Want to use barbellgraph.mat as a mesh

There is a .mat file used as an example for a parition graph. I would like to use it as a 2D mesh to solve PDEs on. I cannot find a way to use it in a model. Could anyone guide me?

[Walter Roberson]

I am far from sure... but I think you might just be able to start by building a triangulation from the Graph, and then passing the triangulation to fegeometry and pass the resulting geometry to femodel

Answer 1

[Malay Agarwal]

Hi Michelle,

I understand that you want to use the data in “barbellgraph.mat” file, used in the example “Partition Graph with Laplacian Matrix”, as a 2D mesh to solve PDEs on.

Please try the following code:

% Load the MAT file
load barbellgraph.mat
% Create a graph using A
G = graph(A, "omitselfloops");
% Extract the endpoints of each edge as two separate variables
x = G.Edges.EndNodes(:, 1);
y = G.Edges.EndNodes(:, 2);
% Create a triangulation
tri = delaunayTriangulation(x, y);
% Create a geometry using the triangulation
geom = fegeometry(tri);
% Create a model using the geometry
% Note: Change AnalysisType according to use case
model = femodel("AnalysisType","structuralStatic","Geometry", geom);

The code above:

• Loads the “.mat” file and creates a graph “G” using the variable “A”.
• Extracts the endpoints for each edge in the graph as two separate variables “x” and “y”. For example, if the graph has edges (a, b) and (c, d), “x” is “[a c]” and “y” is “[b d]”.
• Creates a Delaunay triangulation “tri” using the endpoints of the edges of the graph.
• Uses the triangulation to create a geometry “geom”.
• Creates a model “model” using the geometry, which can now be used to solve PDEs. Please note that you might have to change “AnalysisType” in the call to “femodel” depending on your use case.

The triangulation and the geometry can be plotted as a sanity check using the following code:

triplot(tri);

pdegplot(geom, FaceAlpha=0.3);

The resultant mesh looks as follows:

pdemesh(model);

You can refer to the following resources for more details:

• "graph" documentation: https://www.mathworks.com/help/matlab/ref/graph.html.
• "delaunayTriangulation" documentation: https://www.mathworks.com/help/matlab/ref/delaunay.html.
• "fegeometry" documentation: https://www.mathworks.com/help/pde/ug/fegeometry.html.
• "femodel" documentation: https://www.mathworks.com/help/pde/ug/femodel.html.

Hope this helps!

[Michelle]

Hi Malay,

I already have the .mat file saved. I am looking for a way to use the data in the .mat file as a 2D mesh to solve PDEs on.

[Malay Agarwal]

Hi Michelle,

I have modified the answer based on your requirements

[Michelle]

Thank you for the help. Starting from your code I was able to get the barbell mesh I was looking for. However I now have a new problem. I'll first include my code to get the mesh.

In your code, I first changed

x = G.Edges.EndNodes(:, 1);
y = G.Edges.EndNodes(:, 2);

to

x = xy(:,1);
y = xy(:,2);

and was getting this:

It was close but it had more triangles than should be.

Eventually I ended up with this code:

load barbellgraph.mat

c = []; %empty constraint matrix
coun=1;

for a=1:size(A,1)
    for b=a:size(A,2)
        if A(a,b)==-1
            c(coun,1)=a;
            c(coun,2)=b;
            coun=coun+1;
        end
    end
end
% c contains exactly the edges that should exist

x = xy(:,1);
y = xy(:,2);
tri = delaunayTriangulation(x, y, c); %create triangulation with constraints

conn = tri.ConnectivityList;
E = edges(tri);
nott = ~ismember(E,c,'rows');
Enot = E(nott,:); % edges that should not exist

for i=1:size(Enot,1)
    csize = size(conn,1); % because conn gets smaller as we remove rows
    E1 = mod(find(conn==Enot(i,1)),csize);
    withE2 = (ismember(mod(find(conn==Enot(i,1)),csize),mod(find(conn==Enot(i,2)),csize)));
    removal = E1(withE2); % all triangles containing edges between the two points
    removal = sort(removal,'descend'); % since we're removing by index
    for j=1:size(removal,1)
        conn(removal(j),:)=[];
    end
end

T = conn;
P = tri.Points;
tri = triangulation(T,P);

And the triangulation I wanted:

So then I tried to set up my PDE:

model = createpde;

geometryFromMesh(model,P',T')
specifyCoefficients(model,"m",0,"d",1,"c",1,"a",0,"f",0);
applyBoundaryCondition(model,'dirichlet','Edge',[1,2,3,4],'u',0);
figure(1)
pdeplot(model)

generateMesh(model,"GeometricOrder","linear")
figure(2)
pdeplot(model)

FEM = assembleFEMatrices(model,"nullspace")

But after running generateMesh, the mesh becomes this:

Before there were 2713 nodes, now there are 158. I don't understand why the mesh changes. I need it to be the 2713 node mesh. Setting the GeometricOrder does not make a difference. Any advice would be greatly appreciated.

[Malay Agarwal]

Hi Michelle,

The reason why the number of nodes decreases is due to the call to "generateMesh" with the "GeomtericOrder" name-value argument. The name-value arguments of "generateMesh" are used to modify the mesh generation, as specified in the documentation: https://www.mathworks.com/help/pde/ug/pde.pdemodel.generatemesh.html?s_tid=doc_ta#bupct6k-3:~:text=example-,___,%2CName%2CValue),-modifies%20the%20mesh.

If the model is plotted after these lines:

% Stores the P and T variables
load pt.mat
model = createpde;
geometryFromMesh(model,P',T');
specifyCoefficients(model,"m",0,"d",1,"c",1,"a",0,"f",0);
applyBoundaryCondition(model,'dirichlet','Edge',[1,2,3,4],'u',0);

It has the expected mesh:

pdeplot(model);

I don't think the "generateMesh" call is required here since the model's geometry was created using variables "P" and "T", and the expected mesh is already part of the model. You should be able to solve PDEs after the call to "applyBoundaryCondition".

This can also be confirmed by looking at the "Mesh" property of "model" before and after calling "generateMesh":

• Before the call to "generateMesh":

model.Mesh

ans = 
  FEMesh with properties:

             Nodes: [2×2713 double]
          Elements: [3×5152 double]
    MaxElementSize: 0.0746
    MinElementSize: 0.0074
     MeshGradation: []
    GeometricOrder: 'linear'

• After the call to "generateMesh":

generateMesh(model, "GeometricOrder", "linear");
model.Mesh

ans = 
  FEMesh with properties:

             Nodes: [2×158 double]
          Elements: [3×246 double]
    MaxElementSize: 0.1265
    MinElementSize: 0.0632
     MeshGradation: 1.5000
    GeometricOrder: 'linear'

Hope this helps!

[Michelle]

Hi Malay,

You are correct, removing that line has my code working exactly as expected! Thank you so much for the help.