Orthotropic maps for mesh generation and shape optimization – ORTHOMAP

Optimizing maps is a persistent challenge in computational mathematics, serving as a crucial component for generating high-quality quadrangular and hexahedral meshes essential for simulating highly anisotropic physical phenomena. Additionally, the significance of these maps has resurfaced in the realm of shape optimization, addressing the intricate "dehomogenization" problem. However, in both of these applications, the presence of shear in the mapping process can have catastrophic consequences, rendering a mesh unsuitable for finite element simulations or causing unexpected structural vulnerabilities.

To address these challenges, I propose an investigation into "orthotropic" maps. These maps ensure an absence of shear by restricting deformations to stretching along three orthogonal directions. Rather than relying on conventional soft constraints, as is sometimes done, I draw inspiration from the remarkable success of conformal maps in the domain of texture mapping. As a result, I introduce a infinitesimal characterization of orthotropic maps, establishing a direct connection between stretch directions and their corresponding stretch factors.

This project aims to investigate the effectiveness of this innovative approach for mesh generation and shape optimization. The study places particular emphasis on the computation of singular mappings which are often encountered in these applications and remain poorly understood, particularly in the volumetric case.