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Enabling four-dimensional conformal hybrid meshing with cubic pyramids

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Abstract

The main purpose of this article is to develop a novel refinement strategy for four-dimensional hybrid meshes based on cubic pyramids. This optimal refinement strategy subdivides a given cubic pyramid into a conforming set of congruent cubic pyramids and invariant bipentatopes. The theoretical properties of the refinement strategy are rigorously analyzed and evaluated. In addition, a new class of fully symmetric quadrature rules with positive weights are generated for the cubic pyramid. These rules are capable of exactly integrating polynomials with degrees up to 12. Their effectiveness is successfully demonstrated on polynomial and transcendental functions. Broadly speaking, the refinement strategy and quadrature rules in this paper open new avenues for four-dimensional hybrid meshing, and space-time finite element methods.

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Code Availability

The Polyquad code is available at https://github.com/PyFR/Polyquad.

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Author information

Authors and Affiliations

  1. Department of Material Science and Mechanics of Materials, Technical University of Gabrovo, 5300, Gabrovo, Bulgaria

    Miroslav S. Petrov

  2. Department of Mathematics, Informatics and Natural Sciences, Technical University of Gabrovo, 5300, Gabrovo, Bulgaria

    Todor D. Todorov

  3. Department of Mechanical Engineering, The Pennsylvania State University, University Park, PA, 16802, USA

    Gage S. Walters & David M. Williams

  4. Department of Ocean Engineering, Texas A&M University, College Station, TX, 77843, USA

    Freddie D. Witherden

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  1. Miroslav S. Petrov
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  3. Gage S. Walters
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  5. Freddie D. Witherden
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Correspondence to David M. Williams.

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Appendices

Appendix: A. Supplemental material for Theorem 3

Transitional matrices and translation vectors related to the generic finite element transformations from Theorem 3.

$$ \begin{array}{@{}rcl@{}} A_{1}=\left( \begin{array}{cccc} 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right) B_{1}= \begin{pmatrix} \frac{1}{2}\\-\frac{1}{2}\\ \frac{1}{2}\\-\frac{3}{2} \end{pmatrix}, \qquad A_{2}=\left( \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right) B_{2}= \begin{pmatrix} -\frac{1}{2}\\-\frac{1}{2}\\ \frac{1}{2}\\-\frac{3}{2} \end{pmatrix}, \end{array} $$
$$ \begin{array}{@{}rcl@{}} A_{3}=\left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right) B_{3}= \begin{pmatrix} -\frac{1}{2}\\-\frac{1}{2}\\-\frac{1}{2}\\-\frac{3}{2} \end{pmatrix}, \qquad A_{4}=\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right) B_{4}= \begin{pmatrix} \frac{1}{2}\\-\frac{1}{2}\\ \frac{1}{2}\\-\frac{3}{2} \end{pmatrix}, \end{array} $$
$$ \begin{array}{@{}rcl@{}} A_{5}=\left( \begin{array}{cccc} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right) B_{5}= \begin{pmatrix} \frac{1}{2}\\-\frac{1}{2}\\ \frac{1}{2}\\-\frac{3}{2} \end{pmatrix}, \qquad A_{6}=\left( \begin{array}{cccc} 0 & 0 & -1 & 0 \\ 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right) B_{6}= \begin{pmatrix} \frac{1}{2}\\ \frac{1}{2}\\ -\frac{1}{2}\\ -\frac{3}{2} \end{pmatrix}, \end{array} $$
$$ \begin{array}{@{}rcl@{}} A_{7}=\left( \begin{array}{cccc} 0 & 0 & -1 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right) B_{7} = \begin{pmatrix} \frac{1}{2}\\ \frac{1}{2}\\ \frac{1}{2}\\-\frac{3}{2} \end{pmatrix}, \qquad A_{8}=\left( \begin{array}{cccc} -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right) B_{8}= \begin{pmatrix} -\frac{1}{2}\\\ \frac{1}{2}\\ \frac{1}{2}\\-\frac{3}{2} \end{pmatrix}, \end{array} $$
$$ \begin{array}{@{}rcl@{}} A_{9}=\left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right) B_{9}= \begin{pmatrix} -\frac{1}{2}\\ \frac{1}{2}\\ \frac{1}{2}\\-\frac{3}{2} \end{pmatrix}, \qquad A_{10} = \left( \begin{array}{cccc} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right) B_{10} = \begin{pmatrix} \frac{1}{2}\\ \frac{1}{2}\\ -\frac{1}{2}\\ -\frac{3}{2} \end{pmatrix}, \end{array} $$
$$ \begin{array}{@{}rcl@{}} A_{11}=\left( \begin{array}{cccc} 0 & 0 & -1 & 0 \\ 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right) B_{11} = \begin{pmatrix} -\frac{1}{2}\\ \frac{1}{2}\\ \frac{1}{2}\\-\frac{3}{2} \end{pmatrix}, \qquad A_{12}=\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right) B_{12} = \begin{pmatrix} -\frac{1}{2}\\ \frac{1}{2}\\-\frac{1}{2}\\-\frac{3}{2} \end{pmatrix}, \end{array} $$
$$ \begin{array}{@{}rcl@{}} A_{13}&=&\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right), A_{14}=\left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right), A_{15}=\left( \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right),\\ A_{16}&=&\left( \begin{array}{cccc} 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right), \end{array} $$
$$ \begin{array}{@{}rcl@{}} A_{17}=\left( \begin{array}{cccc} 0 & 0 & -1 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right), A_{18}=\left( \begin{array}{cccc} 0 & 0 & -1 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right), B_{i}= \begin{pmatrix} 0\\ 0\\0\\0 \end{pmatrix}, i=13,14,\ldots,18. \end{array} $$

Appendix: B. Supplemental material for Theorem 4

Transitional matrices and translation vectors related to the generic finite element transformations from Theorem 4.

$$ \begin{array}{@{}rcl@{}} \tilde{A}_{1}=\frac{1}{2}\left( \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \tilde{B}_{1}=\frac{1}{4}\begin{pmatrix}-1\\1\\-1\\-1\\ \end{pmatrix}, \qquad \tilde{A}_{2}=\frac{1}{2}\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \tilde{B}_{2}=\frac{1}{4}\begin{pmatrix}-1\\1\\1\\-1\\ \end{pmatrix}, \end{array} $$
$$ \begin{array}{@{}rcl@{}} \tilde{A}_{3}=\frac{1}{2}\left( \begin{array}{cccc} 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right) \tilde{B}_{3}=\frac{1}{4}\begin{pmatrix}2\\2\\0\\-4\\ \end{pmatrix}, \qquad \tilde{A}_{4}=\frac{1}{2}\left( \begin{array}{cccc} -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right) \tilde{B}_{4}=\frac{1}{4}\begin{pmatrix}0\\2\\2\\-4\\ \end{pmatrix}, \end{array} $$
$$ \begin{array}{@{}rcl@{}} \tilde{A}_{5}=\frac{1}{2}\left( \begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array} \right) \tilde{B}_{5}=\frac{1}{4}\begin{pmatrix}2\\2\\0\\-4\\ \end{pmatrix}, \qquad \tilde{A}_{6}=\frac{1}{2}\left( \begin{array}{cccc} 0 & 0 & -1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 1 & 0 & 0 \end{array} \right) \tilde{B}_{6}=\frac{1}{4}\begin{pmatrix}0\\2\\-2\\-4\\ \end{pmatrix}, \end{array} $$
$$ \begin{array}{@{}rcl@{}} \tilde{A}_{7}=\frac{1}{2}\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right) \tilde{B}_{7}=\frac{1}{4}\begin{pmatrix}0\\2\\-2\\-4\\ \end{pmatrix}, \qquad \tilde{A}_{8}=\frac{1}{2}\left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right) \tilde{B}_{8}=\frac{1}{4}\begin{pmatrix}-2\\2\\0\\-4\\ \end{pmatrix}, \end{array} $$
$$ \begin{array}{@{}rcl@{}} \tilde{A}_{9}=\frac{1}{2}\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array} \right) \tilde{B}_{9}=\frac{1}{4}\begin{pmatrix}0\\4\\0\\-4\\ \end{pmatrix}, \qquad \tilde{A}_{10}=\frac{1}{2}\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array} \right) \tilde{B}_{10}=\frac{1}{4}\begin{pmatrix}0\\0\\0\\-4\\ \end{pmatrix}, \end{array} $$
$$ \begin{array}{@{}rcl@{}} \tilde{A}_{11}=\frac{1}{2}\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \tilde{B}_{11}=\frac{1}{4}\begin{pmatrix}0\\2\\0\\-2\\ \end{pmatrix}, \qquad \tilde{A}_{12}=\frac{1}{2}\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \tilde{B}_{12}=\frac{1}{4}\begin{pmatrix}0\\2\\0\\-2\\ \end{pmatrix}, \end{array} $$
$$ \begin{array}{@{}rcl@{}} \tilde{A}_{13}=\frac{1}{2}\left( \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \end{array} \right) \tilde{B}_{13}=\frac{1}{4}\begin{pmatrix}0\\2\\2\\-4\\ \end{pmatrix}, \qquad \tilde{A}_{14}=\frac{1}{2}\left( \begin{array}{cccc} 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array} \right) \tilde{B}_{14}=\frac{1}{4}\begin{pmatrix}-2\\2\\0\\-4\\ \end{pmatrix}, \end{array} $$
$$ \begin{array}{@{}rcl@{}} \tilde{A}_{15}=\frac{1}{2}\left( \begin{array}{cccc} 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \tilde{B}_{15}=\frac{1}{4}\begin{pmatrix}1\\1\\1\\-1\\ \end{pmatrix}, \qquad A_{16}=\frac{1}{2}\left( \begin{array}{cccc} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) B_{16}=\frac{1}{4}\begin{pmatrix}1\\1\\-1\\-1\\ \end{pmatrix}. \end{array} $$

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Petrov, M.S., Todorov, T.D., Walters, G.S. et al. Enabling four-dimensional conformal hybrid meshing with cubic pyramids. Numer Algor 91, 671–709 (2022). https://doi.org/10.1007/s11075-022-01278-y

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