topology¶

The topology module defines the topology objects, notably the StructuredTopology. Maintaining strict separation of topological and geometrical information, the topology represents a set of elements and their interconnectivity, boundaries, refinements, subtopologies etc, but not their positioning in physical space. The dimension of the topology represents the dimension of its elements, not that of the the space they are embedded in.

The primary role of topologies is to form a domain for nutils.function objects, like the geometry function and function bases for analysis, as well as provide tools for their construction. It also offers methods for integration and sampling, thus providing a high level interface to operations otherwise written out in element loops. For lower level operations topologies can be used as nutils.element iterators.

class nutils.topology.Topology(references, transforms, opposites)¶

Bases: nutils.types.Singleton

topology base class

__str__(self)¶: string representation

basis(self, name, *args, **kwargs)¶: Create a basis.

sample(self, ischeme, degree)¶: Create sample.

integrate_elementwise(self, funcs, *, asfunction=False, **kwargs)¶: element-wise integration

integrate(self, funcs, ischeme='gauss', degree=None, edit=None, *, arguments=None, title='integrate')¶: integrate functions

integral(self, func, ischeme='gauss', degree=None, edit=None)¶

projection(self, fun, onto, geometry, **kwargs)¶: project and return as function

project(self, fun, onto, geometry, ischeme='gauss', degree=None, droptol=1e-12, exact_boundaries=False, constrain=None, verify=None, ptype='lsqr', edit=None, *, arguments=None, **solverargs)¶: L2 projection of function onto function space

refined_by(self, refine)¶: create refined space by refining dofs in existing one

refine(self, n)¶: refine entire topology n times

trim(self, levelset, maxrefine, ndivisions=8, name='trimmed', leveltopo=None, *, arguments=None)¶: trim element along levelset

subset(self, topo, newboundary=None, strict=False)¶: intersection

locate(self, geom, coords, *, ischeme='vertex', scale=1, tol=None, eps=0, maxiter=100, arguments=None)¶

Create a sample based on physical coordinates.

In a finite element application, functions are commonly evaluated in points that are defined on the topology. The reverse, finding a point on the topology based on a function value, is often a nonlinear process and as such involves Newton iterations. The locate function facilitates this search process and produces a nutils.sample.Sample instance that can be used for the subsequent evaluation of any function in the given physical points.

Example:

>>> from . import mesh
>>> domain, geom = mesh.unitsquare(nelems=3, etype='mixed')
>>> sample = domain.locate(geom, [[.9, .4]])
>>> sample.eval(geom).tolist()
[[0.9, 0.4]]

Locate has a long list of arguments that can be used to steer the nonlinear search process, but the default values should be fine for reasonably standard situations.

Parameters

geom (1-dimensional nutils.function.Array) – Geometry function of length ndims.
coords (2-dimensional float array) – Array of coordinates with ndims columns.
tol (float) – Maximum allowed distance between original and located coordinate.
ischeme (str (default: “vertex”)) – Sample points used to determine bounding boxes.
scale (float (default: 1)) – Bounding box amplification factor, useful when element shapes are distorted. Setting this to >1 can increase computational effort but is otherwise harmless.
eps (float (default: 0)) – Epsilon radius around element within which a point is considered to be inside.
maxiter (int (default: 100)) – Maximum allowed number of Newton iterations.
arguments (dict (default: None)) – Arguments for function evaluation.

Returns

located

Return type

nutils.sample.Sample

boundary¶: Topology: The boundary of this topology.

basis_discont(self, degree)¶: discontinuous shape functions

basis_lagrange(self, degree)¶: lagrange shape functions

basis_bernstein(self, degree)¶: bernstein shape functions

basis_std(self, degree)¶: bernstein shape functions

class nutils.topology.WithGroupsTopology(basetopo, vgroups={}, bgroups={}, igroups={}, pgroups={})¶