Bases

In Nutils, a basis is a vector-valued function object that evaluates, in any given point $\xi$ on the topology, to the full array of basis function values ${\phi }_0(\xi ),{\phi }_1(\xi ),\dots ,{\phi }_{n-1}(\xi )$. It must be pointed out that Nutils will in practice operate only on the basis functions that are locally non-zero, a key optimization in Finite Element computations. But as a concept, it helps to think of a basis as evaluating always to the full array.

Several nutils.topology.Topology objects support creating bases via the Topology.basis() method. A nutils.topology.StructuredTopology, as generated by nutils.mesh.rectilinear, can create a spline basis with arbitrary degree and arbitrary continuity. The following generates a degree one spline basis on our previously created unit line topology topo:

basis = topo.basis('spline', degree=1)

The five basis functions are

myplot(topo, geom, basis)

We will use this basis throughout the following sections.

Change the degree argument to 2 for a quadratic spline basis:

myplot(topo, geom, topo.basis('spline', degree=2))

By default the continuity of the spline functions at element edges is the degree minus one. To change this, pass the desired continuity via keyword argument continuity. For example, a quadratic spline basis with $C^0$ continuity is generated with

myplot(topo, geom, topo.basis('spline', degree=2, continuity=0))

$C^0$ continuous spline bases can also be generated by the 'std' basis:

myplot(topo, geom, topo.basis('std', degree=2))

The 'std' basis is supported by topologies with square and/or triangular elements without hanging nodes.

Discontinuous basis functions are generated using the 'discont' type, e.g.

myplot(topo, geom, topo.basis('discont', degree=2))