<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
</head>
<body style="overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">
Replying to myself so that it gets into the googles:
<div><br>
</div>
<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><br>
</div>
<div>Blaise</div>
<div> <br>
<div><br>
<blockquote type="cite">
<div>On Nov 10, 2022, at 6:42 PM, Matthew Knepley <knepley@gmail.com> wrote:</div>
<br class="Apple-interchange-newline">
<div>
<div dir="ltr" style="caret-color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none;">
<div dir="ltr">On Thu, Nov 10, 2022 at 3:46 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-width: 1px; border-left-style: solid; border-left-color: rgb(204, 204, 204); padding-left: 1ex;">
<div>
<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.
<div><br>
</div>
<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>
</div>
<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-width: 1px; border-left-style: solid; border-left-color: 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>
<br>
<div>
<div dir="ltr" style="font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<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>
</div>
<div>
<blockquote style="margin: 0px 0px 0px 0.8ex; border-left-width: 1px; border-left-style: solid; border-left-color: rgb(204, 204, 204); padding-left: 1ex;">
<div>
<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>
<br>
<div>
<div dir="ltr" style="font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<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-width: 1px; border-left-style: solid; border-left-color: 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-width: 1px; border-left-style: solid; border-left-color: rgb(204, 204, 204); padding-left: 1ex;">
<div>
<div>
<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;"> 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>
<div><span style="font-variant-ligatures: no-common-ligatures;"><br>
</span></div>
</span></div>
</div>
<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>
<div><br>
</div>
<div>What am I missing?</div>
<div><br>
</div>
<div>Blaise</div>
<div><br>
</div>
<div><font face="Fira Code"><span style="font-size: 11px;"></span></font></div>
</div>
<div>
<div><font face="Fira Code"><span style="font-size: 11px;">/</span></font></div>
</div>
<div>
<div><br>
<font face="Fira Code"><span style="font-size: 11px;"></span></font></div>
</div>
<div>
<div><font face="Fira Code"><span style="font-size: 11px;"></span></font></div>
<div><br>
<blockquote type="cite">
<div>
<div dir="ltr" style="font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<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-width: 1px; border-left-style: solid; border-left-color: rgb(204, 204, 204); padding-left: 1ex;">
Regards,<br>
Blaise<br>
<br>
<br>
<br>
—<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>
<br>
</blockquote>
</div>
<br clear="all">
<div><br>
</div>
--<span> </span><br>
<div dir="ltr">
<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></div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</blockquote>
</div>
<br>
<div>
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<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>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<br>
</div>
</div>
</blockquote>
</div>
<br clear="all">
<div><br>
</div>
--<span> </span><br>
<div dir="ltr">
<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></div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</blockquote>
</div>
<br>
<div>
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none;">
<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 class="Apple-converted-space"> </span>| +1 (905) 525 9140 ext. 27243</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<br>
</div>
</div>
</div>
</blockquote>
</div>
<br clear="all">
<div><br>
</div>
--<span class="Apple-converted-space"> </span><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></div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</blockquote>
</div>
<br>
<div>
<div dir="auto" style="caret-color: rgb(0, 0, 0); color: rgb(0, 0, 0); letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">
<div dir="auto" style="caret-color: rgb(0, 0, 0); color: rgb(0, 0, 0); letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">
<div dir="auto" style="caret-color: rgb(0, 0, 0); color: rgb(0, 0, 0); letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">
<div dir="auto" style="caret-color: rgb(0, 0, 0); color: rgb(0, 0, 0); letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">
<div dir="auto" style="caret-color: rgb(0, 0, 0); color: rgb(0, 0, 0); letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">
<div dir="auto" style="caret-color: rgb(0, 0, 0); color: rgb(0, 0, 0); letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">
<div dir="auto" style="caret-color: rgb(0, 0, 0); color: rgb(0, 0, 0); letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">
<div dir="auto" style="caret-color: rgb(0, 0, 0); color: rgb(0, 0, 0); letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">
<div dir="auto" style="caret-color: rgb(0, 0, 0); color: rgb(0, 0, 0); letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">
<div dir="auto" style="caret-color: rgb(0, 0, 0); color: rgb(0, 0, 0); letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">
<div dir="auto" style="caret-color: rgb(0, 0, 0); color: rgb(0, 0, 0); letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">
<div dir="auto" style="caret-color: rgb(0, 0, 0); color: rgb(0, 0, 0); letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">
<div dir="auto" style="caret-color: rgb(0, 0, 0); color: rgb(0, 0, 0); letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">
<div dir="auto" style="caret-color: rgb(0, 0, 0); color: rgb(0, 0, 0); letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">
<div dir="auto" style="caret-color: rgb(0, 0, 0); color: rgb(0, 0, 0); letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">
<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>
https://www.math.mcmaster.ca/bourdin | +1 (905) 525 9140 ext. 27243</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<br>
</div>
</body>
</html>