ElementQuad4.h

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#ifndef GOOSEFEM_ELEMENTQUAD4_H
#define GOOSEFEM_ELEMENTQUAD4_H
 
#include "Element.h"
#include "config.h"
#include "detail.h"
 
namespace GooseFEM {
namespace Element {
 
namespace Quad4 {
 
namespace Gauss {
 
inline size_t nip()
{
    return 4;
}
 
inline array_type::tensor<double, 2> xi()
{
    size_t nip = 4;
    size_t ndim = 2;
 
    array_type::tensor<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 array_type::tensor<double, 1> w()
{
    size_t nip = 4;
 
    array_type::tensor<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 array_type::tensor<double, 2> xi()
{
    size_t nip = 4;
    size_t ndim = 2;
 
    array_type::tensor<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 array_type::tensor<double, 1> w()
{
    size_t nip = 4;
 
    array_type::tensor<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 array_type::tensor<double, 2> xi()
{
    size_t nip = 1;
    size_t ndim = 2;
 
    array_type::tensor<double, 2> xi = xt::empty<double>({nip, ndim});
 
    xi(0, 0) = 0.0;
    xi(0, 1) = 0.0;
 
    return xi;
}
 
inline array_type::tensor<double, 1> w()
{
    size_t nip = 1;
 
    array_type::tensor<double, 1> w = xt::empty<double>({nip});
 
    w(0) = 1.0;
 
    return w;
}
 
} // namespace MidPoint
 
class Quadrature : public QuadratureBaseCartesian<Quadrature> {
public:
    Quadrature() = default;
 
    template <class T>
    Quadrature(const T& x) : Quadrature(x, Gauss::xi(), Gauss::w())
    {
    }
 
    template <class T, class X, class W>
    Quadrature(const T& x, const X& xi, const W& w)
    {
        m_x = x;
        m_w = w;
        m_xi = xi;
        m_nip = w.size();
        m_nelem = m_x.shape(0);
        m_N = xt::empty<double>({m_nip, s_nne});
        m_dNxi = xt::empty<double>({m_nip, s_nne, s_ndim});
 
        for (size_t q = 0; q < m_nip; ++q) {
            m_N(q, 0) = 0.25 * (1.0 - xi(q, 0)) * (1.0 - xi(q, 1));
            m_N(q, 1) = 0.25 * (1.0 + xi(q, 0)) * (1.0 - xi(q, 1));
            m_N(q, 2) = 0.25 * (1.0 + xi(q, 0)) * (1.0 + xi(q, 1));
            m_N(q, 3) = 0.25 * (1.0 - xi(q, 0)) * (1.0 + xi(q, 1));
        }
 
        for (size_t q = 0; q < m_nip; ++q) {
            // - dN / dxi_0
            m_dNxi(q, 0, 0) = -0.25 * (1.0 - xi(q, 1));
            m_dNxi(q, 1, 0) = +0.25 * (1.0 - xi(q, 1));
            m_dNxi(q, 2, 0) = +0.25 * (1.0 + xi(q, 1));
            m_dNxi(q, 3, 0) = -0.25 * (1.0 + xi(q, 1));
            // - dN / dxi_1
            m_dNxi(q, 0, 1) = -0.25 * (1.0 - xi(q, 0));
            m_dNxi(q, 1, 1) = -0.25 * (1.0 + xi(q, 0));
            m_dNxi(q, 2, 1) = +0.25 * (1.0 + xi(q, 0));
            m_dNxi(q, 3, 1) = +0.25 * (1.0 - xi(q, 0));
        }
 
        GOOSEFEM_ASSERT(m_x.shape(1) == s_nne);
        GOOSEFEM_ASSERT(m_x.shape(2) == s_ndim);
        GOOSEFEM_ASSERT(xt::has_shape(m_xi, {m_nip, s_ndim}));
        GOOSEFEM_ASSERT(xt::has_shape(m_w, {m_nip}));
        GOOSEFEM_ASSERT(xt::has_shape(m_N, {m_nip, s_nne}));
        GOOSEFEM_ASSERT(xt::has_shape(m_dNxi, {m_nip, s_nne, s_ndim}));
 
        m_dNx = xt::empty<double>({m_nelem, m_nip, s_nne, s_ndim});
        m_vol = xt::empty<double>(this->shape_qscalar());
 
        this->compute_dN_impl();
    }
 
private:
    friend QuadratureBase<Quadrature>;
    friend QuadratureBaseCartesian<Quadrature>;
 
    template <class T, class R>
    void interpQuad_vector_impl(const T& elemvec, R& qvector) const
    {
        GOOSEFEM_ASSERT(xt::has_shape(elemvec, this->shape_elemvec()));
        GOOSEFEM_ASSERT(xt::has_shape(qvector, this->shape_qvector()));
 
        qvector.fill(0.0);
 
#pragma omp parallel for
        for (size_t e = 0; e < m_nelem; ++e) {
 
            auto u = xt::adapt(&elemvec(e, 0, 0), xt::xshape<s_nne, s_ndim>());
 
            for (size_t q = 0; q < m_nip; ++q) {
 
                auto N = xt::adapt(&m_N(q, 0), xt::xshape<s_nne>());
                auto ui = xt::adapt(&qvector(e, q, 0), xt::xshape<s_ndim>());
 
                ui(0) = N(0) * u(0, 0) + N(1) * u(1, 0) + N(2) * u(2, 0) + N(3) * u(3, 0);
                ui(1) = N(0) * u(0, 1) + N(1) * u(1, 1) + N(2) * u(2, 1) + N(3) * u(3, 1);
            }
        }
    }
 
    template <class T, class R>
    void gradN_vector_impl(const T& elemvec, R& qtensor) const
    {
        GOOSEFEM_ASSERT(xt::has_shape(elemvec, this->shape_elemvec()));
        GOOSEFEM_ASSERT(xt::has_shape(qtensor, this->shape_qtensor<2>()));
 
#pragma omp parallel for
        for (size_t e = 0; e < m_nelem; ++e) {
 
            auto u = xt::adapt(&elemvec(e, 0, 0), xt::xshape<s_nne, s_ndim>());
 
            for (size_t q = 0; q < m_nip; ++q) {
 
                auto dNx = xt::adapt(&m_dNx(e, q, 0, 0), xt::xshape<s_nne, s_ndim>());
                auto gradu = xt::adapt(&qtensor(e, q, 0, 0), xt::xshape<s_ndim, s_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);
            }
        }
    }
 
    template <class T, class R>
    void gradN_vector_T_impl(const T& elemvec, R& qtensor) const
    {
        GOOSEFEM_ASSERT(xt::has_shape(elemvec, this->shape_elemvec()));
        GOOSEFEM_ASSERT(xt::has_shape(qtensor, this->shape_qtensor<2>()));
 
#pragma omp parallel for
        for (size_t e = 0; e < m_nelem; ++e) {
 
            auto u = xt::adapt(&elemvec(e, 0, 0), xt::xshape<s_nne, s_ndim>());
 
            for (size_t q = 0; q < m_nip; ++q) {
 
                auto dNx = xt::adapt(&m_dNx(e, q, 0, 0), xt::xshape<s_nne, s_ndim>());
                auto gradu = xt::adapt(&qtensor(e, q, 0, 0), xt::xshape<s_ndim, s_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);
            }
        }
    }
 
    template <class T, class R>
    void symGradN_vector_impl(const T& elemvec, R& qtensor) const
    {
        GOOSEFEM_ASSERT(xt::has_shape(elemvec, this->shape_elemvec()));
        GOOSEFEM_ASSERT(xt::has_shape(qtensor, this->shape_qtensor<2>()));
 
#pragma omp parallel for
        for (size_t e = 0; e < m_nelem; ++e) {
 
            auto u = xt::adapt(&elemvec(e, 0, 0), xt::xshape<s_nne, s_ndim>());
 
            for (size_t q = 0; q < m_nip; ++q) {
 
                auto dNx = xt::adapt(&m_dNx(e, q, 0, 0), xt::xshape<s_nne, s_ndim>());
                auto eps = xt::adapt(&qtensor(e, q, 0, 0), xt::xshape<s_ndim, s_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);
            }
        }
    }
 
    template <class T, class R>
    void int_N_scalar_NT_dV_impl(const T& qscalar, R& elemmat) const
    {
        GOOSEFEM_ASSERT(xt::has_shape(qscalar, this->shape_qscalar()));
        GOOSEFEM_ASSERT(xt::has_shape(elemmat, this->shape_elemmat()));
 
        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<s_nne * s_ndim, s_nne * s_ndim>());
 
            for (size_t q = 0; q < m_nip; ++q) {
 
                auto N = xt::adapt(&m_N(q, 0), xt::xshape<s_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 < s_nne; ++m) {
                    for (size_t n = 0; n < s_nne; ++n) {
                        M(m * s_ndim + 0, n * s_ndim + 0) += N(m) * rho * N(n) * vol;
                        M(m * s_ndim + 1, n * s_ndim + 1) += N(m) * rho * N(n) * vol;
                    }
                }
            }
        }
    }
 
    template <class T, class R>
    void int_gradN_dot_tensor2_dV_impl(const T& qtensor, R& elemvec) const
    {
        GOOSEFEM_ASSERT(xt::has_shape(qtensor, this->shape_qtensor<2>()));
        GOOSEFEM_ASSERT(xt::has_shape(elemvec, this->shape_elemvec()));
 
        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<s_nne, s_ndim>());
 
            for (size_t q = 0; q < m_nip; ++q) {
 
                auto dNx = xt::adapt(&m_dNx(e, q, 0, 0), xt::xshape<s_nne, s_ndim>());
                auto sig = xt::adapt(&qtensor(e, q, 0, 0), xt::xshape<s_ndim, s_ndim>());
                auto& vol = m_vol(e, q);
 
                for (size_t m = 0; m < s_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;
                }
            }
        }
    }
 
    void compute_dN_impl()
    {
#pragma omp parallel
        {
            array_type::tensor<double, 2> J = xt::empty<double>({2, 2});
            array_type::tensor<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<s_nne, s_ndim>());
 
                for (size_t q = 0; q < m_nip; ++q) {
 
                    auto dNxi = xt::adapt(&m_dNxi(q, 0, 0), xt::xshape<s_nne, s_ndim>());
                    auto dNx = xt::adapt(&m_dNx(e, q, 0, 0), xt::xshape<s_nne, s_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 = detail::tensor<2>::inv(J, Jinv);
 
                    // dNx(m,i) += Jinv(i,j) * dNxi(m,j);
                    for (size_t m = 0; m < s_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;
                }
            }
        }
    }
 
    constexpr static size_t s_nne = 4; 
    constexpr static size_t s_ndim = 2; 
    constexpr static size_t s_tdim = 2; 
    size_t m_tdim = 2; 
    size_t m_nelem; 
    size_t m_nip; 
    array_type::tensor<double, 3> m_x; 
    array_type::tensor<double, 1> m_w; 
    array_type::tensor<double, 2> m_xi; 
    array_type::tensor<double, 2> m_N; 
    array_type::tensor<double, 3> m_dNxi; 
    array_type::tensor<double, 4> m_dNx; 
    array_type::tensor<double, 2> m_vol; 
};
 
} // namespace Quad4
} // namespace Element
} // namespace GooseFEM
 
#endif