An approach to investigate the multiple-scattering problems based on the singularity expansion method

Abstract

An approach to perform the multiple-scattering calculations of the two-cylinder acoustic scattering problem is studied here. The coupling factors and the analytical solutions of the poles are extracted by using the singularity expansion method (SEM). Through interpreting the coupling characteristic of the external oscillations, the poles are also obtained which are consistent with the exact results. It is noted that the coupling characteristic of the scattering field is the corresponding coupling between the oscillation modes of the same order, as well as the SEM poles. Based on this, we then infer that by interpreting the physical meaning of the coupling coefficients and explaining the influence of the coupling characteristic on the SEM poles, the pole distributions and the external oscillation characteristics of multiple scatterers can be studied in acoustic scattering problems. The study is based on general models, and then takes a two-same-cylinder model as an example for comparative analysis. In addition, the relationship between the real parts of the SEM poles and the separated distance is also analyzed briefly. The new approach and the related calculation results provide an effective way to perform shape recognition in the acoustic scattering field.

Corrosion shape reconstruction of the mixed boundary in electrostatic imaging

“…In this section, we present a Newton iterative method for solving nonlinear boundary integral equations (14)- (15). Based on the nonlinearity of equations (14)-(15) on the boundary Γ c , we need to linearize equations (14)-(15) respect to the corrosion boundary curve Γ c by solving the Fréchet derivatives of the integral operators on the boundary.…”

“…Compute the solution from the well-posed equation 15, given as the initial value of η. (14) and (15), there are also two feasible methods. From the ill-posed equation 14we get the density η, then linearize the well-posed equation (15), so the approximate boundary can be obtained by iterative updating.…”

“…Then u = 0 in D and R 2 \D. From the integral equations (14) and (15) and the jump relations it follows that the limits of u obtained by approaching ∂D from outside vanish. The uniqueness for the exterior Dirichlet problem, together with some consideration on the behavior of the single-layer potentials at infinity using our assumption that there exists x * ∈ D such that |xx * | = 1 for all x ∈ ∂D, now implies that u vanishes in R 2 \D.…”

“…Hence we can state the following equivalence (see also [13,14]). More specifically, equations (14) and (15) are linear with respect to η, but they are nonlinear with respect to the reconstructed boundary Γ c . Equation 15is well posed, whereas the data equation 14is seriously ill-posed for the inverse problem.…”

“…In many cases, complex scattering problems need to be considered, which usually are described as the scattering problems with absorbing boundary or multiple scatterers [15,19,29] (and references therein). A layer of medium on the surface of a scatterer can be used to protect the coated scatterer from external environment corrosion and also can be used to absorb the incident wave according to its own needs to reduce the intensity of scattering to achieve stealth purpose.…”