--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/+scheme/Beam.m	Fri Sep 07 14:40:58 2018 +0200
@@ -0,0 +1,263 @@
+classdef Beam < scheme.Scheme
+    properties
+        order % Order accuracy for the approximation
+        grid
+
+        D % non-stabalized scheme operator
+        alpha
+
+        h
+        H % Discrete norm
+        Hi
+
+        e_l,  e_r
+        d1_l, d1_r
+        d2_l, d2_r
+        d3_l, d3_r
+        gamm
+        delt
+        alphaII
+        alphaIII
+
+        opt
+    end
+
+    methods
+        function obj = Beam(grid, order, alpha, opsGen, opt)
+            default_arg('alpha', -1);
+
+            % default_arg('opsGen', @sbp.D4);
+            default_arg('opsGen', @sbp.D4Variable); % Supposed to be better
+
+            opt_default.interface_l.tuning = 1.1;
+            opt_default.interface_l.tau = [];
+            opt_default.interface_l.sig = [];
+            opt_default.interface_r.tuning = 1.1;
+            opt_default.interface_r.tau = [];
+            opt_default.interface_r.sig = [];
+            default_struct('opt', opt_default);
+
+            if ~isa(grid, 'grid.Cartesian') || grid.D() ~= 1
+                error('Grid must be 1d cartesian');
+            end
+
+            obj.grid = grid;
+            obj.order = order;
+            obj.alpha = alpha;
+
+            m = grid.m;
+            h = grid.scaling();
+
+            x_lim = {grid.x{1}(1), grid.x{1}(end)};
+            ops = opsGen(m, x_lim, order);
+
+            D4       = ops.D4;
+            obj.H    = ops.H;
+            obj.Hi   = ops.HI;
+            obj.e_l  = ops.e_l;
+            obj.e_r  = ops.e_r;
+            obj.d1_l = ops.d1_l;
+            obj.d1_r = ops.d1_r;
+            obj.d2_l = ops.d2_l;
+            obj.d2_r = ops.d2_r;
+            obj.d3_l = ops.d3_l;
+            obj.d3_r = ops.d3_r;
+
+            obj.D = alpha*D4;
+
+            alphaII  = ops.borrowing.N.S2/2;
+            alphaIII = ops.borrowing.N.S3/2;
+
+            obj.gamm = h*alphaII;
+            obj.delt = h^3*alphaIII;
+            obj.alphaII = alphaII;
+            obj.alphaIII = alphaIII;
+            obj.h = h;
+            obj.opt = opt;
+        end
+
+
+        % Closure functions return the opertors applied to the own doamin to close the boundary
+        % Penalty functions return the opertors 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 string specifying the type of boundary condition if there are several.
+        %       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)
+            default_arg('type','dn');
+
+            [e, d1, d2, d3, s] = obj.get_boundary_ops(boundary);
+            gamm = obj.gamm;
+            delt = obj.delt;
+
+
+            % TODO: Can this be simplifed? Can I handle conditions on u on its own, u_x on its own ...
+
+            switch type
+                case {'dn', 'clamped'} % Dirichlet-neumann boundary condition
+                    alpha = obj.alpha;
+
+                    % tau1 < -alpha^2/gamma
+                    % tuning = 2;
+                    tuning = 1.1;
+
+                    tau1 = tuning * alpha/delt;
+                    tau4 = s*alpha;
+
+                    sig2 = tuning * alpha/gamm;
+                    sig3 = -s*alpha;
+
+                    tau = tau1*e+tau4*d3;
+                    sig = sig2*d1+sig3*d2;
+
+                    closure = obj.Hi*(tau*e' + sig*d1');
+
+                    penalty{1} = -obj.Hi*tau;
+                    penalty{2} = -obj.Hi*sig;
+
+
+                case {'free'}
+                    a = obj.alpha;
+
+                    tau =  s*a*d1;
+                    sig = -s*a*e;
+
+                    closure = obj.Hi*(tau*d2' + sig*d3');
+                    penalty{1} = -obj.Hi*tau;
+                    penalty{1} = -obj.Hi*sig;
+
+                case 'e'
+                    alpha = obj.alpha;
+                    tuning = 1.1;
+
+                    tau1 = tuning * alpha/delt;
+                    tau4 = s*alpha;
+
+                    tau = tau1*e+tau4*d3;
+
+                    closure = obj.Hi*tau*e';
+                    penalty = -obj.Hi*tau;
+                case 'd1'
+                    alpha = obj.alpha;
+
+                    tuning = 1.1;
+
+                    sig2 = tuning * alpha/gamm;
+                    sig3 = -s*alpha;
+
+                    sig = sig2*d1+sig3*d2;
+
+                    closure = obj.Hi*sig*d1';
+                    penalty = -obj.Hi*sig;
+
+                case 'd2'
+                    a = obj.alpha;
+
+                    tau =  s*a*d1;
+
+                    closure = obj.Hi*tau*d2';
+                    penalty = -obj.Hi*tau;
+                case 'd3'
+                    a = obj.alpha;
+
+                    sig = -s*a*e;
+
+                    closure = obj.Hi*sig*d3';
+                    penalty = -obj.Hi*sig;
+
+                otherwise % Unknown, boundary condition
+                    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
+            [e_u,d1_u,d2_u,d3_u,s_u] = obj.get_boundary_ops(boundary);
+            [e_v,d1_v,d2_v,d3_v,s_v] = neighbour_scheme.get_boundary_ops(neighbour_boundary);
+
+
+            alpha_u = obj.alpha;
+            alpha_v = neighbour_scheme.alpha;
+
+
+            switch boundary
+                case 'l'
+                    interface_opt = obj.opt.interface_l;
+                case 'r'
+                    interface_opt = obj.opt.interface_r;
+            end
+
+
+            if isempty(interface_opt.tau) && isempty(interface_opt.sig)
+                gamm_u = obj.gamm;
+                delt_u = obj.delt;
+
+                gamm_v = neighbour_scheme.gamm;
+                delt_v = neighbour_scheme.delt;
+
+                tuning = interface_opt.tuning;
+
+                tau1 = ((alpha_u/2)/delt_u + (alpha_v/2)/delt_v)/2*tuning;
+                sig2 = ((alpha_u/2)/gamm_u + (alpha_v/2)/gamm_v)/2*tuning;
+            else
+                h_u = obj.h;
+                h_v = neighbour_scheme.h;
+
+                switch neighbour_boundary
+                    case 'l'
+                        neighbour_interface_opt = neighbour_scheme.opt.interface_l;
+                    case 'r'
+                        neighbour_interface_opt = neighbour_scheme.opt.interface_r;
+                end
+
+                tau_u = interface_opt.tau;
+                sig_u = interface_opt.sig;
+                tau_v = neighbour_interface_opt.tau;
+                sig_v = neighbour_interface_opt.sig;
+
+                tau1 = tau_u/h_u^3 + tau_v/h_v^3;
+                sig2 = sig_u/h_u   + sig_v/h_v;
+            end
+
+            tau4 = s_u*alpha_u/2;
+            sig3 = -s_u*alpha_u/2;
+            phi2 = s_u*1/2;
+            psi1 = -s_u*1/2;
+
+            tau = tau1*e_u  +                     tau4*d3_u;
+            sig =           sig2*d1_u + sig3*d2_u          ;
+            phi =           phi2*d1_u                      ;
+            psi = psi1*e_u                                 ;
+
+            closure =  obj.Hi*(tau*e_u' + sig*d1_u' + phi*alpha_u*d2_u' + psi*alpha_u*d3_u');
+            penalty = -obj.Hi*(tau*e_v' + sig*d1_v' + phi*alpha_v*d2_v' + psi*alpha_v*d3_v');
+        end
+
+        % Returns the boundary ops and sign for the boundary specified by the string boundary.
+        % The right boundary is considered the positive boundary
+        function [e, d1, d2, d3, s] = get_boundary_ops(obj,boundary)
+            switch boundary
+                case 'l'
+                    e  = obj.e_l;
+                    d1 = obj.d1_l;
+                    d2 = obj.d2_l;
+                    d3 = obj.d3_l;
+                    s = -1;
+                case 'r'
+                    e  = obj.e_r;
+                    d1 = obj.d1_r;
+                    d2 = obj.d2_r;
+                    d3 = obj.d3_r;
+                    s = 1;
+                otherwise
+                    error('No such boundary: boundary = %s',boundary);
+            end
+        end
+
+        function N = size(obj)
+            N = obj.grid.N;
+        end
+
+    end
+end