Curve resampling based high-quality high-order unstructured quadrilateral mesh generation

Abstract

High-order quadrilateral meshes offer superior accuracy and computational efficiency in numerical simulations. However, existing methods struggle to simultaneously preserve boundary/interface features, ensure high quality, and achieve efficient generation, particularly for complex geometries where degenerate and inverted elements frequently occur. To address this issue, this paper proposes a high-quality high-order unstructured quadrilateral mesh generation method based on geometric error-bounded curve reconstruction, which employs an indirect approach to enforce interface consistency. By optimization-based curve reconstruction strategies, our method improves mesh quality while maintaining the validity of high-order elements. Compared to direct high-order mesh optimization techniques, our approach reduces the optimization problem to curve reconstruction problem, significantly lowering computational complexity and enhancing efficiency. Experimental results demonstrate that the proposed method efficiently generates high-quality high-order quadrilateral meshes while preserving boundary/interface geometric features, offering improved adaptability and numerical stability in complex geometries.

Figures (26)

• Figure 1: Quadratic quadrilateral element representations. (b) Reference square and nodes; (a,c) the same element represented using Lagrange basis functions and Bernstein polynomials, respectively.
• Figure 2: Illustration of vector sets and their corresponding sectors. The sectors spanned by the orange vectors $\{\Delta_{ij}^{0}|0 \leq i,j \leq n\}$ and the blue vectors $\{\Delta_{ij}^{1}|0 \leq i,j \leq n\}$ are shown as orange and blue convex cones, respectively, on the right.
• Figure 3: Illustration of input and output. (a) An input region showing interfaces between subregions, and (b) output mesh with consistent interfaces.
• Figure 4: Algorithm pipeline. (a) Input curve; (b) curve reconstruction; (c) linear quadrilateral mesh; (d) high-order quadrilateral mesh. The gray points in (a) and (b) are the endpoints of the input curve, and the red points are the high-order control points.
• Figure 5: Curve approximation with geometric error control and adaptive refinement. (a) Input curves (47 segments), (b) bounded-error approximation result (34 segments), (c) curve reconstruction after adaptive refinement (72 segments), and (d) approximation error versus segment number across three stages for the input curve in (a), showing bounded error, where the geometric tolerance is set to $0.001$ times the bounding-box diagonal.

Theorems & Definitions (10)

• Lemma 1
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