Compact Feature-Aware Hermite-Style High-Order Surface Reconstruction

TL;DR

This work addresses high-order surface reconstruction from triangulated meshes in the absence of CAD models, targeting accurate and stable geometry for high-order PDE discretizations. It extends Continuous Moving Frames (CMF) and Weighted Averaging of Local Fittings (WALF) by introducing Hermite-style polynomial fittings that incorporate normals, enabling compact stencils and fourth- to higher-order accuracy, along with an Iterative Feature-Aware (IFA) parameterization to guarantee $G^{0}$ continuity near sharp features. The key contributions include the formulations of H-CMF and H-WALF, rigorous accuracy analyses (noting $O(h^{p+1})$ for H-CMF and $O(h^{p+1})+O(h^{6})$ for H-WALF), adaptive stencil and weighting strategies using Wendland functions, and the IFA framework that maintains high-order convergence near features for both surfaces and curves; numerical results show superconvergence for even degrees and strong FEM performance with high-order reconstructions. Practically, the approach provides a robust, CAD-free alternative for generating high-order geometry suitable for high-order FEM, enabling near-exact PDE solutions and flexible meshing in scenarios where CAD models are unavailable or impractical.

Abstract

High-order surface reconstruction is an important technique for CAD-free, mesh-based geometric and physical modeling, and for high-order numerical methods for solving partial differential equations (PDEs) in engineering applications. In this paper, we introduce a novel method for accurate and robust reconstructions of piecewise smooth surfaces from a triangulated surface. Our proposed method extends the Continuous Moving Frames (CMF) and the Weighted Averaging of Local Fittings (WALF) methods (Engrg. Comput. 28 (2012)) in two main aspects. First, it utilizes a Hermite-style least squares approximation to achieve fourth and higher-order accuracy with compact support, even if the input mesh is relatively coarse. Second, it introduces an iterative feature-aware parameterization to ensure high-order accurate, G0 continuous reconstructions near sharp features. We present the theoretical framework of the method and compare it against CMF and WALF in terms of accuracy and stability. We also demonstrate that the use of the Hermite-style reconstruction in the solutions of PDEs using finite element methods (FEM), and show that quartic and sextic FEMs using the high-order reconstructed surfaces produce nearly identical results as using exact geometry while providing additional flexibility.

Figures (16)

• Figure 1: Illustration of $k$-ring and $k.5$-ring neighborhoods for stencil selections.
• Figure 2: Wendland's functions before and after scaling.
• Figure 3: 2-D illustration of WALF. The solid curve indicates the exact curve. The dashed and dotted curves indicate the fittings at $\boldsymbol x_{1}$ and $\boldsymbol x_{2}$, respectively. $\boldsymbol q$ is the WALF reconstruction for point $\boldsymbol p$, computed as a weighted average of $\boldsymbol q_{1}$ and $\boldsymbol q_{2}$ from the fittings at $\boldsymbol x_{1}$ and $\boldsymbol x_{2}$, respectively.
• Figure 4: Parametric triangular elements with equally-spaced points in the reference space.
• Figure 5: Parametric triangular elements with Lebesgue-Gauss-Lobatto points.

Theorems & Definitions (8)

• Lemma 1
• Proposition 2
• Proposition 3
• Lemma 4
• Proposition 5
• Proposition 6
• Definition 7
• Theorem 8