Scalable Field-Aligned Reparameterization for Trimmed NURBS

TL;DR

This work presents a semi-automatic and scalable reparameterization pipeline based on a scalable and feature-aligned meshing tool, QuadriFlow, and obtains a watertight spline surface with a simple quadrilateral layout.

Abstract

In engineering design, one of the most daunting problems in the design-through-analysis workflow is to deal with trimmed NURBS (Non-Uniform Rational B-Splines), which often involve topological/geometric issues and lead to inevitable gaps and overlaps in the model. Given the dominance of the trimming technology in CAD systems, reconstructing such a model as a watertight representation is highly desired. While remarkable progress has been made in recent years, especially with the advancement of isogeometric analysis (IGA), there still lack a fully automatic and scalable tool to achieve this reconstruction goal. To address this issue, we present a semi-automatic and scalable reparameterization pipeline based on a scalable and feature-aligned meshing tool, QuadriFlow [1]. On top of it, we provide support for open surfaces to deal with engineering shell structures, and perform sophisticated patch simplification to remove undesired tiny/slender patches. As a result, we obtain a watertight spline surface (multi-patch NURBS or unstructured splines) with a simple quadrilateral layout. Through several challenging models from industry applications, we demonstrate the efficacy and efficiency of the proposed pipeline as well as its integration with IGA. Our source code is publicly available on GitHub [2].

Figures (20)

• Figure 1: The proposed pipeline framework to reconstruct a given trimmed CAD model entirely as a watertight representation.
• Figure 2: Terminology illustration. $\textbf{v}_i$ and $\textbf{v}_j$ are two neighboring vertices in the triangle mesh, where gray lines indicate the uniform local grids in the tangent planes, and red arrows are unit normals $\textbf{n}_i$ and $\textbf{n}_j$. $\textbf{o}_i$ and $\textbf{o}_j$ are representative directions that can be matched in the same direction by rotation matrices $\textbf{k}_{ij}$ and $\textbf{k}_{ji}$. $\textbf{p}_i$ and $\textbf{p}_j$ are origins of local grids that can be made coincide on $\textbf{p}^*$ by integer translations $\textbf{t}_{ij}$ and $\textbf{t}_{ji}$, respectively. The integer offsets $\textbf{d}_{ij}$ is the "distance" from $\textbf{p}_i$ to $\textbf{p}_j$. $\rho$ is the user-defined grid spacing.
• Figure 3: The boundary constraint in the orientation field: the representative direction needs to align with the boundary tangent. When a boundary vertex is a corner (e.g., $\mathbf{v}_2$), its representative direction is prescribed along one of the two boundary edges (e.g., $\mathbf{o}_2^+$ or $\mathbf{o}_2^-$). For a non-corner boundary vertex (e.g., $\mathbf{v}_1$, $\mathbf{v}_3$), its representative direction is prescribed along the boundary tangent and it can only take two possible directions, e.g., $\textbf{o}_{1}^+$ or $\textbf{o}_{1}^-$ for $\mathbf{v}_1$. For interior vertices, their representative directions (e.g., $\textbf{o}_{4}$, $\textbf{o}_{5}$) are free and to be determined by optimization.
• Figure 4: Field matching of representative directions. (a) Inconsistent representative directions $\textbf{o}_i^*$ before field matching, and (b) consistent representative directions after field matching.
• Figure 5: Boundary constraints on the position field. (a) The lattice origin of a boundary vertex that is not a corner can only move along the boundary. (b) The lattice of an interior vertex near the boundary cannot go beyond the boundary; otherwise, the origin will be projected back onto the boundary.