CRTP base class for interpolation and quadrature for a generic element in Cartesian coordinates. More...

#include <GooseFEM/Element.h>

Inheritance diagram for GooseFEM::Element::QuadratureBaseCartesian< D >:

Public Types
using	derived_type = D
	Underlying type.

Public Types inherited from GooseFEM::Element::QuadratureBase< D >
using	derived_type = D
	Underlying type.

Public Member Functions
template<class T >
void	update_x (const T &x)
	Update the nodal positions.

auto	GradN () const -> const array_type::tensor< double, 4 > &
	Shape function gradients (in global coordinates).

auto	dV () const -> const array_type::tensor< double, 2 > &
	Integration volume.

template<class T >
auto	InterpQuad_vector (const T &elemvec) const -> array_type::tensor< double, 3 >
	Interpolate element vector and evaluate at each quadrature point.

template<class T , class R >
void	interpQuad_vector (const T &elemvec, R &qvector) const
	Same as InterpQuad_vector(), but writing to preallocated return.

template<class T >
auto	GradN_vector (const T &elemvec) const -> array_type::tensor< double, 4 >
	Element-by-element: dyadic product of the shape function gradients and a nodal vector.

template<class T , class R >
void	gradN_vector (const T &elemvec, R &qtensor) const
	Same as GradN_vector(), but writing to preallocated return.

template<class T >
auto	GradN_vector_T (const T &elemvec) const -> array_type::tensor< double, 4 >
	The transposed output of GradN_vector().

template<class T , class R >
void	gradN_vector_T (const T &elemvec, R &qtensor) const
	Same as GradN_vector_T(), but writing to preallocated return.

template<class T >
auto	SymGradN_vector (const T &elemvec) const -> array_type::tensor< double, 4 >
	The symmetric output of GradN_vector().

template<class T , class R >
void	symGradN_vector (const T &elemvec, R &qtensor) const
	Same as SymGradN_vector(), but writing to preallocated return.

template<class T >
auto	Int_N_vector_dV (const T &qvector) const -> array_type::tensor< double, 3 >
	Element-by-element: integral of a continuous vector-field.

template<class T , class R >
void	int_N_vector_dV (const T &qvector, R &elemvec) const
	Same as Int_N_vector_dV(), but writing to preallocated return.

template<class T >
auto	Int_N_scalar_NT_dV (const T &qscalar) const -> array_type::tensor< double, 3 >
	Element-by-element: integral of the scalar product of the shape function with a scalar.

template<class T , class R >
void	int_N_scalar_NT_dV (const T &qscalar, R &elemmat) const
	Same as Int_N_scalar_NT_dV(), but writing to preallocated return.

template<class T >
auto	Int_gradN_dot_tensor2_dV (const T &qtensor) const -> array_type::tensor< double, 3 >
	Element-by-element: integral of the dot product of the shape function gradients with a second order tensor.

template<class T , class R >
void	int_gradN_dot_tensor2_dV (const T &qtensor, R &elemvec) const
	Same as Int_gradN_dot_tensor2_dV(), but writing to preallocated return.

template<class T >
auto	Int_gradN_dot_tensor4_dot_gradNT_dV (const T &qtensor) const -> array_type::tensor< double, 3 >
	Element-by-element: integral of the dot products of the shape function gradients with a fourth order tensor.

template<class T , class R >
void	int_gradN_dot_tensor4_dot_gradNT_dV (const T &qtensor, R &elemmat) const
	Same as Int_gradN_dot_tensor4_dot_gradNT_dV(), but writing to preallocated return.

Public Member Functions inherited from GooseFEM::Element::QuadratureBase< D >
auto	nelem () const
	Number of elements.

auto	nne () const
	Number of nodes per element.

auto	ndim () const
	Number of dimensions for node vectors.

auto	tdim () const
	Number of dimensions for integration point tensors.

auto	nip () const
	Number of integration points.

template<class T , class R >
void	asTensor (const T &arg, R &ret) const
	Convert "qscalar" to "qtensor" of certain rank.

template<size_t rank, class T >
auto	AsTensor (const T &arg) const
	Convert "qscalar" to "qtensor" of certain rank.

template<class T >
auto	AsTensor (size_t rank, const T &arg) const
	Convert "qscalar" to "qtensor" of certain rank.

auto	shape_elemvec () const -> std::array< size_t, 3 >
	Get the shape of an "elemvec".

auto	shape_elemvec (size_t arg) const -> std::array< size_t, 3 >
	Get the shape of an "elemvec".

auto	shape_elemmat () const -> std::array< size_t, 3 >
	Get the shape of an "elemmat".

template<size_t rank = 0>
auto	shape_qtensor () const -> std::array< size_t, 2+rank >
	Get the shape of a "qtensor" of a certain rank (0 = scalar, 1, vector, 2 = 2nd-order tensor, etc.).

auto	shape_qtensor (size_t rank) const -> std::vector< size_t >
	Get the shape of a "qtensor" of a certain rank (0 = scalar, 1, vector, 2 = 2nd-order tensor, etc.).

template<size_t trank>
auto	shape_qtensor (size_t rank, size_t arg) const -> std::array< size_t, 2+trank >
	Get the shape of a "qtensor" of a certain rank (0 = scalar, 1, vector, 2 = 2nd-order tensor, etc.).

auto	shape_qtensor (size_t rank, size_t arg) const -> std::vector< size_t >
	Get the shape of a "qtensor" of a certain rank (0 = scalar, 1, vector, 2 = 2nd-order tensor, etc.).

auto	shape_qscalar () const -> std::array< size_t, 2 >
	Get the shape of a "qscalar" (a "qtensor" of rank 0)

auto	shape_qvector () const -> std::array< size_t, 3 >
	Get the shape of a "qvector" (a "qtensor" of rank 1)

auto	shape_qvector (size_t arg) const -> std::array< size_t, 3 >
	Get the shape of a "qvector" (a "qtensor" of rank 1)

template<class R >
auto	allocate_elemvec () const
	Get an allocated `array_type::tensor` to store a "elemvec".

template<class R >
auto	allocate_elemvec (R val) const
	Get an allocated and initialised `xt::xarray` to store a "elemvec".

template<class R >
auto	allocate_elemmat () const
	Get an allocated `array_type::tensor` to store a "elemmat".

template<class R >
auto	allocate_elemmat (R val) const
	Get an allocated and initialised `xt::xarray` to store a "elemmat".

template<size_t rank = 0, class R >
auto	allocate_qtensor () const
	Get an allocated `array_type::tensor` to store a "qtensor" of a certain rank (0 = scalar, 1, vector, 2 = 2nd-order tensor, etc.).

template<size_t rank = 0, class R >
auto	allocate_qtensor (R val) const
	Get an allocated and initialised `array_type::tensor` to store a "qtensor" of a certain rank (0 = scalar, 1, vector, 2 = 2nd-order tensor, etc.).

template<class R >
auto	allocate_qtensor (size_t rank) const
	Get an allocated `xt::xarray` to store a "qtensor" of a certain rank (0 = scalar, 1, vector, 2 = 2nd-order tensor, etc.).

template<class R >
auto	allocate_qtensor (size_t rank, R val) const
	Get an allocated and initialised `xt::xarray` to store a "qtensor" of a certain rank (0 = scalar, 1, vector, 2 = 2nd-order tensor, etc.).

template<class R >
auto	allocate_qscalar () const
	Get an allocated `array_type::tensor` to store a "qscalar" (a "qtensor" of rank 0).

template<class R >
auto	allocate_qscalar (R val) const
	Get an allocated and initialised `xt::xarray` to store a "qscalar" (a "qtensor" of rank 0).

Protected Member Functions
void	compute_dN ()
	Update the shape function gradients (called when the nodal positions are updated).

Friends
class	QuadratureBase< D >

Detailed Description

template<class D>
class GooseFEM::Element::QuadratureBaseCartesian< D >

CRTP base class for interpolation and quadrature for a generic element in Cartesian coordinates.

Naming convention:

elemmat: matrices stored per element, [nelem, nne * ndim, nne * ndim]
elemvec: nodal vectors stored per element, [nelem, nne, ndim]
qtensor: integration point tensor, [nelem, nip, tdim, tdim]
qscalar: integration point scalar, [nelem, nip]

Definition at line 546 of file Element.h.

Member Typedef Documentation

◆ derived_type

template<class D >

using GooseFEM::Element::QuadratureBaseCartesian< D >::derived_type = D

Underlying type.

Definition at line 551 of file Element.h.

Member Function Documentation

◆ compute_dN()

template<class D >

void GooseFEM::Element::QuadratureBaseCartesian< D >::compute_dN ( )

inlineprotected

Update the shape function gradients (called when the nodal positions are updated).

Definition at line 874 of file Element.h.

◆ dV()

template<class D >

auto GooseFEM::Element::QuadratureBaseCartesian< D >::dV ( ) const -> const array_type::tensor<double, 2>&

inline

Integration volume.

Returns: volume stored per element, per integration point [nelem, nip].

Definition at line 584 of file Element.h.

◆ GradN()

template<class D >

auto GooseFEM::Element::QuadratureBaseCartesian< D >::GradN ( ) const -> const array_type::tensor<double, 4>&

inline

Shape function gradients (in global coordinates).

Returns: gradN stored per element, per integration point [nelem, nip, nne, ndim].

Definition at line 575 of file Element.h.

◆ GradN_vector()

template<class D >

template<class T >

auto GooseFEM::Element::QuadratureBaseCartesian< D >::GradN_vector ( const T & elemvec ) const -> array_type::tensor<double, 4>

inline

Element-by-element: dyadic product of the shape function gradients and a nodal vector.

Typical input: nodal displacements. Typical output: quadrature point strains. Within one element::

 for e in range(nelem):
     for q in range(nip):
         for m in range(nne):
             qtensor(e, q, i, j) += dNdx(e, q, m, i) * elemvec(e, m, j)

Note that the functions and their gradients are precomputed upon construction, or updated when calling update_x().

Parameters

elemvec [nelem, nne, ndim]

Returns: qtensor [nelem, nip, tdim, tdim]

Definition at line 635 of file Element.h.

◆ gradN_vector()

template<class D >

template<class T , class R >

void GooseFEM::Element::QuadratureBaseCartesian< D >::gradN_vector	(	const T &	elemvec,
		R &	qtensor ) const

inline

Same as GradN_vector(), but writing to preallocated return.

Parameters

elemvec	[nelem, nne, ndim]
qtensor	overwritten [nelem, nip, tdim, tdim]

Definition at line 649 of file Element.h.

◆ GradN_vector_T()

template<class D >

template<class T >

auto GooseFEM::Element::QuadratureBaseCartesian< D >::GradN_vector_T ( const T & elemvec ) const -> array_type::tensor<double, 4>

inline

The transposed output of GradN_vector().

Within one element::

 for e in range(nelem):
     for q in range(nip):
         for m in range(nne):
             qtensor(e, q, j, i) += dNdx(e, q, m, i) * elemvec(e, m, j)

Parameters

elemvec [nelem, nne, ndim]

Returns: qtensor [nelem, nip, tdim, tdim]

Definition at line 667 of file Element.h.

◆ gradN_vector_T()

template<class D >

template<class T , class R >

void GooseFEM::Element::QuadratureBaseCartesian< D >::gradN_vector_T	(	const T &	elemvec,
		R &	qtensor ) const

inline

Same as GradN_vector_T(), but writing to preallocated return.

Parameters

elemvec	[nelem, nne, ndim]
qtensor	overwritten [nelem, nip, tdim, tdim]

Definition at line 681 of file Element.h.

◆ Int_gradN_dot_tensor2_dV()

template<class D >

template<class T >

auto GooseFEM::Element::QuadratureBaseCartesian< D >::Int_gradN_dot_tensor2_dV ( const T & qtensor ) const -> array_type::tensor<double, 3>

inline

Element-by-element: integral of the dot product of the shape function gradients with a second order tensor.

Typical input: stress. Typical output: nodal force. Within one one element::

 for e in range(nelem):
     for q in range(nip):
         for m in range(nne):
             elemvec(e, m, j) += dNdx(e, q, m, i) * qtensor(e, q, i, j) * dV(e, q)

with i and j tensor dimensions. Note that the functions and their gradients are precomputed upon construction, or updated when calling update_x().

Parameters

qtensor [nelem, nip, ndim, ndim]

Returns: elemvec [nelem, nne. ndim]

Definition at line 808 of file Element.h.

◆ int_gradN_dot_tensor2_dV()

template<class D >

template<class T , class R >

void GooseFEM::Element::QuadratureBaseCartesian< D >::int_gradN_dot_tensor2_dV	(	const T &	qtensor,
		R &	elemvec ) const

inline

Same as Int_gradN_dot_tensor2_dV(), but writing to preallocated return.

Parameters

qtensor	[nelem, nip, ndim, ndim]
elemvec	overwritten [nelem, nne. ndim]

Definition at line 822 of file Element.h.

◆ Int_gradN_dot_tensor4_dot_gradNT_dV()

template<class D >

template<class T >

auto GooseFEM::Element::QuadratureBaseCartesian< D >::Int_gradN_dot_tensor4_dot_gradNT_dV ( const T & qtensor ) const -> array_type::tensor<double, 3>

inline

Element-by-element: integral of the dot products of the shape function gradients with a fourth order tensor.

Typical input: stiffness tensor. Typical output: stiffness matrix. Within one one element::

 for e in range(nelem):
     for q in range(nip):
         for m in range(nne):
             for n in range(nne):
                 elemmat(e, m * ndim + j, n * ndim + k) +=
                     dNdx(e,q,m,i) * qtensor(e,q,i,j,k,l) * dNdx(e,q,n,l) * dV(e,q)

with i, j, k, and l tensor dimensions. Note that the functions and their gradients are precomputed upon construction, or updated when calling update_x().

Parameters

qtensor [nelem, nip, ndim, ndim, ndim, ndim]

Returns: elemmat [nelem, nne * ndim, nne * ndim]

Definition at line 850 of file Element.h.

◆ int_gradN_dot_tensor4_dot_gradNT_dV()

template<class D >

template<class T , class R >

void GooseFEM::Element::QuadratureBaseCartesian< D >::int_gradN_dot_tensor4_dot_gradNT_dV	(	const T &	qtensor,
		R &	elemmat ) const

inline

Same as Int_gradN_dot_tensor4_dot_gradNT_dV(), but writing to preallocated return.

Parameters

qtensor	[nelem, nip, ndim, ndim, ndim, ndim]
elemmat	overwritten [nelem, nne * ndim, nne * ndim]

Definition at line 865 of file Element.h.

◆ Int_N_scalar_NT_dV()

template<class D >

template<class T >

auto GooseFEM::Element::QuadratureBaseCartesian< D >::Int_N_scalar_NT_dV ( const T & qscalar ) const -> array_type::tensor<double, 3>

inline

Element-by-element: integral of the scalar product of the shape function with a scalar.

Within one one element::

 for e in range(nelem):
     for q in range(nip):
         for m in range(nne):
             for n in range(nne):
                 elemmat(e, m * ndim + i, n * ndim + i) +=
                     N(e, q, m) * qscalar(e, q) * N(e, q, n) * dV(e, q)

with i a tensor dimension. Note that the functions and their gradients are precomputed upon construction, or updated when calling update_x().

Parameters

qscalar [nelem, nip]

Returns: elemmat [nelem, nne * ndim, nne * ndim]

Definition at line 771 of file Element.h.

◆ int_N_scalar_NT_dV()

template<class D >

template<class T , class R >

void GooseFEM::Element::QuadratureBaseCartesian< D >::int_N_scalar_NT_dV	(	const T &	qscalar,
		R &	elemmat ) const

inline

Same as Int_N_scalar_NT_dV(), but writing to preallocated return.

Parameters

qscalar	[nelem, nip]
elemmat	overwritten [nelem, nne * ndim, nne * ndim]

Definition at line 785 of file Element.h.

◆ Int_N_vector_dV()

template<class D >

template<class T >

auto GooseFEM::Element::QuadratureBaseCartesian< D >::Int_N_vector_dV ( const T & qvector ) const -> array_type::tensor<double, 3>

inline

Element-by-element: integral of a continuous vector-field.

\( \vec{f}_i^e = \int N_i^e(\vec{x}) \vec{f}(\vec{x}) d\Omega_e \)

which is integration numerically as follows

\( \vec{f}_i^e = \sum\limits_q N_i^e(\vec{x}_q) \vec{f}(\vec{x}_q) \)

Parameters

qvector [nelem, nip. ndim]

Returns: elemvec [nelem, nne. ndim]

Definition at line 732 of file Element.h.

◆ int_N_vector_dV()

template<class D >

template<class T , class R >

void GooseFEM::Element::QuadratureBaseCartesian< D >::int_N_vector_dV	(	const T &	qvector,
		R &	elemvec ) const

inline

Same as Int_N_vector_dV(), but writing to preallocated return.

Parameters

qvector	[nelem, nip. ndim]
elemvec	overwritten [nelem, nne. ndim]

Definition at line 747 of file Element.h.

◆ InterpQuad_vector()

template<class D >

template<class T >

auto GooseFEM::Element::QuadratureBaseCartesian< D >::InterpQuad_vector ( const T & elemvec ) const -> array_type::tensor<double, 3>

inline

Interpolate element vector and evaluate at each quadrature point.

\( \vec{u}(\vec{x}_q) = N_i^e(\vec{x}) \vec{u}_i^e \)

Parameters

elemvec nodal vector stored per element [nelem, nne, ndim].

Returns: qvector [nelem, nip, ndim].

Definition at line 598 of file Element.h.

◆ interpQuad_vector()

template<class D >

template<class T , class R >

void GooseFEM::Element::QuadratureBaseCartesian< D >::interpQuad_vector	(	const T &	elemvec,
		R &	qvector ) const

inline

Same as InterpQuad_vector(), but writing to preallocated return.

Parameters

elemvec	nodal vector stored per element [nelem, nne, ndim].
qvector	[nelem, nip, ndim].

Definition at line 613 of file Element.h.

◆ SymGradN_vector()

template<class D >

template<class T >

auto GooseFEM::Element::QuadratureBaseCartesian< D >::SymGradN_vector ( const T & elemvec ) const -> array_type::tensor<double, 4>

inline

The symmetric output of GradN_vector().

Without one element::

 for e in range(nelem):
     for q in range(nip):
         for m in range(nne):
             qtensor(e, q, i, j) += 0.5 * dNdx(e, q, m, i) * elemvec(e, m, j)
             qtensor(e, q, j, i) += 0.5 * dNdx(e, q, m, i) * elemvec(e, m, j)

Parameters

elemvec [nelem, nne, ndim]

Returns: qtensor [nelem, nip, tdim, tdim]

Definition at line 700 of file Element.h.

◆ symGradN_vector()

template<class D >

template<class T , class R >

void GooseFEM::Element::QuadratureBaseCartesian< D >::symGradN_vector	(	const T &	elemvec,
		R &	qtensor ) const

inline

Same as SymGradN_vector(), but writing to preallocated return.

Parameters

elemvec	[nelem, nne, ndim]
qtensor	overwritten [nelem, nip, tdim, tdim]

Definition at line 714 of file Element.h.

◆ update_x()

template<class D >

template<class T >

void GooseFEM::Element::QuadratureBaseCartesian< D >::update_x ( const T & x )

inline

Update the nodal positions.

This recomputes:

the shape functions,
the shape function gradients (in local and global) coordinates,
the integration points volumes. Under the small deformations assumption this function should not be called.

Parameters

x	nodal coordinates (`elemvec`). Shape should match the earlier definition.

Definition at line 564 of file Element.h.

Friends And Related Symbol Documentation

◆ QuadratureBase< D >

template<class D >

friend class QuadratureBase< D >

friend

Definition at line 885 of file Element.h.

The documentation for this class was generated from the following file:

GooseFEM/Element.h

Public Types

Public Member Functions

Protected Member Functions

Friends

Detailed Description

Member Typedef Documentation

◆ derived_type

Member Function Documentation

◆ compute_dN()

◆ dV()

◆ GradN()

◆ GradN_vector()

◆ gradN_vector()

◆ GradN_vector_T()

◆ gradN_vector_T()

◆ Int_gradN_dot_tensor2_dV()

◆ int_gradN_dot_tensor2_dV()

◆ Int_gradN_dot_tensor4_dot_gradNT_dV()

◆ int_gradN_dot_tensor4_dot_gradNT_dV()

◆ Int_N_scalar_NT_dV()

◆ int_N_scalar_NT_dV()

◆ Int_N_vector_dV()

◆ int_N_vector_dV()

◆ InterpQuad_vector()

◆ interpQuad_vector()

◆ SymGradN_vector()

◆ symGradN_vector()

◆ update_x()

Friends And Related Symbol Documentation

◆ QuadratureBase< D >