--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/+scheme/Elastic2dVariable.m	Fri Feb 09 13:34:27 2018 -0800
@@ -0,0 +1,420 @@
+classdef Elastic2dVariable < scheme.Scheme
+
+% Discretizes the elastic wave equation:
+% rho u_{i,tt} = di lambda dj u_j + dj mu di u_j + dj mu dj u_i 
+% opSet should be cell array of opSets, one per dimension. This
+% is useful if we have periodic BC in one direction.
+
+    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 varible coefficients
+        LAMBDA % Variable coefficient, related to dilation
+        MU     % Shear modulus, variable coefficient
+        RHO, RHOi % Density, variable
+
+        D % Total operator
+        D1 % First derivatives
+
+        % Second derivatives
+        D2_lambda
+        D2_mu
+
+        % Traction operators used for BC
+        T_l, T_r
+        tau_l, tau_r
+
+        H, Hi % Inner products
+        phi % Borrowing constant for (d1 - e^T*D1) from R
+        gamma % Borrowing constant for d1 from M
+        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
+
+        % Kroneckered norms and coefficients
+        RHOi_kron
+        Hi_kron
+    end
+
+    methods
+
+        function obj = Elastic2dVariable(g ,order, lambda_fun, mu_fun, rho_fun, opSet)
+            default_arg('opSet',{@sbp.D2Variable, @sbp.D2Variable});
+            default_arg('lambda_fun', @(x,y) 0*x+1);
+            default_arg('mu_fun', @(x,y) 0*x+1);
+            default_arg('rho_fun', @(x,y) 0*x+1);
+            dim = 2;
+
+            assert(isa(g, 'grid.Cartesian'))
+
+            lambda = grid.evalOn(g, lambda_fun);
+            mu = grid.evalOn(g, mu_fun);
+            rho = grid.evalOn(g, rho_fun);
+            m = g.size();
+            m_tot = g.N();
+
+            h = g.scaling();
+            lim = g.lim;
+
+            % 1D operators
+            ops = cell(dim,1);
+            for i = 1:dim
+                ops{i} = opSet{i}(m(i), lim{i}, order);
+            end
+
+            % Borrowing constants
+            for i = 1:dim
+                beta = ops{i}.borrowing.R.delta_D;
+                obj.H11{i} = ops{i}.borrowing.H11;
+                obj.phi{i} = beta/obj.H11{i};
+                obj.gamma{i} = ops{i}.borrowing.M.d1;
+            end
+
+            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 ========
+            LAMBDA = spdiag(lambda);
+            obj.LAMBDA = LAMBDA;
+            MU = spdiag(mu);
+            obj.MU = MU;
+            RHO = spdiag(rho);
+            obj.RHO = RHO;
+            obj.RHOi = inv(RHO);
+
+            obj.D1 = cell(dim,1);
+            obj.D2_lambda = cell(dim,1);
+            obj.D2_mu = 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_lambda{i} = sparse(m_tot);
+                obj.D2_mu{i} = sparse(m_tot);
+            end
+            ind = grid.funcToMatrix(g, 1:m_tot);
+
+            for i = 1:m(2)
+                D_lambda = D2{1}(lambda(ind(:,i)));
+                D_mu = D2{1}(mu(ind(:,i)));
+
+                p = ind(:,i);
+                obj.D2_lambda{1}(p,p) = D_lambda;
+                obj.D2_mu{1}(p,p) = D_mu;
+            end
+
+            for i = 1:m(1)
+                D_lambda = D2{2}(lambda(ind(i,:)));
+                D_mu = D2{2}(mu(ind(i,:)));
+
+                p = ind(i,:);
+                obj.D2_lambda{2}(p,p) = D_lambda;
+                obj.D2_mu{2}(p,p) = D_mu;
+            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};
+
+            % 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_lambda = obj.D2_lambda;
+            D2_mu = obj.D2_mu;
+            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_lambda{i}*E{j}' +...
+                                            db(i,j)*D1{i}*LAMBDA*D1{j}*E{j}' ...
+                                          );
+                    D = D + E{i}*inv(RHO)*( d(i,j)*D2_mu{i}*E{j}' +...
+                                            db(i,j)*D1{j}*MU*D1{i}*E{j}' + ...
+                                            D2_mu{j}*E{i}' ...
+                                          );
+                end
+            end
+            obj.D = D;
+            %=========================================%
+
+            % Numerical traction operators for BC.
+            % Because d1 =/= e0^T*D1, the numerical tractions are different
+            % at every boundary.
+            T_l = cell(dim,1);
+            T_r = cell(dim,1);
+            tau_l = cell(dim,1);
+            tau_r = cell(dim,1);
+            % tau^{j}_i = sum_k T^{j}_{ik} u_k
+
+            d1_l = obj.d1_l;
+            d1_r = obj.d1_r;
+            e_l = obj.e_l;
+            e_r = obj.e_r;
+            D1 = obj.D1;
+
+            % Loop over boundaries
+            for j = 1:dim
+                T_l{j} = cell(dim,dim);
+                T_r{j} = cell(dim,dim);
+                tau_l{j} = cell(dim,1);
+                tau_r{j} = cell(dim,1);
+
+                % Loop over components
+                for i = 1:dim
+                    tau_l{j}{i} = sparse(m_tot,dim*m_tot);
+                    tau_r{j}{i} = sparse(m_tot,dim*m_tot);
+                    for k = 1:dim
+                        T_l{j}{i,k} = ... 
+                        -d(i,j)*LAMBDA*(d(i,k)*e_l{k}*d1_l{k}' + db(i,k)*D1{k})...
+                        -d(j,k)*MU*(d(i,j)*e_l{i}*d1_l{i}' + db(i,j)*D1{i})... 
+                        -d(i,k)*MU*e_l{j}*d1_l{j}';
+
+                        T_r{j}{i,k} = ... 
+                        d(i,j)*LAMBDA*(d(i,k)*e_r{k}*d1_r{k}' + db(i,k)*D1{k})...
+                        +d(j,k)*MU*(d(i,j)*e_r{i}*d1_r{i}' + db(i,j)*D1{i})... 
+                        +d(i,k)*MU*e_r{j}*d1_r{j}';
+
+                        tau_l{j}{i} = tau_l{j}{i} + T_l{j}{i,k}*E{k}';
+                        tau_r{j}{i} = tau_r{j}{i} + T_r{j}{i,k}*E{k}';
+                    end
+
+                end
+            end
+            obj.T_l = T_l;
+            obj.T_r = T_r;
+            obj.tau_l = tau_l;
+            obj.tau_r = tau_r;
+
+            % Kroneckered norms and coefficients
+            I_dim = speye(dim);
+            obj.RHOi_kron = kron(obj.RHOi, I_dim);
+            obj.Hi_kron = kron(obj.Hi, I_dim);
+
+            % 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.
+        %       boundary            is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'.
+        %       type                is a cell array of strings specifying the type of boundary condition for each 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.
+        function [closure, penalty] = boundary_condition(obj, boundary, type, parameter)
+            default_arg('type',{'free','free'});
+            default_arg('parameter', []);
+
+            % 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;
+                tau = obj.tau_r{j};
+                T = obj.T_r{j};
+            case -1
+                e = obj.e_l;
+                d = obj.d1_l;
+                tau = obj.tau_l{j};
+                T = obj.T_l{j};
+            end
+
+            E = obj.E;
+            Hi = obj.Hi;
+            H_gamma = obj.H_boundary{j};
+            LAMBDA = obj.LAMBDA;
+            MU = obj.MU;
+            RHOi = obj.RHOi;
+
+            dim = obj.dim;
+            m_tot = obj.grid.N();
+
+            RHOi_kron = obj.RHOi_kron;
+            Hi_kron = obj.Hi_kron;
+
+            % Preallocate
+            closure = sparse(dim*m_tot, dim*m_tot);
+            penalty = cell(dim,1);
+            for k = 1:dim
+                penalty{k} = sparse(dim*m_tot, m_tot/obj.m(j));
+            end
+
+            % Loop over components that we (potentially) have different BC on
+            for k = 1:dim
+                switch type{k}
+
+                % Dirichlet boundary condition
+                case {'D','d','dirichlet','Dirichlet'}
+
+                    tuning = 1.2;
+                    phi = obj.phi{j};
+                    h = obj.h(j);
+                    h11 = obj.H11{j}*h;
+                    gamma = obj.gamma{j};
+
+                    a_lambda = dim/h11 + 1/(h11*phi);
+                    a_mu_i = 2/(gamma*h);
+                    a_mu_ij = 2/h11 + 1/(h11*phi);
+
+                    d = @kroneckerDelta;  % Kronecker delta
+                    db = @(i,j) 1-d(i,j); % Logical not of Kronecker delta
+                    alpha = @(i,j) tuning*( d(i,j)* a_lambda*LAMBDA ...
+                                          + d(i,j)* a_mu_i*MU ...
+                                          + db(i,j)*a_mu_ij*MU ); 
+
+                    % Loop over components that Dirichlet penalties end up on
+                    for i = 1:dim
+                        C = T{k,i};
+                        A = -d(i,k)*alpha(i,j);
+                        B = A + C;
+                        closure = closure + E{i}*RHOi*Hi*B'*e{j}*H_gamma*(e{j}'*E{k}' ); 
+                        penalty{k} = penalty{k} - E{i}*RHOi*Hi*B'*e{j}*H_gamma;
+                    end 
+
+                % Free boundary condition
+                case {'F','f','Free','free','traction','Traction','t','T'}
+                        closure = closure - E{k}*RHOi*Hi*e{j}*H_gamma* (e{j}'*tau{k} ); 
+                        penalty{k} = penalty{k} + E{k}*RHOi*Hi*e{j}*H_gamma;
+
+                % Unknown boundary condition
+                otherwise
+                    error('No such boundary condition: type = %s',type);
+                end
+            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