Discretization of non-uniform rational B-spline (NURBS) models for meshless isogeometric analysis

TL;DR

The paper tackles the challenge of generating high-quality, quasi-uniform node sets on CAD geometries for meshless PDE solvers. It introduces NURBS-DIVG, which couples boundary sampling on NURBS CAD surfaces via a surface-adapted sDIVG with interior node filling via DIVG, including multi-patch handling and a supersampled inside/outside test. The approach yields node layouts with favorable local regularity and stability for RBF-FD discretizations of Poisson, Navier-Cauchy, and transient heat equations, demonstrated on complex CAD models. This work bridges meshless node generation and isogeometric CAD representations, enabling practical CAD-based PDE simulations in engineering contexts.

Abstract

We present an algorithm for fast generation of quasi-uniform and variable-spacing nodes on domains whose boundaries are represented as computer-aided design (CAD) models, more specifically non-uniform rational B-splines (NURBS). This new algorithm enables the solution of partial differential equations (PDEs) within the volumes enclosed by these CAD models using (collocation-based) meshless numerical discretizations. Our hierarchical algorithm first generates quasi-uniform node sets directly on the NURBS surfaces representing the domain boundary, then uses the NURBS representation in conjunction with the surface nodes to generate nodes within the volume enclosed by the NURBS surface. We provide evidence for the quality of these node sets by analyzing them in terms of local regularity and separation distances. Finally, we demonstrate that these node sets are well-suited (both in terms of accuracy and numerical stability) for meshless radial basis function generated finite differences (RBF-FD) discretizations of the Poisson, Navier-Cauchy, and heat equations. Our algorithm constitutes an important step in bridging the field of node generation for meshless discretizations with isogeometric analysis.

Figures (20)

• Figure 1: The DIVG expansion scheme (left) and the sDIVG mapping scheme (right).
• Figure 2: Node sets generated by NURBS-DIVG on the famous CAD Utah Teapot (left) and a CAD model of cat (right) based on catmodel. The Utah Teapot model is made of 32 patches and has 7031 boundary nodes; the cat has 211 patches and 3439 boundary nodes.
• Figure 3: Illustration of positioning nodes on a deformed sphere made of five NURBS patches. In the first step, the boundary of the first patch is filled (first), followed by filling of that patch interior (second). Once the first patch is processed, the boundary of the second patch is discretized (third); this process is repeated until all patches are fully populated with nodes (fourth).
• Figure 4: The average normalized distance to $c$ nearest neighbors (see equation Eq. \ref{['eq:dist_def']} for the precise definition) averaged over the whole domain, $\overline{d'}$, for a discretization of a successively subdivided Bezier curve in 2D and an analogous Bezier surface in 3D. In 2D $c = 2$ and in 3D $c = 3$ are used. Ideally, one strives for $\overline d' = 1$.
• Figure 5: Demonstration of the supersampling approach for the inside/outside test in NURBS-DIVG. The figure on the left shows nodes generated by the naive test used in the DIVG algorithm, with nodes escaping the domain boundary. The figure on the right shows the nodes generated using boundary supersampling in NURBS-DIVG; all non-boundary nodes are enclosed within the volume defined by the boundary.