clear; close all; clc;

N_CASE = 6;
h = zeros(1, N_CASE);
err = zeros(3, N_CASE);

for i = 1:N_CASE
    N = 2^i;
    h(i) = 1.0/N;
    
    xMin = 0.0; 
    xMax = 1.0; 
    yMin = 0.0; 
    yMax = 1.0; 
    
    N1 = round((xMax-xMin)/h(i));
    N2 = round((yMax-yMin)/h(i));
    
    fprintf("\nCASE%d: h=1/%d\n", i, N);
    
    err(:, i) = solve_2d_steady_linear_elasticity(xMin, xMax, yMin, yMax, N1, N2);

    fprintf("\n  |err|_inf=%e, |err|_L2=%e, |err|_H1=%e\n", err(1,i), err(2,i), err(3,i));
end

loglog(h, err(1,:), '-s')
hold on
loglog(h, err(2,:), '-s')
hold on
loglog(h, err(3,:), '-s')
grid on
% legend('inf', 'L2', 'semi-H1', 'Location', 'southeast')
loglog([1e0, 1e-2], [1e0, 1e-2])
grid on
loglog([1e0, 1e-2], [1e0, 1e-4])
grid on
loglog([1e0, 1e-2], [1e-1, 1e-7])
grid on
legend('inf', 'L2', 'semi-H1', '1st-order', '2nd-order', '3rd-order', 'Location', 'southeast')

function [errnorm] = solve_2d_steady_linear_elasticity(x_min, x_max, y_min, y_max, N1, N2)
    global P T Pb Tb Jac
    global lambda mu
    
    lambda = 1.0;
    mu = 2.0;
    
    [P, T] = mesh_info_mat(x_min,x_max,y_min,y_max,N1,N2);
    Pb = P; Tb = T;
    [boundary_edge, boundary_node] = boundary_info_mat(N1, N2, T);
    
    Nlb = size(Tb, 1); % Num of local basis functions
    Nb = size(Pb, 2); % Num of global basis functions(FEM unknowns)
    N = size(T,2); % Num of mesh/FEM elements
    nbn = size(boundary_node, 2);
    nbe = size(boundary_edge, 2);
    
    % Element jacobian
    Jac = zeros(N, 1);
    for i = 1:N
        p1 = P(:, T(1, i));
        p2 = P(:, T(2, i));
        p3 = P(:, T(3, i));
        Jac(i) = calc_elem_jacobi(p1, p2, p3);
    end
    
    % Gauss quadrature coordinates & coefficients
    gq_tri_n = 4;
    gq_tri_x0 = [1.0/3, 1.0/5, 3.0/5, 1.0/5];
    gq_tri_y0 = [1.0/3, 1.0/5, 1.0/5, 3.0/5];
    gq_tri_w = [-27.0/96, 25.0/96, 25.0/96, 25.0/96];
    gq_tri_x = zeros(N, gq_tri_n);
    gq_tri_y = zeros(N, gq_tri_n);
    for n = 1:N
        for k = 1:gq_tri_n
            x0 = gq_tri_x0(k); 
            y0 = gq_tri_y0(k);
            [gq_tri_x(n, k), gq_tri_y(n, k)] = affine_mapping_back(n, x0, y0);
        end
    end
    
    % Assemble the stiffness matrix
    A1 = zeros(Nb, Nb);
    A2 = zeros(Nb, Nb);
    A3 = zeros(Nb, Nb);
    A4 = zeros(Nb, Nb);
    A5 = zeros(Nb, Nb);
    A6 = zeros(Nb, Nb);
    A7 = zeros(Nb, Nb);
    A8 = zeros(Nb, Nb);
    for n = 1:N
        for alpha = 1:Nlb % trial
            j = Tb(alpha, n);
            for beta = 1:Nlb % test
                i = Tb(beta, n);
                
                tmp = zeros(8, 1);
                for k = 1:gq_tri_n
                    x = gq_tri_x(n, k);
                    y = gq_tri_y(n, k);                    
                    
                    gpj = grad_trial(alpha, n, x, y);
                    gpi = grad_test(beta, n, x, y);
                    
                    tmp(1) = tmp(1) + gq_tri_w(k) * lambda * gpj(1) * gpi(1);
                    tmp(2) = tmp(2) + gq_tri_w(k) * mu     * gpj(1) * gpi(1);
                    tmp(3) = tmp(3) + gq_tri_w(k) * mu     * gpj(2) * gpi(2);
                    tmp(4) = tmp(4) + gq_tri_w(k) * lambda * gpj(2) * gpi(1);
                    tmp(5) = tmp(5) + gq_tri_w(k) * mu     * gpj(1) * gpi(2);
                    tmp(6) = tmp(6) + gq_tri_w(k) * lambda * gpj(1) * gpi(2);
                    tmp(7) = tmp(7) + gq_tri_w(k) * mu     * gpj(2) * gpi(1);
                    tmp(8) = tmp(8) + gq_tri_w(k) * lambda * gpj(2) * gpi(2);
                end
                tmp = tmp * abs(Jac(n));
                
                A1(i, j) = A1(i, j) + tmp(1);
                A2(i, j) = A2(i, j) + tmp(2);
                A3(i, j) = A3(i, j) + tmp(3);
                A4(i, j) = A4(i, j) + tmp(4);
                A5(i, j) = A5(i, j) + tmp(5);
                A6(i, j) = A6(i, j) + tmp(6);
                A7(i, j) = A7(i, j) + tmp(7);
                A8(i, j) = A8(i, j) + tmp(8);
            end
        end
    end
    A = [A1+2*A2+A3, A4+A5; A6+A7, A8+2*A3+A2];
    
    % Assemble the load vector
    b1 = zeros(Nb, 1);
    b2 = zeros(Nb, 1);
    for n = 1:N
        for beta = 1:Nlb % test
            i = Tb(beta, n);
            
            tmp = zeros(2, 1);
            for k = 1:gq_tri_n
                x0 = gq_tri_x0(k);
                y0 = gq_tri_y0(k);
                x = gq_tri_x(n, k);
                y = gq_tri_y(n, k);  
                
                fval = f(x, y);
                
                tmp(1) = tmp(1) + gq_tri_w(k) * fval(1) * test_ref(beta, x0, y0);
                tmp(2) = tmp(2) + gq_tri_w(k) * fval(2) * test_ref(beta, x0, y0);
            end
            tmp = tmp * abs(Jac(n));
            
            b1(i) = b1(i) + tmp(1);
            b2(i) = b2(i) + tmp(2);
        end
    end
    b = [b1; b2];
    
    % Dirichlet Boundary
    for k = 1:nbn
        if boundary_node(1, k) == -1
            i = boundary_node(2, k);
            g = u(Pb(1, i), Pb(2, i));

            A(i, :) = 0;
            A(i, i) = 1;
            b(i) = g(1);
            
            A(Nb + i, :) = 0;
            A(Nb + i, Nb + i) = 1;
            b(Nb + i) = g(2);
        end
    end
    
    % Solve and Check
    x = A\b;
    u_sol = zeros(2, Nb);
    for i = 1:Nb
        u_sol(1, i) = x(i);
        u_sol(2, i) = x(Nb+i);
    end
    
    errnorm = zeros(1, 3); %inf, L2, semi-H1 respectively
    for n = 1:N
        for k = 1:gq_tri_n
            x0 = gq_tri_x0(k);
            y0 = gq_tri_y0(k);
            x = gq_tri_x(n, k);
            y = gq_tri_y(n, k);  
            
            w = zeros(2, 1);
            for i = 1:Nlb
                w = w + u_sol(:, Tb(i, n)) * trial_ref(i, x0, y0);
            end
            
            err = norm(w - u(x, y), Inf);
            if err > errnorm(1)
                errnorm(1) = err;
            end
        end
    end
    
    for n = 1:N
        res = 0.0;
        for k = 1:gq_tri_n
            x0 = gq_tri_x0(k);
            y0 = gq_tri_y0(k);
            x = gq_tri_x(n, k);
            y = gq_tri_y(n, k);  
            
            w = zeros(2, 1);
            for i = 1:Nlb
                w = w + u_sol(:, Tb(i, n)) * trial_ref(i, x0, y0);
            end
            
            err = norm(w - u(x, y))^2;
            res = res + gq_tri_w(k) * err;
        end
        res = res * abs(Jac(n));
        errnorm(2) = errnorm(2) + res;
    end
    errnorm(2) = sqrt(errnorm(2));
    
    for n = 1:N
        res = 0.0;
        for k = 1:gq_tri_n
            x = gq_tri_x(n, k);
            y = gq_tri_y(n, k);  
            
            w = zeros(2, 2);
            for i = 1:Nlb
                w = w + u_sol(:, Tb(i, n)) * grad_trial(i, n, x, y).';
            end
            
            err = norm(w - grad_u(x, y), 'fro')^2;
            res = res + gq_tri_w(k) * err;
        end
        res = res * abs(Jac(n));
        errnorm(3) = errnorm(3) + res;
    end
    errnorm(3) = sqrt(errnorm(3));
end

function [P, T] = mesh_info_mat(xmin, xmax, ymin, ymax, n1, n2)
    h1 = (xmax-xmin)/n1;
    h2 = (ymax-ymin)/n2;
    
    P = zeros(2, (n1+1)*(n2+1));
    T = zeros(3, 2*n1*n2);
    
    node_idx = zeros(n1+1, n2+1);
    
    for i = 1:n1+1
        x = xmin + (i-1)*h1;
        for j = 1:n2+1
            y = ymin + (j-1)*h2;
            idx = (i-1)*(n2+1)+j;
            P(:,idx) = [x, y];
            node_idx(i, j) = idx;
        end
    end
    
    for i = 1:n1
        for j = 1:n2
            quad_idx = j + (i-1)*n2;
            tri_idx0 = 2*quad_idx-1;
            tri_idx1 = 2*quad_idx;
            idx = [node_idx(i, j), node_idx(i+1, j), node_idx(i+1, j+1), node_idx(i, j+1)];
            T(:,tri_idx0) = [idx(1),idx(2),idx(4)];
            T(:,tri_idx1) = [idx(4),idx(2),idx(3)];
        end
    end
end

function [bdry_edge, bdry_node] = boundary_info_mat(n1, n2, T)
    bdry_edge = zeros(4, 2*(n1+n2));
    bdry_node = zeros(2, 2*(n1+n2));
    
    % Bottom
    for k = 1:n1
        edge_idx = k;
        elem_idx = 1 + (k-1)*n2*2;
        node_idx = edge_idx;

        bdry_edge(1, edge_idx) = -1;
        bdry_edge(2, edge_idx) = elem_idx;
        bdry_edge(3, edge_idx) = T(1, elem_idx);
        bdry_edge(4, edge_idx) = T(2, elem_idx);
        
        bdry_node(1, node_idx) = -1;
        bdry_node(2, node_idx) = T(1, elem_idx);
    end
    
    % Right
    for k = 1:n2
        edge_idx = k+n1;
        elem_idx = 2*n2*(n1-1) + 2*k;
        node_idx = edge_idx;

        bdry_edge(1, edge_idx) = -1;
        bdry_edge(2, edge_idx) = elem_idx;
        bdry_edge(3, edge_idx) = T(2, elem_idx);
        bdry_edge(4, edge_idx) = T(3, elem_idx);
        
        bdry_node(1, node_idx) = -1;
        bdry_node(2, node_idx) = T(2, elem_idx);
    end
    
    % Top
    for k = 1:n1
        edge_idx = k+n2+n1;
        elem_idx = 2*n1*n2 - 2*n2*(k-1);
        node_idx = edge_idx;
        
        bdry_edge(1, edge_idx) = -1;
        bdry_edge(2, edge_idx) = elem_idx;
        bdry_edge(3, edge_idx) = T(3, elem_idx);
        bdry_edge(4, edge_idx) = T(1, elem_idx);
        
        bdry_node(1, node_idx) = -1;
        bdry_node(2, node_idx) = T(3, elem_idx);
    end
    
    % Left
    for k = 1:n2
        edge_idx = k+2*n1+n2;
        elem_idx = 2*n2 - (2*k-1);
        node_idx = edge_idx;

        bdry_edge(1, edge_idx) = -1;
        bdry_edge(2, edge_idx) = elem_idx;
        bdry_edge(3, edge_idx) = T(3, elem_idx);
        bdry_edge(4, edge_idx) = T(1, elem_idx);
        
        bdry_node(1, node_idx) = -1;
        bdry_node(2, node_idx) = T(3, elem_idx);
    end
end

function [ret] = grad_trial(basis, n, x, y)
    ret = grad_test(basis, n, x, y);
end

function [ret] = grad_test(basis, n, x, y)
    global P T Jac

    [x0, y0] = affine_mapping(n, x, y);
    gp = grad_test_ref(basis, x0, y0);
    
    dpx0 = gp(1);
    dpy0 = gp(2);
    
    P1 = P(:, T(1, n));
    P2 = P(:, T(2, n));
    P3 = P(:, T(3, n));
    
    x1 = P1(1); y1 = P1(2);
    x2 = P2(1); y2 = P2(2);
    x3 = P3(1); y3 = P3(2);

    ret = [dpx0 * (y3-y1) + dpy0 * (y1-y2); dpx0 * (x1-x3) + dpy0 * (x2-x1)] / Jac(n);
end

function [ret] = grad_trial_ref(basis, x0, y0)
    ret = grad_test_ref(basis, x0, y0);
end

function [ret] = grad_test_ref(basis, x0, y0)
    switch(basis)
        case 1
            ret = [-1; -1];
        case 2
            ret = [1; 0];
        case 3
            ret = [0; 1];
        otherwise
            ret = [0; 0];
    end
end

function [ret] = trial(basis, n, x, y)
    ret = test(basis, n, x, y);
end

function [ret] = test(basis, n, x, y)
    [x0, y0] = affine_mapping(n, x, y);
    ret = test_ref(basis, x0, y0);
end

function [ret] = trial_ref(basis, x0, y0)
    ret = test_ref(basis, x0, y0);
end

function [ret] = test_ref(basis, x0, y0)
    switch(basis)
        case 1
            ret = 1.0 - x0 - y0;
        case 2
            ret = x0;
        case 3
            ret = y0;
        otherwise
            ret = 0;
    end
end

function [x, y] = affine_mapping_back(n, x0, y0)
    global P T
    
    P1 = P(:, T(1, n));
    P2 = P(:, T(2, n));
    P3 = P(:, T(3, n));

    x1 = P1(1); y1 = P1(2);
    x2 = P2(1); y2 = P2(2);
    x3 = P3(1); y3 = P3(2);
    
    x = (x2-x1)*x0 + (x3-x1)*y0 + x1;
    y = (y2-y1)*x0 + (y3-y1)*y0 + y1;
end

function [x0, y0] = affine_mapping(n, x, y)
    global P T Jac
    
    P1 = P(:, T(1, n));
    P2 = P(:, T(2, n));
    P3 = P(:, T(3, n));
    
    x1 = P1(1); y1 = P1(2);
    x2 = P2(1); y2 = P2(2);
    x3 = P3(1); y3 = P3(2);
    
    x0 = ((y3-y1)*(x-x1)-(x3-x1)*(y-y1))/Jac(n);
    y0 = -((y2-y1)*(x-x1)-(x2-x1)*(y-y1))/Jac(n);
end

function [ret] = calc_elem_jacobi(P1, P2, P3)
    x1 = P1(1); y1 = P1(2);
    x2 = P2(1); y2 = P2(2);
    x3 = P3(1); y3 = P3(2);
    ret = (x2-x1)*(y3-y1)-(x3-x1)*(y2-y1);
end

function [ret] = f(x, y)
    global lambda mu
    ret = zeros(2, 1);
    ret(1) = -(lambda + 2 * mu) * (-pi^2 * sin(pi * x) * sin(pi * y)) - (lambda + mu) * ((2*x-1)*(2*y-1)) - mu * (-pi^2 * sin(pi * x) * sin(pi * y)); 
    ret(2) = -(lambda + 2 * mu) * (2*x*(x-1)) - (lambda + mu) * (pi^2 * cos(pi * x) * cos(pi * y)) - mu * (2*y*(y-1)); 
end

function [ret] = strain(x, y)
    global lambda mu
    ret = zeros(2, 2);
    gu = grad_u(x, y);
    du = gu(1, 1) + gu(2, 2);
    ret(1, 1) = lambda * du + 2 * mu * gu(1, 1);
    ret(1, 2) = mu * (gu(1, 2) + gu(2, 1));
    ret(2, 1) = mu * (gu(2, 1) + gu(1, 2));
    ret(2, 2) = lambda * du + 2 * mu * gu(2, 2);
end

function [ret] = grad_u(x, y)
    ret = zeros(2, 2);
    ret(1, 1) = sin(pi * y) * cos(pi * x) * pi;
    ret(1, 2) = sin(pi * x) * cos(pi * y) * pi;
    ret(2, 1) = y*(y-1)*(2*x-1);
    ret(2, 2) = x*(x-1)*(2*y-1);
end

function [ret] = u(x, y)
    ret = zeros(2, 1);
    ret(1) = sin(pi * x) * sin(pi * y);
    ret(2) = x*(x-1)*y*(y-1);
end