Firedrake can evaluate
Functions at arbitrary physical
points. This feature can be useful for the evaluation of the result
of a simulation. Two APIs are offered to this feature: a
Firedrake-specific one, and one from UFL.
Firedrake offers a convenient API for evaluating functions at
arbitrary points via
# evaluate f at a 1-dimensional point f.at(0.3) # evaluate f at two 1-dimensional points, or at one 2-dimensional point # (depending on f's geometric dimension) f.at(0.2, 0.4) # evaluate f at one 2-dimensional point f.at([0.2, 0.4]) # evaluate f at two 2-dimensional point f.at([0.2, 0.4], [1.2, 0.5]) # evaluate f at two 2-dimensional point (same as above) f.at([[0.2, 0.4], [1.2, 0.5]])
While in these examples we have only shown lists, other iterables
such as tuples and
numpy arrays are also accepted. The following
f.at(0.2, 0.4) f.at((0.2, 0.4)) f.at([0.2, 0.4]) f.at(numpy.array([0.2, 0.4]))
For a single point, the result is a
numpy array, or a tuple of
numpy arrays in case of mixed functions. When evaluating
multiple points, the result is a list of values for each point.
Single point, non-mixed:
Single point, mixed: tuple of
Multiple points, non-mixed: list of
Multiple points, mixed: list of tuples of
Points outside the domain¶
When any point is outside the domain of the function,
PointNotInDomainError exception is raised. If
dont_raise=True is passed to
at(), the result is
None for those points which fall outside the domain.
mesh = UnitIntervalMesh(8) f = mesh.coordinates f.at(1.2) # raises exception f.at(1.2, dont_raise=True) # returns None f.at(0.5, 1.2) # raises exception f.at(0.5, 1.2, dont_raise=True) # returns [0.5, None]
Point evaluation on immersed manifolds is not supported yet, due to the difficulty of specifying a physical point on the manifold.
Evaluation on a moving mesh¶
If you move the mesh, by changing the mesh coordinates, then the bounding box tree that Firedrake
maintains to ensure fast point evaluation must be rebuilt. To do
this, after moving the mesh, call
MeshGeometry.clear_spatial_index on the mesh you have just
Evaluation with a distributed mesh¶
There is limited support for point evaluation when running Firedrake in parallel. There is no special API, but there are some restrictions:
Point evaluation is a collective operation.
Each process must ask for the same list of points.
Each process will get the same values.
UFL reserves the function call operator for evaluation:
will evaluate \(f\) at \((0.2, 0.4)\). UFL does not accept
multiple points at once, and cannot configure what to do with a point
which is not in the domain. The advantage of this syntax is that it
works on any
Expr, for example:
will evaluate \(f \cdot \sin(f)\) at \((0.2, 0.4)\).
The expression itself is not translated into C code. While the evaluation of a function uses the same infrastructure as the Firedrake API, which uses generated C code, the expression tree is evaluated by UFL in Python.