An n-cross field is a locally defined orthogonal coordinate system invariant with respect to the hyperoctahedral symmetry group (cubic for n = 3). Cross fields are finding widespread use in mesh generation, computer graphics, and materials science among many applications. It was recently shown in [A. Chemin et al., International Meshing Roundtable, 2019, pp. 89-108] that 3-cross fields can be embedded into the set of symmetric fourth-order tensors. The concurrent work [D. Palmer, D. Bommes, and J. Solomon, Algebraic Representations for Volumetric Frame Fields, preprint, arXiv:1908.05411 (2019)] further develops a relaxation of this tensor field via a certain set of varieties. In this paper, we consider the problem of generating an arbitrary n-cross field using a fourth-order Q-tensor theory that is constructed out of tensored projection matrices. We establish that by a Ginzburg-Landau relaxation towards a global projection, one can reliably generate an n-cross field on arbitrary Lipschitz domains. Our work provides a rigorous approach that offers several new results including porting the tensor framework to arbitrary dimensions, providing a new relaxation method that embeds the problem into a global steepest descent, and offering a relaxation scheme for aligning the cross field with the boundary. Our approach is designed to fit within the classical Ginzburg-Landau PDE theory, offering a concrete road map for the future careful study of singularities of energy minimizers.
Bibliographical noteFunding Information:
∗Submitted to the journal’s Methods and Algorithms for Scientific Computing section September 17, 2019; accepted for publication (in revised form) June 2, 2021; published electronically September 21, 2021. https://doi.org/10.1137/19M1287857 Funding: The work of the first author was partially supported by NSF grant DMS-1615952. The work of the third author was partially supported by NSF grants DMS-1516565 and DMS-2009352. †Department of Mathematics, The University of Akron, Akron, OH 44325 USA (dmitry@uakron. edu). ‡Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, San Joaquín, Santiago, Chile (firstname.lastname@example.org). §School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA (email@example.com).
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- Cross field
- Frame field
- Hexahedral mesh