2D value interpolation of a polygon

Hi,

We are trying to interpolate an area in the shape of a polygon, based on the values of 3 different edges (E4,E6,E2). See figure_geometry in which the middle area is the one to be interpolated and the other 3 rectangles contain values varying as a function of the y coordinate.

We made a mesh of the problem, see figure_mesh.

We assigned values to the nodes of the polygon at the edges that are known (interfaces with the rectangles). We're however struggling now to execute an interpolation on the polygon.

Here is the code so far:

clear all
close all
model1 = createpde;

rect1 = [3 
    4
    0
    2
    2
    0
    0
    0
    6
    6];

rect2 = [3 
    4
    14
    16
    16
    14
    0
    0
    6
    6];

rect3 = [3 
    4
    5
    11
    11
    5
    -6
    -6
    -4
    -4];


gd = [rect1,rect2,rect3];
ns = char('rect1','rect2','rect3');
ns = ns';

sf = '(rect1+rect2+rect3)';

[dl,bt] = decsg(gd,sf,ns);
geometryFromEdges(model1,dl);
specifyCoefficients(model1,'m',0,...
                          'd',0,...
                          'c',1,...
                          'a',0,...
                          'f',0);

applyBoundaryCondition(model1,'dirichlet','Edge',[10,11,4],'u',60);
applyBoundaryCondition(model1,'dirichlet','Edge',[7,8,2],'u',-60);

%% Model 2 = polygon

model2 = createpde;

poly1 = [2
    6
    2
    5
    11
    14
    14
    2
    0
    -4
    -4
    0
    6
    6
    0];

gd2 = [poly1];
ns2 = char('poly1');
ns2 = ns2';

sf2 = '(poly1)';

f_vector = zeros(17,1);
sixty_vector = repmat(60,17,1);

[dl2,bt2] = decsg(gd2,sf2,ns2);
geometryFromEdges(model2,dl2);
specifyCoefficients(model2,'m',0,...
                          'd',0,...
                          'c',1,...
                          'a',0,...
                          'f',0);
                      


                      
%% plotting figures

figure(1)        % overall geometry
pdegplot(model1,'FaceLabels','on')
hold on
pdegplot(model2,'EdgeLabels','on','FaceLabels','on')
xlim([0 16])
ylim([-6 6])

figure(2)       % mesh for all
mesh = generateMesh(model1,'Hmax',1);
pdemesh(mesh)
hold on
mesh2 = generateMesh(model2,'Hmax',1);
pdemesh(mesh2)
axis equal

results_pde = solvepde(model1);
results = abs(results_pde.NodalSolution);



%% Figure outcome
figure(3)      % solve linear for 3 rectangles
pdeplot(model1,'XYData', results)
hold on 
results_pde2 = solvepde(model2);
results2 = results_pde2.NodalSolution;
pdeplot(model2,'XYData', results2);

axis equal

%% New attempt

Nf_f4 = findNodes(mesh2,'region','Face',1); % nodes of Face 4
values_f4 = results2(Nf_f4);
values_f4 = NaN(1,length(values_f4));

Nf_edge6_f4 = flip(findNodes(mesh2,'region','Edge',6));
Nf_edge4_f4 = flip(findNodes(mesh2,'region','Edge',4));
Nf_edge2_f4 = flip(findNodes(mesh2,'region','Edge',2));

Nf_edge6 = findNodes(mesh,'region','Edge',6); % nodes of E6
values_E6 = results(Nf_edge6);

Nf_edge4 = findNodes(mesh,'region','Edge',1); % nodes of E4
values_E4 = results(Nf_edge4);

Nf_edge3 = findNodes(mesh,'region','Edge',3); % nodes of E6
values_E3 = results(Nf_edge3);

for i = 1:length(values_E6)
    values_f4(Nf_edge6_f4(i)) = values_E6(i);
    values_f4(Nf_edge4_f4(i)) = values_E4(i);
    values_f4(Nf_edge2_f4(i)) = values_E3(i);
end 

Interpolation_coordinates = mesh2.Nodes(:,Nf_f4);

[X,Y] = meshgrid(Interpolation_coordinates(1,:),Interpolation_coordinates(2,:));
values_f4 = repelem(values_f4,[length(values_f4)],[1]);

% How to fill matrix 'values_f4' with the interpolated values?