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

## TL;DR

This paper presents a new high-order surface reconstruction method that improves accuracy and robustness for CAD-free modeling, especially near sharp features, by extending existing techniques with Hermite-style approximation and feature-aware parameterization.

## Contribution

It introduces a Hermite-style least squares approach combined with iterative feature-aware parameterization for high-order, robust surface reconstruction from triangulated meshes.

## Key findings

- Achieves fourth and higher-order accuracy with compact support.
- Ensures high-order G0 continuity near sharp features.
- Provides nearly identical PDE solutions using reconstructed and exact geometries.

## 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.