Triangular Elements. Barycentric Coordinates.

Barycentric Coordinates

For each point

, in this triangle one can define the barycentric coordinades of this point using the value of the 2D shape functions defined for this triangle

satisfying

This way, one can express any point using three coordinates (barycentric)

. In particular

The barycentric coordinates satisfy

. Moreover

unless the point is outside of the triangle.

Compute Barycentric Coordinates

Given a triangle defined by vertices

, compute the barycentric coordinates of point

Verify that

p=[2,2]; %point

vertices=[1,1; %define the triangle vertex coordinates

3,2;

2,4];

vplot=[vertices;vertices(1,:)]; %just to plot a closed triangle, we add the first point at the end

plot(vplot(:,1),vplot(:,2), LineWidth=1.5);

hold on

plot(p(1,1),p(1,2),'o');

hold off

Compute shape functions

A=[ones(3,1), vertices];
b=[1;0;0];
c1=A\b; 
b=[0;1;0];
c2=A\b;
b=[0;0;1];
c3=A\b;
 
Psi1 = @(x,y) c1(1)+c1(2)*x+c1(3)*y;
Psi2 = @(x,y) c2(1)+c2(2)*x+c2(3)*y;
Psi3 = @(x,y) c3(1)+c3(2)*x+c3(3)*y;
%
% Compute barycentric coordinates
%
alpha1 = Psi1(p(1),p(2));
alpha2 = Psi2(p(1),p(2));
alpha3 = Psi3(p(1),p(2));
[alpha1,alpha2,alpha3] %results
ans = 1×3
    0.4000    0.4000    0.2000
alpha1+alpha2+alpha3-1 %check error
ans = 0
%
% Verification
%
p-(alpha1*vertices(1,:)+alpha2*vertices(2,:)+alpha3*vertices(3,:))
ans = 1×2
     0     0

Exercise 1: Build baryCoord function

Build a function file that returns the barycentric coordinates of a point with respect to a triangle. It must return a flag value isInside with value 1 if the point is inside the triangle and 0 if it is outside.

Use the previous example to check if it works.

Exemple: Correspondence with the Reference Triangle

Usually the reference triangular element,

, is the triangle defined by the vertices

We can use barycentric coordinates to make a correspondence between the reference triangle and every other triangle. Let's consider the preivous exemple triangle

Assuming that each vertex has its correspondence

, using the barycentric coordinates we obtain the point correspondences:

verticesTR=[0,0; %define the reference triangle (TR) vertex coordinates

1,0;

0,1];

plot(vplot(:,1),vplot(:,2),LineWidth=1.5);

hold on

axis equal;

plot(p(1,1),p(1,2),'o');

%compute the corresponding point on the reference triangle

pTR= alpha1*verticesTR(1,:)+alpha2*verticesTR(2,:)+alpha3*verticesTR(3,:)

pTR = 1×2

    0.4000    0.2000

vplotTR=[verticesTR;verticesTR(1,:)]; %just to plot a closed triangle, we add the first point at the end

plot(vplotTR(:,1),vplotTR(:,2),LineWidth=1.5);

plot(pTR(1,1),pTR(1,2),'s');

linePlot=[p; pTR]

linePlot = 2×2

    2.0000    2.0000
    0.4000    0.2000

plot(linePlot(:,1),linePlot(:,2),'-.',LineWidth=1);

hold off

Exercise 1.1: Point correspondence

Consider now the obtained point on the Reference Triangle pTR=(0.4,0.2) and compute the associated point on the Original Triangle. Observe that the barycentric coordinates are the same and the point obtained is (2,2)

Exercise 2: Several Triangles

Let's consider now four triangles adding a central vertex (as it is shown in the figure). We can define these polygons by the coordinates of the vertices and the triangles defined by these vertices.

vertex=[

0,0;

1,0;

1,1;

0,1;

0.5,0.5;

];

triang=[

1, 2, 5;

2, 3, 5;

3, 4, 5;

4, 1, 5;

];

plotElements(vertex,triang,1); %find this file at *Numerical Factory*

Exercise 2.1: Several Triangles

Use the function build in the previous exercise

to decide if the point

belongs to each triangle

Solution= It belongs to triangle 2.

ShapeFunctions2D Pract: Interpolate Temperature in a Triangle Mesh

A triangle mesh is defined by a set of nodes (the vertices coordinates) and a set of elements (the triangles) defined by the indices corresponding to the vertices belonging to each triangle (connectivity matrix).

The usual format is

nodes=[ x1,y1; x2,y2; ......; xN,yN]; a Nx2 matrix for N vertices

elements =[ i1,j1,k1; i2,j2,k2; .....; iM,jM,kM]; a Mx3 matrix for M triangles

Compute the temperature at the point

Get the meshFiles file from Numerical Factory

eval('meshHole'); %load data from the file mesHole.m: nodes and elements

numElem = size(elem,1) %number of mesh elements

numNodes = size(nodes,1) %number of mesh nodes

plotElements(nodes,elem,0);

hold on;

p=[0.83,0.7];

plot(p(1),p(2),'*')

hold off;

assing a simple temperature value for each vertex. For now Temp=numNode this way the first nodes are cooler than the last ones.

% assign temperatute equal to the number of the node
uTemp=1:numNodes; %we assume that node 1 has temp=1, node 2 has temp=2, etc.

Final step: Internal point interpolation

Finally find the triangle where the point p belongs to (using the baryCoord function) and interpolate the temperature value.

The idea of the code can be:

for all the elements

compute the barycentric coordinates

if (isInside >= 1), pick the element number; break; end;

end

Take this element and perform the interpolation using the temperatures of the nodes and barycentric coord (see theory presentation).

Solution: Temp= 104.0420; element= 189; nodes=[109, 112, 92];

%
% Look for the element the point p belongs to:
%
for e = 1:numElem     % for each element
    n1 = elem(e,1); % number of the 1st node of the element 
    n2 = elem(e,2); % number of the 2nd node of the element 
    n3 = elem(e,3); % number of the 3th node of the element 
    v1 = nodes(n1,:); % coordinates of the 1st node
    v2 = nodes(n2,:); % coordinates of the 2nd node
    v3 = nodes(n3,:); % coordinates of the 3th node
    vertices = [v1; v2; v3]; 
    % Obs.- In short we can write: 
    %     vertices = nodes(elem(e,:),:) 
    % and it is the same!!
    [alphas, isInside] = baryCoordTriang(vertices, p); 
    if (isInside == 1) % Check if p is inside, compute the results and break the FOR loop
        elemP = e;             %save the present element
        nodElemP = [n1,n2,n3]; %save the number nodes
        alphasP = alphas;      %save the barycentric coordinates
        break   %exit from the for loop
    end 
end
% use barycentric coordinates to interpolate vertices values
uPointInterp = alphasP(1)*uTemp(n1) + alphasP(2)*uTemp(n2) + alphasP(3)*uTemp(n3);
% in short: uPInterpol = dot(alphasP,u(nodElemP))
fprintf('The point belongs to element %d;\n', elemP);
fprintf('Their node numbers are %d, %d, and %d.\n', nodElemP);
fprintf('The interpolated value is uPInterpol = %6.4f\n',uPointInterp);

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