic

Calculates the norm of vector "a" while avoiding underflow and overflow.

Parameters

a	Input vector

Returns: The resulting norm

See also: http://www.netlib.org/lapack/explore-html/da/d7f/dnrm2_8f_source.html

◆ StrainTensorToVector()

template<class TDataType = double>

template<class TMatrixType , class TVector = Vector>

static Vector Kratos::MathUtils< TDataType >::StrainTensorToVector	(	const TMatrixType &	rStrainTensor,
		SizeType	rSize = `0`
	)

inlinestatic

Transforms a given symmetric Strain Tensor to Voigt Notation:

The following cases:

In the 3D case: from a second order tensor (3*3) Matrix to a corresponing (6*1) Vector \( [ e11, e22, e33, 2*e12, 2*e23, 2*e13 ] \) for 3D case and
In the 2D case: from a second order tensor (3*3) Matrix to a corresponing (4*1) Vector \( [ e11, e22, e33, 2*e12 ] \) fir 2D case.
In the 2D case: from a second order tensor (2*2) Matrix to a corresponing (3*1) Vector \( [ e11, e22, 2*e12 ] \) fir 2D case.
Parameters

rStrainTensor the given symmetric second order strain tensor

Returns
the corresponding strain tensor in vector form

Template Parameters

TMatrixType The matrix type considered

TVector The vector returning type

◆ StrainVectorToTensor()

template<class TDataType = double>

template<class TVector , class TMatrixType = MatrixType>

static TMatrixType Kratos::MathUtils< TDataType >::StrainVectorToTensor ( const TVector & rStrainVector )

inlinestatic

Transforms a strain vector into a matrix. Strains are assumed to be stored in the following way: \( [ e11, e22, e33, 2*e12, 2*e23, 2*e13 ] \) for 3D case and \( [ e11, e22, e33, 2*e12 ] \) for 2D case. \( [ e11, e22, 2*e12 ] \) for 2D case.

Hence the deviatoric components of the strain vector are divided by 2 while they are stored into the matrix

Parameters

rStrainVector the given strain vector

Returns: the corresponding strain tensor in matrix form

Template Parameters

TVector	The vector type considered
TMatrixType	The matrix returning type

◆ StressTensorToVector()

template<class TDataType = double>

template<class TMatrixType , class TVector = Vector>

static TVector Kratos::MathUtils< TDataType >::StressTensorToVector	(	const TMatrixType &	rStressTensor,
		SizeType	rSize = `0`
	)

inlinestatic

Transforms a given symmetric Stress Tensor to Voigt Notation:

Components are assumed to be stored in the following way: \( [ s11, s22, s33, s12, s23, s13 ] \) for 3D case and \( [ s11, s22, s33, s12 ] \) for 2D case. \( [ s11, s22, s12 ] \) for 2D case. In the 3D case: from a second order tensor (3*3) Matrix to a corresponing (6*1) Vector In the 3D case: from a second order tensor (3*3) Matrix to a corresponing (4*1) Vector In the 2D case: from a second order tensor (2*2) Matrix to a corresponing (3*1) Vector

Parameters

rStressTensor the given symmetric second order stress tensor

Returns: the corresponding stress tensor in vector form

Template Parameters

TMatrixType	The matrix type considered
TVector	The vector returning type

◆ StressVectorToTensor()

template<class TDataType = double>

template<class TVector , class TMatrixType = MatrixType>

static TMatrixType Kratos::MathUtils< TDataType >::StressVectorToTensor ( const TVector & rStressVector )

inlinestatic

Transforms a stess vector into a matrix. Stresses are assumed to be stored in the following way: \( [ s11, s22, s33, s12, s23, s13 ] \) for 3D case and \( [ s11, s22, s33, s12 ] \) for 2D case. \( [ s11, s22, s12 ] \) for 2D case.

Parameters

rStressVector the given stress vector

Returns: the corresponding stress tensor in matrix form

Template Parameters

TVector	The vector type considered
TMatrixType	The matrix returning type

◆ SubtractMatrix()

template<class TDataType = double>

static void Kratos::MathUtils< TDataType >::SubtractMatrix	(	MatrixType &	rDestination,
		const MatrixType &	rInputMatrix,
		const IndexType	InitialRow,
		const IndexType	InitialCol
	)

inlinestatic

"rInputMatrix" is SUBTRACTED to "rDestination" matrix starting from InitialRow and InitialCol of the destination matrix

"rDestination" is assumed to be able to contain the "input matrix" (no check is performed on the bounds)

Parameters

rDestination	The matric destination
rInputMatrix	The input matrix to be computed
InitialRow	The initial row to compute
InitialCol	The initial column to compute

◆ SymmetricTensorToVector()

template<class TDataType = double>

template<class TMatrixType , class TVector = Vector>

static TVector Kratos::MathUtils< TDataType >::SymmetricTensorToVector	(	const TMatrixType &	rTensor,
		SizeType	rSize = `0`
	)

inlinestatic

Transforms a given symmetric Tensor to Voigt Notation:

The following cases:

In the 3D case: from a second order tensor (3*3) Matrix to a corresponing (6*1) Vector
In the 3D case: from a second order tensor (3*3) Matrix to a corresponing (4*1) Vector
In the 2D case: from a second order tensor (2*2) Matrix to a corresponing (3*1) Vector
Parameters

rTensor the given symmetric second order stress tensor

Returns
the corresponding stress tensor in vector form

Template Parameters

TMatrixType The matrix type considered

TVector The vector returning type

◆ TensorProduct3()

template<class TDataType = double>

static MatrixType Kratos::MathUtils< TDataType >::TensorProduct3	(	const Vector &	a,
		const Vector &	b
	)

inlinestatic

Returns a matrix : A = a.tensorproduct.b.

a,b are assumed to be of order 3, no check is performed on the size of the vectors

Parameters

a	First input vector
b	Second input vector

Returns: Returns A = a.tensorproduct.b

◆ UnitCrossProduct()

template<class TDataType = double>

template<class T1 , class T2 , class T3 >

static void Kratos::MathUtils< TDataType >::UnitCrossProduct	(	T1 &	c,
		const T2 &	a,
		const T3 &	b
	)

inlinestatic

Performs the unitary cross product of the two input vectors a,b.

a,b are assumed to be of size 3 (no check is performed on vector sizes)

Parameters

a	First input vector
b	Second input vector
c	The resulting vector

◆ VecAdd()

template<class TDataType = double>

static void Kratos::MathUtils< TDataType >::VecAdd	(	Vector &	rX,
		const double	coeff,
		Vector &	rY
	)

inlinestatic

Performs rX += coeff*rY. no check on bounds is performed.

Parameters

rX	The vector destination
rY	The vector to be added
coeff	The proportion to be added

◆ VectorsAngle()

template<class TDataType = double>

template<class T1 , class T2 >

static double Kratos::MathUtils< TDataType >::VectorsAngle	(	const T1 &	rV1,
		const T2 &	rV2
	)

inlinestatic

Computes the angle between two vectors in 3D.

Parameters

rV1	First input vector
rV2	Second input vector

◆ VectorToSymmetricTensor()

template<class TDataType = double>

template<class TVector , class TMatrixType = MatrixType>

static TMatrixType Kratos::MathUtils< TDataType >::VectorToSymmetricTensor ( const TVector & rVector )

inlinestatic

Transforms a vector into a symmetric matrix.

Components are assumed to be stored in the following way: \( [ s11, s22, s33, s12, s23, s13 ] \) for 3D case and \( [ s11, s22, s33, s12 ] \) for 2D case. \( [ s11, s22, s12 ] \) for 2D case.

Parameters

rVector the given stress vector

Returns: The corresponding Tensor in matrix form

Template Parameters

TVector	The vector type considered
TMatrixType	The matrix returning type

◆ WriteMatrix()

template<class TDataType = double>

static void Kratos::MathUtils< TDataType >::WriteMatrix	(	MatrixType &	rDestination,
		const MatrixType &	rInputMatrix,
		const IndexType	InitialRow,
		const IndexType	InitialCol
	)

inlinestatic

"rInputMatrix" is WRITTEN on "Destination" matrix starting from InitialRow and InitialCol of the destination matrix

"Destination" is assumed to be able to contain the "input matrix" (no check is performed on the bounds)

Warning: Destination is overwritten!!

Parameters

rDestination	The matric destination
rrInputMatrix	The input matrix to be computed
InitialRow	The initial row to compute
InitialCol	The initial column to compute

Member Data Documentation

◆ MaxIterations

template<class TDataType = double>

TDim BoundedMatrix<double, TDim, TDim> BoundedMatrix<double, TDim, TDim> const double const SizeType Kratos::MathUtils< TDataType >::MaxIterations

Initial value:

= 20
        )
    {
        const bool is_converged = GaussSeidelEigenSystem(rA, rEigenVectorsMatrix, rEigenValuesMatrix, Tolerance, MaxIterations);
 
        const BoundedMatrix<double, TDim, TDim> V_matrix = rEigenVectorsMatrix;
        for(IndexType i = 0; i < TDim; ++i) {
            for(IndexType j = 0; j < TDim; ++j) {
                rEigenVectorsMatrix(i, j) = V_matrix(j, i);
            }
        }
 
        return is_converged;
    }
 
    
    template<class TMatrixType1

◆ rA

template<class TDataType = double>

TDim& Kratos::MathUtils< TDataType >::rA

◆ rEigenValuesMatrix

template<class TDataType = double>

TDim BoundedMatrix<double, TDim, TDim> BoundedMatrix<double, TDim, TDim>& Kratos::MathUtils< TDataType >::rEigenValuesMatrix

◆ rEigenVectorsMatrix

template<class TDataType = double>

TDim BoundedMatrix<double, TDim, TDim>& Kratos::MathUtils< TDataType >::rEigenVectorsMatrix

◆ TDim

template<class TDataType = double>

Kratos::MathUtils< TDataType >::TDim

◆ Tolerance

template<class TDataType = double>

TDim BoundedMatrix<double, TDim, TDim> BoundedMatrix<double, TDim, TDim> const double Kratos::MathUtils< TDataType >::Tolerance = 1.0e-18

◆ ZeroTolerance

template<class TDataType = double>

constexpr double Kratos::MathUtils< TDataType >::ZeroTolerance = std::numeric_limits<double>::epsilon()

staticconstexpr

The machine precision.

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

/home/runner/work/Documentation/Documentation/master/kratos/utilities/math_utils.h
/home/runner/work/Documentation/Documentation/master/kratos/utilities/math_utils.cpp

Type Definitions

Operations

Detailed Description

template<class TDataType = double> class Kratos::MathUtils< TDataType >

Member Typedef Documentation

◆ IndexType

◆ IndirectArrayType

◆ MatrixType

◆ SizeType

◆ VectorType

Member Function Documentation

◆ AddMatrix()

◆ AddVector()

◆ BDBtProductOperation()

◆ BtDBProductOperation()

◆ CalculateExponentialOfMatrix()

◆ CheckConditionNumber()

◆ CheckIsAlias() [1/2]

◆ CheckIsAlias() [2/2]

◆ Cofactor()

◆ CofactorMatrix()

◆ CrossProduct() [1/2]

◆ CrossProduct() [2/2]

◆ DegreesToRadians()

◆ Det()

◆ Det2()

◆ Det3()

◆ Det4()

◆ Dot()

◆ Dot3()

◆ ExpandAndAddReducedMatrix()

◆ ExpandReducedMatrix()

◆ Factorial()

◆ GaussSeidelEigenSystem()

◆ GeneralizedDet()

◆ GeneralizedInvertMatrix()

◆ GetZeroTolerance()

◆ Heron()

◆ InvertMatrix() [1/2]

◆ InvertMatrix() [2/2]

◆ InvertMatrix2()

◆ InvertMatrix3()

◆ InvertMatrix4()

◆ KRATOS_DEPRECATED_MESSAGE()

◆ MatrixSquareRoot()

◆ Norm()

◆ Norm3()

◆ OrthonormalBasis()

◆ OrthonormalBasisFrisvad()

◆ OrthonormalBasisHughesMoeller()

◆ OrthonormalBasisNaive()

◆ Sign()

◆ Solve()

◆ StableNorm()

◆ StrainTensorToVector()

◆ StrainVectorToTensor()

◆ StressTensorToVector()

◆ StressVectorToTensor()

◆ SubtractMatrix()

◆ SymmetricTensorToVector()

◆ TensorProduct3()

◆ UnitCrossProduct()

◆ VecAdd()

◆ VectorsAngle()

◆ VectorToSymmetricTensor()

◆ WriteMatrix()

Member Data Documentation

◆ MaxIterations

◆ rA

◆ rEigenValuesMatrix

◆ rEigenVectorsMatrix

◆ TDim

◆ Tolerance

◆ ZeroTolerance

template<class TDataType = double>
class Kratos::MathUtils< TDataType >