Active Cloaking and Illusion of Electric Potentials in Electrostatics Andreas Helfrich-Schkabarenko Esslingen University of Applied Sciences Alik Ismail-Zadeh ( [email protected] ) Karlsruhe Institute of Technology, Institute of Applied Geosciences, Karlsruhe, Germany; Aron Sommer University of Hannover Research Article Keywords: cloaking, self-potentials, inverse theory, numerical modelling, numerical solutions, electric properties Posted Date: January 7th, 2021 DOI: https://doi.org/10.21203/rs.3.rs-140026/v1 License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License 1 ACTIVE CLOAKING AND ILLUSION OF ELECTRIC POTENTIALS IN ELECTROSTATICS 2 3 Andreas Helfrich-Schkabarenko1, Alik Ismail-Zadeh2,3, Aron Sommer4 4 5 1 University of Applied Sciences, Esslingen, Germany; 6 [email protected] 7 2 Karlsruhe Institute of Technology, Institute of Applied Geosciences, Karlsruhe, Germany; 8 [email protected] 9 3 Russian Academy of Sciences, Institute of Earthquake Prediction Theory and Mathematical 10 Geophysics, Moscow, Russia; [email protected] 11 4 Leibniz University Hannover, Institute of Information Processing, Hannover, Germany; 12 [email protected] 13 14 Summary - Cloaking and illusion has been demonstrated theoretically and experimentally in 15 several research fields. Here we present for the first time an active exterior cloaking device in 16 electrostatics operating in a two-horizontally-layered electroconductive domain, and use the 17 superposition principle to cloak electric potentials. The device uses an additional current source 18 pattern introduced on the interface between two layers to cancel the total electric potential to 19 be measured. Also, we present an active exterior illusion device allowing for detection of a 20 signal pattern corresponding to any arbitrarily chosen current source instead of the existing 21 current source. The performance of the cloaking/illusion devices is demonstrated by three- 22 dimensional models and numerical experiments using synthetic measurements of the electric 23 potential. Sensitivities of numerical results to a noise in measured data and to a size of cloaking 24 devices are analysed. The numerical results show quite reasonable cloaking/illusion 25 performance, which means that a current source can be hidden electrostatically. The developed 26 active cloaking/illusion methodology can be used in subsurface geo-exploration studies, 27 electrical engineering, live sciences, and elsewhere. 28 29 Short title: Cloaking and illusion in electrostatics 30 31 Key words: cloaking, self-potentials, inverse theory, numerical modelling, numerical solutions, 32 electric properties 1 33 34 INTRODUCTION 35 36 Invisibility has been a subject of human fascination for millennia. The basic idea of invisibility is 37 to generate a cloaking device and use it to hide an object. Cloaking devices employ specially 38 designed structures that would make objects ‘invisible’ by detecting devices (e.g. eyes, 39 antennas, airborne or satellite detectors/sensors). Over the last two decades, theoretical and 40 experimental studies on cloaking have been conducted in several research fields such as 41 electromagnetism 1,2,3, acoustics 4,5,6, thermodynamics 7,8,9,10, solid mechanics 11, elastic 12, 13, 14, 42 15, and seismic wave propagation 16, 17, 18, 19. 43 Cloaking devices can differ by its construction (interior and exterior cloaking) and by 44 transforming physical properties of the material surrounding an object (passive cloaking) or 45 adding an active source (active cloaking). An interior cloaking device surrounds an object to be 46 cloaked, so that, the object is located in the interior of the cloaking device 1. An exterior cloaking 47 device hides objects from potential detections without encompassing them 20. A passive 48 cloaking device induces invisibility by a special choice of physical parameters of a designed 49 artificial material (so-called metamaterial) surrounding or partly surrounding an object, so that, 50 an incident wave on the object bypasses it without distortions. A mathematical technique used 51 to develop metamaterials is transformation optics 21,22,23,24,25. In the case of electrostatics, such 52 metamaterial would be a material with an anisotropic electrical conductivity 26. An active 53 cloaking masks emitting objects using active sources 3,27,28,29. 54 In this paper, in horizontally-layered electroconductive domain we use active exterior 55 cloaking devices in the case of electrostatics to mask current source located in the source sub- 56 domain (SSD), e.g. Earth’s ground, so making the source nearly undetectable by measurements 57 in the observational sub-domain (OSD), e.g., seawater (Fig. 1). An “invisibility” in this case is 58 achieved by using the current source networks suitably constructed on the interface between 59 the two sub-domains (hereinafter referred to as ISD), which cancel (cloak) or generate 60 imaginary (illusion) electric potential in the OSD. A mathematical background for developing 61 the active cloaking devices lies in the theory of inverse problems 30 with the use of the 62 superposition principle in terms of active noise control or noise cancellation 31,32. In a three- 63 dimensional model domain comprised of two overlain electroconductive layers, the following 64 direct and inverse problems form essential components of our numerical experiments based 2 65 on an electrostatic model. 66 67 68 Figure 1. Two-dimensional cartoon of the model domain Ω . Dark gray: the area of non-zero current 69 source density; light gray: the SSD Ωl ; Σ is the ISD; Ωu is the OSD; and Γ is a curve (or a set) of 70 measurement points. 71 72 • Direct Problem: To find the generated electrical potential in the entire model domain 73 for a given non-zero current source density located in SSD. 74 • Source Identification Problem: To determine this current source density from its electric 75 potential, which can be measured or inferred from measured electromagnetic data in 76 the OSD. As the source identification problem was analysed by Sommer et al. (ref. [33]), 77 here we describe briefly the results of this study. Applications of the source 78 identification problem are numerous; for example, it is the subject of research in 79 volcanology 34,35 and in geo-explorations 36. 80 • Active Cloaking Problem: To cloak the current source density so that it gets ‘invisible’ for 81 measurements in the OSD. To achieve it, we introduce an additional current source 82 density (thereafter referred to as active cloaking device) on the ISD in order to minimize 83 the total electric potential field in the OSD. 84 • Active Illusion Problem: To generate an illusion in the data measured in the OSD by 85 manipulating the total electric potential field. The manipulation is set up via an 86 additional current source density on the ISD. A similar approach was used in 87 electromagnetics 37. Essentially, an active illusion problem is based on an active cloaking 88 problem. 3 89 In what comes next, we present results of the four interconnected problems mentioned 90 above. Synthetic data (that is, an electric potential) are generated by solving the direct problem 91 (hereinafter we refer to the synthetic data as “measured” data). These data are employed as 92 the input data in the source identification problem to determine the current source density. 93 The active cloaking and illusion devices are then introduced to mask the current source, and 94 the effectiveness of the devices is demonstrated. 95 96 RESULTS 97 98 Electric potential determination 99 The electric potential u (measured in V) is determined from the volumetric current source 100 density f ≠ 0 (also known as the self-potential source 38,39) by solving the boundary value 101 problem for the conductivity equation 102 −∇ ⋅(σ ()xx ∇uf ()) = () x, x∈Ω (1) 103 with the Robin condition at the boundary of the model domain 40 ∂u()x 104 σ ()x+=gu ()() xx 0, x∈∂Ω . (2) ∂nx() −1 x T 105 Here σ is the electrical conductivity (measured in S m ); = (,xxx123 , ) are the Cartesian 106 coordinates; = is the three-dimensional model domain (its description 3 107 can be found Ωin MethodΩ ∪ Σ, and ∪ Ω its⊂ two ℝ -dimensional sketch in Fig. 1); is the SSD, is the OSD, n 108 Σ is the ISD; is the outward unit normal vector at a point on the boundaryΩ ∂Ω , whichΩ restricts 109 to a bounded domain Ω ; and g is a non-negative function defined at the model boundary 3 110 ℝas the reciprocal distance from the boundary to the geometrical centre of the model domain 111 Ω . 112 To solve the problem (1)-(2) numerically, the finite-element method is used 41,42. The 113 solution to a discrete problem corresponding to the weak formulation of the problem (1)-(2) 114 can be presented as: 115 u= Af , (3) 116 4 117 118 Figure 2. Electric potentials generated by two current source densities. The perspective view (a) and top 119 view (c) of the current source density f + ; the perspective view (b) and top view (d) of the electric 120 potential u+ generated by the current source density. Top view of the electric potential u (f) 121 generated by f (e). Here and in Figs. 3-6, a top view image presents a transparent projection of 122 physical quantities at finite element nodes on the plane. The size of the nodes in the images is 123 proportional to the absolute value of the physical quantities it represents, i.e. the nodes with zero-values 124 are not displayed. Dashed lines show the position of the paths, along which synthetic measurements of 125 the electric potential have been made. 126 5 127 where u and f are the discrete representations of the electric potential and the current source 128 density, respectively, and A is the solver operator (see Method).