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I am unfortunately unable to access the attached code.</div>
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Bora: have you defined constraint dofs in your section? In that case the natural SF is wrong, and you can notice duplicate entries for the roots if I remember correctly. I'll do a MR soon about that.</div>
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The natural ordering and distribution is the one implemented by the DM before you call DMPlexDistribute. If your DM is automatically distributed when you load it from your GMSH file thanks to DMPlexCreateFromFile for example, you're not going to find your file
ordering in the natural SF. You may call DMPlexDistributeSetDefault(dm,PETSC_FALE) to disable the automatic distribution.</div>
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Best</div>
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<div id="divRplyFwdMsg" dir="ltr"><font face="Calibri, sans-serif" style="font-size:11pt" color="#000000"><b>From:</b> Blaise Bourdin <bourdin@mcmaster.ca><br>
<b>Sent:</b> July 18, 2022 8:52 PM<br>
<b>To:</b> Bora Jeong <boraj1021@gmail.com><br>
<b>Cc:</b> Matthew Knepley <knepley@gmail.com>; PETSc <petsc-users@mcs.anl.gov>; Alexis Marboeuf <marboeua@mcmaster.ca><br>
<b>Subject:</b> Re: [petsc-users] DMPlex: a Mapping Array between Natural and Distributed Cell Index</font>
<div> </div>
</div>
<div class="" style="word-wrap:break-word; line-break:after-white-space">Alexis noticed a problem with the natural SF when constraints are defined. He has a MR coming.
<div class="">@Alexis: Could it be that what Bora sees is exactly what you fixed?</div>
<div class=""><br class="">
</div>
<div class="">Blaise<br class="">
<div><br class="">
<blockquote type="cite" class="">
<div class="">On Jul 18, 2022, at 8:35 PM, Bora Jeong <<a href="mailto:boraj1021@gmail.com" class="">boraj1021@gmail.com</a>> wrote:</div>
<br class="x_Apple-interchange-newline">
<div class="">
<div dir="ltr" class="">Thank you for the reply.
<div class=""><br class="">
<div class="">I am actually putting 1dof on the vertex points.
<div class="">In the attached sample code, from line 63 to line 99, 1dof on all vertex points is defined for the section. But my problem is, if I print out Natural SF using the viewer, I only have 10 vertex points, instead of 15. Here 15 is the size of the
global vector defined by the aforementioned section. The working example code was attached to the previous email with a small grid. </div>
</div>
<div class=""><br class="">
</div>
<div class="">Best</div>
</div>
</div>
<br class="">
<div class="x_gmail_quote">
<div dir="ltr" class="x_gmail_attr">On Mon, Jul 18, 2022 at 6:50 PM Matthew Knepley <<a href="mailto:knepley@gmail.com" class="">knepley@gmail.com</a>> wrote:<br class="">
</div>
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr" class="">
<div dir="ltr" class="">On Mon, Jul 18, 2022 at 4:30 PM Bora Jeong <<a href="mailto:boraj1021@gmail.com" target="_blank" class="">boraj1021@gmail.com</a>> wrote:<br class="">
</div>
<div class="x_gmail_quote">
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr" class="">
<div dir="ltr" class="">Actually, it seems that my problem is that I do not properly understand what Natural SF means.
<br class="">
<br class="">
What I want to achieve is to extract natural vertex index from Natural SF, and make sure this natural vertex index is the same with what is written in a mesh file from Gmsh. The natural vertex index defined by Gmsh is attached as a snapshot with sample code
and mesh. I want to reproduce this indexing based on the Natural SF that Petsc defines.<br class="">
<br class="">
I am struggling with that because the number of entries defined in the Natural SF that I can see through PetscSFView() is different from the number of vertex points that each processor actually owns. For example, with two processors and with the attached "square.msh",
each processor might have 15 vertex points if I configure Petsc with ParMetis. However, the defined Natural SF for proc=0 only has 10 roots. I expected, for each processor, the Natural SF would show me 15 entries through PetscSFView(), if I define 1 degree
of freedom on vertex when the Petsc Section is defined. <br class="">
<br class="">
When Petsc defines the Natural SF, are there any omitted components since they are trivial to save?<br class="">
</div>
</div>
</blockquote>
<div class=""><br class="">
</div>
<div class="">The naturalSF is mapping the field defined by the Section, not the mesh. You can get the effect you are asking for by putting one</div>
<div class="">degree of freedom (dof) on every vertex for the Section. How are you defining your Section now?</div>
<div class=""><br class="">
</div>
<div class=""> Thanks,</div>
<div class=""><br class="">
</div>
<div class=""> Matt</div>
<div class=""> </div>
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr" class="">
<div dir="ltr" class="">Best<br class="">
</div>
<br class="">
<div class="x_gmail_quote">
<div dir="ltr" class="x_gmail_attr">On Mon, Jul 18, 2022 at 3:50 PM Matthew Knepley <<a href="mailto:knepley@gmail.com" target="_blank" class="">knepley@gmail.com</a>> wrote:<br class="">
</div>
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr" class="">
<div dir="ltr" class="">On Mon, Jul 18, 2022 at 12:14 PM Bora Jeong <<a href="mailto:boraj1021@gmail.com" target="_blank" class="">boraj1021@gmail.com</a>> wrote:<br class="">
</div>
<div class="x_gmail_quote">
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr" class="">Thank you for the corrections. It works. <br class="">
<br class="">
However, I still have a problem. <br class="">
I tested zero stratum for DMGetStratumIS() with the default depth label, since the graph of vertex is needed. Then, PetscSFGetGraph() from the Natural SF is crashed. I checked that pBcCompIS contains proper set of integers via ISView().
<br class="">
<br class="">
On the other hand, it works okay if DMGetStratumIS() is with dim stratum, which is for cell stratum.
<br class="">
The problem is that if the dim stratum is used, the number of entry in ileaves(i)% pointer does not match with the actual number of vertex that each processor has. That is why I tried to put the zero stratum on DMGetStratumIS(), instead of the stratum for cell
(i.e., "dim") to see if any changes on the graph. Can I get comments?<br class="">
</div>
</blockquote>
<div class=""><br class="">
</div>
<div class="">I cannot understand your question. I am not sure what you are using the set of vertices or cell for.</div>
<div class=""><br class="">
</div>
<div class=""> Thanks,</div>
<div class=""><br class="">
</div>
<div class=""> Matt</div>
<div class=""> </div>
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr" class="">Best<br class="">
</div>
<br class="">
<div class="x_gmail_quote">
<div dir="ltr" class="x_gmail_attr">On Sun, Jul 17, 2022 at 9:59 AM Matthew Knepley <<a href="mailto:knepley@gmail.com" target="_blank" class="">knepley@gmail.com</a>> wrote:<br class="">
</div>
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr" class="">
<div dir="ltr" class="">On Fri, Jul 15, 2022 at 7:05 PM Bora Jeong <<a href="mailto:boraj1021@gmail.com" target="_blank" class="">boraj1021@gmail.com</a>> wrote:<br class="">
</div>
<div class="x_gmail_quote">
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr" class="">I found that iroots() and ileaves() have size of nleaves and tested through the attached sample code with the grid previously shared. Then I get the results written in the attached monitor file.
<br class="">
<br class="">
It seems ileaves(i)%rank and ileaves(i)%index at line 127 of the attached code have garbage values, different from displayed values by PetscSFView() at line 113.
<br class="">
<br class="">
It is tough to get it why the garbage values are returned from PetscSFGetGraph(). Any comments will be very appreciated.<br class="">
</div>
</blockquote>
<div class=""><br class="">
</div>
<div class="">Unfortunately, Fortran is not very good at reporting declaration errors. The problem is that you did not include or use the Vec module. I have done this and your example runs for me. I have included the modified code.</div>
<div class=""><br class="">
</div>
<div class=""> Thanks,</div>
<div class=""><br class="">
</div>
<div class=""> Matt</div>
<div class=""> </div>
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr" class="">Best<br class="">
</div>
<br class="">
<div class="x_gmail_quote">
<div dir="ltr" class="x_gmail_attr">On Fri, Jul 15, 2022 at 3:53 PM Matthew Knepley <<a href="mailto:knepley@gmail.com" target="_blank" class="">knepley@gmail.com</a>> wrote:<br class="">
</div>
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<div dir="ltr" class="">
<div dir="ltr" class="">On Fri, Jul 15, 2022 at 2:25 PM Bora Jeong <<a href="mailto:boraj1021@gmail.com" target="_blank" class="">boraj1021@gmail.com</a>> wrote:<br class="">
</div>
<div class="x_gmail_quote">
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr" class="">
<div dir="ltr" class="">Thank you for the comments. Updated sample code is attached here; DMPlexDistribute() is explicitly called and by defining a section before mesh distribution, a Natural SF was able to be defined as found from PetscSFView().
<br class="">
<br class="">
However, I am still struggling to call PetscSFGetGraph() <a href="https://petsc.org/main/docs/manualpages/PetscSF/PetscSFGetGraph/" target="_blank" class="">
https://petsc.org/main/docs/manualpages/PetscSF/PetscSFGetGraph/</a><br class="">
due to data type definition of ilocal and iremote. What is the proper size allocation for those two variables? Does this size for the number of processors? or the number of vertex?<br class="">
</div>
</div>
</blockquote>
<div class=""><br class="">
</div>
<div class="">Since we need to pass back arrays, you need to pass us in F90 pointers. Here is an example of doing such a thing:</div>
<div class=""><br class="">
</div>
<div class=""> <a href="https://gitlab.com/petsc/petsc/-/blob/main/src/vec/is/sf/tutorials/ex1f.F90#L94" target="_blank" class="">https://gitlab.com/petsc/petsc/-/blob/main/src/vec/is/sf/tutorials/ex1f.F90#L94</a></div>
<div class=""><br class="">
</div>
<div class=""> Thanks,</div>
<div class=""><br class="">
</div>
<div class=""> Matt</div>
<div class=""> </div>
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr" class="">
<div dir="ltr" class="">Best<br class="">
</div>
<br class="">
<div class="x_gmail_quote">
<div dir="ltr" class="x_gmail_attr">On Fri, Jul 15, 2022 at 9:09 AM Matthew Knepley <<a href="mailto:knepley@gmail.com" target="_blank" class="">knepley@gmail.com</a>> wrote:<br class="">
</div>
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr" class="">
<div dir="ltr" class="">On Fri, Jul 15, 2022 at 8:46 AM Bora Jeong <<a href="mailto:boraj1021@gmail.com" target="_blank" class="">boraj1021@gmail.com</a>> wrote:<br class="">
</div>
<div class="x_gmail_quote">
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr" class="">
<div class="">Okay, I got it what's happening. First, this very bright functionality of petsc (the natural-global converting) needs to be documented in a better way for sure. Currently, it is very difficult to use/follow this features as an external user. Hope
this will move forward in a better way. </div>
<div class=""><br class="">
</div>
<div class="">Then next question is if I load/distribute mesh just like I am doing right now shown in the sample code, can I assume that my code does not create natural sf during the distribution(always)? In other words, can I always get the natural order of
each node by simply stacking the preceding processor's number of node? For example, for proc=0, natural node ID might be just 1 to nnode_proc_0, </div>
<div class="">for proc=1, it might be {nnode_proc_0 + 1 to nnode_proc_0 + nnode_proc_1} and so on. </div>
<div class=""><br class="">
</div>
<div class="">Does that always make sense in this case? </div>
</div>
</blockquote>
<div class=""><br class="">
</div>
<div class="">No, but if you call DMPlexDistribute() yourself, rather than having it called automatically by DMSetFromOptions(), you can</div>
<div class="">preserve the mapping:</div>
<div class=""><br class="">
</div>
<div class=""> <a href="https://petsc.org/main/docs/manualpages/DMPLEX/DMPlexDistribute/" target="_blank" class="">
https://petsc.org/main/docs/manualpages/DMPLEX/DMPlexDistribute/</a></div>
<div class=""><br class="">
</div>
<div class="">The SF returned maps the original point distribution, which is in the same order as the file, to the redistributed points, which are determined</div>
<div class="">by the mesh partitioner.</div>
<div class=""><br class="">
</div>
<div class=""> Thanks,</div>
<div class=""><br class="">
</div>
<div class=""> Matt </div>
<div class=""> </div>
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr" class="">
<div class="">Best</div>
<div dir="ltr" class=""><br class="">
</div>
<br class="">
<div class="x_gmail_quote">
<div dir="ltr" class="x_gmail_attr">On Fri, Jul 15, 2022 at 8:07 AM Matthew Knepley <<a href="mailto:knepley@gmail.com" target="_blank" class="">knepley@gmail.com</a>> wrote:<br class="">
</div>
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr" class="">
<div dir="ltr" class="">On Fri, Jul 15, 2022 at 7:17 AM Bora Jeong <<a href="mailto:boraj1021@gmail.com" target="_blank" class="">boraj1021@gmail.com</a>> wrote:<br class="">
</div>
<div class="x_gmail_quote">
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr" class="">
<div class="">A sample code for loading dm and declaring Natural SF from a sphere mesh is attached here. PetscSFView() returns NULL from sf_nat. </div>
</div>
</blockquote>
<div class=""><br class="">
</div>
<div class="">The Global-to-Natural mapping relates global dofs to natural dofs. Thus, in order to compute this mapping the DM has to know</div>
<div class="">about the dof layout before distribution. This means you need to setup a local section before distributing, as we do for example in</div>
<div class="">Plex test ex15. This makes things more complicated since everything cannot be packaged up into DMSetFromOptions(), but we need</div>
<div class="">user input so I do not see a simpler way to do things.</div>
<div class=""><br class="">
</div>
<div class=""> Thanks,</div>
<div class=""><br class="">
</div>
<div class=""> Matt</div>
<div class=""> </div>
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr" class="">
<div class="">Best</div>
<br class="">
<div class="x_gmail_quote">
<div dir="ltr" class="x_gmail_attr">On Fri, Jul 15, 2022 at 6:39 AM Matthew Knepley <<a href="mailto:knepley@gmail.com" target="_blank" class="">knepley@gmail.com</a>> wrote:<br class="">
</div>
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr" class="">
<div dir="ltr" class="">On Thu, Jul 14, 2022 at 8:25 PM Bora Jeong <<a href="mailto:boraj1021@gmail.com" target="_blank" class="">boraj1021@gmail.com</a>> wrote:<br class="">
</div>
<div class="x_gmail_quote">
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr" class="">
<div class="">Okay, I checked it and you are correct. In my case, simply, natural node index can be identified by stacking all the preceding processor's numbers of nodes for a particular processor, which is good due to simplicity. However, one serious question
is why this is happening in my code? In other words, why the natural SF is not created during the mesh distribution? My code wants to have consistency in dealing with this natural indexing for several different kinds of mesh files. So there is a necessity
to guarantee of consistency in this weird behavior.<br class="">
</div>
</div>
</blockquote>
<div class=""><br class="">
</div>
<div class="">I can't tell what is going on in your code unless I can run it. Do you have a simple example?</div>
<div class=""><br class="">
</div>
<div class=""> Thanks,</div>
<div class=""><br class="">
</div>
<div class=""> Matt</div>
<div class=""> </div>
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr" class="">
<div class=""></div>
<div class="">Best,</div>
<div class=""><br class="">
</div>
<div class="x_gmail_quote">
<div dir="ltr" class="x_gmail_attr">On Thu, Jul 14, 2022 at 6:43 PM Matthew Knepley <<a href="mailto:knepley@gmail.com" target="_blank" class="">knepley@gmail.com</a>> wrote:<br class="">
</div>
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr" class="">
<div dir="ltr" class="">On Thu, Jul 14, 2022 at 5:47 PM Bora Jeong <<a href="mailto:boraj1021@gmail.com" target="_blank" class="">boraj1021@gmail.com</a>> wrote:<br class="">
</div>
<div class="x_gmail_quote">
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr" class="">
<div dir="ltr" class="">Thank you for the comments. I have these errors when I call PetscSFView() after DMGetNaturalSF()<br class="">
<br class="">
[2]PETSC ERROR: --------------------- Error Message --------------------------------------------------------------<br class="">
[2]PETSC ERROR: Null argument, when expecting valid pointer<br class="">
[2]PETSC ERROR: Null Pointer: Parameter # 1<br class="">
[2]PETSC ERROR: See <a href="https://petsc.org/release/faq/" target="_blank" class="">
https://petsc.org/release/faq/</a> for trouble shooting.<br class="">
[2]PETSC ERROR: Petsc Release Version 3.17.0, unknown <br class="">
[2]PETSC ERROR: #1 PetscSFView() at <br class="">
[2]PETSC ERROR: #2 User provided function() at User file:0<br class="">
Abort(85) on node 2 (rank 0 in comm 16): application called MPI_Abort(MPI_COMM_SELF, 85) - process 0<br class="">
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<div class="">Clearly NULL was returned, which means no natural SF was created.</div>
<div class=""> </div>
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<div dir="ltr" class="">Below is the structure to load a mesh file from gmsh; <br class="">
<br class="">
call DMCreate(PETSC_COMM_WORLD, dm, ierr);CHKERRA(ierr)<br class="">
call DMSetType(dm, plex, ierr);CHKERRA(ierr)<br class="">
call DMSetUseNatural(dm, PETSC_TRUE, ierr);CHKERRA(ierr)<br class="">
call DMSetFromOptions(dm, ierr);CHKERRA(ierr)<br class="">
call DMGetNaturalSF(dm, sf_nat, ierr);CHKERRA(ierr)<br class="">
call PetscSFView(sf_nat, PETSC_VIEWER_STDOUT_WORLD, ierr);CHKERRA(ierr)<br class="">
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<div class="">The natural SF is created during mesh distribution. That has not happened here. This means that</div>
<div class="">the order of cells is identical to the file it was read from.</div>
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<div class=""> Thanks,</div>
<div class=""><br class="">
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<div class=""> Matt</div>
<div class=""> </div>
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<div dir="ltr" class="">Best<br class="">
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<div dir="ltr" class="x_gmail_attr">On Thu, Jul 14, 2022 at 10:49 AM Matthew Knepley <<a href="mailto:knepley@gmail.com" target="_blank" class="">knepley@gmail.com</a>> wrote:<br class="">
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<div dir="ltr" class="">On Wed, Jul 13, 2022 at 10:17 PM Bora Jeong <<a href="mailto:boraj1021@gmail.com" target="_blank" class="">boraj1021@gmail.com</a>> wrote:<br class="">
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<div dir="ltr" class="">Dear petsc team,
<div class=""> <br class="">
I am a user of DMPlex for a finite volume code and there is a necessity to know global index of each cell. Here the global index means the indexing that can be found from a mesh file itself without distribution over processors. It seems petsc community denotes
this indexing term as "natural". <br class="">
<br class="">
What I want to do is to create a local array (not petsc vector but just an array variable in the program) to map distributed cell ID to natual cell ID, for example, an array "A";
<br class="">
A(distributed_node_ID) = natural_node_ID<br class="">
<br class="">
There are some petsc functions to support mapping between global and natural vectors. However, I just need to define the array "A" as above example. To achieve this, what is a proper/smart way? In other words, how can I extract the natural_cell_ID from a distributed
local_cell_ID? <br class="">
<br class="">
I turned on DMSetUseNatural(DM, PETSC_TRUE) before distribution, but after this, defining all the required section and star forest objects to get natural and global vectors seems not that direct way for my purpose, which is just to extract the above mapping
array "A". Can I get any comments about it?<br class="">
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<div class="">There is only one thing created, the sfNatural PetscSF object, which you can get with DMGetNaturalSF(). The roots of this SF are</div>
<div class="">the global numbers of dofs stored in PETSc vectors, and the leaves are natural numbers for these dofs. Thus, when we map global</div>
<div class="">vectors to natural vectors in DMPlexGlobalToNaturalBegin/End(), we call PetscSFBcastBegin/End(). Mapping natural to global we call</div>
<div class="">PetscSFReduceBegin/End(). You could pull the information out of the SF using PetscSFGetGraph() if you want.</div>
<div class=""><br class="">
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<div class=""> Thanks,</div>
<div class=""><br class="">
</div>
<div class=""> Matt</div>
<div class=""> </div>
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<div class="">Regards<br class="">
Mo<br class="">
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<div class=""><a href="http://www.cse.buffalo.edu/~knepley/" target="_blank" class="">https://www.cse.buffalo.edu/~knepley/</a><br class="">
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<div class=""><a href="http://www.cse.buffalo.edu/~knepley/" target="_blank" class="">https://www.cse.buffalo.edu/~knepley/</a><br class="">
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-- Norbert Wiener</div>
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<div class=""><a href="http://www.cse.buffalo.edu/~knepley/" target="_blank" class="">https://www.cse.buffalo.edu/~knepley/</a><br class="">
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-- Norbert Wiener</div>
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<div class=""><a href="http://www.cse.buffalo.edu/~knepley/" target="_blank" class="">https://www.cse.buffalo.edu/~knepley/</a><br class="">
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<div class="">What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br class="">
-- Norbert Wiener</div>
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<div class=""><a href="http://www.cse.buffalo.edu/~knepley/" target="_blank" class="">https://www.cse.buffalo.edu/~knepley/</a><br class="">
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<span id="x_cid:f_l5rfzgvx0"><test.F90></span></div>
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<div>— <br class="">
Tier 1 Canada Research Chair in Mathematical and Computational Aspects of Solid Mechanics</div>
<div>Professor, Department of Mathematics & Statistics<br class="">
Hamilton Hall room 409A, McMaster University<br class="">
1280 Main Street West, Hamilton, Ontario L8S 4K1, Canada <br class="">
<a href="https://www.math.mcmaster.ca/bourdin" class="">https://www.math.mcmaster.ca/bourdin</a> | +1 (905) 525 9140 ext. 27243</div>
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