Boundary‐conforming coordinates with grid line control for acoustic scattering from complexly shaped obstacles

Abstract

The current work sets forth a practical approach to numerically solve two-dimensional direct acoustic scattering problems from complexly shaped scatterers with severe singularities, such as corners and cusps. First, boundary conforming coordinates are generated. This generation is performed through an elliptic grid generator algorithm, including control of the coordinate lines. The grid line control solely depends on the initial distribution of grid points. Following the grid generation process, the initial boundary value problem, modelling the scattering phenomenon, is formulated in terms of the new curvilinear coordinates, and a finite-difference time domain method is implemented. The presence of the boundary singularities causes instability of the numerical method. However, by appropriately controlling the distance between grid lines in the vicinity of these singularities, stability and convergence are achieved. A semianalytical formula for the differential scattering cross-section is obtained from the discrete Fourier transform of the computed scattered pressure field. The method is successfully applied to several interesting scatterers of various shapes.

Generation of smooth grids with line control for scattering from multiple obstacles

“…This incident wave is scattering from a region containing three cylindrical obstacles with their crosssections bounded by the following closed curves: epicycloid, astroid, and four-petal rose. Their parametric equations are given in [13].…”

“…Finally, the values of φ(ξ, η) and ψ(ξ, η) at the interior points of the computational domain D are defined by linear interpolation along the η-curves (η = constant) and the ξ-curves (ξ = constant), respectively. Numerical experiments on several multiply connected domains with a single hole in [12,13] revealed that the desired spacing among the ξ-curves and the η-curves of the final grid is governed by these control functions. Since the control functions are defined from the initial distribution of the grid points at the physical boundaries and at the branch cut, it is this initial distribution of points which ultimately governs the spacing among grid lines.…”

“…Hence, the time step size is determined by those cells with the smallest ratio. As shown in [13], the BCGC grid generator algorithm plays a key role in modifying a given grid to satisfy the stability condition. In many instances, there is no need to reduce the time step t, or to increase the number of cells, to improve the accuracy of the numerical method.…”

“…(13) and the absorbing boundary condition (16). In reference [13], a detailed derivation of this discrete equations is provided. At each time step, Eq.…”

“…N 1 ). Following a procedure detailed in [13], as a first step, a set of coefficientsĉ m areobtained …”