arXiv subjects

Maodong Pan

Publications and source records attributed to Maodong Pan.

4 recordsLinked to original sources

arXiv2026

P2M++: Enhanced Solver for Point-to-Mesh Distance Queries

Point-to-mesh distance queries are fundamental in computer graphics and geometric modeling. While the state-of-the-art P2M method achieves high-speed queries via Voronoi-based localization, it suffers from prohibitive precomputation costs. Its iterative Voronoi sweep for interference detection leads to redundant predicate evaluations and scales poorly on rotationally symmetric structures (e.g., spheres, cones or cylinders), where candidate counts grow quadratically. We propose P2M++ to address these limitations through three key contributions. First, we adaptively augment the set of mesh vertices with auxiliary sites in regions of high Voronoi vertex density to localize complex interference within minimal spatial regions. Second, we reformulate interference detection as a series of sphere-triangle collision tests centered at Voronoi cell corners, which are efficiently resolved using the base mesh's BVH. Finally, we enhance runtime performance by replacing the standard kd-tree search with a faster recursive dynamic programming implementation. Experimental results demonstrate that P2M++ is 3x-10x faster than the original P2M during preprocessing and 1.5x faster in queries, with even more pronounced gains on rotationally symmetric geometries.

cs.GR↗

arXiv2026

Element-based B-spline basis function spaces: construction and application in isogeometric analysis

This paper develops a unified theoretical framework for constructing B-spline basis function spaces with structural equivalence to finite element spaces. The theory rigorously establishes that these bases emerge as explicit linear combinations of B-spline element bases. For any prescribed smoothness requirements, this element-wise formulation enables the Hermite interpolation at nodes, which directly utilizes function values and derivatives without solving global linear systems. By focusing on explicit interpolation properties, element-wise analysis establishes optimal approximation errors, even when the space smoothness attains its theoretical maximum for the space degree. In isogeometric analysis (IgA), the construction naturally decomposes geometric mappings into element-level representations, allowing efficient computations across elements regardless of node distribution. Notably, the same Hermite interpolation framework simultaneously handles domain parameterization and IgA solutions, allowing direct imposition of boundary conditions through function and derivative matching. Numerical tests demonstrate optimal convergence rates and superconvergence properties in 2D IgA under uniform knot configurations, and improved computational efficiency in 3D IgA with non-uniform knot distributions.

math.NA↗

arXiv2019

Volumetric Spline Parameterization for Isogeometric Analysis

Given the spline representation of the boundary of a three dimensional domain, constructing a volumetric spline parameterization of the domain (i.e., a map from a unit cube to the domain) with the given boundary is a fundamental problem in isogeometric analysis. A good domain parameterization should satisfy the following criteria: (1) the parameterization is a bijective map; and (2) the map has lowest possible distortion. However, none of the state-of-the-art volumetric parameterization methods has fully addressed the above issues. In this paper, we propose a three-stage approach for constructing volumetric parameterization satisfying the above criteria. Firstly, a harmonic map is computed between a unit cube and the computational domain. Then a bijective map modeled by a max-min optimization problem is computed in a coarse-to-fine way, and an algorithm based on divide and conquer strategy is proposed to solve the optimization problem efficiently. Finally, to ensure high quality of the parameterization, the MIPS (Most Isometric Parameterizations) method is adopted to reduce the conformal distortion of the bijective map. We provide several examples to demonstrate the feasibility of our approach and to compare our approach with some state-of-the-art methods. The results show that our algorithm produces bijective parameterization with high quality even for complex domains.

cs.GR↗

arXiv2017

A Low-rank Spline Approximation of Planar Domains

Construction of spline surfaces from given boundary curves is one of the classical problems in computer aided geometric design, which regains much attention in isogeometric analysis in recent years and is called domain parameterization. However, for most of the state-of-the-art parameterization methods, the rank of the spline parameterization is usually large, which results in higher computational cost in solving numerical PDEs. In this paper, we propose a low-rank representation for the spline parameterization of planar domains using low-rank tensor approximation technique, and apply quasi-conformal map as the framework of the spline parameterization. Under given correspondence of boundary curves, a quasi-conformal map with low rank and low distortion between a unit square and the computational domain can be modeled as a non-linear optimization problem. We propose an efficient algorithm to compute the quasi-conformal map by solving two convex optimization problems alternatively. Experimental results show that our approach can produce a bijective and low-rank parametric spline representation of planar domains, which results in better performance than previous approaches in solving numerical PDEs.

cs.CG↗