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Reference documentation for deal.II version 9.1.0-pre
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Reference documentation for deal.II version 9.1.0-pre
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#include <deal.II/fe/fe_poly.h>
Classes | |
| class | InternalData |
Public Member Functions | |
| FE_Poly (const PolynomialType &poly_space, const FiniteElementData< dim > &fe_data, const std::vector< bool > &restriction_is_additive_flags, const std::vector< ComponentMask > &nonzero_components) | |
| unsigned int | get_degree () const |
| virtual UpdateFlags | requires_update_flags (const UpdateFlags update_flags) const override |
| std::vector< unsigned int > | get_poly_space_numbering () const |
| std::vector< unsigned int > | get_poly_space_numbering_inverse () const |
| virtual double | shape_value (const unsigned int i, const Point< dim > &p) const override |
| virtual double | shape_value_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override |
| virtual Tensor< 1, dim > | shape_grad (const unsigned int i, const Point< dim > &p) const override |
| virtual Tensor< 1, dim > | shape_grad_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override |
| virtual Tensor< 2, dim > | shape_grad_grad (const unsigned int i, const Point< dim > &p) const override |
| virtual Tensor< 2, dim > | shape_grad_grad_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override |
| virtual Tensor< 3, dim > | shape_3rd_derivative (const unsigned int i, const Point< dim > &p) const override |
| virtual Tensor< 3, dim > | shape_3rd_derivative_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override |
| virtual Tensor< 4, dim > | shape_4th_derivative (const unsigned int i, const Point< dim > &p) const override |
| virtual Tensor< 4, dim > | shape_4th_derivative_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override |
 Public Member Functions inherited from FiniteElement< dim, spacedim > | |
| FiniteElement (const FiniteElementData< dim > &fe_data, const std::vector< bool > &restriction_is_additive_flags, const std::vector< ComponentMask > &nonzero_components) | |
| FiniteElement (FiniteElement< dim, spacedim > &&)=default | |
| FiniteElement (const FiniteElement< dim, spacedim > &)=default | |
| virtual | ~FiniteElement () override=default |
| std::pair< std::unique_ptr< FiniteElement< dim, spacedim > >, unsigned int > | operator^ (const unsigned int multiplicity) const |
| virtual std::unique_ptr< FiniteElement< dim, spacedim > > | clone () const =0 |
| virtual std::string | get_name () const =0 |
| const FiniteElement< dim, spacedim > & | operator[] (const unsigned int fe_index) const |
| virtual bool | operator== (const FiniteElement< dim, spacedim > &fe) const |
| bool | operator!= (const FiniteElement< dim, spacedim > &) const |
| virtual std::size_t | memory_consumption () const |
| virtual bool | has_support_on_face (const unsigned int shape_index, const unsigned int face_index) const |
| virtual const FullMatrix< double > & | get_restriction_matrix (const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const |
| virtual const FullMatrix< double > & | get_prolongation_matrix (const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const |
| bool | prolongation_is_implemented () const |
| bool | isotropic_prolongation_is_implemented () const |
| bool | restriction_is_implemented () const |
| bool | isotropic_restriction_is_implemented () const |
| bool | restriction_is_additive (const unsigned int index) const |
| const FullMatrix< double > & | constraints (const ::internal::SubfaceCase< dim > &subface_case=::internal::SubfaceCase< dim >::case_isotropic) const |
| bool | constraints_are_implemented (const ::internal::SubfaceCase< dim > &subface_case=::internal::SubfaceCase< dim >::case_isotropic) const |
| virtual bool | hp_constraints_are_implemented () const |
| virtual void | get_interpolation_matrix (const FiniteElement< dim, spacedim > &source, FullMatrix< double > &matrix) const |
| virtual void | get_face_interpolation_matrix (const FiniteElement< dim, spacedim > &source, FullMatrix< double > &matrix) const |
| virtual void | get_subface_interpolation_matrix (const FiniteElement< dim, spacedim > &source, const unsigned int subface, FullMatrix< double > &matrix) const |
| virtual std::vector< std::pair< unsigned int, unsigned int > > | hp_vertex_dof_identities (const FiniteElement< dim, spacedim > &fe_other) const |
| virtual std::vector< std::pair< unsigned int, unsigned int > > | hp_line_dof_identities (const FiniteElement< dim, spacedim > &fe_other) const |
| virtual std::vector< std::pair< unsigned int, unsigned int > > | hp_quad_dof_identities (const FiniteElement< dim, spacedim > &fe_other) const |
| virtual FiniteElementDomination::Domination | compare_for_face_domination (const FiniteElement< dim, spacedim > &fe_other) const |
| std::pair< unsigned int, unsigned int > | system_to_component_index (const unsigned int index) const |
| unsigned int | component_to_system_index (const unsigned int component, const unsigned int index) const |
| std::pair< unsigned int, unsigned int > | face_system_to_component_index (const unsigned int index) const |
| unsigned int | adjust_quad_dof_index_for_face_orientation (const unsigned int index, const bool face_orientation, const bool face_flip, const bool face_rotation) const |
| virtual unsigned int | face_to_cell_index (const unsigned int face_dof_index, const unsigned int face, const bool face_orientation=true, const bool face_flip=false, const bool face_rotation=false) const |
| unsigned int | adjust_line_dof_index_for_line_orientation (const unsigned int index, const bool line_orientation) const |
| const ComponentMask & | get_nonzero_components (const unsigned int i) const |
| unsigned int | n_nonzero_components (const unsigned int i) const |
| bool | is_primitive () const |
| bool | is_primitive (const unsigned int i) const |
| unsigned int | n_base_elements () const |
| virtual const FiniteElement< dim, spacedim > & | base_element (const unsigned int index) const |
| unsigned int | element_multiplicity (const unsigned int index) const |
| const FiniteElement< dim, spacedim > & | get_sub_fe (const ComponentMask &mask) const |
| virtual const FiniteElement< dim, spacedim > & | get_sub_fe (const unsigned int first_component, const unsigned int n_selected_components) const |
| std::pair< std::pair< unsigned int, unsigned int >, unsigned int > | system_to_base_index (const unsigned int index) const |
| std::pair< std::pair< unsigned int, unsigned int >, unsigned int > | face_system_to_base_index (const unsigned int index) const |
| types::global_dof_index | first_block_of_base (const unsigned int b) const |
| std::pair< unsigned int, unsigned int > | component_to_base_index (const unsigned int component) const |
| std::pair< unsigned int, unsigned int > | block_to_base_index (const unsigned int block) const |
| std::pair< unsigned int, types::global_dof_index > | system_to_block_index (const unsigned int component) const |
| unsigned int | component_to_block_index (const unsigned int component) const |
| ComponentMask | component_mask (const FEValuesExtractors::Scalar &scalar) const |
| ComponentMask | component_mask (const FEValuesExtractors::Vector &vector) const |
| ComponentMask | component_mask (const FEValuesExtractors::SymmetricTensor< 2 > &sym_tensor) const |
| ComponentMask | component_mask (const BlockMask &block_mask) const |
| BlockMask | block_mask (const FEValuesExtractors::Scalar &scalar) const |
| BlockMask | block_mask (const FEValuesExtractors::Vector &vector) const |
| BlockMask | block_mask (const FEValuesExtractors::SymmetricTensor< 2 > &sym_tensor) const |
| BlockMask | block_mask (const ComponentMask &component_mask) const |
| virtual std::pair< Table< 2, bool >, std::vector< unsigned int > > | get_constant_modes () const |
| const std::vector< Point< dim > > & | get_unit_support_points () const |
| bool | has_support_points () const |
| virtual Point< dim > | unit_support_point (const unsigned int index) const |
| const std::vector< Point< dim-1 > > & | get_unit_face_support_points () const |
| bool | has_face_support_points () const |
| virtual Point< dim-1 > | unit_face_support_point (const unsigned int index) const |
| const std::vector< Point< dim > > & | get_generalized_support_points () const |
| bool | has_generalized_support_points () const |
| const std::vector< Point< dim-1 > > & | get_generalized_face_support_points () const |
| bool | has_generalized_face_support_points () const |
| GeometryPrimitive | get_associated_geometry_primitive (const unsigned int cell_dof_index) const |
| virtual void | convert_generalized_support_point_values_to_dof_values (const std::vector< Vector< double >> &support_point_values, std::vector< double > &nodal_values) const |
| Public Member Functions inherited from Subscriptor | |
| Subscriptor () | |
| Subscriptor (const Subscriptor &) | |
| Subscriptor (Subscriptor &&) noexcept | |
| virtual | ~Subscriptor () |
| Subscriptor & | operator= (const Subscriptor &) |
| Subscriptor & | operator= (Subscriptor &&) noexcept |
| void | subscribe (const char *identifier=nullptr) const |
| void | unsubscribe (const char *identifier=nullptr) const |
| unsigned int | n_subscriptions () const |
| template<typename StreamType > | |
| void | list_subscribers (StreamType &stream) const |
| void | list_subscribers () const |
| template<class Archive > | |
| void | serialize (Archive &ar, const unsigned int version) |
| Public Member Functions inherited from FiniteElementData< dim > | |
| FiniteElementData (const std::vector< unsigned int > &dofs_per_object, const unsigned int n_components, const unsigned int degree, const Conformity conformity=unknown, const BlockIndices &block_indices=BlockIndices()) | |
| unsigned int | n_dofs_per_vertex () const |
| unsigned int | n_dofs_per_line () const |
| unsigned int | n_dofs_per_quad () const |
| unsigned int | n_dofs_per_hex () const |
| unsigned int | n_dofs_per_face () const |
| unsigned int | n_dofs_per_cell () const |
| template<int structdim> | |
| unsigned int | n_dofs_per_object () const |
| unsigned int | n_components () const |
| unsigned int | n_blocks () const |
| const BlockIndices & | block_indices () const |
| unsigned int | tensor_degree () const |
| bool | conforms (const Conformity) const |
| bool | operator== (const FiniteElementData &) const |
Protected Member Functions | |
| virtual std::unique_ptr< typename FiniteElement< dim, spacedim >::InternalDataBase > | get_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &, const Quadrature< dim > &quadrature,::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const override |
| void | correct_third_derivatives (internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data, const internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const unsigned int n_q_points, const unsigned int dof) const |
| Protected Member Functions inherited from FiniteElement< dim, spacedim > | |
| void | reinit_restriction_and_prolongation_matrices (const bool isotropic_restriction_only=false, const bool isotropic_prolongation_only=false) |
| TableIndices< 2 > | interface_constraints_size () const |
| virtual std::unique_ptr< InternalDataBase > | get_face_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &mapping, const Quadrature< dim-1 > &quadrature,::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const |
| virtual std::unique_ptr< InternalDataBase > | get_subface_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &mapping, const Quadrature< dim-1 > &quadrature,::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const |
| virtual void | fill_fe_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const CellSimilarity::Similarity cell_similarity, const Quadrature< dim > &quadrature, const Mapping< dim, spacedim > &mapping, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const InternalDataBase &fe_internal,::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const =0 |
| virtual void | fill_fe_face_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const Quadrature< dim-1 > &quadrature, const Mapping< dim, spacedim > &mapping, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const InternalDataBase &fe_internal,::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const =0 |
| virtual void | fill_fe_subface_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const unsigned int sub_no, const Quadrature< dim-1 > &quadrature, const Mapping< dim, spacedim > &mapping, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const InternalDataBase &fe_internal,::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const =0 |
Protected Attributes | |
| PolynomialType | poly_space |
| Protected Attributes inherited from FiniteElement< dim, spacedim > | |
| std::vector< std::vector< FullMatrix< double > > > | restriction |
| std::vector< std::vector< FullMatrix< double > > > | prolongation |
| FullMatrix< double > | interface_constraints |
| std::vector< Point< dim > > | unit_support_points |
| std::vector< Point< dim-1 > > | unit_face_support_points |
| std::vector< Point< dim > > | generalized_support_points |
| std::vector< Point< dim-1 > > | generalized_face_support_points |
| Table< 2, int > | adjust_quad_dof_index_for_face_orientation_table |
| std::vector< int > | adjust_line_dof_index_for_line_orientation_table |
| std::vector< std::pair< unsigned int, unsigned int > > | system_to_component_table |
| std::vector< std::pair< unsigned int, unsigned int > > | face_system_to_component_table |
| std::vector< std::pair< std::pair< unsigned int, unsigned int >, unsigned int > > | system_to_base_table |
| std::vector< std::pair< std::pair< unsigned int, unsigned int >, unsigned int > > | face_system_to_base_table |
| BlockIndices | base_to_block_indices |
| std::vector< std::pair< std::pair< unsigned int, unsigned int >, unsigned int > > | component_to_base_table |
| const std::vector< bool > | restriction_is_additive_flags |
| const std::vector< ComponentMask > | nonzero_components |
| const std::vector< unsigned int > | n_nonzero_components_table |
| const bool | cached_primitivity |
Additional Inherited Members | |
| Public Types inherited from FiniteElementData< dim > | |
| Static Public Member Functions inherited from FiniteElement< dim, spacedim > | |
| static::ExceptionBase & | ExcShapeFunctionNotPrimitive (int arg1) |
| static::ExceptionBase & | ExcFENotPrimitive () |
| static::ExceptionBase & | ExcUnitShapeValuesDoNotExist () |
| static::ExceptionBase & | ExcFEHasNoSupportPoints () |
| static::ExceptionBase & | ExcEmbeddingVoid () |
| static::ExceptionBase & | ExcProjectionVoid () |
| static::ExceptionBase & | ExcWrongInterfaceMatrixSize (int arg1, int arg2) |
| static::ExceptionBase & | ExcInterpolationNotImplemented () |
| Static Public Member Functions inherited from Subscriptor | |
| static::ExceptionBase & | ExcInUse (int arg1, std::string arg2, std::string arg3) |
| static::ExceptionBase & | ExcNoSubscriber (std::string arg1, std::string arg2) |
| Public Attributes inherited from FiniteElementData< dim > | |
| const unsigned int | dofs_per_vertex |
| const unsigned int | dofs_per_line |
| const unsigned int | dofs_per_quad |
| const unsigned int | dofs_per_hex |
| const unsigned int | first_line_index |
| const unsigned int | first_quad_index |
| const unsigned int | first_hex_index |
| const unsigned int | first_face_line_index |
| const unsigned int | first_face_quad_index |
| const unsigned int | dofs_per_face |
| const unsigned int | dofs_per_cell |
| const unsigned int | components |
| const unsigned int | degree |
| const Conformity | conforming_space |
| const BlockIndices | block_indices_data |
| Static Public Attributes inherited from FiniteElement< dim, spacedim > | |
| static const unsigned int | space_dimension = spacedim |
| Static Public Attributes inherited from FiniteElementData< dim > | |
| static const unsigned int | dimension = dim |
| Static Protected Member Functions inherited from FiniteElement< dim, spacedim > | |
| static std::vector< unsigned int > | compute_n_nonzero_components (const std::vector< ComponentMask > &nonzero_components) |
This class gives a unified framework for the implementation of FiniteElement classes based on polynomial spaces like the TensorProductPolynomials or PolynomialSpace classes.
Every class that has the following public member variables and functions can be used as template parameter PolynomialType.
Example classes are TensorProductPolynomials, PolynomialSpace or PolynomialsP.
This class is not a fully implemented FiniteElement class. Instead there are several pure virtual functions declared in the FiniteElement and FiniteElement classes which cannot be implemented by this class but are left for implementation in derived classes.
| FE_Poly< PolynomialType, dim, spacedim >::FE_Poly | ( | const PolynomialType & | poly_space, |
| const FiniteElementData< dim > & | fe_data, | ||
| const std::vector< bool > & | restriction_is_additive_flags, | ||
| const std::vector< ComponentMask > & | nonzero_components | ||
| ) |
Constructor.
| unsigned int FE_Poly< PolynomialType, dim, spacedim >::get_degree | ( | ) | const |
Return the polynomial degree of this finite element, i.e. the value passed to the constructor.
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overridevirtual |
Given a set of update flags, compute which other quantities also need to be computed in order to satisfy the request by the given flags. Then return the combination of the original set of flags and those just computed.
As an example, if update_flags contains update_gradients a finite element class will typically require the computation of the inverse of the Jacobian matrix in order to rotate the gradient of shape functions on the reference cell to the real cell. It would then return not just update_gradients, but also update_covariant_transformation, the flag that makes the mapping class produce the inverse of the Jacobian matrix.
An extensive discussion of the interaction between this function and FEValues can be found in the How Mapping, FiniteElement, and FEValues work together documentation module.
Implements FiniteElement< dim, spacedim >.
| std::vector<unsigned int> FE_Poly< PolynomialType, dim, spacedim >::get_poly_space_numbering | ( | ) | const |
Return the numbering of the underlying polynomial space compared to lexicographic ordering of the basis functions. Returns PolynomialType::get_numbering().
| std::vector<unsigned int> FE_Poly< PolynomialType, dim, spacedim >::get_poly_space_numbering_inverse | ( | ) | const |
Return the inverse numbering of the underlying polynomial space. Returns PolynomialType::get_numbering_inverse().
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Return the value of the ith shape function at the point p. See the FiniteElement base class for more information about the semantics of this function.
Reimplemented from FiniteElement< dim, spacedim >.
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Return the value of the componentth vector component of the ith shape function at the point p. See the FiniteElement base class for more information about the semantics of this function.
Since this element is scalar, the returned value is the same as if the function without the _component suffix were called, provided that the specified component is zero.
Reimplemented from FiniteElement< dim, spacedim >.
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overridevirtual |
Return the gradient of the ith shape function at the point p. See the FiniteElement base class for more information about the semantics of this function.
Reimplemented from FiniteElement< dim, spacedim >.
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overridevirtual |
Return the gradient of the componentth vector component of the ith shape function at the point p. See the FiniteElement base class for more information about the semantics of this function.
Since this element is scalar, the returned value is the same as if the function without the _component suffix were called, provided that the specified component is zero.
Reimplemented from FiniteElement< dim, spacedim >.
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Return the tensor of second derivatives of the ith shape function at point p on the unit cell. See the FiniteElement base class for more information about the semantics of this function.
Reimplemented from FiniteElement< dim, spacedim >.
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overridevirtual |
Return the second derivative of the componentth vector component of the ith shape function at the point p. See the FiniteElement base class for more information about the semantics of this function.
Since this element is scalar, the returned value is the same as if the function without the _component suffix were called, provided that the specified component is zero.
Reimplemented from FiniteElement< dim, spacedim >.
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overridevirtual |
Return the tensor of third derivatives of the ith shape function at point p on the unit cell. See the FiniteElement base class for more information about the semantics of this function.
Reimplemented from FiniteElement< dim, spacedim >.
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overridevirtual |
Return the third derivative of the componentth vector component of the ith shape function at the point p. See the FiniteElement base class for more information about the semantics of this function.
Since this element is scalar, the returned value is the same as if the function without the _component suffix were called, provided that the specified component is zero.
Reimplemented from FiniteElement< dim, spacedim >.
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Return the tensor of fourth derivatives of the ith shape function at point p on the unit cell. See the FiniteElement base class for more information about the semantics of this function.
Reimplemented from FiniteElement< dim, spacedim >.
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overridevirtual |
Return the fourth derivative of the componentth vector component of the ith shape function at the point p. See the FiniteElement base class for more information about the semantics of this function.
Since this element is scalar, the returned value is the same as if the function without the _component suffix were called, provided that the specified component is zero.
Reimplemented from FiniteElement< dim, spacedim >.
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inlineoverrideprotectedvirtual |
Create an internal data object and return a pointer to it of which the caller of this function then assumes ownership. This object will then be passed to the FiniteElement::fill_fe_values() every time the finite element shape functions and their derivatives are evaluated on a concrete cell. The object created here is therefore used by derived classes as a place for scratch objects that are used in evaluating shape functions, as well as to store information that can be pre-computed once and re-used on every cell (e.g., for evaluating the values and gradients of shape functions on the reference cell, for later re-use when transforming these values to a concrete cell).
This function is the first one called in the process of initializing a FEValues object for a given mapping and finite element object. The returned object will later be passed to FiniteElement::fill_fe_values() for a concrete cell, which will itself place its output into an object of type internal::FEValuesImplementation::FiniteElementRelatedData. Since there may be data that can already be computed in its final form on the reference cell, this function also receives a reference to the internal::FEValuesImplementation::FiniteElementRelatedData object as its last argument. This output argument is guaranteed to always be the same one when used with the InternalDataBase object returned by this function. In other words, the subdivision of scratch data and final data in the returned object and the output_data object is as follows: If data can be pre- computed on the reference cell in the exact form in which it will later be needed on a concrete cell, then this function should already emplace it in the output_data object. An example are the values of shape functions at quadrature points for the usual Lagrange elements which on a concrete cell are identical to the ones on the reference cell. On the other hand, if some data can be pre-computed to make computations on a concrete cell cheaper, then it should be put into the returned object for later re-use in a derive class's implementation of FiniteElement::fill_fe_values(). An example are the gradients of shape functions on the reference cell for Lagrange elements: to compute the gradients of the shape functions on a concrete cell, one has to multiply the gradients on the reference cell by the inverse of the Jacobian of the mapping; consequently, we cannot already compute the gradients on a concrete cell at the time the current function is called, but we can at least pre-compute the gradients on the reference cell, and store it in the object returned.
An extensive discussion of the interaction between this function and FEValues can be found in the How Mapping, FiniteElement, and FEValues work together documentation module. See also the documentation of the InternalDataBase class.
| [in] | update_flags | A set of UpdateFlags values that describe what kind of information the FEValues object requests the finite element to compute. This set of flags may also include information that the finite element can not compute, e.g., flags that pertain to data produced by the mapping. An implementation of this function needs to set up all data fields in the returned object that are necessary to produce the finite- element related data specified by these flags, and may already pre- compute part of this information as discussed above. Elements may want to store these update flags (or a subset of these flags) in InternalDataBase::update_each so they know at the time when FiniteElement::fill_fe_values() is called what they are supposed to compute |
| [in] | mapping | A reference to the mapping used for computing values and derivatives of shape functions. |
| [in] | quadrature | A reference to the object that describes where the shape functions should be evaluated. |
| [out] | output_data | A reference to the object that FEValues will use in conjunction with the object returned here and where an implementation of FiniteElement::fill_fe_values() will place the requested information. This allows the current function to already pre-compute pieces of information that can be computed on the reference cell, as discussed above. FEValues guarantees that this output object and the object returned by the current function will always be used together. |
Implements FiniteElement< dim, spacedim >.
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protected |
Correct the shape third derivatives by subtracting the terms corresponding to the Jacobian pushed forward gradient and second derivative.
Before the correction, the third derivatives would be given by
\[ D_{ijkl} = \frac{d^3\phi_i}{d \hat x_J d \hat x_K d \hat x_L} (J_{jJ})^{-1} (J_{kK})^{-1} (J_{lL})^{-1}, \]
where \(J_{iI}=\frac{d x_i}{d \hat x_I}\). After the correction, the correct third derivative would be given by
\[ \frac{d^3\phi_i}{d x_j d x_k d x_l} = D_{ijkl} - H_{mjl} \frac{d^2 \phi_i}{d x_k d x_m} - H_{mkl} \frac{d^2 \phi_i}{d x_j d x_m} - H_{mjk} \frac{d^2 \phi_i}{d x_l d x_m} - K_{mjkl} \frac{d \phi_i}{d x_m}, \]
where \(H_{ijk}\) is the Jacobian pushed-forward derivative and \(K_{ijkl}\) is the Jacobian pushed-forward second derivative.
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protected |
1.8.11
