Reference-node generation (internal)

Reference-node generators that the basis families build on. More...

Collaboration diagram for Reference-node generation (internal):

Classes
struct	svmp::FE::basis::LagrangeNodeLayout
	Reference Lagrange node coordinates paired with their integer lattice index. More...
class	svmp::FE::basis::ReferenceNodeLayout
	Reference-node coordinate and count lookups for an element type. More...

Functions
double	svmp::FE::basis::line_coord_pm_one (int i, int order)
	The i-th 1D tensor-axis reference node on [-1, 1] at the given order.

Detailed Description

Reference-node generators that the basis families build on.

Warning

Internal implementation detail. Do not use these directly: obtain a basis through basis_factory and read its nodes via BasisFunction::nodes(). These declarations are part of the internal node-ordering machinery and their interface may change without notice.

This is the reference-node generator the basis families build on, not a consumer entry point. It is documented for FE core developers; model-level code never calls it directly.

Function Documentation

line_coord_pm_one()

double svmp::FE::basis::line_coord_pm_one	(	int	i,
		int	order
	)

The i-th 1D tensor-axis reference node on [-1, 1] at the given order.

Returns the Gauss-Lobatto-Legendre (GLL) node of index i for a degree-order distribution: the endpoints are -1 and +1 and the interior nodes are the roots of \(P'_{order}\), so high-order tensor interpolation stays well-conditioned (a logarithmic Lebesgue constant instead of the exponential growth of equispaced nodes). At order 1 the nodes are \(\{-1, +1\}\) and at order 2 \(\{-1, 0, +1\}\), so they coincide with the equispaced layout for the production orders and differ only for order >= 3. Returns 0 for order <= 0 when i is 0. Invalid indices throw.

This is the single definition of the tensor-axis node distribution: the reference-node layout generators, the Lagrange tensor-axis initialization, and the serendipity edge/face/interior strata all source their 1D nodes here. The LagrangeBasis and SerendipityBasis docs point back to this description of the GLL distribution and its conditioning rather than restating it.

Parameters

i	Node index in [0, order] for positive orders, or 0 for order <= 0.
order	Polynomial order of the 1D distribution.

Returns

GLL node coordinate on [-1, 1].

Exceptions

BasisNodeOrderingException

If i is outside the valid range.