Topology concepts

Topology concepts#

Domain#

Mathematicaly speaking, a domain \(\Omega\) is defined as an interior \(\a{\Omega}\) and a boundary \(\partial \Omega\). A domain is defined by its interior and its boundaries. For instance, a domain without a boundary is periodic, since, sympde does not consider infinite domains.

In addition to these two attributes, a Domain can also be defined as the image of a logical domain using a mapping.

Finally, a Domain can have subdomains, in which case, a connectivity must be provided as well.

The simplest Domain is the NCube object. For instance, Line, Square and Cube are specific cases of NCube.

Let’s take a look at the Line object.

>>> from sympde import topology
>>> domain = topology.Line()
>>> print(type(domain))
sympde.topology.domain.Line

The parent class of Line is NCube as we can see in our example

>>> print(type(topology.Line.__bases__))
sympde.topology.domain.NCube

In fact, the objects Line, Square and Cube are subclasses of NCube which is a subclasse of Domain that is an extension of the BasicDomain object.

Properties#

  • name: returns the domain’s name

  • interior: returns the interior of the domain

  • boundary: returns the boundary of the domain

  • mapping: returns the mapping, if associated to the domain

  • logical_domain: returns the logical domain if a mapping is used

  • connectivity: returns information about the interfaces of the subdomains

  • dim: dimension of the space

  • dtype: a dictionary that contains information about the domain

  • interfaces: returns the interfaces of the subdomains

  • corners: returns the corners of the domain

TODO add inherited properties

Examples#

Example 1#

>>> from sympde.topology import InteriorDomain, Boundary, Domain

>>> Omega_1 = InteriorDomain('Omega_1', dim=2)
>>> Omega_2 = InteriorDomain('Omega_2', dim=2)

>>> Gamma_1 = Boundary('Gamma_1', Omega_1)
>>> Gamma_2 = Boundary('Gamma_2', Omega_2)
>>> Gamma_3 = Boundary('Gamma_3', Omega_2)

>>> Omega = Domain('Omega',
                   interiors=[Omega_1, Omega_2],
                   boundaries=[Gamma_1, Gamma_2, Gamma_3])

>>> print( Omega.dim )
2
>>> print( len(Omega.interior) )
2
>>> print( len(Omega.boundary) )
3

Example 2#

>>> from sympde.topology import InteriorDomain, Boundary, Domain

>>> Omega_1 = InteriorDomain('Omega_1', dim=2)

>>> Gamma_1 = Boundary('Gamma_1', Omega_1)
>>> Gamma_2 = Boundary('Gamma_2', Omega_1)

>>> Omega = Domain('Omega',
                   interiors=[Omega_1],
                   boundaries=[Gamma_1, Gamma_2])

>>> assert(Omega.boundary == Union(Gamma_1, Gamma_2))
>>> assert(Omega.boundary.complement(Gamma_1) == Gamma_2)
>>> assert(Omega.boundary - Gamma_1 == Gamma_2)

PeriodicDomain#

Similar to the Domain object, a PeriodicDomain is defined by giving a Domain and the associated periods.

Properties#

  • domain: returns the domain

  • periods: returns the flags for each axis

  • boundary: return the boundary of the domain

  • dim returns the dimension of the domain

  • coordinates return the coordinates symbols

NCube#

An NCube is a Domain that models hypercubes in \(\mathbb{R}^n\). It is defined by specifying its name, dim, min_coords and max_coords.

The interior of a NCube is defined as NCubeInterior. To make things easier, we provide specific classes in the case where \(n \in \{ 1, 2, 3 \}\).

Line#

This corresponds to the case where \(n=1\), meaning it is \((x_{min}, x_{max})\).

Properties#

  • bounds: the bounds for the line, i.e. \(x_{min}\) and \(x_{max}\).

Square#

This corresponds to the case where \(n=2\), meaning it is \((x_{min}, x_{max}) \times (y_{min}, y_{max})\).

Properties#

  • bounds1: the bounds in the first axis, i.e. \(x_{min}\) and \(x_{max}\).

  • bounds2: the bounds in the first axis, i.e. \(y_{min}\) and \(y_{max}\).

Cube#

This corresponds to the case where \(n=3\), meaning it is \((x_{min}, x_{max}) \times (y_{min}, y_{max}) \times (z_{min}, z_{max})\).

Properties#

  • bounds1: the bounds in the first axis, i.e. \(x_{min}\) and \(x_{max}\).

  • bounds2: the bounds in the first axis, i.e. \(y_{min}\) and \(y_{max}\).

  • bounds3: the bounds in the first axis, i.e. \(z_{min}\) and \(z_{max}\).

InteriorDomain#

TODO

Examples#

Example 1#

In this example, we create two interior domains and then we build their union.

>>> from sympde.topology import InteriorDomain, Union

>>> D1 = InteriorDomain('D1', dim=2)
>>> D2 = InteriorDomain('D2', dim=2)

>>> # Union is commutative
>>> assert( Union(D2, D1) == Union(D1, D2) )

>>> D = Union(D1, D2)
>>> # the union will be of dimension 2, and has a length of 2
>>> print(D.dim)
2
>>> print(len(D))
2
>>> # we see it clearly when converting it to a dictionary
>>> print(D.todict())
[{'name': 'D1', 'mapping': 'None'}, {'name': 'D2', 'mapping': 'None'}]

NCubeInterior#

This class defines the interior of a domain which is NCube.

Properties#

  • min_coords

  • max_coords

  • boundary

Additional geometry objects#

Vectors#

BoundaryVector#

is an extension of the IndexedBase object from sympy.

NormalVector#

defines the normal vector to boundaries. It is an extension of the BoundaryVector object.

MinusNormalVector#

defines the inward normal vector to a boundary. It is an extension of the NormalVector object.

MinusNormalVector#

defines the outward normal vector to a boundary. It is an extension of the NormalVector object.

TangentVector#

defines the tangent vector to boundaries. It is an extension of the BoundaryVector object.

ElementDomain#

defines an element of the domain.

BasicArea#

A basic class defining the concept of an area. It is an extension of the AtomicExpr object from sympy, this way we can use it in arithmetic expressions.

DomainArea#

defines the area of a Domain.

ElementArea#

defines the area of an element of a Domain.

Geometry Operations#

TODO

BasicGeometryOperator#

TODO

Area#

TODO

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