fe_utils package¶
Subpackages¶
- fe_utils.scripts package
- Submodules
- fe_utils.scripts.plot_function_space_nodes module
- fe_utils.scripts.plot_interpolate_lagrange module
- fe_utils.scripts.plot_lagrange_basis_functions module
- fe_utils.scripts.plot_lagrange_points module
- fe_utils.scripts.plot_mesh module
- fe_utils.scripts.plot_sin_function module
- Module contents
- fe_utils.solvers package
Submodules¶
fe_utils.finite_elements module¶
- class fe_utils.finite_elements.FiniteElement(cell, degree, nodes, entity_nodes=None)[source]¶
Bases:
object
A finite element defined over cell.
- Parameters
cell – the
ReferenceCell
over which the element is defined.degree – the polynomial degree of the element. We assume the element spans the complete polynomial space.
nodes – a list of coordinate tuples corresponding to point evaluation node locations on the element.
entity_nodes – a dictionary of dictionaries such that entity_nodes[d][i] is the list of nodes associated with entity (d, i) of dimension d and index i.
Most of the implementation of this class is left as exercises.
- cell¶
The
ReferenceCell
over which the element is defined.
- degree¶
The polynomial degree of the element. We assume the element spans the complete polynomial space.
- entity_nodes¶
A dictionary of dictionaries such that
entity_nodes[d][i]
is the list of nodes associated with entity (d, i).
- interpolate(fn)[source]¶
Interpolate fn onto this finite element by evaluating it at each of the nodes.
- Parameters
fn – A function
fn(X)
which takes a coordinate vector and returns a scalar value.- Returns
A vector containing the value of
fn
at each node of this element.
The implementation of this method is left as an exercise.
- node_count¶
The number of nodes in this element.
- nodes¶
The list of coordinate tuples corresponding to the nodes of the element.
- nodes_per_entity¶
nodes_per_entity[d]
is the number of entities associated with an entity of dimension d.
- tabulate(points, grad=False)[source]¶
Evaluate the basis functions of this finite element at the points provided.
- Parameters
points – a list of coordinate tuples at which to tabulate the basis.
grad – whether to return the tabulation of the basis or the tabulation of the gradient of the basis.
- Result
an array containing the value of each basis function at each point. If grad is True, the gradient vector of each basis vector at each point is returned as a rank 3 array. The shape of the array is (points, nodes) if
grad
isFalse
and (points, nodes, dim) ifgrad
isTrue
.
The implementation of this method is left as an exercise.
- class fe_utils.finite_elements.LagrangeElement(cell, degree)[source]¶
Bases:
fe_utils.finite_elements.FiniteElement
An equispaced Lagrange finite element.
- Parameters
cell – the
ReferenceCell
over which the element is defined.degree – the polynomial degree of the element. We assume the element spans the complete polynomial space.
The implementation of this class is left as an exercise.
- fe_utils.finite_elements.lagrange_points(cell, degree)[source]¶
Construct the locations of the equispaced Lagrange nodes on cell.
- Parameters
cell – the
ReferenceCell
degree – the degree of polynomials for which to construct nodes.
- Returns
a rank 2
array
whose rows are the coordinates of the nodes.
The implementation of this function is left as an exercise.
- fe_utils.finite_elements.vandermonde_matrix(cell, degree, points, grad=False)[source]¶
Construct the generalised Vandermonde matrix for polynomials of the specified degree on the cell provided.
- Parameters
cell – the
ReferenceCell
degree – the degree of polynomials for which to construct the matrix.
points – a list of coordinate tuples corresponding to the points.
grad – whether to evaluate the Vandermonde matrix or its gradient.
- Returns
the generalised Vandermonde matrix
The implementation of this function is left as an exercise.
fe_utils.function_spaces module¶
- class fe_utils.function_spaces.Function(function_space, name=None)[source]¶
Bases:
object
A function in a finite element space. The main role of this object is to store the basis function coefficients associated with the nodes of the underlying function space.
- Parameters
function_space – The
FunctionSpace
in which thisFunction
lives.name – An optional label for this
Function
which will be used in output and is useful for debugging.
- function_space¶
The
FunctionSpace
in which thisFunction
lives.
- interpolate(fn)[source]¶
Interpolate a given Python function onto this finite element
Function
.- Parameters
fn – A function
fn(X)
which takes a coordinate vector and returns a scalar value.
- plot(subdivisions=None)[source]¶
Plot the value of this
Function
. This is quite a low performance plotting routine so it will perform poorly on larger meshes, but it has the advantage of supporting higher order function spaces than many widely available libraries.- Parameters
subdivisions – The number of points in each direction to use in representing each element. The default is \(2d+1\) where \(d\) is the degree of the
FunctionSpace
. Higher values produce prettier plots which render more slowly!
- class fe_utils.function_spaces.FunctionSpace(mesh, element)[source]¶
Bases:
object
A finite element space.
- Parameters
mesh – The
Mesh
on which this space is built.element – The
FiniteElement
of this space.
Most of the implementation of this class is left as an exercise.
- cell_nodes¶
The global cell node list. This is a two-dimensional array in which each row lists the global nodes incident to the corresponding cell. The implementation of this member is left as an exercise
- element¶
The
FiniteElement
of this space.
- node_count¶
The total number of nodes in the function space.
fe_utils.mesh module¶
- class fe_utils.mesh.Mesh(vertex_coords, cell_vertices)[source]¶
Bases:
object
A one or two dimensional mesh composed of intervals or triangles respectively.
- Parameters
vertex_coords – a vertex_count x dim array of the coordinates of the vertices in the mesh.
cell_vertices – a cell_count x (dim+1) array of the indices of the vertices of which each cell is made up.
- adjacency(dim1, dim2)[source]¶
Return the set of dim2 entities adjacent to each dim1 entity. For example if dim1==2 and dim2==1 then return the list of edges (1D entities) adjacent to each triangle (2D entity).
The return value is a rank 2
numpy.array
such thatadjacency(dim1, dim2)[e1, :]
is the list of dim2 entities adjacent to entity(dim1, e1)
.This operation is only defined where self.dim >= dim1 > dim2.
This method is simply a more systematic way of accessing
edge_vertices
,cell_edges
andcell_vertices
.
- cell¶
The
ReferenceCell
of which thisMesh
is composed.
- cell_edges¶
The indices of the edges incident to each cell (only for 2D meshes).
- cell_vertices¶
The indices of the vertices incident to cell.
- dim¶
The geometric and topological dimension of the mesh. Immersed manifolds are not supported.
- edge_vertices¶
The indices of the vertices incident to edge (only for 2D meshes).
- entity_counts¶
The number of entities of each dimension in the mesh. So
entity_counts[0]
is the number of vertices in the mesh.
- jacobian(c)[source]¶
Return the Jacobian matrix for the specified cell.
- Parameters
c – The number of the cell for which to return the Jacobian.
- Result
The Jacobian for cell
c
.
- vertex_coords¶
The coordinates of all the vertices in the mesh.
- class fe_utils.mesh.UnitIntervalMesh(nx)[source]¶
Bases:
fe_utils.mesh.Mesh
A mesh of the unit interval.
- Parameters
nx – The number of cells.
- class fe_utils.mesh.UnitSquareMesh(nx, ny)[source]¶
Bases:
fe_utils.mesh.Mesh
A triangulated
Mesh
of the unit square.- Parameters
nx – The number of cells in the x direction.
ny – The number of cells in the y direction.
fe_utils.quadrature module¶
- class fe_utils.quadrature.QuadratureRule(cell, degree, points, weights)[source]¶
Bases:
object
A quadrature rule implementing integration of the reference cell provided.
- Parameters
cell – the
ReferenceCell
over which this quadrature rule is defined.degree – the degree of precision of this quadrature rule.
- Points
a list of the position vectors of the quadrature points.
- Weights
the corresponding vector of quadrature weights.
- cell¶
The
ReferenceCell
over which this quadrature rule is defined.
- degree¶
The degree of precision of the quadrature rule.
- integrate(function)[source]¶
Integrate the function provided using this quadrature rule.
- Parameters
function – A Python function taking a position vector as its single argument and returning a scalar value.
The implementation of this method is left as an exercise.
- points¶
Two dimensional array, the rows of which form the position vectors of the quadrature points.
- weights¶
The corresponding array of quadrature weights.
- fe_utils.quadrature.gauss_quadrature(cell, degree)[source]¶
Return a Gauss-Legendre
QuadratureRule
.- Parameters
cell – the
ReferenceCell
over which this quadrature rule is defined.degree – the degree of precision of this quadrature rule.
fe_utils.reference_elements module¶
- class fe_utils.reference_elements.ReferenceCell(vertices, topology, name)[source]¶
Bases:
object
An object storing the geometry and topology of the reference cell.
- Parameters
vertices – a list of coordinate vectors corresponding to the coordinates of the vertices of the cell.
topology – a dictionary of dictionaries such that topology[d][i] is the list of vertices incident to the i-th entity of dimension d.
- dim¶
The geometric and topological dimension of the reference cell.
- entity_counts¶
The number of entities of each dimension.
- point_in_entity(x, e)[source]¶
Return true if the point x lies on the entity e.
- Parameters
x – The coordinate vector of the point.
e – The (d, i) pair describing the entity of dimension d and index i.
- topology¶
The vertices making up each topological entity of the reference cell.
- vertices¶
The list of coordinate veritices of the reference cell.
- fe_utils.reference_elements.ReferenceInterval = ReferenceInterval¶
A
ReferenceCell
storing the geometry and topology of the interval [0, 1].
- fe_utils.reference_elements.ReferenceTriangle = ReferenceTriangle¶
A
ReferenceCell
storing the geometry and topology of the triangle with vertices [[0., 0.], [1., 0.], [0., 1.]].