Planewave density interpolation methods for 3D Helmholtz boundary integral equations

This paper introduces planewave density interpolation methods for the regularization of weakly singular, strongly singular, hypersingular and nearly singular integral kernels present in 3D Helmholtz surface layer potentials and associated integral operators. Relying on Green's third identity and pointwise interpolation of density functions in the form of planewaves, these methods allow layer potentials and integral operators to be expressed in terms of integrand functions that remain smooth (at least bounded) regardless the location of the target point relative to the surface sources. Common challenging integrals that arise in both Nystr\"om and boundary element discretization of boundary integral equation, can then be numerically evaluated by standard quadrature rules that are irrespective of the kernel singularity. Closed-form and purely numerical planewave density interpolation procedures are presented in this paper, which are used in conjunction with Chebyshev-based Nystr\"om and Galerkin boundary element methods. A variety of numerical examples---including problems of acoustic scattering involving multiple touching and even intersecting obstacles, demonstrate the capabilities of the proposed technique

General-purpose kernel regularization of boundary integral equations via density interpolation

This paper presents a general high-order kernel regularization technique applicable to all four integral operators of Calder\'on calculus associated with linear elliptic PDEs in two and three spatial dimensions. Like previous density interpolation methods, the proposed technique relies on interpolating the density function around the kernel singularity in terms of solutions of the underlying homogeneous PDE, so as to recast singular and nearly singular integrals in terms of bounded (or more regular) integrands. We present here a simple interpolation strategy which, unlike previous approaches, does not entail explicit computation of high-order derivatives of the density function along the surface. Furthermore, the proposed approach is kernel- and dimension-independent in the sense that the sought density interpolant is constructed as a linear combination of point-source fields, given by the same Green's function used in the integral equation formulation, thus making the procedure applicable, in principle, to any PDE with known Green's function. For the sake of definiteness, we focus here on Nystr\"om methods for the (scalar) Laplace and Helmholtz equations and the (vector) elastostatic and time-harmonic elastodynamic equations. The method's accuracy, flexibility, efficiency, and compatibility with fast solvers are demonstrated by means of a variety of large-scale three-dimensional numerical examples

Periodic fast multipole method

Applications in electrostatics, magnetostatics, fluid mechanics, and elasticity often involve sources contained in a unit cell C, centered at the origin, on which periodic boundary condition are imposed. The free-space Green’s functions for many classical partial differential equations (PDE), such as the modified Helmholtz equation, are well-known. Among the existing schemes for imposing the periodicity, three common approaches are: direct discretization of the governing PDE including boundary conditions to yield a large sparse linear system of equations, spectral methods which solve the governing PDE using Fourier analysis, and the method of images based on tiling the plane with copies of the unit cell and computing the formal solution. In the method of images, the lattice of image cells is divided into a “near” region consisting of the unit source cell and its nearest images and an infinite “far” region covered by the remaining images. Recently, two new approaches were developed to carry out calculation of the free-space Green’s function over sources in the near region and correct for the lack of periodicity using an integral representation or a representation in terms of discrete auxiliary Green’s functions. Both of these approaches are effective even for unit cells of high aspect ratio, but require the solution of a possibly ill-conditioned linear system of equations in the correction step. In this dissertation, a new scheme is proposed to treat periodic boundary conditions within the framework of the fast multipole method (FMM). The scheme is based on an explicit, low-rank representation for the influence of all far images. It avoids the lattice sum/Taylor series formalism altogether and is insensitive to the aspect ratio of the unit cell. The periodizing operators are formulated with plane-wave factorizations that are valid for half spaces, leading to a simple fast algorithm. When the rank is large, a more elaborate algorithm using the Non-Uniform Fast Fourier Transform (NUFFT) can further reduce the computational cost. The computation for modified Helmholtz case is explained in detail. The Poisson equation is discussed, with charge neutrality as a necessary constraint. Both the Stokes problem and the modified Stokes problem are formulated and solved. The full scheme including the NUFFT acceleration is described in detail and the performance of the method is illustrated with extensive numerical examples. In the last chapter, another project about boundary integral equations is presented. Boundary integral equations and Nystrom discretization methods provide a powerful tool for computing the solution of Laplace and Helmholtz boundary value problems (BVP). Using the fundamental solution (free-space Green’s function) for these equations, such problems can be converted into boundary integral equations, thereby reducing the dimension of the problem by one. The resulting geometric simplicity and reduced dimensionality allow for high-order accurate numerical solutions with greater efficiency than standard finite-difference or finite-element discretizations. Integral equation methods require appropriate quadrature rules for evaluating the singular and nearly singular integrals involved. A standard approach uses a panel-based discretization of the curve and Generalized Gaussian Quadrature (GGQ) rules for treating singular and nearly-singular integrals separately, which correspond to a panel’s interaction with itself and its neighbors, respectively. In this dissertation, a new panel-based scheme is developed which circumvents the difficulties of the nearly-singular integrals. The resulting rule is more efficient than standard GGQ in terms of the number of required kernel evaluations

Fast, high-order numerical evaluation of volume potentials via polynomial density interpolation

This article presents a high-order accurate numerical method for the evaluation of singular volume integral operators, with attention focused on operators associated with the Poisson and Helmholtz equations in two dimensions. Following the ideas of the density interpolation method for boundary integral operators, the proposed methodology leverages Green's third identity and a local polynomial interpolant of the density function to recast the volume potential as a sum of single- and double-layer potentials and a volume integral with a regularized (bounded or smoother) integrand. The layer potentials can be accurately and efficiently evaluated everywhere in the plane by means of existing methods (e.g.\ the density interpolation method), while the regularized volume integral can be accurately evaluated by applying elementary quadrature rules. We describe the method both for domains meshed by mapped quadrilaterals and triangles, introducing for each case (i) well-conditioned methods for the production of certain requisite source polynomial interpolants and (ii) efficient translation formulae for polynomial particular solutions. Compared to straightforwardly computing corrections for every singular and nearly-singular volume target, the method significantly reduces the amount of required specialized quadrature by pushing all singular and near-singular corrections to near-singular layer-potential evaluations at target points in a small neighborhood of the domain boundary. Error estimates for the regularization and quadrature approximations are provided. The method is compatible with well-established fast algorithms, being both efficient not only in the online phase but also to set-up. Numerical examples demonstrate the high-order accuracy and efficiency of the proposed methodology

Windowed Green function method for wave scattering by periodic arrays of 2D obstacles

This paper introduces a novel boundary integral equation (BIE) method for the numerical solution of problems of planewave scattering by periodic line arrays of two-dimensional penetrable obstacles. Our approach is built upon a direct BIE formulation that leverages the simplicity of the free-space Green function but in turn entails evaluation of integrals over the unit-cell boundaries. Such integrals are here treated via the window Green function method. The windowing approximation together with a finite-rank operator correction -- used to properly impose the Rayleigh radiation condition -- yield a robust second-kind BIE that produces super-algebraically convergent solutions throughout the spectrum, including at the challenging Rayleigh-Wood anomalies. The corrected windowed BIE can be discretized by means of off-the-shelf Nystr\"om and boundary element methods, and it leads to linear systems suitable for iterative linear-algebra solvers as well as standard fast matrix-vector product algorithms. A variety of numerical examples demonstrate the accuracy and robustness of the proposed methodolog