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Cloaking laminate design based on GPT-vanishing structures

Eleanor Gemida, Mikyoung Lim

TL;DR

This work advances near-cloaking by marrying GPT-vanishing multicoatings with homogenization-based laminates. By canceling leading GPTs up to order $N$, the authors achieve enhanced invisibility with an enlarged pre-transformation inclusion size, while using isotropic materials arranged in radial laminates. They derive sharp bounds for the DtN map under small GPT-vanishing inclusions and provide constructive designs for isotropic laminates that approximate the anisotropic push-forward cloak. The framework extends to arbitrary inclusions through an additional low-conductivity layer, and the authors illustrate the approach with detailed 2D and 3D numerical examples, highlighting practical feasibility and design trade-offs.

Abstract

We propose a near-cloaking design which is a lamination of a finite number of layers of isotropic materials. The proposed design is an approximation of a cloaking material obtained by pushing forward a multi-coated structure for which the coating cancels the generalized polarization tensors (GPTs) up to several leading orders. The enhanced cloaking effect achieved by the GPT-vanishing structure permits a coarser microscale requirement and reduces the contrast in the constituent isotropic materials, thereby improving constructibility of cloaking laminates compared with designs based on a non-coated structure.

Cloaking laminate design based on GPT-vanishing structures

TL;DR

This work advances near-cloaking by marrying GPT-vanishing multicoatings with homogenization-based laminates. By canceling leading GPTs up to order , the authors achieve enhanced invisibility with an enlarged pre-transformation inclusion size, while using isotropic materials arranged in radial laminates. They derive sharp bounds for the DtN map under small GPT-vanishing inclusions and provide constructive designs for isotropic laminates that approximate the anisotropic push-forward cloak. The framework extends to arbitrary inclusions through an additional low-conductivity layer, and the authors illustrate the approach with detailed 2D and 3D numerical examples, highlighting practical feasibility and design trade-offs.

Abstract

We propose a near-cloaking design which is a lamination of a finite number of layers of isotropic materials. The proposed design is an approximation of a cloaking material obtained by pushing forward a multi-coated structure for which the coating cancels the generalized polarization tensors (GPTs) up to several leading orders. The enhanced cloaking effect achieved by the GPT-vanishing structure permits a coarser microscale requirement and reduces the contrast in the constituent isotropic materials, thereby improving constructibility of cloaking laminates compared with designs based on a non-coated structure.
Paper Structure (23 sections, 10 theorems, 198 equations, 4 figures)

This paper contains 23 sections, 10 theorems, 198 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: GPT-vanishing structures
  3. The concept of GPT-vanishing structures
  4. Construction of GPT-vanishing structures with multicoated balls
  5. Two-dimensional case
  6. Three-dimensional case
  7. Boundary value problem with a small GPT-vanishing structure
  8. Small volume expansion for the Dirichlet-to-Neumann map
  9. Design of a laminate cloaking structure for an insulated core
  10. Cloaking design with anisotropic materials via pushforward of GPT-vanishing structures
  11. Cloaking design with isotropic materials via homogenization
  12. Proof of the main result
  13. A remark on the case with arbitrary inclusion
  14. Review of cloaking arbitrary an inclusion
  15. Near-cloaking laminate structure for arbitrary objects
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\sigma$ be a conductivity profile exhibiting the GPT-vanishing property of order $N$ and containing an insulating core in $\mathbb{R}^d$ ($d=2,3$), as detailed in Propositions prop:radial:2D and prop:radial:3D. Let $A_\varepsilon$ be given by (A_ep:intro) for a lamination scale $\varepsilon$ sa Then, the following estimate holds: where $C$ is a positive constant independent of $\rho$ but dep

Figures (4)

  • Figure 7.1: Cloaking laminate using a non-coated insulated core in $\mathbb{R}^2$.
  • Figure 7.2: Cloaking laminate based on a GPT-vanishing structure of order $N=4$ in $\mathbb{R}^2$.
  • Figure 7.3: Cloaking laminate based on a GPT-vanishing structure of order $N=6$ in $\mathbb{R}^2$.
  • Figure 7.4: Cloaking laminate based on a GPT-vanishing structure of order $N=3$ in $\mathbb{R}^3$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Proposition 2.1: Ammari:2013:ENC_1
  • Proposition 2.2
  • Lemma 3.1
  • proof
  • Theorem 4.1
  • Remark 1
  • Lemma 4.2
  • proof
  • Lemma 5.1: Capdeboscq:2025:CCC
  • ...and 5 more