The DtN nonreflecting boundary condition for multiple scattering problems in the half-plane

Abstract

The multiple-Dirichlet-to-Neumann (multiple-DtN) non-reflecting boundary condition is adapted to acoustic scattering from obstacles embedded in the half-plane. The multiple-DtN map is coupled with the method of images as an alternative model for multiple acoustic scattering in the presence of acoustically soft and hard plane boundaries. As opposed to the current practice of enclosing all obstacles with a large semicircular artificial boundary that contains portion of the plane boundary, the proposed technique uses small artificial circular boundaries that only enclose the immediate vicinity of each obstacle in the half-plane. The adapted multiple-DtN condition is simultaneously imposed in each of the artificial circular boundaries. As a result the computational effort is significantly reduced. A computationally advantageous boundary value problem is numerically solved with a finite difference method supported on boundary-fitted grids. Approximate solutions to problems involving two scatterers of arbitrary geometry are presented. The proposed numerical method is validated by comparing the approximate and exact far-field patterns for the scattering from a single and from two circular obstacles in the half-plane.

High order methods for acoustic scattering: Coupling farfield expansions ABC with deferred-correction methods

“…These grids correspond to generalized curvilinear coordinates conforming to the boundaries of the scatterers. Moreover, this grids were used by the same authors to solve single and multiple scattering problems from complexly shaped obstacles in the following articles [44,45,33]. In particular in [33], they obtained second order convergence for a star shaped scatter with smooth boundary by using second order finite difference approximation based on curvilinear coordinantes conforming to the scatterer boundary.…”

“…Then, we will use the multiple-KFE condition and symmetry relations between the outgoing waves to construct the KFE condition for the half-plane. Our purpose is to imitate the procedure employed by Acosta and Villamizar in [45] for the construction of the Dirichlet to Neumann (DtN) condition for a single obstacle in the half-plane from the multiple DtN condition for the full-plane [50,44]. However, a formulation of the KFE for waveguides or more arbitrary geometries may not be possible.…”

An efficient iterative method for solving multiple scattering in locally inhomogeneous media

“…The framework therein is well-suited for numerical discretization, but it requires to solve coupled systems. Indeed, Acosta and Villamizar [16,17] discussed the multiple acoustic scattering from scatterers of complex shape using coupling of Dirichlet-to-Neumann boundary condition and the finite difference method. In practice, the iterative method is more desirable.…”

“…Note that the final discrete systems resulted from the discretisation proposed by [15,16,17] have block structure, so the block iterative methods can be directly applied (e.g., block Gauss-Seidel iterative method [23,24]). Although the pursuit of "decoupling" between scatterers is similar to these works, the derivation of our iterative algorithm is from the boundary integral equations that leads to the use of purely outgoing waves rather than the whole scattering field on the boundaries of the scatterers (homogeneous media case) or the artificial boundary (inhomogeneous media case) for the communication between scatterers.…”

Perfect absorbing layers for wave scattering problems and related topics

“…The framework therein is well-suited for numerical discretization, but it requires to solve coupled systems. Indeed, Acosta and Villamizar [2,3] discussed the multiple acoustic scattering from scatterers of complex shapes using coupling of Dirichlet-to-Neumann boundary condition and the finite difference method. In practice, the iterative method is more desirable.…”