Namespaces
namespace	Frame
	Indicates the Frame that a TensorIndexType is in.

namespace	TensorMetafunctions
	Contains all metafunctions related to Tensor manipulations.

namespace	tnsr
	Type aliases to construct common Tensors.

Classes
struct	Frame::FrameIsPhysical
	Marks a Frame as being "physical" in the sense that it is meaningful to evaluate an analytic solution in that frame. More...

struct	tt::is_tensor_index_type< T >
	Inherits from std::true_type if T is a TensorIndexType. More...

class	Tensor< X, Symm, IndexList< Indices... > >
	Represents an object with multiple components. More...

Typedefs
template<typename CheckFrame >
using	Frame::is_frame_physical = std::integral_constant< bool, std::is_base_of< FrameIsPhysical, CheckFrame >::value >
	Returns std::true_type if the frame is "physical" in the sense that it is meaningful to evaluate an analytic solution in that frame. More...

template<size_t SpatialDim, UpLo Ul, typename Fr >
using	SpatialIndex = Tensor_detail::TensorIndexType< SpatialDim, Ul, Fr, IndexType::Spatial >
	A SpatialIndex holds information about the number of spatial dimensions, whether the index is covariant or contravariant (UpLo), and the Frame the index is in. More...

template<size_t SpatialDim, UpLo Ul, typename Fr >
using	SpacetimeIndex = Tensor_detail::TensorIndexType< SpatialDim, Ul, Fr, IndexType::Spacetime >
	A SpacetimeIndex holds information about the number of spatial dimensions, whether the index is covariant or contravariant (UpLo), and the Frame the index is in. More...

template<typename Index >
using	change_index_up_lo = Tensor_detail::TensorIndexType< Index::index_type==IndexType::Spatial ? Index::value :Index::value - 1, Index::ul==UpLo::Up ? UpLo::Lo :UpLo::Up, typename Index::Frame, Index::index_type >
	Change the TensorIndexType to be covariant if it's contravariant and vice-versa. More...

template<typename Index >
using	change_index_type = Tensor_detail::TensorIndexType< Index::index_type==IndexType::Spatial ? Index::dim :Index::dim - 1, Index::ul, typename Index::Frame, Index::index_type==IndexType::Spatial ? IndexType::Spacetime :IndexType::Spatial >
	Change the TensorIndexType to be spacetime if it's spatial and vice versa. More...

template<typename Symm , typename... IndexPack>
using	TensorMetafunctions::check_index_symmetry = typename detail::check_index_symmetry_impl< tmpl::size< Symm >::value==0 or tmpl::size< Symm >::value==1 ? 0 :2 >::template f< Symm, tmpl::map<>, IndexPack... >
	Check that each of symmetric indices is in the same frame and have the same dimensionality.

template<typename Tensor , std::size_t VolumeDim, UpLo Ul, typename Fr = Frame::Grid>
using	TensorMetafunctions::prepend_spatial_index = ::Tensor< typename Tensor::type, tmpl::push_front< typename Tensor::symmetry, tmpl::int32_t< 1+tmpl::fold< typename Tensor::symmetry, tmpl::int32_t< 0 >, tmpl::max< tmpl::_state, tmpl::_element > >::value > >, tmpl::push_front< typename Tensor::index_list, SpatialIndex< VolumeDim, Ul, Fr > > >
	Add a spatial index to the front of a Tensor. More...

template<typename Tensor , std::size_t VolumeDim, UpLo Ul, typename Fr = Frame::Grid>
using	TensorMetafunctions::prepend_spacetime_index = ::Tensor< typename Tensor::type, tmpl::push_front< typename Tensor::symmetry, tmpl::int32_t< 1+tmpl::fold< typename Tensor::symmetry, tmpl::int32_t< 0 >, tmpl::max< tmpl::_state, tmpl::_element > >::value > >, tmpl::push_front< typename Tensor::index_list, SpacetimeIndex< VolumeDim, Ul, Fr > > >
	Add a spacetime index to the front of a Tensor. More...

template<typename Tensor >
using	TensorMetafunctions::remove_first_index = ::Tensor< typename Tensor::type, tmpl::pop_front< typename Tensor::symmetry >, tmpl::pop_front< typename Tensor::index_list > >
	remove the first index of a tensor More...

template<typename NewType , typename Tensor >
using	TensorMetafunctions::swap_type = ::Tensor< NewType, typename Tensor::symmetry, typename Tensor::index_list >
	Swap the data type of a tensor for a new type. More...

template<typename Tensor , typename Frame >
using	TensorMetafunctions::any_index_in_frame = tmpl::any< typename Tensor::index_list, tmpl::bind< detail::frame_is_the_same, tmpl::_1, Frame > >
	Return tmpl::true_type if any indices of the Tensor are in the frame Frame.

template<std::int32_t... T>
using	Symmetry = typename detail::SymmetryImpl< std::make_index_sequence< sizeof...(T)>, tmpl::integral_list< std::int32_t, T... > >::type
	Computes the canonical symmetry from the integers `T` More...

template<typename T >
using	Scalar = Tensor< T, Symmetry<>, index_list<> >
	Scalar type.

Enumerations
enum class	UpLo { UpLo::Up , UpLo::Lo }
	Whether a TensorIndexType is covariant or contravariant. More...

enum class	IndexType : char { IndexType::Spatial , IndexType::Spacetime }
	Indicates whether the TensorIndexType is Spatial or Spacetime. More...

Functions
template<typename DataType , typename Index >
Tensor< DataType, Symmetry< 1 >, index_list< Index > >	cross_product (const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &vector_a, const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &vector_b) noexcept
	Compute the Euclidean cross product of two vectors or one forms. More...

template<typename DataType , typename Index >
Tensor< DataType, Symmetry< 1 >, index_list< change_index_up_lo< Index > > >	cross_product (const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &vector_a, const Tensor< DataType, Symmetry< 1 >, index_list< change_index_up_lo< Index > > > &vector_b) noexcept
	Compute the Euclidean cross product of a vector and a one form. More...

template<typename DataType , typename Index >
Tensor< DataType, Symmetry< 1 >, index_list< Index > >	cross_product (const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &vector_a, const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &vector_b, const Tensor< DataType, Symmetry< 1, 1 >, index_list< Index, Index > > &metric_or_inverse_metric, const Scalar< DataType > &metric_determinant) noexcept
	Compute the cross product of two vectors or one forms. More...

template<typename DataType , typename Index >
Tensor< DataType, Symmetry< 1 >, index_list< change_index_up_lo< Index > > >	cross_product (const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &vector_a, const Tensor< DataType, Symmetry< 1 >, index_list< change_index_up_lo< Index > > > &vector_b, const Tensor< DataType, Symmetry< 1, 1 >, index_list< Index, Index > > &metric_or_inverse_metric, const Scalar< DataType > &metric_determinant) noexcept
	Compute the cross product of a vector and a one form. More...

template<typename DataType , typename Symm , typename IndexList >
double	l2_norm (const Tensor< DataType, Symm, IndexList > &tensor) noexcept
	Compute Euclidean \(L^2\)-norm of arbitrary Tensors reduced over an element. More...

template<size_t Dim, typename DataType >
tnsr::Ij< DataType, Dim, Frame::NoFrame >	identity (const DataType &used_for_type) noexcept
	returns the Identity matrix

template<typename Fr >
std::string	Frame::prefix () noexcept
	The frame-dependent prefix used when constructing the string returned by the name function of a tag. More...

template<int I, class... Ts>
constexpr size_t	index_dim (const Tensor< Ts... > &) noexcept
	Get dimensionality of i'th tensor index. More...

Variables
template<typename CheckFrame >
constexpr bool	Frame::is_frame_physical_v = is_frame_physical<CheckFrame>::value
	Returns true if the frame is "physical" in the sense that it is meaningful to evaluate an analytic solution in that frame. More...

template<typename Tensor , typename Frame >
constexpr bool	TensorMetafunctions::any_index_in_frame_v = any_index_in_frame<Tensor, Frame>::value
	Return true if any indices of the Tensor are in the frame Frame.

template<typename T , typename Symm , typename Index0 , typename Index1 >
void	determinant (const gsl::not_null< Scalar< T > * > det_tensor, const Tensor< T, Symm, index_list< Index0, Index1 > > &tensor) noexcept
	Computes the determinant of a rank-2 Tensor `tensor`. More...

template<typename T , typename Symm , typename Index0 , typename Index1 >
Scalar< T >	determinant (const Tensor< T, Symm, index_list< Index0, Index1 > > &tensor) noexcept
	Computes the determinant of a rank-2 Tensor `tensor`. More...

template<typename T , typename Symm , typename Index0 , typename Index1 >
void	determinant_and_inverse (const gsl::not_null< Scalar< T > * > det, const gsl::not_null< Tensor< T, Symm, tmpl::list< change_index_up_lo< Index1 >, change_index_up_lo< Index0 > > > * > inv, const Tensor< T, Symm, tmpl::list< Index0, Index1 > > &tensor) noexcept
	Computes the determinant and inverse of a rank-2 Tensor. More...

template<typename T , typename Symm , typename Index0 , typename Index1 >
auto	determinant_and_inverse (const Tensor< T, Symm, tmpl::list< Index0, Index1 > > &tensor) noexcept -> std::pair< Scalar< T >, Tensor< T, Symm, tmpl::list< change_index_up_lo< Index1 >, change_index_up_lo< Index0 > > > >
	Computes the determinant and inverse of a rank-2 Tensor. More...

template<typename DetTag , typename InvTag , typename T , typename Symm , typename Index0 , typename Index1 >
void	determinant_and_inverse (const gsl::not_null< Variables< tmpl::list< DetTag, InvTag > > * > det_and_inv, const Tensor< T, Symm, tmpl::list< Index0, Index1 > > &tensor) noexcept
	Computes the determinant and inverse of a rank-2 Tensor. More...

template<typename DetTag , typename InvTag , typename T , typename Symm , typename Index0 , typename Index1 >
auto	determinant_and_inverse (const Tensor< T, Symm, tmpl::list< Index0, Index1 > > &tensor) noexcept -> Variables< tmpl::list< DetTag, InvTag > >
	Computes the determinant and inverse of a rank-2 Tensor. More...

template<typename DataType , typename Index >
void	dot_product (const gsl::not_null< Scalar< DataType > * > dot_product, const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &vector_a, const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &vector_b) noexcept
	Compute the Euclidean dot product of two vectors or one forms. More...

template<typename DataType , typename Index >
Scalar< DataType >	dot_product (const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &vector_a, const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &vector_b) noexcept
	Compute the Euclidean dot product of two vectors or one forms. More...

template<typename DataType , typename Index >
void	dot_product (const gsl::not_null< Scalar< DataType > * > dot_product, const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &vector_a, const Tensor< DataType, Symmetry< 1 >, index_list< change_index_up_lo< Index > > > &vector_b) noexcept
	Compute the dot product of a vector and a one form. More...

template<typename DataType , typename Index >
Scalar< DataType >	dot_product (const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &vector_a, const Tensor< DataType, Symmetry< 1 >, index_list< change_index_up_lo< Index > > > &vector_b) noexcept
	Compute the dot product of a vector and a one form. More...

template<typename DataType , typename Index >
void	dot_product (const gsl::not_null< Scalar< DataType > * > dot_product, const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &vector_a, const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &vector_b, const Tensor< DataType, Symmetry< 1, 1 >, index_list< change_index_up_lo< Index >, change_index_up_lo< Index > > > &metric) noexcept
	Compute the dot_product of two vectors or one forms. More...

template<typename DataType , typename Index >
Scalar< DataType >	dot_product (const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &vector_a, const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &vector_b, const Tensor< DataType, Symmetry< 1, 1 >, index_list< change_index_up_lo< Index >, change_index_up_lo< Index > > > &metric) noexcept
	Compute the dot_product of two vectors or one forms. More...

template<typename DataType , typename Index >
Scalar< DataType >	magnitude (const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &vector) noexcept
	Compute the Euclidean magnitude of a rank-1 tensor. More...

template<typename DataType , typename Index >
void	magnitude (const gsl::not_null< Scalar< DataType > * > magnitude, const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &vector) noexcept
	Compute the Euclidean magnitude of a rank-1 tensor. More...

template<typename DataType , typename Index >
Scalar< DataType >	magnitude (const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &vector, const Tensor< DataType, Symmetry< 1, 1 >, index_list< change_index_up_lo< Index >, change_index_up_lo< Index > > > &metric) noexcept
	Compute the magnitude of a rank-1 tensor. More...

template<typename DataType , typename Index >
void	magnitude (const gsl::not_null< Scalar< DataType > * > magnitude, const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &vector, const Tensor< DataType, Symmetry< 1, 1 >, index_list< change_index_up_lo< Index >, change_index_up_lo< Index > > > &metric) noexcept
	Compute the magnitude of a rank-1 tensor. More...

template<typename DataType >
Scalar< DataType >	sqrt_magnitude (const Scalar< DataType > &input) noexcept
	Compute square root of the Euclidean magnitude of a rank-0 tensor. More...

template<typename DataType >
void	sqrt_magnitude (const gsl::not_null< Scalar< DataType > * > sqrt_magnitude, const Scalar< DataType > &input) noexcept
	Compute square root of the Euclidean magnitude of a rank-0 tensor. More...

template<typename DataType , typename Symm , typename IndexList >
Scalar< DataType >	pointwise_l2_norm (const Tensor< DataType, Symm, IndexList > &tensor) noexcept
	Compute point-wise Euclidean \(L^2\)-norm of arbitrary Tensors. More...

template<typename DataType , typename Symm , typename IndexList >
void	pointwise_l2_norm (const gsl::not_null< Scalar< DataType > * > norm, const Tensor< DataType, Symm, IndexList > &tensor) noexcept
	Compute point-wise Euclidean \(L^2\)-norm of arbitrary Tensors. More...

template<typename DataType , size_t VolumeDim, typename Frame >
void	orthonormal_oneform (gsl::not_null< tnsr::i< DataType, VolumeDim, Frame > * > orthonormal_form, const tnsr::i< DataType, VolumeDim, Frame > &unit_form, const tnsr::II< DataType, VolumeDim, Frame > &inv_spatial_metric) noexcept
	Compute a spatial one-form orthonormal to the given unit form. More...

template<typename DataType , size_t VolumeDim, typename Frame >
tnsr::i< DataType, VolumeDim, Frame >	orthonormal_oneform (const tnsr::i< DataType, VolumeDim, Frame > &unit_form, const tnsr::II< DataType, VolumeDim, Frame > &inv_spatial_metric) noexcept
	Compute a spatial one-form orthonormal to the given unit form. More...

template<typename DataType , typename Frame >
void	orthonormal_oneform (gsl::not_null< tnsr::i< DataType, 3, Frame > * > orthonormal_form, const tnsr::i< DataType, 3, Frame > &first_unit_form, const tnsr::i< DataType, 3, Frame > &second_unit_form, const tnsr::ii< DataType, 3, Frame > &spatial_metric, const Scalar< DataType > &det_spatial_metric) noexcept
	Compute a spatial one-form orthonormal to two given unit forms. More...

template<typename DataType , typename Frame >
tnsr::i< DataType, 3, Frame >	orthonormal_oneform (const tnsr::i< DataType, 3, Frame > &first_unit_form, const tnsr::i< DataType, 3, Frame > &second_unit_form, const tnsr::ii< DataType, 3, Frame > &spatial_metric, const Scalar< DataType > &det_spatial_metric) noexcept
	Compute a spatial one-form orthonormal to two given unit forms. More...

Detailed Description

Tensor use documentation.

Typedef Documentation

◆ change_index_type

template<typename Index >

using change_index_type = Tensor_detail::TensorIndexType< Index::index_type == IndexType::Spatial ? Index::dim : Index::dim - 1, Index::ul, typename Index::Frame, Index::index_type == IndexType::Spatial ? IndexType::Spacetime : IndexType::Spatial>

Change the TensorIndexType to be spacetime if it's spatial and vice versa.

Template Parameters

Index the TensorIndexType to change

◆ change_index_up_lo

template<typename Index >

using change_index_up_lo = Tensor_detail::TensorIndexType< Index::index_type == IndexType::Spatial ? Index::value : Index::value - 1, Index::ul == UpLo::Up ? UpLo::Lo : UpLo::Up, typename Index::Frame, Index::index_type>

Change the TensorIndexType to be covariant if it's contravariant and vice-versa.

Here is an example of how to use change_index_up_lo

using Index = SpatialIndex<3, UpLo::Lo, Frame::Grid>;
using UpIndex = change_index_up_lo<Index>;
static_assert(std::is_same_v<UpIndex, SpatialIndex<3, UpLo::Up, Frame::Grid>>,
              "Failed testing change_index_up_lo");

Template Parameters

Index the TensorIndexType to change

◆ is_frame_physical

template<typename CheckFrame >

using Frame::is_frame_physical = typedef std::integral_constant<bool, std::is_base_of<FrameIsPhysical, CheckFrame>::value>

Returns std::true_type if the frame is "physical" in the sense that it is meaningful to evaluate an analytic solution in that frame.

Example

static_assert(not Frame::is_frame_physical_v<Frame::Logical>,
              "Failed testing Frame::is_frame_physical");
static_assert(not Frame::is_frame_physical_v<Frame::Distorted>,
              "Failed testing Frame::is_frame_physical");
static_assert(not Frame::is_frame_physical_v<Frame::Grid>,
              "Failed testing Frame::is_frame_physical");
static_assert(Frame::is_frame_physical_v<Frame::Inertial>,
              "Failed testing Frame::is_frame_physical");

◆ prepend_spacetime_index

template<typename Tensor , std::size_t VolumeDim, UpLo Ul, typename Fr = Frame::Grid>

using TensorMetafunctions::prepend_spacetime_index = typedef ::Tensor< typename Tensor::type, tmpl::push_front< typename Tensor::symmetry, tmpl::int32_t< 1 + tmpl::fold<typename Tensor::symmetry, tmpl::int32_t<0>, tmpl::max<tmpl::_state, tmpl::_element> >::value> >, tmpl::push_front<typename Tensor::index_list, SpacetimeIndex<VolumeDim, Ul, Fr> >>

Add a spacetime index to the front of a Tensor.

Template Parameters

Tensor	the tensor type to which the new index is prepended
VolumeDim	the volume dimension of the tensor index to prepend
Fr	the Frame of the tensor index to prepend

◆ prepend_spatial_index

template<typename Tensor , std::size_t VolumeDim, UpLo Ul, typename Fr = Frame::Grid>

using TensorMetafunctions::prepend_spatial_index = typedef ::Tensor< typename Tensor::type, tmpl::push_front< typename Tensor::symmetry, tmpl::int32_t< 1 + tmpl::fold<typename Tensor::symmetry, tmpl::int32_t<0>, tmpl::max<tmpl::_state, tmpl::_element> >::value> >, tmpl::push_front<typename Tensor::index_list, SpatialIndex<VolumeDim, Ul, Fr> >>

Add a spatial index to the front of a Tensor.

Template Parameters

Tensor	the tensor type to which the new index is prepended
VolumeDim	the volume dimension of the tensor index to prepend
Fr	the Frame of the tensor index to prepend

◆ remove_first_index

template<typename Tensor >

using TensorMetafunctions::remove_first_index = typedef ::Tensor<typename Tensor::type, tmpl::pop_front<typename Tensor::symmetry>, tmpl::pop_front<typename Tensor::index_list> >

remove the first index of a tensor

Template Parameters

Tensor the tensor type whose first index is removed

◆ SpacetimeIndex

template<size_t SpatialDim, UpLo Ul, typename Fr >

using SpacetimeIndex = Tensor_detail::TensorIndexType<SpatialDim, Ul, Fr, IndexType::Spacetime>

A SpacetimeIndex holds information about the number of spatial dimensions, whether the index is covariant or contravariant (UpLo), and the Frame the index is in.

Template Parameters

SpatialDim	the spatial dimensionality of the TensorIndexType
Ul	either UpLo::Up or UpLo::Lo for contra or covariant
Fr	the Frame the TensorIndexType is in

◆ SpatialIndex

template<size_t SpatialDim, UpLo Ul, typename Fr >

using SpatialIndex = Tensor_detail::TensorIndexType<SpatialDim, Ul, Fr, IndexType::Spatial>

A SpatialIndex holds information about the number of spatial dimensions, whether the index is covariant or contravariant (UpLo), and the Frame the index is in.

Template Parameters

SpatialDim	the spatial dimensionality of the TensorIndexType
Ul	either UpLo::Up or UpLo::Lo for contra or covariant
Fr	the Frame the TensorIndexType is in

◆ swap_type

template<typename NewType , typename Tensor >

using TensorMetafunctions::swap_type = typedef ::Tensor<NewType, typename Tensor::symmetry, typename Tensor::index_list>

Swap the data type of a tensor for a new type.

Template Parameters

NewType	the new data type
Tensor	the tensor from which to keep symmetry and index information

◆ Symmetry

template<std::int32_t... T>

using Symmetry = typename detail::SymmetryImpl< std::make_index_sequence<sizeof...(T)>, tmpl::integral_list<std::int32_t, T...> >::type

Computes the canonical symmetry from the integers T

Details

Compute the canonical symmetry typelist given a set of integers, T. The resulting typelist is in ascending order of the integers, from right to left. For example, the result of Symmetry<1, 2, 1, 3> is integral_list<int32_t, 2, 3, 2, 1>. Anti-symmetries are not currently supported.

Template Parameters

T	the integers denoting the symmetry of the Tensor

Enumeration Type Documentation

◆ IndexType

enum class IndexType : char

strong

Indicates whether the TensorIndexType is Spatial or Spacetime.

Enumerator
Spatial	The TensorIndexType is purely spatial.
Spacetime	The TensorIndexType is a spacetime index.

◆ UpLo

enum class UpLo

strong

Whether a TensorIndexType is covariant or contravariant.

Enumerator
Up	Contravariant, or Upper index.
Lo	Covariant, or Lower index.

Function Documentation

◆ cross_product() [1/4]

template<typename DataType , typename Index >

Tensor< DataType, Symmetry< 1 >, index_list< change_index_up_lo< Index > > > cross_product	(	const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &	vector_a,
		const Tensor< DataType, Symmetry< 1 >, index_list< change_index_up_lo< Index > > > &	vector_b
	)

noexcept

Compute the Euclidean cross product of a vector and a one form.

Details

Returns \(A^j B_l \delta^{lk} \epsilon_{ijk}\) for input vector \(A^j\) and input one form \(B_l\) or \(A_j B^l \delta_{lk} \epsilon^{ijk}\) for input one form \(A_j\) and input vector \(B^l\). Note that this function returns a vector if vector_b is a vector and a one form if vector_b is a one form.

◆ cross_product() [2/4]

template<typename DataType , typename Index >

Tensor< DataType, Symmetry< 1 >, index_list< change_index_up_lo< Index > > > cross_product	(	const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &	vector_a,
		const Tensor< DataType, Symmetry< 1 >, index_list< change_index_up_lo< Index > > > &	vector_b,
		const Tensor< DataType, Symmetry< 1, 1 >, index_list< Index, Index > > &	metric_or_inverse_metric,
		const Scalar< DataType > &	metric_determinant
	)

noexcept

Compute the cross product of a vector and a one form.

Details

Returns \(\sqrt{g} A^j B_l g^{lk} \epsilon_{ijk}\) for input vector \(A^j\) and input one form \(B_l\). In this case, the argument vector_a should be a vector, vector_b should be a one form, metric_or_inverse_metric should be the inverse spatial metric \(g^{ij}\), and metric_determinant should be the determinant of the spatial metric \(\det(g_{ij})\). Or, returns \(\sqrt{g}^{-1} A_j B^l g_{lk} \epsilon^{ijk}\) for input one form \(A_j\) and input vector \(B^l\). In this case, the argument vector_a should be a one form, vector_b should be a vector, metric_or_inverse_metric should be the spatial metric \(g_{ij}\), and metric_determinant should be the determinant of the spatial metric \(\det(g_{ij})\). Note that this function returns a vector if vector_b is a vector and a one form if vector_b is a one form.

◆ cross_product() [3/4]

template<typename DataType , typename Index >

Tensor< DataType, Symmetry< 1 >, index_list< Index > > cross_product	(	const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &	vector_a,
		const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &	vector_b
	)

noexcept

Compute the Euclidean cross product of two vectors or one forms.

Details

Returns \(A^j B^k \epsilon_{ljk} \delta^{il}\) for input vectors \(A^j\) and \(B^k\) or \(A_j B_k \epsilon^{ljk} \delta_{il}\) for input one forms \(A_j\) and \(B_k\).

◆ cross_product() [4/4]

template<typename DataType , typename Index >

Tensor< DataType, Symmetry< 1 >, index_list< Index > > cross_product	(	const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &	vector_a,
		const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &	vector_b,
		const Tensor< DataType, Symmetry< 1, 1 >, index_list< Index, Index > > &	metric_or_inverse_metric,
		const Scalar< DataType > &	metric_determinant
	)

noexcept

Compute the cross product of two vectors or one forms.

Details

Returns \(\sqrt{g} g^{li} A^j B^k \epsilon_{ljk}\), where \(A^j\) and \(B^k\) are vectors and \(g^{li}\) and \(g\) are the inverse and determinant, respectively, of the spatial metric (computed via determinant_and_inverse). In this case, the arguments vector_a and vector_b should be vectors, the argument metric_or_inverse_metric should be the inverse spatial metric \(g^{ij}\), and the argument metric_determinant should be the determinant of the spatial metric \(\det(g_{ij})\). Or, returns \(\sqrt{g}^{-1} g_{li} A_j B_k \epsilon^{ljk}\), where \(A_j\) and \(B_k\) are one forms and \(g_{li}\) and \(g\) are the spatial metric and its determinant. In this case, the arguments vector_a and vector_b should be one forms, the argument metric_or_inverse_metric should be the spatial metric \(g_{ij}\), and the argument metric_determinant should be the determinant of the spatial metric \(\det(g_{ij})\).

◆ determinant() [1/2]

template<typename T , typename Symm , typename Index0 , typename Index1 >

void determinant	(	const gsl::not_null< Scalar< T > * >	det_tensor,
		const Tensor< T, Symm, index_list< Index0, Index1 > > &	tensor
	)

noexcept

Computes the determinant of a rank-2 Tensor tensor.

Requires: That tensor be a rank-2 Tensor, with both indices sharing the same dimension and type.

◆ determinant() [2/2]

template<typename T , typename Symm , typename Index0 , typename Index1 >

Scalar< T > determinant ( const Tensor< T, Symm, index_list< Index0, Index1 > > & tensor )

noexcept

Computes the determinant of a rank-2 Tensor tensor.

Requires: That tensor be a rank-2 Tensor, with both indices sharing the same dimension and type.

◆ determinant_and_inverse() [1/4]

template<typename T , typename Symm , typename Index0 , typename Index1 >

void determinant_and_inverse	(	const gsl::not_null< Scalar< T > * >	det,
		const gsl::not_null< Tensor< T, Symm, tmpl::list< change_index_up_lo< Index1 >, change_index_up_lo< Index0 > > > * >	inv,
		const Tensor< T, Symm, tmpl::list< Index0, Index1 > > &	tensor
	)

noexcept

Computes the determinant and inverse of a rank-2 Tensor.

Computes the determinant and inverse together, because this leads to fewer operations compared to computing the determinant independently.

Details

Treats the input rank-2 tensor as a matrix. The first (second) index of the tensor corresponds to the rows (columns) of the matrix. The determinant is a scalar tensor. The inverse is a rank-2 tensor whose indices are reversed and of opposite valence relative to the input tensor, i.e. given \(T_a^b\) returns \((Tinv)_b^a\).

Note: When inverting a 4x4 spacetime metric, it is typically more efficient to use the 3+1 decomposition of the 4-metric in terms of lapse, shift, and spatial 3-metric, in which only the spatial 3-metric needs to be inverted.

◆ determinant_and_inverse() [2/4]

template<typename DetTag , typename InvTag , typename T , typename Symm , typename Index0 , typename Index1 >

void determinant_and_inverse	(	const gsl::not_null< Variables< tmpl::list< DetTag, InvTag > > * >	det_and_inv,
		const Tensor< T, Symm, tmpl::list< Index0, Index1 > > &	tensor
	)

noexcept

Computes the determinant and inverse of a rank-2 Tensor.

Computes the determinant and inverse together, because this leads to fewer operations compared to computing the determinant independently.

Template Parameters

DetTag	the Tag for the determinant of input Tensor.
InvTag	the Tag for the inverse of input Tensor.

Details

See determinant_and_inverse().

◆ determinant_and_inverse() [3/4]

template<typename T , typename Symm , typename Index0 , typename Index1 >

auto determinant_and_inverse ( const Tensor< T, Symm, tmpl::list< Index0, Index1 > > & tensor ) -> std::pair<Scalar<T>, Tensor<T, Symm, tmpl::list<change_index_up_lo<Index1>, change_index_up_lo<Index0>>>>

noexcept

Computes the determinant and inverse of a rank-2 Tensor.

Computes the determinant and inverse together, because this leads to fewer operations compared to computing the determinant independently.

Details

Treats the input rank-2 tensor as a matrix. The first (second) index of the tensor corresponds to the rows (columns) of the matrix. The determinant is a scalar tensor. The inverse is a rank-2 tensor whose indices are reversed and of opposite valence relative to the input tensor, i.e. given \(T_a^b\) returns \((Tinv)_b^a\).

Note: When inverting a 4x4 spacetime metric, it is typically more efficient to use the 3+1 decomposition of the 4-metric in terms of lapse, shift, and spatial 3-metric, in which only the spatial 3-metric needs to be inverted.

◆ determinant_and_inverse() [4/4]

template<typename DetTag , typename InvTag , typename T , typename Symm , typename Index0 , typename Index1 >

auto determinant_and_inverse ( const Tensor< T, Symm, tmpl::list< Index0, Index1 > > & tensor ) -> Variables<tmpl::list<DetTag, InvTag>>

noexcept

Computes the determinant and inverse of a rank-2 Tensor.

Computes the determinant and inverse together, because this leads to fewer operations compared to computing the determinant independently.

Template Parameters

DetTag	the Tag for the determinant of input Tensor.
InvTag	the Tag for the inverse of input Tensor.

Details

See determinant_and_inverse().

◆ dot_product() [1/6]

template<typename DataType , typename Index >

void dot_product	(	const gsl::not_null< Scalar< DataType > * >	dot_product,
		const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &	vector_a,
		const Tensor< DataType, Symmetry< 1 >, index_list< change_index_up_lo< Index > > > &	vector_b
	)

noexcept

Compute the dot product of a vector and a one form.

Details

Returns \(A^a B_b \delta_{a}^b\) for input vector \(A^a\) and input one form \(B_b\) or \(A_a B^b \delta^a_b\) for input one form \(A_a\) and input vector \(B^b\).

◆ dot_product() [2/6]

template<typename DataType , typename Index >

void dot_product	(	const gsl::not_null< Scalar< DataType > * >	dot_product,
		const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &	vector_a,
		const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &	vector_b
	)

noexcept

Compute the Euclidean dot product of two vectors or one forms.

Details

Returns \(A^a B^b \delta_{ab}\) for input vectors \(A^a\) and \(B^b\) or \(A_a B_b \delta^{ab}\) for input one forms \(A_a\) and \(B_b\).

◆ dot_product() [3/6]

template<typename DataType , typename Index >

void dot_product	(	const gsl::not_null< Scalar< DataType > * >	dot_product,
		const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &	vector_a,
		const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &	vector_b,
		const Tensor< DataType, Symmetry< 1, 1 >, index_list< change_index_up_lo< Index >, change_index_up_lo< Index > > > &	metric
	)

noexcept

Compute the dot_product of two vectors or one forms.

Details

Returns \(g_{ab} A^a B^b\), where \(g_{ab}\) is the metric, \(A^a\) is vector_a, and \(B^b\) is vector_b. Or, returns \(g^{ab} A_a B_b\) when given one forms \(A_a\) and \(B_b\) with an inverse metric \(g^{ab}\).

◆ dot_product() [4/6]

template<typename DataType , typename Index >

Scalar< DataType > dot_product	(	const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &	vector_a,
		const Tensor< DataType, Symmetry< 1 >, index_list< change_index_up_lo< Index > > > &	vector_b
	)

noexcept

Compute the dot product of a vector and a one form.

Details

Returns \(A^a B_b \delta_{a}^b\) for input vector \(A^a\) and input one form \(B_b\) or \(A_a B^b \delta^a_b\) for input one form \(A_a\) and input vector \(B^b\).

◆ dot_product() [5/6]

template<typename DataType , typename Index >

Scalar< DataType > dot_product	(	const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &	vector_a,
		const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &	vector_b
	)

noexcept

Compute the Euclidean dot product of two vectors or one forms.

Details

Returns \(A^a B^b \delta_{ab}\) for input vectors \(A^a\) and \(B^b\) or \(A_a B_b \delta^{ab}\) for input one forms \(A_a\) and \(B_b\).

◆ dot_product() [6/6]

template<typename DataType , typename Index >

Scalar< DataType > dot_product	(	const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &	vector_a,
		const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &	vector_b,
		const Tensor< DataType, Symmetry< 1, 1 >, index_list< change_index_up_lo< Index >, change_index_up_lo< Index > > > &	metric
	)

noexcept

Compute the dot_product of two vectors or one forms.

Details

Returns \(g_{ab} A^a B^b\), where \(g_{ab}\) is the metric, \(A^a\) is vector_a, and \(B^b\) is vector_b. Or, returns \(g^{ab} A_a B_b\) when given one forms \(A_a\) and \(B_b\) with an inverse metric \(g^{ab}\).

◆ index_dim()

template<int I, class... Ts>

constexpr size_t index_dim ( const Tensor< Ts... > & )

constexprnoexcept

Get dimensionality of i'th tensor index.

    using T = Tensor<double, Symmetry<1, 2, 3>,
                     index_list<SpacetimeIndex<2, UpLo::Up, Frame::Inertial>,
                                SpatialIndex<1, UpLo::Up, Frame::Inertial>,
                                SpatialIndex<2, UpLo::Up, Frame::Inertial>>>;
    const T t{};
    CHECK(index_dim<0>(t) == 3);
    CHECK(index_dim<1>(t) == 1);
    CHECK(index_dim<2>(t) == 2);
    CHECK(T::index_dim(0) == 3);
    CHECK(T::index_dim(1) == 1);
    CHECK(T::index_dim(2) == 2);
    CHECK(T::index_dims() == std::array<size_t, 3>{{3, 1, 2}});

◆ l2_norm()

template<typename DataType , typename Symm , typename IndexList >

double l2_norm ( const Tensor< DataType, Symm, IndexList > & tensor )

noexcept

Compute Euclidean \(L^2\)-norm of arbitrary Tensors reduced over an element.

Details

Computes the RMS value of the point-wise Frobenius norm of a given Tensor with arbitrary rank over all grid points in an element. If the Tensor \(A\) has rank \(n\) and dimensionality \(D\), and the element (of order \(N\)) has \(N+1\) points, then its element-reduced Frobenius norm is computed as:

\begin{equation} ||A||_2 = \left(\frac{1}{N+1}\sum^{N}_{p=0} \left(\sum^{D-1}_{i_1=0}\sum^{D-1}_{i_2=0}\cdots \sum^{D-1}_{i_n=0} |A^p_{i_1 i_2 \cdots i_n}|^2 \right) \right)^{1/2}, \end{equation}

where both contra-variant and co-variant indices are shown as lower indices, and \(p\) indexes grid points in the element.

Warning: This function reduces the Frobenius norm over the element, not the whole domain.

◆ magnitude() [1/4]

template<typename DataType , typename Index >

void magnitude	(	const gsl::not_null< Scalar< DataType > * >	magnitude,
		const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &	vector
	)

noexcept

Compute the Euclidean magnitude of a rank-1 tensor.

Details

Computes the square root of the sum of the squares of the components of the rank-1 tensor.

◆ magnitude() [2/4]

template<typename DataType , typename Index >

void magnitude	(	const gsl::not_null< Scalar< DataType > * >	magnitude,
		const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &	vector,
		const Tensor< DataType, Symmetry< 1, 1 >, index_list< change_index_up_lo< Index >, change_index_up_lo< Index > > > &	metric
	)

noexcept

Compute the magnitude of a rank-1 tensor.

Details

Returns the square root of the input tensor contracted twice with the given metric.

◆ magnitude() [3/4]

template<typename DataType , typename Index >

Scalar< DataType > magnitude ( const Tensor< DataType, Symmetry< 1 >, index_list< Index > > & vector )

noexcept

Compute the Euclidean magnitude of a rank-1 tensor.

Details

Computes the square root of the sum of the squares of the components of the rank-1 tensor.

◆ magnitude() [4/4]

template<typename DataType , typename Index >

Scalar< DataType > magnitude	(	const Tensor< DataType, Symmetry< 1 >, index_list< Index > > &	vector,
		const Tensor< DataType, Symmetry< 1, 1 >, index_list< change_index_up_lo< Index >, change_index_up_lo< Index > > > &	metric
	)

noexcept

Compute the magnitude of a rank-1 tensor.

Details

Returns the square root of the input tensor contracted twice with the given metric.

◆ orthonormal_oneform() [1/4]

template<typename DataType , typename Frame >

tnsr::i< DataType, 3, Frame > orthonormal_oneform	(	const tnsr::i< DataType, 3, Frame > &	first_unit_form,
		const tnsr::i< DataType, 3, Frame > &	second_unit_form,
		const tnsr::ii< DataType, 3, Frame > &	spatial_metric,
		const Scalar< DataType > &	det_spatial_metric
	)

noexcept

Compute a spatial one-form orthonormal to two given unit forms.

Given a unit spatial one-form \(s_i\) and another form \(t_i\) that is orthonormal to \(s_i\), compute a new form \(u_i\) which is orthonormal to both \(s_i\) and \(t_i\), in the sense that \(\gamma^{ij}s_i u_j = \gamma^{ij}t_i u_j = 0\), for the given inverse spatial metric \(\gamma^{ij}\). The normalization of \(u_i\) is such that \(\gamma^{ij}u_iu_j = 1\).

Details

The new form is obtained by taking the covariant cross product of \(s_i\) and \( t_i\), for which the spatial metric as well as its determinant must be provided.

◆ orthonormal_oneform() [2/4]

template<typename DataType , size_t VolumeDim, typename Frame >

tnsr::i< DataType, VolumeDim, Frame > orthonormal_oneform	(	const tnsr::i< DataType, VolumeDim, Frame > &	unit_form,
		const tnsr::II< DataType, VolumeDim, Frame > &	inv_spatial_metric
	)

noexcept

Compute a spatial one-form orthonormal to the given unit form.

Given a unit spatial one-form \(s_i\), compute a new form \(t_i\) which is orthonormal to \(s_i\), in the sense that \(\gamma^{ij}s_i t_j = 0\), for the given inverse spatial metric \(\gamma^{ij}\). The normalization of \(t_i\) is such that \(\gamma^{ij}t_it_j = 1\).

Details

The new form is obtained via Gram-Schmidt process, starting from a form whose components are \(t_i = \delta_i^I\), where \(I\) is the index of the component of \(s_i\) with the smallest absolute value.

◆ orthonormal_oneform() [3/4]

template<typename DataType , typename Frame >

void orthonormal_oneform	(	gsl::not_null< tnsr::i< DataType, 3, Frame > * >	orthonormal_form,
		const tnsr::i< DataType, 3, Frame > &	first_unit_form,
		const tnsr::i< DataType, 3, Frame > &	second_unit_form,
		const tnsr::ii< DataType, 3, Frame > &	spatial_metric,
		const Scalar< DataType > &	det_spatial_metric
	)

noexcept

Compute a spatial one-form orthonormal to two given unit forms.

Given a unit spatial one-form \(s_i\) and another form \(t_i\) that is orthonormal to \(s_i\), compute a new form \(u_i\) which is orthonormal to both \(s_i\) and \(t_i\), in the sense that \(\gamma^{ij}s_i u_j = \gamma^{ij}t_i u_j = 0\), for the given inverse spatial metric \(\gamma^{ij}\). The normalization of \(u_i\) is such that \(\gamma^{ij}u_iu_j = 1\).

Details

The new form is obtained by taking the covariant cross product of \(s_i\) and \( t_i\), for which the spatial metric as well as its determinant must be provided.

◆ orthonormal_oneform() [4/4]

template<typename DataType , size_t VolumeDim, typename Frame >

void orthonormal_oneform	(	gsl::not_null< tnsr::i< DataType, VolumeDim, Frame > * >	orthonormal_form,
		const tnsr::i< DataType, VolumeDim, Frame > &	unit_form,
		const tnsr::II< DataType, VolumeDim, Frame > &	inv_spatial_metric
	)

noexcept

Compute a spatial one-form orthonormal to the given unit form.

Given a unit spatial one-form \(s_i\), compute a new form \(t_i\) which is orthonormal to \(s_i\), in the sense that \(\gamma^{ij}s_i t_j = 0\), for the given inverse spatial metric \(\gamma^{ij}\). The normalization of \(t_i\) is such that \(\gamma^{ij}t_it_j = 1\).

Details

The new form is obtained via Gram-Schmidt process, starting from a form whose components are \(t_i = \delta_i^I\), where \(I\) is the index of the component of \(s_i\) with the smallest absolute value.

◆ pointwise_l2_norm() [1/2]

template<typename DataType , typename Symm , typename IndexList >

void pointwise_l2_norm	(	const gsl::not_null< Scalar< DataType > * >	norm,
		const Tensor< DataType, Symm, IndexList > &	tensor
	)

noexcept

Compute point-wise Euclidean \(L^2\)-norm of arbitrary Tensors.

Details

At each grid point \(p\) in the element, this function computes the point-wise Frobenius norm of a given Tensor with arbitrary rank. If the Tensor \(A\) has rank \(n\) and dimensionality \(D\), then its Frobenius norm at point \(p\) is computed as:

\begin{equation} ||A||_2(p) = \left(\sum^{D-1}_{i_1=0}\sum^{D-1}_{i_2=0}\cdots \sum^{D-1}_{i_n=0} |A_{i_1 i_2 \cdots i_n}(p)|^2 \right)^{1/2}, \end{equation}

where both contra-variant and co-variant indices are shown as lower indices.

◆ pointwise_l2_norm() [2/2]

template<typename DataType , typename Symm , typename IndexList >

Scalar< DataType > pointwise_l2_norm ( const Tensor< DataType, Symm, IndexList > & tensor )

noexcept

Compute point-wise Euclidean \(L^2\)-norm of arbitrary Tensors.

Details

At each grid point \(p\) in the element, this function computes the point-wise Frobenius norm of a given Tensor with arbitrary rank. If the Tensor \(A\) has rank \(n\) and dimensionality \(D\), then its Frobenius norm at point \(p\) is computed as:

\begin{equation} ||A||_2(p) = \left(\sum^{D-1}_{i_1=0}\sum^{D-1}_{i_2=0}\cdots \sum^{D-1}_{i_n=0} |A_{i_1 i_2 \cdots i_n}(p)|^2 \right)^{1/2}, \end{equation}

where both contra-variant and co-variant indices are shown as lower indices.

◆ prefix()

template<typename Fr >

std::string Frame::prefix ( )

inlinenoexcept

The frame-dependent prefix used when constructing the string returned by the name function of a tag.

For Frame::Inertial it is the empty string, otherwise, it is the name of the Frame followed by an underscore (as the name will be used in I/O).

Example

  TestHelpers::db::test_simple_tag<
      hydro::Tags::SpatialVelocity<DataVector, 3, Frame::Grid>>(
      "Grid_SpatialVelocity");
  TestHelpers::db::test_simple_tag<
      hydro::Tags::SpatialVelocityOneForm<DataVector, 3, Frame::Logical>>(
      "Logical_SpatialVelocityOneForm");

◆ sqrt_magnitude() [1/2]

template<typename DataType >

void sqrt_magnitude	(	const gsl::not_null< Scalar< DataType > * >	sqrt_magnitude,
		const Scalar< DataType > &	input
	)

noexcept

Compute square root of the Euclidean magnitude of a rank-0 tensor.

Details

Computes the square root of the absolute value of the scalar.

◆ sqrt_magnitude() [2/2]

template<typename DataType >

Scalar< DataType > sqrt_magnitude ( const Scalar< DataType > & input )

noexcept

Compute square root of the Euclidean magnitude of a rank-0 tensor.

Details

Computes the square root of the absolute value of the scalar.

Variable Documentation

◆ is_frame_physical_v

template<typename CheckFrame >

constexpr bool Frame::is_frame_physical_v = is_frame_physical<CheckFrame>::value

constexpr

Returns true if the frame is "physical" in the sense that it is meaningful to evaluate an analytic solution in that frame.

Example

static_assert(not Frame::is_frame_physical_v<Frame::Logical>,
              "Failed testing Frame::is_frame_physical");
static_assert(not Frame::is_frame_physical_v<Frame::Distorted>,
              "Failed testing Frame::is_frame_physical");
static_assert(not Frame::is_frame_physical_v<Frame::Grid>,
              "Failed testing Frame::is_frame_physical");
static_assert(Frame::is_frame_physical_v<Frame::Inertial>,
              "Failed testing Frame::is_frame_physical");

Namespaces

Classes

Typedefs

Enumerations

Functions

Variables

Detailed Description

Typedef Documentation

◆ change_index_type

◆ change_index_up_lo

◆ is_frame_physical

Example

◆ prepend_spacetime_index

◆ prepend_spatial_index

◆ remove_first_index

◆ SpacetimeIndex

◆ SpatialIndex

◆ swap_type

◆ Symmetry

Details

Enumeration Type Documentation

◆ IndexType

◆ UpLo

Function Documentation

◆ cross_product() [1/4]

Details

◆ cross_product() [2/4]

Details

◆ cross_product() [3/4]

Details

◆ cross_product() [4/4]

Details

◆ determinant() [1/2]

◆ determinant() [2/2]

◆ determinant_and_inverse() [1/4]

Details

◆ determinant_and_inverse() [2/4]

Details

◆ determinant_and_inverse() [3/4]

Details

◆ determinant_and_inverse() [4/4]

Details

◆ dot_product() [1/6]

Details

◆ dot_product() [2/6]

Details

◆ dot_product() [3/6]

Details

◆ dot_product() [4/6]

Details

◆ dot_product() [5/6]

Details

◆ dot_product() [6/6]

Details

◆ index_dim()

◆ l2_norm()

Details

◆ magnitude() [1/4]

Details

◆ magnitude() [2/4]

Details

◆ magnitude() [3/4]

Details

◆ magnitude() [4/4]

Details

◆ orthonormal_oneform() [1/4]

Details

◆ orthonormal_oneform() [2/4]

Details

◆ orthonormal_oneform() [3/4]

Details

◆ orthonormal_oneform() [4/4]

Details

◆ pointwise_l2_norm() [1/2]

Details

◆ pointwise_l2_norm() [2/2]

Details

◆ prefix()

Example

◆ sqrt_magnitude() [1/2]