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Thank you for the clarification. Are there references specifically for this tabulation method and its construction? I have seen some references about the "FIAT" algorithm, but from a quick look I could not find all details. ---On a related
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<div style="font-family: Arial, sans-serif; font-size: 14px;">Thank you for the clarification.</div><div style="font-family: Arial, sans-serif; font-size: 14px;"><br></div><div style="font-family: Arial, sans-serif; font-size: 14px;">Are there references specifically for this tabulation method and its construction? I have seen some references about the "FIAT" algorithm, but from a quick look I could not find all details.<br></div><div style="font-family: Arial, sans-serif; font-size: 14px;"><br></div><div style="font-family: Arial, sans-serif; font-size: 14px;">---</div><div style="font-family: Arial, sans-serif; font-size: 14px;"><br></div><div style="font-family: Arial, sans-serif; font-size: 14px;">On a related note, I stated the values of Nq, Nc and Nb, as they can be checked. But to be sure; for the given 2D example:</div><div style="font-family: Arial, sans-serif; font-size: 14px;"><ul data-editing-info="{"orderedStyleType":1,"unorderedStyleType":2}"><li style="list-style-type: "- ";"><span>Nc = 2 refers to the two compoents as in x/y in 2D<br></span></li></ul></div><div style="font-family: Arial, sans-serif; font-size: 14px;"><ul data-editing-info="{"orderedStyleType":1,"unorderedStyleType":2}"><li style="list-style-type: "- ";"><span>Nb = 3 * 2 i.e. 3 shape functions (or nodes) times 2 components</span></li></ul><div><br></div><div>Testing with a 3D mesh (e.g. a 4-node linear tetrahedron), Nc = 3 and Nb = 12, so the same math seems to work, but perhaps there is a different idea behind it.</div><div><br></div><div>Thanks.<br></div></div><div style="font-family: Arial, sans-serif; font-size: 14px;">Noam<br></div><div class="protonmail_quote">
On Tuesday, March 26th, 2024 at 11:17 PM, Matthew Knepley <knepley@gmail.com> wrote:<br>
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<div dir="ltr"><div dir="ltr">On Tue, Mar 26, 2024 at 2:23 PM Noam T. via petsc-users <<a target="_blank" rel="noreferrer nofollow noopener" href="mailto:petsc-users@mcs.anl.gov">petsc-users@mcs.anl.gov</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 class="msg6357488074775657635">
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Hello, I am trying to understand the FE Tabulation data obtained from e. g . PetscFEComputeTabulation. Using a 2D mesh with a single triangle, first order, with vertices (0,0), (0,1), (1,0) (see msh file attached), and a single quadrature point
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<div style="font-family:Arial,sans-serif;font-size:14px">Hello,</div><div style="font-family:Arial,sans-serif;font-size:14px"><br></div><div style="font-family:Arial,sans-serif;font-size:14px">I am trying to understand the FE Tabulation data obtained from e.g . PetscFEComputeTabulation. Using a 2D mesh with a single triangle, first order, with vertices (0,0), (0,1), (1,0) (see msh file attached), and a single quadrature point at (1/3, 1/3), one gets Nb = 6, Nc = 2, Nq = 1, and the arrays for the basis and first derivatives are of sizes [Nq x Nb x Nc] = 12 and[Nq x Nb x Nc x dim] = 24, respectively</div></div></blockquote><div><br></div><div>The tabulations from PetscFE are recorded on the reference cell. For triangles, the reference cell is</div><div>(-1, -1) -- (1, -1) -- (-1, 1). The linear basis functions at these nodes are</div><div><br></div><div>phi_0: -(x <a target="_blank" rel="noopener noreferrer" class="gmail_plusreply" id="plusReplyChip-1">+</a> y) / 2</div><div>phi_1: (x + 1) / 2</div><div>phi_2: (y + 1) / 2</div><div><br></div><div>and then you use the tensor product for Nc = 2.</div><div><br></div><div>/ phi_0 \ / 0 \ etc.</div><div>\ 0 / \ phi_0 /</div><div><br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div class="msg6357488074775657635"><div style="font-family:Arial,sans-serif;font-size:14px">The values of these two arrays are:</div><div style="font-family:Arial,sans-serif;font-size:14px">basis (T->T[0])<br></div><div style="font-family:Arial,sans-serif;font-size:14px">[-1/3, 0, 0, -1/3, 2/3, 0, </div><div style="font-family:Arial,sans-serif;font-size:14px"> 0, 2/3, 2/3, 0, 0, 2/3]</div></div></blockquote><div><br></div><div>So these values are indeed the evaluations of those basis functions at (1/3, 1/3). The derivatives are similar.</div><div><br></div><div>These are the evaluations you want if you are integrating in reference space, as we do for the finite element integrals, and also the only way we could use a single tabulation for the mesh.</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 class="msg6357488074775657635"><div style="font-family:Arial,sans-serif;font-size:14px">deriv (T->T[1])<br></div><div style="font-family:Arial,sans-serif;font-size:14px">[-1/2, -1/2, 0, 0, 0, 0, </div><div style="font-family:Arial,sans-serif;font-size:14px"> -1/2, -1/2, 1/2, 0, 0, 0, </div><div style="font-family:Arial,sans-serif;font-size:14px"> 0, 0, 1/2, 0, 0, 1/2,</div><div style="font-family:Arial,sans-serif;font-size:14px"> 0, 0, 0, 0, 0, 1/2] <br></div><div style="font-family:Arial,sans-serif;font-size:14px"><br></div><div style="font-family:Arial,sans-serif;font-size:14px">How does one get these values? I can't quite find a way to relate them to evaluating the basis functions of a P1 triangle in the given quadrature point.<br></div><div style="font-family:Arial,sans-serif;font-size:14px"><br></div><div style="font-family:Arial,sans-serif;font-size:14px">Thanks,</div><div style="font-family:Arial,sans-serif;font-size:14px">Noam<br></div><div style="font-family:Arial,sans-serif;font-size:14px"><br></div><div style="font-family:Arial,sans-serif;font-size:14px"><br></div></div></blockquote></div><br clear="all"><div><br></div><span class="gmail_signature_prefix">-- </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 rel="noreferrer nofollow noopener" href="https://urldefense.us/v3/__http://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!d_p8b8_VRd0sM4TjKw7CDB2HbOn9JfMTaOrTqoazk2UvhfJwWRwR8i6S7cmZrXu_arDmY1F-Oyq1J5v30G8ZxjPmwkXUbNgo$" target="_blank">https://www.cse.buffalo.edu/~knepley/</a><br></div></div></div></div></div></div></div></div>
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