--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/+scheme/Elastic2dVariableAnisotropicUpwind.m	Sat Nov 16 14:26:06 2019 -0800
@@ -0,0 +1,642 @@
+classdef Elastic2dVariableAnisotropicUpwind < scheme.Scheme
+
+% Discretizes the elastic wave equation:
+% rho u_{i,tt} = dj C_{ijkl} dk u_j
+% 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
+        RHO, RHOi, RHOi_kron % Density
+        C                    % Elastic stiffness tensor
+
+        D  % Total operator
+        Dp, Dm % First derivatives
+
+        % 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, Hi, Hi_kron, H_1D
+
+        % Boundary inner products (for scalar field)
+        H_w, H_e, H_s, H_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
+
+        % E{i}^T picks out component i
+        E
+
+        % Borrowing constants of the form gamma*h, where gamma is a dimensionless constant.
+        h11 % First entry in norm matrix
+
+    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 = Elastic2dVariableAnisotropicUpwind(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 + 1;
+                        end
+                    end
+                end
+            end
+            default_arg('C', C_default);
+            assert(isa(g, 'grid.Cartesian'))
+
+            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();
+            lim = g.lim;
+            if isempty(lim)
+                x = g.x;
+                lim = cell(length(x),1);
+                for i = 1:length(x)
+                    lim{i} = {min(x{i}), max(x{i})};
+                end
+            end
+
+            % 1D operators
+            ops = cell(dim,1);
+            h = zeros(dim,1);
+            for i = 1:dim
+                ops{i} = opSet{i}(m(i), lim{i}, order);
+                h(i) = ops{i}.h;
+            end
+
+            % Borrowing constants
+            for i = 1:dim
+                obj.h11{i} = ops{i}.H(1,1);
+            end
+
+            I = cell(dim,1);
+            Dp = cell(dim,1);
+            Dm = cell(dim,1);
+            H = cell(dim,1);
+            Hi = cell(dim,1);
+            e_0 = cell(dim,1);
+            e_m = cell(dim,1);
+            d1_0 = cell(dim,1);
+            d1_m = cell(dim,1);
+
+            for i = 1:dim
+                I{i} = speye(m(i));
+                Dp{i} = ops{i}.Dp;
+                Dm{i} = ops{i}.Dm;
+                H{i} =  ops{i}.H;
+                Hi{i} = ops{i}.HI;
+                e_0{i} = ops{i}.e_l;
+                e_m{i} = ops{i}.e_r;
+                d1_0{i} = (ops{i}.e_l' * Dm{i})';
+                d1_m{i} = (ops{i}.e_r' * Dm{i})';
+            end
+
+            %====== Assemble full operators ========
+            I_dim = speye(dim, dim);
+            RHO = spdiag(rho);
+            obj.RHO = RHO;
+            obj.RHOi = inv(RHO);
+            obj.RHOi_kron = kron(obj.RHOi, I_dim);
+
+            obj.Dp = cell(dim,1);
+            obj.Dm = cell(dim,1);
+
+            % D1
+            obj.Dp{1} = kron(Dp{1},I{2});
+            obj.Dp{2} = kron(I{1},Dp{2});
+            obj.Dm{1} = kron(Dm{1},I{2});
+            obj.Dm{2} = kron(I{1},Dm{2});
+
+            % Boundary restriction operators
+            e_l = cell(dim,1);
+            e_r = cell(dim,1);
+            e_l{1} = kron(e_0{1}, I{2});
+            e_l{2} = kron(I{1}, e_0{2});
+            e_r{1} = kron(e_m{1}, I{2});
+            e_r{2} = kron(I{1}, e_m{2});
+
+            e_scalar_w = e_l{1};
+            e_scalar_e = e_r{1};
+            e_scalar_s = e_l{2};
+            e_scalar_n = e_r{2};
+
+            e_w = kron(e_scalar_w, I_dim);
+            e_e = kron(e_scalar_e, I_dim);
+            e_s = kron(e_scalar_s, I_dim);
+            e_n = kron(e_scalar_n, I_dim);
+
+            % 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;
+
+            e1_w = (e_scalar_w'*E{1}')';
+            e1_e = (e_scalar_e'*E{1}')';
+            e1_s = (e_scalar_s'*E{1}')';
+            e1_n = (e_scalar_n'*E{1}')';
+
+            e2_w = (e_scalar_w'*E{2}')';
+            e2_e = (e_scalar_e'*E{2}')';
+            e2_s = (e_scalar_s'*E{2}')';
+            e2_n = (e_scalar_n'*E{2}')';
+
+            % Quadratures
+            obj.H = kron(H{1},H{2});
+            obj.Hi = inv(obj.H);
+            obj.H_w = H{2};
+            obj.H_e = H{2};
+            obj.H_s = H{1};
+            obj.H_n = H{1};
+            obj.H_1D = {H{1}, H{2}};
+
+            % Differentiation matrix D (without SAT)
+            Dp = obj.Dp;
+            Dm = obj.Dm;
+            D = sparse(dim*m_tot,dim*m_tot);
+            for i = 1:dim
+                for j = 1:dim
+                    for k = 1:dim
+                        for l = 1:dim
+                            D = D + E{j}*Dp{i}*C_mat{i,j,k,l}*Dm{k}*E{l}';
+                        end
+                    end
+                end
+            end
+            D = obj.RHOi_kron*D;
+            obj.D = D;
+            %=========================================%'
+
+            % Numerical traction operators for BC.
+            %
+            % Formula at boundary j: % tau^{j}_i = sum_l T^{j}_{il} u_l
+            %
+            T_l = cell(dim,1);
+            T_r = cell(dim,1);
+            tau_l = cell(dim,1);
+            tau_r = cell(dim,1);
+
+            D1 = obj.Dm;
+
+            % Boundary j
+            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);
+
+                [~, n_l] = size(e_l{j});
+                [~, n_r] = size(e_r{j});
+
+                % Traction component i
+                for i = 1:dim
+                    tau_l{j}{i} = sparse(dim*m_tot, n_l);
+                    tau_r{j}{i} = sparse(dim*m_tot, n_r);
+
+                    % Displacement component l
+                    for l = 1:dim
+                        T_l{j}{i,l} = sparse(m_tot, n_l);
+                        T_r{j}{i,l} = sparse(m_tot, n_r);
+
+                        % Derivative direction k
+                        for k = 1:dim
+                            T_l{j}{i,l} = T_l{j}{i,l} ...
+                                        - (e_l{j}'*C_mat{j,i,k,l}*D1{k})';
+                            T_r{j}{i,l} = T_r{j}{i,l} ...
+                                        + (e_r{j}'*C_mat{j,i,k,l}*D1{k})';
+                        end
+                        tau_l{j}{i} = tau_l{j}{i} + (T_l{j}{i,l}'*E{l}')';
+                        tau_r{j}{i} = tau_r{j}{i} + (T_r{j}{i,l}'*E{l}')';
+                    end
+                end
+            end
+
+            % Traction tensors, T_ij
+            obj.T_w = T_l{1};
+            obj.T_e = T_r{1};
+            obj.T_s = T_l{2};
+            obj.T_n = T_r{2};
+
+            % Restriction operators
+            obj.e_w = e_w;
+            obj.e_e = e_e;
+            obj.e_s = e_s;
+            obj.e_n = e_n;
+
+            obj.e1_w = e1_w;
+            obj.e1_e = e1_e;
+            obj.e1_s = e1_s;
+            obj.e1_n = e1_n;
+
+            obj.e2_w = e2_w;
+            obj.e2_e = e2_e;
+            obj.e2_s = e2_s;
+            obj.e2_n = e2_n;
+
+            obj.e_scalar_w = e_scalar_w;
+            obj.e_scalar_e = e_scalar_e;
+            obj.e_scalar_s = e_scalar_s;
+            obj.e_scalar_n = e_scalar_n;
+
+            % First component of traction
+            obj.tau1_w = tau_l{1}{1};
+            obj.tau1_e = tau_r{1}{1};
+            obj.tau1_s = tau_l{2}{1};
+            obj.tau1_n = tau_r{2}{1};
+
+            % Second component of traction
+            obj.tau2_w = tau_l{1}{2};
+            obj.tau2_e = tau_r{1}{2};
+            obj.tau2_s = tau_l{2}{2};
+            obj.tau2_n = tau_r{2}{2};
+
+            % Traction vectors
+            obj.tau_w = (e_w'*e1_w*obj.tau1_w')' + (e_w'*e2_w*obj.tau2_w')';
+            obj.tau_e = (e_e'*e1_e*obj.tau1_e')' + (e_e'*e2_e*obj.tau2_e')';
+            obj.tau_s = (e_s'*e1_s*obj.tau1_s')' + (e_s'*e2_s*obj.tau2_s')';
+            obj.tau_n = (e_n'*e1_n*obj.tau1_n')' + (e_n'*e2_n*obj.tau2_n')';
+
+            % Kroneckered norms and coefficients
+            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'.
+        %       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' );
+            comp = bc{1};
+            type = bc{2};
+            if ischar(comp)
+                comp = obj.getComponent(comp, boundary);
+            end
+
+            e       = obj.getBoundaryOperatorForScalarField('e', boundary);
+            tau     = obj.getBoundaryOperator(['tau' num2str(comp)], boundary);
+            T       = obj.getBoundaryTractionOperator(boundary);
+            h11     = obj.getBorrowing(boundary);
+            H_gamma = obj.getBoundaryQuadratureForScalarField(boundary);
+            nu      = obj.getNormal(boundary);
+
+            E = obj.E;
+            Hi = obj.Hi;
+            RHOi = obj.RHOi;
+            C = obj.C;
+
+            dim = obj.dim;
+            m_tot = obj.grid.N();
+
+            % Preallocate
+            [~, col] = size(tau);
+            closure = sparse(dim*m_tot, dim*m_tot);
+            penalty = sparse(dim*m_tot, col);
+
+            j = comp;
+            switch type
+
+            % Dirichlet boundary condition
+            case {'D','d','dirichlet','Dirichlet','displacement','Displacement'}
+
+                if numel(bc) >= 3
+                    dComps = bc{3};
+                else
+                    dComps = 1:dim;
+                end
+
+                % Loops over components that Dirichlet penalties end up on
+                % Y: symmetrizing part of penalty
+                % Z: symmetric part of penalty
+                % X = Y + Z.
+
+                % Nonsymmetric part goes on all components to
+                % yield traction in discrete energy rate
+                for i = 1:dim
+                    Y = T{j,i}';
+                    X = e*Y;
+                    closure = closure + E{i}*RHOi*Hi*X'*e*H_gamma*(e'*E{j}' );
+                    penalty = penalty - E{i}*RHOi*Hi*X'*e*H_gamma;
+                end
+
+                % Symmetric part only required on components with displacement BC.
+                % (Otherwise it's not symmetric.)
+                for i = dComps
+                    Z = sparse(m_tot, m_tot);
+                    for l = 1:dim
+                        for k = 1:dim
+                            Z = Z + nu(l)*C{l,i,k,j}*nu(k);
+                        end
+                    end
+                    Z = -tuning*dim/h11*Z;
+                    X = Z;
+                    closure = closure + E{i}*RHOi*Hi*X'*e*H_gamma*(e'*E{j}' );
+                    penalty = penalty - E{i}*RHOi*Hi*X'*e*H_gamma;
+                end
+
+            % Free boundary condition
+            case {'F','f','Free','free','traction','Traction','t','T'}
+                    closure = closure - E{j}*RHOi*Hi*e*H_gamma*tau';
+                    penalty = penalty + E{j}*RHOi*Hi*e*H_gamma;
+
+            % Unknown boundary condition
+            otherwise
+                error('No such boundary condition: type = %s',type);
+            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);
+
+            switch type.interpolation
+            case {'none', ''}
+                [closure, penalty] = interfaceStandard(obj,boundary,neighbour_scheme,neighbour_boundary,type);
+            case {'op','OP'}
+                [closure, penalty] = interfaceNonConforming(obj,boundary,neighbour_scheme,neighbour_boundary,type);
+            otherwise
+                error('Unknown type of interpolation: %s ', type.interpolation);
+            end
+        end
+
+        function [closure, penalty] = interfaceStandard(obj,boundary,neighbour_scheme,neighbour_boundary,type)
+            tuning = type.tuning;
+
+            % u denotes the solution in the own domain
+            % v denotes the solution in the neighbour domain
+
+            u = obj;
+            v = neighbour_scheme;
+
+            % Operators, u side
+            e_u       = u.getBoundaryOperatorForScalarField('e', boundary);
+            tau_u     = u.getBoundaryOperator('tau', boundary);
+            h11_u     = u.getBorrowing(boundary);
+            nu_u      = u.getNormal(boundary);
+
+            E_u = u.E;
+            C_u = u.C;
+            m_tot_u = u.grid.N();
+
+            % Operators, v side
+            e_v       = v.getBoundaryOperatorForScalarField('e', neighbour_boundary);
+            tau_v     = v.getBoundaryOperator('tau', neighbour_boundary);
+            h11_v     = v.getBorrowing(neighbour_boundary);
+            nu_v      = v.getNormal(neighbour_boundary);
+
+            E_v = v.E;
+            C_v = v.C;
+            m_tot_v = v.grid.N();
+
+            % Operators that are only required for own domain
+            Hi      = u.Hi_kron;
+            RHOi    = u.RHOi_kron;
+            e_kron  = u.getBoundaryOperator('e', boundary);
+            T_u     = u.getBoundaryTractionOperator(boundary);
+
+            % Shared operators
+            H_gamma         = u.getBoundaryQuadratureForScalarField(boundary);
+            H_gamma_kron    = u.getBoundaryQuadrature(boundary);
+            dim             = u.dim;
+
+            % Preallocate
+            [~, m_int] = size(H_gamma);
+            closure = sparse(dim*m_tot_u, dim*m_tot_u);
+            penalty = sparse(dim*m_tot_u, dim*m_tot_v);
+
+            % ---- Continuity of displacement ------
+
+            % Y: symmetrizing part of penalty
+            % Z: symmetric part of penalty
+            % X = Y + Z.
+
+            % Loop over components to couple across interface
+            for j = 1:dim
+
+                % Loop over components that penalties end up on
+                for i = 1:dim
+                    Y = 1/2*T_u{j,i}';
+                    Z_u = sparse(m_int, m_int);
+                    Z_v = sparse(m_int, m_int);
+                    for l = 1:dim
+                        for k = 1:dim
+                            Z_u = Z_u + e_u'*nu_u(l)*C_u{l,i,k,j}*nu_u(k)*e_u;
+                            Z_v = Z_v + e_v'*nu_v(l)*C_v{l,i,k,j}*nu_v(k)*e_v;
+                        end
+                    end
+                    Z = -tuning*dim*( 1/(4*h11_u)*Z_u + 1/(4*h11_v)*Z_v );
+                    X = Y + Z*e_u';
+                    closure = closure + E_u{i}*X'*H_gamma*e_u'*E_u{j}';
+                    penalty = penalty - E_u{i}*X'*H_gamma*e_v'*E_v{j}';
+                end
+            end
+
+            % ---- Continuity of traction ------
+            closure = closure - 1/2*e_kron*H_gamma_kron*tau_u';
+            penalty = penalty - 1/2*e_kron*H_gamma_kron*tau_v';
+
+            % ---- Multiply by inverse of density x quadraure ----
+            closure = RHOi*Hi*closure;
+            penalty = RHOi*Hi*penalty;
+
+        end
+
+        function [closure, penalty] = interfaceNonConforming(obj,boundary,neighbour_scheme,neighbour_boundary,type)
+            error('Non-conforming interfaces not implemented yet.');
+        end
+
+        % Returns the component number that is the tangential/normal component
+        % at the specified boundary
+        function comp = getComponent(obj, comp_str, boundary)
+            assertIsMember(comp_str, {'normal', 'tangential'});
+            assertIsMember(boundary, {'w', 'e', 's', 'n'});
+
+            switch boundary
+            case {'w', 'e'}
+                switch comp_str
+                case 'normal'
+                    comp = 1;
+                case 'tangential'
+                    comp = 2;
+                end
+            case {'s', 'n'}
+                switch comp_str
+                case 'normal'
+                    comp = 2;
+                case 'tangential'
+                    comp = 1;
+                end
+            end
+        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.h11{1};
+            case {'s', 'n'}
+                h11 = obj.h11{2};
+            end
+        end
+
+        % Returns the outward unit normal vector for the boundary specified by the string boundary.
+        function nu = getNormal(obj, boundary)
+            assertIsMember(boundary, {'w', 'e', 's', 'n'})
+
+            switch boundary
+            case 'w'
+                nu = [-1,0];
+            case 'e'
+                nu = [1,0];
+            case 's'
+                nu = [0,-1];
+            case 'n'
+                nu = [0,1];
+            end
+        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'})
+
+            switch op
+                case {'e', 'e1', 'e2', 'tau', 'tau1', 'tau2'}
+                    o = obj.([op, '_', boundary]);
+            end
+
+        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