A FEniCS Tutorial - FEniCS Project

numerical u at the center point: [ 1.83333333]
exact u at the center point: [ 1.75]
The discrepancy is due to the fact that the center point is not a node in this
particular mesh, but a point in the interior of a cell, and u varies linearly over
the cell while u0 is a quadratic function.
We have seen how to extract the nodal values in a numpy array. If desired,
we can adjust the nodal values too. Say we want to normalize the solution such
that max j U j = 1. Then we must divide all U j values by max j U j . The following
snippet performs the task:
max_u = u_array.max()
u_array /= max_u
u.vector()[:] = u_array
u.vector().set_local(u_array) # alternative
print u.vector().array()
That is, we manipulate u_array as desired, and then we insert this array into
‘u‘’s Vector object. The /= operator implies an in-place modification of the
object on the left-hand side: all elements of the u_array are divided by the
value max_u. Alternatively, one could write u_array = u_array/max_u, which
implies creating a new array on the right-hand side and assigning this array to
the name u_array.
A call like u.vector().array() returns a copy of the data in u.vector().
Onemustthereforeneverperformassignmentslikeu.vector.array()[:] = ...,
but instead extract the numpy array (i.e., a copy), manipulate it, and insert it
back with u.vector()[:] = or u.set_local(...).
All the code in this subsection can be found in the file d4_p2D.py in the
stationary/poisson directory. We have commented out the plotting statements
in this version of the program, but if you want plotting to happen, make
sure that interactive is called at the very end of the program.
1.7 Solving a Real Physical Problem
Perhaps you are not particularly amazed by viewing the simple surface of u in
the test problem from Section 1.3. However, solving a real physical problem
with a more interesting and amazing solution on the screen is only a matter of
specifying a more exciting domain, boundary condition, and/or right-hand side
f.
One possible physical problem regards the deflection D(x,y) of an elastic
circular membrane with radius R, subject to a localized perpendicular pressure
force, modeled as a Gaussian function. The appropriate PDE model is
−T∇ 2 D = p(x,y) in Ω = {(x,y)|x 2 +y 2 ≤ R}, (14)
with
p(x,y) = A
2πσ exp (− 1 2
( ) 2 x−x0
− 1 ( ) ) 2 y −y0
. (15)
σ 2 σ
20