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

TL;DR

The paper tackles the challenge of achieving high-order mesh quality while guaranteeing mesh validity and CAD-free boundary handling in r-adaptivity. It extends TMOP to include tangential relaxation on curved surfaces using only the discrete mesh and enforces a positive Jacobian determinant through provable lower bounds, including shifted-barrier techniques. Key methodological advances include closest-point projection with Laplace blending for boundary relaxation, piecewise-linear Jacobian bounds derived from control points, and optional $q$-refinement to selectively raise quadrature order. The proposed framework is validated on turbine blade, untangling, and ALE simulations, demonstrating robust mesh improvements, preserved geometric fidelity, and guaranteed validity across challenging scenarios.

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.

Figures (10)

• Figure 1: Blade mesh example illustrating (a) initial high-order mesh and (b) mesh after optimization.
• Figure 2: Schematic representation of the major TMOP matrices.
• Figure 3: Bounding a 4th-order function using the approach in mittal2025generaldzanic2025method with (left) 6 control points and (right) 10 control points.
• Figure 4: (a) Inverted element from the blade example in Figure \ref{['fig:blade_example']}. (b) Jacobian determinant in the reference configuration. (c) Lower piecewise linear bound on the jacobian determinant with $8 \times 8$ control points. (d) Comparison of the lower bound using different number of control points and Bernstein polynomials with exact minimum.
• Figure 5: Impact of tangential relaxation near the leading edge of the blade after one $r$-adaptivity iteration. The blade surface for tangential relaxation is shown in red. (a) Initial mesh. (b) Optimized mesh with boundary nodes moved due to the solver update. (c) Boundary nodes projected back to the original mesh surface and corresponding displacement blended into the interior. (d) Final mesh (gray) and input mesh (black) shown together for comparison.