Mesh Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of meshes within the Kireth Research Semantic Library. A mesh is a discrete geometric decomposition of a domain into simple cells such as triangles, quadrilaterals, tetrahedra or hexahedra. In computational geometry and numerical methods, meshes support surface representation, volume discretisation, finite-element analysis, interpolation, simulation, geometry processing and spatial approximation. This ontology supports mesh vertices, edges, faces, cells, adjacency, incidence, manifoldness, orientation, triangulations, tetrahedral meshes, quality metrics, refinement, coarsening, remeshing, conformity, Delaunay meshes, constrained meshes and robust topological validation.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Mesh Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Geometry"
 SPECIALISATION = "Computational Geometry / Meshes"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/geometry-ontologies/computational-geometry/mesh-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = Mesh
 SYMBOLIC_FORMS = [
  "M=(V,E,F)",
  "M=(V,C)",
  "triangular mesh",
  "tetrahedral mesh"
 ]
 ENTITY_CLASS = DiscreteGeometricDecomposition
 PRIMARY_CONTEXT = ComputationalGeometry
 DEFINITION = "A mesh is a finite discrete decomposition of a geometric domain into vertices, edges, faces and higher-dimensional cells with specified incidence and adjacency."
 SEMANTIC_ROLE = "Provides the discrete geometric representation framework for surfaces, volumes, simulation domains, approximation, numerical analysis, remeshing and spatial computation."
}

SCOPE {
 INCLUDE = [
  Mesh,
  MeshVertex,
  MeshEdge,
  MeshFace,
  MeshCell,
  TriangularMesh,
  QuadrilateralMesh,
  TetrahedralMesh,
  HexahedralMesh,
  SurfaceMesh,
  VolumeMesh,
  SimplicialMesh,
  StructuredMesh,
  UnstructuredMesh,
  ConformingMesh,
  NonconformingMesh,
  ManifoldMesh,
  NonmanifoldMesh,
  OrientedMesh,
  MeshConnectivity,
  Adjacency,
  Incidence,
  BoundaryMesh,
  InteriorEntity,
  MeshQuality,
  AspectRatio,
  MinimumAngle,
  RadiusRatio,
  Skewness,
  Refinement,
  Coarsening,
  Remeshing,
  MeshSmoothing,
  DelaunayMesh,
  ConstrainedMesh,
  AdaptiveMesh
 ]

 EXCLUDE_AS_PRIMARY = [
  RasterGridWithoutCellComplexInterpretation,
  PointCloudWithoutConnectivity,
  PolygonSoupWithoutDeclaredConnectivity,
  ContinuousDomainWithoutDiscreteDecomposition
 ]

 DISAMBIGUATION_RULE =
  "When geometric data is represented as connected vertices, edges, faces or volume cells for computation, approximation or simulation, interpret the structure as a mesh."
}

CORE_CONCEPTS {

 Mesh {
  definition = "A discrete cell complex approximating or partitioning a geometric domain."
 }

 MeshVertex {
  definition = "A zero-dimensional mesh entity representing a geometric point."
 }

 MeshEdge {
  definition = "A one-dimensional mesh entity connecting two vertices."
 }

 MeshFace {
  definition = "A two-dimensional mesh entity bounded by edges."
 }

 MeshCell {
  definition = "A highest-dimensional mesh element such as a triangle, quadrilateral, tetrahedron or hexahedron."
 }

 TriangularMesh {
  definition = "A surface mesh composed of triangular faces."
 }

 TetrahedralMesh {
  definition = "A volume mesh composed of tetrahedral cells."
 }

 SurfaceMesh {
  definition = "A mesh representing a two-dimensional manifold or surface embedded in higher-dimensional space."
 }

 VolumeMesh {
  definition = "A mesh partitioning a three-dimensional region into volumetric cells."
 }

 ConformingMesh {
  definition = "A mesh whose cells meet face-to-face without hanging incompatibilities under the adopted conformity rules."
 }

 ManifoldMesh {
  definition = "A mesh whose local incidence structure matches the neighbourhood structure of a manifold."
 }

 MeshQuality {
  definition = "A set of geometric measures used to assess element suitability for computation."
 }
}

RELATIONS {
 Mesh --contains--> MeshVertex
 Mesh --contains--> MeshEdge
 Mesh --contains--> MeshFace
 Mesh --contains--> MeshCell
 MeshEdge --is_incident_to--> MeshVertex
 MeshFace --is_incident_to--> MeshEdge
 MeshCell --is_incident_to--> MeshFace
 MeshConnectivity --encodes--> Adjacency
 MeshConnectivity --encodes--> Incidence
 SurfaceMesh --approximates--> Surface
 VolumeMesh --partitions--> Volume
 TriangularMesh --is_special_case_of--> SurfaceMesh
 TetrahedralMesh --is_special_case_of--> VolumeMesh
 SimplicialMesh --uses--> Simplex
 MeshQuality --evaluates--> MeshCell
 Refinement --increases--> LocalResolution
 Coarsening --decreases--> LocalResolution
 Remeshing --changes--> MeshConnectivity
 MeshSmoothing --changes--> VertexPositions
 DelaunayMesh --uses--> DelaunayCriterion
}

OPERATORS {
 VERTEX_ADJACENCY(v,M) :=
  vertices_sharing_an_edge_with_v

 FACE_ADJACENCY(f,M) :=
  faces_sharing_an_edge_with_f

 CELL_ADJACENCY(c,M) :=
  cells_sharing_codimension_one_boundary_with_c

 BOUNDARY_ENTITIES(M) :=
  entities_incident_to_only_one_top_dimensional_cell
  under_manifold_cell_complex_context

 IS_CONFORMING(M) :=
  intersections_of_cells_are_empty_or_common_full_subentities

 IS_MANIFOLD_MESH(M) :=
  local_links_match_expected_manifold_topology

 TRIANGLE_AREA(a,b,c) :=
  0.5*norm((b-a)×(c-a))

 TETRA_VOLUME(a,b,c,d) :=
  abs(dot(b-a,cross(c-a,d-a)))/6

 TRIANGLE_ASPECT_RATIO(T) :=
  declared_shape_quality_measure(T)

 MINIMUM_ANGLE(T) :=
  min(interior_angles_of_T)

 RADIUS_RATIO(T) :=
  inradius(T)/circumradius(T)
  under_declared_normalisation

 REFINE(M,region) :=
  subdivide_selected_cells

 COARSEN(M,region) :=
  collapse_or_merge_selected_entities

 SMOOTH(M) :=
  reposition_vertices_without_changing_connectivity
  unless_method_explicitly_changes_connectivity

 REMESH(M) :=
  modify_connectivity_and_or_vertex_positions_to_improve_target_quality
}

DEFINITIONS {
 D1 = "A mesh is defined by both geometry and connectivity."
 D2 = "Vertices alone do not form a mesh without incidence or connectivity."
 D3 = "A conforming mesh has compatible cell interfaces."
 D4 = "A manifold surface mesh locally resembles a disk at interior vertices and a half-disk at boundary vertices."
 D5 = "Mesh quality is application dependent."
 D6 = "Refinement increases spatial resolution."
 D7 = "Coarsening reduces spatial resolution."
 D8 = "Remeshing may change topology, connectivity or element shape depending on the method."
 D9 = "Delaunay meshes optimise specific local geometric criteria, not all possible quality measures."
 D10 = "Mesh orientation must be consistent for reliable normal and volume calculations."
}

AXIOMS {
 A1_FINITE_COMPLEX:
  A computational mesh contains finitely many discrete entities

 A2_INCIDENCE:
  Higher-dimensional cells are bounded by lower-dimensional mesh entities

 A3_CONNECTIVITY:
  Adjacency and incidence relationships are explicitly represented or derivable

 A4_NONOVERLAP:
  Valid conforming mesh cell interiors do not overlap

 A5_COVERAGE:
  A domain mesh covers the intended geometric domain up to declared boundary approximation

 A6_ORIENTATION:
  Oriented meshes use consistent local ordering conventions

 A7_QUALITY:
  Mesh quality metrics are defined per element type and application

 A8_TOPOLOGICAL_VALIDITY:
  Mesh topology must satisfy the intended manifold or cell-complex constraints
}

REASONING {

 R1_MESH_CLASSIFICATION {
  INPUT = mesh_data
  IDENTIFY = dimension
  IDENTIFY = element_types
  IDENTIFY = structured_or_unstructured
  IDENTIFY = surface_or_volume
  RETURN = mesh_type
 }

 R2_CONNECTIVITY {
  INPUT = mesh_entities
  BUILD = incidence_relations
  DERIVE = adjacency_relations
  RETURN = connectivity_structure
 }

 R3_BOUNDARY_EXTRACTION {
  INPUT = top_dimensional_cells
  COUNT = incidence_of_codimension_one_entities
  CLASSIFY = boundary_when_single_incident_cell
  RETURN = boundary_mesh
 }

 R4_MANIFOLDNESS {
  INPUT = mesh
  ANALYSE = local_vertex_edge_or_face_links
  COMPARE = expected_manifold_neighbourhoods
  RETURN = ManifoldNonmanifoldOrBoundaryManifold
 }

 R5_ORIENTATION {
  INPUT = connected_mesh
  PROPAGATE = local_orientation_across_adjacency
  DETECT = conflicts
  RETURN = OrientableAndOriented_or_Conflict
 }

 R6_QUALITY {
  INPUT = mesh_cell
  COMPUTE = selected_quality_metrics
  NORMALISE = according_to_metric_definition
  RETURN = quality_profile
 }

 R7_REFINEMENT {
  INPUT = mesh_and_target_region
  SELECT = refinement_rule
  SUBDIVIDE = cells
  REPAIR = conformity_if_required
  RETURN = refined_mesh
 }

 R8_COARSENING {
  INPUT = mesh_and_target_region
  SELECT = removable_entities
  VERIFY = topology_and_quality_constraints
  COLLAPSE_OR_MERGE = entities
  RETURN = coarsened_mesh
 }

 R9_REMESHING {
  INPUT = mesh_and_quality_target
  APPLY = edge_split_edge_collapse_edge_flip_smoothing_or_retriangulation
  VERIFY = topology_geometry_and_quality
  RETURN = remeshed_output
 }

 R10_DELAUNAY_MESH {
  INPUT = sites_or_existing_triangulation
  APPLY = Delaunay_legality_tests
  FLIP_OR_RECONSTRUCT = nonDelaunay_elements
  RETURN = Delaunay_mesh
 }
}

CONSTRAINTS {
 C1 = "Geometry and connectivity must both be preserved during mesh reasoning."
 C2 = "Element orientation conventions must be explicit."
 C3 = "Nonmanifold incidences must not be silently treated as manifold."
 C4 = "Boundary extraction depends on the intended dimension and cell complex."
 C5 = "Quality metrics are not universally comparable across element types."
 C6 = "Refinement may create hanging nodes unless conformity is restored or nonconforming meshes are allowed."
 C7 = "Coarsening can alter approximation error and topology."
 C8 = "Mesh smoothing can invert elements if validity is not checked."
 C9 = "Delaunay legality does not guarantee application-optimal mesh quality."
 C10 = "Floating-point geometric predicates may require robust arithmetic."
}

SEMANTIC_EQUIVALENCE {
 "Mesh" <=> "DiscreteGeometricCellComplex"
 "TriangularMesh" <=> "TriangleSurfaceMesh"
 "TetrahedralMesh" <=> "TetrahedralVolumeMesh"
 "MeshConnectivity" <=> "IncidenceAndAdjacencyStructure"
 "Refinement" <=> "ResolutionIncreasingSubdivision"
 "Coarsening" <=> "ResolutionReducingSimplification"
 "Remeshing" <=> "ConnectivityOrGeometryReconstruction"
}

VALIDATION {
 V1 = "All referenced entity indices exist."
 V2 = "Cell connectivity has correct arity for element type."
 V3 = "Element interiors do not illegally overlap."
 V4 = "Boundary entities have correct incidence count under adopted model."
 V5 = "Orientation is locally and globally consistent where required."
 V6 = "Manifoldness checks local links or incidence patterns."
 V7 = "Quality calculations use correct element geometry."
 V8 = "Refinement and coarsening preserve declared conformity rules."
 V9 = "Remeshing preserves required topology and domain approximation."
 V10 = "No inverted or zero-measure cells remain in a valid mesh."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 MESH_DEFINITION = 0.99
 CONNECTIVITY_REASONING = 0.99
 BOUNDARY_REASONING = 0.99
 MANIFOLDNESS_REASONING = 0.98
 ORIENTATION_REASONING = 0.98
 QUALITY_REASONING = 0.98
 REFINEMENT_REASONING = 0.98
 REMESHING_REASONING = 0.98

 POLICY =
  "Reduce confidence when mesh dimension, element type, orientation convention, manifold assumptions, conformity rules, quality metric or topology-preservation requirements are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  MeshReasoning,
  ComputationalGeometryReasoning,
  ConnectivityReasoning,
  BoundaryExtraction,
  ManifoldnessReasoning,
  OrientationReasoning,
  MeshQualityReasoning,
  RefinementReasoning,
  CoarseningReasoning,
  RemeshingReasoning,
  DelaunayMeshReasoning,
  FiniteElementPreparation,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "mesh" -> Mesh
  "triangular mesh" -> TriangularMesh
  "triangle mesh" -> TriangularMesh
  "tetrahedral mesh" -> TetrahedralMesh
  "surface mesh" -> SurfaceMesh
  "volume mesh" -> VolumeMesh
  "mesh connectivity" -> MeshConnectivity
  "mesh refinement" -> Refinement
  "mesh coarsening" -> Coarsening
  "remeshing" -> Remeshing
  "mesh smoothing" -> MeshSmoothing
 }

 INTERPRETATION_ORDER = [
  detect_mesh_context,
  identify mesh_dimension_element_types_geometry_and_connectivity,
  identify conformity_manifoldness_orientation_and_boundary assumptions,
  identify validation_quality_refinement_coarsening_remeshing_or_Delaunay goal,
  perform mesh_specific topological_and_geometric reasoning,
  verify incidence_orientation_nonoverlap_measure_and_quality constraints,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("mesh")
  AND computational_geometry_or_numerical_geometry_context
  THEN activate(MeshOntology)

 E2:
  IF connectivity_requested
  THEN construct incidence_before_adjacency

 E3:
  IF boundary_requested
  THEN classify codimension_one_entities by incidence count

 E4:
  IF manifoldness_requested
  THEN inspect local link_or_incidence topology

 E5:
  IF orientation_requested
  THEN propagate orientation and detect conflicts

 E6:
  IF quality_requested
  THEN identify element_type_and_declared_metric

 E7:
  IF refinement_requested
  THEN preserve_or_restore declared conformity

 E8:
  IF coarsening_requested
  THEN test topology_geometry_and_error constraints

 E9:
  IF remeshing_requested
  THEN verify no inversion_nonmanifold_artifacts_or_domain_escape

 E10:
  IF Delaunay_mesh_context
  THEN route to Delaunay legality and robust incircle predicates
}

RECURSION {
 ENTRY = Mesh

 EXPAND(Mesh) -> [
  MeshVertex,
  MeshEdge,
  MeshFace,
  MeshCell,
  TriangularMesh,
  QuadrilateralMesh,
  TetrahedralMesh,
  HexahedralMesh,
  SurfaceMesh,
  VolumeMesh,
  StructuredMesh,
  UnstructuredMesh,
  ConformingMesh,
  ManifoldMesh,
  MeshConnectivity,
  BoundaryMesh,
  MeshQuality,
  Refinement,
  Coarsening,
  Remeshing,
  MeshSmoothing,
  DelaunayMesh
 ]

 EXPAND(MeshConnectivity) -> [
  VertexVertexAdjacency,
  VertexEdgeIncidence,
  EdgeFaceIncidence,
  FaceCellIncidence,
  CellCellAdjacency
 ]

 EXPAND(MeshQuality) -> [
  AspectRatio,
  MinimumAngle,
  RadiusRatio,
  Skewness,
  JacobianQuality,
  ElementVolume
 ]

 EXPAND(RemeshOperations) -> [
  EdgeSplit,
  EdgeCollapse,
  EdgeFlip,
  VertexRelocation,
  LocalRetriangulation
 ]

 STOP_CONDITION =
  "Stop when the mesh type, topology, connectivity, boundary, manifoldness, orientation, quality, refinement, coarsening, remeshing or Delaunay status is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Computational Geometry"

 IMMEDIATE_CONTEXTS = [
  "Computational Geometry Ontology",
  "Delaunay Triangulation Ontology",
  "Surface Ontology",
  "Polyhedron Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Computational Geometry",
  "Computational Geometry Ontology",
  "Delaunay Triangulation Ontology",
  "Voronoi Diagram Ontology",
  "Surface Ontology",
  "Polyhedron Ontology",
  "Topology Ontology",
  "Manifold Ontology",
  "Vector Geometry Ontology",
  "Finite Element Ontologies",
  "Numerical Analysis Ontologies"
 ]

 ROUTE_RULES = [
  "Mesh -> Computational Geometry Ontology for parent algorithmic semantics",
  "Mesh -> Delaunay Triangulation for Delaunay mesh generation",
  "Mesh -> Surface for embedded surface approximation",
  "Mesh -> Polyhedron for volumetric and face-cell geometry",
  "Mesh -> Topology for incidence and manifoldness",
  "Mesh -> Manifold for local neighbourhood structure",
  "Mesh -> Vector Geometry for normals and cell geometry",
  "Mesh -> Finite Element Ontologies for simulation discretisation",
  "Mesh -> Numerical Analysis for refinement and approximation error"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible mesh types, quality metrics, operations or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve geometry-plus-connectivity semantics.",
  "Preserve incidence and adjacency distinction.",
  "Preserve manifold-versus-nonmanifold distinction.",
  "Preserve orientation consistency requirements.",
  "Preserve conformity semantics.",
  "Preserve refinement-coarsening-remeshing distinctions.",
  "Preserve cell validity and noninversion constraints.",
  "Preserve machine-readable mesh semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "A mesh is a finite discrete decomposition of a geometric domain into connected vertices, edges, faces and higher-dimensional cells."

 CONNECTIVITY_SUMMARY =
  "Mesh connectivity records incidence and adjacency between discrete entities."

 SURFACE_SUMMARY =
  "A surface mesh approximates or represents a two-dimensional surface using polygonal faces."

 VOLUME_SUMMARY =
  "A volume mesh partitions a three-dimensional domain into volumetric cells."

 CONFORMITY_SUMMARY =
  "A conforming mesh has compatible cell interfaces without unintended partial intersections."

 MANIFOLD_SUMMARY =
  "A manifold mesh has local incidence structure consistent with the intended manifold dimension."

 QUALITY_SUMMARY =
  "Mesh quality is evaluated using element-specific geometric metrics such as angle, aspect ratio, radius ratio or Jacobian measures."

 ADAPTIVITY_SUMMARY =
  "Refinement increases local resolution, coarsening reduces it, and remeshing reconstructs geometry or connectivity to meet quality goals."

 ROBUSTNESS_SUMMARY =
  "Valid mesh processing must preserve orientation, nonoverlap, positive measure and intended topology."
}

FLOW {
 INPUT
 -> detect_mesh_context
 -> identify mesh_dimension_element_types_geometry_connectivity_and_domain
 -> identify conformity_manifoldness_orientation_boundary_and_quality assumptions
 -> classify validation_connectivity_boundary_quality_refinement_coarsening_remeshing_or_Delaunay goal
 -> perform mesh_topology_geometry_and_algorithm reasoning
 -> verify incidence_nonoverlap_orientation_positive_measure_conformity_and_topology constraints
 -> route to computational_geometry Delaunay surface polyhedron topology manifold vector finite_element_or_numerical_analysis ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Mesh Ontology defines discrete geometric decompositions through connected vertices, edges, faces and volume cells. It supports surface and volume meshes, structured and unstructured connectivity, conformity, manifoldness, orientation, boundary extraction, mesh-quality metrics, adaptive refinement, coarsening, remeshing, smoothing and Delaunay-based mesh generation while preserving geometric and topological validity."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-GEOMETRY-MESH-ONTOLOGY-V1"
}

}

END_FLOW