Namespace GooseFEM::Element::Hex8

namespace Hex8

8-noded hexahedral element in 3d (GooseFEM::Mesh::ElementType::Hex8).

class Quadrature : public GooseFEM::Element::QuadratureBaseCartesian<Quadrature>

#include <GooseFEM/ElementHex8.h>

Interpolation and quadrature.

Fixed dimensions:

• ndim = 3: number of dimensions.

• nne = 8: number of nodes per element.

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, ndim, ndim]

• qscalar: integration point scalar, [nelem, nip]

Public Functions

template<class T>
inline Quadrature(const T &x)

Constructor: use default Gauss integration.

The following is pre-computed during construction:

• the shape functions,

• the shape function gradients (in local and global) coordinates,

• the integration points volumes. They can be reused without any cost. They only have to be recomputed when the nodal position changes (note that they are assumed to be constant under a small-strain assumption). In that case use update_x() to update the nodal positions and to recompute the above listed quantities.

Parameters

x – nodal coordinates (elemvec).

template<class T, class X, class W>
inline Quadrature(const T &x, const X &xi, const W &w)

Constructor with custom integration.

The following is pre-computed during construction:

• the shape functions,

• the shape function gradients (in local and global) coordinates,

• the integration points volumes. They can be reused without any cost. They only have to be recomputed when the nodal position changes (note that they are assumed to be constant under a small-strain assumption). In that case use update_x() to update the nodal positions and to recompute the above listed quantities.

Parameters

• x – nodal coordinates (elemvec).

• xi – Integration point coordinates (local coordinates) [nip].

• w – Integration point weights [nip].

namespace Gauss

gauss quadrature: quadrature points such that integration is exact for these bi-linear elements::

Functions

inline size_t nip()

Number of integration points:

 nip = nne = 8

Returns

unsigned int

inline array_type::tensor<double, 2> xi()

Integration point coordinates (local coordinates).

Returns

Coordinates [nip, ndim], with ndim = 3.

inline array_type::tensor<double, 1> w()

Integration point weights.

Returns

Coordinates [nip].

namespace Nodal

nodal quadrature: quadrature points coincide with the nodes.

The order is the same as in the connectivity.

Functions

inline size_t nip()

Number of integration points:

 nip = nne = 8

Returns

unsigned int

inline array_type::tensor<double, 2> xi()

Integration point coordinates (local coordinates).

Returns

Coordinates [nip, ndim], with ndim = 3.

inline array_type::tensor<double, 1> w()

Integration point weights.

Returns

Coordinates [nip].