JuliaFEM.jl/src/topology

src/topology/

Reference shapes for finite elements: parametric coordinates, edge and face connectivity, topological entities. Topology owns geometry; basis functions and quadrature are separate concerns and live in src/basis/ and src/quadrature/.

Files

• api.jl — AbstractTopology{N}, the topological-entity hierarchy (TopologicalEntity, Vertex, Edge, Face, Cell), the entity-counting helpers (nvertices, nedges, nfaces) and the generic interface (nnodes, dim, reference_coordinates, edges, faces).
• topology.jl — small compatibility shim included by JuliaFEM.jl.
• segments.jl — Segment{N} plus the Seg2, Seg3 aliases.
• triangles.jl — Triangle{N} plus Tri3, Tri6, Tri7, Tri10.
• quadrilaterals.jl — Quadrilateral{N} plus Quad4, Quad8, Quad9.
• tetrahedra.jl — Tetrahedron{N} plus Tet4, Tet10.
• hexahedra.jl — Hexahedron{N} plus Hex8, Hex20, Hex27.
• pyramids.jl — Pyramid{N} plus Pyr5.
• wedges.jl — Wedge{N} plus Wedge6, Wedge15.

Topologies provided

The full set of currently exported aliases:

Family	Type signature	Aliases
Segment (1D)	Segment{N}	Seg2, Seg3
Triangle (2D)	Triangle{N}	Tri3, Tri6, Tri7, Tri10
Quad (2D)	Quadrilateral{N}	Quad4, Quad8, Quad9
Tetrahedron	Tetrahedron{N}	Tet4, Tet10
Hexahedron	Hexahedron{N}	Hex8, Hex20, Hex27
Pyramid	Pyramid{N}	Pyr5
Wedge / Prism	Wedge{N}	Wedge6, Wedge15

Interface

Every topology implements:

nnodes(::Triangle{6})              # 6
dim(::Triangle{6})                 # 2
reference_coordinates(::Tri6)      # SVector{6, Vec{2, Float64}}
edges(::Tri6)                      # NTuple of (i, j) corner pairs
faces(::Tet4)                      # NTuple of (i, j, k) corner triples

The functions return compile-time constants (nnodes, dim) or static data (SVector, NTuple) so they are inlined at the call site and add no allocations to the assembly hot path.

Reference coordinate conventions:

• Segments — xi in [-1, 1].
• Triangles — (u, v) with u, v >= 0 and u + v <= 1.
• Quadrilaterals — (u, v) in [-1, 1]^2.
• Tetrahedra — (u, v, w) >= 0 with u + v + w <= 1.
• Hexahedra — (u, v, w) in [-1, 1]^3.

Pairing topology and basis

Topology and basis are independent type parameters of the element template. The basis dispatches on its own type parameter only:

get_basis_functions(::Type{Triangle{6}}, ::Type{Lagrange{2}}, xi)

so the same Triangle{6} can be combined with any compatible basis family (Lagrange{2}, Serendipity{2}, …) without redefining the topology.

Adding a new topology

1. Create the file under src/topology/ (or extend an existing family with a new N).
2. Define the struct and any aliases.
3. Implement nnodes, dim, reference_coordinates, edges and faces. Use SVector / NTuple returns; keep them allocation free.
4. Add the export to src/JuliaFEM.jl.
5. Add unit tests under test/topology/.
6. If you also want a Lagrange basis on this topology, add the matching VandermondeBasisDescription to src/basis/basis_descriptions.jl and re-run julia --project=. src/basis/basis_generator.jl.

Related code

• src/basis/ — basis functions and the symbolic generator.
• src/quadrature/ — integration points for each topology.
• src/mesh/ — Mesh{N, T<:AbstractTopology{N}}.
• src/elements/ — Element{K, P, S, N} template.
• test/topology/ — public test suite.