Design of elliptical underwater acoustic cloak with truss-latticed pentamode materials

Yuanyuan Ge, Xiaoning Liu , Gengkai Hu

School of Aerospace Engineering, Beijing Institute of Technology, Beijing 10 0 081, China

Keywords: Elliptical acoustic cloak Pentamode material Microstructure design Truss lattice

ABSTRACT Pentamode acoustic cloak is promising for underwater sound control due to its solid nature and broad- band efficiency, however its realization is only limited to simple cylindrical shape. In this work, we estab- lished a set of techniques for the microstructure design of elliptical pentamode acoustic cloak based on truss lattice model, including the inverse design of unit cell and algorithms for latticed cloak assembly. The designed cloak was numerically validated by the well wave concealing performance. The work proves that more general pentamode acoustic wave devices beyond simple cylindrical geometry are theoretically feasible, and sheds light on more practical design for waterborne sound manipulation.

Invisible cloaks and other devices aiming to freely manipulate physical fields have been fascinating subjects these years. Pendry et al. [1] and Leonhardt [2] first came up with the concept of trans- formation electromagnetics (EM), based on the pioneering work of Dolin [3] , and the EM cloak was soon demonstrated with meta- material techniques [4] . Transformation acoustics based on meta- fluid of anisotropic density was first proposed by observing the analogy between acoustic equation and Maxwell’s equation [ 5 , 6 ]. A variety of meta-fluids realizing anisotropic density have been sug- gested, such as alternating fluid layers, perforated plates immersed in fluid, etc., however the working media are basically fluidic in nature [7–9] .

Besides the meta-fluids with anisotropic mass density, there is however an alternative route for acoustic cloak making use of solid-based pentamode material (PM) with anisotropic modulus. PM is degenerated elastic material with elastic tensor having a sin- gle nonzero eigenvalue [10] . By microstructure design, PM can sup- port a more general stress state other than the hydrostatic of con- ventional fluids [11–13] . The milestone for transformation acoustics based on PM is due to Norris, he proved that, under curvilinear co- ordinate transformation, conventional acoustic equation possesses the same form as that of PM wave equation [14] . Acoustic cloak using PM has advantages of broadband efficiency and solid na- ture, thus is more promising for practical applications. These mer- its stimulated intense researches on PM transformation acoustics and a number of wave manipulation functions have been designed and experimentally demonstrated for underwater sound [15–19] . It is worth mentioning that some active schemes have been success- fully used in elastic waves steering and wideband cloaking [ 20 , 21 ], whether these active techniques could be employed in tuning solid materials towards desired PM behavior for waterborne acoustic control is also an interesting problem.

At present, design of PM acoustic devices was mostly limited to regular configurations. The origin is that, in accordance with the divergence-free characteristic stress of the graded PM, the coordi- nate mapping must be curl-free in order to produce a symmet- ric deformation gradient tensor [22] . This requirement is only easy to achieve with axisymmetric shape such as the cylindrical and the spherical cloak. By interpreting the mapping as displacement field of a special elastostatic problem, Chen et al. [22] proposed a solution to construct quasi-curl-free mapping for arbitrary shaped PM cloak, with which PM properties for irregular cloak can be ob- tained. Recently, Quadrelli et al. [23] derived quasi-curl-free map- ping as well as the necessary PM properties for a double-elliptical cloak making use of the elliptic coordinates. However, microstruc- ture design of cloak other than cylindrical shape is not reported so far. In comparison with the asymmetric cloak, an essential dif- ference is that the PM unit cell must be designed and assembled cell by cell and no preferred orientation can be presupposed, thus necessitating substantial updating of design techniques. PM based acoustic wave control, though possessing much advantages over other metafluids, their engineering applications would be quite limited if the device configuration can only be cylindrical or spher- ical. The techniques devised in this work will be very meaning- ful for extending PM based wave functionality to applications with general geometry, e.g. cloaking of irregular underwater vehicles, ar- bitrary wave bender, carpet cloak, etc.

In this letter, we will present a systematic microstructure de- sign scheme for an elliptical PM cloak based on ideal truss lattice model [24] . The algorithm proposed in Ref. [22] is adopted at first to determine the required gradient PM distribution for a typical el- liptical cloak. Then analytical homogenization of PM for a general unit cell is given in closed form, with which efficient inverse de- sign of PM cell can be implemented. Finally, algorithms are estab- lished to segment the cloak domain and to assemble the PM cells up to an integral latticed cloak. The performance of the designed cloak will be numerically validated via finite element simulation.

Gradient PM properties for an elliptical cloak

PMs are characterized by elastic tensor with only one nonzero eigenvalue, hence the elastic tensor can be expressed as C =KS ?S [ 10 , 14 ], where the second order symmetric tensor S is called char- acteristic stress. The constitutive relation then reads:

The material can only withstand stressσproportional to S , i.e.,σ= -pS , andpis called pseudo pressure in analogy to the acous- tic pressure in ordinary fluids. Expressed in pseudo pressure, wave equations of PMs are

where v is the particle velocity,ρthe density, and time harmonic convention exp(?iωt)is adopted. Note that the S tensor has to be divergence free, i.e.,·S = 0 , in order that the material has to be at equilibrium [14] . For the trivial isotropic case of S = I , Eq. (2) re- duces to the traditional acoustic wave equation.

Basic ingredients of transformation acoustics via PM are out- lined here in brief with an example of elliptical cloak, as depicted in Fig. 1 . Consider a virtual space X occupied by homogeneous acoustic fluid with densityρ0and bulk modulusK0, pressurep′(X)and velocity v′(X)are governed by

Fig. 1. Illustration of transformation acoustics based on PM material: a Virtual space; b Physical space.

where F =?x /?X is the deformation gradient tensor, andJ= det F .

There is an additional condition for S at the outer boundary of the cloak required by fields continuity and S = I inγout. At?γ+, principal directions of S (x) must be in parallel with the normal and tangent directions (en, et, see Fig. 1 b) of the boundary and its normal components must be unity, i.e., S = enen+Stte t e t . So far, there is no general method for constructing divergence free S (x) under these stringent constraints. One solution is to find acurl-freecoordinate mapping such that F ≈FTand let S =J?1 F, then S is nearly symmetric and naturally divergence free because of the identity·(J?1F)= 0 , and the boundary constraint is automat- ically satisfied. An extra benefit of this choice is that, c.f. Eq. (4) , the tensorial density reduces to isotropic one,ρ=ρI . In this case, the graded PM properties in the cloak shell can be summarized as,

Fig. 2. Contour plots of cloak properties: a Symmetry deviation of F (F 12 - F 21); b Normalized density ρ/ ρ0 ; c Principal orientation angle of S; d Anisotropy degree S t / S n .

Note that here and henceforth, the scalar coefficient is absorbed into S in the expression of C . If such material pattern could be im- plemented, the cloak will conceal objects in the cavity and result- ing scattering would in theory be the same with that in the virtual space.

Curl-free coordinate mapping can only be constructed intu- itively for the cylindrical and spherical cloak. Here for the ellip- tical cloak, a numerical scheme proposed in Ref. [22] for finding quasi-curl-free mapping for irregular geometry is employed. De- fine inverse mapping X (x) = x + u (x) where u represents displace- ment taken a point x back to its corresponding point X in the vir- tual space. Thus, u is prescribed at the inner and outer boundaries of the cloak:

Once u is determined F can be calculated as (?X /?x)?1, and it is obvious that F is symmetric if×u = 0 . Though strictly curl- free u is hardly to get, it has been proved that quasi-curl-free u field can be obtained by solving

where |ξ|1 , in conjunction with the boundary conditions Eq. (6) . Actually, Eq. (7) is equivalent to the elastostatic equation of isotropic elastic material with special chosen Laméconstants sat- isfyingλ=(ξ?2)μ, which can be solved easily using standard fi- nite element method (FEM) software. Once u (x) is determined, the distribution of cloak material properties can be obtained through Eq. (5) .

For illustration, we consider an elliptical PM cloak immersed in the background water characterized byρ0= 10 0 0 kg/m3andK0= 2.25 GPa [13] . The half-long and half-short axes of the cloak shell are 2m and 1.6m, respectively. The radius of the inner cav- ity isb= 1 m, and the radius of tiny void in the virtual space isδ= 10?3m. In solving the quasi-curl-free u via Eq. (7) ,ξ= 10?4is used and Laméconstants are chosen asλ= -1.9999 Pa andμ= 1 Pa [22] . Firstly, the symmetry deviation of F tensor derived from u field, which is quantified byF12 -F21 , is checked and shown in Fig. 2 a. It is seen the F tensor is overall very symmetric, the asym- metric part is lower than 10?3in the most region and becomes a little bit larger only near the inner boundary. Safely, S tensor is de- fined using the symmetric part of F , i.e., S = (K0/J)1/2 sym(F). Fig. 2 b shows normalized densityρ/ρ0of the cloak shell, and it follows that 0<ρ/ρ0<2.

The symmetric S tensor can be diagonalized in its principal frame (e n , e t) as

and the angleφbetween the principal axis e n andxaxis (see Fig. 2 c) is determined as

The anisotropy of the PM modulus, i.e. the ratio of principal moduliSt/Sn, and the principal orientation angleφare the most important properties of the cloak for guiding wave around an ob- stacle, and they are contoured in Fig. 2 c and 2 d, respectively. It is seen that unlike the cylindrical case, the principal orientation is not axisymmetrically distributed and possesses a complex pattern. Fig. 2 d shows theSt/Sncontour, from which it is seen that the PM anisotropy is high and varies sharply near the inner boundary, and actually the anisotropy is also not distributed regularly. Since there is no symmetry in the PM pattern can be utilized, the microstruc- ture design and assembly of non-circular cloak have to be done cell by cell, calling for a smarter and more automatic design scheme.

Fig. 3. Distorted honeycomb truss lattice model of PM.

Truss based PMs with honeycomb lattice

The required PM properties of the cloak can be realized with distorted honeycomb truss lattices as shown in Fig. 3 , in which all bars are ideally hinged. A unit cell defined by lattice vectors a = (a1,a2)Tand b = (b1,b2)Tis indicated by the dashed paral- lelogram. The unit cell includes three bars characterized by ten- sion stiffnessEiAiand densityρi, whereEiandAiare the Young’s modulus and the section area of the bari, respectively. Bar geom- etry is defined by nodes (1 ～3) on the lattice sites and one internal node 4. Without loss of generality, place the coordinate origin at the node 1, the node positions are then

In microstructure design and assembling stage, suppose that the cloak domain is divided into many quadrilateral cells, three bars should be embedded into each cell and the internal node will be precisely adjusted to meet the conditions forSt/Snand the princi- pal orientation angleφcorresponding to the cell location.

As shown in Fig. 3 , we define three unit vectors along each bar

wherel1= | p |,l2= | a –p |,l3= | a –p | are the bar lengths. Under any loading, tension forces t = (t1,t2,t3)Tin the three bars cannot be arbitrary and their relative ratios have to balance at node 4 in absence of external load [10] , i.e.,t1 e 1 +t2 e 2 +t3 e 3 = 0 , or equiva- lently

Therefore, the final bar tensions can be expressed as

where coefficientαis to be determined and s can be solved via Eq. (13) :

The s vector is called self-stress state of the truss lattice sub- jected to periodic boundary. It represents also the non-compatible bar elongations which cannot be generated by nodal displacements [25] . The macroscopic stress is defined by integrating equilibrated microscopic stressσover the cell domainΩ. Using the identity·(r ?σ)=σin absence of body forces, the macroscopic stress is defined via tractions on the cell boundary?Ω, which in this case of truss lattice is just a sum including the forcest1 e 1 ,t2 e 2 andt3 e 3 :

whereVcell is the area of unit cell. Substituting Eqs. (11) , (12) and (14) in to Eq. (16) , it can be shown thatΣdepends only on the internal node position (p1,p2) and the undetermined coefficientα,

where:

To determine the unknownαunder macroscopic strain E , suppose the bars undergo firstly an affine elongation eaff=, where

Bar tension induced by affine elongations are not balanced, therefore eaffmust be relaxed by an extraΔe to reach the true elongations, i.e.

SinceΔe is compatible, it must not overlap with the non- compatible elongation s , that is, sT(e –eaff) = 0 , with which it can be solved that

where

Combining Eqs. (17) and (21) gives

where

is just the characteristic stress tensor of PM as required by Eq. (5) . In the inverse design, given a unit cell specified by vectors (a, b) as well as desiredSt/Snand angleφ, the internal node location (p1 ,p2) can be conveniently solved out using Eqs. (9) , (10) , (15) and (18) . Then the densityρ= (ΣρiliAi)/Vcell and the absolute magni- tude of S will be matched byρiandEiAiof bars. In the following, for the three bars in the same cell indexed byc, bar properties are set as the sameEiAi=EcAcandρi=ρc.

Design and validation of integral latticed cloak

In order to build the cloak with an assemblage of graded PM lattices, the cloak region needs to be reasonably meshed into a set of cells which is inevitably non-uniform, then for each cell a three-bar PM model can be appropriately designed and housed ac- cording to the previous section. Due to the mirror symmetry, the procedure is illustrated by a quarter of the cloak shown in Fig. 4 . At first, a quasi-conformal mesh lines are generated using an algo- rithm described in the following, so as the cloak region is divided into an almost rectangle mesh with reasonable mesh density, as shown in Fig. 4 a. Second, the midpoints of rectangular edges are used as vertices to general whole set of rhombic cells which will be as regular as possible (Fig. 4 b), then the PM design procedure given in the previous section can be executed repeatedly for each rhombic PM cell. Third, for each PM cellc, corresponding vectors (ac, bc) and S (xc) are then obtained with xcbeing the cell center, thus the internal node location as well as the bar properties (EcAc,ρc) can be solved. This design procedure for the integral latticed cloak has been automated by appropriate programming.

The algorithm for producing meshes in Fig. 4 a is borrowed from Ref. [26] , which is originally developed for building quasi- conformal coordinate mapping for arbitrary geometry. In accor- dance with the elliptical shell, suppose that the mesh are specified by a set of circumferential grid linesr(x,y) =rm, (m= 1 ～mc), and a set of radial grid linesθ(x,y) =θn, (n= 1 ～nc), as selectively exemplified in red and blue in Fig. 4 a. The functionsrandθcan be obtained by solving Laplace equations2r= 0 and2θ= 0 on the quarter cloak domain, with appropriate applied boundary con- straints. In particular, forr(x,y):

Fig. 4. Procedure of cloak design with truss lattice: a Generation of quasi-quadrilateral mesh; b Gradient rhombic cells for housing the three-bar PM model.

Fig. 5. Configuration and parameters of latticed cloak: a Color plot of bar stiffness EA ; b Color plot of line density ρA of bars.

Dirichlet conditionr= 0 on?D,r= 1 on?B,

while forθ(x,y):

Dirichlet conditionθ= 0 on?A,θ= π/2 on?C

Here n is the unit normal vector on domain boundary. The physical meaning of this boundary conditions is that: bounded byr∈ [0,1] , i.e.?Band?D,r=rmgrid lines are equipotential lines ruled by the Laplace equation, but the ends of grid line are al- lowed to slide along?Aand?Cin order to be orthogonal to them. Similar explanation applies to theθlines. The Laplace equation to- gether with the proposed boundary condition is actually equiva- lent to minimization of the Winslow functional which keeps the grid lines as mutually orthogonal as possible [26] . The scheme is quite general can be used to automatically generate for irregular domain a regular mesh division, so as to ease the PM design stage. In Fig. 4 a,mc= 29 circumferential grid lines andnc= 84 radial grid lines are used to mesh the cloak, and lines ofr= 0.83 andθ= 0.096πare highlighted. The discretermandθnvalues can be adjusted to control the mesh density, e.g. that, denser division is desired where material properties possess high gradient.

Finally, the design of cloak lattice realizing the previously de- termined material distribution in Fig. 2 is completed and displayed in Fig. 5 . The colors in Fig. 5 a and 5 b indicate tension stiffnessEAand line densityρAof bars, respectively. There are totally 4358 PM cells in a quarter of the cloak. The zoomed view of Fig. 5 a details the gradually changed irregular honeycomb lattice, which is irreg- ularly distorted and very different with an axisymmetric circular cloak.

In order to check the wave concealing performance of the de- signed cloak, full wave simulation is performed using commer- cial FEM software ANSYS, and the results are presented in Fig. 6 . Figure 6 a shows the pressure field in the background water when the immersed cloak lattice is illuminated by sound wave excited by a monopole acoustic point source placed upper left, and the op- eration frequency is 2.25 kHz. In the calculation, a fluid-structure interface is used to deal with bridging of the lattice bars and the acoustic fluid, while non-reflecting condition is set at the exter- nal boundary of background water. For comparison, Fig. 6 b and 6 c display the pressure fields of an un-cloaked void obstacle and a void protected by continuum cloak (with the material distribu- tion described by Fig. 2), respectively, under the same wave load- ing. Inside the cloak shell of Fig. 6 c, pseudo pressurep= -Jσ11/F11is displayed instead. It is observed that, though the discretization of continuously varied material property and the approximation of PM by truss lattice bring a certain amount of error, the pressure fields of Fig. 6 a and 6 c exhibit high similarity. The scattering in case of the lattice cloak is significantly reduced, and the wave front passed through the target remains almost unperturbed as shown in Fig. 6 a. Conversely in Fig. 6 b, obvious reverberation and shadow are observed in case of a bare obstable. An advantage of PM cloak is its broadband effectivity since the metamaterial mechanism is not resonance based. As long as quasi-static homogenization is jus- tified under the wavelength of the operating frequency, the cloak will work well. We conducted a supplementary simulation at a higher frequency 3.25 kHz, and the corresponding results are plot- ted in Fig. 6 d, 6 e and 6 f for case of latticed cloak, bare obstacle and continuum cloak, respectively, and as expected good cloaking performance is observed as well.

Fig. 6. Simulated pressure fields for different cases at frequency 2.25 kHz: a Latticed cloak; b Bare void; c Continuum cloak, and at frequency 3.25 kHz; d Latticed cloak; e Bare void; f Continuum cloak.

In summary, in this work, we have developed a complete set of techniques necessary for design of truss-based PM acoustic de- vices with irregular shape, including the analytical homogeniza- tion of PM for a general unit cell, its inverse design scheme, and algorithms for assembling the PM cells into a graded lattice rep- resenting a domain induced by PM transformation acoustics. The developed methods were gone through by the design of latticed elliptical PM cloak for the first time, and effectiveness was nu- merically validated by the well wave concealing performance. The work proves that, together with the algorithm of numerical quasi- curl-free coordinate mapping [22] , more general PM underwater acoustic wave devices beyond the simple geometry can be ex- pected to be brought into reality. Of course, the ideal truss model used here is still very theoretical, and it is noticed that the bar parameters and lattice discretization here is very stringent. More future works, including optimization and compromise between the cloaking performance and material sharpness as well as the feasi- ble continuum-based PM design are in order to put them forward to experimental demonstration.

Declaration of Competing Interest

The authors declare that they have no known competing finan- cial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the National Natural Science Foun- dation of China (Grant Nos. 11972080 , 11972083 and 11991030) and the Innovation Foundation of Maritime Defense Technologies Inno- vation Center (Grant No. JJ-2021-719-06).