<div dir="ltr"><div dir="ltr">On Mon, Nov 14, 2022 at 4:19 PM Blaise Bourdin <<a href="mailto:bourdin@mcmaster.ca">bourdin@mcmaster.ca</a>> wrote:<br></div><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">



<div>
Replying to myself so that it gets into the googles:
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<div>The reference tetrahedron used by DMPlexComputeCellGeometryAffineFEM has vertices at (-1,1,-1), (-1,-1,-1), (1, -1, -1) and (-1,-1,1).</div></div></blockquote><div><br></div><div>It is </div><div><br></div><div>  (-1, -1, -1) -- (-1, 1, -1) -- (1, -1, -1) -- (-1, -1, 1)</div><div><br></div><div>that way the first face has an outward normal.</div><div><br></div><div>   Matt</div><div> </div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div>
<div>Blaise</div>
<div> <br>
<div><br>
<blockquote type="cite">
<div>On Nov 10, 2022, at 6:42 PM, Matthew Knepley <<a href="mailto:knepley@gmail.com" target="_blank">knepley@gmail.com</a>> wrote:</div>
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<div dir="ltr">On Thu, Nov 10, 2022 at 3:46 PM Blaise Bourdin <<a href="mailto:bourdin@mcmaster.ca" target="_blank">bourdin@mcmaster.ca</a>> wrote:<br>
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<div>I am not sure I am buying this… If the tet was inverted, detJ would be negative, but it is always 1/8, as expected.
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<div>The attached mesh is a perfectly valid tet generated by Cubit, with orientation matching the exodus documentation (ignore the mid-edge dof since this is a tet4).</div>
<div>Here is what I get out of the code I attached in my previous email:</div>
</div>
</div>
</blockquote>
<div><br>
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<div>Yes, I use the opposite convention from ExodusII. In my opinion, orienting face (1, 2, 3) to have an inward normal is sacrilegious.</div>
<div><br>
</div>
<div>  Thanks,</div>
<div><br>
</div>
<div>     Matt</div>
<div> </div>
<blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
<div>
<div>
<div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"><b>SiMini</b>:Tests (dmplex)$ ./TestDMPlexComputeCellGeometryAffineFEM -i ../TestMeshes/TetCubit.gen </span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"> filename ../TestMeshes/TetCubit.gen</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">Vec Object: coordinates 1 MPI process</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">  type: seq</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">0.</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">0.</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">0.</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">1.</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">0.</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">0.</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">0.</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">1.</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">0.</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">0.</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">0.</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">1.</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">v0</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"> 0:   1.0000e+00   0.0000e+00   0.0000e+00</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">J</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"> 0:  -5.0000e-01  -5.0000e-01  -5.0000e-01</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"> 0:   5.0000e-01   0.0000e+00   0.0000e+00</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"> 0:   0.0000e+00   0.0000e+00   5.0000e-01</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">invJ</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"> 0:   0.0000e+00   2.0000e+00   0.0000e+00</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"> 0:  -2.0000e+00  -2.0000e+00  -2.0000e+00</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"> 0:   0.0000e+00   0.0000e+00   2.0000e+00</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">detJ : 0.125</span></div>
</div>
<div><span style="font-variant-ligatures:no-common-ligatures"><br>
</span></div>
<div>From J, invJ, and v0, I still can’t reconstruct a reasonable reference tet which I was naively assuming was either the unit simplex, or the simplex with vertices at (-1,-1,-1), (-1,0,-1), (0, -1, -1), and (-1,-1,1) not necessarily in this order. In order
 to build my FE basis functions on the reference element, I really need to know what this element is…</div>
<div><br>
</div>
<div>Blaise</div>
<div></div>
</div>
<div>
<div></div>
<div><br>
</div>
<div> </div>
<div></div>
</div>
<div>
<div></div>
<div><br>
<div><br>
<blockquote type="cite">
<div>On Nov 9, 2022, at 6:56 PM, Matthew Knepley <<a href="mailto:knepley@gmail.com" target="_blank">knepley@gmail.com</a>> wrote:</div>
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<div dir="ltr">On Wed, Nov 9, 2022 at 10:46 AM Blaise Bourdin <<a href="mailto:bourdin@mcmaster.ca" target="_blank">bourdin@mcmaster.ca</a>> wrote:<br>
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<div><br>
<div><br>
<blockquote type="cite">
<div>On Nov 9, 2022, at 10:04 AM, Matthew Knepley <<a href="mailto:knepley@gmail.com" target="_blank">knepley@gmail.com</a>> wrote:</div>
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<div dir="ltr">On Tue, Nov 8, 2022 at 9:14 PM Blaise Bourdin <<a href="mailto:bourdin@mcmaster.ca" target="_blank">bourdin@mcmaster.ca</a>> wrote:<br>
</div>
<div>
<blockquote style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
Hi,<br>
<br>
What reference simplex is DMPlexComputeCellGeometryAffineFEM using in 2 and 3D?<br>
I am used to computing my shape functions on the unit simplex (vertices at the origin and each e_i), but it does not look to be the reference simplex in this function:<br>
<br>
In 3D, for the unit simplex with vertices at (0,0,0) (1,0,0) (0,1,0) (0,0,1) (in this order), I get J = 1 / 2 . [[-1,-1,-1],[1,0,0],[0,0,1]] and v0 = [0,0,1]<br>
<br>
In 2D, for the unit simplex with vertices at (0,0), (1,0), and (0,1), I get J = 1 / 2. I and v0 = [0,0], which does not make any sense to me (I was assuming that the 2D reference simplex had vertices at (-1,-1), (1, -1) and (-1,1), but if this were the case,
 v0 would not be 0).<br>
<br>
I can build a simple example with meshes consisting only of the unit simplex in 2D and 3D if that would help.<br>
</blockquote>
<div><br>
</div>
<div>I need to rewrite the documentation on geometry, but I was waiting until I rewrite the geometry calculations to fit into libCEED. Toby found a nice</div>
<div>way to express them in BLAS form which I need to push through everything.</div>
<div><br>
</div>
<div>I always think of operating on the cell with the first vertex at the origin (I think it is easier), so I have a xi0 that translates the first vertex</div>
<div>of the reference to the origin, and a v0 that translates the first vertex of the real cell to the origin. You can see this here</div>
<div><br>
</div>
<div>  <a href="https://gitlab.com/petsc/petsc/-/blob/main/include/petsc/private/petscfeimpl.h#L251" target="_blank">https://gitlab.com/petsc/petsc/-/blob/main/include/petsc/private/petscfeimpl.h#L251</a></div>
<div><br>
</div>
<div>This explains the 2D result. I cannot understand your 3D result, unless the vertices are in another order.</div>
</div>
</div>
</div>
</blockquote>
<div><br>
</div>
That makes two of us, then… I am attaching a small example and test meshes (one cell being the unit simplex starting with the origin and numbered in direct order when looking from (1,1,1)</div>
</div>
</div>
</blockquote>
<div><br>
</div>
<div>Oh, it is probably inverted. All faces are oriented for outward normals. It is in the Orientation chapter in the book :)</div>
<div><br>
</div>
<div>  Thanks,</div>
<div><br>
</div>
<div>      Matt</div>
<div> </div>
<blockquote style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
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<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"> filename ../TestMeshes/1Tri.gen</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">Vec Object: coordinates 1 MPI process</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">  type: seq</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">0.</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">0.</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">1.</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">0.</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">0.</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">1.</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">v0</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"> 0:   0.0000e+00   0.0000e+00</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">J</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"> 0:   5.0000e-01   0.0000e+00</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"> 0:   0.0000e+00   5.0000e-01</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">invJ</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"> 0:   2.0000e+00  -0.0000e+00</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"> 0:  -0.0000e+00   2.0000e+00</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">detJ : 0.25</span></div>
<div><span style="font-variant-ligatures:no-common-ligatures"><br>
</span></div>
<div><span style="font-variant-ligatures:no-common-ligatures">And </span></div>
<div><span style="font-variant-ligatures:no-common-ligatures">
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"> filename ../TestMeshes/1Tet.gen</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">Vec Object: coordinates 1 MPI process</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">  type: seq</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">0.</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">0.</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">0.</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">1.</span></div>
<div style="margin:0px;font-stretch:normal;line-height:normal"><span style="font-family:"Fira Code";font-size:11px;font-variant-ligatures:no-common-ligatures">0.</span></div>
<p style="margin:0px;font-stretch:normal;line-height:normal"></p>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">0.</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">0.</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">1.</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">0.</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">0.</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">0.</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">1.</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">v0</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"> 0:   1.0000e+00   0.0000e+00   0.0000e+00</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">J</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"> 0:  -5.0000e-01  -5.0000e-01  -5.0000e-01</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"> 0:   5.0000e-01   0.0000e+00   0.0000e+00</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"> 0:   0.0000e+00   0.0000e+00   5.0000e-01</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">invJ</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"> 0:   0.0000e+00   2.0000e+00   0.0000e+00</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"> 0:  -2.0000e+00  -2.0000e+00  -2.0000e+00</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures"> 0:   0.0000e+00   0.0000e+00   2.0000e+00</span></div>
<div style="margin:0px;font-stretch:normal;font-size:11px;line-height:normal;font-family:"Fira Code"">
<span style="font-variant-ligatures:no-common-ligatures">detJ : 0.125</span></div>
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<div>I don’t understand why v0=(0,0) in 2D and (1,0,0) in 3D (but don’t really care) since I only want J. J makes no sense to me in 3D. In particular, one does not seem to have X~ = invJ.X + v0 (X = J.(X~-v0) as stated in CoordinatesRefToReal (it works in 2D
 if V0 = (1,1), which is consistent with a reference simplex with vertices at (-1,-1), (1,-1) and (-1,1)).</div>
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<div>What am I missing?</div>
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<div>Blaise</div>
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<div>  Thanks,</div>
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Regards,<br>
Blaise<br>
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—<span> </span><br>
Canada Research Chair in Mathematical and Computational Aspects of Solid Mechanics (Tier 1)<br>
Professor, Department of Mathematics & Statistics<br>
Hamilton Hall room 409A, McMaster University<br>
1280 Main Street West, Hamilton, Ontario L8S 4K1, Canada<span> </span><br>
<a href="https://www.math.mcmaster.ca/bourdin" rel="noreferrer" target="_blank">https://www.math.mcmaster.ca/bourdin</a><span> </span>| +1 (905) 525 9140 ext. 27243<br>
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<div>What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br>
-- Norbert Wiener</div>
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<div><a href="http://www.cse.buffalo.edu/~knepley/" target="_blank">https://www.cse.buffalo.edu/~knepley/</a></div>
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<div>— <br>
Canada Research Chair in Mathematical and Computational Aspects of Solid Mechanics (Tier 1)</div>
<div>Professor, Department of Mathematics & Statistics<br>
Hamilton Hall room 409A, McMaster University<br>
1280 Main Street West, Hamilton, Ontario L8S 4K1, Canada <br>
<a href="https://www.math.mcmaster.ca/bourdin" target="_blank">https://www.math.mcmaster.ca/bourdin</a><span> </span>| +1 (905) 525 9140 ext. 27243</div>
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<div>What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br>
-- Norbert Wiener</div>
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<div><a href="http://www.cse.buffalo.edu/~knepley/" target="_blank">https://www.cse.buffalo.edu/~knepley/</a></div>
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<div>— <br>
Canada Research Chair in Mathematical and Computational Aspects of Solid Mechanics (Tier 1)</div>
<div>Professor, Department of Mathematics & Statistics<br>
Hamilton Hall room 409A, McMaster University<br>
1280 Main Street West, Hamilton, Ontario L8S 4K1, Canada <br>
<a href="https://www.math.mcmaster.ca/bourdin" target="_blank">https://www.math.mcmaster.ca/bourdin</a><span> </span>| +1 (905) 525 9140 ext. 27243</div>
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<div>What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br>
-- Norbert Wiener</div>
<div><br>
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<div><a href="http://www.cse.buffalo.edu/~knepley/" target="_blank">https://www.cse.buffalo.edu/~knepley/</a></div>
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<div>— <br>
Canada Research Chair in Mathematical and Computational Aspects of Solid Mechanics (Tier 1)</div>
<div>Professor, Department of Mathematics & Statistics<br>
Hamilton Hall room 409A, McMaster University<br>
1280 Main Street West, Hamilton, Ontario L8S 4K1, Canada <br>
<a href="https://www.math.mcmaster.ca/bourdin" target="_blank">https://www.math.mcmaster.ca/bourdin</a> | +1 (905) 525 9140 ext. 27243</div>
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</blockquote></div><br clear="all"><div><br></div>-- <br><div dir="ltr" class="gmail_signature"><div dir="ltr"><div><div dir="ltr"><div><div dir="ltr"><div>What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br>-- Norbert Wiener</div><div><br></div><div><a href="http://www.cse.buffalo.edu/~knepley/" target="_blank">https://www.cse.buffalo.edu/~knepley/</a><br></div></div></div></div></div></div></div></div>