A FEniCS Tutorial - FEniCS Project

mesh = UnitCircle(n)
where n is the typical number of elements in the radial direction.
The function f is represented by an Expression object. There are many
physical parameters in the formula for f that enter the expression string and
these parameters must have their values set by keyword arguments:
f = Expression(’4*exp(-0.5*(pow((R*x[0] - x0)/sigma, 2)) ’
’ - 0.5*(pow((R*x[1] - y0)/sigma, 2)))’,
R=R, x0=x0, y0=y0, sigma=sigma)
ThecoordinatesinExpressionobjectsmust beavectorwithindices0, 1, and2,
and with the namex. Otherwise we are free to introduce names of parameters as
long as these are given default values by keyword arguments. All the parameters
initialized by keyword arguments can at any time have their values modified.
For example, we may set
f.sigma = 50
f.x0 = 0.3
It would be of interest to visualize f along with w so that we can examine
the pressure force and its response. We must then transform the formula
(Expression) to a finite element function (Function). The most natural approach
is to construct a finite element function whose degrees of freedom (values
at the nodes in this case) are calculated from f. That is, we interpolate f (see
Section 1.6):
f = interpolate(f, V)
Callingplot(f) will produce a plot of f. Note that the assignment tofdestroys
the previous Expression object f, so if it is of interest to still have access to
this object, another name must be used for the Function object returned by
interpolate.
We need some evidence that the program works, and to this end we may use
the analytical solution listed above for the case σ → ∞. In scaled coordinates
the solution reads
w(x,y) = 1−x 2 −y 2 .
Practical values for an infinite σ may be 50 or larger, and in such cases the
program will report the maximum deviation between the computed w and the
(approximate) exact w e .
Note that the variational formulation remains the same as in the program
from Section 1.3, except that u is replaced by w and u 0 = 0. The final program
is found in the file membrane1.py, located in the stationary/poisson directory,
and also listed below. We have inserted capabilities for iterative solution
methods and hence large meshes (Section 1.4), used objects for the variational
problem and solver (Section 1.5), and made numerical comparison of the numerical
and (approximate) analytical solution (Section 1.6).
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