%--------------------------
 % @Author: Jingqiao Hu
 % @Date: 2021-01-25 20:51:51
 % @LastEditTime: 2021-01-27 11:21:48
%--------------------------
function [U1, converged] = line_search_nonlinear(du, rhs, fext, U0, dNh_e, dN_c, ...
        edofMat_mi, edofMat_ma, fixeddofs, dx, global_u, global_l, opt_rotation, ...
        regulated_matrix, sort_u, dFdx)

    maxloop = 10; loop = 0;
    alldofs = size(U0, 1);
    num_vdofs_ma = size(edofMat_ma, 2);
    eleNum_mi = size(edofMat_mi, 1);

    step = 1;
    approxE0 = -sum(dot(rhs, U0));
    norm0 = max(max(abs(rhs)));
    found_step = 0;

    while loop < maxloop
        loop = loop+1;

        U1 = step*du + U0;   
        if opt_rotation
            R = update_corotation_matrix(edofMat_mi, edofMat_ma, U1, dN_c);
        else
            % update deform gradient
            F = deform_grad(sort_u, edofMat_ma, dNh_e, eleNum_mi, U1, regulated_matrix);
        end

        inverted = check_inversion(F);
        step = step*0.5;
        if (inverted)
            continue;
            % regard as not converged,
            % it is advised that the deformation gradient F be temporarily replaced by
            % the nearest physically plausible value ~F
        end

        fint = multi_elastic_force(num_vdofs_ma, alldofs, dx, F, dFdx, edofMat_ma, global_u, global_l);
        rhs = fint + fext;
        rhs(fixeddofs) = 0;

        approxE1 = -sum(dot(rhs, U1));
        norm1 = max(max(abs(rhs)));
        if( (norm1<norm0) || (approxE1<approxE0) )
            found_step=1;
            break;
        end
    end

    converged = 0;
    if(~found_step || (abs(norm1-norm0)<1e-5))
        converged = 1;
    end

    U1(fixeddofs) = U0(fixeddofs);
end

function inverted = check_inversion(F)
    inverted = false;
    eps = 1e-5;
    eleNum_ma = size(F, 2);

    for ele_ma = 1 : eleNum_ma
        for gp = 1:4
            F1 = F{gp, ele_ma}; % 4 * eleNum_mi
            Fdet = F1(1, :).*F1(4, :) - F1(2, :).*F1(3, :);
            if (sum(Fdet <= eps) > 0)
                inverted = true;
                    return;
            end
        end
    end
end