Merge pull request #109 from tristangdwl/demo_Neumann_combined

Add Neumann combined field demo

chunkie/@chunkgraph/chunkgraph.m

@@ -265,7 +265,19 @@
                     else
                         cploc.tb = tb;
                     end
+
                     vs =fchnks{i}([ta,tb]);
+                    if isfield(cploc,'maxchunklen')
+                        if ~isnan(i1) && ~isnan(i2)
+                            if i1 ~= i2 
+                                vfin0 = verts(:,i1);
+                                vfin1 = verts(:,i2);
+                                scale = norm(vfin1-vfin0,'fro')/norm(vs(:,2)-vs(:,1),'fro');
+                                cploc.maxchunklen = cploc.maxchunklen/scale;
+                            end
+                        end
+                    end
+
                     chnkr = chunkerfunc(fchnks{i}, cploc, pref);
                     chnkr = sort(chnkr);
                     if ~isnan(i1) && ~isnan(i2)

chunkie/demo/demo_Neumann_combined.m

@@ -0,0 +1,148 @@
+%DEMO_NEUMANN_COMBINED
+% Solve the exterior Helmholtz scattering problem on a domain with corners
+% and the preconditioned combined field representation
+%
+% Demonstrates kernel interleaving
+% Demonstrates a mixture of straight and curved edges
+
+%% Define geometry
+% planewave direction
+kvec = [0;8];
+zk = norm(kvec);
+
+% make vertices
+narms = 5;
+rots = exp(1i*2*pi*(1:narms)/narms);
+verts = zeros(2,2*narms);
+radi = 1;
+rado = 5;
+for i = 1:narms
+    verts(:,2*i-1) = radi*[real(rots(i));imag(rots(i))];
+    verts(:,2*i  ) = rado*[real(rots(i));imag(rots(i))];
+end
+nverts = size(verts,2);
+edge2verts = [1:nverts;circshift(1:nverts,-1)];
+
+% curve parameters
+amp = -0.5;
+frq = 5;
+
+% define curve for each edge
+fchnks = {};
+for i = 1:narms
+    % odd edges are straight
+    fchnks{2*i-1} = [];
+    % even edges are curved
+    fchnks{2*i  } = @(t) sinearc(t,amp,frq);
+end
+
+cparams = [];
+cparams.maxchunklen = min(4.0/zk,.5);
+[cgrph] = chunkgraph(verts,edge2verts,fchnks,cparams);
+
+
+% plot
+figure(1);clf
+plot(cgrph)
+% hold on
+% quiver(cgrph)
+% hold off
+axis equal
+
+%% Define system
+
+% define kernels
+Sk     = kernel('helmholtz', 's', zk);
+Skp    = kernel('helmholtz', 'sprime', zk);
+Dk     = kernel('helmholtz', 'd', zk);
+
+% combined field uses modified-Helmholtz kernels
+Sik    = kernel('helmholtz', 's', 1i*zk);
+Sikp   = kernel('helmholtz', 'sprime', 1i*zk);
+Dkdiff = kernel('helmdiff', 'dprime', [zk 1i*zk]);
+
+Z = kernel.zeros();
+
+alpha = 1;
+c1 = -1/(0.5 + 1i*alpha*0.25);
+c2 = -1i*alpha/(0.5 + 1i*alpha*0.25);
+c3 = -1;
+
+% define a matrix valued kernel, which corresponds to unpacking the
+% composition of operators
+K = [ c1*Skp  c2*Dkdiff c2*Sikp ;
+   c3*Sik  Z        Z        ;
+   c3*Sikp Z        Z        ];
+K = kernel(K);
+Keval = c1*kernel([Sk 1i*alpha*Dk Z]);
+
+npts = cgrph.npt;
+nsys = K.opdims(1)*npts;
+start = tic;
+A = chunkermat(cgrph, K) + eye(nsys); 
+tmat = toc(start)
+
+%% solve system
+% Set up boundary data
+sources = [4;1];
+strengths = 1;
+
+rhs = zeros(nsys,1);
+% get Neumann boundary data
+rhs_n = -sum(kvec(:).*cgrph.n(:,:),1).*planewave(kvec(:),cgrph.r(:,:));
+
+% only first row of K is physical
+rhs(1:K.opdims(1):end) = rhs_n(:);
+
+% solve
+start = tic;
+sol = gmres(A, rhs, [], 1e-13, 200);
+tsolve = toc(start)
+
+%% compute field
+
+L = 2*rado;
+x1 = linspace(-L,L,300);
+[xx,yy] = meshgrid(x1,x1);
+targs = [xx(:).'; yy(:).'];
+ntargs = size(targs,2);
+
+% identify points in computational domain
+in = chunkerinterior(cgrph,{x1,x1});
+out = ~in;
+
+% get incoming solution
+uin = nan(size(xx));
+uin(out) = planewave(kvec(:),targs(:,out));
+
+% get solution
+opts = [];
+opts.forcesmooth = false;
+uscat = nan(size(xx));
+uscat(out) = chunkerkerneval(cgrph,Keval,sol,targs(:,out),opts);
+
+utot = uin + uscat;
+
+%% make plots
+umax = max(abs(utot(:))); 
+figure(2);clf
+h = pcolor(xx,yy,imag(utot)); set(h,'EdgeColor','none'); colorbar
+colormap(redblue); clim([-umax,umax]);
+hold on 
+plot(cgrph,'k')
+axis equal
+title('$u^{\textrm{tot}}$','Interpreter','latex','FontSize',12)
+
+
+function [r,d,d2] = sinearc(t,amp,frq)
+xs = t;
+ys = amp*sin(frq*t);
+xp = ones(size(t));
+yp = amp*frq*cos(frq*t);
+xpp = zeros(size(t));
+ypp = -frq*frq*amp*sin(t);
+
+r = [(xs(:)).'; (ys(:)).'];
+d = [(xp(:)).'; (yp(:)).'];
+d2 = [(xpp(:)).'; (ypp(:)).'];
+end