Schema-Aware Single-Mesh Generation

• Schema-aware single-mesh generation is a technique that explicitly integrates topological schemas into a single mesh structure, ensuring accurate representation of high-genus and multi-feature domains.
• It employs advanced local refinement operators, including vertex-split and hybrid splits, to balance mesh growth and geometric fidelity while managing complex topological invariants.
• The approach leverages SPH-based feature encoding for simultaneous curve, surface, and volume discretization, significantly enhancing simulation and mapping in computational geometry.

Schema-aware single-mesh generation refers to geometric mesh construction techniques that explicitly encode topological schemas—such as canonical polygonal schemas or feature-based domain decompositions—directly into the mesh structure or its underlying sampling algorithm. These approaches are foundational in scenarios where parameterization, simulation, or discretization must respect topological invariants or feature hierarchies (e.g., handles in surfaces of genus $g$, singular feature curves, or volumetric features). Two primary strands of research exemplify schema-aware mesh generation: the canonical polygonal schema extraction from high-genus surfaces with minimal mesh growth (Livesu, 2020), and feature-aware smoothed-particle hydrodynamics (SPH) for isotropic unstructured mesh generation (Ji et al., 2020). Both paradigms address the challenge of coordinating topology, geometry, and element quality in automated mesh generation workflows.

1. Formal Topological and Algorithmic Foundations

Given a closed, orientable 2-manifold $M$ of genus $g$, the canonical polygonal schema corresponds to a 2D $4g$-gon derived by cutting $M$ along a system of $2g$ non-contractible loops, each based at a common origin $p$ and otherwise pairwise disjoint. In mesh terms, $M$ is represented as a triangulation $(V, E, F)$ with $n_v=|V|$ vertices, $M$0 edges, $M$1 faces, and Euler characteristic $M$2. A cut-graph is a minimal set of edges whose removal renders $M$3 simply connected (a disk). The canonical schema's sides correspond to a homotopically independent loop system $M$4, with only basepoint intersection at $M$5.

The polynomial-time greedy homotopy basis algorithm of Erickson & Whittlesey constructs these loops by selecting $M$6 cycles formed from non-tree edges of a shortest-path tree $M$7 rooted at $M$8, but, critically, the resulting loops may overlap except at $M$9 (Livesu, 2020). Therefore, schema-aware single-mesh extraction requires postprocessing to physically separate these loops in the mesh, creating a true $g$0-gon cut.

For feature-aware unstructured mesh generation, mesh-vertices are modeled as SPH particles whose evolution is governed by isothermal compressible Lagrangian equations, subject to boundary correction terms that enable simultaneous curve, surface, and volume feature handling without explicit sequencing (Ji et al., 2020). This unifies multi-dimensional meshing and preserves topological consistency across feature hierarchies.

2. Local Mesh-Refinement Operators for Topological Separation

Schema-aware single-mesh canonical polygonal schema realization demands localized mesh refinement to ensure all loops are disjoint except at the origin. Three primary mesh-splitting primitives enable this detachment:

Operator	Growth per Application	Geometric Deviation
Edge split	$g$1, $g$2	None
Vertex split	$g$3, $g$4	Possible; none only if local fan planar
Triangle split	$g$5, $g$6	Possible; limited use
Hybrid split (novel)	Min$g$7 (planarity conditioned)	Bounded by planarity $g$8 parameter

The edge split (Li et al. 2008) preserves geometric fidelity but can induce super-quadratic mesh growth in genus $g$9, making this operator prohibitive for extreme topology. The vertex split, introduced to minimize mesh complexity, duplicates a merging vertex and reattaches loop bundles with only two extra triangles per event, limiting mesh growth to $4g$0 even up to genus $4g$1 (Livesu, 2020). The triangle split is rarely applicable in practice. The hybrid operator adaptively chooses between vertex- and edge-splits based on a planarity test (maximal inter-fan triangle normal angle), guaranteeing deviation from the input surface is bounded by a threshold $4g$2 (e.g., $4g$3), thereby providing a principled tradeoff between mesh complexity and geometric faithfulness.

3. Algorithmic Pipeline and Complexity Control

The schema-aware meshing pipeline proceeds as follows:

1. Given a triangulated mesh $4g$4 and genus $4g$5, compute a greedy homotopy basis of $4g$6 loops $4g$7 rooted at $4g$8 using the algorithm of Erickson & Whittlesey, with $4g$9 time at a single basepoint or $M$0 for all-points search.
2. Identify all merging vertices $M$1—locations (other than $M$2) where at least two loops share an edge.
3. Place all $M$3 in a FIFO queue $M$4.
4. While $M$5 is nonempty, pop $M$6, apply the appropriate local split operator to separate a pair of merging loops, and, if the new merge point $M$7, push $M$8 into $M$9.
5. Continue until all loops meet only at $2g$0; cut along the now-disjoint system to obtain the canonical $2g$1-gon schema.

The per-merge-event complexity is bounded by $2g$2, and the total number of events is $2g$3, $2g$4 the average loop length. For practical meshes, this yields near-linear total complexity in input size after greedy basis extraction, and empirical mesh growth remains modest using vertex-split or hybrid (Livesu, 2020).

In SPH-based feature-aware meshing (Ji et al., 2020), initialization involves feature encoding (cell tags for curves, surfaces, volumes), computing target densities, and concurrent evolution of particles via two-phase (feature-size adaptation, mesh-quality optimization) Lagrangian updates subject to boundary correction. This approach obviates the need for sequence-ordered meshing (surface→volume), enabling single-pass multi-feature discretization.

4. Experimental Validation and Performance Metrics

Empirical studies on 12 meshes with $2g$5 ranging from $2g$6 to $2g$7 and up to $2g$8 vertices corroborate the theoretical analysis with respect to mesh complexity and geometric deviation. Exclusive use of edge splits yields pronounced super-quadratic mesh growth (up to $2g$9 vertex increase, e.g., $p$0k→$p$1M for a genus-245 model). In contrast, exclusive vertex-split and hybrid-split remain substantially sublinear ($p$2 and $p$3 for $p$4) and consistently limit geometric error (average Hausdorff distance $p$5) (Livesu, 2020).

SPH-based feature-aware meshing demonstrates significant acceleration over previous variational and centroidal methods: speedups of $p$6 to $p$7 observed across complex 2D/3D cases, with no deterioration (often slight improvement) in mesh element quality, as measured by geometric regularity ($p$8 ratio for triangles, dihedral angles and radius ratio for tets) (Ji et al., 2020). Convergence criteria based on normalized total volume and pressure residuals guarantee rapid adaptation and global improvement in element isotropy.

5. Implementation Guidelines and Numerical Considerations

Effective implementation of schema-aware single-mesh methods entails explicit half-edge or corner-table data structures to allow constant-time access to one-rings and rapid local modification. Robust planarity testing for hybrid splits requires precomputing triangle normals and thresholding maximal angles (default $p$9). Vertex-split positioning should use careful interpolation $M$0; $M$1 reduced to prevent triangle flipping, and ultimate fallback to edge-split to avoid degeneracy. Careful marking and management of the merge-event queue is required to avoid duplicate processing.

In SPH schemes, particle-based neighbor data structures must efficiently support computation of renormalization factors $M$2 and boundary-proxy corrections. Feature-tagged cell indexing, adaptive timestep selection, and phase-conditional velocity treatments ensure consistent convergence of all features. Boundary particles must properly encode normals and reference areas to ensure completeness of kernel support.

A plausible implication is that implementation details—especially local operation complexity, memory usage, and geometric error management—dominate the scalability and robustness of schema-aware mesh generators for extreme topological and multi-feature scenarios.

6. Broader Significance and Applications

Schema-aware single-mesh generation enables direct parameterization, simulation, and analysis of high-genus surfaces, multiply-connected domains, and geometric modeling problems where topological constraints are central. Canonical polygonal schemas underlie surface parameterization, cut graph extraction, and topology-aware mapping in computer graphics and applied geometry. Concurrent feature-aware meshing supports automatic discretization for finite element and particle-based simulation, where curve, surface, and volumetric domain fidelity is required in a unified mesh structure. The demonstrated scalability to high genus and rapid multi-feature convergence indicates suitability for large-scale scientific and engineering applications, avoiding the pathologies of excessive mesh growth or loss of geometric regularity (Livesu, 2020, Ji et al., 2020).