arXiv subjects

Jose Alberto Montero

Publications and source records attributed to Jose Alberto Montero.

3 recordsLinked to original sources

arXiv2023

Tetrahedral frame fields via constrained third order symmetric tensors

Tetrahedral frame fields have applications to certain classes of nematic liquid crystals and frustrated media. We consider the problem of constructing a tetrahedral frame field in three dimensional domains in which the boundary normal vector is included in the frame on the boundary. To do this we identify an isomorphism between a given tetrahedral frame and a symmetric, traceless third order tensor under a particular nonlinear constraint. We then define a Ginzburg-Landau-type functional which penalizes the associated nonlinear constraint. Using gradient descent, one retrieves a globally defined limiting tensor outside of a singular set. The tetrahedral frame can then be recovered from this tensor by a determinant maximization method, developed in this work. The resulting numerically generated frame fields are smooth outside of one dimensional filaments that join together at triple junctions.

math.AP↗

arXiv2021

Refined Asymptotics for Landau-de Gennes Minimizers on Planar Domains

In our previous work,, we studied asymptotic behavior of minimizers of the Landau-de Gennes energy functional on planar domains as the nematic correlation length converges to zero. Here we improve upon those results, in particular by sharpening the description of the limiting map of the minimizers. We also provide an expression for the energy valid for a small, but fixed value of the nematic correlation length.

math.AP↗

arXiv2020

A variational method for generating $n$-cross fields using higher-order $Q$-tensors

An $n$-cross field is a locally-defined orthogonal coordinate system invariant with respect to the cubic symmetry group. Cross fields are finding wide-spread use in mesh generation, computer graphics, and materials science among many applications. It was recently by other authors that $3$-cross fields can be embedded into the set of symmetric $4$th-order tensors. Another concurrent work 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.

math.AP↗