2019/09/02 by Dmitry Golovaty, Golovaty, Dmitry, José Alberto Montero +3 · 1 citation
Engineering · Mathematics · Physics and Astronomy · #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #Computational Geometry (cs.CG) #Electromagnetic Scattering and Analysis #FOS: Computer and information sciences #FOS: Mathematics #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.1909.00922
openalex publication_date 2019/09/02 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
An n-cross field is a locally-defined orthogonal coordinate system\ninvariant with respect to the cubic symmetry group. Cross fields are finding\nwide-spread use in mesh generation, computer graphics, and materials science\namong many applications. It was recently by other authors that 3-cross fields\ncan be embedded into the set of symmetric 4th-order tensors. Another\nconcurrent work further develops a relaxation of this tensor field via a\ncertain set of varieties. In this paper, we consider the problem of generating\nan arbitrary n-cross field using a fourth-order Q-tensor theory that is\nconstructed out of tensored projection matrices. We establish that by a\nGinzburg-Landau relaxation towards a global projection, one can reliably\ngenerate an n-cross field on arbitrary Lipschitz domains. Our work provides a\nrigorous approach that offers several new results including porting the tensor\nframework to arbitrary dimensions, providing a new relaxation method that\nembeds the problem into a global steepest descent, and offering a relaxation\nscheme for aligning the cross field with the boundary. Our approach is designed\nto fit within the classical Ginzburg-Landau PDE theory, offering a concrete\nroad map for the future careful study of singularities of energy minimizers.\n