arXiv subjects

Kendrick M. Shepherd

Publications and source records attributed to Kendrick M. Shepherd.

5 recordsLinked to original sources

arXiv2026

Combinatorial maps for hierarchical splines

Hierarchical splines are an important part of multiscale and adaptive isogeometric analysis formulations. The Bézier meshes of these splines are an essential part of their definition and of several important hierarchical spline algorithms, such as adaptive refinement and Bézier extraction. Topological data associated with the Bézier mesh-such as adjacency information-can be used to improve the performance of many of these algorithms as well as downstream applications of the splines, but typical hierarchical spline formulations do not compute the topological data, storing instead just a list of elements. In this work we present algorithms to build a performant topological data structure, namely the combinatorial map, to represent Bézier meshes of hierarchical splines over cubical cell complexes where the refinement levels have conforming Bézier meshes. This includes hierarchical and truncated hierarchical B-splines, as well as subsets of other hierarchical spline formulations. We show the performance characteristics of the construction algorithms of these hierarchical combinatorial maps, as well as an example use case, showing that the topological information can provide up to an order of magnitude reduction in computation time in downstream applications of the splines.

cs.CG↗

arXiv2026

Construction of exact refinements for the two-dimensional hierarchical B-spline de Rham complex

The de Rham complex arises naturally when studying problems in electromagnetism and fluid mechanics. Stable numerical methods to solve these problems can be obtained by using a discrete de Rham complex that preserves the structure of the continuous one. This property is not necessarily guaranteed when the discrete function spaces are hierarchical B-splines, and research shows that an arbitrary choice of refinement domains may give rise to spurious harmonic fields that ruin the accuracy of the solution. We will focus on the two-dimensional de Rham complex over the unit square $Ω\subseteq \mathbb{R}^2$, and provide theoretical results and a constructive algorithm to ensure that the structure of the complex is preserved: when a pair of functions are in conflict some additional functions, forming an L-chain between the pair, are also refined. Another crucial aspect to consider in the hierarchical setting is the notion of admissibility, as it is possible to obtain optimal convergence rates of numerical solutions and improved stability by limiting the multi-level interaction of basis functions. We show that, under a common restriction, the admissibility class of the first space of the discrete complex persists throughout the remaining spaces. As such, admissible refinement can be combined with our new algorithm to obtain admissible meshes that also respect the structure of the de Rham complex. Moreover, we detail how our algorithm can be easily included in standard adaptive mesh refinement schemes. Finally, we include numerical results that motivate the importance of the previous concerns for the vector Laplace and Maxwell eigenvalue problems.

math.NA↗

arXiv2025

Counterexamples to Proofs for Volumetric Parameterization of Topological Sweeps

Harmonic maps are important in generating parameterizations for various domains, particularly in two and three dimensions. General extensions of two-dimensional harmonic parameterizations for volumetric parameterizations are known to fail in a variety of contexts, though more specialized volumetric parameterizations have been proposed. This work provides and contextualizes a counterexample to various proposed proofs that employ harmonic maps to sweep a parameterization from a base surface, $Γ_0$, to the entire domain of a geometry that is homeomorphic to $Γ_0\times[0,1]$ or $Γ_0\times S^1$. While this does not negate the potential value of such topological sweep parameterizations, it does clarify that these swept parameterizations come with no inherent guarantees of bijectivity, as they may in two dimensions.

cs.CG↗

arXiv2025

Tusqh: Topological Control of Volume-Fraction Meshes Near Small Features and Dirty Geometry

This work develops a framework to create meshes with user-specified homology from potentially dirty geometry by coupling background grids, persistent homology, and a generalization of volume fractions. For a mesh with fixed grid size, the topology of the output mesh changes predictably and monotonically as its volume-fraction threshold decreases. Topological anti-aliasing methods are introduced to resolve pinch points and disconnected regions that are artifacts of user choice of grid size and orientation, making the output meshes suitable for downstream processes including analysis. The methodology is demonstrated on geographical, mechanical, and graphics models in 2D and 3D using a custom-made software called Tusqh. The work demonstrates that the proposed framework is viable for generating meshes on topologically invalid geometries and for automatic defeaturing of small geometric artifacts. Finally, the work shows that although subdividing the background grid frequently improves the topological and geometrical fidelity of the output mesh, there are simple 2D examples for which the topology does not converge under refinement for volume-fraction codes.

cs.CG↗

arXiv2020

The Quad Layout Immersion: A Mathematically Equivalent Representation of a Surface Quadrilateral Layout

Quadrilateral layouts on surfaces are valuable in texture mapping, and essential in generation of quadrilateral meshes and in fitting splines. Previous work has characterized such layouts as a special metric on a surface or as a meromorphic quartic differential with finite trajectories. In this work, a surface quadrilateral layout is alternatively characterized as a special immersion of a cut representation of the surface into the Euclidean plane. We call this a quad layout immersion. This characterization, while posed in smooth topology, naturally generalizes to piecewise-linear representations. As such, it mathematically describes and generalizes integer grid maps, which are common in computer graphics settings. Finally, the utility of the representation is demonstrated by computationally extracting quadrilateral layouts on surfaces of interest.

cs.CG↗