Tensor Utilities

Collection of several utilities for dealing with Eigen tensors. More...

Collaboration diagram for Tensor Utilities:

Functions
template<typename Derived , typename T , auto rank>
Eigen::Tensor< typename Derived::Scalar, rank >	Ikarus::tensorView (const Eigen::EigenBase< Derived > &matrix, const std::array< T, rank > &dims)
	View an Eigen matrix as an Eigen Tensor with specified dimensions. More...
auto	Ikarus::dyadic (const auto &A_ij, const auto &B_kl)
	Computes the dyadic product of two Eigen tensors. More...
template<typename ScalarType = double, int dim = 3>
auto	Ikarus::symmetricIdentityFourthOrder ()
	Generates a symmetric identity fourth-order tensor. More...
template<typename ScalarType = double, int dim = 3>
auto	Ikarus::symmetricFourthOrder (const auto &A, const auto &B)
	Generates a symmetric fourth-order tensor based on two second-order input tensors. More...
template<typename ScalarType = double, int dim = 3>
auto	Ikarus::identityFourthOrder ()
	Generates an identity fourth-order tensor. More...
template<typename AType , typename BType >
auto	Ikarus::fourthOrderIKJL (const Eigen::MatrixBase< AType > &A, const Eigen::MatrixBase< BType > &B)
	Computes the IKJL product of two matrices. More...
template<typename ScalarType , long int dim>
auto	Ikarus::symTwoSlots (const Eigen::TensorFixedSize< ScalarType, Eigen::Sizes< dim, dim, dim, dim > > &t, const std::array< size_t, 2 > &slots)
	Creates a symmetric fourth-order tensor in the two specified slots of the input tensor. More...
constexpr Eigen::Index	Ikarus::toVoigt (Eigen::Index i, Eigen::Index j) noexcept
	Converts 2D indices to Voigt notation index. More...
template<typename ScalarType = double>
Eigen::Matrix< ScalarType, 6, 6 >	Ikarus::toVoigt (const Eigen::TensorFixedSize< ScalarType, Eigen::Sizes< 3, 3, 3, 3 > > &ft)
	Converts a fourth-order tensor of fixed size 3x3x3x3 to a Voigt notation matrix of size 6x6. More...
template<typename ST , int size, int Options, int maxSize> requires ((size > 0 and size <= 3) or (maxSize > 0 and maxSize <= 3 and size == Eigen::Dynamic))
auto	Ikarus::toVoigt (const Eigen::Matrix< ST, size, size, Options, maxSize, maxSize > &E, bool isStrain=true)
	Converts a square 2x2 or 3x3 matrix to a Voigt notation vector. More...
template<typename ST , int size, int Options, int maxSize> requires ((size == 1 or size == 3 or size == 6) or ((maxSize == 1 or maxSize == 3 or maxSize == 6) and size == Eigen::Dynamic))
auto	Ikarus::fromVoigt (const Eigen::Matrix< ST, size, 1, Options, maxSize, 1 > &EVoigt, bool isStrain=true)
	Converts a vector given in Voigt notation to a matrix. More...
constexpr std::array< size_t, 2 >	Ikarus::fromVoigt (size_t i)
	Converts a Voigt notation index to matrix indices. More...
template<typename ScalarType >
auto	Ikarus::fromVoigt (const Eigen::Matrix< ScalarType, 6, 6 > &CVoigt)
	Converts a matrix in Voigt notation to a Fourth-order tensor. More...

Detailed Description

Function Documentation

dyadic()

auto Ikarus::dyadic	(	const auto &	A_ij,
		const auto &	B_kl
	)

The components of the result read

\[ \CI_{ijkl} = A_{ij}B_{kl}. \]

Parameters

A_ij	First tensor.
B_kl	Second tensor.

Returns

Resulting tensor after the dyadic product.

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fourthOrderIKJL()

template<typename AType , typename BType >

auto Ikarus::fourthOrderIKJL	(	const Eigen::MatrixBase< AType > &	A,
		const Eigen::MatrixBase< BType > &	B
	)

The components of the result read

\[ \CI_{ijkl} = A_{ik}B_{jl}, \]

which simply swaps the inner slots jand k

Template Parameters

AType	Type of the first matrix.
BType	Type of the second matrix.

Parameters

A	First matrix.
B	Second matrix.

Returns

Resulting tensor of the IKJL product.

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fromVoigt() [1/3]

template<typename ScalarType >

auto Ikarus::fromVoigt

(

const Eigen::Matrix< ScalarType, 6, 6 > &

CVoigt

)

Template Parameters

ScalarType

Scalar type of the matrix elements.

Parameters

CVoigt

Voigt notation matrix.

Returns

Fourth-order tensor corresponding to the matrix in Voigt notation.

This function converts a Voigt notation matrix to the corresponding 4th-order tensor. The function uses the fromVoigt function to map matrix indices to tensor indices. The resulting tensor is symmetric due to symmetry considerations.

fromVoigt() [2/3]

template<typename ST , int size, int Options, int maxSize>
requires ((size == 1 or size == 3 or size == 6) or ((maxSize == 1 or maxSize == 3 or maxSize == 6) and size == Eigen::Dynamic))

auto Ikarus::fromVoigt	(	const Eigen::Matrix< ST, size, 1, Options, maxSize, 1 > &	EVoigt,
		bool	isStrain = true
	)

Template Parameters

ST	Scalar type of the vector elements.
size	Size of the Voigt notation vector.

Parameters

EVoigt	Voigt notation vector.
isStrain	Flag indicating whether the vector represents a strain (default is true).

Returns

Matrix corresponding to the vector in Voigt notation.

This function converts a vector given in Voigt notation to the corresponding matrix. The conversion depends on the size The parameter isStrain is used to determine the conversion factor for off-diagonal components, which need to be divided by 2 in the matrix representation if the quantity is a strain tensor.

The function requires that the size of the Voigt notation vector is valid (1, 3, or 6).

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fromVoigt() [3/3]

constexpr std::array< size_t, 2 > Ikarus::fromVoigt ( size_t  i )

constexpr

Parameters

i	Voigt notation index.

Returns

Matrix indices corresponding to the Voigt notation index.

This function converts a Voigt notation index to the corresponding matrix indices. The mapping is based on the assumption that the Voigt notation indices 0, 1, and 2 represent the diagonal components 00, 11, and 22, respectively. The remaining Voigt notation indices (3, 4, and 5) correspond to the off-diagonal components (12 and 21, 02 and 20, 01 and 10).

The function asserts that the input index is within the valid range for Voigt notation (0 to 5).

identityFourthOrder()

template<typename ScalarType = double, int dim = 3>

auto Ikarus::identityFourthOrder

(

)

The components of the result read

\[ \CI_{ijkl} = \de_{ij}\de_{kl}. \]

Template Parameters

ScalarType	Type of the elements in the tensor.
dim	Dimension of the tensor.

Returns

Identity fourth-order tensor.

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symmetricFourthOrder()

template<typename ScalarType = double, int dim = 3>

auto Ikarus::symmetricFourthOrder	(	const auto &	A,
		const auto &	B
	)

The components of the result read

\[ \CI_{ijkl} = \frac{1}{2} \left(A_{ik}B_{jl}+A_{il}B_{jk} \right) . \]

Template Parameters

ScalarType	Type of the elements in the tensors.
dim	Dimension of the tensors.

Parameters

A	First tensor.
B	Second tensor.

Returns

Symmetric fourth-order tensor.

symmetricIdentityFourthOrder()

template<typename ScalarType = double, int dim = 3>

auto Ikarus::symmetricIdentityFourthOrder

(

)

Template Parameters

ScalarType	Type of the elements in the tensor.
dim	Dimension of the tensor.

Returns

Symmetric identity fourth-order tensor.

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symTwoSlots()

template<typename ScalarType , long int dim>

auto Ikarus::symTwoSlots	(	const Eigen::TensorFixedSize< ScalarType, Eigen::Sizes< dim, dim, dim, dim > > &	t,
		const std::array< size_t, 2 > &	slots
	)

Template Parameters

ScalarType

Type of the elements in the tensor.

Parameters

t	Input tensor.
slots	Indices of the slots to be swapped.

Returns

Fourth-order Tensor which is symmetric in the given slots.

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tensorView()

template<typename Derived , typename T , auto rank>

Eigen::Tensor< typename Derived::Scalar, rank > Ikarus::tensorView	(	const Eigen::EigenBase< Derived > &	matrix,
		const std::array< T, rank > &	dims
	)

Template Parameters

Derived	Type of the input Eigen matrix.
T	Type of the elements in the matrix.
rank	Rank of the resulting Tensor.

Parameters

matrix	Input Eigen matrix to be cast.
dims	Dimensions of the resulting Tensor.

Returns

Eigen::Tensor<typename Derived::Scalar, rank> The casted Eigen Tensor.

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toVoigt() [1/3]

template<typename ST , int size, int Options, int maxSize>
requires ((size > 0 and size <= 3) or (maxSize > 0 and maxSize <= 3 and size == Eigen::Dynamic))

auto Ikarus::toVoigt	(	const Eigen::Matrix< ST, size, size, Options, maxSize, maxSize > &	E,
		bool	isStrain = true
	)

Template Parameters

ST	Data type of the matrix elements.
size	Number of rows and columns of the square matrix.
Options	Eigen matrix options.

Parameters

E	Input matrix of size (size x size).
isStrain	Flag indicating whether the conversion is for strain (true) or not (false).

Returns

Vector with components in Voigt notation vector.

This function converts a square matrix to a Voigt notation vector, which contains the unique components of the input matrix. The mapping from the matrix indices to the Voigt notation indices is performed by the toVoigt function.

The optional isStrain parameter allows the user to specify whether the conversion is intended for strain calculations. If isStrain is true, the off-diagonal components are multiplied by 2, providing the correct Voigt notation for symmetric strain tensors.

toVoigt() [2/3]

template<typename ScalarType = double>

Eigen::Matrix< ScalarType, 6, 6 > Ikarus::toVoigt

(

const Eigen::TensorFixedSize< ScalarType, Eigen::Sizes< 3, 3, 3, 3 > > &

ft

)

Template Parameters

ScalarType

Data type of the tensor elements.

Parameters

ft	Fourth-order tensor .

Returns

Voigt notation matrix.

This function converts a fourth-order tensor to a Voigt notation matrix, which is a symmetric 6x6 matrix containing the unique components of the input tensor. The mapping from the tensor indices to the Voigt notation indices is performed by the toVoigt function.

Remarks

The current implementation does not take advantage of this symmetry.

toVoigt() [3/3]

constexpr Eigen::Index Ikarus::toVoigt ( Eigen::Index  i, Eigen::Index  j  )

constexprnoexcept

Parameters

i	Row index.
j	Column index.

Returns

Eigen::Index Voigt notation index.

This function converts 2D indices (i, j) to a Voigt notation index. The Voigt notation is used to represent the six unique components of a symmetric 3x3 matrix in a one-dimensional array.

If the input indices are not within the valid range (0, 1, 2), an assertion failure is triggered.

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