Hi.
If my element is not aligned with the axes, [...]
is this what meant by the "perfect Cartesian" situation?
Yes, exactly. If this is not what you have, I'm afraid the idea of generating
the quadratures in real space is not an option.
Another option I was contemplating is to apply the
NonMatching::DiscreteQuadratureGenerator<dim> on a fake triangulation
with just one element. Then I simply would need to interpolate
(evaluate) my real-space function at this one element and let the
class' implementation do the magic for me. This is probably cheaper
and about equally accurate?
That sounds doable and cheaper. If you are going for this, FEFieldFunction
might be useful:
https://www.dealii.org/developer/doxygen/deal.II/classFunctions_1_1FEFieldFunction.html
Regarding accuracy, you typically get the same order of accuracy. That is, say
you that you use Lagrange elements of order p in the finite element space for
you pde solution, and compare using the exact level set function with one that
has been interpolated (also onto a space of elements of order p), then you
typically see that
Exact level set function => error ~ C_1 h^q
Interpolated level set function => error ~ C_2 h^q
(error in some global norm)
However, I know one expert who have claimed that the difference between the
constants C_1 and C_2 can be surprisingly large for some problems. But I am
unaware of any publication that has actually compared this.
Just out of curiosity (and if it is not a research secret), what is your use
case? Why do you need to generate many quadratures for the same cell?
Best,
Simon
On 04/08/2023 12:39, Anton Evgrafov wrote:
Dear Simon,
Thank you very much for your reply. The wrapper code looks
embarrassingly simple!
If you are working with a perfectly Cartesian background I would expect that it
would actually be faster to generate the quadrature in real space and then
transform the quadrature to reference space before passing it to FEValues. If you
want to do this the functions cell->bounding_box() and
BoundingBox::real_to_unit(..) are useful.
That I did not think about for some reason :)
If my element is not aligned with the axes, won't I risk getting
quadrature points outside of the element though? Or is this what you
meant by the "perfect Cartesian" situation? Also I am not 100% clear
at the moment about how to change the quadrature weights after the
physical->unit transformation, as I would need access to Jacobian
determinants.
Another option I was contemplating is to apply the
NonMatching::DiscreteQuadratureGenerator<dim> on a fake triangulation
with just one element. Then I simply would need to interpolate
(evaluate) my real-space function at this one element and let the
class' implementation do the magic for me. This is probably cheaper
and about equally accurate?
/Anton
On Fri, Aug 4, 2023 at 12:07 PM Simon Sticko <simonsti...@gmail.com> wrote:
Hi.
I have done this in the past. I dug up the wrapper function I had for it, in
case we decide that we should add it to the library:
https://github.com/dealii/dealii/pull/15838
The problem with this approach is that evaluating the reference space level set
function gets very expensive. QuadratureGenerator uses many function calls and
we have to use the Jacobian and its derivative in the transformation. This
makes this approach significantly slower than using an interpolated level set
function.
If you are working with a perfectly Cartesian background I would expect that it
would actually be faster to generate the quadrature in real space and then
transform the quadrature to reference space before passing it to FEValues. If you
want to do this the functions cell->bounding_box() and
BoundingBox::real_to_unit(..) are useful.
Best,
Simon
On 03/08/2023 23:11, Anton wrote:
Hello,
I would like to use the NonMatching::QuadratureGenerator<dim> class directly,
as I need to generate multiple (as in many!, unfortunately) nonmatching quadratures
for the same cell. Therefore I would like to avoid the route of interpolating the
level set function onto the whole mesh etc, as described in CutFEM tutorial.
Therefore I am thinking about calling the "generate" function, which only needs
a level set function and a bounding box as inputs. According to the documentation the
level set function should provide the gradient and the Hessian. I would like to do this
in the reference cell coordinates, so that I can simply use this quadrature later via
FEView as one normally does.
* The bounding box I presume is the hypercube [0,1]^{dim} for the meshes I
am interested in. (Can one request them for the reference cell and not the
triangulation cell?)
* The level set function should also be reasonably straightforward. I know the
function in the physical coordinates (say, f(x)) and all its derivatives. I can also get
the map from reference to physical coordinates via FEView, let us call this map x=M(y).
The issue is how to extract the derivatives of this map so that the chain rule can be
applied? I am mostly working with Q1 maps, so of course I could figure out the
derivatives "by hand". I am just wondering if there is a more intelligent
(code-wise) way of doing this?
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