Analysis of an approximate cloaking for acoustic scattering problems in R3

摘要

In this paper, we consider the acoustic cloaking based on singular transformations and its approximation in the context of scattering problems in R3. The cloaking based on singular transformations can not avoid a singularity of the resulting Laplace–Beltrami operator because singular transformations blow up one point to a sphere. In order to treat the singularity properly, we use the variational method proposed in Greenleaf et al. (2007) [1] and prove the possibility of the perfect cloaking. To design an approximation scheme that avoids the singularity of the perfect cloaking, we regularize the singular transformation and show that transmitted fields into the cloaked region in H1-norm and scattered fields on any compact set outside of the cloaked region in L∞-norm converge to zero as a regularization (approximation) parameter approaches zero provided that wave number is not a Neumann eigenvalue of the Helmholtz equation in the cloaked region.