Cylindrical and spherical versions of derived quantities

[RemDelaporteMathurin]

Following what was done in #689 we need an implementation of the cylindrical and spherical versions of

• TotalVolume Total Volume for Cylindrical and Spherical meshes #882
• TotalSurface
• AverageVolume Average surface and volume derived quantities in Cylindrical and Spherical #820
• AverageSurface Average surface and volume derived quantities in Cylindrical and Spherical #820

2D axisymmetric

In 2D axisymmetric, the surface element ds is
ds = r dr dtheta or ds = r dz dtheta
In both cases, in fenics integrating a function f is assemble(f * r * ds) where r is the radius.

dV = r dr dtheta dz
Integrating a function f in fenics is therefore assemble(f * r * dx) where r is the radius.

1D spherical

The surface element in 1D spherical is:
dS_r = r^2 sin(theta) dtheta dphi
The integral of f on a spherical surface is therefore
integral(f dS_r) = integral(f r^2 sin(theta) dtheta dphi)
= (phi2 - phi1) * (-cos(theta2) + cos(theta1)) * f r^2
The volume element is

dV = r^2 sin(theta) dr dtheta dphi
the integral of f on a spherical volume is therefore

integral(f dV) = (phi2 - phi1) * (-cos(theta2) + cos(theta1)) * integral(f r^2 dr)

phi is the polar angle and theta is the azimuthal angle

[gabriele-ferrero]

This is going to be very useful, I'm planning to help soon

[jhdark]

@gabriele-ferrero If you wanna do the TotalVolume and TotalSurface ones I can do the Average ones

[RemDelaporteMathurin]

@jhdark @gabriele-ferrero I "assigned" you to this issue. Let me know if you need help and if you want to have this in 1.3

[KulaginVladimir]

I suppose, HydrogenFlux and ThermalFlux also require cylindrical/spherical versions.

[RemDelaporteMathurin]

@rekomodo is having a go at implementing the remaining classes