Regularized Transformation-Optics Cloaking for the Helmholtz Equation: From Partial Cloak to Full Cloak

We develop a very general theory on the regularized approximate invisibility cloaking for the wave scattering governed by the Helmholtz equation in any space dimensions via the approach of transformation optics. There are four major ingredients in our proposed theory: 1) The non-singular cloaking medium is obtained by the push-forwarding construction through a transformation which blows up a subset $K_\varepsilon$ in the virtual space, where $\varepsilon$ is a small positive asymptotic regularization parameter. $K_\varepsilon$ will degenerate to $K_0$ as $\varepsilon$ goes to $0$, and in our theory $K_0$ could be any convex compact set in $\mathbb{R}^N$, or any set whose boundary consists of Lipschitz hypersurfaces, or a finite combination of those sets. 2) A general lossy layer with the material parameters satisfying certain compatibility integral conditions is employed right between the cloaked and cloaking regions. 3)The contents being cloaked could also be extremely general, possibly including, at the same time, generic mediums and, sound-soft, sound-hard and impedance-type obstacles, as well as some sources or sinks. 4) In order to achieve a cloaking device of compact size, particularly for the case when $K_\varepsilon$ is not ``uniformly small", an assembly-by-components, the (ABC) geometry is developed for both the virtual and physical spaces and the blow-up construction is based on concatenating different components. Within the proposed framework, we show that the scattered wave field $u_\varepsilon$ corresponding to a cloaking problem will converge to $u_0$ as $\varepsilon$ goes to $0$, with $u_0$ being the scattered wave field corresponding to a sound-hard $K_0$. The convergence result is used to theoretically justify the approximate full and partial invisibility cloaks, depending on the geometry of $K_0$. On the other hand, the convergence results are conducted in a much more general setting than what is needed for the invisibility cloaking, so they are of significant mathematical interest for their own sake. As for applications, we construct three types of full and partial cloaks. Some numerical experiments are also conducted to illustrate our theoretical results.