Vibration isolation in plates using flexural cloaking

 

Aidin Nojavan¹

Institute for Acoustics, Technische Universität Braunschweig

Langer Kamp 19, 38106 Braunschweig, Germany

 

 

Sabine C. Langer²

Institute for Acoustics, Technische Universität Braunschweig

Langer Kamp 19, 38106 Braunschweig, Germany

 

ABSTRACT

 

Cloaking is a novel idea, where the incident waves are guided around a space called the core. The cloaking concept can be realized by engineering the material matrix around the core. Cloaks can be used to guide waves around an obstacle with minimum reflection and scattering. On the other hand, they can be also used to isolate the core space, which makes cloaking an innovative potential noise control measure at challenging lower frequencies. Space isolation using cloaks can be beneficial for different applications such as architectural noise control, or soundproofing the cabin in automotive and aerospace industries. However, this interesting aspect of cloaking is to the best knowledge of the authors, not investigated in the literature. In this study, a flexural cloak in a thin plate is modeled using the finite element method to investigate the impact of cloaking on isolation of the core numerically. The effect is examined using plane waves as well as concentrated force excitation. Moreover, the dependency of the isolation effect on the geometrical size of the core and the cloak is investigated with regard to the wavelength.

 

1. MOTIVATION

 

Cloaking is a special case of wave guidance, where the waves are guided around a space or object through appropriate material distribution. A schematic of a cloak is shown in the right panel of Figure 1, where the guidance of the waves around the central core space is illustrated by ray paths. In the figure, the waves are guided around the core space with radius R₁ by manipulating the material properties in the cloak region with radius R₂ . The left panel in the figure illustrates how the space seems for the waves as a result of perfect cloaking. Figure 1 can represent the cross section of a 2-D cylindrical or a fully 3-D spherical cloak. The core space can represent an obstacle with a very high impedance, which causes the waves to be scattered and reflected. The guidance of the waves around the obstacle reduces the wave scattering and reflection. This is the main aspect of cloaking, which is investigated in several studies in different fields [1] [4]. From this perspective, a cloak can be used, for example, to guide the waves around the pillars in a hall for a better acoustic experience, or for stealth technology.

 

In Figure 2, the performance of a cloak in a plate is shown. In the figure, the plane waves propagate from right to left. On the left side of the figure, a homogeneous plate without any obstacles is shown, which is called the reference plate in this study. In the reference plate, the waves can propagate without any disturbance. In the middle, the waves are shown in a homogeneous plate with an obstacle but without a cloak. The waves are scattered after meeting the obstacle and a shadow region is resulted after the obstacle. On the right side of the figure, the cloak guides the waves around the obstacle, which reduces the wave scattering and shadow region.

 

Figure 1: Wave propagation in the initial (left) and distorted space (right). The waves are bent over the core space by experiencing a transformation in the cloak.

 

Figure 2: A cloak guides the waves around an obstacle, which reduces the scattering of the waves [3].

 

The cloaking concept was firstly introduced in electromagnetism and later followed in other fields such as acoustics [4], and elastodynamics [5]. Later, researchers attempted to bring the advantages of thin plates and cloaking together [6]. Thin plates can bear large loads despite their light weight, which makes them in addition to their compactness, beneficial in the industries. By the vibration of thin plates, flexural wave is the main concern. Accordingly, cloaking in a thin plate is about manipulation of flexural waves and usually the term flexural cloaking is used to refer to elastic cloaking in thin plates. Flexural cloaking can be used, for example, on the neighborhood of a discontinuity in a plate, such as a hole, or on a position with abrupt impedance change to avoid scattering of the waves.

 

Besides using cloaking to reduce wave scattering, cloaking can be used from another perspective to isolate the core space from the incident wave. This interesting aspect of cloaking can benefit a variety of applications and improve our future living environment. The space isolation by a cloak can be used, for instance, for noise reduction in buildings, isolation of the cabin in planes and cars, protection from seismic waves, protect sensible equipment, and isolate laboratory and measurement devices. Many of these applications can be theoretically realized by flexural cloaking. A flexural cloak can be more easily integrated in different systems regarding its compactness and light weight.

 

Despite the potential applications regarding space isolation through cloaking, this aspect is to the best knowledge of the authors not investigated in the literature yet. Though the possible applications are referred in some literature, no studies are performed to verify the possibility of the concept. In the literature, normally the core region is clamped, and the reduction of the wave scattering through cloaking is studied. Clamping the core region means ideal reflective boundary conditions on its edge ( R₁ in Figure 1). In fact, the proposed transformations in the literature for flexural cloaking are based on the assumption that the core’s edge is ideally reflective and possess boundless impedance so that the waves can not propagate through the core region. In this sense, the core region is normally clamped, to represent a very stiff obstacle or a discontinuity such as a hole in the plate. Our initial investigations implied that in the presence of a free core, the waves propagate through the core region and the cloak stops to function as expected. However, it is of interest to know, whether cloaks can also isolate the core space, if the perfect reflective boundary conditions are not applied on its edge. Hence, in this study, the possibility of isolating the core space in a flexural cloak is studied numerically using the finite element method (FEM). As an initial study, we use simply supported boundary conditions on the core’s edge, which allows the cloak to function normally. On the other hand, the simply supported boundary conditions allow the moments to be transferred to the core region and vibrate it. As a result, the potentials of cloaking in isolating the core region can be primarily examined with the present transformation. In further studies, there is the need for modified transformations for a more realistic solution.

 

Another important topic in this study is the excitation mechanism and its influence on cloaking. In the literature, the cloaking effect is normally examined with respect to plane waves similar to Figure 2. It is investigated, whether plane waves maintain their form after passing through a cloak or the plane pattern of the waves is disturbed due to scattering. This approach is understandable and reasonable in studying the performance of cloaks in reducing wave scattering. However, plane wave propagation is not an often case in real applications. Hence, this study is performed for plane waves as well as concentrated force excitation as a more realistic scenario.

 

One of the principle motivations in studying any system is to learn about the influencing parameters on its behavior. The achieved knowledge helps to comprehend the system more deeply, and be able to predict its behavior more precisely. Hence, it can be designed more accurately and efficiently. In this regard, the operation of the modeled cloak in this study in isolation of the core space is explored with regard to the dimensions of its core and cloak relative to the wavelength. The main aim is to investigate, whether the cloak’s performance is related to its geometrical specifications. It is of interest to know the influencing parameters and limitations of the cloak’s performance.

 

Following the introduction to the topic in this section, the theoretical basis for designing a cloak is explained in section 2. Afterwards, the methodology of the numerical studies, including the general information on the modeling and simulation procedures, are explained in section 3. The results are discussed in section 4.

 

2. THEORY

 

The concept of cloaking has to overcome different challenges before revealing itself. In a cloak, the propagation of the waves is influenced by engineering the material properties, for example in the doughnut region with radii R₁ and R₂ in Figure 1. In most of the engineering problems, a physical field is solved in a medium with properties, which are either already known or are determined previously by other examinations. In a cloak, however, the distribution of the physical field is known or desired over a specific space, and based on that, the required material properties are searched. From this point of view, designing a cloak is an inverse problem. There are only few techniques to solve an inverse problem [7, p. 197]. One of them is the transformation method , which is explained in section 2.1.

 

Another main challenge of the cloaking concept is its realization. Normally, the calculated material matrix for a cloak is continuously varying, anisotropic and inhomogeneous. One of the promising techniques to realize such a complex material matrix is the multilayer method , which is explained in section 2.2.

 

2.1. Transformation Method

 

In a cloak, such as the one shown in Figure 1, the waves that normally propagate all over the space with radius R₂ , are transformed to the doughnut-shaped region between R₁ and R₂ . Transformation is a general term for manipulating the position of a set of points and correspondingly manipulate the position or shape of an object by shifting, scaling, or reflection. In the context of cloaking, the waves are squeezed, stretched, or shifted in the doughnut region by engineering the material properties in this region. The transformation method is a technique to find the appropriate material distribution for transformation of a physical field in space.

 

In the transformation method, the physical space and the material properties are joint together in space. This is usually already present in the governing equations of the physical field. Then, coordinate transformation is used to describe the desired transformation of the physical field. Considering Figure 1, a coordinate transformation from the space within R 2 to the doughnut-shaped space between R₁ and R₂ is used to describe the physical field transformation. Regarding that the physical field and material properties are already associated together in space, the coordinate transformation is used to inversely transform the physical field and calculate the related material property distribution.

 

Concerning the topic of this study, the transformation method is used to transform the flexural wave propagation in a thin plate. It is supposed that Figure 1 depicts the cross section of a cylindrical cloak in a thin plate. The vibrations in a thin plate are described by the Kichhoff ’s theory and determined by the following equation in polar coordinates:

 

 

where D , w , ρ , h , and ω denote the flexural rigidity, deflection, mass density, thickness, and angular frequency of the plate, respectively. In fact, the Kirchhoff ’s equation describes the flexural wave propagation as the physical field in space and in relationship to the material properties. Hence, the flexural wave propagation in a thin plate can be transformed by applying the respective coordinate transformation on the Kirchhoff ’s equation. One possible transformation for the discussed cylindrical cloak is the following transformation, which is originally proposed by [8] for flexural cloaking:

 

 

where r and θ refer to the radius and angle in the untransformed space. The terms with tilde symbol (˜) in this study belong to the transformed space.

 

In order to find the transformed material distribution, the above transformation is applied on the Kirchhoff ’s equation. For more details on the derivation of the relations, it can be referred to the work of Zareei et al. [8]. Considering that the Kirchhoff ’s equation is formulated in relation to the flexural rigidity D , the transformed relations are also achieved for orthotropic flexural rigidity in radial (  D˜ r˜ ) and azimuthal (  D˜  θ˜ ) directions. However, the adjustment of the flexural rigidity for the complex material matrix of a cloak is challenging, as flexural rigidity depends on the elasticity modulus E , Poisson’s ratio ν , and the plate’s thickness h . Therefore, the calculated relations for the transformed flexural rigidity are often assigned to the elasticity modulus. In fact, Stenger et al. [9] designed and experimentally validated their cloak successfully based on this assumption. The transformed orthotropic elasticity moduli based on transformation (2) can be simplified to the following relations:

 

 

As previously mentioned, further details on derivation of the relations can be found in [8]. By applying the material distribution function (3) in a cloak, the waves can be guided around the core space. In the next section, it is discussed, how the calculated property distribution can be realized.

 

2.2. Multilayer Technique

 

The required material properties for a cloak is normally complex and challenging. The elasticity modulus in relation (3) is anisotropic and inhomegeneous. In fact, it changes for each value of  r˜ . In order to make the cloak’s realization feasible, the continuous material distribution in relation (3) is discretized by using the multilayer technique. The technique was firstly adopted by Farhat et al. [6] for flexural cloaking and inspired by the previous work of Torrent and Sánchez-Dehesa [10] in the field of acoustics. In the multilayer technique, multiple layers are used to discretize the material properties distribution. The technique benefits from the effective medium theory and uses a mixture of at least two materials in each layer to adjust the required properties. Based on the effective medium theory, if the size of a microstructure is much smaller than the wavelength, its macro-structural behavior can be described by its effective properties and regardless of the micro-structural interactions. A schematic view of the layers and the respective materials in a multilayered cloak is shown in Figure 3 left.

 

 

Usually, a mixture of a softer and a harder material is used in each layer, which are denoted in this study by (s) and (h), respectively. The use of a mixture in each layer facilitates to tailor the properties and also to achieve orthotropic properties in each layer. The effective properties of the mixture in each layer is calculated based on the rule of mixture:

 

 

where d and j refer to the width of each ring or layer, and the layer number, respectively.

 

In Figure 3, the distribution of the elasticity modulus for a cloak designed based on relation (3) is shown. In the figure, the x-axis denotes the distance from the core’s edge as a factor of R 1 . The y-axis shows the ratio of elasticity moduli to the elasticity modulus of the host plate ( E_host ). The cloak shown in the diagram is designed with R₁ = 15 mm and R₂ = 70 mm. Accordingly, the ratios in the diagram are equal to unity for  r˜ / R₁ > 4.67, which refers to the host plate. The distributions in the radial and azimuthal directions are shown with solid and dashed lines, respectively. The continuous distributions from relation (3) are depicted with   in radial and  in azimuthal directions. The corresponding discretized values are shown with  and , respectively.

 

In the design procedure of a cloak, a specific number of layers is assumed for the cloak. For each of these layers, a reference value  r˜ is assumed. Then, the effective orthotropic elasticity moduli of each layer is calculated based on equation (3). At the end, the fraction volume of the soft and hard rings in each layer are calculated by equation (4). In this study, the same fraction volume is assumed for the rings, which means that γ^j = 1. Based on this assumption, the elasticity moduli of the soft and hard rings in each layer are calculated by equation (4). The calculated properties are set into model as artificial materials regardless of their availability. Using this approach makes the design procedure very convenient while still physically correct. It facilitates focusing on the effects of the cloaking concept regardless of the material availability.

 

3. MODELING AND SIMULATION

 

The numerical investigations are carried out by the FEM and using the commercial software Abaqus 2018. The host plate has the dimensions of 900 × 1100 × 1 mm. It is modeled by quadratic thick shell elements (S8R) in Abaqus with 8 nodes, which includes contrary to the Kirchhoff theory also transverse shear deformation [11]. The elasticity modulus and Poisson’s ratio of the host material are equal to E_host = 0.5 GPa and ν_host = 0.45. The plate is simply supported on two edges in accordance to the schematics in Figure 4.

 

The cloak’s rings are modeled as circular element sets on the plate. Regarding the explanations in section 2, artificial materials are used for the cloak in this study. The elasticity moduli of the rings are calculated by equation (4) and assuming γ^j = 1. The Poisson’s ratio of all the materials is set to ν_host . The calculated material properties are assigned to the circular element sets to model the cloak. In the study, different cloak configurations with different dimensions are investigated, which are explained in section 4. The simulation of cloaks demands a fine geometrical discretization. In this study, a global mesh size of 0.0095 is set for all the models, which is based on a convergence study at 800 Hz.

 

For the investigations with plane waves, the plate is excited on its right side using pressure excitation on the shown surface. Moreover, the nodes on the excitation surface are tied together using the multipoint constraint in Abaqus to make them move in phase and generate plane waves in the plate. However, in most of the real applications, the system is not excited with plane waves. Hence, the investigations are performed also with a concentrated force excitation. The concentrated force is applied on the point shown in the figure, which is a slightly above the horizontal midline of the plate to avoid symmetric wave propagation.

 

The models are simulated for 0.03 s using the implicit transient analysis in Abaqus. The number of time steps is set to the frequency so that the time discretization is finer at higher frequencies. The simulation time suffices the waves to reach the opposite edge of the plate and hence, pass by the cloak. The propagating waves are dissipated on the edges of the plate using dashpots. Using this approach, the edges of the plate are non-reflecting, which is essential for analyzing the cloaking effect. The dashpots are arranged in rows on each edge. Their damping constant increases linearly in each row to avoid impedance jump and reflection of the waves. In the figure, higher damping constants are shown with darker colors.

 

In order to avoid numerical errors, a very small structural damping of 0.1% is assumed for all the materials in this study. Choosing a small identical damping for all the materials is necessary for a fair comparison of the cloaking effect. In reality, different materials with different damping constants are present in a cloak. However, the concern of this study is to investigate the influence of a cloak on isolation of the core space just by guiding the waves around it and independent of the damping.

 

One of the main issues in investigating the cloak for space isolation is the boundary conditions

 

Figure 4: Schematics of the modeled cloak (after [3])

 

on the core region. In the present literature, the influence of cloaking on reducing wave scattering is examined, for which the core space is clamped to mimic an obstacle or a hole in a plate. Clamping the core is clearly not an option for investigations of space isolation with cloaking. Ideally, the core region must be free and without any boundary conditions. However, the transformations introduced in the literature, including transformation (2), are defined to transform the space with 0 ≤ r ≤ R₂ to the space with R₁ ≤  r˜ ≤ R₂ . It is noticed from relation (3) that the transformed properties on the core’s edge with  r˜ = R₁ are zero and unrealistic. In other words, the transformation causes the region inside the core to appear as an infinitesimal point ( r = 0) to the waves and therefore minimize scattering of the waves. The transformation is based on the assumption that the core’s edge with radius R₁ is fixed and reflects the waves perfectly. If the core’s region, and particularly its edge is free, the waves pass through the core’s space and the cloak does not work.

 

In this study, the edge of the core ( r˜ = R₁ ) is simply supported in the deflection’s direction, which makes the transformation in equation (2) valid and applicable for isolation. On the other hand, the moments can still be transferred to the core’s space and vibrate it. Hence, the effect of the cloak in space isolation can be investigated. In a further step, however, it is necessary for the study of realistic applications to modify and extend the transformation to enable the isolation of a free core region through cloaking or in a general point of view, regardless of the applied boundary conditions on it.

 

4. RESULTS AND DISCUSSION.

 

In order to assess the influence of the cloak on isolation of the core space, the surface weighted root square mean velocity of the elements in the core region is calculated and averaged over a specific time. The calculated values are compared for cloak configurations and respective homogeneous plates with identical specifications but without cloak.

 

In Figure 5, the results are shown for a cloak with dimensions of R₁ = 15 mm and R₂ = 70 mm. The results for the cloak configurations are shown in orange and compared with the same plates without cloaking, which are shown in gray. The results for plane wave excitation are depicted with solid lines, whereas the results with the concentrated excitation are shown with dashed lines. It is noticed that cloaking isolates the core space remarkably for both types of excitation. This indicates that the performance of the cloak is not limited to the special case of plane incident wave or symmetric excitation.

 

The cloak reduces the velocity level in the core region up to 15 dB. It shows an impressive performance at frequencies below 500 Hz. In fact, the performance of the cloak in isolation decreases as the frequency increases. The vibration attenuation through cloaking diminishes gradually for the concentrated excitation as the frequency increases. For the plane wave incident, however, the effect disappears abruptly. While the cloaking effect reduces the velocity level over 10 dB at 700 Hz, it increases the velocity level at 750 Hz.

 

Figure 5: Wave isolation through cloaking. The guiding of the waves around the core reduces the vibrations inside the core.

 

In order to understand the reason of this behavior, the ratio of the core’s size to the wavelength is calculated. A cloak consists of different materials. However, as explained previously, the present flexural cloaking transformations are based on the assumption of perfect reflection on the core’s edge. Therefore, the design of the respective cloaks is theoretically independent of the core’s material. In this study, the core region has the host material. Accordingly, the bending wavelength in the core region can be denoted by λ_b,host . Therefore, the ratio of R₁ /λ_b,host is calculated to achieve more knowledge on the wave behavior in the core region. A larger ratio indicates that the waves have more space to build modes in the core region, which enables low frequency waves to cause a larger deflection. Accordingly and in accordance to the results in Figure 5, the cloak’s performance in isolation of the core decreases as ratio R₁ /λ_b,host increases.

 

On the other hand, the performance of a multilayered cloak decreases in general as the frequency increases. In fact, we show in our recent paper [3] that the performance of the same cloak in reducing wave scattering diminishes as the frequency increases. A multilayered cloak is designed based on the effective medium theory, which requires the size of the microstructure to be much smaller than the wavelength. As frequency increases and correspondingly the wavelength decreases, the validity of the effective medium theory fades and the performance of the cloak decreases. It is shown in our previous study [3] that using narrower layers can enhance the performance of the cloak over a broader frequency range. Using narrower layers means that the ratio of the microstructure’s dimension to the wavelength is decreased and the validity of the effective properties is increased. Hence, the decrease of the cloak’s performance in Figure 5 can also be connected to the effective medium theory and width of the layers. In this regard and in order to understand the behavior of the cloak better, the investigation is performed on cloak configurations with different R₁ .

 

In Figure 6, the performance of three cloak configurations with different radii of 15, 22.5, and 30 mm are compared for plane incident wave and concentrated excitation. The radius R 2 of the configurations is equal to 70, 105, and 140 mm, respectively, to ensure the same ratio of R₂ / R₁ = 4.67 for all of them. Moreover, the width of the layers is equal to 5.5 mm for all the configurations to maintain the same condition from the perspective of the effective medium theory. In the figure, the reduction of the velocity level for each configuration with regard to its counterpart homogeneous plate is depicted. The configurations show different behavior, which is reasonable as they are different systems with different conditions and material distributions. The different radius R₁ enables different modes to be formed and hence, the core region in each of the configurations vibrates differently. Therefore, each configuration is compared with its respective homogeneous plate.

 

However, the main aim of the comparison is to show that the cloaking effect can isolate also larger spaces remarkably. In this regard, the ratio of R₁ /λ_b,host is noted for each configuration and frequency in the left panel of the figure. It is shown that the cloak is able to reduce the vibrations in a core larger than the wavelength to a great degree.

 

It is also noticed that the configurations show different behaviors at higher frequencies. While the vibration attenuation by the configuration with R₁ = 15 mm is diminished at higher frequencies, it is improved for the configuration with R₁ = 22.5 mm. This indicates that the abrupt change at 750 Hz in Figure 5 is not necessarily related to the width of the cloak’s layers and the effective medium theory. This encourages further investigations on the topic, including the limitation by the effective medium theory and the modal behavior of the system.

 

Figure 6: Comparison of core isolation by cloak with different R₁ . In the left panel, the ratio of R₁ /λ_b,host is noted for each configuration and frequency. The values show that cloaking is able to reduces the vibrations in core regions of different sizes, including core spaces larger than the wavelength, to a great degree.

 

In the next step of this study, it is investigated, whether the size of the cloak influences its performance in isolation of the core space. For this aim, three configurations with identical boundary conditions, identical R₁ = 15 mm, but different R₂ of 48, 70, and 103 mm are compared together. The configurations are designed with the same width for the layers. The results are shown in Figure 7. It is noticed that the size of the cloak has minor influence on the performance of the investigated cloak in isolating the core region. The results are in good accordance to the observations by [3]. Considering the distribution of the elasticity module in Figure 3 for transformation (2), it is noticed that the elasticity modulus changes very smoothly for larger radii close to R₂ , whereas it changes drastically in the neighborhood of R₁ . Therefore, a change of R₂ has little influence on the performance of the investigated cloak. However, this conclusion is only valid for a cloak designed based on transformation (2).

 

Figure 7: The influence of the cloak’s radius ( R₂ ) on its performance in isolation of the core. The size of the cloak has minor influence on the performance of the investigated cloak. This lies on transformation 2, which is very smooth in the neighborhood of R₂ .

 

5. SUMMARY

 

Cloaking is a novel concept, where the waves are guided around a central space, called the core, by engineering material properties in its neighboring region, which is called the cloak. In cloaking, the transformation of a physical field is described by coordinate transformation and realized by appropriate material distribution around the core. Cloaking can be used to reduce the scattering and reflection of waves by an obstacle. For this aim, a transformation, e.g. equation (2) transforms the whole space inside the cloak region ( r ≤ R₂ ) to the doughnut-shaped space of R₁ ≤  r˜ ≤ R₂ . This aspect of cloaking is investigated in different fields, including flexural cloaking in thin plates [6]. Flexural cloaking enables to benefit from cloaking concept and compactness and light weight of thin plates simultaneously.

 

Another interesting aspect of cloaking is to isolate the core space from the incident waves through guiding the waves around it. This aspect of cloaking has been to the best knowledge of the authors not investigated yet. In the literature related to flexural cloaking, the core region ( r ≤ R₁ ) is normally clamped to mimic an obstacle with a very high impedance or a discontinuity such as a hole in a plate. It is then investigated, how cloaking can reduce the resulted wave scattering. Accordingly, the cloaking transformation is also defined based on the assumption of a perfect wave reflection on the core’s edge. The fixed boundary conditions on the core’s edge is not realistic for isolation applications. In an ideal case, the cloak should be able to isolate a free core, independent of its boundary conditions. For this aim, the cloaking transformation must be modified.

 

In this study, however, the possibility of using flexural cloaking to isolate the core space in a thin plate is investigated by a minor modification of the boundary conditions of the core region. More exactly, the core’s edge is simply supported in the deflection’s direction, where the moments can still be transferred to the core space and vibrate it. Therefore, the possibility of space isolation with cloaking can be investigated by the existing transformation. The results of this study can be a motivation for further studies on the topic. In further studies, there is the need for modified transformations for a more realistic solution.

 

The investigations in this study show an impressive potential of flexural cloaks in isolation of the core. The vibrations of the core can be reduced remarkably at a broad frequency range by the cloak. The interesting fact about isolation with multilayered cloaks is that they show a better performance at lower frequencies, where isolation is challenging by the conventional materials and solutions. This is due to the e ff ective medium theory as the basis for multilayered cloaks and the size of the cloak’s layers with respect to the wavelength. Flexural cloaking shows capability of isolating the core space with di ff erent dimensions with regard to the wavelength. Therefore, it can be an innovative solution for isolation for small applications such as isolating a small measurement device, or large applications like isolation in buildings. Nevertheless, further studies are encouraged on the topic.

 

For the investigated cloak in this study, little influence of the cloak’s radius ( R₂ ) on its performance is observed. This is due to transformation (2) used for the investigated cloak, which is smooth in the neighborhood of R₂ and drastic in the close neighborhood of R₁ . As a result, an increase of R₂ influences the cloak’s transformation slightly. It should be reminded that this observation is only valid for the transformation, accordingly for the cloak, used in this study.

 

While cloaking shows theoretically an impressive potential in isolation, it should overcome different challenges to become a practical solution in urban and industrial applications. There is the need for modifications on the transformation to isolate the waves around the core with more practical boundary conditions or ideally independent of the boundary conditions on the core’s edge. Besides the theoretical work on the transformation, the mechanical realization of such a cloak can be challenging. While the transformation method is a powerful tool for theoretical work on cloaking, it does not include solutions for the realization challenges. The complex anisotropic material distribution of cloaking needs simplifications and innovative solutions in terms of manufacturing engineering to be integrated in the industry.

 

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