Akash Anand

Analysis of multiple scattering iterations for high-frequency scattering problems. II: The three-dimensional scalar case

In this paper, we continue our analysis of the treatment of multiple scattering effects within a recently proposed methodology, based on integral-equations, for the numerical solution of scattering problems at high frequencies. In more detail, here we extend the two-dimensional results in part I of this work to fully three-dimensional geometries. As in the former case, our concern here is the determination of the rate of convergence of the multiple-scattering iterations for a collection of three-dimensional convex obstacles that are inherent in the aforementioned high-frequency schemes. To this end, we follow a similar strategy to that we devised in part I: first, we recast the (iterated, Neumann) multiple-scattering series in the form of a sum of periodic orbits (of increasing period) corresponding to multiple reflections that periodically bounce off a series of scattering sub-structures; then, we proceed to derive a high-frequency recurrence that relates the normal derivatives of the fields induced on these structures as the waves reflect periodically; and, finally, we analyze this recurrence to provide an explicit rate of convergence associated with each orbit. While the procedure is analogous to its two-dimensional counterpart, the actual analysis is significantly more involved and, perhaps more interestingly, it uncovers new phenomena that cannot be distinguished in two-dimensional configurations (e.g. the further dependence of the convergence rate on the relative orientation of interacting structures). As in the two-dimensional case, and beyond their intrinsic interest, we also explain here how the results of our analysis can be used to accelerate the convergence of the multiple-scattering series and, thus, to provide significant savings in computational times.

Well conditioned boundary integral equations for two-dimensional sound-hard scattering problems in domains with corners

We present several well-posed, well-conditioned direct and indirect integral equation formulations for the solution of two-dimensional acoustic scattering problems with Neumann boundary conditions in domains with corners. We focus mainly on Direct Regularized Combined Field Integral Equations (DCFIE-R) formulations whose name reflects that (1) they consist of combinations of direct boundary integral equations of the second-kind and first-kind integral equations which are preconditioned on the left by coercive boundary single-layer operators, and (2) their unknowns are physical quantities, i.e the total field on the boundary of the scatterer. The DCFIE-R equations are shown to be uniquely solvable in appropriate function spaces under certain assumptions on the coupling parameter. Using Calderon’s identities and the fact that the unknowns are bounded in the neighborhood of the corners, the integral operators that enter the DCFIE-R formulations are recast in a form that involves integral operators that are expressed by convergent integrals only. The polynomially-graded mesh quadrature introduced by Kress [30] enables the high-order resolution of the weak singularities of the kernels of the integral operators and the singularities in the derivatives of the unknowns in the vicinity of the corners. This approach is shown to lead to an efficient, high-order Nystrom method capable of producing solutions of sound-hard scattering problems in domains with corners which require small numbers of Krylov subspace iterations throughout the frequency spectrum. We present a variety of numerical results that support our claims.

Contact of a Rigid Cylindrical Punch with an Adhesive Elastic Layer

We investigate the contact of a cylindrical punch with an adhesive elastic layer or film bonded to a rigid substrate. The classical assumption of Hertzian contact are coupled with tools of fracture mechanics to resolve the contact problem. The contact edges singularity is resolved by introducing a Dugdale-Barenblatt model for the adhesive zone extending in front of the contact region. Exact governing equations so obtained are solved by a semi-analytical technique employing Chebyshev polynomials. We also extend the useful JKR approximation to the indentation of elastic layers as a particular case of our general solution.

Improved convergence of fast integral equation solvers for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface

Abstract In recent years, several fast solvers for the solution of the Lippmann–Schwinger integral equation that mathematically models the scattering of time-harmonic acoustic waves by penetrable inhomogeneous obstacles, have been proposed. While many of these fast methodologies exhibit rapid convergence for smoothly varying scattering configurations, the rate for most of them reduce to either linear or quadratic when material properties are allowed to jump across the interface. A notable exception to this is a recently introduced Nystrom scheme (Anand et al., 2016 [22] ) that utilizes a specialized quadrature in the boundary region for a high-order treatment of the material interface. In this text, we present a solution framework that relies on the specialized boundary integrator to enhance the convergence rate of other fast, low order methodologies without adding to their computational complexity of O ( N log ⁡ N ) for an N-point discretization. In particular, to demonstrate the efficacy of the proposed framework, we explain its implementation to enhance the order to convergence of two schemes, one introduced by Duan and Rokhlin (2009) [13] that is based on a pre-corrected trapezoidal rule while the other by Bruno and Hyde (2004) [12] which relies on a suitable decomposition of the Greens function via Addition theorem. In addition to a detailed description of these methodologies, we also present a comparative performance study of the improved versions of these two and the Nystrom solver in Anand et al. (2016) [22] through a wide range of numerical experiments.

A Nyström-based finite element method on polygonal elements

Abstract We consider families of finite elements on polygonal meshes, that are defined implicitly on each mesh cell as solutions of local Poisson problems with polynomial data. Functions in the local space on each mesh cell are evaluated via Nystrom discretizations of associated integral equations, allowing for curvilinear polygons and non-polynomial boundary data. Several experiments demonstrate the approximation quality of interpolated functions in these spaces.