Once we have solved the PDE is solved we usually would like to visualise the solution and possibly other data or fields associated with the problem. Feel++ provides a very powerful framework for post-processing allowing to:
To achieve this, Feel++ defines a so-called Exporter
object
The library Feel itself does not have any visualization capabilities. However, it provides tools to export both scalar and vector fields into two formats: EnSight
and Gmsh
. The EnSight format can be read by the visualization software EnSight http://www.ensight.com or, for instance, by the open source package Paraview http://www.paraview.org.
The choice of format depends on several factors, some of them being the robustness/capabilities of the visualization packages associated and the type of data to be plotted.
For first (at most second) degree piecewise polynomials defined in straight edge/faces meshes, Paraview (or any other software that reads the EnSight format) is a very good choice. A trick to use Paraview (or any other visualization software that plots first order geometrical and finite elements) to visualize high degree polynomials in curved meshes, is to use an interpolation operator built on high order nodes associated with the polynomial space, see~[10]. However, for high degree polynomials or meshes with curved elements, Gmsh http://geuz.org/gmsh is prefered due to its adaptive visualization algorithm for this type of finite/geometrical element, see [5].
To illustrate both approaches in visualizing high degree polynomials in curved meshes, we plot, in the Figures below, the nodal projection of the function in the unit circle onto several function spaces. Notice the improved
look
of the projection using high degree polynomials instead of linear projections in a finer mesh. We remark also that the difference between the two approaches fades away as we increase the degree of the polynomials and the order of the geometrical elements. However, the additional cost of building the piecewise first order finite element space associated with the finer mesh, the calculation of the projection onto this space and the smoothness of the graphics makes Gmsh's approach more appealing to visualize this type of finite elements. We highlight that these algorithms are available for meshes composed only of simplices or quadrilaterals (see Mathematical Concepts and Notations), in 1D, 2D and 3D.
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To further illustrate the capabilities of the Feel's exporter to Gmsh, we plot in Figures 1D and 3D functions defined in the interval and the unit sphere, respectively.
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We would like to visualise the function over
.
is approximated by
.
To define and
, the code reads
We start with an Exporter
object that allows to visualise the interpolant of
over