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Abstract
Various strategies have been proposed to achieve invisibility cloaking, but usually only one phenomenon is controlled by each device. Cloaking an object from two different waves, such as electromagnetic and acoustic waves, is a challenging problem, if not impossible, to be achieved using transformation theory and metamaterials, which are the major approaches in physics. Here, by developing topology optimization for controlling both electromagnetic and acoustic waves, we present a multidisciplinary attempt for designing biphysical cloaks with triple-wave cloaking capabilities, specifically for Ez- and Hz-polarized waves and acoustic wave. The topology-optimized biphysical cloak cancels the scattering of the three waves and reproduces the original propagating waves as if nothing is present, thus instilling the desired cloaking capability. In addition, we describe cloaking structures for multiple incident directions of the three waves and structures that work for both electromagnetic waves and sound waves of different wavelengths.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Cloaking an object from two different physical flows has been a focus since the pioneering work to develop a biphysical cloak manipulating both heat flux and direct current (DC) was proposed [1]. The theoretical concept behind biphysical cloaking in diffusive systems was, after a while, experimentally demonstrated using bifunctional metamaterials [2], bilayer shell systems [3] and a combination of passive and active schemes [4]. These technologies have been extended to invisible sensing [5] and independent thermal and electrical fields manipulations [6] in multiple-physical diffusive fields. Recently, computational inverse design based on topology optimization was demonstrated for thermal-electrical biphysical cloaks [7]. The two diffusive systems are widely known to exhibit very similar phenomena because both thermal and DC electrical conduction satisfy Laplace’s equation in steady states, and insulators of both are modeled using Neumann boundary conditions.
While development of biphysical cloaking in diffusive systems has been successful, cloaking an object from two different wave types is much more difficult. The performances of cloaking devices based on metamaterials [8,9] for a wave system are far from ideal because transformation optics-based methods [10–12] require unrealistic physical parameters and inhomogeneous media with complex material profiles. Constructing an inhomogeneous anisotropic device of this type remains very difficult, offering little advantage for practical applications. The earliest attempt for biphysical cloaking in wave systems proposed a structured cloak made of aluminum [13]; shortly thereafter, a carpet cloak based on a metasurface controlling local reflection phases was developed [14]. Recently, multi-physics surface transformations [15] were formulated offering simultaneous manipulations of polarized electromagnetic and acoustic waves. These three theories are based on aluminum structures used in constructing biphysical cloaks; as a perfect electric conductor (PEC) to $H_z$-polarized waves and hard wall to acoustic waves in air, both are governed by the Helmholtz equation applying Neumann boundary conditions. Therefore, the three theories can not be applied when cloaking an object from $E_z$-polarized wave, for which Dirichlet boundary conditions need to be imposed on the PEC. Additionally, the PEC is not suitable for modeling materials with low electrical conductivity, and very few materials (e.g., metals) can thus be used to construct the cloak. To achieve an ideal biphysical cloak, cloaking an object is essential for not only $H_z$-polarized and acoustic waves but also $E_z$-polarized waves. That is, triple-wave cloaking is required but is such cloaking possible for electromagnetic and acoustic waves?
Using transformation theory and metamaterials, solving this problem is very challenging, if not impossible. In recent years, remarkable progress has been made in informatics, which involves data analysis, optimization, and machine learning, and is seen as a powerful tool for overcoming such challenging problems. These innovative algorithms are not limited to information science and are being applied to the design of wave-controlling devices in the fields of optics [16–18] and electromagnetism. Topology optimization, the most flexible structural optimization in allowing creation of pores, has performed particularly well in designing high-performance metastructures for cloaks in not only diffusive systems [19,20] but also single-propagating-wave systems for both electromagnetic cloaks [21–25] and acoustic cloaks [26–29]. However, existing topology optimization approaches produce only metastructures designed to cloak objects for either electromagnetic waves or acoustic waves.
In this work, we propose a topology optimization for biphysical cloaks with triple-wave cloaking capabilities, specifically for $E_z$- and $H_z$-polarized waves and acoustic wave. For this purpose, we set up three objective functions, one for each of the three waves. For polarization-independent electromagnetic cloaking, structures of optimal topology made of acrylonitrile-butadiene-styrene (ABS) resin that surrounded a PEC obstacle were explored by solving the Helmholtz equations; Dirichlet and Neumann boundary conditions were imposed on the propagating $E_z$- and $H_z$-polarized waves, respectively. In evaluating acoustic cloaking, the acoustic-elastic coupled problem were solved taking the elasticity into account. To optimize the structure as a biphysical cloak, the covariance matrix adaptation evolution strategy (CMA-ES) [30] is implemented; using level-set methods, the aim is to suppress scattering of all $E_z$- and $H_z$-polarized waves and the acoustic wave.
2. Formulation
2.1 Optimization scheme for triple wave cloaking
To attain biphysical cloaking in the wave system [Fig. 1(a)], topology optimization involves suppressing both electromagnetic and acoustic scattering produced by an obstacle $\Omega _\mathrm {obs}$, here made of aluminum; the fixed design domain $\Omega _\mathrm {D}$, in which the ABS structure $\Omega _\mathrm {s}$ is transformed to affect biphysical cloaking, is set to cover $\Omega _\mathrm {obs}$. Structural symmetries are imposed on $x$ and $y$ axes. To establish wave scattering in the open region, we configure a perfectly matched layer (PML), $\Omega _\mathrm {PML}$, imposing on it the absorbing boundary condition. The interfaces between free space, $\Omega _\mathrm {D}\backslash \Omega _\mathrm {s}$, and the structure, $\Omega _\mathrm {s}$, are expressed as iso-surfaces of a level-set function [Fig. 1(b)], from which finite elements are generated. We note that, by using a level set method, no grayscale is included in the structure, resulting in a fully binarized and optimized structure with crisp structural boundaries; hence, there is no need to use a filtering scheme to remove a grayscale.
Table 1 lists the material parameters used in our topology optimization. The relative permittivity of the ABS is set assuming the cloaking of an object from microwaves.
Fig. 1. Schematic of a topology optimization for an electromagnetic-acoustic biphysical cloak. Domain sizes are $R_\mathrm {obs}=R_\mathrm {D}/3$, $L_x=R_\mathrm {D}\times 2$, $L_y=R_\mathrm {D}\times 3/2$, and $L_\mathrm {grid}=R_\mathrm {D}/70$.
Table 1. Material Properties in the Optimization.
2.2 Objective functions
The materials occupying $\Omega _\mathrm {s}$ that constitute the cloak are both dielectric and elastic and minimize the scattering of electromagnetic and acoustic waves in the outer domain $\Omega _\mathrm {out}$. The objective function determining the design of the electromagnetic-acoustic biphysical cloak is defined as
2.3 Governing equations and boundary conditions
Although the objective functions that determine the canceling of the scattered $E_z$- and $H_z$-polarized waves and the acoustic wave may be expressed in the same form (1), the electromagnetic waves and acoustic wave satisfy different governing equations and boundary conditions. The $E_z$- and $H_z$-polarized waves satisfy the following Helmholtz equations;
The governing equations for sound pressure $p$ in an acoustic medium, and the stress $\sigma _{ij}$ and displacements $u_{i}$ in the elastic media, are
2.4 Level set method and regularized fitness
The free space and structural domains, $\Omega _\mathrm {D}\backslash \Omega _\mathrm {s}$ and $\Omega _\mathrm {s}$, are outlined using a level-set function $\phi (\boldsymbol {x})$ defined as
Here, the interfaces between the free space and structural domains, $\Gamma _\mathrm {D}^\mathrm {s}$, are expressible as iso-surfaces of the level-set function [Fig. 1(b)]. The optimal $\boldsymbol {\phi }=\{ \phi _1, \ldots, \phi _j, \ldots, \phi _n \}$, the set of discretized level-set functions, is explored using CMA-ES [30] (samplings $\lambda =400$) with box-constraint handling [31], $-1\leq \phi _j \leq 1$, for infimizing regularized fitness,
where $L$ denotes the normalized perimeter length of the structure and $\tau$ the coefficient of regularization, specifically, the ratio between the maximum of the three objective functions and $L$.2.5 CMA-ES
CMA-ES [30,32] is one of the most powerful evolutionary computation approaches for searching optimal design variables $\boldsymbol{\phi }$, and the algorithm of the CMA-ES is based on the adaptation of the distribution parameters for normally distributed sampling (i.e., the shape, size, and center of the distribution) to the landscape of the minimized fitness $F$. The CMA-ES is known to be robust against optimization problems having difficult properties [33], such as multimodality and the interdependence of design variables. The CMA-ES enables designers to realize well-performing configurations [34–36] in topology optimizations with the use of box constraint handling [31,37] without implementing a trial-and-error approach to make initial guesses. The default values of strategy parameters of CMA-ES have been well studied [38] and the parameters do not need to be adjusted. The only hyper-parameter that is advised to tune is the population size, $\lambda$, which is the number of candidate solutions. It is empirically known that a greater population size tends to be more robust against local minima [39]. Because we can parallelize the fitness evaluation of $\lambda$ candidate solutions and the number of generations of the algorithm reduces as we increase $\lambda$ [40], we can accelerate the optimization wall clock time through parallel implementation. We used the following CMA-ES algorithm.
The distribution parameters are initialized at the first generation $g=0$ as
The candidate solutions are ranked using $F$ with an adaptive penalty function for handling the box constraint [31], $-1\leq \phi _j \leq 1$. The differences between the mean vector and $i$-th-ranked solution denoted $\boldsymbol{\phi }_{i:\lambda }^{(g+1)}$ are scaled as
The distribution parameters are updated following the scheme
The weighted average of $\boldsymbol{y}_{i:\lambda }^{(g+1)}$, $\boldsymbol{m}_{\boldsymbol y}$, is expressed as
and $h_\sigma ^{(g+1)}$ is the Heaviside step function defined asThe convergence of the optimization is assessed according to the error of the mean vector $\boldsymbol{m}^{(g)}$, defined as
The optimization process continues until the above convergence criteria are satisfied.
The internal time complexity of the CMA-ES per generation is $O(\lambda n^2)$, where $n$ is the number of design variables; i.e., the dimension of design space. The number of the fitness evaluations required is usually $O(n)$. Therefore, roughly speaking, the computational cost for the optimization algorithm itself scales as $O(n^3)$ as the dimension increases. See [40] for the scalability analysis of the CMA-ES.
3. Results
3.1 Wave scattering by a bare cylinder
A bare cylinder made of aluminum [Fig. 2(a)] is modeled as a PEC; both $E_z$- and $H_z$-polarized waves are intensively scattered from the cylinder’s surface [Fig. 2(b,c)]. The difference in interference patterns between $E_z$-polarized wave [Fig. 2(b)] and $H_z$-polarized wave [Fig. 2(c)] arise from differences in the PEC boundary conditions in Eqs. (5) and (6). Although acoustic scattering [Fig. 2(d)] is modeled as an acoustic-elastic coupled system with elasticity corresponding to aluminum, the scattering of the $H_z$-polarized wave by the PEC with Neumann boundary conditions corresponds well to that of an acoustic wave, thus supporting this model’s suitability as a hard wall.
Fig. 2. (a) Bare cylinder without a cloak and its scattering of (b) $E_z$-polarized wave, (c) $H_z$-polarized wave, and (d) acoustic wave around the cylinder; (d) mean stress $\Re [\sigma _\mathrm {m}]=\Re [\sigma _{ii}]/2$ in the cylinder. The normalized frequency is set to $\omega _\mathrm {e} R_\mathrm {D}/(2\pi c)=\omega _\mathrm {a} R_\mathrm {D}/(2\pi \sqrt {\kappa _\mathrm {a}/\rho _\mathrm {a}}) =2.5$.
3.2 Topology optimization for unidirectional and multidirectional cloaking
Topology optimization has been demonstrated for minimizing $F$ and optimal level set functions are explored by the CMA-ES under the perimeter constraint. Figure 3 shows the improvements of the fitness [Fig. 3(a)], the values of objective functions [Fig. 3(c)–(e)], and the perimeter [Fig. 3(f)] of the structure that exhibits the smallest $F$ from $\lambda$-candidate solutions in each generation $g$. CMA-ES generates candidate solutions based on the distribution parameters with random numbers in Eq. (14), then as shown in Fig. 3(a), the fitness value oscillates up and down slightly, but decreases as $g$ is updated. The optimization computation is considered to have converged when $\Delta \boldsymbol{m}_\mathrm {error}$ in Fig. 3(b) is less than or equal to 0.01, the calculation is terminated, and the structure that records the smallest $F$ in the process is the optimized structure. By imposing structural symmetries about not only $x$ and $y$ axes but also diagonal axes $y=\pm x$, multidirectional cloaking is also attempted to be design. In Figs. 3(c)–(e), all the value of objective functions reaches less than one (i.e. $\max \big ( \Psi _E, \Psi _H, \Psi _p \big )<1.00$) at generation $g=352$ ($\tau =0.01$), $g=336$ ($\tau =0.001$), $g=256$ ($\tau =0.01$ with $y=\pm x$ symmetries), and $g=241$ ($\tau =0.001$ with $y=\pm x$ symmetries) for the first time. The smaller regularization coefficient, $\tau =0.001$, prioritize the improvement of performances for cloaking over reducing the perimeter [Fig. 3(c)–(e)], and the performances have been improved even with less $g$ compared to the topology optimization with strict perimeter constraints, $\tau =0.01$, while the value of perimeter, implying the simplicity and good manufacturability of configuration, is not aggressively improved [Fig. 3(f)]. Because of the additional structural symmetry about $y=\pm x$, the number of design variables can be reduced by about half, which results in convergence with less $g$ [Fig. 3(b)].
Fig. 3. Histories of (a) the fitness and (b) the convergence error and objective functions of cloaking for (c) the $E_z$-polarized wave, (d) the $H_z$-polarized wave, and (e) the acoustic wave, and (f) the structural perimeter. The optimization process was judged to have converged at $\Delta \boldsymbol{m}_\mathrm {error} < 0.01$ (b).
Figure 4 shows the results of topology optimizations minimizing fitness, Eq. (12). The numerical demonstration obtains an optimal design [Fig. 4(b)] that realizes both polarization-independent [Fig. 4(c)(d)] and acoustic [Fig. 4(e)] cloaking. With the weaker constraint $\tau =0.001$ prioritizing performance improvement [Fig. 4(f)–(j)], the three objective functions fall below $0.0467$ in value [Fig. 4(i)], corresponding to a reduction in scattering to $4.67\%$ of that for a bare obstacle present without a cloak [Fig. 2]. Nearly perfect performances for triple-wave cloaking capabilities are thus obtained numerically. By additionally imposing structural symmetries on $y=\pm x$ [Fig. 5], triple-wave cloaking capabilities are demonstrated for multidirectional wave propagations [Fig. 6(b,d,f)]. Several dips in the three objective functions [Fig. 6(b,d,f)] in the results obtained with $y=\pm x$ symmetry, that indicate the emergence of cloaking are observed not only when $\theta =-\pi, 0$, and $\pi$ but also when $\theta =-\pi /2$ and $\pi /2$. Because the level set functions are discretized on points of square cells as shown in Fig. 1(b), it is difficult to strictly impose higher $s$-fold discrete rotational symmetry ($s>8$) in the structure. It is not impossible to impose additional rotational symmetries through the linear interpolation of the level set function in the square cells, but a structure with a slightly broken symmetry appears. In further increasing the number of incident angles at which cloaking can be achieved, it is practical to impose symmetry about the diagonal axis $y=\pm x$ as shown in Fig. 5 on the structure and then compute the objective functions for several plane waves propagating at different incidence angles, and minimize all the objective functions simultaneously.
Fig. 4. (a,f) Transitions in structural topology exhibiting lowest $F$ at each $g$ during the optimization process, (b,g) optimal configuration of a biphysical cloak and its performance for (c,h) $E_z$-polarized wave, (d,i) $H_z$-polarized wave, and (e,j) an acoustic wave with (a)–(e) $\tau =0.01$ and (f)–(j) $\tau =0.001$. Structural perimeter in the optimal configuration becomes (b) $L = 37.4$ and (g) $L = 47.0$. The normalized frequency is set to $\omega _\mathrm {e} R_\mathrm {D}/(2\pi c)=\omega _\mathrm {a} R_\mathrm {D}/(2\pi \sqrt {\kappa _\mathrm {a}/\rho _\mathrm {a}}) =2.5$. The number of design variables is $n=3463$; the number of samplings is set to $\lambda =400$.
Fig. 5. (a,f) Transitions in structural topology exhibiting lowest $F$ at each $g$ with diagonal symmetries during the optimization process, (b,g) Optimal configuration of a multidirectional biphysical cloaks and their performances for (c,h) $E_z$-polarized wave, (d,i) $H_z$-polarized wave, and (e,j) acoustic wave with (a)–(e) $\tau =0.01$ and (f)–(j) $\tau =0.001$. Structural perimeter in the optimal configuration becomes (b) $L = 37.9$ and (g) $L = 72.2$. The normalized frequency is set to $\omega _\mathrm {e} R_\mathrm {D}/(2\pi c)=\omega _\mathrm {a} R_\mathrm {D}/(2\pi \sqrt {\kappa _\mathrm {a}/\rho _\mathrm {a}}) =2.5$. The number of design variables is $n=1748$; the number of samplings is set to $\lambda =400$.
Fig. 6. Incident-angle dependences of cloaking performances of optimal configurations shown in [(a,c,e)] Fig. 4(b)(g) and [(b,d,f)] Fig. 5(b)(g) with $y=\pm x$ symmetries for (a,b) the $E_z$-polarized wave, (c,d) the $H_z$-polarized wave, and (e,f) the acoustic wave.
Having discussed and demonstrated topology optimization for biphysical cloaks for electromagnetic and acoustic waves of the same wavelength in air, sound waves usually have wavelengths much longer than those of electromagnetic waves. Therefore, we examined whether a biphysical cloak can be designed that works with electromagnetic and sound waves of different wavelengths by the topology optimization. Assuming an incident sound wave of longer wavelength than that of the electromagnetic wave, results of topology optimization and frequency response for the cloaking performance of the optimal structure [Fig. 8] show that biphysical cloaking is achievable at least under the conditions we had investigated.
STL data for the optimal configurations in Fig. 4(b,g), Fig. 5(b,g) and Fig. 8(a)–(d) can be downloaded from Dataset 1, Ref. [42]. See Appendix B for details.
4. Discussion
We described topology-optimized biphysical cloaks that perform scattering cancelations for $E_z$- and $H_z$-polarized waves and acoustic wave. The frequency responses of the cloaking performances [Fig. 7] confirm the simultaneous cloaking of the three waves having the same wavelength in air despite the waves having different boundary conditions. We note that the normalized frequency is equal to the inverse of the normalized wavelength, e.g., $\omega _\mathrm {e} R_\mathrm {D}/(2\pi c)=R_\mathrm {D}/\lambda _\mathrm {e}$, where $\lambda _\mathrm {e}$ is the wavelength of the electromagnetic wave in air.
Fig. 7. Frequency dependences of cloaking performances of optimal configurations shown in [(a,c,e)] Fig. 4(b)(g) and [(b,d,f)] Fig. 5(b)(g) with $y=\pm x$ symmetries for (a,b) the $E_z$-polarized wave, (c,d) the $H_z$-polarized wave, and (e,f) the acoustic wave.
This optimization suppresses the scattering of the electromagnetic and acoustic waves independently; no physical interaction between the two waves, i.e., photon-phonon interaction, occurs [43,44]. How does the elastic vibration arising from the acoustic wave influence the cloaking performance for electromagnetic waves? We estimate the Doppler shift of the electromagnetic wave impinging on an optimized structure. The normalized shift in frequency is
5. Conclusions
We showed numerically that both electromagnetic and acoustic propagating waves can be manipulated to offer biphysical cloaking and a triple-wave cloaking capability exhibiting near-perfect cloaking performance of an isotropic and homogeneous material. Concerning the optimal configurations obtained, the scattering of both electromagnetic waves and acoustic waves were suppressed and biphysical cloaking for both wave types was achieved after topology optimization. Additionally, the presented method addressed to incorporate some physical attributes in the structural optimization such as polarization for electromagnetic waves and material elasticity for sound waves. The effect of the interaction between electromagnetic and acoustic waves on the cloaking performance is limited and negligible. The performances of our optimal biphysical cloaks as well as the electromagnetic cloaks [45,46] are expected to be demonstrated in future experiments. Our method may be extended to design other biphysical devices by replacing an objective function with an appropriate function that evaluates device performance.
Appendix A
Acoustic performance of the topology-optimized electromagnetic cloak
Figures 7(e,f) and 8(i,j) show that the objective function for acoustic waves at frequencies not taken into account in the optimization is higher than that for electromagnetic waves. It is interesting to see what happens if topology optimizations are demonstrated for infimizing only the objective functions in electromagnetic cloaking and to see how the resulting device will perform for acoustic waves. Additionally, we can compare the results obtained with and without considering acoustic waves in our topology optimizations and make sure that our optimization approach does not ignore that term.
Fig. 8. Optimal configurations for different acoustic wave frequencies: $\omega _\mathrm {a} R_\mathrm {D}/(2\pi \sqrt {\kappa _\mathrm {a}/\rho _\mathrm {a}})$ equal to (a) $0.5$, (b) $1.0$, (c) $1.5$, and (d) $2.0$, for fixed frequency of the electromagnetic wave, $\omega _\mathrm {e} R_\mathrm {D}/(2\pi c)=2.5$ and $\tau =0.01$. (a)–(d) The number of design variables is $n=1748$; the number of samplings is set to $\lambda =400$. Frequency responses of the cloaking performances for (e,h) $E_z$-polarized electromagnetic wave, (f,i) $H_z$-polarized electromagnetic wave and (g,j) acoustic cloaks.
We then investigate how the topology-optimized electromagnetic cloak, designed without considering the objective function for acoustic cloaking, performs for acoustic waves. A topology optimization has been demonstrated for a polarization-independent cloak by infimizing the fitness without the objective function for acoustic cloaking:
We also investigate the cloaking performance of the acoustic cloak for electromagnetic waves presented in Ref. [29], for which the fitness is infimized as
Figures 9(a)–(d) show the optimization results of infimizing $F_{EH}$ without incorporating $\Psi _p$ whereas Figs. 9(e)–(h) show those of infimizing $F_{p}$ without incorporating $\Psi _E$ and $\Psi _H$ obtained from Ref. [29]. These results show that the unconsidered objective function performs poorly as shown in Fig. 9(d) and Fig. 9(f,g), and we confirm that the our optimization approach does not ignore every single objective function incorporated in the fitness $F$ in Eq. (12).
Fig. 9. (a) Polarization-independent electromagnetic cloak obtained by infimizing $F_{EH}$ and its cloaking performance for (b) the $E_z$-polarized wave, (c) the $H_z$-polarized wave, and (d) the acoustic wave. (e) Topology-optimized acoustic cloak obtained by infimizing $F_{p}$ in Ref. [29] and its cloaking performance for (f) the $E_z$-polarized wave, (g) the $H_z$-polarized wave, and (h) the acoustic wave. The regularization coefficient is set at $\tau =0.001$ and diagonal structural symmetries are imposed. (a)–(d) The number of design variables is $n=1748$; the number of samplings is set to $\lambda =400$. (e)–(h) The number of design variables is $n=2009$; the number of samplings is set to $\lambda =160$ in Ref. [29]. (e,h) Reprinted from [29] with the permission of AIP Publishing.
Appendix B
STL data of the presented optimal configurations
We provide stereolithography (STL) data of the optimal configurations shown in Fig. 4(b,g), Fig. 5(b,g), and Fig. 8(a)–(d) for experimental demonstrations of the biphysical cloaks designed through topology optimization as we show in Dataset 1, Ref. [42]. Optimal structures made from ABS resin are provided as shown in Fig. 10. Each STL file name corresponds to the figure number of optimal configurations in the manuscript; e.g., STL of the optimal configuration in Fig. 4(b) is denoted “Fig 4b.stl".
Fig. 10. (a) Optimal configuration of electromagnetic-acoustic biphysical cloak in Fig. 5(b) in the manuscript. (b) Corresponding STL file named “Fig5b.stl”.
In the optimizations in Figs. 4 and 5, we assume that the wavelengths of the electromagnetic and acoustic waves are $\lambda _\mathrm {e}=125$mm and $\lambda _\mathrm {a}=125$mm respectively and that the frequencies of electromagnetic and acoustic waves to be approximately $f_\mathrm {e}=2.40$GHz and $f_\mathrm {a}=2.72$kHz respectively. The radius of the fixed design domain becomes $R_\mathrm {D}=312.5$mm. The height in the $z$ direction is set at $20$ mm. We keep in mind that when the aluminum cylinder is in contact with the ABS, the two structures are simulated as adhering to each other. Operating wavelengths and frequencies are listed in Table 2.
Table 2. Optimum Structures, STL Data, Operating Wavelengths, and Frequency.
Roughly speaking, the smallest feature becomes the discretized cell size $L_\mathrm {grid}$ estimated from the wavelength, $\lambda _\mathrm {e}$, and radius of the fixed design domain, $R_\mathrm {D}$. The ratio between $\lambda _\mathrm {e}$ and $R_\mathrm {D}$ is obtained from the normalized frequency as $\omega _\mathrm {e} R_\mathrm {D}/(2\pi c)= R_\mathrm {D}/\lambda _\mathrm {e}=2.5$. The cell size is then
The smallest size of a structural feature is estimated as being approximately 4.46 mm. Meanwhile, by regarding $L_\mathrm {grid}$ as the fabrication tolerance of the patterned ABS, the wavelength is inversely calculated as
If we assume that the manufacturing tolerance of recent 3D printers is $10\mu$m–$25\mu$m, the wavelength $\lambda _\mathrm {e}$ becomes $280\mu$m–$700\mu$m. We note that this minimal structural feature can be improved by increasing the regularization coefficient $\tau$, as described in this paper. If the minimum structural features are desired to be sufficiently larger than $L_\mathrm {grid}$, it is recommended to set the coefficient $\tau$ larger.
Acknowledgments
We thank Richard Haase, Ph.D., from Edanz (https://jp.edanz.com/ac) for editing a draft of this manuscript.
Disclosures
The authors declare that there are no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request. STL data for the optimal configurations in Fig. 4(b,g), Fig. 5(b,g), and Fig. 8(a)–(d) can be downloaded from Dataset 1, Ref. [42] for supporting content.
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