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. To illustrate the effect of parity and the need for adapted methods in the presence of high aspect ratio particles, we consider the scattering in two dimensions by a sound-hard, high aspect ratio ellipse. This fundamental problem highlights the main challenge and provides valuable insights to tackle plasmonic problems and general high aspect ratio particles. 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\u27s 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

Computational aspects of spectral invariants

The spectral theory of the Laplace operator has long been studied in connection with physics. It appears in the wave equation, the heat equation, Schroedinger's equation and in the expression of quantum effects such as the Casimir force. The Casimir effect can be studied in terms of spectral invariants computed entirely from the spectrum of the Laplace operator. It is these spectral invariants and their computation that are the object of study in the present work. The objective of this thesis is to present a computational framework for the spectral zeta function $\zeta(s)$ and its derivative on a Euclidean domain in $\mathbb{R}^2$, with rigorous theoretical error bounds when this domain is polygonal. To obtain error bounds that remain practical in applications an improvement to existing heat trace estimates is necessary. Our main result is an original estimate and proof of a heat trace estimate for polygons that improves the one of van den Berg and Srisatkunarajah, using finite propagation speed of the corresponding wave kernel. We then use this heat trace estimate to obtain a rigorous error bound for $\zeta(s)$ computations. We will provide numerous examples of our computational framework being used to calculate $\zeta(s)$ for a variety of situations involving a polygonal domain, including examples involving cutouts and extrusions that are interesting in applications. Our second result is the development a new eigenvalue solver for a planar polygonal domain using a partition of unity decomposition technique. Its advantages include multiple precision and ease of use, as well as reduced complexity compared to Finite Elemement Method. While we hoped that it would be able to contend with existing packages in terms of speed, our implementation was many times slower than MPSPack when dealing with the same problem (obtaining the first 5 digits of the principal eigenvalue of the regular unit hexagon). Finally, we present a collection of numerical examples where we compute the spectral determinant and Casimir energy of various polygonal domains. We also use our numerical tools to investigate extremal properties of these spectral invariants. For example, we consider a square with a small square cut out of the interior, which is allowed to rotate freely about its center

The Reciprocity Gap Functional for Identifying Defects and Cracks

International audienceThe recovery of defects and cracks in solids using overdetermined boundary data, both the Dirichlet and the Neumann types, is considered in this paper. A review of the method for solving these inverse problems is given, focusing particularly on linearized inverse problems. It is shown how the reciprocity gap functional can solve nonlinear inverse problems involving identification of cracks and distributed defects in bounded solids. Exact solutions for planar cracks in 3D solids are given for static elasticity, heat diffusion and transient acoustics

The viscous Cahn-Hilliard equation. I. Computations

The viscous Cahn-Hilliard equation arises as a singular limit of the phase-field model of phase transitions. It contains both the Cahn-Hilliard and Allen-Cahn equations as particular limits. The equation is in gradient form and possesses a compact global attractor A, comprising heteroclinic orbits between equilibria. Two classes of computation are described. First heteroclinic orbits on the global attractor are computed; by using the viscous Cahn-Hilliard equation to perform a homotopy, these results show that the orbits, and hence the geometry of the attractors, are remarkably insensitive to whether the Allen-Cahn or Cahn-Hilliard equation is studied. Second, initial-value computations are described; these computations emphasize three differing mechanisms by which interfaces in the equation propagate for the case of very small penalization of interfacial energy. Furthermore, convergence to an appropriate free boundary problem is demonstrated numerically

Aerodynamic Analyses Requiring Advanced Computers, part 2

Papers given at the conference present the results of theoretical research on aerodynamic flow problems requiring the use of advanced computers. Topics discussed include two-dimensional configurations, three-dimensional configurations, transonic aircraft, and the space shuttle

The numerical simulation of nonlinear waves in a hydrodynamic model test basin

This thesis describes the development of a numerical algorithm for the fully nonlinear simulation of free-surface waves. The aim of the research is to develop, implement and investigate an algorithm for the deterministic and accurate simulation of twodimensional nonlinear water waves in a model test basin. The simulated wave field may have a broad-banded spectrum and the simulations should be carried out by an efficient algorithm in order to be applicable in practical situations. The algorithm is based on a combination of Runge-Kutta (for time integration), Finite Element (boundary value problem) and Finite Difference (velocity recovery) methods. The scheme is further refined and investigated using different models for wave generation, propagation and absorption of waves