Tensors and tensor operations for a(n) array of 3d tensors of different rank, defined in a Cartesian coordinate system. More...

Namespaces
namespace	pointer
	API for individual tensors with pointer-only input.

Classes
class	Array
	Array of tensors: More...

Functions
array_type::tensor< double, 2 >	Random2 ()
	Random 2nd-order tensor (for example for use in testing). More...

array_type::tensor< double, 4 >	Random4 ()
	Random 4th-order tensor (for example for use in testing). More...

array_type::tensor< double, 2 >	O2 ()
	2nd-order null tensor (all components equal to zero). More...

array_type::tensor< double, 4 >	O4 ()
	4th-order null tensor (all components equal to zero). More...

array_type::tensor< double, 2 >	I2 ()
	2nd-order identity tensor. More...

array_type::tensor< double, 4 >	II ()
	Result of the dyadic product of two 2nd-order identity tensors (see I2()). More...

array_type::tensor< double, 4 >	I4 ()
	Fourth order unit tensor. More...

array_type::tensor< double, 4 >	I4rt ()
	Right-transposed fourth order unit tensor. More...

array_type::tensor< double, 4 >	I4s ()
	Fourth order symmetric projection. More...

array_type::tensor< double, 4 >	I4d ()
	Fourth order deviatoric projection. More...

template<class T >
auto	Trace (const T &A)
	Trace or 2nd-order tensor. More...

template<class T , class R >
void	trace (const T &A, R &ret)
	Same as Trace() but writes to externally allocated output. More...

template<class T >
auto	Hydrostatic (const T &A)
	Hydrostatic part of a tensor. More...

template<class T , class R >
void	hydrostatic (const T &A, R &ret)
	Same as Hydrostatic() but writes to externally allocated output. More...

template<class T >
auto	Det (const T &A)
	Determinant. More...

template<class T , class R >
void	det (const T &A, R &ret)
	Same as Det() but writes to externally allocated output. More...

template<class T >
auto	A2_ddot_B2 (const T &A, const T &B)
	Double tensor contraction. More...

template<class T , class R >
void	A2_ddot_B2 (const T &A, const T &B, R &ret)
	Same as A2_ddot_B2(const T& A, const T& B) but writes to externally allocated output. More...

template<class T >
auto	A2s_ddot_B2s (const T &A, const T &B)
	Same as A2_ddot_B2(const T& A, const T& B, R& ret) but for symmetric tensors. More...

template<class T , class R >
void	A2s_ddot_B2s (const T &A, const T &B, R &ret)
	Same as A2s_ddot_B2s(const T& A, const T& B) but writes to externally allocated output. More...

template<class T >
auto	Norm_deviatoric (const T &A)
	Norm of the tensor's deviator: More...

template<class T , class R >
void	norm_deviatoric (const T &A, R &ret)
	Same as Norm_deviatoric() but writes to externally allocated output. More...

template<class T >
auto	Deviatoric (const T &A)
	Deviatoric part of a tensor: More...

template<class T , class R >
void	deviatoric (const T &A, R &ret)
	Same as Deviatoric() but writes to externally allocated output. More...

template<class T >
auto	Sym (const T &A)
	Symmetric part of a tensor: More...

template<class T , class R >
void	sym (const T &A, R &ret)
	Same as Sym() but writes to externally allocated output. More...

template<class T >
auto	Inv (const T &A)
	Inverse. More...

template<class T , class R >
void	inv (const T &A, R &ret)
	Same as Inv() but writes to externally allocated output. More...

template<class T >
auto	Logs (const T &A)
	Logarithm. More...

template<class T , class R >
void	logs (const T &A, R &ret)
	Same as Logs() but writes to externally allocated output. More...

template<class T >
auto	A2_dot_A2T (const T &A)
	Dot-product (single tensor contraction) More...

template<class T , class R >
void	A2_dot_A2T (const T &A, R &ret)
	Same as A2_dot_A2T(const T& A) but writes to externally allocated output. More...

template<class T >
auto	A2_dot_B2 (const T &A, const T &B)
	Dot-product (single tensor contraction) More...

template<class T , class R >
void	A2_dot_B2 (const T &A, const T &B, R &ret)
	Same as A2_dot_B2(const T& A, const T& B) but writes to externally allocated output. More...

template<class T >
auto	A2_dyadic_B2 (const T &A, const T &B)
	Dyadic product. More...

template<class T , class R >
void	A2_dyadic_B2 (const T &A, const T &B, R &ret)
	Same as A2_dyadic_B2(const T& A, const T& B) but writes to externally allocated output. More...

template<class T , class U >
auto	A4_ddot_B2 (const T &A, const U &B)
	Double tensor contraction. More...

template<class T , class U , class R >
void	A4_ddot_B2 (const T &A, const U &B, R &ret)
	Same as A4_ddot_B2(const T& A, const U& B) but writes to externally allocated output. More...

template<class T , class U >
auto	A4_dot_B2 (const T &A, const U &B)
	Tensor contraction. More...

template<class T , class U , class R >
void	A4_dot_B2 (const T &A, const U &B, R &ret)
	Same as A4_dot_B2(const T& A, const U& B) but writes to externally allocated output. More...

template<class T >
size_t	underlying_size_A2 (const T &A)
	Size of the underlying array. More...

template<class T >
size_t	underlying_size_A4 (const T &A)
	Size of the underlying array. More...

template<class T >
auto	underlying_shape_A2 (const T &A) -> std::array< size_t, detail::impl_A2< T, 3 >::rank >
	Shape of the underlying array. More...

template<class T >
auto	underlying_shape_A4 (const T &A) -> std::array< size_t, detail::impl_A4< T, 3 >::rank >
	Shape of the underlying array. More...

Detailed Description

Tensors and tensor operations for a(n) array of 3d tensors of different rank, defined in a Cartesian coordinate system.

Function Documentation

◆ A2_ddot_B2() [1/2]

template<class T >

auto GMatTensor::Cartesian3d::A2_ddot_B2	(	const T &	A,
		const T &	B
	)

inline

Double tensor contraction.

\( c = A : B \)

or in index notation

\( c = A_{ij} A_{ji} \)

To write to allocated data use A2_ddot_B2(const T& A, const T& B, R& ret).

Parameters

A	[..., 3, 3] array.
B	[..., 3, 3] array.

Returns: [...] array.

Definition at line 1024 of file Cartesian3d.h.

◆ A2_ddot_B2() [2/2]

template<class T , class R >

void GMatTensor::Cartesian3d::A2_ddot_B2	(	const T &	A,
		const T &	B,
		R &	ret
	)

inline

Same as A2_ddot_B2(const T& A, const T& B) but writes to externally allocated output.

Parameters

A	[..., 3, 3] array.
B	[..., 3, 3] array.
ret	output [...] array.

Definition at line 1038 of file Cartesian3d.h.

◆ A2_dot_A2T() [1/2]

template<class T >

auto GMatTensor::Cartesian3d::A2_dot_A2T ( const T & A )

inline

Dot-product (single tensor contraction)

\( C = A \cdot A^T \)

or in index notation

\( C_{ik} = A_{ij} A_{kj} \)

To write to allocated data use A2_dot_A2T(const T& A, R& ret).

Parameters

A	[..., 3, 3] array.

Returns: [..., 3, 3] array.

Definition at line 1239 of file Cartesian3d.h.

◆ A2_dot_A2T() [2/2]

template<class T , class R >

void GMatTensor::Cartesian3d::A2_dot_A2T	(	const T &	A,
		R &	ret
	)

inline

Same as A2_dot_A2T(const T& A) but writes to externally allocated output.

Parameters

A	[..., 3, 3] array.
ret	output [..., 3, 3] array.

Definition at line 1252 of file Cartesian3d.h.

◆ A2_dot_B2() [1/2]

template<class T >

auto GMatTensor::Cartesian3d::A2_dot_B2	(	const T &	A,
		const T &	B
	)

inline

Dot-product (single tensor contraction)

\( C = A \cdot B \)

or in index notation

\( C_{ik} = A_{ij} B_{jk} \)

To write to allocated data use A2_dot_B2(const T& A, const T& B, R& ret).

Parameters

A	[..., 3, 3] array.
B	[..., 3, 3] array.

Returns: [..., 3, 3] array.

Definition at line 1274 of file Cartesian3d.h.

◆ A2_dot_B2() [2/2]

template<class T , class R >

void GMatTensor::Cartesian3d::A2_dot_B2	(	const T &	A,
		const T &	B,
		R &	ret
	)

inline

Same as A2_dot_B2(const T& A, const T& B) but writes to externally allocated output.

Parameters

A	[..., 3, 3] array.
B	[..., 3, 3] array.
ret	output [..., 3, 3] array.

Definition at line 1289 of file Cartesian3d.h.

◆ A2_dyadic_B2() [1/2]

template<class T >

auto GMatTensor::Cartesian3d::A2_dyadic_B2	(	const T &	A,
		const T &	B
	)

inline

Dyadic product.

\( C = A \otimes B \)

or in index notation

\( C_{ijkl} = A_{ij} B_{kl} \)

To write to allocated data use A2_dyadic_B2(const T& A, const T& B, R& ret).

Parameters

A	[..., 3, 3] array.
B	[..., 3, 3] array.

Returns: [..., 3, 3, 3, 3] array.

Definition at line 1312 of file Cartesian3d.h.

◆ A2_dyadic_B2() [2/2]

template<class T , class R >

void GMatTensor::Cartesian3d::A2_dyadic_B2	(	const T &	A,
		const T &	B,
		R &	ret
	)

inline

Same as A2_dyadic_B2(const T& A, const T& B) but writes to externally allocated output.

Parameters

A	[..., 3, 3] array.
B	[..., 3, 3] array.
ret	output [..., 3, 3, 3, 3] array.

Definition at line 1327 of file Cartesian3d.h.

◆ A2s_ddot_B2s() [1/2]

template<class T >

auto GMatTensor::Cartesian3d::A2s_ddot_B2s	(	const T &	A,
		const T &	B
	)

inline

Same as A2_ddot_B2(const T& A, const T& B, R& ret) but for symmetric tensors.

This function is slightly faster. There is no assertion to check the symmetry. To write to allocated data use A2s_ddot_B2s(const T& A, const T& B, R& ret).

Parameters

A	[..., 3, 3] array.
B	[..., 3, 3] array.

Returns: [...] array.

Definition at line 1055 of file Cartesian3d.h.

◆ A2s_ddot_B2s() [2/2]

template<class T , class R >

void GMatTensor::Cartesian3d::A2s_ddot_B2s	(	const T &	A,
		const T &	B,
		R &	ret
	)

inline

Same as A2s_ddot_B2s(const T& A, const T& B) but writes to externally allocated output.

Parameters

A	[..., 3, 3] array.
B	[..., 3, 3] array.
ret	output [...] array.

Definition at line 1069 of file Cartesian3d.h.

◆ A4_ddot_B2() [1/2]

template<class T , class U >

auto GMatTensor::Cartesian3d::A4_ddot_B2	(	const T &	A,
		const U &	B
	)

inline

Double tensor contraction.

\( C = A : B \)

or in index notation

\( C_{ij} = A_{ijkl} A_{lk} \)

To write to allocated data use A4_ddot_B2(const T& A, const U& B, R& ret).

Parameters

A	[..., 3, 3, 3, 3] array.
B	[..., 3, 3] array.

Returns: [..., 3, 3] array.

Definition at line 1350 of file Cartesian3d.h.

◆ A4_ddot_B2() [2/2]

template<class T , class U , class R >

void GMatTensor::Cartesian3d::A4_ddot_B2	(	const T &	A,
		const U &	B,
		R &	ret
	)

inline

Same as A4_ddot_B2(const T& A, const U& B) but writes to externally allocated output.

Parameters

A	[..., 3, 3, 3, 3] array.
B	[..., 3, 3] array.
ret	output [..., 3, 3] array.

Definition at line 1365 of file Cartesian3d.h.

◆ A4_dot_B2() [1/2]

template<class T , class U >

auto GMatTensor::Cartesian3d::A4_dot_B2	(	const T &	A,
		const U &	B
	)

inline

Tensor contraction.

\( C = A \cdot B \)

or in index notation

\( C_{ijkm} = A_{ijkl} A_{lm} \)

To write to allocated data use A4_dot_B2(const T& A, const U& B, R& ret).

Parameters

A	[..., 3, 3, 3, 3] array.
B	[..., 3, 3] array.

Returns: [..., 3, 3, 3, 3] array.

Definition at line 1388 of file Cartesian3d.h.

◆ A4_dot_B2() [2/2]

template<class T , class U , class R >

void GMatTensor::Cartesian3d::A4_dot_B2	(	const T &	A,
		const U &	B,
		R &	ret
	)

inline

Same as A4_dot_B2(const T& A, const U& B) but writes to externally allocated output.

Parameters

A	[..., 3, 3, 3, 3] array.
B	[..., 3, 3] array.
ret	output [..., 3, 3, 3, 3] array.

Definition at line 1403 of file Cartesian3d.h.

◆ Det()

template<class T >

auto GMatTensor::Cartesian3d::Det ( const T & A )

inline

Determinant.

To write to allocated output use det().

Parameters

A	[..., 3, 3] array.

Returns: [...] array.

Definition at line 991 of file Cartesian3d.h.

◆ det()

template<class T , class R >

void GMatTensor::Cartesian3d::det	(	const T &	A,
		R &	ret
	)

inline

Same as Det() but writes to externally allocated output.

Parameters

A	[..., 3, 3] array.
ret	output [...] array.

Definition at line 1003 of file Cartesian3d.h.

◆ Deviatoric()

template<class T >

auto GMatTensor::Cartesian3d::Deviatoric ( const T & A )

inline

Deviatoric part of a tensor:

A - Hydrostatic(A) * I2

See Hydrostatic(). To write to allocated data use deviatoric().

Parameters

A	[..., 3, 3] array.

Returns: [..., 3, 3] array.

Definition at line 1116 of file Cartesian3d.h.

◆ deviatoric()

template<class T , class R >

void GMatTensor::Cartesian3d::deviatoric	(	const T &	A,
		R &	ret
	)

inline

Same as Deviatoric() but writes to externally allocated output.

Parameters

A	[..., 3, 3] array.
ret	output [..., 3, 3] array.

Definition at line 1129 of file Cartesian3d.h.

◆ Hydrostatic()

template<class T >

auto GMatTensor::Cartesian3d::Hydrostatic ( const T & A )

inline

Hydrostatic part of a tensor.

== trace(A) / 3 == trace(A) / d

where d = 3. To write to allocated output use hydrostatic().

Parameters

A	[..., 3, 3] array.

Returns: [...] array.

Definition at line 966 of file Cartesian3d.h.

◆ hydrostatic()

template<class T , class R >

void GMatTensor::Cartesian3d::hydrostatic	(	const T &	A,
		R &	ret
	)

inline

Same as Hydrostatic() but writes to externally allocated output.

Parameters

A	[..., 3, 3] array.
ret	output [...] array.

Definition at line 978 of file Cartesian3d.h.

◆ I2()

array_type::tensor< double, 2 > GMatTensor::Cartesian3d::I2 ( )

inline

2nd-order identity tensor.

By definition

\( I_{ij} = \delta_{ij} \)

such that

\( I \cdot A = A \)

or in index notation

\( I_{ij} A_{jk} = A_{ik} \)

See A2_dot_B2().

Returns: [3, 3] array.

Definition at line 798 of file Cartesian3d.h.

◆ I4()

array_type::tensor< double, 4 > GMatTensor::Cartesian3d::I4 ( )

inline

Fourth order unit tensor.

By definition

\( I_{ijkl} = \delta_{il} \delta_{jk} \)

such that

\( I : A = A \)

or in index notation

\( I_{ijkl} A_{lk} = A_{ij} \)

See A4_ddot_B2().

Returns: [3, 3, 3, 3] array.

Definition at line 848 of file Cartesian3d.h.

◆ I4d()

array_type::tensor< double, 4 > GMatTensor::Cartesian3d::I4d ( )

inline

Fourth order deviatoric projection.

By definition

I = I4s() - 1.0 / 3.0 * II()

such that

\( I : A = sym(A) - tr(A) / 3 \)

See A4_ddot_B2(), Deviatoric().

Returns: [3, 3, 3, 3] array.

Definition at line 919 of file Cartesian3d.h.

◆ I4rt()

array_type::tensor< double, 4 > GMatTensor::Cartesian3d::I4rt ( )

inline

Right-transposed fourth order unit tensor.

By definition

\( I_{ijkl} = \delta_{ik} \delta_{jl} \)

such that

\( I : A = A^T \)

or in index notation

\( I_{ijkl} A_{lk} = A_{ji} \)

See A4_ddot_B2().

Returns: [3, 3, 3, 3] array.

Definition at line 873 of file Cartesian3d.h.

◆ I4s()

array_type::tensor< double, 4 > GMatTensor::Cartesian3d::I4s ( )

inline

Fourth order symmetric projection.

By definition

I = 0.5 * (I4() + I4rt())

such that

\( I : A = sym(A) \)

or in index notation

\( I_{ijkl} A_{lk} = (A_{ij} + A_{ji}) / 2 \)

See A4_ddot_B2(), Sym().

Returns: [3, 3, 3, 3] array.

Definition at line 898 of file Cartesian3d.h.

◆ II()

array_type::tensor< double, 4 > GMatTensor::Cartesian3d::II ( )

inline

Result of the dyadic product of two 2nd-order identity tensors (see I2()).

By definition

\( (II)_{ijkl} = \delta_{ij} \delta_{kl} \)

such that

\( II : A = tr(A) I \)

or in index notation

\( (II)_{ijkl} A_{lk} = tr(A) I_{ij} \)

See A4_ddot_B2(), Trace(), I2().

Returns: [3, 3, 3, 3] array.

Definition at line 823 of file Cartesian3d.h.

◆ Inv()

template<class T >

auto GMatTensor::Cartesian3d::Inv ( const T & A )

inline

Inverse.

To write to allocated output use inv().

Parameters

A	[..., 3, 3] array.

Returns: [..., 3, 3] array.

Definition at line 1177 of file Cartesian3d.h.

◆ inv()

template<class T , class R >

void GMatTensor::Cartesian3d::inv	(	const T &	A,
		R &	ret
	)

inline

Same as Inv() but writes to externally allocated output.

Parameters

A	[..., 3, 3] array.
ret	output [..., 3, 3] array.

Definition at line 1190 of file Cartesian3d.h.

◆ Logs()

template<class T >

auto GMatTensor::Cartesian3d::Logs ( const T & A )

inline

Logarithm.

Symmetric tensors only, no assertion. To write to allocated output use logs().

Parameters

A	[..., 3, 3] array.

Returns: [..., 3, 3] array.

Definition at line 1205 of file Cartesian3d.h.

◆ logs()

template<class T , class R >

void GMatTensor::Cartesian3d::logs	(	const T &	A,
		R &	ret
	)

inline

Same as Logs() but writes to externally allocated output.

Parameters

A	[..., 3, 3] array.
ret	output [..., 3, 3] array, may be the same reference as `A`.

Definition at line 1218 of file Cartesian3d.h.

◆ Norm_deviatoric()

template<class T >

auto GMatTensor::Cartesian3d::Norm_deviatoric ( const T & A )

inline

Norm of the tensor's deviator:

\( \sqrt{(dev(A))_{ij} (dev(A))_{ji}} \)

To write to allocated data use norm_deviatoric().

Parameters

A	[..., 3, 3] array.

Returns: [...] array.

Definition at line 1086 of file Cartesian3d.h.

◆ norm_deviatoric()

template<class T , class R >

void GMatTensor::Cartesian3d::norm_deviatoric	(	const T &	A,
		R &	ret
	)

inline

Same as Norm_deviatoric() but writes to externally allocated output.

Parameters

A	[..., 3, 3] array.
ret	output [...] array

Definition at line 1099 of file Cartesian3d.h.

◆ O2()

array_type::tensor< double, 2 > GMatTensor::Cartesian3d::O2 ( )

inline

2nd-order null tensor (all components equal to zero).

Returns: [3, 3] array.

Definition at line 765 of file Cartesian3d.h.

◆ O4()

array_type::tensor< double, 4 > GMatTensor::Cartesian3d::O4 ( )

inline

4th-order null tensor (all components equal to zero).

Returns: [3, 3, 3, 3] array.

Definition at line 775 of file Cartesian3d.h.

◆ Random2()

array_type::tensor< double, 2 > GMatTensor::Cartesian3d::Random2 ( )

inline

Random 2nd-order tensor (for example for use in testing).

Returns: [3, 3] array.

Definition at line 743 of file Cartesian3d.h.

◆ Random4()

array_type::tensor< double, 4 > GMatTensor::Cartesian3d::Random4 ( )

inline

Random 4th-order tensor (for example for use in testing).

Returns: [3, 3, 3, 3] array.

Definition at line 754 of file Cartesian3d.h.

◆ Sym()

template<class T >

auto GMatTensor::Cartesian3d::Sym ( const T & A )

inline

Symmetric part of a tensor:

\( (A + A^T) / 2 \)

of in index notation

\( (A_{ij} + A_{ji}) / 2 \)

To write to allocated data use sym().

Parameters

A	[..., 3, 3] array.

Returns: [..., 3, 3] array.

Definition at line 1150 of file Cartesian3d.h.

◆ sym()

template<class T , class R >

void GMatTensor::Cartesian3d::sym	(	const T &	A,
		R &	ret
	)

inline

Same as Sym() but writes to externally allocated output.

Parameters

A	[..., 3, 3] array.
ret	output [..., 3, 3] array, may be the same reference as `A`.

Definition at line 1163 of file Cartesian3d.h.

◆ Trace()

template<class T >

auto GMatTensor::Cartesian3d::Trace ( const T & A )

inline

Trace or 2nd-order tensor.

\( tr(A) = A_{ii} \)

To write to allocated data use trace().

Parameters

A	[..., 3, 3] array.

Returns: [...] array.

Definition at line 937 of file Cartesian3d.h.

◆ trace()

template<class T , class R >

void GMatTensor::Cartesian3d::trace	(	const T &	A,
		R &	ret
	)

inline

Same as Trace() but writes to externally allocated output.

Parameters

A	[..., 3, 3] array.
ret	output [...] array.

Definition at line 949 of file Cartesian3d.h.

◆ underlying_shape_A2()

template<class T >

auto GMatTensor::Cartesian3d::underlying_shape_A2 ( const T & A ) -> std::array<size_t, detail::impl_A2<T, 3>::rank>

inline

Shape of the underlying array.

Parameters

A	[..., 2, 2] array.

Returns: [...].

Definition at line 1441 of file Cartesian3d.h.

◆ underlying_shape_A4()

template<class T >

auto GMatTensor::Cartesian3d::underlying_shape_A4 ( const T & A ) -> std::array<size_t, detail::impl_A4<T, 3>::rank>

inline

Shape of the underlying array.

Parameters

A	[..., 2, 2] array.

Returns: [...].

Definition at line 1453 of file Cartesian3d.h.

◆ underlying_size_A2()

template<class T >

size_t GMatTensor::Cartesian3d::underlying_size_A2 ( const T & A )

inline

Size of the underlying array.

Parameters

A	[..., 2, 2] array.

Returns: prod([...]).

Definition at line 1417 of file Cartesian3d.h.

◆ underlying_size_A4()

template<class T >

size_t GMatTensor::Cartesian3d::underlying_size_A4 ( const T & A )

inline

Size of the underlying array.

Parameters

A	[..., 2, 2] array.

Returns: prod([...]).

Definition at line 1429 of file Cartesian3d.h.

Namespaces

Classes

Functions

Detailed Description

Function Documentation

◆ A2_ddot_B2() [1/2]

◆ A2_ddot_B2() [2/2]

◆ A2_dot_A2T() [1/2]

◆ A2_dot_A2T() [2/2]

◆ A2_dot_B2() [1/2]

◆ A2_dot_B2() [2/2]

◆ A2_dyadic_B2() [1/2]

◆ A2_dyadic_B2() [2/2]

◆ A2s_ddot_B2s() [1/2]

◆ A2s_ddot_B2s() [2/2]

◆ A4_ddot_B2() [1/2]

◆ A4_ddot_B2() [2/2]

◆ A4_dot_B2() [1/2]

◆ A4_dot_B2() [2/2]

◆ Det()

◆ det()

◆ Deviatoric()

◆ deviatoric()

◆ Hydrostatic()

◆ hydrostatic()

◆ I2()

◆ I4()

◆ I4d()

◆ I4rt()

◆ I4s()

◆ II()

◆ Inv()

◆ inv()

◆ Logs()

◆ logs()

◆ Norm_deviatoric()

◆ norm_deviatoric()

◆ O2()

◆ O4()

◆ Random2()

◆ Random4()

◆ Sym()

◆ sym()

◆ Trace()

◆ trace()

◆ underlying_shape_A2()

◆ underlying_shape_A4()

◆ underlying_size_A2()

◆ underlying_size_A4()