%--------------------------
 % @Author: Jingqiao Hu
 % @Date: 2021-12-07 09:13:42
 % @LastEditTime: 2021-12-12 12:36:07
%--------------------------
function [dc, dv] = sensitivity_cvt_mmc(dH_cell, area_cell, rho_cell, psi_cell, sK_cell, ...
    v_local, edofMat, U, e0, e1, penal, optVcons, area0, nc_local, nd)

    ncvt = size(rho_cell, 1);

    pn = 16; % p-norm for volume fraction
    dpnorm = 1/ncvt * (mean(v_local.^pn))^(1/pn-1) * (v_local.^(pn-1)); % [N, 1]

    nc_global = sum(nc_local) * nd;
    dc = zeros(nc_global, 1);
    dv = zeros(nc_global, 1);
    
    % for local fine mesh 
    nc_local(2:ncvt+1) = nc_local;
    nc_local(1) = 0;

    for ele = 1 : ncvt
        %% compliance sensitivity
        psi1 = psi_cell{ele};          % [6, 2v, m]
        psi2 = permute(psi1, [2,1,3]); % [2v, 6, m]
        nvdofs = size(psi2, 1);

        sK0 = sK_cell{ele}; % [36, m]
        rho = rho_cell{ele};

        sK1 = sK0 .* (penal * rho(:)'.^(penal-1) * (e1 - e0));
        sK2 = reshape(sK1, 6,6,[]); % [6,6,m]

        Ke0 = mtimesx(mtimesx(psi2, sK2), psi1); % [2v,2v,m]
        Ke1 = reshape(Ke0, nvdofs, []); % [2v, 2v*m]

        edof = edofMat{ele};
        ue = U(edof); % [2v,1]

        dc1 = ue' * Ke1; % [1, 2v*m]
        dc2 = reshape(dc1, nvdofs, [])'; % [m,2v]
        dc3 = dc2 * ue; % [m, 1]

        m = length(dc3);
        dH0 = dH_cell{ele};          % [nc, 3, m]
        dH1 = permute(dH0, [2,1,3]); % [3, nc, m]
        dH2 = reshape(dH1, [], m);   % [3*nc, m]

        dc4 = dH2 * dc3; % dc_dmmc, [3*nc]

        idx_begin = sum(nc_local(1:ele))*nd + 1;
        idx_end = sum(nc_local(1:ele+1))*nd; 
        dc(idx_begin : idx_end) = dc4;

        %% volume
        area = area_cell{ele};
        switch optVcons
        case 'pnorm'
            dv0 = dpnorm(ele) * area / sum(area);
        case 'overall'
            dv0 = area / area0;
        end
        dv(idx_begin : idx_end) = dH2 * dv0(:);
    end
end