Boundary Integral Equation Methods for Simulation and Design of Photonic Devices

This thesis presents novel boundary integral equation (BIE) and associated optimization methodologies for photonic devices. The simulation and optimization of such structures is a vast and rapidly growing engineering area, which impacts on design of optical devices such as waveguide splitters, tapers, grating couplers, and metamaterial structures, all of which are commonly used as elements in the field of integrated photonics. The design process has been significantly facilitated in recent years on the basis of a variety of methods in computational electromagnetic (EM) simulation and design. Unfortunately, however, the expense required by previous simulation tools has limited the extent and complexity of the structures that can be treated. The methods presented in this thesis represent the results of our efforts towards accomplishing the dual goals of 1) Accurate and efficient EM simulation for general, highly-complex three-dimensional problems, and 2) Development of effective optimization methods leading to an improved state of the art in EM design. One of the main proposed elements utilizes BIE in conjunction with a modified-search algorithm to obtain the modes of uniform waveguides with arbitrary cross sections. This method avoids spurious solutions by means of a certain normalization procedure for the fields within the waveguides. In order to handle problems including nonuniform waveguide structures, we introduce the windowed Green function (WGF) method, which used in conjunction with auxiliary integral representations for bound mode excitations, has enabled accurate simulation of a wide variety of waveguide problems on the basis of highly accurate and efficient BIE, in two and three spatial dimensions. The "rectangular-polar" method provides the basic high-order singular-integration engine. Based on non-overlapping Chebyshev-discretized patches, the rectangular-polar method underlies the accuracy and efficiency of the proposed general-geometry three-dimensional BIE approach. Finally, we introduce a three-dimensional BIE framework for the efficient computation of sensitivities — i.e. gradients with respect to design parameters — via adjoint techniques. This methodology is then applied to the design of metalenses including up to a thousand parameters, where the overall optimization process takes in the order of three hours using five hundred computing cores. Forthcoming work along the lines of this effort seeks to extend and apply these methodologies to some of the most challenging and exciting design problems in electromagnetics in general, and photonics in particular.</p

The numerical solution of the dynamic fluid-structure interaction problem.

Merged with duplicate record 10026.1/2055 on 12.04.2017 by CS (TIS)In this thesis we consider the problem of the dynamic fluid-structure interaction between a finite elastic structure and the acoustic field in an unbounded fluid-filled exterior domain. We formulate the exterior acoustic problem as an integral equation over the structure surface. However, the classical boundary integral equation formulations of this problem do not have unique solutions at certain characteristic frequencies (which depend on the surface) and it is necessary to employ modified boundary integral equation formulations which are valid for all frequencies. The modified integral equation formulation used here involves certain arbitrary parameters and we shall study the effect of these parameters on the stability and accuracy of the numerical methods used to solve the integral equation. We then couple the boundary element analysis of the exterior acoustic problem with a finite element analysis of the elastic structure to investigate the interaction between the structure and the acoustic field. Recently there has been some controversy over whether or not the coupled problem suffers from the non-uniqueness problems associated with the classical integral equation formulations of the exterior acoustic problem. We resolve this question by demonstrating that the solution to the coupled problem is not unique at the characteristic frequencies and that we need to employ an integral equation formulation valid for all frequencies. We discuss the accuracy of our numerical results for both the acoustic problem and the coupled problem, for a number of axisymmetric and fully three-dimensional problems. Finally, we apply our method to the problem of a piezoelectric sonar transducer transmitting an acoustic signal in water, and observe reasonable agreement between our theoretical predictions and some experimental results.Admiralty Research Establishment, Portlan

Pluripotential-theoretic methods in K-stability and the space of K\ue4hler metrics

It is a natural problem, dating back to Calabi, to find canonical metrics on complex manifolds. In the case of polarized compact K\ue4hler manifolds, a natural candidate is a metric with constant scalar curvature (cscK metric).Since the 80s, Yau, Tian, Donaldson among others proposed that the existence of these special metrics are equivalent to an algebrico-geometric notion of K-stability. There are several known approaches to the study of K-stability and canonical metrics, using various tools from the theory of PDEs, algebraic geometry and non-Archimedean geometry for example. In this thesis, we study a different approach, based on pluripotential theory. In geometric terms, pluripotential theory is the study of positively curved metrics on vector bundles. For the purpose of K-stability, we only need pluripotential theory on an ample line bundle. In this case, pluripotential theory can be identified with the study of quasi-plurisubharmonic functions on the manifold. The application of pluripotential theory in K-stability is not completely new, but previously, people are principally interested in the regular (or mildly singular) quasi-plurisubharmonic functions. In this thesis, we put more emphasis on the role of singular\ua0quasi-plurisubharmonic functions and their singularities. In Paper 1 and Paper 2, we prove a criterion for the existence of canonical metrics on Fano manifolds in terms of quasi-plurisubharmonic functions. In Paper 3, we are concerned with the case when there are no canonical metrics, we prove that there is always an optimal destabilizer to the K-stability

Implicit boundary integral methods

Boundary integral methods (BIMs) solve constant coefficient, linear partial differential equations (PDEs) which have been formulated as integral equations. Implicit BIMs (IBIMs) transform these boundary integrals in a level set framework, where the boundaries are described implicitly as the zero level set of a Lipschitz function. The advantage of IBIMs is that they can work on a fixed Cartesian grid without having to parametrize the boundaries. This dissertation extends the IBIM model and develops algorithms for problems in two application areas. The first part of this dissertation considers nonlinear interface dynamics driven by bulk diffusion, which involves solving Dirichlet Laplace Problems for multiply connected regions and propagating the interface according to the solutions of the PDE at each time instant. We develop an algorithm that inherits the advantages of both level set methods (LSMs) and BIMs to simulate the nonlocal front propagation problem with possible topological changes. Simulation results in both 2D and 3D are provided to demonstrate the effectiveness of the algorithm. The second part considers wave scattering problems in unbounded domains. To obtain solutions at eigenfrequencies, boundary integral formulations use a combination of double and single layer potentials to cover the null space of the single layer integral operator. However, the double layer potential leads to a hypersingular integral in Neumann problems. Traditional schemes involve an interpretation of the integral as its Hadamard's Finite Part or a complicated process of element kernel regularization. In this thesis, we introduce an extrapolatory implicit boundary integral method (EIBIM) that evaluates the natural definition of the BIM. It is able to solve the Helmholtz problems at eigenfrequencies and requires no extra complication in different dimensions. We illustrate numerical results in both 2D and 3D for various boundary shapes, which are implicitly described by level set functions.Mathematic

Gratings: Theory and Numeric Applications, Second Revisited Edition

International audienceThe second Edition of the Book contains 13 chapters, written by an international team of specialist in electromagnetic theory, numerical methods for modelling of light diffraction by periodic structures having one-, two-, or three-dimensional periodicity, and aiming numerous applications in many classical domains like optical engineering, spectroscopy, and optical telecommunications, together with newly born fields such as photonics, plasmonics, photovoltaics, metamaterials studies, cloaking, negative refraction, and super-lensing. Each chapter presents in detail a specific theoretical method aiming to a direct numerical application by university and industrial researchers and engineers.In comparison with the First Edition, we have added two more chapters (ch.12 and ch.13), and revised four other chapters (ch.6, ch.7, ch.10, and ch.11

Fast, High-order Algorithms for Simulating Vesicle Flows Through Periodic Geometries.

This dissertation presents a new boundary integral equation (BIE) method for simulating vesicle flows through periodic geometries. We begin by describing the periodization scheme, in the absence of vesicles, for singly and doubly periodic geometries in 2 dimensions and triply periodic geometries in three dimensions. Later, the periodization scheme will be expanded to include multiple vesicles confined by singly periodic channels of arbitrary shape. Rather than relying on the periodic Green’s function as classical BIE methods do, the method combines the free-space Green’s function with a small auxiliary basis and imposes periodicity as an extra linear condition. As a result, we can exploit existing free-space solver libraries, quadratures, and fast algorithms to handle a large number of vesicles in a geometrically complex domain. Spectral accuracy in space is achieved using the periodic trapezoid rule and product quadratures, while a first-order semi-implicit scheme evolves particles by treating the vesicle-channel interactions explicitly. New constraint-correction formulas are introduced that preserve reduced areas of vesicles, independent of the number of time steps taken. By using two types of fast algorithms, (i) the fast multipole method (FMM) for the computation of the vesicle-vesicle and the vesicle-channel hydrodynamic interaction, and (ii) a fast direct solver for the BIE on the fixed channel geometry, the computational cost is reduced to O(N) per time step where N is the spatial discretization size. We include two example applications that utilize BIE methods with periodic boundary conditions. The first seeks to determine the equilibrium shapes of periodic planar elastic membranes. The second models the opening and closing of mechanosensitive (MS) channels on the membrane of a vesicle when exposed to shear stress while passing through a constricting channel.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/135778/1/gmarple_1.pd