23 June 2014

Function evaluation

As said in the previous post, my latest result is the possibility to evaluate a fem-fenics function on a point of its domain. This way it is possible to generalise methods that, otherwise, would rely on analytical solutions. In [1] we have an example of such a method.

The paper deals with the deposition of nanoparticles in tissues, for the treatment of cancer. The phenomenon is described with a Monte Carlo simulation of these particles' trajectories, assuming that the velocity field of the carrying fluid is known. In this study, some simplifying hypotheses about the geometry of the cells and the fluid layer nearby allow for an analytical solution of the Stokes equation. Unfortunately, these assumptions do not hold generally in human tissues: for instance, in the liver cells have cubic shape, contrasting to the spherical one used in this paper. Now, if this method is implemented in Octave, we can solve numerically the Stokes equation on a realistic domain and obtain right away a more general approach to this significant application.

The evaluation of a fem-fenics function was already possible via the feval method, but it had some glitches. One aspect is that the solution of a differential problem could not be used as if it was a regular Octave function, then a user should have adapted his/her algorithms to take advantage of it. One more critical issue is that the previous implementation did not handle the exception raised by the underlying FEniCS method when it gets as argument the coordinates of a point outside of the domain, thus leading to a crash of Octave.

In order to address these problems, I added the subsref method to the function class and implemented the proper exception handling in feval. To avoid code duplication, the former relies on the latter for the real computation, so it basically just forwards the parameters after checking that the right type of indexing was used. As a result, it is now possible to solve the equations:

- ν Δu + ∇p = 0
∇ ⋅ u = 0

with relevant border conditions, on a proper mesh and finite element space, and then evaluate the solution with the Octave expression  values = u (points), where points is a matrix holding the coordinates of every point where to do so, one per column. Moreover, a careless evaluation will not result in your Octave session crashing any more.

Even if this feature of the package underwent some improvement, there is still room for more. Two issues I have not addressed yet are the somehow weird interface and the possibility to create a function handle to perform evaluations with. Regarding the former, we might observe that the above mentioned expression remains exactly the same no matter what the geometrical dimension of the domain is. I should modify the implementation so that a vectorial function on a 3D space is evaluated with [ux, uy, uz] = velocity (x, y, z). Moving to the latter, in my understanding the class design should be modified to allow the exploitation of the Octave internals managing functions, so this would require a careful reflection on all the possible collateral effects of such a change.

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