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ElementQuad4.hpp
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/*
(c - GPLv3) T.W.J. de Geus (Tom) | tom@geus.me | www.geus.me | github.com/tdegeus/GooseFEM
*/
#ifndef GOOSEFEM_ELEMENTQUAD4_HPP
#define GOOSEFEM_ELEMENTQUAD4_HPP
#include "ElementQuad4.h"
namespace GooseFEM {
namespace Element {
namespace Quad4 {
template <class T>
inline double inv(const T& A, T& Ainv)
{
double det = A(0, 0) * A(1, 1) - A(0, 1) * A(1, 0);
Ainv(0, 0) = A(1, 1) / det;
Ainv(0, 1) = -1.0 * A(0, 1) / det;
Ainv(1, 0) = -1.0 * A(1, 0) / det;
Ainv(1, 1) = A(0, 0) / det;
return det;
}
namespace Gauss {
inline size_t nip()
{
return 4;
}
inline xt::xtensor<double, 2> xi()
{
size_t nip = 4;
size_t ndim = 2;
xt::xtensor<double, 2> xi = xt::empty<double>({nip, ndim});
xi(0, 0) = -1.0 / std::sqrt(3.0);
xi(0, 1) = -1.0 / std::sqrt(3.0);
xi(1, 0) = +1.0 / std::sqrt(3.0);
xi(1, 1) = -1.0 / std::sqrt(3.0);
xi(2, 0) = +1.0 / std::sqrt(3.0);
xi(2, 1) = +1.0 / std::sqrt(3.0);
xi(3, 0) = -1.0 / std::sqrt(3.0);
xi(3, 1) = +1.0 / std::sqrt(3.0);
return xi;
}
inline xt::xtensor<double, 1> w()
{
size_t nip = 4;
xt::xtensor<double, 1> w = xt::empty<double>({nip});
w(0) = 1.0;
w(1) = 1.0;
w(2) = 1.0;
w(3) = 1.0;
return w;
}
} // namespace Gauss
namespace Nodal {
inline size_t nip()
{
return 4;
}
inline xt::xtensor<double, 2> xi()
{
size_t nip = 4;
size_t ndim = 2;
xt::xtensor<double, 2> xi = xt::empty<double>({nip, ndim});
xi(0, 0) = -1.0;
xi(0, 1) = -1.0;
xi(1, 0) = +1.0;
xi(1, 1) = -1.0;
xi(2, 0) = +1.0;
xi(2, 1) = +1.0;
xi(3, 0) = -1.0;
xi(3, 1) = +1.0;
return xi;
}
inline xt::xtensor<double, 1> w()
{
size_t nip = 4;
xt::xtensor<double, 1> w = xt::empty<double>({nip});
w(0) = 1.0;
w(1) = 1.0;
w(2) = 1.0;
w(3) = 1.0;
return w;
}
} // namespace Nodal
namespace MidPoint {
inline size_t nip()
{
return 1;
}
inline xt::xtensor<double, 2> xi()
{
size_t nip = 1;
size_t ndim = 2;
xt::xtensor<double, 2> xi = xt::empty<double>({nip, ndim});
xi(0, 0) = 0.0;
xi(0, 1) = 0.0;
return xi;
}
inline xt::xtensor<double, 1> w()
{
size_t nip = 1;
xt::xtensor<double, 1> w = xt::empty<double>({nip});
w(0) = 1.0;
return w;
}
} // namespace MidPoint
inline Quadrature::Quadrature(const xt::xtensor<double, 3>& x)
: Quadrature(x, Gauss::xi(), Gauss::w())
{
}
inline Quadrature::Quadrature(
const xt::xtensor<double, 3>& x,
const xt::xtensor<double, 2>& xi,
const xt::xtensor<double, 1>& w)
: m_x(x), m_w(w), m_xi(xi)
{
GOOSEFEM_ASSERT(m_x.shape(1) == m_nne);
GOOSEFEM_ASSERT(m_x.shape(2) == m_ndim);
m_nelem = m_x.shape(0);
m_nip = m_w.size();
GOOSEFEM_ASSERT(m_xi.shape(0) == m_nip);
GOOSEFEM_ASSERT(m_xi.shape(1) == m_ndim);
GOOSEFEM_ASSERT(m_w.size() == m_nip);
m_N = xt::empty<double>({m_nip, m_nne});
m_dNxi = xt::empty<double>({m_nip, m_nne, m_ndim});
m_dNx = xt::empty<double>({m_nelem, m_nip, m_nne, m_ndim});
m_vol = xt::empty<double>({m_nelem, m_nip});
for (size_t q = 0; q < m_nip; ++q) {
m_N(q, 0) = 0.25 * (1.0 - m_xi(q, 0)) * (1.0 - m_xi(q, 1));
m_N(q, 1) = 0.25 * (1.0 + m_xi(q, 0)) * (1.0 - m_xi(q, 1));
m_N(q, 2) = 0.25 * (1.0 + m_xi(q, 0)) * (1.0 + m_xi(q, 1));
m_N(q, 3) = 0.25 * (1.0 - m_xi(q, 0)) * (1.0 + m_xi(q, 1));
}
for (size_t q = 0; q < m_nip; ++q) {
// - dN / dxi_0
m_dNxi(q, 0, 0) = -0.25 * (1.0 - m_xi(q, 1));
m_dNxi(q, 1, 0) = +0.25 * (1.0 - m_xi(q, 1));
m_dNxi(q, 2, 0) = +0.25 * (1.0 + m_xi(q, 1));
m_dNxi(q, 3, 0) = -0.25 * (1.0 + m_xi(q, 1));
// - dN / dxi_1
m_dNxi(q, 0, 1) = -0.25 * (1.0 - m_xi(q, 0));
m_dNxi(q, 1, 1) = -0.25 * (1.0 + m_xi(q, 0));
m_dNxi(q, 2, 1) = +0.25 * (1.0 + m_xi(q, 0));
m_dNxi(q, 3, 1) = +0.25 * (1.0 - m_xi(q, 0));
}
compute_dN();
}
inline size_t Quadrature::nelem() const
{
return m_nelem;
}
inline size_t Quadrature::nne() const
{
return m_nne;
}
inline size_t Quadrature::ndim() const
{
return m_ndim;
}
inline size_t Quadrature::nip() const
{
return m_nip;
}
inline xt::xtensor<double, 4> Quadrature::GradN() const
{
return m_dNx;
}
template <size_t rank>
inline void
Quadrature::asTensor(const xt::xtensor<double, 2>& arg, xt::xtensor<double, 2 + rank>& ret) const
{
GOOSEFEM_ASSERT(xt::has_shape(arg, {m_nelem, m_nne}));
GooseFEM::asTensor<2, rank>(arg, ret);
}
inline xt::xtensor<double, 2> Quadrature::dV() const
{
return m_vol;
}
inline void Quadrature::update_x(const xt::xtensor<double, 3>& x)
{
GOOSEFEM_ASSERT(x.shape() == m_x.shape());
xt::noalias(m_x) = x;
compute_dN();
}
inline void Quadrature::compute_dN()
{
#pragma omp parallel
{
xt::xtensor<double, 2> J = xt::empty<double>({2, 2});
xt::xtensor<double, 2> Jinv = xt::empty<double>({2, 2});
#pragma omp for
for (size_t e = 0; e < m_nelem; ++e) {
auto x = xt::adapt(&m_x(e, 0, 0), xt::xshape<m_nne, m_ndim>());
for (size_t q = 0; q < m_nip; ++q) {
auto dNxi = xt::adapt(&m_dNxi(q, 0, 0), xt::xshape<m_nne, m_ndim>());
auto dNx = xt::adapt(&m_dNx(e, q, 0, 0), xt::xshape<m_nne, m_ndim>());
// J(i,j) += dNxi(m,i) * x(m,j);
J(0, 0) = dNxi(0, 0) * x(0, 0) + dNxi(1, 0) * x(1, 0) + dNxi(2, 0) * x(2, 0) +
dNxi(3, 0) * x(3, 0);
J(0, 1) = dNxi(0, 0) * x(0, 1) + dNxi(1, 0) * x(1, 1) + dNxi(2, 0) * x(2, 1) +
dNxi(3, 0) * x(3, 1);
J(1, 0) = dNxi(0, 1) * x(0, 0) + dNxi(1, 1) * x(1, 0) + dNxi(2, 1) * x(2, 0) +
dNxi(3, 1) * x(3, 0);
J(1, 1) = dNxi(0, 1) * x(0, 1) + dNxi(1, 1) * x(1, 1) + dNxi(2, 1) * x(2, 1) +
dNxi(3, 1) * x(3, 1);
double Jdet = inv(J, Jinv);
// dNx(m,i) += Jinv(i,j) * dNxi(m,j);
for (size_t m = 0; m < m_nne; ++m) {
dNx(m, 0) = Jinv(0, 0) * dNxi(m, 0) + Jinv(0, 1) * dNxi(m, 1);
dNx(m, 1) = Jinv(1, 0) * dNxi(m, 0) + Jinv(1, 1) * dNxi(m, 1);
}
m_vol(e, q) = m_w(q) * Jdet;
}
}
}
}
inline void Quadrature::gradN_vector(
const xt::xtensor<double, 3>& elemvec, xt::xtensor<double, 4>& qtensor) const
{
GOOSEFEM_ASSERT(xt::has_shape(elemvec, {m_nelem, m_nne, m_ndim}));
GOOSEFEM_ASSERT(xt::has_shape(qtensor, {m_nelem, m_nip, m_ndim, m_ndim}));
#pragma omp parallel for
for (size_t e = 0; e < m_nelem; ++e) {
auto u = xt::adapt(&elemvec(e, 0, 0), xt::xshape<m_nne, m_ndim>());
for (size_t q = 0; q < m_nip; ++q) {
auto dNx = xt::adapt(&m_dNx(e, q, 0, 0), xt::xshape<m_nne, m_ndim>());
auto gradu = xt::adapt(&qtensor(e, q, 0, 0), xt::xshape<m_ndim, m_ndim>());
// gradu(i,j) += dNx(m,i) * u(m,j)
gradu(0, 0) = dNx(0, 0) * u(0, 0) + dNx(1, 0) * u(1, 0) + dNx(2, 0) * u(2, 0) +
dNx(3, 0) * u(3, 0);
gradu(0, 1) = dNx(0, 0) * u(0, 1) + dNx(1, 0) * u(1, 1) + dNx(2, 0) * u(2, 1) +
dNx(3, 0) * u(3, 1);
gradu(1, 0) = dNx(0, 1) * u(0, 0) + dNx(1, 1) * u(1, 0) + dNx(2, 1) * u(2, 0) +
dNx(3, 1) * u(3, 0);
gradu(1, 1) = dNx(0, 1) * u(0, 1) + dNx(1, 1) * u(1, 1) + dNx(2, 1) * u(2, 1) +
dNx(3, 1) * u(3, 1);
}
}
}
inline void Quadrature::gradN_vector_T(
const xt::xtensor<double, 3>& elemvec, xt::xtensor<double, 4>& qtensor) const
{
GOOSEFEM_ASSERT(xt::has_shape(elemvec, {m_nelem, m_nne, m_ndim}));
GOOSEFEM_ASSERT(xt::has_shape(qtensor, {m_nelem, m_nip, m_ndim, m_ndim}));
#pragma omp parallel for
for (size_t e = 0; e < m_nelem; ++e) {
auto u = xt::adapt(&elemvec(e, 0, 0), xt::xshape<m_nne, m_ndim>());
for (size_t q = 0; q < m_nip; ++q) {
auto dNx = xt::adapt(&m_dNx(e, q, 0, 0), xt::xshape<m_nne, m_ndim>());
auto gradu = xt::adapt(&qtensor(e, q, 0, 0), xt::xshape<m_ndim, m_ndim>());
// gradu(j,i) += dNx(m,i) * u(m,j)
gradu(0, 0) = dNx(0, 0) * u(0, 0) + dNx(1, 0) * u(1, 0) + dNx(2, 0) * u(2, 0) +
dNx(3, 0) * u(3, 0);
gradu(1, 0) = dNx(0, 0) * u(0, 1) + dNx(1, 0) * u(1, 1) + dNx(2, 0) * u(2, 1) +
dNx(3, 0) * u(3, 1);
gradu(0, 1) = dNx(0, 1) * u(0, 0) + dNx(1, 1) * u(1, 0) + dNx(2, 1) * u(2, 0) +
dNx(3, 1) * u(3, 0);
gradu(1, 1) = dNx(0, 1) * u(0, 1) + dNx(1, 1) * u(1, 1) + dNx(2, 1) * u(2, 1) +
dNx(3, 1) * u(3, 1);
}
}
}
inline void Quadrature::symGradN_vector(
const xt::xtensor<double, 3>& elemvec, xt::xtensor<double, 4>& qtensor) const
{
GOOSEFEM_ASSERT(xt::has_shape(elemvec, {m_nelem, m_nne, m_ndim}));
GOOSEFEM_ASSERT(xt::has_shape(qtensor, {m_nelem, m_nip, m_ndim, m_ndim}));
#pragma omp parallel for
for (size_t e = 0; e < m_nelem; ++e) {
auto u = xt::adapt(&elemvec(e, 0, 0), xt::xshape<m_nne, m_ndim>());
for (size_t q = 0; q < m_nip; ++q) {
auto dNx = xt::adapt(&m_dNx(e, q, 0, 0), xt::xshape<m_nne, m_ndim>());
auto eps = xt::adapt(&qtensor(e, q, 0, 0), xt::xshape<m_ndim, m_ndim>());
// gradu(i,j) += dNx(m,i) * u(m,j)
// eps(j,i) = 0.5 * (gradu(i,j) + gradu(j,i))
eps(0, 0) = dNx(0, 0) * u(0, 0) + dNx(1, 0) * u(1, 0) + dNx(2, 0) * u(2, 0) +
dNx(3, 0) * u(3, 0);
eps(1, 1) = dNx(0, 1) * u(0, 1) + dNx(1, 1) * u(1, 1) + dNx(2, 1) * u(2, 1) +
dNx(3, 1) * u(3, 1);
eps(0, 1) = 0.5 * (dNx(0, 0) * u(0, 1) + dNx(1, 0) * u(1, 1) + dNx(2, 0) * u(2, 1) +
dNx(3, 0) * u(3, 1) + dNx(0, 1) * u(0, 0) + dNx(1, 1) * u(1, 0) +
dNx(2, 1) * u(2, 0) + dNx(3, 1) * u(3, 0));
eps(1, 0) = eps(0, 1);
}
}
}
inline void Quadrature::int_N_scalar_NT_dV(
const xt::xtensor<double, 2>& qscalar, xt::xtensor<double, 3>& elemmat) const
{
GOOSEFEM_ASSERT(xt::has_shape(qscalar, {m_nelem, m_nip}));
GOOSEFEM_ASSERT(xt::has_shape(elemmat, {m_nelem, m_nne * m_ndim, m_nne * m_ndim}));
elemmat.fill(0.0);
#pragma omp parallel for
for (size_t e = 0; e < m_nelem; ++e) {
auto M = xt::adapt(&elemmat(e, 0, 0), xt::xshape<m_nne * m_ndim, m_nne * m_ndim>());
for (size_t q = 0; q < m_nip; ++q) {
auto N = xt::adapt(&m_N(q, 0), xt::xshape<m_nne>());
auto& vol = m_vol(e, q);
auto& rho = qscalar(e, q);
// M(m*ndim+i,n*ndim+i) += N(m) * scalar * N(n) * dV
for (size_t m = 0; m < m_nne; ++m) {
for (size_t n = 0; n < m_nne; ++n) {
M(m * m_ndim + 0, n * m_ndim + 0) += N(m) * rho * N(n) * vol;
M(m * m_ndim + 1, n * m_ndim + 1) += N(m) * rho * N(n) * vol;
}
}
}
}
}
inline void Quadrature::int_gradN_dot_tensor2_dV(
const xt::xtensor<double, 4>& qtensor, xt::xtensor<double, 3>& elemvec) const
{
GOOSEFEM_ASSERT(xt::has_shape(qtensor, {m_nelem, m_nip, m_ndim, m_ndim}));
GOOSEFEM_ASSERT(xt::has_shape(elemvec, {m_nelem, m_nne, m_ndim}));
elemvec.fill(0.0);
#pragma omp parallel for
for (size_t e = 0; e < m_nelem; ++e) {
auto f = xt::adapt(&elemvec(e, 0, 0), xt::xshape<m_nne, m_ndim>());
for (size_t q = 0; q < m_nip; ++q) {
auto dNx = xt::adapt(&m_dNx(e, q, 0, 0), xt::xshape<m_nne, m_ndim>());
auto sig = xt::adapt(&qtensor(e, q, 0, 0), xt::xshape<m_ndim, m_ndim>());
auto& vol = m_vol(e, q);
for (size_t m = 0; m < m_nne; ++m) {
f(m, 0) += (dNx(m, 0) * sig(0, 0) + dNx(m, 1) * sig(1, 0)) * vol;
f(m, 1) += (dNx(m, 0) * sig(0, 1) + dNx(m, 1) * sig(1, 1)) * vol;
}
}
}
}
inline void Quadrature::int_gradN_dot_tensor4_dot_gradNT_dV(
const xt::xtensor<double, 6>& qtensor, xt::xtensor<double, 3>& elemmat) const
{
GOOSEFEM_ASSERT(xt::has_shape(qtensor, {m_nelem, m_nip, m_ndim, m_ndim, m_ndim, m_ndim}));
GOOSEFEM_ASSERT(xt::has_shape(elemmat, {m_nelem, m_nne * m_ndim, m_nne * m_ndim}));
elemmat.fill(0.0);
#pragma omp parallel for
for (size_t e = 0; e < m_nelem; ++e) {
auto K = xt::adapt(&elemmat(e, 0, 0), xt::xshape<m_nne * m_ndim, m_nne * m_ndim>());
for (size_t q = 0; q < m_nip; ++q) {
auto dNx = xt::adapt(&m_dNx(e, q, 0, 0), xt::xshape<m_nne, m_ndim>());
auto C = xt::adapt(&qtensor(e, q, 0, 0, 0, 0), xt::xshape<m_ndim, m_ndim, m_ndim, m_ndim>());
auto& vol = m_vol(e, q);
for (size_t m = 0; m < m_nne; ++m) {
for (size_t n = 0; n < m_nne; ++n) {
for (size_t i = 0; i < m_ndim; ++i) {
for (size_t j = 0; j < m_ndim; ++j) {
for (size_t k = 0; k < m_ndim; ++k) {
for (size_t l = 0; l < m_ndim; ++l) {
K(m * m_ndim + j, n * m_ndim + k) +=
dNx(m, i) * C(i, j, k, l) * dNx(n, l) * vol;
}
}
}
}
}
}
}
}
}
template <size_t rank>
inline xt::xtensor<double, 2 + rank>
Quadrature::AsTensor(const xt::xtensor<double, 2>& qscalar) const
{
return GooseFEM::AsTensor<2, rank>(qscalar, m_ndim);
}
inline xt::xarray<double>
Quadrature::AsTensor(size_t rank, const xt::xtensor<double, 2>& qscalar) const
{
return GooseFEM::AsTensor(rank, qscalar, m_ndim);
}
inline xt::xtensor<double, 4> Quadrature::GradN_vector(const xt::xtensor<double, 3>& elemvec) const
{
xt::xtensor<double, 4> qtensor = xt::empty<double>({m_nelem, m_nip, m_ndim, m_ndim});
this->gradN_vector(elemvec, qtensor);
return qtensor;
}
inline xt::xtensor<double, 4>
Quadrature::GradN_vector_T(const xt::xtensor<double, 3>& elemvec) const
{
xt::xtensor<double, 4> qtensor = xt::empty<double>({m_nelem, m_nip, m_ndim, m_ndim});
this->gradN_vector_T(elemvec, qtensor);
return qtensor;
}
inline xt::xtensor<double, 4>
Quadrature::SymGradN_vector(const xt::xtensor<double, 3>& elemvec) const
{
xt::xtensor<double, 4> qtensor = xt::empty<double>({m_nelem, m_nip, m_ndim, m_ndim});
this->symGradN_vector(elemvec, qtensor);
return qtensor;
}
inline xt::xtensor<double, 3>
Quadrature::Int_N_scalar_NT_dV(const xt::xtensor<double, 2>& qscalar) const
{
xt::xtensor<double, 3> elemmat = xt::empty<double>({m_nelem, m_nne * m_ndim, m_nne * m_ndim});
this->int_N_scalar_NT_dV(qscalar, elemmat);
return elemmat;
}
inline xt::xtensor<double, 3>
Quadrature::Int_gradN_dot_tensor2_dV(const xt::xtensor<double, 4>& qtensor) const
{
xt::xtensor<double, 3> elemvec = xt::empty<double>({m_nelem, m_nne, m_ndim});
this->int_gradN_dot_tensor2_dV(qtensor, elemvec);
return elemvec;
}
inline xt::xtensor<double, 3>
Quadrature::Int_gradN_dot_tensor4_dot_gradNT_dV(const xt::xtensor<double, 6>& qtensor) const
{
xt::xtensor<double, 3> elemmat = xt::empty<double>({m_nelem, m_ndim * m_nne, m_ndim * m_nne});
this->int_gradN_dot_tensor4_dot_gradNT_dV(qtensor, elemmat);
return elemmat;
}
template <size_t rank>
inline xt::xtensor<double, rank + 2> Quadrature::AllocateQtensor() const
{
std::array<size_t, rank + 2> shape;
shape[0] = m_nelem;
shape[1] = m_nip;
size_t n = m_ndim;
std::fill(shape.begin() + 2, shape.end(), n);
xt::xtensor<double, rank + 2> ret = xt::empty<double>(shape);
return ret;
}
template <size_t rank>
inline xt::xtensor<double, rank + 2> Quadrature::AllocateQtensor(double val) const
{
xt::xtensor<double, rank + 2> ret = this->AllocateQtensor<rank>();
ret.fill(val);
return ret;
}
inline xt::xarray<double> Quadrature::AllocateQtensor(size_t rank) const
{
std::vector<size_t> shape(rank + 2);
shape[0] = m_nelem;
shape[1] = m_nip;
size_t n = m_ndim;
std::fill(shape.begin() + 2, shape.end(), n);
xt::xarray<double> ret = xt::empty<double>(shape);
return ret;
}
inline xt::xarray<double> Quadrature::AllocateQtensor(size_t rank, double val) const
{
xt::xarray<double> ret = this->AllocateQtensor(rank);
ret.fill(val);
return ret;
}
inline xt::xtensor<double, 2> Quadrature::AllocateQscalar() const
{
return this->AllocateQtensor<0>();
}
inline xt::xtensor<double, 2> Quadrature::AllocateQscalar(double val) const
{
return this->AllocateQtensor<0>(val);
}
} // namespace Quad4
} // namespace Element
} // namespace GooseFEM
#endif

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