Interpolation ¶

Firedrake offers various ways to interpolate expressions onto fields (Functions). Interpolation is often used to set up initial conditions and/or boundary conditions. The basic syntax for interpolation is:

# create new function f on function space V
f = interpolate(expression, V)

# alternatively:
f = Function(V).interpolate(expression)

# setting the values of an existing function
f.interpolate(expression)

Note

Interpolation is supported for most, but not all, of the elements that Firedrake provides. In particular, higher-continuity elements such as Argyris and Hermite do not presently support interpolation.

The recommended way to specify the source expression is UFL. UFL produces clear error messages in case of syntax or type errors, yet UFL expressions have good run-time performance, since they are translated to C interpolation kernels using TSFC technology. Moreover, UFL offers a rich language for describing expressions, including:

The coordinates: in physical space as SpatialCoordinate, and in reference space as ufl.geometry.CellCoordinate.
Firedrake Functions, derivatives of Functions, and Constants.
Literal numbers, basic arithmetic operations, and also mathematical functions such as sin, cos, sqrt, abs, etc.
Conditional expressions using UFL conditional.
Compound expressions involving any of the above.

Here is an example demonstrating some of these features:

# g is a vector-valued Function, e.g. on an H(div) function space
f = interpolate(sqrt(3.2 * div(g)), V)

This also works as expected when interpolating into a a space defined on the facets of the mesh:

# where trace is a trace space on the current mesh:
f = interpolate(expression, trace)

Interpolator objects ¶

Firedrake is also able to generate reusable Interpolator objects which provide caching of the interpolation operation. The following line creates an interpolator which will interpolate the current value of expression into the space V:

interpolator = Interpolator(expression, V)

If expression does not contain a TestFunction() then the interpolation can be performed with:

f = interpolator.interpolate()

Alternatively, one can use the interpolator to set the value of an existing Function:

f = Function(V)
interpolator.interpolate(output=f)

If expression contains a TestFunction() then the interpolator acts to interpolate Functions in the test space to those in the target space. For example:

w = TestFunction(W)
interpolator = Interpolator(w, V)

Here, interpolator acts as the interpolation matrix from the FunctionSpace() W into the FunctionSpace() V. Such that if f is a Function in W then g = interpolator.interpolate(f) is its interpolation into a function g in V. As before, the output parameter can be used to write into an existing Function. Passing the transpose=True option to interpolate() will cause the transpose interpolation to occur. This is equivalent to the multigrid restriction operation which interpolates assembled 1-forms in the dual space to V to assembled 1-forms in the dual space to W.

Interpolation across meshes ¶

The interpolation API supports interpolation between meshes where the target function space has finite elements (as given in the list of supported elements)

Lagrange/CG (also known a Continuous Galerkin or P elements),
Q (i.e. Lagrange/CG on lines, quadrilaterals and hexahedra),
Discontinuous Lagrange/DG (also known as Discontinuous Galerkin or DP elements) and
DQ (i.e. Discontinuous Lagrange/DG on lines, quadrilaterals and hexahedra).

Vector, tensor and mixed function spaces can also be interpolated into from other meshes as long as they are constructed from these spaces.

Note

The list of supported elements above is only for target function spaces. Function spaces on the source mesh can be built from most of the supported elements.

There are few constraints on the meshes involved: the target mesh can have a different cell shape, topological dimension, or resolution to the source mesh. There are many use cases for this: For example, two solutions to the same problem calculated on meshes with different resolutions or cell shapes can be interpolated onto one another, or onto a third, finer mesh, and be directly compared.

Interpolating onto sub-domain meshes ¶

The target mesh for a cross-mesh interpolation need not cover the full domain of the source mesh. Volume, surface and line integrals can therefore be calculated by interpolating onto the mesh or immersed manifold which defines the volume, surface or line of interest in the domain. The integral itself is calculated by calling assemble() on an approriate form over the target mesh function space:

# Start with a simple field exactly represented in the function space over
# the unit square domain.
m = UnitSquareMesh(2, 2)
V = FunctionSpace(m, "CG", 2)
x, y = SpatialCoordinate(m)
f = Function(V).interpolate(x * y)

# We create a 1D mesh immersed 2D from (0, 0) to (1, 1) which we call "line".
# We want to calculate the line integral of f along it. To do this we
# create a function space on the line mesh...
V_line = FunctionSpace(line, "CG", 2)

# ... and interpolate our function f onto it.
f_line = interpolate(f, V_line)

# The integral of f along the line is then a simple form expression which
# we assemble:
assemble(f_line * dx)  # this outputs sqrt(2) / 3

For more on forms, see this section of the manual.

Interpolating onto other meshes ¶

Note

Interpolation from high-order meshes is currently not supported.

If the target mesh extends outside the source mesh domain, then cross-mesh interpolation will raise a DofNotDefinedError.

# These meshes only share some of their domain
src_mesh = UnitSquareMesh(2, 2)
dest_mesh = UnitSquareMesh(3, 3, quadrilateral=True)
dest_mesh.coordinates.dat.data_wo[:] *= 2

# We consider a simple function on our source mesh...
x_src, y_src = SpatialCoordinate(src_mesh)
V_src = FunctionSpace(src_mesh, "CG", 2)
f_src = Function(V_src).interpolate(x_src**2 + y_src**2)

# ... and want to interpolate into a function space on our target mesh ...
V_dest = FunctionSpace(dest_mesh, "Q", 2)

# ... but get a DofNotDefinedError if we try
f_dest = interpolate(f_src, V_dest)  # raises DofNotDefinedError

This can be overriden with the optional allow_missing_dofs keyword argument:

f_dest = interpolate(f_src, V_dest, allow_missing_dofs=True)

f_dest.at(0.5, 0.5)  # returns 0.5**2 + 0.5**2 = 0.5

In this case, the missing degrees of freedom (DoFs, the global basis function coefficients which could not be set) are, by default, set to zero:

f_dest.at(1.5, 1.5)  # returns 0.0

If we specify an output Function then the missing DoFs are unmodified.

We can optionally specify a value to use for our missing DoFs. Here we set them to be nan (‘not a number’) for easy identification:

f_dest = interpolate(
    f_src, V_dest, allow_missing_dofs=True, default_missing_val=np.nan
)
f_dest.at(1.5, 1.5)  # returns np.nan

If we specify an output Function, this overwrites the missing DoFs.

When using Interpolators, the allow_missing_dofs keyword argument is set at construction:

interpolator = Interpolator(f_src, V_dest, allow_missing_dofs=True)

The default_missing_val keyword argument is then set whenever we call interpolate():

f_dest = interpolator.interpolate(default_missing_val=np.nan)

If we supply an output Function and don’t set default_missing_val then any missing DoFs are left as they were prior to interpolation:

x_dest, y_dest = SpatialCoordinate(dest_mesh)
f_dest = Function(V_dest).interpolate(x_dest + y_dest)
f_dest.at(0.5, 0.5)  # returns x_dest + y_dest = 1.0

interpolator.interpolate(output=f_dest)
f_dest.at(0.5, 0.5)  # now returns x_src^2 + y_src^2 = 0.5

# Similarly, using the interpolate method on a function will not overwrite
# the pre-existing values if default_missing_val is not set
f_dest.interpolate(f_src, allow_missing_dofs=True)

Interpolation from external data ¶

Unfortunately, UFL interpolation is not applicable if some of the source data is not yet available as a Firedrake Function or UFL expression. Here we describe a recipe for moving external to Firedrake fields.

Let us assume that there is some function mydata(X) which takes as input an \(n \times d\) array, where \(n\) is the number of points at which the data values are needed, and \(d\) is the geometric dimension of the mesh. mydata(X) shall return a \(n\) long vector of the scalar values evaluated at the points provided. (Assuming that the target FunctionSpace is scalar valued, although this recipe can be extended to vector or tensor valued fields.) Presumably mydata works by interpolating the external data source, but the precise details are not relevant now. In this case, interpolation into a target function space V proceeds as follows:

# First, grab the mesh.
m = V.mesh()

# Now make the VectorFunctionSpace corresponding to V.
W = VectorFunctionSpace(m, V.ufl_element())

# Next, interpolate the coordinates onto the nodes of W.
X = interpolate(m.coordinates, W)

# Make an output function.
f = Function(V)

# Use the external data function to interpolate the values of f.
f.dat.data[:] = mydata(X.dat.data_ro)

This will also work in parallel, as the interpolation will occur on each process, and Firedrake will take care of the halo updates before the next operation using f.

For interaction with external point data, see the corresponding manual section.

C string expressions ¶

Warning

C string expressions were a FEniCS compatibility feature which has now been removed. Users should use UFL expressions instead. This section only remains to assist in the transition of existing code.

Here are a couple of old-style C string expressions, and their modern replacements.

# Expression:
f = interpolate(Expression("sin(x[0]*pi)"), V)

# UFL equivalent:
x = SpatialCoordinate(V.mesh())
f = interpolate(sin(x[0] * math.pi), V)

# Expression with a Constant parameter:
f = interpolate(Expression('sin(x[0]*t)', t=t), V)

# UFL equivalent:
x = SpatialCoordinate(V.mesh())
f = interpolate(sin(x[0] * t), V)

Python expression classes ¶

Warning

Python expression classes were a FEniCS compatibility feature which has now been removed. Users should use UFL expressions instead. This section only remains to assist in the transition of existing code.

Since Python Expression classes expressions are deprecated, below are a few examples on how to replace them with UFL expressions:

# Python expression:
class MyExpression(Expression):
    def eval(self, value, x):
        value[:] = numpy.dot(x, x)

    def value_shape(self):
        return ()

f.interpolate(MyExpression())

# UFL equivalent:
x = SpatialCoordinate(f.function_space().mesh())
f.interpolate(dot(x, x))

Generating Functions with randomised values ¶

The randomfunctiongen module wraps the external numpy package numpy.random, which gives Firedrake users an easy access to many stochastically sound random number generators, including PCG64, Philox, and SFC64, which are parallel-safe. All distribution methods defined in numpy.random, are made available, and one can pass a FunctionSpace to most of these methods to generate a randomised Function.

mesh = UnitSquareMesh(2,2)
V = FunctionSpace(mesh, "CG", 1)
# PCG64 random number generator
pcg = PCG64(seed=123456789)
rg = RandomGenerator(pcg)
# beta distribution
f_beta = rg.beta(V, 1.0, 2.0)

print(f_beta.dat.data)

# produces:
# [0.56462514 0.11585311 0.01247943 0.398984 0.19097059 0.5446709 0.1078666 0.2178807 0.64848515]