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SUMMARY:A Hausdorff-measure BEM for acoustic scattering by fractal screens
  - part 2 - Andrew Gibbs (University College London)
DTSTART:20230418T104500Z
DTEND:20230418T113000Z
UID:TALK198715@talks.cam.ac.uk
DESCRIPTION:In part 1 (the prequel to this talk)\, it will be shown that s
 ound-soft scattering by fractal screens (such as the Cantor Set or Sierpin
 ski Triangle) can be modelled by generalising the Boundary Element Method 
 to obstacles with non-integer dimension\, the so-called "Hausdorff BEM".\n
 In Hausdorff BEM\, like the scatterer\, the mesh elements are self-similar
  fractals with non-integer dimension and zero Lebesgue measure. Therefore\
 , implementation of this non-standard Galerkin BEM presents an interesting
  new challenge: approximation of double integrals of singular Green's kern
 els\, over fractal domains with respect to Hausdorff measure. This motivat
 ed the research I will present in this talk.\nI will begin by discussing n
 umerical methods for approximating smooth integrals with respect to Hausdo
 rff measure\, summarising our recent contributions and existing methods. N
 ext\, for integrals over a self-similar domain\, where the integrand has a
  singularity of logarithmic or algebraic type\, I will present a novel alg
 orithm which exploits this self-similarity to reformulate this singular in
 tegral as a sum of smooth integrals (which are easier to approximate). Thi
 s technique forms an essential part of our Hausdorff BEM implementation. I
  will conclude by presenting fully discrete estimates for Hausdorff BEM\, 
 alongside numerical results.
LOCATION:Seminar Room 1\, Newton Institute
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