A Simple Method for Computing Singular or Nearly Singular Integrals on Closed Surfaces

We present a simple, accurate method for computing singular or nearly singular integrals on a smooth, closed surface, such as layer potentials for harmonic functions evaluated at points on or near the surface. The integral is computed with a regularized kernel and corrections are added for regularization and discretization, which are found from analysis near the singular point. The surface integrals are computed from a new quadrature rule using surface points which project onto grid points in coordinate planes. The method does not require coordinate charts on the surface or special treatment of the singularity other than the corrections. The accuracy is about $O(h^3)$, where $h$ is the spacing in the background grid, uniformly with respect to the point of evaluation, on or near the surface. Improved accuracy is obtained for points on the surface. The treecode of Duan and Krasny for Ewald summation is used to perform sums. Numerical examples are presented with a variety of surfaces.Comment: to appear in Commun. Comput. Phy

Extrapolated regularization of nearly singular integrals on surfaces

We present a method for computing nearly singular integrals that occur when single or double layer surface integrals, for harmonic potentials or Stokes flow, are evaluated at points nearby. Such values could be needed in solving an integral equation when one surface is close to another or to obtain values at grid points. We replace the singular kernel with a regularized version having a length parameter $\delta$ in order to control discretization error. Analysis near the singularity leads to an expression for the error due to regularization which has terms with unknown coefficients multiplying known quantities. By computing the integral with three choices of $\delta$ we can solve for an extrapolated value that has regularization error reduced to $O(\delta^5)$. In examples with $\delta/h$ constant and moderate resolution we observe total error about $O(h^5)$. For convergence as $h \to 0$ we can choose $\delta$ proportional to $h^q$ with $q < 1$ to ensure the discretization error is dominated by the regularization error. With $q = 4/5$ we find errors about $O(h^4)$. For harmonic potentials we extend the approach to a version with $O(\delta^7)$ regularization; it typically has smaller errors but the order of accuracy is less predictable.Comment: submitted to Adv. Comput. Mat

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

Harmonic density interpolation methods for high-order evaluation of Laplace layer potentials in 2D and 3D

We present an effective harmonic density interpolation method for the numerical evaluation of singular and nearly singular Laplace boundary integral operators and layer potentials in two and three spatial dimensions. The method relies on the use of Green's third identity and local Taylor-like interpolations of density functions in terms of harmonic polynomials. The proposed technique effectively regularizes the singularities present in boundary integral operators and layer potentials, and recasts the latter in terms of integrands that are bounded or even more regular, depending on the order of the density interpolation. The resulting boundary integrals can then be easily, accurately, and inexpensively evaluated by means of standard quadrature rules. A variety of numerical examples demonstrate the effectiveness of the technique when used in conjunction with the classical trapezoidal rule (to integrate over smooth curves) in two-dimensions, and with a Chebyshev-type quadrature rule (to integrate over surfaces given as unions of non-overlapping quadrilateral patches) in three-dimensions

A fast integral equation method for solid particles in viscous flow using quadrature by expansion

Boundary integral methods are advantageous when simulating viscous flow around rigid particles, due to the reduction in number of unknowns and straightforward handling of the geometry. In this work we present a fast and accurate framework for simulating spheroids in periodic Stokes flow, which is based on the completed double layer boundary integral formulation. The framework implements a new method known as quadrature by expansion (QBX), which uses surrogate local expansions of the layer potential to evaluate it to very high accuracy both on and off the particle surfaces. This quadrature method is accelerated through a newly developed precomputation scheme. The long range interactions are computed using the spectral Ewald (SE) fast summation method, which after integration with QBX allows the resulting system to be solved in M log M time, where M is the number of particles. This framework is suitable for simulations of large particle systems, and can be used for studying e.g. porous media models