High-Order Mesh r-Adaptivity with Tangential Relaxation and Guaranteed Mesh Validity

TL;DR

This paper extends high-order mesh r-adaptivity techniques by introducing tangential relaxation on curved surfaces without geometric access and ensuring mesh validity through positive Jacobian determinants, enhancing robustness for simulations.

Contribution

It introduces a novel tangential relaxation method on curved surfaces using only mesh data and guarantees mesh validity by maintaining positive Jacobian determinants.

Findings

Robust high-order mesh adaptation demonstrated in various numerical experiments.

Mesh validity ensured by continuous Jacobian positivity.

Effective tangential relaxation without geometric model access.

Abstract

High-order meshes are crucial for achieving optimal convergence rates in curvilinear domains, preserving symmetry, and aligning with key flow features in moving mesh simulations, but their quality is challenging to control. In prior work, we have developed techniques based on Target-Matrix Optimization Paradigm (TMOP) to adapt a given high-order mesh to the geometry and solution of the partial differential equation (PDE). Here, we extend this framework to address two key gaps in the literature for high-order mesh r-adaptivity. First, we introduce tangential relaxation on curved surfaces using solely the discrete mesh representation, eliminating the need for access to underlying geometry (e.g., CAD model). Second, we ensure a continuously positive Jacobian determinant throughout the domain. This determinant positivity is essential for using the high-order mesh resulting from r-adaptivity with arbitrary quadrature schemes in simulations. The proposed approach is demonstrated to be robust using a variety of numerical experiments.