Quadrature by parity asymptotic expansions (QPAX) for scattering by high aspect ratio particles

We study scattering by a high aspect ratio particle using boundary integral equation methods. This problem has important applications in nanophotonics problems, including sensing and plasmonic imaging. Specifically, we consider scattering in two dimensions by a sound-hard, high aspect ratio ellipse. For this problem, we find that the boundary integral operator is nearly singular due to the collapsing geometry from an ellipse to a line segment. We show that this nearly singular behavior leads to qualitatively different asymptotic behaviors for solutions with different parities. Without explicitly taking this nearly singular behavior and this parity into account, computed solutions incur a large error. To address these challenges, we introduce a new method called Quadrature by Parity Asymptotic eXpansions (QPAX) that effectively and efficiently addresses these issues. We first develop QPAX to solve the Dirichlet problem for Laplace’s equation in a high aspect ratio ellipse. Then, we extend QPAX for scattering by a sound-hard, high aspect ratio ellipse. We demonstrate the effectiveness of QPAX through several numerical examples

Spectrally-Accurate Close Evaluation Schemes for Stokes Boundary Integral Operators

Dense particulate flow simulations using integral equation methods demand accurate evaluation of Stokes layer potentials on arbitrarily close interfaces. In this thesis, two spectrally-accurate integration schemes for close evaluation of 2D Stokes layer potentials are developed -- a global quadrature for the moving particles (e.g., blood cells, vesicles) represented as smooth closed curves, and an adaptive panel quadrature for the stationary boundaries (e.g., vessel walls, microfluidic channels) which are more complex curves that can be non-smooth. Both schemes rely on expressing Stokes layer potentials in terms of Laplace potentials and related complex contour integrals, which are then evaluated accurately either through a singularity cancellation technique or using analytic expressions. Numerical examples are presented to demonstrate the robustness and super-algebraic convergence of both schemes. Finally, as an application of the integration schemes, we investigate the electrohydrodynamic interactions between (possibly deflated) vesicles, where interesting behaviors unique to vesicles, such as circulatory and oscillatory motions, are observed and analyzed.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/153404/1/boweiwu_1.pd

Asymptotic analysis for close evaluation of layer potentials

International audienceAccurate evaluation of layer potentials near boundaries is needed in many applications, including fluid-structure interactions and near-field scattering in nano-optics. When numerically evaluating layer potentials, it is natural to use the same quadrature rule as the one used in the Nyström method to solve the underlying boundary integral equation. However, this method is problematic for evaluation points close to boundaries. For a fixed number of quadrature points, N , this method incurs O(1) errors in a boundary layer of thickness O(1/N). Using an asymp-totic expansion for the kernel of the layer potential, we remove this O(1) error. We demonstrate the effectiveness of this method for interior and exterior problems for Laplace's equation in two dimensions