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The study of scattering by a high aspect ratio particle has important applications in sensing and plasmonic imaging. To illustrate the effect of particle’s narrowness (that can be related to parity properties) and the need for adapted methods (in the context of boundary integral methods), we consider the scattering by a penetrable, high aspect ratio ellipse. This problem highlights the main challenge and provides valuable insights to tackle general high aspect ratio particles. We find that boundary integral operators are nearly singular due to the collapsing geometry to a line segment. We show that these nearly singular behaviors lead to qualitatively different asymptotic behaviors for solutions with different parities. Without explicitly taking this into account, computed solutions incur large errors. We introduce the Quadrature by Parity Asymptotic eXpansions (QPAX) that effectively and efficiently addresses these issues. We demonstrate the effectiveness of QPAX through several numerical examples.
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Decomposition and conformal mapping techniques for the quadrature of nearly singular integrals
Abstract Gauss–Legendre quadrature, Clenshaw–Curtis quadrature and the trapezoid rule are powerful tools for numerical integration of analytic functions. For nearly singular problems, however, these standard methods become unacceptably slow. We discuss and generalize some existing methods for improving on these schemes when the location of the nearby singularity is known. We conclude with an application to some nearly singular surface integrals that arise in three-dimensional viscous fluid flow.
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- Award ID(s):
- 1907796
- PAR ID:
- 10434528
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- BIT Numerical Mathematics
- Volume:
- 63
- Issue:
- 3
- ISSN:
- 0006-3835
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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