A framework for generating nonconforming triangular meshes with multiple discretization layers

Abstract: We present a framework for generating nonconforming triangular meshes with multiple discretization layers. The framework exploits characteristic structural properties of meshes produced by a frontal Delaunay algorithm with uniform element size. The bulk region of such meshes exhibits a structured pattern resembling a regular triangular lattice. Owing to this structure, the bulk region can be coarsened by grouping elements into connected subsets of larger composite elements. When applied repeatedly, this procedure produces a mesh with multiple discretization layers. For complex geometries, the framework can be used to create composite multidomain meshes, where multiple discretization layers are generated within each subdomain. We present examples of meshes generated by the proposed framework and discuss postprocessing strategies for their refinement and coarsening. The resulting meshes are well suited for the application of adaptive mesh refinement techniques. The proposed framework can be readily integrated with existing mesh generators and finite-element solvers that support nonconforming triangular meshes.

• The paper introduces a combinatorial framework that represents hanging-node interfaces with zero-area void elements and coarsens structured bulk regions through compact supertriangulations.
• Repeated coarsening creates progressively larger elements from the boundary toward the interior, and demonstrations on squares, circles, ellipses, and composite domains produce usable multilevel meshes.
• The method reverses ten frontal red-refinement steps exactly in a unit-circle test, but remains dependent on runtime geometric checks, mesh quality, and parameter choices without formal guarantees.

Overview and motivation

This paper by Semenov and Becker (Leibniz Institute for Plasma Science and Technology, Greifswald) presents a framework for constructing nonconforming triangular meshes that possess multiple discretization layers, i.e., a hierarchy of element-size levels transitioning from a fine, unstructured boundary layer to a coarsely discretized interior bulk. The work is motivated by adaptive mesh refinement (AMR) on unstructured meshes. While nonconforming AMR is well established on logically structured meshes via quadtree/octree hierarchies (e.g., p4est, Afivo), the analogous capability on unstructured meshes—typically realized through red refinement of a background mesh—requires the background mesh itself to possess a suitable hierarchy of discretization layers. The central contribution is an algorithmic procedure that manufactures such a hierarchy from an otherwise uniform background mesh, thereby enabling red-refinement-based AMR in a fully unstructured setting.

The framework builds on a structural observation: triangular meshes produced by the frontal Delaunay algorithm with uniform element size exhibit a characteristic two-region structure—an unstructured layer along the domain boundary that resolves geometric complexity, connected to a bulk region that closely resembles a regular triangular lattice. Because the bulk is nearly regular, it can be coarsened by grouping fine elements into connected subsets forming larger composite elements. Applying this grouping repeatedly yields a nonconforming mesh with multiple discretization layers.

Unified representation of conforming and nonconforming meshes

The framework's data-structure foundation is a deliberate reinterpretation of nonconformity. The authors treat a nonconforming mesh as the limiting case of a conforming mesh in which nonconforming interfaces are represented by triangles of zero area, termed void elements. A hanging node $\mathbf{p}_k$ is placed at the midpoint of an edge, satisfying $$\mathbf{p}_k = \tfrac{1}{2}(\mathbf{p}_i + \mathbf{p}_j)$$, and the degenerate triangle $$\triangle(\mathbf{p}_i, \mathbf{p}_j, \mathbf{p}_k)$$ is appended to the connectivity matrix as an ordinary row.

This construction has two practical consequences. First, the mesh remains combinatorially conforming at all times, so standard combinatorial operations on mesh elements apply uniformly to both conforming and nonconforming configurations. Second, all mesh manipulations are performed purely at the level of the connectivity matrix, with the geometric representation used only for visualization. The authors are explicit that a comprehensive theoretical formulation of the continuous conforming-to-nonconforming transition lies beyond the scope of the work; the justification offered is practical rather than analytic. This is a genuine limitation: no error analysis or theoretical guarantee accompanies the representation, and its compatibility with constrained finite-element approximation spaces is assumed rather than proven.

The supertriangulation data structure

The second structural component is the supertriangulation, a data structure that groups elements of the background mesh into top-level complexes suitable for hierarchical management. Given a triangulation $\mathcal{T}$, the core subset $\mathcal{T}_{\mathrm{core}}$ consists of triangles whose edges are all shared with adjacent triangles. For each core triangle $\tau_\alpha$, the associated supertriangle $\tau_\alpha^{\mathrm{sup}}$ is formed by the three vertices of the neighboring triangles opposite their shared edges—geometrically, a coarse triangle covering a four-element patch of the background mesh.

Supertriangles generally intersect geometrically, so the framework operates on them in purely combinatorial terms. Each supertriangle carries a combinatorial stencil—the set of indices of the original mesh elements used to construct it—and two supertriangles are combinatorially disjoint if their stencils share no indices. A supertriangulation is compact if all its supertriangles are pairwise combinatorially disjoint.

Several filtering operations are defined on supertriangulations: smoothing (removal of supertriangles with fewer than two connections, single-pass or iterative), stripping (single-pass removal of doubly connected supertriangles), cleaning (stripping followed by iterative smoothing), and detachment (removal of supertriangles with boundary vertices). A reduction procedure extracts a connected subset via breadth-first search on the adjacency matrix, seeded from a supertriangle closest to a user-specified anchor point. Because reduction alone does not guarantee compactness, the authors combine reduction with cleaning in an iterative procedure that terminates when the supertriangulation becomes either compact or empty. Notably, the empty outcome is a possible termination state, meaning the coarsening procedure can fail outright for a given parameter choice.

The mesh-coarsening algorithm

The mesh construction algorithm is formalized as an operator $$F^{\rm coarsen}(\alpha_{\rm detach}, \mathbf{c}_0, n_{\rm shrink})$$ on the set of meshes, with three control parameters: a boolean governing boundary detachment of the background supertriangulation, an anchor point selecting the reduction seed, and an integer number of post-reduction cleaning steps. The algorithm proceeds in three stages.

Input processing. The background supertriangulation is constructed from the mesh core, optionally detached from the boundary, reduced iteratively to a compact connected subset, and optionally cleaned further. Failure at any stage terminates the procedure.

Mesh decomposition. The triangulation is split into an inner mesh (the reduced supertriangulation itself) and an outer mesh (the complement of the covered elements). The algorithm restricts attention to configurations in which the outer mesh has annular topology. The critical structural assumption enters here: the algorithm requires that the inner boundary path $\Gamma_{\rm out}^{\rm in}$ of the outer mesh contain exactly twice as many points as the outer boundary path $\Gamma_{\rm in}^{\rm out}$ of the inner mesh, with the correspondence $$\mathbf{p}_k = \tfrac{1}{2}(\mathbf{p}_i + \mathbf{p}_j)$$0 holding at the level of global point indices. The authors state plainly that a theoretical guarantee for the existence of a configuration satisfying this condition is beyond the scope of the work; the condition is checked explicitly, and the procedure is terminated if it fails. This is the principal point at which the method depends on the regularity of the frontal Delaunay bulk rather than on a proven property.

Mesh reconnection. The inner and outer meshes are merged, and void elements are inserted along the contact region: each hanging node $$\mathbf{p}_k = \tfrac{1}{2}(\mathbf{p}_i + \mathbf{p}_j)$$1 is repositioned to the midpoint $$\mathbf{p}_k = \tfrac{1}{2}(\mathbf{p}_i + \mathbf{p}_j)$$2, and the row $$\mathbf{p}_k = \tfrac{1}{2}(\mathbf{p}_i + \mathbf{p}_j)$$3 is appended to the connectivity matrix. The result is a simply connected nonconforming mesh with a single outer boundary, suitable as input for the next coarsening iteration.

Because the operator accepts nonconforming meshes as input, it can be applied iteratively, with each application producing a coarser central region and hence an additional discretization layer. The element size increases monotonically from the domain boundary toward the bulk.

Postprocessing: frontal refinement and coarsening

The generated meshes admit frontal refinement and coarsening driven by the void elements marking nonconforming interfaces. The refinement front consists of triangles sharing with a void element the edge opposite the hanging node; each is refined by red subdivision into four congruent triangles, with the old void element removed and new void elements inserted at the newly created hanging nodes. The coarsening front consists of supertriangles containing exactly two subtriangles adjacent to void elements, subject to the additional requirement that the central subtriangle's vertices coincide with edge midpoints—guaranteeing that coarsening is the exact inverse of red refinement.

The authors demonstrate this reversibility empirically: starting from a nonconforming mesh in the unit circle, ten frontal refinement steps followed by ten de-refinement steps reproduce the initial mesh exactly. This exact reversibility is the strongest concrete result in the paper and confirms that the meshes support a well-defined refinement–coarsening cycle, a prerequisite for conservative AMR.

The framework also accommodates defects—supertriangles not satisfying the red-refinement condition, most commonly those whose central subtriangle has two hanging nodes as vertices. Such defects arise at interfaces between discretization layers and can be coarsened in a non-reversible postprocessing step.

Numerical examples

The algorithm is implemented in the finite-element package triellipt and demonstrated on several domains, with background meshes generated in Gmsh using the frontal Delaunay algorithm at a uniform element size of 0.02. In the unit square, unit circle, and an ellipse with axes 2 and 1, repeated application of the coarsening operator (two to three iterations, with parameters such as $$\mathbf{p}_k = \tfrac{1}{2}(\mathbf{p}_i + \mathbf{p}_j)$$4, $$\mathbf{p}_k = \tfrac{1}{2}(\mathbf{p}_i + \mathbf{p}_j)$$5 at the domain center, and $$\mathbf{p}_k = \tfrac{1}{2}(\mathbf{p}_i + \mathbf{p}_j)$$6) consistently produces the expected layered structure: an unstructured boundary layer separated from a structured bulk that is progressively coarsened.

The framework also applies to structured triangular meshes, specifically grids with uniform diagonal orientation (a $$\mathbf{p}_k = \tfrac{1}{2}(\mathbf{p}_i + \mathbf{p}_j)$$7 point grid is shown), but not to cross-triangulated meshes, where the supertriangulation consists mostly of disconnected supertriangles. This is a concrete boundary of applicability: the method is sensitive to the diagonal orientation pattern of structured inputs.

For complex geometries, the paper proposes composite multidomain meshes: the domain is decomposed into geometrically simple subdomains, each discretized independently with multiple layers. The motivating application is the simulation of atmospheric-pressure gas discharges in rod-to-plane configurations, where thin ionization waves require AMR. In the composite setting, interpolation operators can be constructed independently per subdomain, and the coarse bulk regions support controlled red refinement and de-refinement. The authors note, however, that frontal Delaunay meshes in complex geometries have lower bulk quality, which reduces algorithm performance and requires more careful parameter adjustment—an acknowledged limitation rather than a solved problem.

Limitations and open questions

Several constraints on the framework should be stated plainly. First, the key geometric assumption—that the inner and outer boundary paths of the decomposed mesh satisfy the two-to-one node correspondence—is checked at runtime rather than guaranteed, and no theoretical characterization of the mesh classes for which it holds is provided. Second, the iterative reduction can return an empty supertriangulation, so the coarsening operator is partial: termination without a mesh is an admissible outcome, and the parameter space $$\mathbf{p}_k = \tfrac{1}{2}(\mathbf{p}_i + \mathbf{p}_j)$$8 governing success is not systematically explored. Third, the unified void-element representation is justified only practically; a rigorous formulation of the conforming-to-nonconforming limit, and its consequences for finite-element approximation and hanging-node constraint enforcement, remain open. Fourth, the method is demonstrated only on domains with relatively simple geometry or on composite assemblies of such domains; its behavior on general complex geometries with lower-quality frontal Delaunay bulks is acknowledged to require further tuning. Finally, no quantitative element-quality or element-count statistics are reported, so the claim that the resulting discretizations minimize the total number of elements remains qualitative.

Conclusion

The paper contributes a practical, combinatorially clean framework for generating nonconforming triangular meshes with multiple discretization layers, built on two ideas: the representation of nonconforming interfaces as zero-area void elements within an otherwise conforming connectivity matrix, and the supertriangulation data structure for extracting and coarsening connected element groups from the structured bulk of frontal Delaunay meshes. The demonstrated exact reversibility of frontal refinement and coarsening, together with the composite multidomain construction for complex geometries, makes the framework a viable basis for red-refinement-based AMR on unstructured meshes. Its applicability rests on the empirical regularity of the frontal Delaunay bulk and on runtime-checkable geometric conditions whose theoretical guarantees are left open, and its quantitative performance relative to alternative unstructured AMR discretizations remains to be established.