A robust and high precision algorithm for elastic scattering problems from cornered domains

The Navier equation is the governing equation of elastic waves, and computing its solution accurately and rapidly has a wide range of applications in geophysical exploration, materials science, etc. In this paper, we focus on the efficient and high-precision numerical algorithm for the time harmonic elastic wave scattering problems from cornered domains via the boundary integral equations in two dimensions. The approach is based on the combination of Nystr\"om discretization, analytical singular integrals and kernel-splitting method, which results in a high-order solver for smooth boundaries. It is then combined with the recursively compressed inverse preconditioning (RCIP) method to solve elastic scattering problems from cornered domains. Numerical experiments demonstrate that the proposed approach achieves high accuracy, with stabilized errors close to machine precision in various geometric configurations. The algorithm is further applied to investigate the asymptotic behavior of density functions associated with boundary integral operators near corners, and the numerical results are highly consistent with the theoretical formulas

Boundary integral equation methods for superhydrophobic flow and integrated photonics

This dissertation presents fast integral equation methods (FIEMs) for solving two important problems encountered in practical engineering applications. The first problem involves the mixed boundary value problem in two-dimensional Stokes flow, which appears commonly in computational fluid mechanics. This problem is particularly relevant to the design of microfluidic devices, especially those involving superhydrophobic (SH) flows over surfaces made of composite solid materials with alternating solid portions, grooves, or air pockets, leading to enhanced slip. The second problem addresses waveguide devices in two dimensions, governed by the Helmholtz equation with Dirichlet conditions imposed on the boundary. This problem serves as a model for photonic devices, and the systematic investigation focuses on the scattering matrix formulation, in both analysis and numerical algorithms. This research represents an important step towards achieving efficient and accurate simulations of more complex photonic devices with straight waveguides as input and output channels, and Maxwell\u27s equations in three dimensions as the governing equations. Numerically, both problems pose significant challenges due to the following reasons. First, the problems are typically defined in infinite domains, necessitating the use of artificial boundary conditions when employing volumetric methods such as finite difference or finite element methods. Second, the solutions often exhibit singular behavior, characterized by corner singularities in the geometry or abrupt changes in boundary conditions, even when the underlying geometry is smooth. Analyzing the exact nature of these singularities at corners or transition points is extremely difficult. Existing methods often resort to adaptive refinement, resulting in large linear systems, numerical instability, low accuracy, and extensive computational costs. Under the hood, fast integral equation methods serve as the common engine for solving both problems. First, by utilizing the constant-coefficient nature of the governing partial differential equations (PDEs) in both problems and the availability of free-space Green\u27s functions, the solutions are represented via proper combination of layer potentials. By construction, the representation satisfies the governing PDEs within the volumetric domain and appropriate conditions at infinity. The combination of boundary conditions and jump relations of the layer potentials then leads to boundary integral equations (BIEs) with unknowns defined only on the boundary. This reduces dimensionality of the problem by one in the solve phase. Second, the kernels of the layer potentials often contain logarithmic, singular, and hypersingular terms. High-order kernel-split quadratures are employed to handle these weakly singular, singular, and hypersingular integrals for self-interactions, as well as nearly weakly singular, nearly singular, and nearly hypersingular integrals for near-interactions and close evaluations. Third, the recursively compressed inverse preconditioning (RCIP) method is applied to treat the unknown singularity in the density around corners and transition points. Finally, the celebrated fast multipole method (FMM) is applied to accelerate the scheme in both the solve and evaluation phases. In summary, high-order numerical schemes of linear complexity have been developed to solve both problems often with ten digits of accuracy, as illustrated by extensive numerical examples

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

On the use of rational-function fitting methods for the solution of 2D Laplace boundary-value problems

A computational scheme for solving 2D Laplace boundary-value problems using rational functions as the basis functions is described. The scheme belongs to the class of desingularized methods, for which the location of singularities and testing points is a major issue that is addressed by the proposed scheme, in the context of the 2D Laplace equation. Well-established rational-function fitting techniques are used to set the poles, while residues are determined by enforcing the boundary conditions in the least-squares sense at the nodes of rational Gauss-Chebyshev quadrature rules. Numerical results show that errors approaching the machine epsilon can be obtained for sharp and almost sharp corners, nearly-touching boundaries, and almost-singular boundary data. We show various examples of these cases in which the method yields compact solutions, requiring fewer basis functions than the Nystr\"{o}m method, for the same accuracy. A scheme for solving fairly large-scale problems is also presented

A new integral representation for quasiperiodic fields and its application to two-dimensional band structure calculations

In this paper, we consider band-structure calculations governed by the Helmholtz or Maxwell equations in piecewise homogeneous periodic materials. Methods based on boundary integral equations are natural in this context, since they discretize the interface alone and can achieve high order accuracy in complicated geometries. In order to handle the quasi-periodic conditions which are imposed on the unit cell, the free-space Green's function is typically replaced by its quasi-periodic cousin. Unfortunately, the quasi-periodic Green's function diverges for families of parameter values that correspond to resonances of the empty unit cell. Here, we bypass this problem by means of a new integral representation that relies on the free-space Green's function alone, adding auxiliary layer potentials on the boundary of the unit cell itself. An important aspect of our method is that by carefully including a few neighboring images, the densities may be kept smooth and convergence rapid. This framework results in an integral equation of the second kind, avoids spurious resonances, and achieves spectral accuracy. Because of our image structure, inclusions which intersect the unit cell walls may be handled easily and automatically. Our approach is compatible with fast-multipole acceleration, generalizes easily to three dimensions, and avoids the complication of divergent lattice sums.Comment: 25 pages, 6 figures, submitted to J. Comput. Phy

Nystrom methods for high-order CQ solutions of the wave equation in two dimensions

An investigation of high order Convolution Quadratures (CQ) methods for the solution of the wave equation in unbounded domains in two dimensions is presented. These rely on Nystrom discretizations for the solution of the ensemble of associated Laplace domain modified Helmholtz problems. Two classes of CQ discretizations are considered: one based on linear multistep methods and the other based on Runge-Kutta methods. Both are used in conjunction with Nystrom discretizations based on Alpert and QBX quadratures of Boundary Integral Equation (BIE) formulations of the Laplace domain Helmholtz problems with complex wavenumbers. CQ in conjunction with BIE is an excellent candidate to eventually explore numerical homogenization to replace a chaff cloud by a dispersive lossy dielectric that produces the same scattering. To this end, a variety of accuracy tests are presented that showcase the high-order in time convergence (up to and including fifth order) that the Nystrom CQ discretizations are capable of delivering for a variety of two dimensional single and multiple scatterers. Particular emphasis is given to Lipschitz boundaries and open arcs with both Dirichlet and Neumann boundary conditions