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ElementQuad4.h
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ElementQuad4.h
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/**
Quadrature for 4-noded quadrilateral element (GooseFEM::Mesh::ElementType::Quad4).
\file ElementQuad4.h
\copyright Copyright 2017. Tom de Geus. All rights reserved.
\license This project is released under the GNU Public License (GPLv3).
*/
#ifndef GOOSEFEM_ELEMENTQUAD4_H
#define GOOSEFEM_ELEMENTQUAD4_H
#include "config.h"
#include "Element.h"
namespace GooseFEM {
namespace Element {
namespace Quad4 {
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
namespace MidPoint {
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 MidPoint
class Quadrature : public GooseFEM::Element::QuadratureBase<4, 2, 2> {
public:
// Fixed dimensions:
// ndim = 2 - number of dimensions
// nne = 4 - 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 shape function gradients
xt::xtensor<double, 4> GradN() const;
// Return integration volume
xt::xtensor<double, 2> dV() const;
/**
Interpolate element vector.
\param elemvec Nodal vector stored per element (shape: ``[nelem, nne, ndim]``).
\param qvector Output: integration point vector (shape: ``[nelem, nip, ndim]``).
*/
template <class T>
void interp_N_vector(const xt::xtensor<T, 3>& elemvec, xt::xtensor<T, 3>& qvector) const;
/**
Same as interp_N_vector(), but returns auto-allocated data.
\param elemvec Nodal vector stored per element (shape: ``[nelem, nne, ndim]``).
\return integration point vector (shape: ``[nelem, nip, ndim]``).
*/
template <class T>
xt::xtensor<T, 3> Interp_N_vector(const xt::xtensor<T, 3>& elemvec) 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;
private:
// Compute "vol" and "dNdx" based on current "x"
void compute_dN();
private:
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 Quad4
} // namespace Element
} // namespace GooseFEM
#include "ElementQuad4.hpp"
#endif

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