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ElementHex8.h
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ElementHex8.h
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/* =================================================================================================
(c - GPLv3) T.W.J. de Geus (Tom) | tom@geus.me | www.geus.me | github.com/tdegeus/GooseFEM
================================================================================================= */
#ifndef XGOOSEFEM_ELEMENTHEX8_H
#define XGOOSEFEM_ELEMENTHEX8_H
// -------------------------------------------------------------------------------------------------
#include "GooseFEM.h"
// ==================================== GooseFEM::Element::Hex8 ====================================
namespace xGooseFEM {
namespace Element {
namespace Hex8 {
// ======================================== tensor algebra =========================================
using T2 = xt::xtensor_fixed<double, xt::xshape<3,3>>;
inline double inv(const T2 &A, T2 &Ainv);
// ================================ GooseFEM::Element::Hex8::Gauss =================================
namespace Gauss {
inline size_t nip(); // number of integration points
inline xt::xtensor<double,2> xi(); // integration point coordinates (local coordinates)
inline xt::xtensor<double,1> w(); // integration point weights
}
// ================================ GooseFEM::Element::Hex8::Nodal =================================
namespace Nodal {
inline size_t nip(); // number of integration points
inline xt::xtensor<double,2> xi(); // integration point coordinates (local coordinates)
inline xt::xtensor<double,1> w(); // integration point weights
}
// =================================================================================================
// ------------------------------------------ quadrature -------------------------------------------
class Quadrature
{
private:
// dimensions (flexible)
size_t m_nelem; // number of elements
size_t m_nip; // number of integration points
// dimensions (fixed for this element type)
static const size_t m_nne=8; // number of nodes per element
static const size_t m_ndim=3; // number of dimensions
// data arrays
xt::xtensor<double,3> m_x; // nodal positions stored per element [nelem, nne, ndim]
xt::xtensor<double,1> m_w; // weight of each integration point [nip]
xt::xtensor<double,2> m_xi; // local coordinate of each integration point [nip, ndim]
xt::xtensor<double,2> m_N; // shape functions [nip, nne]
xt::xtensor<double,3> m_dNxi; // shape function gradients w.r.t. local coordinate [nip, nne, ndim]
xt::xtensor<double,4> m_dNx; // shape function gradients w.r.t. global coordinate [nelem, nip, nne, ndim]
xt::xtensor<double,2> m_vol; // integration point volume [nelem, nip]
private:
// compute "vol" and "dNdx" based on current "x"
void compute_dN();
public:
// 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]
// constructor: integration point coordinates and weights are optional (default: Gauss)
Quadrature() = default;
Quadrature(const xt::xtensor<double,3> &x);
Quadrature(const xt::xtensor<double,3> &x, const xt::xtensor<double,2> &xi, const xt::xtensor<double,1> &w);
// update the nodal positions (shape of "x" should match the earlier definition)
void update_x(const xt::xtensor<double,3> &x);
// return dimensions
size_t nelem() const; // number of elements
size_t nne() const; // number of nodes per element
size_t ndim() const; // number of dimension
size_t nip() const; // number of integration points
// return integration volume
// - in-place
void dV(xt::xtensor<double,2> &qscalar) const;
void dV(xt::xtensor<double,4> &qtensor) const;
// - return qscalar/qtensor
xt::xtensor<double,2> dV() const;
xt::xtensor<double,4> dVtensor() const;
// dyadic product "qtensor(i,j) += dNdx(m,i) * elemvec(m,j)", its transpose and its symmetric part
// - in-place
void gradN_vector (const xt::xtensor<double,3> &elemvec, xt::xtensor<double,4> &qtensor) const;
void gradN_vector_T (const xt::xtensor<double,3> &elemvec, xt::xtensor<double,4> &qtensor) const;
void symGradN_vector(const xt::xtensor<double,3> &elemvec, xt::xtensor<double,4> &qtensor) const;
// - return qtensor
xt::xtensor<double,4> gradN_vector (const xt::xtensor<double,3> &elemvec) const;
xt::xtensor<double,4> gradN_vector_T (const xt::xtensor<double,3> &elemvec) const;
xt::xtensor<double,4> symGradN_vector(const xt::xtensor<double,3> &elemvec) const;
// integral of the scalar product "elemmat(m*ndim+i,n*ndim+i) += N(m) * qscalar * N(n) * dV"
// - in-place
void int_N_scalar_NT_dV(const xt::xtensor<double,2> &qscalar, xt::xtensor<double,3> &elemmat) const;
// - return elemmat
xt::xtensor<double,3> int_N_scalar_NT_dV(const xt::xtensor<double,2> &qscalar) const;
// integral of the dot product "elemvec(m,j) += dNdx(m,i) * qtensor(i,j) * dV"
// - in-place
void int_gradN_dot_tensor2_dV(const xt::xtensor<double,4> &qtensor, xt::xtensor<double,3> &elemvec) const;
// - return elemvec
xt::xtensor<double,3> int_gradN_dot_tensor2_dV(const xt::xtensor<double,4> &qtensor) const;
};
// -------------------------------------------------------------------------------------------------
}}} // namespace ...
// =================================================================================================
#endif

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