--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/+scheme/elasticDilationVariable.m	Tue Jan 16 17:41:55 2018 -0800
@@ -0,0 +1,415 @@
+classdef elasticDilationVariable < scheme.Scheme
+    properties
+        m % Number of points in each direction, possibly a vector
+        h % Grid spacing
+
+        grid
+        dim
+
+        order % Order accuracy for the approximation
+
+        A % Variable coefficient lambda of the operator (as diagonal matrix here)
+        RHO % Density (as diagonal matrix here)
+
+        D % Total operator
+        D1 % First derivatives
+        D2 % Second derivatives
+        Div % Divergence operator used for BC
+        H, Hi % Inner products
+        phi % Borrowing constant for (d1 - e^T*D1) from R
+        H11 % First element of H
+        e_l, e_r
+        d1_l, d1_r % Normal derivatives at the boundary
+        E % E{i}^T picks out component i
+        
+        H_boundary % Boundary inner products
+
+        A_boundary_l % Variable coefficient at boundaries
+        A_boundary_r % 
+    end
+
+    methods
+        % Implements the shear part of the elastic wave equation, i.e.
+        % rho u_{i,tt} = d_i a d_j u_j + d_j a d_j u_i
+        % where a = mu.
+
+        function obj = elasticDilationVariable(g ,order, a_fun, rho_fun, opSet)
+            default_arg('opSet',@sbp.D2Variable);
+            default_arg('a_fun', @(x,y) 0*x+1);
+            default_arg('rho_fun', @(x,y) 0*x+1);
+            dim = 2;
+
+            assert(isa(g, 'grid.Cartesian'))
+
+            a = grid.evalOn(g, a_fun);
+            rho = grid.evalOn(g, rho_fun);
+            m = g.size();
+            m_tot = g.N();
+
+            h = g.scaling();
+            L = (m-1).*h;
+
+            % 1D operators
+            ops = cell(dim,1);
+            for i = 1:dim
+                ops{i} = opSet(m(i), {0, L(i)}, order);
+            end
+
+            % Borrowing constants
+            beta = ops{1}.borrowing.R.delta_D;
+            obj.H11 = ops{1}.borrowing.H11;
+            obj.phi = beta/obj.H11;
+
+            I = cell(dim,1);
+            D1 = cell(dim,1);
+            D2 = cell(dim,1);
+            H = cell(dim,1);
+            Hi = cell(dim,1);
+            e_l = cell(dim,1);
+            e_r = cell(dim,1);
+            d1_l = cell(dim,1);
+            d1_r = cell(dim,1);
+
+            for i = 1:dim
+                I{i} = speye(m(i));
+                D1{i} = ops{i}.D1;
+                D2{i} = ops{i}.D2;
+                H{i} =  ops{i}.H;
+                Hi{i} = ops{i}.HI;
+                e_l{i} = ops{i}.e_l;
+                e_r{i} = ops{i}.e_r;
+                d1_l{i} = ops{i}.d1_l;
+                d1_r{i} = ops{i}.d1_r;
+            end
+
+            %====== Assemble full operators ========
+            A = spdiag(a);
+            obj.A = A;
+            RHO = spdiag(rho);
+            obj.RHO = RHO;
+
+
+            obj.D1 = cell(dim,1);
+            obj.D2 = cell(dim,1);
+            obj.e_l = cell(dim,1);
+            obj.e_r = cell(dim,1);
+            obj.d1_l = cell(dim,1);
+            obj.d1_r = cell(dim,1);
+
+            % D1
+            obj.D1{1} = kron(D1{1},I{2});
+            obj.D1{2} = kron(I{1},D1{2});
+
+            % Boundary operators
+            obj.e_l{1} = kron(e_l{1},I{2});
+            obj.e_l{2} = kron(I{1},e_l{2});
+            obj.e_r{1} = kron(e_r{1},I{2});
+            obj.e_r{2} = kron(I{1},e_r{2});
+
+            obj.d1_l{1} = kron(d1_l{1},I{2});
+            obj.d1_l{2} = kron(I{1},d1_l{2});
+            obj.d1_r{1} = kron(d1_r{1},I{2});
+            obj.d1_r{2} = kron(I{1},d1_r{2});
+
+            % D2
+            for i = 1:dim
+                obj.D2{i} = sparse(m_tot);
+            end
+            ind = grid.funcToMatrix(g, 1:m_tot);
+
+            for i = 1:m(2)
+                D = D2{1}(a(ind(:,i)));
+                p = ind(:,i);
+                obj.D2{1}(p,p) = D;
+            end
+
+            for i = 1:m(1)
+                D = D2{2}(a(ind(i,:)));
+                p = ind(i,:);
+                obj.D2{2}(p,p) = D;
+            end
+
+            % Quadratures
+            obj.H = kron(H{1},H{2});
+            obj.Hi = inv(obj.H);
+            obj.H_boundary = cell(dim,1);
+            obj.H_boundary{1} = H{2};
+            obj.H_boundary{2} = H{1};
+
+            % Boundary coefficient matrices and quadratures
+            obj.A_boundary_l = cell(dim,1);
+            obj.A_boundary_r = cell(dim,1);
+            for i = 1:dim
+                obj.A_boundary_l{i} = obj.e_l{i}'*A*obj.e_l{i};
+                obj.A_boundary_r{i} = obj.e_r{i}'*A*obj.e_r{i};
+            end
+
+            % E{i}^T picks out component i.
+            E = cell(dim,1);
+            I = speye(m_tot,m_tot);
+            for i = 1:dim
+                e = sparse(dim,1);
+                e(i) = 1;
+                E{i} = kron(I,e);
+            end
+            obj.E = E;
+
+            % Differentiation matrix D (without SAT)
+            D2 = obj.D2;
+            D1 = obj.D1;
+            D = sparse(dim*m_tot,dim*m_tot);
+            d = @kroneckerDelta;    % Kronecker delta
+            db = @(i,j) 1-d(i,j); % Logical not of Kronecker delta
+            for i = 1:dim
+                for j = 1:dim
+                    D = D + E{i}*inv(RHO)*( d(i,j)*D2{i}*E{j}' +...
+                                            db(i,j)*D1{i}*A*D1{j}*E{j}' ...
+                                          );
+                end
+            end
+            obj.D = D;
+            %=========================================%
+
+            % Divergence operator for BC
+            Div = cell(dim,1);
+            for i = 1:dim
+                Div{i} = sparse(m_tot,dim*m_tot);
+                for j = 1:dim
+                    Div{i} = Div{i} + d(i,j)*(obj.e_l{i}*obj.d1_l{i}' + obj.e_r{i}*obj.d1_r{i}')*E{j}' ...
+                              + db(i,j)*obj.D1{j}*E{j}';
+                end
+            end
+            obj.Div = Div;
+
+            % Misc.
+            obj.m = m;
+            obj.h = 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.
+        % Here penalty{i,j} enforces data component j on solution component i
+        %       boundary            is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'.
+        %       type                is a string specifying the type of boundary condition if there are several.
+        %       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.
+        function [closure, penalty] = boundary_condition(obj, boundary, type, parameter)
+            default_arg('type','free');
+            default_arg('parameter', []);
+
+            delta = @kroneckerDelta;    % Kronecker delta
+            delta_b = @(i,j) 1-delta(i,j); % Logical not of Kronecker delta
+
+            % j is the coordinate direction of the boundary
+            % nj: outward unit normal component. 
+            % nj = -1 for west, south, bottom boundaries
+            % nj = 1  for east, north, top boundaries
+            [j, nj] = obj.get_boundary_number(boundary);
+            switch nj
+            case 1
+                e = obj.e_r;
+                d = obj.d1_r;
+            case -1
+                e = obj.e_l;
+                d = obj.d1_l;
+            end
+
+            E = obj.E;
+            Hi = obj.Hi;
+            H_gamma = obj.H_boundary{j};
+            A = obj.A;
+            RHO = obj.RHO;
+            D1 = obj.D1;
+
+            phi = obj.phi;
+            H11 = obj.H11;
+            h = obj.h;
+
+            switch type
+                % Dirichlet boundary condition
+                case {'D','d','dirichlet'}
+                    error('Not implemented');
+                    tuning = 1.2;
+                    phi = obj.phi;
+
+                    closures = cell(obj.dim,1);
+                    penalties = cell(obj.dim,obj.dim);
+                    % Loop over components
+                    for i = 1:obj.dim
+                        H_gamma_i = obj.H_boundary{i};
+                        sigma_ij = tuning*delta(i,j)*2/(gamma*h(j)) +...
+                                   tuning*delta_b(i,j)*(2/(H11*h(j)) + 2/(H11*h(j)*phi));
+
+                        ci = E{i}*inv(RHO)*nj*Hi*...
+                                ( (e{j}*H_gamma*e{j}'*A*e{j}*d{j}')'*E{i}' + ...
+                                  delta(i,j)*(e{j}*H_gamma*e{j}'*A*e{j}*d{j}')'*E{j}' ...
+                                   ) ...
+                                - sigma_ij*E{i}*inv(RHO)*Hi*A*e{j}*H_gamma*e{j}'*E{i}';
+
+                        cj = E{j}*inv(RHO)*nj*Hi*...
+                                ( delta_b(i,j)*(e{j}*H_gamma*e{j}'*A*D1{i})'*E{i}' ...
+                                   );
+
+                        if isempty(closures{i})
+                            closures{i} = ci;
+                        else
+                            closures{i} = closures{i} + ci;
+                        end
+
+                        if isempty(closures{j})
+                            closures{j} = cj;
+                        else
+                            closures{j} = closures{j} + cj;
+                        end
+   
+                        pii = - E{i}*inv(RHO)*nj*Hi*...
+                                ( (H_gamma*e{j}'*A*e{j}*d{j}')' + ...
+                                  delta(i,j)*(H_gamma*e{j}'*A*e{j}*d{j}')' ...
+                                   ) ...
+                                + sigma_ij*E{i}*inv(RHO)*Hi*A*e{j}*H_gamma;
+
+                        pji = - E{j}*inv(RHO)*nj*Hi*...
+                                ( delta_b(i,j)*(H_gamma*e{j}'*A*D1{i})' );
+
+                        % Dummies
+                        pij = - 0*E{i}*e{j};
+                        pjj = - 0*E{j}*e{j};
+
+                        if isempty(penalties{i,i})
+                            penalties{i,i} = pii;
+                        else
+                            penalties{i,i} = penalties{i,i} + pii;
+                        end
+
+                        if isempty(penalties{j,i})
+                            penalties{j,i} = pji;
+                        else
+                            penalties{j,i} = penalties{j,i} + pji;
+                        end
+
+                        if isempty(penalties{i,j})
+                            penalties{i,j} = pij;
+                        else
+                            penalties{i,j} = penalties{i,j} + pij;
+                        end
+
+                        if isempty(penalties{j,j})
+                            penalties{j,j} = pjj;
+                        else
+                            penalties{j,j} = penalties{j,j} + pjj;
+                        end
+                    end
+                    [rows, cols] = size(closures{1});
+                    closure = sparse(rows, cols);
+                    for i = 1:obj.dim
+                        closure = closure + closures{i};
+                    end
+                    penalty = penalties;
+
+                % Free boundary condition
+                case {'F','f','Free','free'}
+                    closures = cell(obj.dim,1);
+                    penalties = cell(obj.dim,obj.dim);
+
+                    % Divergence operator
+                    Div = obj.Div{j};
+
+                    % Loop over components
+                    %for i = 1:obj.dim
+                        closure = -nj*E{j}*inv(RHO)*Hi*e{j} ...
+                                     *H_gamma*e{j}'*A*e{j}*e{j}'*Div;
+                        penalty = nj*E{j}*inv(RHO)*Hi*e{j} ...
+                                     *H_gamma*e{j}'*A*e{j};
+                    %end
+                    % [rows, cols] = size(closures{1});
+                    % closure = sparse(rows, cols);
+                    % for i = 1:obj.dim
+                    %     closure = closure + closures{i};
+                    %     for j = 1:obj.dim
+                    %         if i~=j
+                    %             [rows cols] = size(penalties{j,j});
+                    %             penalties{i,j} = sparse(rows,cols);
+                    %         end
+                    %     end
+                    % end
+                    % penalty = penalties;
+
+
+                % Unknown boundary condition
+                otherwise
+                    error('No such boundary condition: type = %s',type);
+            end
+        end
+
+        function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary)
+            % u denotes the solution in the own domain
+            % v denotes the solution in the neighbour domain
+            tuning = 1.2;
+            % tuning = 20.2;
+            error('Interface not implemented');
+        end
+
+        % Returns the coordinate number and outward normal component for the boundary specified by the string boundary.
+        function [j, nj] = get_boundary_number(obj, boundary)
+
+            switch boundary
+                case {'w','W','west','West', 'e', 'E', 'east', 'East'}
+                    j = 1;
+                case {'s','S','south','South', 'n', 'N', 'north', 'North'}
+                    j = 2;
+                otherwise
+                    error('No such boundary: boundary = %s',boundary);
+            end
+
+            switch boundary
+                case {'w','W','west','West','s','S','south','South'}
+                    nj = -1;
+                case {'e', 'E', 'east', 'East','n', 'N', 'north', 'North'}
+                    nj = 1;
+            end
+        end
+
+        % Returns the coordinate number and outward normal component for the boundary specified by the string boundary.
+        function [return_op] = get_boundary_operator(obj, op, boundary)
+
+            switch boundary
+                case {'w','W','west','West', 'e', 'E', 'east', 'East'}
+                    j = 1;
+                case {'s','S','south','South', 'n', 'N', 'north', 'North'}
+                    j = 2;
+                otherwise
+                    error('No such boundary: boundary = %s',boundary);
+            end
+
+            switch op
+                case 'e'
+                    switch boundary
+                        case {'w','W','west','West','s','S','south','South'}
+                            return_op = obj.e_l{j};
+                        case {'e', 'E', 'east', 'East','n', 'N', 'north', 'North'}
+                            return_op = obj.e_r{j};
+                    end
+                case 'd'
+                    switch boundary
+                        case {'w','W','west','West','s','S','south','South'}
+                            return_op = obj.d_l{j};
+                        case {'e', 'E', 'east', 'East','n', 'N', 'north', 'North'}
+                            return_op = obj.d_r{j};
+                    end
+                otherwise
+                    error(['No such operator: operatr = ' op]);
+            end
+
+        end
+
+        function N = size(obj)
+            N = prod(obj.m);
+        end
+    end
+end