function [Q,dQ] = homogenization_energy(nelx,nely,nelz,D0,xPhys,Emin,penal)
Emin=0;
nele = nelx*nely*nelz;
% KE = elementMatVec3D(1.0/nelx, 1.0/nely, 1.0/nelz, D0);
KE = elementMatVec3D(1.0/2, 1.0/2, 1.0/2, D0);
Num_node = (1+nely)*(1+nelx)*(1+nelz);
nodenrs = reshape(1:Num_node,1+nely,1+nelx,1+nelz);
edofVec = reshape(3*nodenrs(1:end-1,1:end-1,1:end-1)+1,nelx*nely*nelz,1);
edofMat = repmat(edofVec,1,24)+repmat([0 1 2 3*nely+[3 4 5 0 1 2] -3 -2 -1 3*(nelx+1)*(nely+1)+[0 1 2 3*nely+[3 4 5 0 1 2] -3 -2 -1]], nele, 1);
iK = reshape(kron(edofMat,ones(24,1))',24*24*nele,1);
jK = reshape(kron(edofMat,ones(1,24))',24*24*nele,1);
n1 = [nodenrs(end, [1 end], 1) nodenrs(1, [end 1], 1) nodenrs(end, [1 end], end) nodenrs(1, [end 1], end)];
d1 = reshape([3*n1-2; 3*n1-1; 3*n1],3*numel(n1),1);
n3 = [reshape(squeeze(nodenrs(end,1,2:end-1)),1,numel(squeeze(nodenrs(end,1,2:end-1))))...
    reshape(squeeze(nodenrs(1, 1, 2:end-1)),1,numel(squeeze(nodenrs(1, 1, 2:end-1))))...
    reshape(squeeze(nodenrs(end,2:end-1,1)),1,numel(squeeze(nodenrs(end,2:end-1,1))))...
    reshape(squeeze(nodenrs(1, 2:end-1, 1)),1,numel(squeeze(nodenrs(1, 2:end-1, 1))))...
    reshape(squeeze(nodenrs(2:end-1, 1, 1)),1,numel(squeeze(nodenrs(2:end-1, 1, 1))))...
    reshape(squeeze(nodenrs(2:end-1,1,end)),1,numel(squeeze(nodenrs(2:end-1,1,end))))...
    reshape(squeeze(nodenrs(2:end-1, 2:end-1, 1)),1,numel(squeeze(nodenrs(2:end-1, 2:end-1, 1))))...
    reshape(squeeze(nodenrs(2:end-1, 1, 2:end-1)),1,numel(squeeze(nodenrs(2:end-1, 1, 2:end-1))))...
    reshape(squeeze(nodenrs(end,2:end-1,2:end-1)),1,numel(squeeze(nodenrs(end,2:end-1,2:end-1))))];
d3 = reshape([3*n3-2; 3*n3-1; 3*n3],3*numel(n3),1);
n4 = [reshape(squeeze(nodenrs(1, end, 2:end-1)),1,numel(squeeze(nodenrs(1, end, 2:end-1))))...
    reshape(squeeze(nodenrs(end,end,2:end-1)),1,numel(squeeze(nodenrs(end,end,2:end-1))))...
    reshape(squeeze(nodenrs(1, 2:end-1, end)),1,numel(squeeze(nodenrs(1, 2:end-1, end))))...
    reshape(squeeze(nodenrs(end,2:end-1,end)),1,numel(squeeze(nodenrs(end,2:end-1,end))))...
    reshape(squeeze(nodenrs(2:end-1,end,end)),1,numel(squeeze(nodenrs(2:end-1,end,end))))...
    reshape(squeeze(nodenrs(2:end-1, end, 1)),1,numel(squeeze(nodenrs(2:end-1, end, 1))))...
    reshape(squeeze(nodenrs(2:end-1,2:end-1,end)),1,numel(squeeze(nodenrs(2:end-1,2:end-1,end))))...
    reshape(squeeze(nodenrs(2:end-1,end,2:end-1)),1,numel(squeeze(nodenrs(2:end-1,end,2:end-1))))...
    reshape(squeeze(nodenrs(1, 2:end-1, 2:end-1)),1,numel(squeeze(nodenrs(1, 2:end-1, 2:end-1))))];
d4 = reshape([3*n4-2; 3*n4-1; 3*n4],3*numel(n4),1);
n2 = setdiff(nodenrs(:),[n1(:);n3(:);n4(:)]);
d2 = reshape([3*n2-2; 3*n2- 1; 3*n2],3*numel(n2),1);
e = eye(6); ufixed = zeros(24,6);
vert_cor = [0, nelx, nelx, 0, 0, nelx, nelx, 0;
    0, 0, nely, nely, 0, 0, nely, nely;
    0, 0, 0, 0, nelz, nelz, nelz, nelz];
for i = 1:6
    epsilon = [ e(i,1), e(i,4)/2, e(i,6)/2;
        e(i,4)/2, e(i,2), e(i,5)/2;
        e(i,6)/2, e(i,5)/2, e(i,3)];
    ufixed(:,i) = reshape(epsilon*vert_cor,24,1);
end
% 3D boundary constraint equations
wfixed = [repmat(ufixed(7:9,:),numel(squeeze(nodenrs(end,1,2:end-1))),1);
    repmat(ufixed( 4:6,:)-ufixed(10:12,:),numel(squeeze(nodenrs(1, 1, 2:end-1))),1);
    repmat(ufixed(22:24,:),numel(squeeze(nodenrs(end,2:end-1,1))),1);
    repmat(ufixed(13:15,:)-ufixed(10:12,:),numel(squeeze(nodenrs(1, 2:end-1, 1))),1);
    repmat(ufixed(16:18,:),numel(squeeze(nodenrs(2:end-1, 1, 1))),1);
    repmat(ufixed( 4:6,:)-ufixed(13:15,:),numel(squeeze(nodenrs(2:end-1,1,end))),1);
    repmat(ufixed(13:15,:),numel(squeeze(nodenrs(2:end-1, 2:end-1,1))),1);
    repmat(ufixed(4:6,:),numel(squeeze(nodenrs(2:end-1, 1, 2:end- 1))),1);
    repmat(ufixed(10:12,:),numel(squeeze(nodenrs(end,2:end-1,2:end-1))),1)];
% sK = reshape(KE(:)*(Emin+xPhys(:)'.^penal*(1-Emin)),24*24*nele,1);
sK = reshape(KE(:)*(Emin+xPhys(:)'.^penal*(1-Emin)),24*24*nele,1);
K = sparse(iK,jK,sK); K = (K+K')/2;
Kr = [K(d2,d2), K(d2,d3)+K(d2,d4);
    K(d3,d2)+K(d4,d2),K(d3,d3)+K(d4,d3)+K(d3,d4)+K(d4,d4)];
U(d1,:) = ufixed;
U([d2;d3],:) = Kr\(-[K(d2,d1);K(d3,d1)+K(d4,d1)]*ufixed-[K(d2,d4);K(d3,d4)+K(d4,d4)]*wfixed);
U(d4,:) = U(d3,:)+wfixed;
cellVolume = nele;
% cellVolume = 1;
qe = cell(6,6);
Q  = zeros(6,6);dQ = cell(6,6);
for i = 1:6
    for j = 1:6
        U1 = U(:,i);
        U2 = U(:,j);
        qe{i,j} = reshape(sum((U1(edofMat)*KE).*U2(edofMat),2),nely,nelx,nelz);
        Q(i,j) = 1/cellVolume*sum(sum(sum((Emin+xPhys.^penal*(1-Emin)).*qe{i,j})));
        dQ{i,j} = 1/cellVolume*(penal*(1-Emin)*xPhys.^(penal-1).*qe{i,j});
    end
end
end

function Ke = elementMatVec3D(a, b, c, DH)
GN_x=[-1/sqrt(3),1/sqrt(3)];
GN_y=GN_x;
GN_z=GN_x;
GaussWeigh=[1,1];
Ke = zeros(24,24); L = zeros(6,9);
L(1,1) = 1; L(2,5) = 1; L(3,9) = 1;
L(4,2) = 1; L(4,4) = 1; L(5,6) = 1;
L(5,8) = 1; L(6,3) = 1; L(6,7) = 1;
for i=1:length(GN_x)
    for j=1:length(GN_y)
        for k=1:length(GN_z)
            x = GN_x(i);y = GN_y(j);z = GN_z(k);
            dNx = 1/8*[-(1-y)*(1-z), (1-y)*(1-z), (1+y)*(1-z), -(1+y)*(1-z), -(1-y)*(1+z), (1-y)*(1+z), (1+y)*(1+z) -(1+y)*(1+z)];
            dNy = 1/8*[-(1-x)*(1-z) -(1+x)*(1-z), (1+x)*(1-z), (1-x)*(1-z), -(1-x)*(1+z), -(1+x)*(1+z), (1+x)*(1+z), (1-x)*(1+z)];
            dNz = 1/8*[-(1-x)*(1-y) -(1+x)*(1-y) -(1+x)*(1+y) -(1-x)*(1+y), (1-x)*(1-y), (1+x)*(1-y), (1+x)*(1+y), (1-x)*(1+y)];
            J = [dNx;dNy;dNz]*[ -a a a -a -a a a -a ;
                -b -b b b -b -b b b;
                -c -c -c -c c c c c]';
            G = [inv(J) zeros(3) zeros(3);zeros(3) inv(J) zeros(3);zeros(3) zeros(3) inv(J)];
            dN(1,1:3:24) = dNx; dN(2,1:3:24) = dNy; dN(3,1:3:24) = dNz;
            dN(4,2:3:24) = dNx; dN(5,2:3:24) = dNy; dN(6,2:3:24) = dNz;
            dN(7,3:3:24) = dNx; dN(8,3:3:24) = dNy; dN(9,3:3:24) = dNz;
            Be = L*G*dN;
            Ke = Ke + GaussWeigh(i)*GaussWeigh(j)*GaussWeigh(k)*det(J)*(Be'*DH*Be);
        end
    end
end
end