DEMO_febio_0029_contact_friction_benchmark_boxes

Below is a demonstration for a FEBio demo presented at the FEBio workshop at WCB 2018. Four cubes are stacked on top of each other. Sliding-elastic contact with friction is defined between them. The top cube is forced downwards in a rotation motion thereby compressing the other cubes and also transferring some twist due to frictional forces.

The demo includes * Building geometry for 4 cubes with hexahedral elements * Defining the boundary conditions * Coding the febio structure * Running the model * Importing and visualizing the displacement results

Contents

Keywords

clear; close all; clc;

Plot settings

fontSize=15;
faceAlpha1=0.8;
faceAlpha2=0.3;
markerSize=40;
markerSize2=20;
lineWidth=3;

Control parameters

% Path names
defaultFolder = fileparts(fileparts(mfilename('fullpath')));
savePath=fullfile(defaultFolder,'data','temp');

% Defining file names
febioFebFileNamePart='tempModel';
febioFebFileName=fullfile(savePath,[febioFebFileNamePart,'.feb']); %FEB file name
febioLogFileName=fullfile(savePath,[febioFebFileNamePart,'.txt']); %FEBio log file name
febioLogFileName_disp=[febioFebFileNamePart,'_disp_out.txt']; %Log file name for exporting displacement
febioLogFileName_strainEnergy=[febioFebFileNamePart,'_fsed_out.txt']; %Log file name for exporting strain energy density

%Specifying dimensions and number of elements for slab
sampleHeight=2; %Height
sampleWidth=sampleHeight; %Width
sampleThickness=sampleHeight; %Thickness

numElementsWidth=[4 3 5 4]; %Number of elemens in dir 1
numElementsThickness=numElementsWidth; %Number of elemens in dir 2
numElementsHeight=numElementsWidth; %Number of elemens in dir 3

%Material parameter set
youngsModuli   = [0.3 10  0.3 10 ];
poissonsRatios = [0.4 0.1 0.4 0.1];

% FEA control settings
numTimeSteps=10; %Number of time steps desired
max_refs=25; %Max reforms
max_ups=0; %Set to zero to use full-Newton iterations
opt_iter=25; %Optimum number of iterations
max_retries=10; %Maximum number of retires
dtmin=(1/numTimeSteps)/10; %Minimum time step size
dtmax=1/numTimeSteps; %Maximum time step size
symmetric_stiffness=0;

runMode='external';% 'internal' or 'external'

%Contact parameters
contactInitialOffset=0;
boxOffsets=sampleHeight+contactInitialOffset;

%Prescribed displacement
prescribedDisplacement_Z=-2;
prescribedRotation_Z=pi/2;

Creating model geometry and mesh

A box is created with tri-linear hexahedral (hex8) elements using the hexMeshBox function. The function offers the boundary faces with seperate labels for the top, bottom, left, right, front, and back sides. As such these can be used to define boundary conditions on the exterior.

E=[];
elementMaterialID=[];
V=[];
Fb=[];
Cb=[];
for q=1:1:4

    % Create a box with hexahedral elements
    beamDimensions=[sampleWidth sampleThickness sampleHeight]; %Dimensions
    beamElementNumbers=[numElementsWidth(q) numElementsThickness(q) numElementsHeight(q)]; %Number of elements
    outputStructType=2; %A structure compatible with mesh view
    [meshStruct]=hexMeshBox(beamDimensions,beamElementNumbers,outputStructType);

    %Access elements, nodes, and faces from the structure
    E1=meshStruct.elements; %The elements
    V1=meshStruct.nodes; %The nodes (vertices)
    Fb1=meshStruct.facesBoundary; %The boundary faces
    Cb1=meshStruct.boundaryMarker; %The "colors" or labels for the boundary faces
    elementMaterialIndices=ones(size(E1,1),1); %Element material indices

    V1(:,3)=V1(:,3)+(q-1)*boxOffsets;

    E=[E;E1+size(V,1)];
    Fb=[Fb;Fb1+size(V,1)];
    V=[V;V1];
    colorOffset=max(Cb(:));
    if isempty(colorOffset)
        colorOffset=0;
    end
    Cb=[Cb;Cb1+colorOffset];

    elementMaterialID=[elementMaterialID;q*ones(size(E1,1),1)];

end
V(:,3)=V(:,3)-min(V(:,3)); %Shift so bottom is at 0

Plotting model boundary surfaces and a cut view

hFig=cFigure;
hold on;
title('Model boundary surfaces and labels','FontSize',fontSize);
gpatch(Fb,V,Cb,'k',faceAlpha1);
colormap(gjet(250)); icolorbar;
axisGeom(gca,fontSize);
drawnow;

logicTops=false(size(Cb,1),4);
logicBottoms=false(size(Cb,1),4);
for q=1:1:4
    logicTops(:,q)=Cb==6+(6*(q-1));
    logicBottoms(:,q)=Cb==5+(6*(q-1));
end

Plotting model boundary surfaces

hFig=cFigure;
hold on;
title('Contact faces','FontSize',fontSize);
gpatch(Fb,V,'kw','none',0.2);
for q=1:1:size(logicTops,2)-1
    gpatch(Fb(logicTops(:,q),:),V,q*ones(nnz(logicTops(:,q)),1),'g',1);
    gpatch(Fb(logicBottoms(:,q+1),:),V,q*ones(nnz(logicBottoms(:,q+1)),1),'y',1);
end

colormap(gjet(250)); icolorbar;
axisGeom(gca,fontSize);
drawnow;

Define boundary conditions

F_support=Fb(logicBottoms(:,1),:);

F_rigidBody=Fb(logicTops(:,end),:);
indNodesRigidBody=unique(F_rigidBody(:));
center_of_mass=mean(V(indNodesRigidBody,:),1);

%Supported nodes
bcSupportList=unique(F_support(:));

%Visualize BC's
hf=cFigure;
title('Boundary conditions model','FontSize',fontSize);
xlabel('X','FontSize',fontSize); ylabel('Y','FontSize',fontSize); zlabel('Z','FontSize',fontSize);
hold on;

gpatch(Fb,V,'kw','none',faceAlpha2);
hl(1)=gpatch(F_rigidBody,V,'rw','k',1);
hl(2)=plotV(V(bcSupportList,:),'k.','MarkerSize',markerSize);
hl(3)=plotV(center_of_mass,'r.','MarkerSize',50);

legend(hl,{'Rigid body','BC support','Rigid body center of mass'});

axisGeom(gca,fontSize);
camlight headlight;
drawnow;

Defining the FEBio input structure

See also febioStructTemplate and febioStruct2xml and the FEBio user manual.

%Get a template with default settings
[febio_spec]=febioStructTemplate;

%febio_spec version
febio_spec.ATTR.version='4.0';

%Module section
febio_spec.Module.ATTR.type='solid';

%Control section
febio_spec.Control.analysis='STATIC';
febio_spec.Control.time_steps=numTimeSteps;
febio_spec.Control.step_size=1/numTimeSteps;
febio_spec.Control.solver.max_refs=max_refs;
febio_spec.Control.solver.qn_method.max_ups=max_ups;
febio_spec.Control.solver.symmetric_stiffness=symmetric_stiffness;
febio_spec.Control.time_stepper.dtmin=dtmin;
febio_spec.Control.time_stepper.dtmax=dtmax;
febio_spec.Control.time_stepper.max_retries=max_retries;
febio_spec.Control.time_stepper.opt_iter=opt_iter;


%Material section
for q=1:1:numel(youngsModuli)
    materialNameNow=['Material',num2str(q)];
    febio_spec.Material.material{q}.ATTR.name=materialNameNow;
    febio_spec.Material.material{q}.ATTR.type='neo-Hookean';
    febio_spec.Material.material{q}.ATTR.id=q;
    febio_spec.Material.material{q}.E=youngsModuli(q);
    febio_spec.Material.material{q}.v=poissonsRatios(q);
end

materialNameRigid='RigidMaterial';
matIdRigid=numel(youngsModuli)+1;
febio_spec.Material.material{matIdRigid}.ATTR.name=materialNameRigid;
febio_spec.Material.material{matIdRigid}.ATTR.type='rigid body';
febio_spec.Material.material{matIdRigid}.ATTR.id=numel(youngsModuli)+1;
febio_spec.Material.material{matIdRigid}.density=1;
febio_spec.Material.material{matIdRigid}.center_of_mass=center_of_mass;

%Mesh section
% -> Nodes
febio_spec.Mesh.Nodes{1}.ATTR.name='nodeSet_all'; %The node set name
febio_spec.Mesh.Nodes{1}.node.ATTR.id=(1:size(V,1))'; %The node id's
febio_spec.Mesh.Nodes{1}.node.VAL=V; %The nodel coordinates

% -> Elements
n=1;
for q=1:1:numel(youngsModuli)
    logicMaterialNow=(elementMaterialID==q);
    partNameNow=['Part',num2str(q)];
    febio_spec.Mesh.Elements{q}.ATTR.name=partNameNow; %Name of this part
    febio_spec.Mesh.Elements{q}.ATTR.type='hex8'; %Element type of this set
    febio_spec.Mesh.Elements{q}.elem.ATTR.id=(n:1:(n-1+nnz(logicMaterialNow)))'; %Element id's
    febio_spec.Mesh.Elements{q}.elem.VAL=E(logicMaterialNow,:);
    n=n+nnz(logicMaterialNow);
end

partNameRigid='RigidPart';
n=max(febio_spec.Mesh.Elements{end}.elem.ATTR.id);
febio_spec.Mesh.Elements{numel(youngsModuli)+1}.ATTR.name=partNameRigid; %Name of this part
febio_spec.Mesh.Elements{numel(youngsModuli)+1}.ATTR.type='quad4'; %Element type of this set
febio_spec.Mesh.Elements{numel(youngsModuli)+1}.ATTR.mat=numel(youngsModuli)+1; %material index for this set
febio_spec.Mesh.Elements{numel(youngsModuli)+1}.elem.ATTR.id=(n:1:(n-1+size(F_rigidBody,1)))'; %Element id's
febio_spec.Mesh.Elements{numel(youngsModuli)+1}.elem.VAL=F_rigidBody;

% -> NodeSets
nodeSetName1='bcSupportList';
febio_spec.Mesh.NodeSet{1}.ATTR.name=nodeSetName1;
febio_spec.Mesh.NodeSet{1}.VAL=mrow(bcSupportList);

%MeshDomains section
for q=1:1:numel(youngsModuli)
    partNameNow=['Part',num2str(q)];
    materialNameNow=['Material',num2str(q)];
    febio_spec.MeshDomains.SolidDomain{q}.ATTR.name=partNameNow;
    febio_spec.MeshDomains.SolidDomain{q}.ATTR.mat=materialNameNow;
end

febio_spec.MeshDomains.ShellDomain.ATTR.name=partNameRigid;
febio_spec.MeshDomains.ShellDomain.ATTR.mat=materialNameRigid;

% -> Surfaces
febio_spec.Mesh.Surface=[];
for q=1:1:numel(youngsModuli)-1
    F_contact_now=Fb(logicTops(:,q),:);
    c=numel(febio_spec.Mesh.Surface)+1;
    febio_spec.Mesh.Surface{c}.ATTR.name=['contact_',num2str(c)];
    febio_spec.Mesh.Surface{c}.quad4.ATTR.id=(1:1:size(F_contact_now,1))';
    febio_spec.Mesh.Surface{c}.quad4.VAL=F_contact_now;

    F_contact_now=Fb(logicBottoms(:,q+1),:);
    c=numel(febio_spec.Mesh.Surface)+1;
    febio_spec.Mesh.Surface{c}.ATTR.name=['contact_',num2str(c)];
    febio_spec.Mesh.Surface{c}.quad4.ATTR.id=(1:1:size(F_contact_now,1))';
    febio_spec.Mesh.Surface{c}.quad4.VAL=F_contact_now;
end

% -> Surface pairs
surfaceIndices=reshape(1:(2*(numel(youngsModuli)-1)),2,3)'; %Indices for surface pairs
for q=1:1:numel(youngsModuli)-1
    febio_spec.Mesh.SurfacePair{q}.ATTR.name=['Contact_',num2str(q)];

    numElements1=size(febio_spec.Mesh.Surface{surfaceIndices(q,1)}.quad4.VAL,1);
    numElements2=size(febio_spec.Mesh.Surface{surfaceIndices(q,2)}.quad4.VAL,1);
    if numElements1>numElements2
        febio_spec.Mesh.SurfacePair{q}.primary = febio_spec.Mesh.Surface{surfaceIndices(q,1)}.ATTR.name;
        febio_spec.Mesh.SurfacePair{q}.secondary = febio_spec.Mesh.Surface{surfaceIndices(q,2)}.ATTR.name;
    else
        febio_spec.Mesh.SurfacePair{q}.primary = febio_spec.Mesh.Surface{surfaceIndices(q,2)}.ATTR.name;
        febio_spec.Mesh.SurfacePair{q}.secondary = febio_spec.Mesh.Surface{surfaceIndices(q,1)}.ATTR.name;
    end
end

%Boundary condition section
% -> Fix boundary conditions
febio_spec.Boundary.bc{1}.ATTR.name='zero_displacement_xyz';
febio_spec.Boundary.bc{1}.ATTR.type='zero displacement';
febio_spec.Boundary.bc{1}.ATTR.node_set=nodeSetName1;
febio_spec.Boundary.bc{1}.x_dof=1;
febio_spec.Boundary.bc{1}.y_dof=1;
febio_spec.Boundary.bc{1}.z_dof=1;

%Rigid section
% ->Rigid body fix boundary conditions
febio_spec.Rigid.rigid_bc{1}.ATTR.name='RigidFix';
febio_spec.Rigid.rigid_bc{1}.ATTR.type='rigid_fixed';
febio_spec.Rigid.rigid_bc{1}.rb=matIdRigid;
febio_spec.Rigid.rigid_bc{1}.Rx_dof=1;
febio_spec.Rigid.rigid_bc{1}.Ry_dof=1;
febio_spec.Rigid.rigid_bc{1}.Ru_dof=1;
febio_spec.Rigid.rigid_bc{1}.Rv_dof=1;

% ->Rigid body prescribe boundary conditions
febio_spec.Rigid.rigid_bc{2}.ATTR.name='RigidPrescribe_disp';
febio_spec.Rigid.rigid_bc{2}.ATTR.type='rigid_displacement';
febio_spec.Rigid.rigid_bc{2}.rb=matIdRigid;
febio_spec.Rigid.rigid_bc{2}.dof='z';
febio_spec.Rigid.rigid_bc{2}.value.ATTR.lc=1;
febio_spec.Rigid.rigid_bc{2}.value.VAL=prescribedDisplacement_Z;
febio_spec.Rigid.rigid_bc{2}.relative=0;

febio_spec.Rigid.rigid_bc{3}.ATTR.name='RigidPrescribe_rot';
febio_spec.Rigid.rigid_bc{3}.ATTR.type='rigid_rotation';
febio_spec.Rigid.rigid_bc{3}.rb=matIdRigid;
febio_spec.Rigid.rigid_bc{3}.dof='Rw';
febio_spec.Rigid.rigid_bc{3}.value.ATTR.lc=1;
febio_spec.Rigid.rigid_bc{3}.value.VAL=prescribedRotation_Z;
febio_spec.Rigid.rigid_bc{3}.relative=0;

%Contact section
for q=1:1:numel(youngsModuli)-1
    febio_spec.Contact.contact{q}.ATTR.surface_pair=febio_spec.Mesh.SurfacePair{q}.ATTR.name;
    febio_spec.Contact.contact{q}.ATTR.type='sliding-elastic';
    febio_spec.Contact.contact{q}.two_pass=1;
    febio_spec.Contact.contact{q}.laugon=1;
    febio_spec.Contact.contact{q}.tolerance=0.2;
    febio_spec.Contact.contact{q}.gaptol=0;
    febio_spec.Contact.contact{q}.minaug=1;
    febio_spec.Contact.contact{q}.maxaug=10;
    febio_spec.Contact.contact{q}.search_tol=0.01;
    febio_spec.Contact.contact{q}.search_radius=0.1*sqrt(sum((max(V,[],1)-min(V,[],1)).^2,2));
    febio_spec.Contact.contact{q}.symmetric_stiffness=0;
    switch q
        case 1
            febio_spec.Contact.contact{q}.auto_penalty=1;
            febio_spec.Contact.contact{q}.penalty=5;
            febio_spec.Contact.contact{q}.fric_coeff=1;
        case 2
            febio_spec.Contact.contact{q}.auto_penalty=1;
            febio_spec.Contact.contact{q}.penalty=0.1;
            febio_spec.Contact.contact{q}.fric_coeff=0.08;
        case 3
            febio_spec.Contact.contact{q}.auto_penalty=1;
            febio_spec.Contact.contact{q}.penalty=0.1;
            febio_spec.Contact.contact{q}.fric_coeff=0.08;
    end
end

%LoadData section
% -> load_controller
febio_spec.LoadData.load_controller{1}.ATTR.name='LC_1';
febio_spec.LoadData.load_controller{1}.ATTR.id=1;
febio_spec.LoadData.load_controller{1}.ATTR.type='loadcurve';
febio_spec.LoadData.load_controller{1}.interpolate='LINEAR';
%febio_spec.LoadData.load_controller{1}.extend='CONSTANT';
febio_spec.LoadData.load_controller{1}.points.pt.VAL=[0 0; 1 1];

%Output section
% -> log file
febio_spec.Output.logfile.ATTR.file=febioLogFileName;
febio_spec.Output.logfile.node_data{1}.ATTR.file=febioLogFileName_disp;
febio_spec.Output.logfile.node_data{1}.ATTR.data='ux;uy;uz';
febio_spec.Output.logfile.node_data{1}.ATTR.delim=',';

febio_spec.Output.logfile.element_data{1}.ATTR.file=febioLogFileName_strainEnergy;
febio_spec.Output.logfile.element_data{1}.ATTR.data='sed';
febio_spec.Output.logfile.element_data{1}.ATTR.delim=',';

% Plotfile section
febio_spec.Output.plotfile.compression=0;

Quick viewing of the FEBio input file structure

The febView function can be used to view the xml structure in a MATLAB figure window.

febView(febio_spec); %Viewing the febio file

Exporting the FEBio input file

Exporting the febio_spec structure to an FEBio input file is done using the febioStruct2xml function.

febioStruct2xml(febio_spec,febioFebFileName); %Exporting to file and domNode

Running the FEBio analysis

To run the analysis defined by the created FEBio input file the runMonitorFEBio function is used. The input for this function is a structure defining job settings e.g. the FEBio input file name. The optional output runFlag informs the user if the analysis was run succesfully.

febioAnalysis.run_filename=febioFebFileName; %The input file name
febioAnalysis.run_logname=febioLogFileName; %The name for the log file
febioAnalysis.disp_on=1; %Display information on the command window
febioAnalysis.runMode=runMode;

[runFlag]=runMonitorFEBio(febioAnalysis);%START FEBio NOW!!!!!!!!

 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-------->    RUNNING/MONITORING FEBIO JOB    <-------- 20-Apr-2023 10:47:02
FEBio path: /home/kevin/FEBioStudio2/bin/febio4
# Attempt removal of existing log files                20-Apr-2023 10:47:02
 * Removal succesful                                   20-Apr-2023 10:47:02
# Attempt removal of existing .xplt files              20-Apr-2023 10:47:02
 * Removal succesful                                   20-Apr-2023 10:47:02
# Starting FEBio...                                    20-Apr-2023 10:47:02
  Max. total analysis time is: Inf s
 * Waiting for log file creation                       20-Apr-2023 10:47:03
   Max. wait time: 30 s
 * Log file found.                                     20-Apr-2023 10:47:03
# Parsing log file...                                  20-Apr-2023 10:47:03
    number of iterations   : 17                        20-Apr-2023 10:47:03
    number of reformations : 0                         20-Apr-2023 10:47:03
------- converged at time : 0.1                        20-Apr-2023 10:47:03
    number of iterations   : 10                        20-Apr-2023 10:47:04
    number of reformations : 0                         20-Apr-2023 10:47:04
------- converged at time : 0.2                        20-Apr-2023 10:47:04
    number of iterations   : 11                        20-Apr-2023 10:47:04
    number of reformations : 0                         20-Apr-2023 10:47:04
------- converged at time : 0.3                        20-Apr-2023 10:47:04
    number of iterations   : 11                        20-Apr-2023 10:47:04
    number of reformations : 0                         20-Apr-2023 10:47:04
------- converged at time : 0.4                        20-Apr-2023 10:47:04
    number of iterations   : 11                        20-Apr-2023 10:47:04
    number of reformations : 0                         20-Apr-2023 10:47:04
------- converged at time : 0.5                        20-Apr-2023 10:47:04
    number of iterations   : 10                        20-Apr-2023 10:47:05
    number of reformations : 0                         20-Apr-2023 10:47:05
------- converged at time : 0.6                        20-Apr-2023 10:47:05
    number of iterations   : 13                        20-Apr-2023 10:47:05
    number of reformations : 0                         20-Apr-2023 10:47:05
------- converged at time : 0.7                        20-Apr-2023 10:47:05
    number of iterations   : 13                        20-Apr-2023 10:47:05
    number of reformations : 0                         20-Apr-2023 10:47:05
------- converged at time : 0.8                        20-Apr-2023 10:47:05
    number of iterations   : 10                        20-Apr-2023 10:47:06
    number of reformations : 0                         20-Apr-2023 10:47:06
------- converged at time : 0.9                        20-Apr-2023 10:47:06
    number of iterations   : 12                        20-Apr-2023 10:47:06
    number of reformations : 0                         20-Apr-2023 10:47:06
------- converged at time : 1                          20-Apr-2023 10:47:06
 Elapsed time : 0:00:03                                20-Apr-2023 10:47:06
 N O R M A L   T E R M I N A T I O N
# Done                                                 20-Apr-2023 10:47:06
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Import FEBio results

if runFlag==1 %i.e. a succesful run

Importing nodal displacements from a log file

    dataStruct=importFEBio_logfile(fullfile(savePath,febioLogFileName_disp),0,1);

    %Access data
    N_disp_mat=dataStruct.data; %Displacement
    timeVec=dataStruct.time; %Time

    %Create deformed coordinate set
    V_DEF=N_disp_mat+repmat(V,[1 1 size(N_disp_mat,3)]);

Importing element stress from a log file

    dataStruct=importFEBio_logfile(fullfile(savePath,febioLogFileName_strainEnergy),0,1);

    %Access data
    E_energy=dataStruct.data;

Plotting the simulated results using anim8 to visualize and animate deformations

    [CV]=faceToVertexMeasure(E,V,E_energy(:,:,end));

    % Create basic view and store graphics handle to initiate animation
    hf=cFigure; %Open figure
    gtitle([febioFebFileNamePart,': Press play to animate']);
    title('$\sigma_{zz}$ [MPa]','Interpreter','Latex')
    hp=gpatch(Fb,V_DEF(:,:,end),CV,'k',1); %Add graphics object to animate
    hp.Marker='.';
    hp.MarkerSize=markerSize2;
    hp.FaceColor='interp';
    hp2=gpatch(F_rigidBody,V_DEF(:,:,end),'w','k',1); %A static graphics object

    axisGeom(gca,fontSize);
    colormap(gjet(250)); colorbar;
    caxis([min(E_energy(:)) max(E_energy(:))]/2);
    axis(axisLim(V_DEF)); %Set axis limits statically
    camlight headlight;

    % Set up animation features
    animStruct.Time=timeVec; %The time vector
    for qt=1:1:size(N_disp_mat,3) %Loop over time increments

        [CV]=faceToVertexMeasure(E,V,E_energy(:,:,qt));

        %Set entries in animation structure
        animStruct.Handles{qt}=[hp hp hp2]; %Handles of objects to animate
        animStruct.Props{qt}={'Vertices','CData','Vertices'}; %Properties of objects to animate
        animStruct.Set{qt}={V_DEF(:,:,qt),CV,V_DEF(:,:,qt)}; %Property values for to set in order to animate
    end
    anim8(hf,animStruct); %Initiate animation feature
    drawnow;

end

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) 2006-2022 Kevin Mattheus Moerman and the GIBBON contributors

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.

You should have received a copy of the GNU General Public License along with this program. If not, see http://www.gnu.org/licenses/.

Published with MATLAB® R2022b