Symfem¶

Welcome to the documention of Symfem: a symbolic finite element definition library.

Symfem can be used to create a very wide range of finite element spaces. The basis functions of these spaces can be computed symbolically, allowing them to easily be further manipulated.

Installing Symfem¶

Symfem can be installed from GitHub repo by running:

git clone https://github.com/mscroggs/symfem.git
cd symfem
python3 setup.py install

Alternatively, the latest release can be installed from PyPI by running:

pip3 install symfem

… or from Conda by running:

conda install symfem

Using Symfem¶

Finite elements¶

Finite elements can be created in Symfem using the symfem.create_element() function. For example, some elements are created in the following snippet:

import symfem

lagrange = symfem.create_element("triangle", "Lagrange", 1)
rt = symfem.create_element("tetrahedron", "Raviart-Thomas", 2)
nedelec = symfem.create_element("triangle", "N2curl", 1)
qcurl = symfem.create_element("quadrilateral", "Qcurl", 2)

create_element will create a symfem.finite_element.FiniteElement object. The basis functions spanning the finite element space can be obtained, or tabulated at a set of points:

import symfem

lagrange = symfem.create_element("triangle", "Lagrange", 1)
print(lagrange.get_basis_functions())

points = [[0, 0], [0.5, 0], [1, 0], [0.25, 0.25]]
print(lagrange.tabulate_basis(points))

[-x - y + 1, x, y]
[(1, 0, 0), (0.500000000000000, 0.500000000000000, 0), (0, 1, 0), (0.500000000000000, 0.250000000000000, 0.250000000000000)]

Each basis function will be a Sympy symbolic expression.

The majority of the elements in Symfem are defined using Ciarlet’s [Ciarlet] definition of a finite element (symfem.finite_element.CiarletElement): these elements are using a polynomial set, and a set of functionals. In Symfem, the polynomial set of an element can be obtained by:

import symfem

lagrange = symfem.create_element("triangle", "Lagrange", 1)
print(lagrange.get_polynomial_basis())

[1, x, y]

The functionals of the finite element space can be obtained with the following snippet.

import symfem

lagrange = symfem.create_element("triangle", "Lagrange", 1)
print(lagrange.dofs)

[<symfem.functionals.PointEvaluation object at 0x{ADDRESS}>, <symfem.functionals.PointEvaluation object at 0x{ADDRESS}>, <symfem.functionals.PointEvaluation object at 0x{ADDRESS}>]

Each functional will be a functional defined in symfem.functionals.

Reference cells¶

Reference cells can be obtained from a symfem.finite_element.FiniteElement:

import symfem

lagrange = symfem.create_element("triangle", "Lagrange", 1)
reference = lagrange.reference

Alternatively, reference cells can be created using the symfem.create_reference() function. For example:

import symfem

triangle = symfem.create_reference("triangle")
interval = symfem.create_reference("interval")
tetrahedron = symfem.create_reference("tetrahedron")
triangle2 = symfem.create_reference("triangle", ((0, 0, 0), (0, 1, 0), (1, 0, 1)))

In the final example, the vertices of the reference have been provided, so a reference with these three vertices will be created.

Various information about the reference can be accessed. The reference cell’s subentities can be obtained:

import symfem

triangle = symfem.create_reference("triangle")
print(triangle.vertices)
print(triangle.edges)

((0, 0), (1, 0), (0, 1))
((1, 2), (0, 2), (0, 1))

The origin and axes of the element can be obtained:

import symfem

triangle = symfem.create_reference("triangle")
print(triangle.origin)
print(triangle.axes)

(0, 0)
((1, 0), (0, 1))

The topological and geometric dimensions of the element can be obtained:

import symfem

triangle = symfem.create_reference("triangle")
print(triangle.tdim)
print(triangle.gdim)

triangle2 = symfem.create_reference("triangle", ((0, 0, 0), (0, 1, 0), (1, 0, 1)))
print(triangle2.tdim)
print(triangle2.gdim)

The reference types of the subentities can be obtained. This can be used to create a reference representing a subentity:

import symfem

triangle = symfem.create_reference("triangle")
print(triangle.sub_entity_types)

vertices = []
for i in triangle.edges[0]:
    vertices.append(triangle.vertices[i])
edge0 = symfem.create_reference(
    triangle.sub_entity_types[1], vertices)
print(edge0.vertices)

['point', 'interval', 'triangle', None]
((1, 0), (0, 1))

Documentation index¶

References¶

[Ciarlet]

Ciarlet, P. G., The Finite Element Method for Elliptic Problems (2002, first published 1978) DOI: 10.1137/1.9780898719208