This study starts with a simple question: can the vibration of plates or beams be efficiently reduced using a lightweight structure that occupies a small space? As an efficient technique to damp vibration, the concept of an acoustic black hole (ABH) is adopted with a simple modification of the geometry. The original shape of an ABH is a straight wedge-type profile with power-law thickness, with the reduction of vibration in beams or plates increasing as the length of the ABH increases. However, in real-world applications, there exists an upper bound of the length of an ABH due to space limitations. Therefore, in this study, the authors propose a curvilinear shaped ABH using the simple mathematical geometry of an Archimedean spiral, which allows a uniform gap distance between adjacent baselines of the spiral. In numerical simulations, the damping performance increases as the arc length of the Archimedean spiral increases, regardless of the curvature of the spiral in the mid- and high-frequency ranges. Adding damping material to an ABH can also strongly enhance the damping performance while not significantly increasing the weight. In addition, the radiated sound power of a spiral ABH is similar to that of a standard ABH.

Vibration damping using a spiral acoustic black hole

Maxwell's fish-eye lens (MFEL) with positive refraction has been shown to achieve perfect imaging, but with the cost of drain assistance. This has led to ongoing heated debates about the rigor of the physics of super-resolution phenomena in MFEL. In this work, we report that a MFEL embedded in an exterior coating, inspired by the solid immersion concept, can realize super-resolution imaging without a drain. Such a solution mitigates and bypasses the corresponding criticisms and debates of the past decades. We find that the total reflection at the outer solid-immersion interface and the native perfect focusing of MFEL synthetically contribute to a super-resolution image formed in the air. Moreover, this intuitive yet simple recipe can be robustly applied to other absolute instruments, such as the general Luneburg lens and more versatile superimaging systems are anticipated. We demonstrate the imaging performance in a solid immersion general Luneburg lens both numerically and experimentally, which indirectly verifies the imaging validity of the solid immersion MFEL without a drain. 

Solid Immersion Maxwell's Fish-Eye Lens Without Drain

• A novel approach is proposed to design flexural wave gradient index lens based on Rays Inserting Method. • A lens was designed to act as a collimator with a point source excitation. • A lens was designed to act as a waveguide which can rotate the direction of incident wave with 45°. • Lenses were achieved with variable thickness structures. The ability to control and manipulate elastic waves is important for applications such as structural health monitoring, signal processing, and vibration isolations. In this paper, we investigated the feasibility of using the Rays Inserting Method (RIM), an approach originally proposed for optical elements, to design structural components for flexural wave manipulation. The RIM entails a simple process that allows to design thickness variations in a thin plate with a desirable refractive index distribution for an intended wave path. Based on this method, a focusing/collimating lens and a waveguide that rotates the wavefront by 45° were designed and studied. Frequency domain simulations and time-based experimental characterizations were carried out. The results demonstrated that the effectiveness of the RIM for designing variable thickness structures for manipulation of flexural wave propagation along desired paths. 

Designing Flexural Wave Gradient Index Lens based on the Rays Inserting Method

Emergence of negative-dispersion passbands below the ring frequency of a piezoelectric meta-shell

Metamaterials, together with transformation optics, have been exploited to study the effects of curved spacetime predicted by Einstein's general theory of relativity in the laboratory. However, the stringent requirements in metamaterial fabrications and the narrow operating frequency make their realistic implementations challenging. In this work, we propose and demonstrate experimentally a paradigm to mimic the gravitational lensing effect using flexural waves in an elastic plate without using metamaterials. The needed refractive index field is fulfilled just by slightly varying the local thickness of plates. The generated focusing effect has a broad band from 5 to 100 kHz and shows robustness against changes on structural shapes and thicknesses of the lens. This work provides insight to study nontrivial gravitational phenomena analogically using conventional solid structures and may suggest a possible route to regulate elastic waves arbitrarily. 

Mimicking gravitational lens on a thin elastic plate

Transformation optics, which is generically applicable to other classical waves such as acoustic and elastic waves, provides an emerging design paradigm to manipulate waves. However, some lenses and optical-transformation devices require a singular refractive index; meeting this requirement is a significant challenge. A method called transmutation can relax some types of index singularity into finite anisotropy around the singularity. Here, we show that such lenses with a singularity for flexural waves can be obtained by approaching a near-zero thickness of the plate precisely at the location of the singularity. As examples, we demonstrate a series of Eaton lenses theoretically and experimentally by projecting the refractive index in space onto the thickness in plates and by working in a broad frequency range in which impedance mismatch is negligible. This framework offers an insight into feasible methods that can be used to develop singular devices such as cloaking devices on thin flexible curved plates and can be further extended to a general methodology for shaping elastic waves. We hope that this elastic platform can also be a test bed to indirectly study unprecedented phenomena enabled by gravitational and quantum fields in terms of analog models.

Singular Lenses for Flexural Waves on Elastic Thin Curved Plates

The concept of birefringence, commonly treated as a property resulting from more than one polarization, is associated with different refractive indices, depending on polarization. Here, we propose a two-dimensional elastic metamaterial engineered to demonstrate birefringence of single polarization for flexural waves. By adopting a network of beams with ``overpass'' structures to upset the topology of unit cells from a conventional elastic plate, birefringence of single polarization at low frequencies can be realized and confirmed through measured band structures and double refraction of the flexural waves; this is further supported by full-wave simulations and models. Our approach can engineer birefringence at low frequencies in which the wavelength is at least a few times larger than the lattice constant, further allowing efficient control of beam splitting, filtering, and multiplexing with applications for elastic wave devices. 

Birefringence of Single Polarization for Flexural-Wave Metamaterials

Experimental demonstration of Willis coupling for elastic torsional waves

We present a black hole eﬀect by strategically leveraging a conformal mapping in elastic continuum with curved-space framework, which is less stringent compared to a Schwarzschild model transformed to isotropic refractive index proﬁles. In the conformal map approach, the 2D point singularity associated to the black hole eﬀect is accomplished by physical plates with near-to-zero thickness. The analog gravity around the singularity results in highly conﬁned energy and lagged timings within a branch cut of the conformal map. These eﬀects are quantiﬁed both numerically and experimentally in reference to control trials in which the thickness is not modulated. The ﬁndings would deepen our understanding of the elastic analog in mimicking gravitational phenomena, as well as establish the elastic continuum framework for developing a generic design recipe in the presence of the index singularity. Geometric landscapes with elastically curved surfaces would be applicable in a variety of applications such as sensing, imaging, vibration isolation, and energy harvesting.

Yiran Hao

Papers

Singular Lenses for Flexural Waves on Elastic Thin Curved Plates

Birefringence of Single Polarization for Flexural-Wave Metamaterials

Experimental demonstration of Willis coupling for elastic torsional waves

Conformally Mapped Black Hole Effect in Elastic Curved Continuum