Dear deal.II community
I have attempted to implement a Galerkin BEM method by adapting the
collocation method in Step-34
<https://dealii.org/current/doxygen/deal.II/step_34.html>. This first
implementation is quite crude. I compute the singular integrals over
identical elements i.e.
int_T1 psi(x) int_T1 phi(y) G(x,y) dy dx
by wrapping the existing QGaussOneOverR
<https://dealii.org/current/doxygen/deal.II/classQGaussOneOverR.html> in an
outer loop over a QGauss
<https://dealii.org/current/doxygen/deal.II/classQGauss.html> quadrature.
In all other cases I use a tensor QGauss quadrature. Although slow, the
method does converges. However, I do not observe higher-order convergence
when increasing the order of the basis functions. I assume this is due to
the poor handling of the singular and near singular integrals.
I now want to improve this method focusing on the 3D problem with
quadrilateral elements. The plan is to use one quadrature for the double
integral itself through a series of coordinate transforms as proposed in
[2, Sec 5.2]. Note that this is also the approach taken in BEM++ [1]. This
method provides four different integration schemes (x_i,y_i,w_i),
corresponding to the different cases
- T1 and T2 are identical.
- T1 and T2 share an edge, i.e. T1(s,0) = T2(s,0).
- T1 and T2 share a vertex, i.e. T1(0,0) = T2(0,0).
- T1 and T2 do not overlap.
In each case, the quadrature can be evaluated as
sum_i w_i tilde(psi)(x_i) tilde(phi)(y_i) tilde(G)(x_i,y_i).
While reading the documentation, I came across FECouplingValues
<https://dealii.org/current/doxygen/deal.II/classFECouplingValues.html>,
which seems intended to support this type of element pair integration.
However, I am struggling to understand how to practically handle the
different geometric cases (identity, shared edge, shared vertex) and the
required rotations within this framework.
Concretely, I would appreciate pointers on the following:
1. Is FECouplingValues the right tool for implementing these four
quadrature cases for Galerkin BEM?
2. If so, is the intended workflow to explicitly construct paired
(x_i,sqrt(w_i)) and (y_i,sqrt(w_i)) for each case (and possibly for each
rotation), and then instantiate FECouplingValues separately for each
configuration? This seems rather heavy, and I am wondering if there is a
cleaner approach.
3. Do you know of any examples or code fragments in deal.II (or user
projects) that implement Galerkin BEM?
References:
[1] BEM++ Documentation, http://www.bempp.org/quadrature.html
[2] Sauter and Schwab, Boundary Element Methods, 2011
Best regards,
Jakob
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