Post processing: computing integrals over boundary subsets.

[RadostW]

Hello,

I managed to solve a problem I'm interested in (got a nice picture too 😄 ) but I'd like to compute some quantity from the solution field over some subset of the boundary. A simplest example could be an average.

Is there some util that can handle this already or does it have to be implemented each time?

The actual integral that I'd like to compute is

$$ \int_0^{R_{max}} ( u(r,Z) \mathrm{Pe} + \partial_z ) \phi 2\pi r \mathrm{d}r $$

at the top boundary for $u(r,Z)$ the advection field as given in example_50.py (this would be total heat flux at the top surface).

[kinnala]

It is a combination of FacetBasis and Functional wrapper.

fbasis = FacetBasis(mesh, ElementTriP1(), facets='top')

@Functional
def integral(w):
    return … # w['u'] is the solution

print(integral.assemble(fbasis))

[RadostW]

I assume coordinate fields are w['x'] and w['y']?

Is @Functional wrapper documented somewhere?

[kinnala]

I think not exactly, in the sense that you mean. Perhaps it should be documented more clearly, for example, in the context of postprocessing. I will think about adding another section to the documentation.

[RadostW]

Adding

fbasis = FacetBasis(mesh, ElementTriP1(), facets='top')

@Functional
def total(w):
    return w['u']
    
@Functional
def unit(w):
    return 1
    
print(total.assemble(fbasis) / unit.assemble(fbasis))

a the end of ex50.py just gives KeyError like this

Traceback (most recent call last):
  File "/home/r/stokes_flow_cylindric_metric.py", line 125, in <module>
    print(total.assemble(fbasis) / total.assemble(fbasis))
  File "/home/r/venv/lib/python3.10/site-packages/skfem/assembly/form/form.py", line 76, in assemble
    out = (COOData(*self._assemble(*args, **kwargs))  # type: ignore
  File "/home/r/venv/lib/python3.10/site-packages/skfem/assembly/form/functional.py", line 58, in _assemble
    np.array([self.elemental(vbasis, **kwargs).sum(-1)]),
  File "/home/r/venv/lib/python3.10/site-packages/skfem/assembly/form/functional.py", line 34, in elemental
    return self._kernel(w, v.dx)
  File "/home/r/venv/lib/python3.10/site-packages/skfem/assembly/form/functional.py", line 24, in _kernel
    return (self.form(w) * dx).sum(-1)
  File "/home/r/stokes_flow_cylindric_metric.py", line 119, in total
    return w['u']
KeyError: 'u'

[RadostW]

I have found a solution in other examples

fbasis = FacetBasis(mesh, ElementTriP1(), facets='top')

@Functional
def total(w):
    return w['u']
    
@Functional
def unit(w):
    return 1

print('Mean value is')    
print(asm(total, fbasis, u=u) / asm(unit, fbasis, u=u))