A multi-level framework for type-safe topological structures with efficient connectivity queries

We present a framework for representing and querying multi-type topological structures with applications to finite element meshes, hypergraphs, and hierarchical data. The framework introduces a four-level architecture—O2M (One-to-Many), M2M (Many-to-Many), MM2M (Multi-type Many-to-Many), and Topology (type-safe generic API)—that provides progressively richer operations while maintaining bounded complexity. Key contributions include: (1) a type-safe API with compile-time entity disambiguation and low runtime overhead; (2) lazy cache synchronization for transpose and position lookups achieving O(1) amortized query cost; (3) symmetry-based canonical forms enabling automatic element deduplication with O(|G|·k) complexity where |G| is the symmetry group size and k the element node count; and (4) algebraic operations on sparse adjacency structures supporting transitive connectivity computation via symbolic matrix multiplication. The framework's topological operations are demonstrated through two mesh modification algorithms that operate entirely at the connectivity level: edge bisection refinement with tetrahedral closure, and crack insertion by node duplication using level-set functions, both applicable in 2D and 3D without explicit crack geometry representation. We establish complexity bounds for all operations and demonstrate through benchmarks that common mesh connectivity queries achieve 40–625× speedup over naive implementations without caching through cache-aware traversal algorithms. On a 1M element mesh, our framework achieves 50M queries/sec for direct lookups and 400K queries/sec for neighbor traversals, with construction throughput of 8M elements/sec and near-linear parallel scaling for transpose operations up to 16 cores. Supplementary materials comprising a tutorial with complete FEA implementations, a technical supplement, and an API reference are provided to facilitate reproducibility.