classdef Elastic2dCurvilinearAnisotropicUpwind < scheme.Scheme

% Discretizes the elastic wave equation:
% rho u_{i,tt} = dj C_{ijkl} dk u_j
% in curvilinear coordinates.
% opSet should be cell array of opSets, one per dimension. This
% is useful if we have periodic BC in one direction.
% Assumes fully compatible operators.

    properties
        m % Number of points in each direction, possibly a vector
        h % Grid spacing

        grid
        dim

        order % Order of accuracy for the approximation

        % Diagonal matrices for variable coefficients
        J, Ji
        RHO % Density
        C   % Elastic stiffness tensor

        D  % Total operator

        Dx, Dy % Physical derivatives
        sigma % Cell matrix of physical stress operators
        n_w, n_e, n_s, n_n % Physical normals

        % Boundary operators in cell format, used for BC
        T_w, T_e, T_s, T_n

        % Traction operators
        tau_w, tau_e, tau_s, tau_n      % Return vector field
        tau1_w, tau1_e, tau1_s, tau1_n  % Return scalar field
        tau2_w, tau2_e, tau2_s, tau2_n  % Return scalar field

        % Inner products
        H

        % Boundary inner products (for scalar field)
        H_w, H_e, H_s, H_n

        % Surface Jacobian vectors
        s_w, s_e, s_s, s_n

        % Boundary restriction operators
        e_w, e_e, e_s, e_n      % Act on vector field, return vector field at boundary
        e1_w, e1_e, e1_s, e1_n  % Act on vector field, return scalar field at boundary
        e2_w, e2_e, e2_s, e2_n  % Act on vector field, return scalar field at boundary
        e_scalar_w, e_scalar_e, e_scalar_s, e_scalar_n; % Act on scalar field, return scalar field
        en_w, en_e, en_s, en_n  % Act on vector field, return normal component

        % E{i}^T picks out component i
        E

        % Elastic2dVariableAnisotropic object for reference domain
        refObj
    end

    methods

        % The coefficients can either be function handles or grid functions
        % optFlag -- if true, extra computations are performed, which may be helpful for optimization.
        function obj = Elastic2dCurvilinearAnisotropicUpwind(g, order, rho, C, opSet, optFlag)
            default_arg('rho', @(x,y) 0*x+1);
            default_arg('opSet',{@sbp.D1Upwind, @sbp.D1Upwind});
            default_arg('optFlag', false);
            dim = 2;

            C_default = cell(dim,dim,dim,dim);
            for i = 1:dim
                for j = 1:dim
                    for k = 1:dim
                        for l = 1:dim
                            C_default{i,j,k,l} = @(x,y) 0*x;
                        end
                    end
                end
            end
            default_arg('C', C_default);

            assert(isa(g, 'grid.Curvilinear'));

            if isa(rho, 'function_handle')
                rho = grid.evalOn(g, rho);
            end

            C_mat = cell(dim,dim,dim,dim);
            for i = 1:dim
                for j = 1:dim
                    for k = 1:dim
                        for l = 1:dim
                            if isa(C{i,j,k,l}, 'function_handle')
                                C{i,j,k,l} = grid.evalOn(g, C{i,j,k,l});
                            end
                            C_mat{i,j,k,l} = spdiag(C{i,j,k,l});
                        end
                    end
                end
            end
            obj.C = C_mat;

            m = g.size();
            m_tot = g.N();

            % 1D operators
            opSetMetric = {@sbp.D2VariableCompatible, @sbp.D2VariableCompatible};
            orderMetric = ceil(order/2)*2;
            m_u = m(1);
            m_v = m(2);
            ops_u = opSetMetric{1}(m_u, {0, 1}, orderMetric);
            ops_v = opSetMetric{2}(m_v, {0, 1}, orderMetric);

            h_u = ops_u.h;
            h_v = ops_v.h;

            I_u = speye(m_u);
            I_v = speye(m_v);

            D1_u = ops_u.D1;
            H_u =  ops_u.H;
            Hi_u = ops_u.HI;
            e_l_u = ops_u.e_l;
            e_r_u = ops_u.e_r;
            d1_l_u = ops_u.d1_l;
            d1_r_u = ops_u.d1_r;

            D1_v = ops_v.D1;
            H_v =  ops_v.H;
            Hi_v = ops_v.HI;
            e_l_v = ops_v.e_l;
            e_r_v = ops_v.e_r;
            d1_l_v = ops_v.d1_l;
            d1_r_v = ops_v.d1_r;


            % Logical operators
            Du = kr(D1_u,I_v);
            Dv = kr(I_u,D1_v);

            e_w  = kr(e_l_u,I_v);
            e_e  = kr(e_r_u,I_v);
            e_s  = kr(I_u,e_l_v);
            e_n  = kr(I_u,e_r_v);

            % Metric coefficients
            coords = g.points();
            x = coords(:,1);
            y = coords(:,2);

            x_u = Du*x;
            x_v = Dv*x;
            y_u = Du*y;
            y_v = Dv*y;

            J = x_u.*y_v - x_v.*y_u;

            K = cell(dim, dim);
            K{1,1} = y_v./J;
            K{1,2} = -y_u./J;
            K{2,1} = -x_v./J;
            K{2,2} = x_u./J;

            % Physical derivatives
            obj.Dx = spdiag( y_v./J)*Du + spdiag(-y_u./J)*Dv;
            obj.Dy = spdiag(-x_v./J)*Du + spdiag( x_u./J)*Dv;

            % Wrap around Aniosotropic Cartesian
            rho_tilde = J.*rho;

            PHI = cell(dim,dim,dim,dim);
            for i = 1:dim
                for j = 1:dim
                    for k = 1:dim
                        for l = 1:dim
                            PHI{i,j,k,l} = 0*C{i,j,k,l};
                            for m = 1:dim
                                for n = 1:dim
                                    PHI{i,j,k,l} = PHI{i,j,k,l} + J.*K{m,i}.*C{m,j,n,l}.*K{n,k};
                                end
                            end
                        end
                    end
                end
            end

            gRef = grid.equidistant([m_u, m_v], {0,1}, {0,1});
            refObj = scheme.Elastic2dVariableAnisotropicUpwind(gRef, order, rho_tilde, PHI, opSet);

            %---- Set object properties ------
            obj.RHO = spdiag(rho);

            % Volume quadrature
            obj.J = spdiag(J);
            obj.Ji = spdiag(1./J);
            obj.H = obj.J*refObj.H;

            % Boundary quadratures
            s_w = sqrt((e_w'*x_v).^2 + (e_w'*y_v).^2);
            s_e = sqrt((e_e'*x_v).^2 + (e_e'*y_v).^2);
            s_s = sqrt((e_s'*x_u).^2 + (e_s'*y_u).^2);
            s_n = sqrt((e_n'*x_u).^2 + (e_n'*y_u).^2);
            obj.s_w = s_w;
            obj.s_e = s_e;
            obj.s_s = s_s;
            obj.s_n = s_n;

            obj.H_w = H_v*spdiag(s_w);
            obj.H_e = H_v*spdiag(s_e);
            obj.H_s = H_u*spdiag(s_s);
            obj.H_n = H_u*spdiag(s_n);

            % Restriction operators
            obj.e_w = refObj.e_w;
            obj.e_e = refObj.e_e;
            obj.e_s = refObj.e_s;
            obj.e_n = refObj.e_n;

            % Adapt things from reference object
            obj.D = refObj.D;
            obj.E = refObj.E;

            obj.e1_w = refObj.e1_w;
            obj.e1_e = refObj.e1_e;
            obj.e1_s = refObj.e1_s;
            obj.e1_n = refObj.e1_n;

            obj.e2_w = refObj.e2_w;
            obj.e2_e = refObj.e2_e;
            obj.e2_s = refObj.e2_s;
            obj.e2_n = refObj.e2_n;

            obj.e_scalar_w = refObj.e_scalar_w;
            obj.e_scalar_e = refObj.e_scalar_e;
            obj.e_scalar_s = refObj.e_scalar_s;
            obj.e_scalar_n = refObj.e_scalar_n;

            e1_w = obj.e1_w;
            e1_e = obj.e1_e;
            e1_s = obj.e1_s;
            e1_n = obj.e1_n;

            e2_w = obj.e2_w;
            e2_e = obj.e2_e;
            e2_s = obj.e2_s;
            e2_n = obj.e2_n;

            obj.tau1_w = (spdiag(1./s_w)*refObj.tau1_w')';
            obj.tau1_e = (spdiag(1./s_e)*refObj.tau1_e')';
            obj.tau1_s = (spdiag(1./s_s)*refObj.tau1_s')';
            obj.tau1_n = (spdiag(1./s_n)*refObj.tau1_n')';

            obj.tau2_w = (spdiag(1./s_w)*refObj.tau2_w')';
            obj.tau2_e = (spdiag(1./s_e)*refObj.tau2_e')';
            obj.tau2_s = (spdiag(1./s_s)*refObj.tau2_s')';
            obj.tau2_n = (spdiag(1./s_n)*refObj.tau2_n')';

            obj.tau_w = (refObj.e_w'*obj.e1_w*obj.tau1_w')' + (refObj.e_w'*obj.e2_w*obj.tau2_w')';
            obj.tau_e = (refObj.e_e'*obj.e1_e*obj.tau1_e')' + (refObj.e_e'*obj.e2_e*obj.tau2_e')';
            obj.tau_s = (refObj.e_s'*obj.e1_s*obj.tau1_s')' + (refObj.e_s'*obj.e2_s*obj.tau2_s')';
            obj.tau_n = (refObj.e_n'*obj.e1_n*obj.tau1_n')' + (refObj.e_n'*obj.e2_n*obj.tau2_n')';

            % Physical normals
            e_w = obj.e_scalar_w;
            e_e = obj.e_scalar_e;
            e_s = obj.e_scalar_s;
            e_n = obj.e_scalar_n;

            e_w_vec = obj.e_w;
            e_e_vec = obj.e_e;
            e_s_vec = obj.e_s;
            e_n_vec = obj.e_n;

            nu_w = [-1,0];
            nu_e = [1,0];
            nu_s = [0,-1];
            nu_n = [0,1];

            obj.n_w = cell(2,1);
            obj.n_e = cell(2,1);
            obj.n_s = cell(2,1);
            obj.n_n = cell(2,1);

            n_w_1 = (1./s_w).*e_w'*(J.*(K{1,1}*nu_w(1) + K{1,2}*nu_w(2)));
            n_w_2 = (1./s_w).*e_w'*(J.*(K{2,1}*nu_w(1) + K{2,2}*nu_w(2)));
            obj.n_w{1} = spdiag(n_w_1);
            obj.n_w{2} = spdiag(n_w_2);

            n_e_1 = (1./s_e).*e_e'*(J.*(K{1,1}*nu_e(1) + K{1,2}*nu_e(2)));
            n_e_2 = (1./s_e).*e_e'*(J.*(K{2,1}*nu_e(1) + K{2,2}*nu_e(2)));
            obj.n_e{1} = spdiag(n_e_1);
            obj.n_e{2} = spdiag(n_e_2);

            n_s_1 = (1./s_s).*e_s'*(J.*(K{1,1}*nu_s(1) + K{1,2}*nu_s(2)));
            n_s_2 = (1./s_s).*e_s'*(J.*(K{2,1}*nu_s(1) + K{2,2}*nu_s(2)));
            obj.n_s{1} = spdiag(n_s_1);
            obj.n_s{2} = spdiag(n_s_2);

            n_n_1 = (1./s_n).*e_n'*(J.*(K{1,1}*nu_n(1) + K{1,2}*nu_n(2)));
            n_n_2 = (1./s_n).*e_n'*(J.*(K{2,1}*nu_n(1) + K{2,2}*nu_n(2)));
            obj.n_n{1} = spdiag(n_n_1);
            obj.n_n{2} = spdiag(n_n_2);

            % Operators that extract the normal component
            obj.en_w = (obj.n_w{1}*obj.e1_w' + obj.n_w{2}*obj.e2_w')';
            obj.en_e = (obj.n_e{1}*obj.e1_e' + obj.n_e{2}*obj.e2_e')';
            obj.en_s = (obj.n_s{1}*obj.e1_s' + obj.n_s{2}*obj.e2_s')';
            obj.en_n = (obj.n_n{1}*obj.e1_n' + obj.n_n{2}*obj.e2_n')';

            % Stress operators
            sigma = cell(dim, dim);
            D1 = {obj.Dx, obj.Dy};
            E = obj.E;
            N = length(obj.RHO);
            for i = 1:dim
                for j = 1:dim
                    sigma{i,j} = sparse(N,2*N);
                    for k = 1:dim
                        for l = 1:dim
                            sigma{i,j} = sigma{i,j} + obj.C{i,j,k,l}*D1{k}*E{l}';
                        end
                    end
                end
            end
            obj.sigma = sigma;

            % Misc.
            obj.refObj = refObj;
            obj.m = refObj.m;
            obj.h = refObj.h;
            obj.order = order;
            obj.grid = g;
            obj.dim = dim;

        end


        % Closure functions return the operators applied to the own domain to close the boundary
        % Penalty functions return the operators to force the solution. In the case of an interface it returns the operator applied to the other doamin.
        %       boundary            is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'.
        %       bc                  is a cell array of component and bc type, e.g. {1, 'd'} for Dirichlet condition
        %                           on the first component. Can also be e.g.
        %                           {'normal', 'd'} or {'tangential', 't'} for conditions on
        %                           tangential/normal component.
        %       data                is a function returning the data that should be applied at the boundary.
        %       neighbour_scheme    is an instance of Scheme that should be interfaced to.
        %       neighbour_boundary  is a string specifying which boundary to interface to.

        % For displacement bc:
        % bc = {comp, 'd', dComps},
        % where
        % dComps = vector of components with displacement BC. Default: 1:dim.
        % In this way, we can specify one BC at a time even though the SATs depend on all BC.
        function [closure, penalty] = boundary_condition(obj, boundary, bc, tuning)
            default_arg('tuning', 1.0);
            assert( iscell(bc), 'The BC type must be a 2x1 or 3x1 cell array' );

            [closure, penalty] = obj.refObj.boundary_condition(boundary, bc, tuning);

            type = bc{2};

            switch type
            case {'F','f','Free','free','traction','Traction','t','T'}
                s = obj.(['s_' boundary]);
                s = spdiag(s);
                penalty = penalty*s;
            end
        end

        % type     Struct that specifies the interface coupling.
        %          Fields:
        %          -- tuning:           penalty strength, defaults to 1.0
        %          -- interpolation:    type of interpolation, default 'none'
        function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary,type)

            defaultType.tuning = 1.0;
            defaultType.interpolation = 'none';
            default_struct('type', defaultType);

            [closure, penalty] = obj.refObj.interface(boundary,neighbour_scheme.refObj,neighbour_boundary,type);
        end

        % Returns h11 for the boundary specified by the string boundary.
        % op -- string
        function h11 = getBorrowing(obj, boundary)
            assertIsMember(boundary, {'w', 'e', 's', 'n'})

            switch boundary
            case {'w','e'}
                h11 = obj.refObj.h11{1};
            case {'s', 'n'}
                h11 = obj.refObj.h11{2};
            end
        end

        % Returns the outward unit normal vector for the boundary specified by the string boundary.
        % n is a cell of diagonal matrices for each normal component, n{1} = n_1, n{2} = n_2.
        function n = getNormal(obj, boundary)
            assertIsMember(boundary, {'w', 'e', 's', 'n'})

            n = obj.(['n_' boundary]);
        end

        % Returns the boundary operator op for the boundary specified by the string boundary.
        % op -- string
        function o = getBoundaryOperator(obj, op, boundary)
            assertIsMember(boundary, {'w', 'e', 's', 'n'})
            assertIsMember(op, {'e', 'e1', 'e2', 'tau', 'tau1', 'tau2', 'en'})

            o = obj.([op, '_', boundary]);

        end

        % Returns the boundary operator op for the boundary specified by the string boundary.
        % op -- string
        function o = getBoundaryOperatorForScalarField(obj, op, boundary)
            assertIsMember(boundary, {'w', 'e', 's', 'n'})
            assertIsMember(op, {'e'})

            switch op

                case 'e'
                    o = obj.(['e_scalar', '_', boundary]);
            end

        end

        % Returns the boundary operator T_ij (cell format) for the boundary specified by the string boundary.
        % Formula: tau_i = T_ij u_j
        % op -- string
        function T = getBoundaryTractionOperator(obj, boundary)
            assertIsMember(boundary, {'w', 'e', 's', 'n'})

            T = obj.(['T', '_', boundary]);
        end

        % Returns square boundary quadrature matrix, of dimension
        % corresponding to the number of boundary unknowns
        %
        % boundary -- string
        function H = getBoundaryQuadrature(obj, boundary)
            assertIsMember(boundary, {'w', 'e', 's', 'n'})

            H = obj.getBoundaryQuadratureForScalarField(boundary);
            I_dim = speye(obj.dim, obj.dim);
            H = kron(H, I_dim);
        end

        % Returns square boundary quadrature matrix, of dimension
        % corresponding to the number of boundary grid points
        %
        % boundary -- string
        function H_b = getBoundaryQuadratureForScalarField(obj, boundary)
            assertIsMember(boundary, {'w', 'e', 's', 'n'})

            H_b = obj.(['H_', boundary]);
        end

        function N = size(obj)
            N = obj.dim*prod(obj.m);
        end
    end
end