NuMRI/codes/MATLAB/leo/meshStructTess.m

function [nodes, tets, faces, P] = meshStructTess(nodes, dx, dy, dz, check_mesh, plot_mesh)
%% [nodes, tets, faces] = meshStructTess(nodes, dx, dy, dz, check_mesh, plot_mesh)
%  Generate a tessalation from a list of structured nodes.
%  input:   nodes:  n times 3 matrix with on the rows the coordinates of
%                   the n points in the mesh
%           dx, dy, dz: the mesh-size in the directions x, y and z
%           check_mesh: if true, then it solves a Poisson problem
%           plot_mesh: if true, then it plots the mesh
%  output:  nodes:  m times 3 matrix with on the rows the coordinates of
%                   the m <= n points in the triangulationedi
%           tets:   l times 4 matrix with on the rows the tetrahedra 
%           faces:  k times 3 matrix with on the rows the triangles of the
%                   boundary of the mesh
%           P:      Transformation matrix from input nodes to output nodes.
%                   Useful also for transforming node-valued functions on
%                   the input nodes to node-valued functions on the output
%                   nodes
%
%  The triangulation can be plotted using tetramesh(tets,nodes)


% compute the minimum and number of points in each direction
if size(nodes,1) < 4
    error('Triangulation needs at least 4 points')
end
mn = min(nodes);
xmin = mn(1);
ymin = mn(2);
zmin = mn(3);

mn = max(nodes);
xmax = mn(1);
ymax = mn(2);
zmax = mn(3);

nx = round((xmax-xmin)/dx +1);
ny = round((ymax-ymin)/dy +1);
nz = round((zmax-zmin)/dz +1);

Nnodes = size(nodes,1);


% Define tensor which consist of nodes indices, used for the creation of
% the tetrahedra

nodes3d = zeros(nx,ny,nz); % preallocate
for i=1:Nnodes
        nodes3d(round((nodes(i,1)-xmin)/dx)+1,round((nodes(i,2)-ymin)/dy)+1,round((nodes(i,3)-zmin)/dz)+1)=i;
end


disp('Creating Tetrahedra')

% create tetrahedral mesh in cube, which we will reuse.
ii = 1;
X = zeros(8,3);
for i=0:1
    for j=0:1
        for k=0:1
            X(ii,:) = [i,j,k];
            ii = ii+1;
        end
    end
end
cubetet = delaunay(X);

% Run through the mesh
el = 1;
Tetrahedra = zeros(6*(nnz(nodes3d)),4); % preallocate

for i=1:nx-1
    for j=1:ny-1
        for k=1:nz-1
            % take [i:i+1,j:j+1,k:k+1] as cube
            nod = zeros(1,8); % perallocate
            
            for l = 1:8
                % nod is vector with node indices of cube
                nod(l) = nodes3d(i + X(l,1), j + X(l,2), k + X(l,3));
            end
            
            if nnz(nod) == 8 % then the cube is inside the mesh
                tet = nod(cubetet);
            else % then there is at least one point of the cube outside the mesh
                Xs = X(logical(nod),:); % take only nodes inside the mesh
                nodx = nod(logical(nod));
                if nnz(nod) == 4 % 4 nodes, check if points are coplanar
                    C = cross(Xs(2,:)-Xs(1,:), Xs(3,:)-Xs(1,:));
                    cop = logical(dot(C,Xs(4,:)-Xs(1,:)));
                    % if cop = 0, then points are coplanar end thus no
                    % tetrahedra exists.
                end
                if (nnz(nod)>4) || (nnz(nod) == 4 && cop)
                    % create tetrahedra
                    tet1 = delaunay(Xs);
                    tet = nodx(tet1);
                else % no tetrahedra exists
                    tet = [];
                end
            end
            
            % add new tetrahedra to list
            Tetrahedra(el:el+size(tet,1)-1,:) = tet;
            el = el+size(tet,1);
        end
    end
end

tets = Tetrahedra(1:el-1,:); % Delete extra preallocated rows.
clear Tetrahedra

disp([num2str(size(tets,1)), ' tetrahedra created'])

% Delete nodes which are not in any tetrahedra.
disp('Update mesh') 
contr = zeros(size(nodes,1),1);
for i=1:size(tets,1)
    for j=1:4
        contr(tets(i,j))=1;
    end
end

nodes = nodes(logical(contr),:);

% compute P
P = speye(Nnodes);
P = P(logical(contr),:);

disp([num2str(nnz(~contr)), ' unused nodes in triangulation deleted.']) 

disp('Update tetrahedra') 

% make tetrahedra compatible with new node indices
cumcon = cumsum(~contr)';
tets = tets - cumcon(tets);

% create triangles
if size(tets,1) == 0
    warning('No tetrahedra created')
    faces = zeros(0,3);
else
    disp('Create Triangles')
    faces = freeBoundary(triangulation(tets,nodes));
    disp([num2str(size(faces,1)), ' triangles created'])
end

% checking the mesh by solving a Poisson problem
if check_mesh
    % Builds the P1 stiffness matrix from tets and nodes
    [A,volumes]=stifness_matrixP1_3D(tets,nodes);
    % Check if element volumes may be negative
    if any(volumes<=0)
        warning('Some elements have zero or negative volume')
    end
    % solve the Poisson problem with Dirichlet BC
    A(2:end,2:end)\ones(size(A(2:end,2:end),1),1); 
    disp('If there are no warnings, it probably means that the mesh is fine')
end

% Plots mesh
if plot_mesh
    tetramesh(tets,nodes)
    xlabel('x')
    ylabel('y')
    zlabel('z')
end

end