tensor.h

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/*******************************************************************************
 * Copyright (C) 2017-2023 Theodore Chang
 *
 * This program is free software: you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation, either version 3 of the License, or
 * (at your option) any later version.
 *
 * This program is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU General Public License for more details.
 *
 * You should have received a copy of the GNU General Public License
 * along with this program.  If not, see <http://www.gnu.org/licenses/>.
 ******************************************************************************/
#ifndef TENSOR_H
#define TENSOR_H
 
#include <Toolbox/utility.h>
 
template<typename T> class Quaternion;
 
namespace tensor {
    mat isotropic_stiffness(double, double);
    mat orthotropic_stiffness(const vec&, const vec&);
 
    mat unit_deviatoric_tensor4();
    mat unit_deviatoric_tensor4v2();
    mat unit_symmetric_tensor4();
 
    static const vec unit_tensor2{1., 1., 1., 0., 0., 0.};
 
    namespace stress {
        // applies to 3D tensor only, either principal or not
        double invariant1(const vec&);
        // applies to 3D tensor only, either principal or not
        double invariant2(const vec&);
        // applies to 3D tensor only, either principal or not
        double invariant3(const vec&);
 
        // compute lode angle based on input stress
        double lode(vec);
        // compute derivative of lode angle based on input stress
        vec lode_der(vec);
 
        static const vec norm_weight{1., 1., 1., 2., 2., 2.};
    } // namespace stress
    namespace strain {
        // applies to 3D tensor only, either principal or not
        double invariant1(const vec&);
        // applies to 3D tensor only, either principal or not
        double invariant2(const vec&);
        // applies to 3D tensor only, either principal or not
        double invariant3(const vec&);
 
        // compute lode angle based on input deviatoric strain
        double lode(vec);
 
        static const vec norm_weight{1., 1., 1., .5, .5, .5};
    } // namespace strain
    double trace2(const vec&);
    double trace3(const vec&);
    double mean3(const vec&);
    vec dev(const vec&);
    vec dev(vec&&);
 
    mat dev(const mat&);
    mat dev(mat&&);
 
    namespace strain {
        mat to_green(mat&&);
        mat to_green(const mat&);
        mat to_tensor(const vec&);
        vec to_voigt(const mat&);
        double norm(const vec&);
        double norm(vec&&);
        double double_contraction(const vec&, const vec&);
        double double_contraction(vec&&, vec&&);
    } // namespace strain
    namespace stress {
        mat to_tensor(const vec&);
        vec to_voigt(const mat&);
        double norm(const vec&);
        double norm(vec&&);
        double double_contraction(const vec&, const vec&);
        double double_contraction(vec&&, vec&&);
    } // namespace stress
}     // namespace tensor
 
namespace transform {
    double atan2(const vec&);
    mat compute_jacobian_nominal_to_principal(const mat&);
    mat compute_jacobian_principal_to_nominal(const mat&);
 
    template<typename T> Mat<T> skew_symm(const Mat<T>& R) {
        suanpan_assert([&] { if(R.n_elem != 3) throw invalid_argument("need 3 element vector"); });
 
        Mat<T> S(3, 3, fill::zeros);
 
        S(0, 1) = -(S(1, 0) = R(2));
        S(2, 0) = -(S(0, 2) = R(1));
        S(1, 2) = -(S(2, 1) = R(0));
 
        return S;
    }
 
    template<typename T> Mat<T> skew_symm(const subview_col<T>& R) { return skew_symm(R.eval()); }
 
    template<typename T> Mat<T> rodrigues(const Mat<T>& R) { return arma::expmat(transform::skew_symm(R)); }
 
    template<typename T> Mat<T> rodrigues(const subview_col<T>& R) { return arma::expmat(transform::skew_symm(R)); }
 
    template<typename T> Quaternion<T> to_quaternion(const Mat<T>& R) {
        if(3 == R.n_elem) {
            const auto angle = arma::norm(R);
 
            if(suanpan::approx_equal(angle, 0.)) return {0., 0., 0., 0.};
 
            return {std::cos(.5 * angle), std::sin(.5 * angle) / angle * R};
        }
 
        if(9 == R.n_elem) {
            const auto tr_r = arma::trace(R);
            const auto max_r = arma::max(R.diag());
            if(tr_r >= max_r) {
                const auto q0 = .5 * std::sqrt(tr_r + 1.);
                const auto q1 = .25 / q0 * (R(2, 1) - R(1, 2));
                const auto q2 = .25 / q0 * (R(0, 2) - R(2, 0));
                const auto q3 = .25 / q0 * (R(1, 0) - R(0, 1));
                return {q0, q1, q2, q3};
            }
 
            for(auto I = 0; I < 3; ++I)
                if(suanpan::approx_equal(R(I, I), max_r)) {
                    const auto J = (I + 1) % 3;
                    const auto K = (J + 1) % 3;
                    vec q(3, fill::none);
                    q(I) = std::sqrt(.5 * max_r + .25 * (1. - tr_r));
                    const double q0 = .25 / q(I) * (R(K, J) - R(J, K));
                    q(J) = .25 / q(I) * (R(J, I) + R(I, J));
                    q(K) = .25 / q(I) * (R(K, I) + R(I, K));
 
                    return {q0, std::move(q)};
                }
        }
 
        throw invalid_argument("need either rotation vector or matrix");
    }
 
    template<typename T> Col<T> to_pseudo(const Mat<T>& R) {
        const Mat<T> S = arma::real(arma::logmat(R));
 
        return {S(2, 1), S(0, 2), S(1, 0)};
    }
 
    namespace strain {
        double angle(const vec&);
        mat trans(double);
        vec principal(const vec&);
        vec rotate(const vec&, double);
    } // namespace strain
    namespace stress {
        double angle(const vec&);
        mat trans(double);
        vec principal(const vec&);
        vec rotate(const vec&, double);
    } // namespace stress
    namespace beam {
        mat global_to_local(double, double, double);
        mat global_to_local(const vec&, double);
    } // namespace beam
    namespace triangle {
        vec to_area_coordinate(const vec&, const mat&);
    }
} // namespace transform
 
namespace suanpan {
    template<typename T> T ramp(const T in) { return in > T(0) ? in : T(0); }
} // namespace suanpan
 
#endif