Innov. Infrastruct. Solut. (2017) 2:20 DOI 10.1007/s41062-017-0063-x TECHNICAL NOTE Metamaterial-like transformed urbanism 1 1 1 1 Ste´phane Bruˆle´ • Bogdan Ungureanu • Younes Achaoui • Andre´ Diatta • 1 2 3 1 Ronald Aznavourian • Tryfon Antonakakis • Richard Craster • Stefan Enoch • Se´bastien Guenneau1 Received: 9 January 2017 / Accepted: 24 April 2017 / Published online: 7 June 2017 Ó Springer International Publishing Switzerland 2017 Abstract Viewed from the sky, the urban fabric pattern Introduction appears similar to the geometry of structured devices called metamaterials; these were developed by Physicists to In this short communication, we propose an analysis of the interact with waves that have wavelengths in the range dynamic response of surface structures. These buildings are from nanometers to meters (from electromagnetic to seis- not studied as single elements with their own characteris- mic metamaterials). Visionary research in the late 1980s tics and modeled by a single degree of freedom oscillator based on the interaction of big cities with seismic signals (SDOFO) but as an interacting set of above-ground res- and more recent studies on seismic metamaterials, made of onators clamped in a common slab, at the scale of a district holes or vertical inclusions in the soil, has generated in a city. interest in exploring the multiple interaction effects of In this article, we analyze a perfect elastic system con- seismic waves in the ground and the local resonances of sisting of soil and buildings. We do not introduce soil– both buried pillars and buildings. Here, we use techniques foundation–structure interaction [4] and damping, which are from transformational optics and theoretically validate, by major aspects in earthquake engineering. We simply wish to numerical experiments, that a district of buildings could be investigate, given that communities have built highly considered as a set of above-ground resonators, purely structured cities, whether the design of a district could elastic, interacting with an incident seismic signal. We benefit from results obtained for waves interacting with hope that our proposal will contribute to all theoretical and structured metamaterials that are built at a smaller scale. experimental efforts in design of cities of the future, from a Most of the time in earthquake engineering, the dynamic metamaterial standpoint. response of a building is formulated as an elastic response spectrum SðT; nÞ, in acceleration Sa, velocity Sv, or hori- Keywords Metamaterial Á Urban fabric Á Metacity Á zontal displacement Sd of an SDOF system (mass m, Transformational physics Á Homogenization stiffness k) under seismic acceleration of the soil x€gðtÞ.This graph represents all the maximum value of acceleration, velocity, or displacement, for a relative displacement xðtÞ in time domain, for given damping ratio n and a funda- mental period T of the structure (Fig. 1a). xT ðtÞ is the overall displacement and xgðtÞ is the displacement of the soil recorded at the surface by seismograph. & Ste´phane Bruˆle´ [email protected] The novel approach illustrated in this paper suggests to take into account the effects of the overall dynamic 1 CNRS, Centrale Marseille, Institut Fresnel, Aix-Marseille response of the system composed of a group of buildings University, Marseille, France clamped in a common slab (Fig. 1b). The system is not 2 Multiwave Technologies AG, Geneva, Switzerland tested under a true seismic disturbance but by varying the 3 Department of Mathematics, Imperial College London, frequency step by step. London, UK 123 20 Page 2 of 11 Innov. Infrastruct. Solut. (2017) 2:20 Fig. 1 Principle of usual single Surface structures degree of freedom oscillator model for a building (a). Global b z a approach with a set of xt(t) resonators modeling a group of buildings of different heights x (t) g m (b) and vibrating in the bending Local resonance mode in this figure k x(t) Slab O x ̈( ) Overall dynamic response of the system The plan of the paper is as follows: First, we show the especially on soft soils [24, 25, 50], and the presence of similarity of the urban fabric with the geometry of structures at the surface of a homogeneous half-space can metamaterials (‘‘The concept of metacity’’ and ‘‘From significantly modify the ground motion [12, 20, 47, 51]. urbanism to metamaterials’’), and then, we recall the Most of the time, the problem of ground response is dis- concept of multi-scale resonant structures and the main connected from that of the resonant response of buildings characteristics of seismic ground motion (‘‘Earthquakes or group of buildings. and resonance structures’’). This allows us to propose the On the basis of studies carried out on the interaction of concept of a metacity on the basis of resonant structures cities with the seismic signal [2, 5, 6, 21, 22] and on the having their foundations in a common slab (‘‘Meta city interaction of buildings with each other [46], we propose and analogy with urban fabric’’). We make the assump- further extensions of this concept (Fig. 2) based on tion of elastic behaviour for each component of the analogies with electromagnetic and seismic metamaterials mechanical system (soil, foundation, and buildings). To [11]. Actually, the field of transformational optics, which is illustrate the idea of a city district envisioned as a 10 years old [41], can help explain metamaterial-like megastructure interacting with the seismic signal, we urbanism: When viewed from the sky, some cities look develop a numerical approach based on finite elements similar to invisibility cloaks. Numerical simulations (implemented in the commercial software Comsol Mul- backup our claim of control of seismic waves via artificial tiphysics). Starting with a square network of streets and anisotropy in metacities. buildings, we map it onto a quasi-conformal network of The high density of deep foundation or ground rein- streets with square buildings (‘‘Meta city and analogy forcement techniques in the urban area leads researchers with urban fabric’’), deduced from some spatially varying to believe that there is a significant interaction of these (but isotropic) material parameters given by a set of buried structures with a certain component of the seismic transformed elastodynamic equations. In ‘‘Discussion and signal. A promising way to cause a modification of the perspectives’’, numerical results show Rayleigh waves seismic disturbance is to create complete artificial aniso- (0.5 Hz) detoured around the center of a seismic cloak, tropy by implementing geometrical elements, full or which is 1 km in diameter. empty, in the soil [11]. The physical process is the interference of waves (body or surface waves) scattered from surfaces or objects. The effects of the anisotropy are reinforced by the local resonance of implemented ele- The concept of metacity ments which are placed along a grid according to trans- formation elastodynamics and morphing tools; these could The metacity is a concept first put forward in the late 1980s theoretically lead to an ideal cloak detouring waves on the basis of experimental observations of ground around a protected area. In these periodic or non-periodic response and its devastating effects. It is known, since the media, the desired effects are total reflection (Bragg’s 1950s, that the natural frequencies of any man-made effect), bandgaps, wave-path control, and mitigation by structure are influenced by soil–structure interaction, energy-dissipation, amongst others. 123 Innov. Infrastruct. Solut. (2017) 2:20 Page 3 of 11 20 Fig. 2 From left to right (arbitrary scale): a buried columns of deep co-authors demonstrated in 1998 the first acoustical band-gap foundation (piles) or rigid inclusion or holey-ground. b Phononic structure in the audible frequency range [38]. Analogy of b for above-ground structure made of vertical metallic elements, inspired seismic waves illustrated by a diagram of a neighbourhood with from the sculpture of Eusebio Sempere in Madrid. Meseguer and his different heights of building and various underground levels (c) From urbanism to metamaterials and how they all work together in an interrelated manner to enable movement and access. If we describe in a factual In urbanism, the urban fabric (Fig. 3) is the physical form way these structures, observed from the sky, we note a of towns and cities and the urban grain defined as the distribution of constructions reminiscent of many man- combined pattern of blocks and streets, taking into account made structures at different scales, and in particular, the character of street blocks and building height and size, metamaterials. An early example, going back in the Fig. 3 Above illustration of the urban fabric of Paris and New York (sky view). Below plan of the ideal and virtual city of Chaux in France [35] 123 20 Page 4 of 11 Innov. Infrastruct. Solut. (2017) 2:20 ab z Sol 1 y ρ1, v1 x Sol 2 ρ2, v2 ρ2v2 > ρ1v1 Fig. 4 Schematic view of principle (side view) of vertical columns low frequencies (i.e., long wavelengths get reflected by arrays of buried within a soft soil (soil 1) and clamped within a bedrock (soil columns with a deeply subwavelength cross section). Inspired from 2). These columns undergo bending (a) and longitudinal (b) motions [10] under earthquake. These local resonances create elastic stop bands at Fig. 5 Deformation of building (a). Elastic deformation for the adb c first fundamental mode (b). Ductile deformation (c) and brittle deformation (d) historical record, is the Royal Saltworks (1779) at Arc-et- to phononic crystals, which are artificial man-made struc- Senans in France, designed and built by the visionary tures whose size ranges from a few meters down to hun- Architect Claude-Nicolas Ledoux (1736–1806); this semi- dreds of nanometers or less. circular complex of buildings (Unesco World Heritage) Briefly, the first application suggested by Kushwaha formed the starting point for his project of a utopian city et al.