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);
camlight headlight;
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);
camlight headlight;
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
----> Adding node field
----> Adding facet field
----> Adding holes specification
----> Adding region specification
--- 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);
camlight headlight;
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);
camlight headlight;
set(gca,'FontSize',fontSize);
drawnow;

GIBBON www.gibboncode.org

Kevin Mattheus Moerman, [email protected]

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License: https://github.com/gibbonCode/GIBBON/blob/master/LICENSE

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

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

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