Concentric Spherical Shells

In this example, two concentric spherical shells with different dielectric behaviors are considered. Moreover, on the innermost and outermost spheres fixed potentials are applied. This constellation gives rise to a simple analytic expression for the potential. In order to see how to generate the concentricSpheres.yaml input file with the preprocessor of POLARIS, check the Preprocessor Tutorial.

Model parameters

Main run

In order to compute the surface density and potential, run the command

polaris -i concentricSpheres.mesh -s concentricSpheres.yaml

in order to obtain the result file concentricSpheres.vtkhdf.

Posterior run

With

polaris.post -i concentricSpheres.vtkhdf -p concentricSpheres.xyz -s concentricSpheres.yaml 

the potential and E-field along a radial line are calculated and tabulated in the file concentricSpheres.csv

Results

This problem with spherical symmetry can be reduced to two second-order ODEs in the radial direction. Their respective integration leads to solutions of the form

\[ u_1(r) = A + \frac{B}{r} \quad r_1 \leq r \leq r_2 \]

for the inner shell and

\[ u_2(r) = C + \frac{D}{r} \quad r_2 \leq r \leq r_3 \]

for the outer shell. The integration constants \(A\), \(B\), \(C\) and \(D\) can be obtained from the boundary conditions at \(r=r_1\) and \(r=r_3\), that is the prescribed potentials \(V_1\) and \(V_3\) and from the transmission conditions at \(r=r_2\). The final result of these constants can be found in the provided gnuplot script.

With the following commands

gnuplot
gnuplot> load 'concentricSpheres.gp'

one obtains a plot of the computed results along the radial line and the corresponding values of the analytic solution.

Below are screenshots from ParaView and the gnuplot image

Surface mesh

Potential along radial line