In Basix, it is possible to create custom finite elements. This demo describes how to do this and what data you need to provide when creating a custom element.
First, we import Basix and Numpy.
import basix
import numpy as np
from basix import CellType, MapType, PolynomialType, LatticeType
As a first example, we create a degree 1 Lagrange element with a quadratic bubble on a quadrilateral cell. This element will span the following set of polynomials:
We will define the degrees of freedom (DOFs) of this element by placing a point evaluation at each vertex, plus one at the midpoint of the cell.
When creating a custom element, we must input the coefficients that define a basis of the set of polynomials that our element spans. In this example, we will represent the 5 functions above in terms of the 9 orthogonal polynomials of degree \(\leqslant2\) on a quadrilateral, so we create a 5 by 9 matrix.
wcoeffs = np.zeros((5, 9))
The degree 3 orthonormal polynomials for a quadrilateral will have their highest degree terms in the following order:
The order in which the polynomials appear in the orthonormal polynomial sets for each cell are documented at https://docs.fenicsproject.org/basix/main/polyset-order.html.
As our polynomial space contains 1, \(y\), \(x\) and \(xy\). The first four rows of the matrix contain a single 1 for the four orthogonal polynomials with these are their highest degree terms.
wcoeffs[0, 0] = 1
wcoeffs[1, 1] = 1
wcoeffs[2, 3] = 1
wcoeffs[3, 4] = 1
The final row of the matrix defines the polynomials \(x(1-x)y(1-y)\). As the polynomials are orthonormal, we can represent this as
where $p_0$ to $p_8$ are the orthonormal polynomials. Therefore the coefficients we want to put in the final row of our matrix are:
We compute these integrals using a degree 4 quadrature rule (this is the largest degree that the integrand will be, so these integrals will be exact).
pts, wts = basix.make_quadrature(CellType.quadrilateral, 4)
poly = basix.tabulate_polynomials(PolynomialType.legendre, CellType.quadrilateral, 2, pts)
x = pts[:, 0]
y = pts[:, 1]
f = x * (1 - x) * y * (1 - y)
for i in range(9):
wcoeffs[4, i] = sum(f * poly[:, i] * wts)
Next, we compute the points and matrices that define how functions can be interpolated into this space. These are representations of the functionals that are used in the Ciarlet definition of the finite element – in this example, these are evaluations at the points described above.
First, we define the points. We create an array of points for each entity of each dimension. For each vertex of the cell, we include the coordinates of that vertex. For the interior of the cell, we include a point at \((0.5,0.5)\).
The shape of each of the point lists is (number of points, dimension).
x = [[], [], [], []]
x[0].append(np.array([[0.0, 0.0]]))
x[0].append(np.array([[1.0, 0.0]]))
x[0].append(np.array([[0.0, 1.0]]))
x[0].append(np.array([[1.0, 1.0]]))
x[2].append(np.array([[0.5, 0.5]]))
There are no DOFs associates with the edges for this element, so we add an empty array of points for each edge.
for _ in range(4):
x[1].append(np.zeros((0, 2)))
We then define the interpolation matrices that define how the evaluations at the points are combined to evaluate the functionals. As all the DOFs are point evaluations in this example, the matrices are all identity matrices for the entities that have a point.
The shape of each matrix is (number of DOFs, value size, number of points).
M = [[], [], [], []]
for _ in range(4):
M[0].append(np.array([[[1.]]]))
M[2].append(np.array([[[1.]]]))
There are no DOFs associates with the edges for this element, so we add an empty matrix for each edge.
for _ in range(4):
M[1].append(np.zeros((0, 1, 0)))
We now create the custom element. The inputs into basix.create_custom_element are:
The cell type. In this example, this is a quadrilateral.
The polynomial degree of the element. In this example, this is 2.
The value shape of the element. In this example, this is [] as the element is scalar.
The coefficients that define the polynomial set. In this example, this is wcoeffs.
The points used to define interpolation into the element. In this example, this is x.
The matrix used to define interpolation into the element. In this example, this is M.
The map type. In this example, this is the identity map.
A bool indicating whether the element is discontinuous. In this example, this is False.
The highest degree \(n\) such that all degree \(n\) polynomials are contained in this set. In this example, this is 1.
element = basix.create_custom_element(
CellType.quadrilateral, 2, [], wcoeffs, x, M, MapType.identity, False, 1)
We can now use this element in the same way we can use a built-in element. For example, we can tabulate the element at a set of points. If the points we use are the same as the points we used to define the DOFs, we see that each basis function is equal to 1 at one of these points, and equal to zero at all the other points.
points = np.array([[0.0, 0.0], [1.0, 0.0], [0.0, 1.0], [1.0, 1.0], [0.5, 0.5]])
print(element.tabulate(0, points))
As a second example, we create a degree 1 Raviart–Thomas element on a triangle. Details of the definition of this element can be found at https://defelement.com/elements/raviart-thomas.html. This element spans:
The DOFs of this element are integrals along each edge of the cell of the dot product of the function with the normal to the edge.
In contrast to the scalar-valued element above, this element has a value size of 2.
In this example, we will represent the 3 functions above in terms of the 3 orthogonal polynomials of degree \(\leqslant1\) on a triangle in each of the two coordinate directions. We therefore create a 3 by 6 matrix.
wcoeffs = np.zeros((3, 6))
The highest degree terms in each polynomial will be:
We include \((1,0)\) and \((0,1)\) as the first two rows of the matrix, and use integrals to represent \((x,y)\) as in the previous example.
wcoeffs[0, 0] = 1
wcoeffs[1, 3] = 1
pts, wts = basix.make_quadrature(CellType.triangle, 2)
poly = basix.tabulate_polynomials(PolynomialType.legendre, CellType.triangle, 1, pts)
x = pts[:, 0]
y = pts[:, 1]
for i in range(3):
wcoeffs[2, i] = sum(x * poly[:, i] * wts)
wcoeffs[2, 3 + i] = sum(y * poly[:, i] * wts)
For this element, there will be multiple points used per DOF, as the functionals that define the element are integrals. We begin by defining a degree 1 quadrature rule on an interval. This quadrature rule will be used to integrate on the edges of the triangle.
pts, wts = basix.make_quadrature(CellType.interval, 1)
The points associated with each edge are calculated by mapping the quadrature points to each edge.
x = [[], [], [], []]
for _ in range(3):
x[0].append(np.zeros((0, 2)))
x[1].append(np.array([[1 - p[0], p[0]] for p in pts]))
x[1].append(np.array([[0, p[0]] for p in pts]))
x[1].append(np.array([[p[0], 0] for p in pts]))
x[2].append(np.zeros((0, 2)))
The interpolation matrices for the edges in this example will be have shape (1, 2, len(pts)), as there is one DOF per edge, the value size is 2, and we have len(pts) quadrature points on each edge. The entries of these matrices are the quadrature weights multiplied by the normal directions.
M = [[], [], [], []]
for _ in range(3):
M[0].append(np.zeros((0, 2, 0)))
for normal in [[-1, -1], [-1, 0], [0, 1]]:
M[1].append(np.array([[normal[0] * wts, normal[1] * wts]]))
M[2].append(np.zeros((0, 2, 0)))
element = basix.create_custom_element(
CellType.triangle, 1, [2], wcoeffs, x, M, MapType.contravariantPiola, False, 0)
To confirm that we have defined this element correctly, we compare it to the built-in Raviart–Thomas element.
rt = basix.create_element(basix.ElementFamily.RT, CellType.triangle, 1)
points = basix.create_lattice(CellType.triangle, 1, LatticeType.equispaced, True)
assert np.allclose(rt.tabulate(0, points), element.tabulate(0, points))