47template <
int dim,
int spacedim>
67template <
int dim,
int spacedim>
72 const unsigned int n_original_q_points)
76 this->update_each = update_flags;
86 if (this->update_each &
88 compute_manifold_quadrature_weights(q);
91 covariant.resize(n_original_q_points);
94 contravariant.resize(n_original_q_points);
97 volume_elements.resize(n_original_q_points);
101template <
int dim,
int spacedim>
106 const unsigned int n_original_q_points)
108 initialize(update_flags, q, n_original_q_points);
120 unit_tangentials[i].resize(n_original_q_points);
121 std::fill(unit_tangentials[i].begin(),
122 unit_tangentials[i].end(),
127 .resize(n_original_q_points);
142template <
int dim,
int spacedim>
149template <
int dim,
int spacedim>
150std::unique_ptr<Mapping<dim, spacedim>>
153 return std::make_unique<MappingManifold<dim, spacedim>>(*this);
158template <
int dim,
int spacedim>
170template <
int dim,
int spacedim>
177 std::array<double, GeometryInfo<dim>::vertices_per_cell> weights;
184 return cell->get_manifold().get_new_point(
204template <
int dim,
int spacedim>
216 for (
unsigned int i = 0; i < 5; ++i)
266template <
int dim,
int spacedim>
267std::unique_ptr<typename Mapping<dim, spacedim>::InternalDataBase>
271 std::unique_ptr<typename Mapping<dim, spacedim>::InternalDataBase> data_ptr =
272 std::make_unique<InternalData>();
281template <
int dim,
int spacedim>
282std::unique_ptr<typename Mapping<dim, spacedim>::InternalDataBase>
289 std::unique_ptr<typename Mapping<dim, spacedim>::InternalDataBase> data_ptr =
290 std::make_unique<InternalData>();
294 ReferenceCells::get_hypercube<dim>(), quadrature[0]),
295 quadrature[0].size());
302template <
int dim,
int spacedim>
303std::unique_ptr<typename Mapping<dim, spacedim>::InternalDataBase>
308 std::unique_ptr<typename Mapping<dim, spacedim>::InternalDataBase> data_ptr =
309 std::make_unique<InternalData>();
313 ReferenceCells::get_hypercube<dim>(), quadrature),
323 namespace MappingManifoldImplementation
333 template <
int dim,
int spacedim>
335 maybe_compute_q_points(
337 const typename ::MappingManifold<dim, spacedim>::InternalData
348 for (
unsigned int point = 0; point < quadrature_points.size();
351 quadrature_points[point] = data.manifold->get_new_point(
354 data.cell_manifold_quadrature_weights[point + data_set]));
367 template <
int dim,
int spacedim>
369 maybe_update_Jacobians(
370 const typename ::QProjector<dim>::DataSetDescriptor data_set,
371 const typename ::MappingManifold<dim, spacedim>::InternalData
378 const unsigned int n_q_points = data.contravariant.size();
380 std::fill(data.contravariant.begin(),
381 data.contravariant.end(),
385 data.vertices.size());
386 for (
unsigned int point = 0; point < n_q_points; ++point)
390 const Point<dim> &p = data.quad.point(point + data_set);
396 data.cell_manifold_quadrature_weights[point + data_set]));
405 for (
unsigned int i = 0; i < dim; ++i)
408 const double pi = p[i];
409 Assert(pi >= 0 && pi <= 1.0,
411 "Was expecting a quadrature point "
412 "inside the unit reference element."));
416 const double L = pi > .5 ? -pi : 1 - pi;
421 for (
const unsigned int j :
423 data.vertex_weights[j] =
431 data.manifold->get_tangent_vector(P, NP);
433 for (
unsigned int d = 0; d < spacedim; ++d)
434 data.contravariant[point][d][i] = T[d] / L;
440 const unsigned int n_q_points = data.contravariant.size();
441 for (
unsigned int point = 0; point < n_q_points; ++point)
443 data.covariant[point] =
444 (data.contravariant[point]).covariant_form();
450 const unsigned int n_q_points = data.contravariant.size();
451 for (
unsigned int point = 0; point < n_q_points; ++point)
452 data.volume_elements[point] =
453 data.contravariant[point].determinant();
463template <
int dim,
int spacedim>
478 const unsigned int n_q_points = quadrature.
size();
481 data.
manifold = &(cell->get_manifold());
483 internal::MappingManifoldImplementation::maybe_compute_q_points<dim,
489 internal::MappingManifoldImplementation::maybe_update_Jacobians<dim,
494 const std::vector<double> &weights = quadrature.
get_weights();
509 for (
unsigned int point = 0; point < n_q_points; ++point)
520 Assert(det > 1e-12 * Utilities::fixed_power<dim>(
521 cell->diameter() /
std::sqrt(
double(dim))),
523 cell->center(), det, point)));
525 output_data.
JxW_values[point] = weights[point] * det;
533 for (
unsigned int i = 0; i < spacedim; ++i)
534 for (
unsigned int j = 0; j < dim; ++j)
538 for (
unsigned int i = 0; i < dim; ++i)
539 for (
unsigned int j = 0; j < dim; ++j)
540 G[i][j] = DX_t[i] * DX_t[j];
547 Assert(spacedim == dim + 1,
549 "There is no (unique) cell normal for " +
551 "-dimensional cells in " +
553 "-dimensional space. This only works if the "
554 "space dimension is one greater than the "
555 "dimensionality of the mesh cells."));
559 cross_product_2d(-DX_t[0]);
562 cross_product_3d(DX_t[0], DX_t[1]);
567 if (cell->direction_flag() ==
false)
580 for (
unsigned int point = 0; point < n_q_points; ++point)
588 for (
unsigned int point = 0; point < n_q_points; ++point)
600 namespace MappingManifoldImplementation
613 template <
int dim,
int spacedim>
615 maybe_compute_face_data(
616 const ::MappingManifold<dim, spacedim> &mapping,
617 const typename ::Triangulation<dim, spacedim>::cell_iterator
619 const unsigned int face_no,
620 const unsigned int subface_no,
621 const unsigned int n_q_points,
622 const std::vector<double> &weights,
623 const typename ::MappingManifold<dim, spacedim>::InternalData
641 for (
unsigned int d = 0; d != dim - 1; ++d)
644 data.unit_tangentials.size(),
647 data.aux[d].size() <=
649 .unit_tangentials[face_no +
657 .unit_tangentials[face_no +
668 for (
unsigned int i = 0; i < n_q_points; ++i)
677 (face_no == 0 ? -1 : +1);
681 cross_product_2d(data.aux[0][i]);
685 cross_product_3d(data.aux[0][i], data.aux[1][i]);
701 for (
unsigned int point = 0;
point < n_q_points; ++
point)
709 data.contravariant[
point].transpose()[0];
711 (face_no == 0 ? -1. : +1.) *
723 cross_product_3d(DX_t[0], DX_t[1]);
724 cell_normal /= cell_normal.
norm();
729 cross_product_3d(data.aux[0][point], cell_normal);
751 const double area_ratio =
753 cell->subface_case(face_no), subface_no);
765 for (
unsigned int point = 0;
point < n_q_points; ++
point)
766 output_data.
jacobians[point] = data.contravariant[point];
769 for (
unsigned int point = 0;
point < n_q_points; ++
point)
771 data.covariant[point].transpose();
782 template <
int dim,
int spacedim>
785 const ::MappingManifold<dim, spacedim> &mapping,
786 const typename ::Triangulation<dim, spacedim>::cell_iterator
788 const unsigned int face_no,
789 const unsigned int subface_no,
792 const typename ::MappingManifold<dim, spacedim>::InternalData
797 data.store_vertices(cell);
799 data.manifold = &cell->face(face_no)->get_manifold();
801 maybe_compute_q_points<dim, spacedim>(data_set,
804 maybe_update_Jacobians<dim, spacedim>(data_set, data);
816 template <
int dim,
int spacedim,
int rank>
825 Assert((
dynamic_cast<const typename ::
826 MappingManifold<dim, spacedim>::InternalData *
>(
827 &mapping_data) !=
nullptr),
829 const typename ::MappingManifold<dim, spacedim>::InternalData
831 static_cast<const typename ::MappingManifold<dim, spacedim>::
832 InternalData &
>(mapping_data);
834 switch (mapping_kind)
841 "update_contravariant_transformation"));
843 for (
unsigned int i = 0; i < output.size(); ++i)
855 "update_contravariant_transformation"));
859 "update_volume_elements"));
864 for (
unsigned int i = 0; i < output.size(); ++i)
868 output[i] /= data.volume_elements[i];
880 "update_covariant_transformation"));
882 for (
unsigned int i = 0; i < output.size(); ++i)
894 template <
int dim,
int spacedim,
int rank>
903 Assert((
dynamic_cast<const typename ::
904 MappingManifold<dim, spacedim>::InternalData *
>(
905 &mapping_data) !=
nullptr),
907 const typename ::MappingManifold<dim, spacedim>::InternalData
909 static_cast<const typename ::MappingManifold<dim, spacedim>::
910 InternalData &
>(mapping_data);
912 switch (mapping_kind)
919 "update_covariant_transformation"));
923 "update_contravariant_transformation"));
926 for (
unsigned int i = 0; i < output.size(); ++i)
943 "update_covariant_transformation"));
946 for (
unsigned int i = 0; i < output.size(); ++i)
963 "update_covariant_transformation"));
967 "update_contravariant_transformation"));
971 "update_volume_elements"));
974 for (
unsigned int i = 0; i < output.size(); ++i)
983 output[i] /= data.volume_elements[i];
996 template <
int dim,
int spacedim>
1005 Assert((
dynamic_cast<const typename ::
1006 MappingManifold<dim, spacedim>::InternalData *
>(
1007 &mapping_data) !=
nullptr),
1009 const typename ::MappingManifold<dim, spacedim>::InternalData
1011 static_cast<const typename ::MappingManifold<dim, spacedim>::
1012 InternalData &
>(mapping_data);
1014 switch (mapping_kind)
1021 "update_covariant_transformation"));
1025 "update_contravariant_transformation"));
1027 for (
unsigned int q = 0; q < output.size(); ++q)
1028 for (
unsigned int i = 0; i < spacedim; ++i)
1030 double tmp1[dim][dim];
1031 for (
unsigned int J = 0; J < dim; ++J)
1032 for (
unsigned int K = 0;
K < dim; ++
K)
1035 data.contravariant[q][i][0] * input[q][0][J][
K];
1036 for (
unsigned int I = 1; I < dim; ++I)
1038 data.contravariant[q][i][I] * input[q][I][J][K];
1040 for (
unsigned int j = 0; j < spacedim; ++j)
1043 for (
unsigned int K = 0;
K < dim; ++
K)
1045 tmp2[
K] = data.covariant[q][j][0] * tmp1[0][
K];
1046 for (
unsigned int J = 1; J < dim; ++J)
1047 tmp2[K] += data.covariant[q][j][J] * tmp1[J][K];
1049 for (
unsigned int k = 0; k < spacedim; ++k)
1051 output[q][i][j][k] =
1052 data.covariant[q][k][0] * tmp2[0];
1053 for (
unsigned int K = 1;
K < dim; ++
K)
1054 output[q][i][j][k] +=
1055 data.covariant[q][k][K] * tmp2[K];
1067 "update_covariant_transformation"));
1069 for (
unsigned int q = 0; q < output.size(); ++q)
1070 for (
unsigned int i = 0; i < spacedim; ++i)
1072 double tmp1[dim][dim];
1073 for (
unsigned int J = 0; J < dim; ++J)
1074 for (
unsigned int K = 0;
K < dim; ++
K)
1077 data.covariant[q][i][0] * input[q][0][J][
K];
1078 for (
unsigned int I = 1; I < dim; ++I)
1080 data.covariant[q][i][I] * input[q][I][J][K];
1082 for (
unsigned int j = 0; j < spacedim; ++j)
1085 for (
unsigned int K = 0;
K < dim; ++
K)
1087 tmp2[
K] = data.covariant[q][j][0] * tmp1[0][
K];
1088 for (
unsigned int J = 1; J < dim; ++J)
1089 tmp2[K] += data.covariant[q][j][J] * tmp1[J][K];
1091 for (
unsigned int k = 0; k < spacedim; ++k)
1093 output[q][i][j][k] =
1094 data.covariant[q][k][0] * tmp2[0];
1095 for (
unsigned int K = 1;
K < dim; ++
K)
1096 output[q][i][j][k] +=
1097 data.covariant[q][k][K] * tmp2[K];
1110 "update_covariant_transformation"));
1114 "update_contravariant_transformation"));
1118 "update_volume_elements"));
1120 for (
unsigned int q = 0; q < output.size(); ++q)
1121 for (
unsigned int i = 0; i < spacedim; ++i)
1124 for (
unsigned int I = 0; I < dim; ++I)
1126 data.contravariant[q][i][I] / data.volume_elements[q];
1127 double tmp1[dim][dim];
1128 for (
unsigned int J = 0; J < dim; ++J)
1129 for (
unsigned int K = 0;
K < dim; ++
K)
1131 tmp1[J][
K] = factor[0] * input[q][0][J][
K];
1132 for (
unsigned int I = 1; I < dim; ++I)
1133 tmp1[J][K] += factor[I] * input[q][I][J][K];
1135 for (
unsigned int j = 0; j < spacedim; ++j)
1138 for (
unsigned int K = 0;
K < dim; ++
K)
1140 tmp2[
K] = data.covariant[q][j][0] * tmp1[0][
K];
1141 for (
unsigned int J = 1; J < dim; ++J)
1142 tmp2[K] += data.covariant[q][j][J] * tmp1[J][K];
1144 for (
unsigned int k = 0; k < spacedim; ++k)
1146 output[q][i][j][k] =
1147 data.covariant[q][k][0] * tmp2[0];
1148 for (
unsigned int K = 1;
K < dim; ++
K)
1149 output[q][i][j][k] +=
1150 data.covariant[q][k][K] * tmp2[K];
1165 template <
int dim,
int spacedim,
int rank>
1174 Assert((
dynamic_cast<const typename ::
1175 MappingManifold<dim, spacedim>::InternalData *
>(
1176 &mapping_data) !=
nullptr),
1178 const typename ::MappingManifold<dim, spacedim>::InternalData
1180 static_cast<const typename ::MappingManifold<dim, spacedim>::
1181 InternalData &
>(mapping_data);
1183 switch (mapping_kind)
1190 "update_covariant_transformation"));
1192 for (
unsigned int i = 0; i < output.size(); ++i)
1207template <
int dim,
int spacedim>
1211 const unsigned int face_no,
1224 internal::MappingManifoldImplementation::do_fill_fe_face_values(
1230 ReferenceCells::get_hypercube<dim>(),
1232 cell->face_orientation(face_no),
1233 cell->face_flip(face_no),
1234 cell->face_rotation(face_no),
1235 quadrature[0].
size()),
1243template <
int dim,
int spacedim>
1247 const unsigned int face_no,
1248 const unsigned int subface_no,
1259 internal::MappingManifoldImplementation::do_fill_fe_face_values(
1265 ReferenceCells::get_hypercube<dim>(),
1268 cell->face_orientation(face_no),
1269 cell->face_flip(face_no),
1270 cell->face_rotation(face_no),
1272 cell->subface_case(face_no)),
1280template <
int dim,
int spacedim>
1288 internal::MappingManifoldImplementation::transform_fields(input,
1296template <
int dim,
int spacedim>
1304 internal::MappingManifoldImplementation::transform_differential_forms(
1305 input, mapping_kind, mapping_data, output);
1310template <
int dim,
int spacedim>
1318 switch (mapping_kind)
1321 internal::MappingManifoldImplementation::transform_fields(input,
1330 internal::MappingManifoldImplementation::transform_gradients(
1331 input, mapping_kind, mapping_data, output);
1340template <
int dim,
int spacedim>
1353 switch (mapping_kind)
1359 "update_covariant_transformation"));
1361 for (
unsigned int q = 0; q < output.size(); ++q)
1362 for (
unsigned int i = 0; i < spacedim; ++i)
1363 for (
unsigned int j = 0; j < spacedim; ++j)
1366 for (
unsigned int K = 0; K < dim; ++K)
1368 tmp[K] = data.
covariant[q][j][0] * input[q][i][0][K];
1369 for (
unsigned int J = 1; J < dim; ++J)
1370 tmp[K] += data.
covariant[q][j][J] * input[q][i][J][K];
1372 for (
unsigned int k = 0; k < spacedim; ++k)
1374 output[q][i][j][k] = data.
covariant[q][k][0] * tmp[0];
1375 for (
unsigned int K = 1; K < dim; ++K)
1376 output[q][i][j][k] += data.
covariant[q][k][K] * tmp[K];
1389template <
int dim,
int spacedim>
1397 switch (mapping_kind)
1402 internal::MappingManifoldImplementation::transform_hessians(
1403 input, mapping_kind, mapping_data, output);
1411#include "mapping_manifold.inst"
ArrayView< typename std::remove_reference< typename std::iterator_traits< Iterator >::reference >::type, MemorySpaceType > make_array_view(const Iterator begin, const Iterator end)
void store_vertices(const typename Triangulation< dim, spacedim >::cell_iterator &cell) const
std::vector< double > volume_elements
std::array< std::vector< Tensor< 1, dim > >, GeometryInfo< dim >::faces_per_cell *(dim - 1)> unit_tangentials
std::vector< DerivativeForm< 1, dim, spacedim > > contravariant
void initialize(const UpdateFlags update_flags, const Quadrature< dim > &quadrature, const unsigned int n_original_q_points)
SmartPointer< const Manifold< dim, spacedim > > manifold
std::vector< std::vector< Tensor< 1, spacedim > > > aux
virtual std::size_t memory_consumption() const override
std::vector< std::vector< double > > cell_manifold_quadrature_weights
void initialize_face(const UpdateFlags update_flags, const Quadrature< dim > &quadrature, const unsigned int n_original_q_points)
std::vector< Point< spacedim > > vertices
std::vector< double > vertex_weights
std::vector< DerivativeForm< 1, dim, spacedim > > covariant
Triangulation< dim, spacedim >::cell_iterator cell
virtual Point< spacedim > transform_unit_to_real_cell(const typename Triangulation< dim, spacedim >::cell_iterator &cell, const Point< dim > &p) const override
MappingManifold()=default
virtual Point< dim > transform_real_to_unit_cell(const typename Triangulation< dim, spacedim >::cell_iterator &cell, const Point< spacedim > &p) const override
virtual void transform(const ArrayView< const Tensor< 1, dim > > &input, const MappingKind kind, const typename Mapping< dim, spacedim >::InternalDataBase &internal, const ArrayView< Tensor< 1, spacedim > > &output) const override
virtual void fill_fe_subface_values(const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const unsigned int subface_no, const Quadrature< dim - 1 > &quadrature, const typename Mapping< dim, spacedim >::InternalDataBase &internal_data, internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &output_data) const override
virtual void fill_fe_face_values(const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const hp::QCollection< dim - 1 > &quadrature, const typename Mapping< dim, spacedim >::InternalDataBase &internal_data, internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &output_data) const override
virtual UpdateFlags requires_update_flags(const UpdateFlags update_flags) const override
virtual std::unique_ptr< typename Mapping< dim, spacedim >::InternalDataBase > get_data(const UpdateFlags, const Quadrature< dim > &quadrature) const override
virtual std::unique_ptr< Mapping< dim, spacedim > > clone() const override
virtual std::unique_ptr< typename Mapping< dim, spacedim >::InternalDataBase > get_face_data(const UpdateFlags flags, const hp::QCollection< dim - 1 > &quadrature) const override
virtual std::unique_ptr< typename Mapping< dim, spacedim >::InternalDataBase > get_subface_data(const UpdateFlags flags, const Quadrature< dim - 1 > &quadrature) const override
virtual CellSimilarity::Similarity fill_fe_values(const typename Triangulation< dim, spacedim >::cell_iterator &cell, const CellSimilarity::Similarity cell_similarity, const Quadrature< dim > &quadrature, const typename Mapping< dim, spacedim >::InternalDataBase &internal_data, internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &output_data) const override
Abstract base class for mapping classes.
static Point< dim, Number > unit_vector(const unsigned int i)
static DataSetDescriptor cell()
const std::vector< double > & get_weights() const
unsigned int size() const
numbers::NumberTraits< Number >::real_type norm() const
unsigned int size() const
#define DEAL_II_NAMESPACE_OPEN
#define DEAL_II_NAMESPACE_CLOSE
static ::ExceptionBase & ExcNotImplemented()
#define Assert(cond, exc)
#define AssertDimension(dim1, dim2)
static ::ExceptionBase & ExcInternalError()
static ::ExceptionBase & ExcDimensionMismatch(std::size_t arg1, std::size_t arg2)
static ::ExceptionBase & ExcMessage(std::string arg1)
@ update_jacobian_pushed_forward_2nd_derivatives
@ update_volume_elements
Determinant of the Jacobian.
@ update_contravariant_transformation
Contravariant transformation.
@ update_jacobian_pushed_forward_grads
@ update_jacobian_grads
Gradient of volume element.
@ update_normal_vectors
Normal vectors.
@ update_JxW_values
Transformed quadrature weights.
@ update_covariant_transformation
Covariant transformation.
@ update_jacobians
Volume element.
@ update_inverse_jacobians
Volume element.
@ update_quadrature_points
Transformed quadrature points.
@ update_jacobian_pushed_forward_3rd_derivatives
@ update_boundary_forms
Outer normal vector, not normalized.
@ mapping_covariant_gradient
@ mapping_contravariant_hessian
@ mapping_covariant_hessian
@ mapping_contravariant_gradient
CGAL::Exact_predicates_exact_constructions_kernel_with_sqrt K
std::enable_if_t< std::is_fundamental< T >::value, std::size_t > memory_consumption(const T &t)
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
std::string int_to_string(const unsigned int value, const unsigned int digits=numbers::invalid_unsigned_int)
void transform_differential_forms(const ArrayView< const DerivativeForm< rank, dim, spacedim > > &input, const MappingKind mapping_kind, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_data, const ArrayView< Tensor< rank+1, spacedim > > &output)
void do_fill_fe_face_values(const ::MappingQ< dim, spacedim > &mapping, const typename ::Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const unsigned int subface_no, const typename QProjector< dim >::DataSetDescriptor data_set, const Quadrature< dim - 1 > &quadrature, const typename ::MappingQ< dim, spacedim >::InternalData &data, const std::vector< Polynomials::Polynomial< double > > &polynomials_1d, const std::vector< unsigned int > &renumber_lexicographic_to_hierarchic, internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &output_data)
void transform_fields(const ArrayView< const Tensor< rank, dim > > &input, const MappingKind mapping_kind, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_data, const ArrayView< Tensor< rank, spacedim > > &output)
void transform_gradients(const ArrayView< const Tensor< rank, dim > > &input, const MappingKind mapping_kind, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_data, const ArrayView< Tensor< rank, spacedim > > &output)
void transform_hessians(const ArrayView< const Tensor< 3, dim > > &input, const MappingKind mapping_kind, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_data, const ArrayView< Tensor< 3, spacedim > > &output)
void maybe_compute_face_data(const ::MappingQ< dim, spacedim > &mapping, const typename ::Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const unsigned int subface_no, const unsigned int n_q_points, const std::vector< double > &weights, const typename ::MappingQ< dim, spacedim >::InternalData &data, internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &output_data)
static const unsigned int invalid_unsigned_int
::VectorizedArray< Number, width > sqrt(const ::VectorizedArray< Number, width > &)
static std_cxx20::ranges::iota_view< unsigned int, unsigned int > face_indices()
static double subface_ratio(const internal::SubfaceCase< dim > &subface_case, const unsigned int subface_no)
static double d_linear_shape_function(const Point< dim > &xi, const unsigned int i)
static std_cxx20::ranges::iota_view< unsigned int, unsigned int > vertex_indices()
DEAL_II_HOST constexpr Number determinant(const SymmetricTensor< 2, dim, Number > &)