JuliaFEM.jl/src/quadrature

src/quadrature/

Gauss-Legendre quadrature rules for every supported topology. The API returns static data (SVector{N, QuadraturePoint{D, Float64}}) so integration loops in the assemblers are inlined and allocation-free.

Files

• api.jl — public types and interface (AbstractQuadratureRule, GaussLegendre{N, V}, QuadraturePoint{D, T}, get_quadrature_points, default_quadrature).
• quaddata.jl — 1D Gauss-Legendre point and weight tables.
• gl_tensor_product.jl — segments, quadrilaterals, hexahedra (tensor products of quaddata).
• gl_triangles.jl — triangle rules.
• gl_tetrahedra.jl — tetrahedron rules.
• gl_wedges.jl — wedge / prism rules.
• gl_pyramids.jl — pyramid rules.
• gauss.jl — integration_points(topology_instance) and npoints(topology_instance) thin wrappers over default_quadrature + get_quadrature_points, used by kernels that iterate with a concrete topology value rather than a type.

Quick start

using JuliaFEM
using Tensors

points = get_quadrature_points(Hexahedron, GaussLegendre{2}())
# SVector{8, QuadraturePoint{3, Float64}}

for qp in points
    xi = qp.coords    # Vec{3, Float64}
    w  = qp.weight    # Float64
    # ... evaluate basis at xi and accumulate ...
end

Type parameters

GaussLegendre{N, V}:

• N is the order in the mathematical sense — points per dimension on tensor-product elements, or a rule index on simplices.
• V is a variant tag (:default, :B, …) used where a topology has more than one rule of the same order.

QuadraturePoint{D, T} carries a Vec{D, T} of parametric coordinates (deliberately a Tensors.jl Vec, for compatibility with the basis functions and Jacobian code) and a scalar weight.

Default rules

default_quadrature(1)                 # GaussLegendre{2}()
default_quadrature(Hexahedron, 1)     # GaussLegendre{2}()

The convention is GaussLegendre{basis_order + 1}(), which integrates the stiffness contribution of the corresponding Lagrange basis exactly.

Available rules

Tensor-product elements (segments, quads, hexes) provide GaussLegendre{1} through GaussLegendre{5} (1, 2, 3, 4, 5 points per direction). Triangles provide rules of total degrees 1 through 6 with some :B variants. Tetrahedra provide rules of degrees 1 through 4 (degree 4 keeps all weights positive). Wedges and pyramids each provide a small handful of rules; see the corresponding gl_*.jl file for the exact list.

Adding a rule

1. Implement the rule in the appropriate gl_*.jl file. Keep the method @inline, return an SVector of QuadraturePoint, and use Vec{D} (not SVector{D}) for the coordinates.
2. Add coverage under test/quadrature/ for point count, weight sum and polynomial exactness.
3. If you are extending default_quadrature, update the dispatch in api.jl.

Related code

• src/topology/ — the topologies these rules are attached to.
• src/basis/ — basis functions evaluated at the quadrature points.
• src/geometry/ — Jacobian-weighted integration.
• src/assemblers/ — every assembler iterates over these point lists.
• test/quadrature/ — public test suite.