Reduced Quadrilateralization Approach

• Reduced quadrilateralization is a framework that transforms meshes and polynomial programs into simplified quadrilateral or quadratic forms while preserving essential structural features.
• It utilizes techniques such as centroid-based pairing, integer linear programming, and chord simplification to optimize mesh regularity and polynomial degree reduction.
• This approach enhances computational efficiency in high-order spectral solvers, CAD mesh generation, and large-scale polynomial optimization with scalable performance.

The reduced quadrilateralization approach encompasses a collection of algorithms and techniques for transforming mathematical or geometric entities—typically surface meshes or polynomial programs—into quadrilateral-dominant structures or quadratic forms, while minimizing complexity and preserving essential structure. These techniques have become foundational in high-order spectral solvers, quadrilateral mesh generation, and polynomial program simplification. This entry surveys the principal reduced quadrilateralization methodologies, their algorithmic workflows, mathematical underpinnings, and the contexts of CAD geometry, triangular-to-quad mesh conversion, and global polynomial optimization.

1. Definition and Conceptual Framework

Reduced quadrilateralization refers to either (a) geometric strategies that convert non-quadrilateral (often triangulated) surface representations into coarse quadrilateral mesh layouts by minimizing the requisite number of quads or (b) algebraic procedures that reduce general polynomial programs to quadratic (or lower-degree) formulations for tractable optimization. In both settings, the goal is to maintain fidelity to the original structure (e.g., CAD edges, cross-field singularities, polynomial constraints) while achieving a simpler or more uniform representation that enables efficient downstream computation or analysis (Zavalani, 30 Dec 2025, Couplet et al., 2021, González-Rodríguez et al., 2024, Viertel et al., 2019).

2. Reduced Quadrilateralization in Spectral Element Solvers

In high-order spectral element methods for PDEs on surfaces, the classical hierarchical Poincaré–Steklov (HPS) framework is quadrilateral-centric. To accommodate triangulated input geometries, a reduced quadrilateralization scheme is employed that pairs adjacent triangles into rhombic quadrilaterals via centroid-based construction. Specifically, for each interior shared edge $\{v_a,v_b\}$ between triangles $\hat{T}_i, \hat{T}_j$, with centroids $c_i, c_j$, a new quadrilateral patch is created as $[\;v_a, c_j, v_b, c_i\;]$. This minimal pairing closely matches the original surface and preserves adjacency, allowing the deployment of tensor-product Chebyshev-Lobatto grids and spectral collocation within each patch (Zavalani, 30 Dec 2025).

The resulting mesh supports efficient Chebyshev interpolation, explicit local reference-to-physical mappings (with Jacobian and metric tensor assembly), and the construction of Dirichlet-to-Neumann maps at the quad-patch level. These patches are then merged hierarchically via the HPS solver with binary-tree domain decomposition. Representative empirical results demonstrate spectral accuracy (error decay like $\rho^{-n}$ with $\rho\approx 9.3$) and favorable computational complexity: $O(K^{3/2} n^3)$ per direct solve, with $K$ the number of rhombi and $n$ the polynomial degree.

3. Integer Linear Programming Approaches for CAD Quad Mesh Simplification

In CAD mesh generation, reduced quadrilateralization for coarse, high-order quad mesh extraction is formulated as a topology modification problem on T-meshes defined over initial fine quad meshes. Here, irregular vertices induce a motorcycle graph (traced directions) and the intersections define a T-mesh network $(\mathcal{N},\mathcal{A},\mathcal{P})$. Each arc $a\in\mathcal{A}$ is assigned an integer length variable $q_a$, and the objective is to minimize a weighted sum $\sum_{a} w_a q_a$ (favoring the collapse of fine arcs to achieve the coarsest possible layout) (Couplet et al., 2021).

The constraints enforce:

• Non-negativity ($q_a \ge 0$),
• Mesh consistency (opposite sides of each patch sum to equal arc lengths),
• Singularity separation (to avoid geometric pathologies),
• Geometric distortion bounds.

The resultant integer linear program, solvable efficiently via standard MIP solvers, yields provably valid quad partitions that preserve CAD features strictly and support block-structured subdivision and smoothing (e.g., Winslow smoothing). Reduction ratios up to $11\times$ in patch count have been observed for real-world CAD models, with pipeline timings under two minutes.

4. Cross Field Separatrix Partition and Chord-Based Simplification

Another geometric setting for reduced quadrilateralization is the simplification of cross-field separatrix partitions on manifolds. Starting from a streamlines-induced partition (via eigenfield propagation on curved surfaces), the domain is divided into quads with T-junctions by tracing cross field separatrices. Subsequent simplification targets maximal collapse of chords (chains of patches) while guaranteeing that singularities remain properly placed and geometric degeneracies are avoided (Viertel et al., 2019).

Algorithmic steps include:

• Identification of collapsible chords (zip / non-zip),
• Greedy cost-based selection (patch energies: $e_{\rm patch}=\frac{\pi}{8} - \arctan(w/\ell)$),
• Structural invariants that guarantee monotonic reduction of regions and T-junctions.

Extensive benchmarking on 100 CAD-derived models corroborates the approach’s efficiency, typically reducing T-junctions to zero (92% of models), and outperforming prior pipeline stages by an order of magnitude in runtime.

5. Degree Reduction for Polynomial Optimization Problems

In polynomial programming, reduced quadrilateralization is realized algebraically as "quadrification"—rewriting degree-$\delta$ polynomial programs as degree-two (quadratic) or lower-degree instances, which are amenable to modern LP/QP and relaxation techniques. The QUAD-RLT scheme proceeds by systematically introducing lifted variables $X_J=\prod_{j\in J} x_j$ for high-degree monomials and imposing recursive constraint chains that minimize the size of the resultant reformulation, exploiting shared monomial structure (González-Rodríguez et al., 2024).

The workflow entails:

• Sorting and decomposing monomials by maximal reuse of existing quadratic submonomials,
• Applying McCormick convex envelopes for bilinear constraints,
• Optionally reducing to arbitrary degree $d$ ($d=4,6,8$),
• Solving the resultant relaxation with state-of-the-art LP/QP solvers.

Computation results (RAPOSa implementation) indicate that, for problem degrees $\delta\geq 10$, QUAD-RLT recovers best-in-class tradeoffs between solution tightness and tractability, outperforming earlier chain-based quadrification schemes in both relaxation quality and model size.

6. Comparative Discussion, Advantages, and Limitations

Reduced quadrilateralization methods provide provable guarantees of topological validity and, when formulated as optimization problems (ILP for meshes, QUAD-RLT for polynomials), often produce minimal representations within their respective frameworks. These methods

• strictly preserve user-specified or natural features,
• enable mesh and algebraic simplification without global parameterization,
• exhibit excellent computational performance on practical datasets (Zavalani, 30 Dec 2025, Couplet et al., 2021, González-Rodríguez et al., 2024, Viertel et al., 2019).

Principal limitations include the linear-objective-induced residual density (for mesh ILP), the sensitivity to B-Rep quality (geometry) in CAD, and the combinatorial scaling of auxiliary variables in algebraic degree reduction. Proposed improvements encompass nonlinear (total-area) objectives, conditional constraints for deeper topological simplifications, and pre-simplification or repair mechanisms for challenging CAD and algebraic instances.

7. Applications and Future Directions

Reduced quadrilateralization has immediate utility in:

• High-order spectral solvers (especially spectral/hp methods for PDEs on arbitrary surfaces),
• CAD mesh generation for isogeometric and $p$-FEM analysis,
• Mesh decimation and simplification (e.g., for level-of-detail rendering),
• Large-scale polynomial optimization with high-degree monomials.

Future research directions include improving coarseness and patch regularity via nonlinear optimization, robust handling of degenerate geometries, hybridization with global parameterization strategies, and further algorithmic acceleration for large-scale high-dimensional regimes (González-Rodríguez et al., 2024, Zavalani, 30 Dec 2025, Couplet et al., 2021).