tetVolMeanEst

Below is a demonstration of the features of the tetVolMeanEst function

Contents

```clear; close all; clc;
```

Plot settings

```fontSize=15;
faceAlpha1=0.5;
faceAlpha2=1;
edgeColor=0.25*ones(1,3);
edgeWidth=1.5;
patchColor=[1 0.5 0];
```

Estimaging tetrahedral volume based on mean edge length and regular face assumption

```% Get a regular tentrahedron
[V,F]=platonic_solid(1,1);

% Calculate true volume
VE=tetVol([1 2 3 4],V)

%Estimated volume for regular tets based on mean edge lengths
[VE_est]=tetVolMeanEst(F,V)
```
```VE =

0.5132

VE_est =

0.5132

```

Plotting model

```hf=cFigure;
title('A regular tetrahedron','FontSize',fontSize);
xlabel('X','FontSize',fontSize); ylabel('Y','FontSize',fontSize); zlabel('Z','FontSize',fontSize);
hold on;

hp=patch('Faces',F,'Vertices',V);
set(hp,'FaceColor',patchColor,'FaceAlpha',faceAlpha1,'lineWidth',edgeWidth,'edgeColor',edgeColor);
set(gca,'FontSize',fontSize);
view(3); axis tight;  axis equal;  grid on;
```

The two metrics coincide in this case but for irregular meshes they may divergerge.

Using tetVolMeanEst to set desired mesh volume for tetgen meshing

For tetrahedral meshing schemes surface geometry is usually provided. For instance triangulated surface data. If the desired element volume can be specified then in this case tetVolMeanEst can be used to estimate the desired element volume given the input surface mesh (provided the surface mesh is not remeshed). This is highlighted in the following example.

Building a geodesic dome surface model

```[F,V,~]=geoSphere(2,1);
```

Plotting model

```hf=cFigure;
title('A triangulated surface model','FontSize',fontSize);
xlabel('X','FontSize',fontSize); ylabel('Y','FontSize',fontSize); zlabel('Z','FontSize',fontSize);
hold on;

hp=patch('Faces',F,'Vertices',V);
set(hp,'FaceColor',patchColor,'FaceAlpha',faceAlpha1,'lineWidth',edgeWidth,'edgeColor',edgeColor);
set(gca,'FontSize',fontSize);
view(3); axis tight;  axis equal;  grid on;
```

The triangles are quite regular and can be used to estimate desired tetrahedral element volume

```[regionA]=tetVolMeanEst(F,V); %Volume for regular tets
```

Defining input structure

```inputStruct.stringOpt='-pq1.2AaYQ';
inputStruct.Faces=F;
inputStruct.Nodes=V;
inputStruct.holePoints=[];
inputStruct.faceBoundaryMarker=ones(size(F,1),1); %Face boundary markers
inputStruct.regionPoints=[0 0 0]; %region points
inputStruct.regionA=regionA;
inputStruct.minRegionMarker=2; %Minimum region marker
```

Mesh model using tetrahedral elements using tetGen

```[meshOutput]=runTetGen(inputStruct); %Run tetGen
```
```
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
--- TETGEN Tetrahedral meshing --- 20-Jun-2019 15:11:06

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
--- Writing SMESH file --- 20-Jun-2019 15:11:06
--- Done --- 20-Jun-2019 15:11:06
--- Running TetGen to mesh input boundary--- 20-Jun-2019 15:11:06
Opening /mnt/data/MATLAB/GIBBON/data/temp/temp.smesh.
--- Done --- 20-Jun-2019 15:11:06

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
--- Importing TetGen files --- 20-Jun-2019 15:11:06
--- Done --- 20-Jun-2019 15:11:06
```

Access model element and patch data

```F=meshOutput.faces;
V=meshOutput.nodes;
C=meshOutput.faceMaterialID;
E=meshOutput.elements;
```

PLOTTING MODEL

```%Selecting half of the model to see interior
Y=V(:,2); YE=mean(Y(E),2);
L=YE>mean(Y);
[Fs,Cs]=element2patch(E(L,:),C(L));

hf1=cFigure;
subplot(1,2,1);
title('Solid tetrahedral mesh model','FontSize',fontSize);
xlabel('X','FontSize',fontSize); ylabel('Y','FontSize',fontSize); zlabel('Z','FontSize',fontSize); hold on;
hps=patch('Faces',F,'Vertices',V,'FaceColor','flat','CData',C,'lineWidth',edgeWidth,'edgeColor',edgeColor);
view(3); axis tight;  axis equal;  grid on;
colormap(autumn);
set(gca,'FontSize',fontSize);
subplot(1,2,2);
title('Cut view of Solid tetrahedral mesh model','FontSize',fontSize);
xlabel('X','FontSize',fontSize); ylabel('Y','FontSize',fontSize); zlabel('Z','FontSize',fontSize); hold on;
hps=patch('Faces',Fs,'Vertices',V,'FaceColor','flat','CData',Cs,'lineWidth',edgeWidth,'edgeColor',edgeColor);
view(3); axis tight;  axis equal;  grid on;
colormap(autumn);
set(gca,'FontSize',fontSize);
drawnow;
```

GIBBON www.gibboncode.org

Kevin Mattheus Moerman, gibbon.toolbox@gmail.com

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GIBBON: The Geometry and Image-based Bioengineering add-On. A toolbox for image segmentation, image-based modeling, meshing, and finite element analysis.

Copyright (C) 2019 Kevin Mattheus Moerman

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