% Quadratic boundary conditions
function [ufixed,border, freedofs] = microQBC(nelx,nely,cx,cy)

    nodenrs = reshape(1:(1 + nelx) * (1 + nely), 1 + nely, 1 + nelx);
    
    e0 = eye(9);   
    testNum = size(e0,1);
    ufixed = zeros(2*(nely + 1)*(nelx + 1), testNum);    
    
    alldofs = 1:2 * (nely + 1) * (nelx + 1);
    n1 = [nodenrs(end, [1, end]), nodenrs(1, [end, 1])];
    d1 = reshape([2 * n1 - 1; 2 * n1], 1, 8); % four corner points
    n3 = [nodenrs(2:end - 1, 1)', nodenrs(end, 2:end - 1)];
    d3 = reshape([2 * n3 - 1; 2 * n3], 1, 2 * (nelx + nely - 2)); % left+bottom boundaries
    n4 = [nodenrs(2:end - 1, end)', nodenrs(1, 2:end - 1)];
    d4 = reshape([2 * n4 - 1; 2 * n4], 1, 2 * (nelx + nely - 2)); % right+up boundaries
    d2 = setdiff(alldofs, [d1, d4, d3]);   
    
    coor1m = [cx(end, [1, end]), cx(1, [end, 1]);
             cy(end, [1, end]), cy(1, [end, 1])];
    coor3m = [cx(2:end - 1, 1)', cx(end, 2:end - 1);
             cy(2:end - 1, 1)', cy(end, 2:end - 1)];
    coor4m = [cx(2:end - 1, end)', cx(1, 2:end - 1);
             cy(2:end - 1, end)', cy(1, 2:end - 1)];
    coor1 = coor1m(:);
    coor3 = coor3m(:);
    coor4 = coor4m(:);
    
    
    for j = 1:3
        e = [e0(1, j), e0(3, j) / 2; e0(3, j) / 2, e0(2, j)];
        
        % d1, corner
        for i = 1:2:length(d1)
            ufixed(d1(i:i+1), j) = e * coor1(i:i+1,1);
        end
        
        % border
        for i = 1:2:length(d3)
            ufixed(d3(i:i+1), j) = e * coor3(i:i+1,1);
            ufixed(d4(i:i+1), j) = e * coor4(i:i+1,1);
        end
    end
    
    if testNum>3
        for j = 4:testNum
            [ex,ey] = strainGradient(e0(:,j));

            % d1, corner
            for i = 1:2:length(d1)
                coor = [coor1(i,1), coor1(i+1,1)];
                ufixed(d1(i), j) = 1/2 * coor * ex * coor';
                ufixed(d1(i+1), j) = 1/2 * coor * ey * coor';
            end

            % d3, left+bottom
            for i = 1:2:length(d3)
                coor = [coor3(i,1),coor3(i+1,1)];
                ufixed(d3(i), j) = 1/2 * coor * ex * coor';
                ufixed(d3(i+1), j) = 1/2 * coor * ey * coor';

                coor = [coor4(i,1),coor4(i+1,1)];
                ufixed(d4(i), j) = 1/2 * coor * ex * coor';
                ufixed(d4(i+1), j) = 1/2 * coor * ey * coor';
            end
        end
    end
    
    border = [d1,d3,d4];
    freedofs = d2;

    % test
%     for j = 1:testNum
%         figure;
%         for i = 1:2:length(d1)
%             coor = ufixed(d1(i:i+1),j)+coor1(i:i+1,1);
%             scatter(coor(1:2:end),coor(2:2:end),'filled'); hold on;
%         end
%         
%         for i = 1:2:length(d3)
%             coor = ufixed(d3(i:i+1), j) + coor3(i:i+1,1);
%             scatter(coor(1:2:end),coor(2:2:end)); hold on;
%             coor = ufixed(d4(i:i+1), j) + coor4(i:i+1,1);
%             scatter(coor(1:2:end),coor(2:2:end)); hold on;
%         end
%         axis equal;
%     end
end

function u = bilinearU(x,y,lx,ly,u0)
    N1 = (1-x/lx).*(1-y/ly);
    N2 = x/lx.*(1-y/ly);
    N3 = x/lx.*y/ly;
    N4 = (1-x/lx).*y/ly;
    
    u1 = N1*u0(1) + N2*u0(3) + N3*u0(5) + N4*u0(7);
    u2 = N1*u0(2) + N2*u0(4) + N3*u0(6) + N4*u0(8);
    u = [u1,u2];
end

function [gx,gy] = strainGradient(e0)
    e = zeros(2,2,2);
    e(1,1,1) = e0(4);
    e(2,2,1) = e0(5);
    e(1,2,2) = e0(6); e(2,1,2) = e(1,2,2);
    e(2,2,2) = e0(7);
    e(1,1,2) = e0(8);
    e(1,2,1) = e0(9); e(2,1,1) = e(1,2,1);
    
    g111 = secondGradient(e,1,1,1);
    g121 = secondGradient(e,1,2,1);
    g112 = secondGradient(e,1,1,2);
    g122 = secondGradient(e,1,2,2);
    gx = [g111, g121;
          g112, g122];
    g211 = secondGradient(e,2,1,1);
    g221 = secondGradient(e,2,2,1);
    g212 = secondGradient(e,2,1,2);
    g222 = secondGradient(e,2,2,2);
    gy = [g211, g221;
          g212, g222];
end

function g = secondGradient(e,i,j,k)
    g = e(i,j,k)+e(i,k,j) - e(j,k,i);
end