Redo the quadrilateral measure function.

[drwells]

Supersedes #10419.

This is a lot simpler and somewhat more efficient:

• master, best of 10, summing hyper ball face measures: 2434 ms
• feature, same setup: 2124 ms

Lets not use complex checks for planarity - either we have a parallelogram (up to the last few significant digits) or we do not. If people are reading in CAD data then they probably don't have uniform meshes anyway. I don't think we should lower the tolerance in this case either: we should compute the most accurate answer we can.

benchmark source:

#include <deal.II/grid/grid_generator.h>
#include <deal.II/grid/tria.h>

#include <chrono>
#include <iostream>

int
main()
{
  using namespace dealii;

  Triangulation<3> tria;
  GridGenerator::hyper_ball(tria);

  tria.refine_global(3);

  double area = 0.0;
  auto   t0   = std::chrono::high_resolution_clock::now();
  for (unsigned int i = 0; i < 1000; ++i)
    for (const auto &face : tria.active_face_iterators())
      area += face->measure();
  auto t1 = std::chrono::high_resolution_clock::now();
  std::cout
    << "time elapsed = "
    << std::chrono::duration_cast<std::chrono::milliseconds>(t1 - t0).count()
    << std::endl;
}

[marcfehling]

This looks way simpler, I like it!

[kronbichler]

@drwells I am curious about the order of the quadrature formula. As far as I see, we have the cross product, which contains the square of some bilinear terms, but then we take the norm and thus a square root. So the integrand is non-polynomial, but not near the singular spot for sqrt because we have positive determinants for the cells. This means that using 4 points should give a really good approximation. Is your reasoning along those points?

[kronbichler]

I like the simpler formula. Thank you.

[drwells]

Is your reasoning along those points?

Yes - to expand a little, the normal vector is constant when a = 0 and so for faces that are not extremely distorted (if I recall correctly, Strang and Fix prove that things won't work if cells have curvature on the same order of their diameter, so in that case we have bigger problems) we are integrating a function which varies very little. Similarly, the two tangential components correspond to the two reference cell independent variables: i.e., the tangent vectors should be close to 90 degrees apart unless we have a pinched quadrilateral (and in that case we also have bad problems outside of this function) which avoids a near-zero cross product.

More simply, though, the previous version of this function used the same 16-point quadrature formula - it just unrolled all the loops manually. This version is compatible and works with the test suite. I don't want to break people's stuff by picking a different quadrature rule.