A FEniCS Tutorial - FEniCS Project

It is possible to restrict the integration to a part of the boundary using a
mesh function to mark the relevant part, as explained in Section 5.3. Assuming
that the part corresponds to subdomain number i, the relevant form for the
flux is -p*inner(grad(u), n)*ds(i).
1.12 Visualization of Structured Mesh Data
When finite element computations are done on a structured rectangular mesh,
maybe with uniform partitioning, VTK-based tools for completely unstructured
2D/3D meshes are not required. Instead we can use visualization and data
analysis tools for structured data. Such data typically appear in finite difference
simulations and image analysis. Analysis and visualization of structured data
arefasterandeasierthandoingthesamewithdataonunstructuredmeshes, and
the collection of tools to choose among is much larger. We shall demonstrate the
potential of such tools and how they allow for tailored and flexible visualization
and data analysis.
A necessary first step is to transform our mesh object to an object representing
a rectangle with equally-shaped rectangular cells. The Python package
scitools (code.google.com/p/scitools) has this type of structure, called a
UniformBoxGrid. The second step is to transform the one-dimensional array
of nodal values to a two-dimensional array holding the values at the corners of
the cells in the structured grid. In such grids, we want to access a value by its
i and j indices, i counting cells in the x direction, and j counting cells in the y
direction. This transformation is in principle straightforward, yet it frequently
leads to obscure indexing errors. The BoxField object in scitools takes conveniently
care of the details of the transformation. With a BoxField defined on
a UniformBoxGrid it is very easy to call up more standard plotting packages
to visualize the solution along lines in the domain or as 2D contours or lifted
surfaces.
Let us go back to the vcp2D.py code from Section 1.10 and map u onto a
BoxField object:
import scitools.BoxField
u2 = u if u.ufl_element().degree() == 1 else \
interpolate(u, FunctionSpace(mesh, ’Lagrange’, 1))
u_box = scitools.BoxField.dolfin_function2BoxField(
u2, mesh, (nx,ny), uniform_mesh=True)
The function dolfin_function2BoxField can only work with finite element
fields with linear (degree 1) elements, so for higher-degree elements we here
simply interpolate the solution onto a mesh with linear elements. We could also
interpolate/project onto a finer mesh in the higher-degree case. Such transformations
to linear finite element fields are very often needed when calling up
plotting packages or data analysis tools. The u.ufl_element() method returns
an object holding the element type, and this object has a method degree() for
returning the element degree as an integer. The parameters nx and ny are
the number of divisions in each space direction that were used when calling
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