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ElementHex8.h
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rGOOSEFEM GooseFEM
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 GOOSEFEM_ELEMENTHEX8_H
#define GOOSEFEM_ELEMENTHEX8_H
#include "config.h"
namespace GooseFEM {
namespace Element {
namespace Hex8 {
template <class T>
inline double inv(const T& A, T& Ainv);
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
} // namespace Gauss
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
} // namespace Nodal
class Quadrature {
public:
// 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]
// 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 shape function gradients
xt::xtensor<double, 4> GradN() const;
// Convert "qscalar" to "qtensor" of certain rank
template <size_t rank = 0>
void asTensor(const xt::xtensor<double, 2>& qscalar, xt::xtensor<double, 2 + rank>& qtensor) const;
// Return integration volume
xt::xtensor<double, 2> dV() const;
// Dyadic product (and its transpose and symmetric part)
// qtensor(i,j) += dNdx(m,i) * elemvec(m,j)
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;
// Integral of the scalar product
// elemmat(m*ndim+i,n*ndim+i) += N(m) * qscalar * N(n) * dV
void int_N_scalar_NT_dV(
const xt::xtensor<double, 2>& qscalar, xt::xtensor<double, 3>& elemmat) const;
// Integral of the dot product
// elemvec(m,j) += dNdx(m,i) * qtensor(i,j) * dV
void int_gradN_dot_tensor2_dV(
const xt::xtensor<double, 4>& qtensor, xt::xtensor<double, 3>& elemvec) const;
// Integral of the dot product
// elemmat(m*2+j, n*2+k) += dNdx(m,i) * qtensor(i,j,k,l) * dNdx(n,l) * dV
void int_gradN_dot_tensor4_dot_gradNT_dV(
const xt::xtensor<double, 6>& qtensor, xt::xtensor<double, 3>& elemmat) const;
// Auto-allocation of the functions above
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;
xt::xtensor<double, 3> Int_N_scalar_NT_dV(const xt::xtensor<double, 2>& qscalar) const;
xt::xtensor<double, 3> Int_gradN_dot_tensor2_dV(const xt::xtensor<double, 4>& qtensor) const;
xt::xtensor<double, 3> Int_gradN_dot_tensor4_dot_gradNT_dV(const xt::xtensor<double, 6>& qtensor) const;
// Convert "qscalar" to "qtensor" of certain rank
template <size_t rank = 0>
xt::xtensor<double, 2 + rank> AsTensor(const xt::xtensor<double, 2>& qscalar) const;
xt::xarray<double> AsTensor(size_t rank, const xt::xtensor<double, 2>& qscalar) const;
template <size_t rank = 0>
xt::xtensor<double, rank + 2> AllocateQtensor() const;
template <size_t rank = 0>
xt::xtensor<double, rank + 2> AllocateQtensor(double val) const;
xt::xarray<double> AllocateQtensor(size_t rank) const;
xt::xarray<double> AllocateQtensor(size_t rank, double val) const;
xt::xtensor<double, 2> AllocateQscalar() const;
xt::xtensor<double, 2> AllocateQscalar(double val) const;
private:
// Compute "vol" and "dNdx" based on current "x"
void compute_dN();
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 grad. wrt local coor. [nip, nne, ndim]
xt::xtensor<double, 4> m_dNx; // shape function grad. wrt global coor. [nelem, nip, nne, ndim]
xt::xtensor<double, 2> m_vol; // integration point volume [nelem, nip]
};
} // namespace Hex8
} // namespace Element
} // namespace GooseFEM
#include "ElementHex8.hpp"
#endif
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