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Band structures in Sierpinski triangle fractal porous phononic crystals
Physica B 498 (2016) 33–42
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Physica B journal homepage: www.elsevier.com/locate/physb
Band structures in Sierpinski triangle fractal porous phononic crystals Kai Wang, Ying Liu n, Tianshu Liang Department of Mechanics, School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 8 December 2015 Received in revised form 31 May 2016 Accepted 19 June 2016 Available online 23 June 2016
In this paper, the band structures in Sierpinski triangle fractal porous phononic crystals (FPPCs) are studied with the aim to clarify the effect of fractal hierarchy on the band structures. Firstly, one kind of FPPCs based on Sierpinski triangle routine is proposed. Then the influence of the porosity on the elastic wave dispersion in Sierpinski triangle FPPCs is investigated. The sensitivity of the band structures to the fractal hierarchy is discussed in detail. The results show that the increase of the hierarchy increases the sensitivity of ABG (Absolute band gap) central frequency to the porosity. But further increase of the fractal hierarchy weakens this sensitivity. On the same hierarchy, wider ABGs could be opened in Sierpinski equilateral triangle FPPC; whilst, a lower ABG could be opened at lower porosity in Sierpinski right-angled isosceles FPPCs. These results will provide a meaningful guidance in tuning band structures in porous phononic crystals by fractal design. & 2016 Elsevier B.V. All rights reserved.
Keywords: A. Sierpinski triangle fractal B. Porous phononic crystal C. Band structure D. FEM
1. Introduction Since the proposing of the concept [1], phononic crystals (PCs) have become a prominent research topic due to their unique physical properties and many promising applications [1–7]. One of the distinctive characteristics of PCs is the existence of absolute band gaps (ABGs) that the elastic/acoustic wave at a certain frequency range could not propagate, which leads to a variety of important applications, such as wave guides, sound filters, etc. [8– 12]. Lately, Liu et al. [13] proposed the concept of ‘porous phononic crystals’ (PPCs), and since then more and more attention [14–21] has been attracted due to the advantages of the lower density and potential ABGs tuning through the pore topology design. Up to now, a lot of efforts have been paid to explore the band structures in different PPCs with the aim to find an optimized strategy in pore topology design for optimized ABGs. Liu et al. [22] discussed the influence of the lattice structures and pore shapes, as well as the lattice transformation, on the band structures in PPCs. Wang et al. [23] proposed a grading strategy in pore design and studied the influence of grading on the band structures. Liu et al. [24] analyzed the wave localization in Kagome honeycombs, which is constituted by hexagonal component and triangular cells. For the three-dimensional lattice, Liu et al. [25] studied acoustic wave propagation in a novel three-dimensional porous phononic crystal-Kagome lattice. Yang et al. [26] investigated the ABGs in a nano-scale three-dimensional (3D) Silicon PC with spherical pores. n
Corresponding author. E-mail address: [email protected] (Y. Liu).
http://dx.doi.org/10.1016/j.physb.2016.06.018 0921-4526/& 2016 Elsevier B.V. All rights reserved.
These works shed some lights on the possibility of the band tuning through the pore topology design. However, more efforts have to be paid to find more optimized pore design strategy. As an optimized structure of natural evolution, fractal structures, which have self-similarity topology, wisely exist in the nature, such as mountains, coastlines, trees, and seashells. It has been a hot topic to use fractals to describe or model the universe since Mandelbrot put forward the mathematical concept [27]. The fractal structures have never-ending patterns, which are created by repeating a simple process in an ongoing feedback loop, and treated as a very clever solution to the problems of the nature. There has been ongoing interest in developing strategies based on fractal structures to obtain optimized ABGs. Kuo and Piazza [28] obtained ultra high frequency bandgaps by the design of fractal phononic crystals in aluminum nitride. Yan et al. [29] proposed layered phononic crystals with different fractal super-lattices and studied elastic wave localization in them. Liu et al. [30] presented a T-square fractal two-dimensional phononic crystal and studied the local resonant and Bragg scattering in it. Gao and Wu et al. [31,32] investigated the band gaps of two-dimensional (2D) PCs which is constituted by self-similarity shape cells. Although these efforts can broaden the ABGs to a certain degree, the relationship between geometrical parameter of fractal structure and ABGs has still not been clarified. Serving as an example of a more general route, the fractal structures based on Sierpinski triangle have attracted growing attention in many fields [33–35]. In the present discussion, one kind of fractal PPCs (FPPCs) based on Sierpinski triangle routine is firstly proposed. Through controlling the pore size, the variation of
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K. Wang et al. / Physica B 498 (2016) 33–42
ABGs in Sierpinski triangle FPPCs is investigated. The influence of the fractal hierarchy on the formation of the ABGs is clarified with the aim to provide some guidance on the optimizing design of FPPCs.
2. Theory A fractal is a mathematical set or a natural phenomenon that exhibits a repeating pattern at every level. If the replication is exactly the same at each level, it is called a self-similar pattern. The way in which they scale makes fractals be different from each other. So fractal dimension is an important indicator in describing the fractal patterns with various degrees of self-similarity. According to Hausdorff-Becikovich [34], we have
D = ln N (ε)/ln (1/ε) ,
(1)
where D is the dimension of the structure, ε is the size of the basic unit, N(ε) is the number of units we obtained based on ε. In the present study, we introduce Sierpinski triangle, a typical fractal pattern, following which the porous phononic crystals are established. Let us start with a triangular plate, as shown in Fig. 1, n ¼0, where n is the fractal hierarchy. Connecting the midpoint of each side of the triangle, we can get a new small triangle in the graphic center, and three entity triangles. At this time, we have the first order fractal, n ¼ 1. Repeat this process for each entity triangle, we can get Sierpinski triangle structure with n ¼ 2 and n ¼3, respectively, as shown in Fig. 1. It is seen that the triangle side is reduced by half. The new triangle can be seen as a repeat of the unit in next level of fractal. If we measure the structure by the triangle with the side length 1/2n, the amount of the units is 3n. Then the Hausdorff-Becikovich dimension of the Sierpinski triangle fractal is n
D = ln 3n /ln( 1/2) = 1.585.
(2)
Fig. 2 is a diagrammatic sketch of the FPPCs following Sierpinski triangle strategy. Here equilateral and right angled isosceles triangle structures are considered, which are shown in Fig. 2a and b, respectively. In the figures, the white part represents the pores. The pores are treated as vacuum. Porosity is the ratio between the pore area and the surface, which is an important factor in the description of PPCs. The porosities of these two types of Sierpinski
Fig. 2. Diagrammatic sketch of 2D fractal PPCs. (a) Sierpinski equilateral triangle structure; (b) Sierpinski right angled isosceles triangle structure. The red line marks out the unit cell used in the calculation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 1. Sierpinski triangle fractal up to three hierarchies. (a) Sierpinski equilateral triangle structure; (b) Sierpinski right angled isosceles triangle structure.
K. Wang et al. / Physica B 498 (2016) 33–42
where r¼ (x, y) denotes the position vector, ρ is the mass density, ω is the angular frequency, μ and λ are the shear modulus and Lamé constant. u¼ (ux, uy) is the displacement vector in the transverse plane, and is the 2D vector differential operator. Based on the Bloch theorem, for the periodic structure, the displacement field can be expressed as
triangle PPCs are calculated as
f = A pore /A unit ⎛ 3 3 2 3 2 3 2⎞ =⎜ b2 + ic + 3 jd + 9 ke ⎟ 4 4 4 4 ⎝ ⎠ =
b2 ⎛ 1 1 3 9 ⎞ ⎜ + i+ j+ k⎟. 8 32 128 ⎠ a2 ⎝ 2
⎛ 3 ⎞ ⎜ a2⎟ ⎝ 2 ⎠ (3)
In this manuscript, we have nmax ¼3. According to Eq. (3), when n ¼0, i¼j¼ k¼ 0; when n ¼1, i ¼1, j¼ 0, k ¼0; when n ¼2, i ¼j¼1, k ¼0. As shown in Fig. 2, the z-coordinate is set perpendicular to the vacuum holes. The elastic waves can be decoupled into the mixed anti-plane shear mode and the in-plane mode since they propagate in the transverse plane (x0y plane), which means that the displacement vectors are independent of the z-coordinate. Accordingly, the in-plane frequency-domain wave equations are expressed as
⎛ ∂ ⎞ −ρ( r)ω2ui = ∇⋅ μ( r)∇ui + ∇⋅⎜ μ( r) u⎟ ∂xi ⎠ ⎝ ∂ + λ( r)∇⋅u , (i = x, y) ∂ xi
(
(
)
35
)
(4)
u( r) = ei( k ⋅ r)uk ( r) ,
(5)
where uk(r) is a periodical vector function with the same periodicity as the crystal lattice and k ¼(kx, ky) is the wave vector within the first Brillouin zone of the reciprocal lattice. COMSOL Multiphysics is utilized to solve the governing equations under the 2D plane strain application mode (ACPN). In the unit cell, the eigenvalue equations in the discrete form can be written as
( K − ω M)U = 0 , 2
(6)
where K and M are the stiffness and mass matrices of the unit cell and U is the displacement at the nodes. On the surface of the pore, the free boundary condition is applied. The Bloch theorem should be applied on the two opposite boundaries of the unit cell, yielding
Fig. 3. Variation of the band structures of Sierpinski equilateral triangle FPPCs with respect to the porosity in the first fractal level. (a) f¼ 0.2; (b) f¼ 0.4; (c) f¼ 0.5; (d) f ¼0.6.
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K. Wang et al. / Physica B 498 (2016) 33–42
U( r + a) = ei( k ⋅ a)U( r) ,
(7)
where a is the lattice basis vector associated with the PPCs and r is located at the boundary nodes. The unit cell is meshed by using the triangular Lagrange quadratic elements provided by COMSOL. The direct SPOOLES is chosen as the linear system solver and the eigen-frequency analysis is selected as the solver mode. In addition, parameter settings in symmetry direction in the advanced solver and the Hermitian transpose of constraint matrix are required. The model built in CAD is imported into COMSOL and it is saved as a MATLAB-compatible ‘.m’ file. This ‘.m’ file is programmed to let the wave vector k sweep the edges of the first irreducible Brillouin zone to obtain the whole dispersion relations. Since the present results have compared with the ones given in Ref. [13], and well coincidence has been obtained, we do not give the detailed convergence analysis of the calculation here anymore.
3. Numerical examples and discussion Seen as Fig. 2, the geometrical parameters of the Sierpinski triangle FPPCs could be changed under certain fractal hierarchy or porosity. In this section, based on the structure given in Fig. 2, we will discuss the effects of the porosity and fractal hierarchy on the band structures in FPPCs, respectively. In the calculation, the
matrix material is Aluminum, and the pore is vacuum. The elastic parameters of the matrix are taken as (in units of GPa): λ ¼68.3, μ ¼28.3, and ρ ¼2700 kg/m3. The lattice constant a ¼0.02 m. As shown in Fig. 2, fixing the central positions, we change the pore areas simultaneously to obtain different porosities by considering the geometrical limitation of the lattice. Fig. 3 shows the band structures in Sierpinski equilateral triangle FPPCs with different porosities in the first fractal level, that is, n ¼1. In the figures, the orange rectangle marks out the possible ABGs. The dimensionless frequency ω a /2πVt is used in the y axis, with Vt ¼3110 m/s, the velocity of the elastic transverse wave in Al matrix. Seen as Fig. 3a, when f¼ 0.2, one ABG is opened between the 10th and the 11st bands (Fig. 3a). The central frequency of the ABG is 1.160 with the bandwidth 1.183 1.136 ¼0.047. Along with the increase of the porosity, the central frequency of the ABG is lowered down and more ABGs are opened. As shown in Fig. 3b, when f ¼0.4, four more ABGs are opened at 4th–5th, 12nd–13rd, 18th–19th and 20th–21th bands, respectively. The 18th–19th band has the widest bandwidth, that is, 1.52 1.31 ¼0.21. Along with the further increase of the porosity, the central frequencies of the ABGs are decreased with wider gap width. When f ¼0.6, it is noticed that the central frequency of the 18th–19th ABG is dropped to 1.226 with the largest bandwidth 1.660–0.793 ¼0.867. In the same way, the ABGs in the Sierpinski right-angled isosceles triangle FPPCs are also investigated and shown in Fig. 4.
Fig. 4. Variation of the band structures of Sierpinski right angled isosceles triangle FPPCs with respect to the porosity. (a) f ¼0.25; (b) f ¼0.3; (c) f¼ 0.35; (d) f ¼0.4.
K. Wang et al. / Physica B 498 (2016) 33–42
Seen as Fig. 4a, when f¼ 0.25, three ABGs are opened at the 2nd– 3rd, 4th–5th, and 5th–6th bands. The central frequency of the first ABG is 0.47 with the gap width 0.478 0.462 ¼0.016. The bandwidth of the 4th–5th band is 0.031 with the central frequency 0.762. The central frequency of the 5th–6th band is 0.923 with the bandwidth 0.936 0.908 ¼0.028. Along with the increase of the porosity, the central frequency of the ABG is lowered down and the 5th–6th bandwidth is increased. When f¼ 0.35 (Fig. 4c), a new ABG is opened at 8th–9th band and the 4th–5th band is closed. The central frequency of the 8th–9th band is 1.216 with the bandwidth 1.252 1.179 ¼0.073. Along with the further increase of the porosity, see f ¼0.4 (Fig. 4d), the 5th–6th band is also closed.
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The variation of ABGs in FPPCs with different hierarchies is also discussed. The detailed variation of the ABGs with respect to the porosity is plotted in Figs. 6 and 7, respectively. In order to verify the accuracy of the numerical calculation, the transmission spectra through the Sierpinski triangle FPPCs composed by a finite array of 8 1 units are calculated and showed in Fig. 5. A line wave source is located at the leftmost boundary of the structure and the response at the rightmost boundary is computed. For the in-plane mode, only the translational (displacement) wave sources are considered. The red lines represent the x-polarized translational (displacement) wave sources with unit amplitude meanwhile the black lines represent the y-polarized direction.
Fig. 5. (a) The band structure and (b) transmission spectra of the Sierpinski equilateral triangle structure; (c) the band structure and (d) transmission spectra of Sierpinski right angled isosceles triangle structure. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
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Fig. 6. Eigenmode shapes and displacement vector fields of Sierpinski equilateral triangle FPPC structure at (a) the upper band edge at n ¼1, f¼ 0.4; (b) the lower band edge at n¼1, f¼ 0.4; (c) the upper band at n ¼1, f¼ 0.2; and (d) the lower band at n¼ 1, f ¼0.2.
K. Wang et al. / Physica B 498 (2016) 33–42
The band structure and the transmission spectra along ΓX direction for the Sierpinski equilateral triangle lattice at n ¼1, f ¼0.6 are given in Fig. 5a and b. A very obvious attenuation appears between 0.793 and 1.660. Correspondingly, the band structure and the transmission spectra along ΓX direction for the Sierpinski right angled isosceles triangle structure at n ¼1, and f ¼0.3 are also given in Fig. 5c and d. It can be noticed that attenuations are conspicuously nearby 1.35 and 0.55 which gets a good match with the band structure. The results indicate the good agreement in the directional band structure and the transmission spectra. Fig. 6 gives the vibration eigen-modes of unit cell at the upper and lower edges of the 12nd–13rd ABG when n ¼1, f ¼0.4 (Fig. 3 Au and Ab). Color contour represents the magnitude of displacement. It can be found that for the upper-edge mode (Fig. 6a), a slight torsion occurred in the central of the matrix. The unit cell vibrates centrally symmetrically to the three red dash dot lines as plotted; whilst for the lower-edge mode (Fig. 6b), there is a slightly longitudinal deformation in the central of the matrix and the unit cell vibrates symmetrically to the vertical dash dot line. In order to understand the mechanism of the bandgap generation, the vibration eigenmodes of unit cell at n ¼1, f ¼0.2 with the same wave vector (Fig. 3, marked out by Bu and Bb) are given comparatively in Fig. 6c and d. At this porosity, the 12nd–13rd ABG is not opened. It is seen that although the unit cell makes the same symmetrically
39
vibration at the 12nd band as that when f¼ 0.4, its vibration mode at 13rd band is not symmetry. That is, the unit cell makes the harmonic resonance at the adjacent bands simultaneously, the complete bandgaps will be opened. Fig. 7 shows the variation of the ABGs with respect to the porosity at different hierarchies in Sierpinski equilateral triangle FPPCs. Seen as Fig. 7a, for a normal one with n ¼0 (Fig. 7a and Fig. 8a), an ABG is firstly opened at the 6th–7th band at f ¼0.18. Along with the increase of the porosity, two more ABGs are opened at 10th–11th and 4th–5th bands. When f¼ 0.34, a higher order ABG is opened at the 14th–15th band. Along with the increase of the porosity, the bandwidth is gradually increased, especially for the 6th–7th band. The bandwidth is increased from 0.016 at f ¼0.2 to 0.4128 at f ¼0.45. Conversely, the central frequencies of the ABGs are dropped. When n ¼1 (Fig. 7b), an ABG is firstly opened at the 10th–11st band at f ¼0.2. Along with the increase of the porosity, two ABGs are opened at 18th–19th and 20th–21st bands. When f¼ 0.4, except for an ABG at the 12nd–13rd band, a lower band ABG is also opened at the 4th–5th band with lower central frequency. It is seen that the increase of the porosity leads the wideness of the bandwidth, especially for the 18th–19th band. The bandwidth is increased from 0.025 at f ¼0.3 to 0.867 at f ¼0.6. It is noticed that the central frequencies of the ABGs are dropped sharply at higher
Fig. 7. Variation of the ABGs with respect to the porosity in the Sierpinski equilateral triangle FPPC when (a) n ¼0; (b) n¼1; (c) n ¼2; and (d) n¼ 3.
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K. Wang et al. / Physica B 498 (2016) 33–42
porosity, except for the higher order ABG, see, the 20th–21st band. Its central frequency is a little increased along with the increase of the porosity. Comparison between Fig. 7a and b shows that the critical porosity for the opening of the first ABG is increased, see, from 0.18 at n ¼0 to 0.2 at n¼ 1. Fractal rises up the bands of ABGs and the central frequency is more sensitive to the variation of the porosity. When n ¼ 2 (Figs. 1a and 7c), an ABG is firstly opened at 10th– 11st bands at f ¼0.28, which is larger than f ¼0.2 when n ¼ 1. The higher band, see, the 20th–21st ABG, is not opened again. Along with the increase of the porosity, three ABGs are opened at 18th– 19th, 12nd–13rd, and 4th–5th bands at the porosities f ¼0.32, f ¼0.43, and f ¼0.45, which are larger than the corresponding values at n ¼1, see, f ¼0.3 and 0.4, respectively. Along with the further increase of the fractal hierarchy, see, n ¼3, the critical porosities for the opening of ABGs at different bands are all postponed, especially for the 12nd–13rd band, extended to f ¼0.53. The bandwidth of higher ABGs tends to be narrower, and the central frequencies are decreased. Comparing Fig. 7c and d to Fig. 7b indicates that lower fractal hierarchy is easier to form wider ABG. When n ¼1, the bandwidth of the 18th– 19th ABG is 0.867 at f ¼0.6, which is far wider than the one when n ¼2 at the same porosity, see 0.089, but when n Z2, further increase of the fractal hierarchy has limited effects on the band structures in FPPCs.
Correspondingly, Fig. 8 shows the variation of the ABGs with respect to the porosity in different hierarchies in Sierpinski rightangled isosceles triangle FPPCs. It is seen that when n ¼0 (Fig. 1b and Fig. 8a), a 3rd–4th ABG is firstly opened at f ¼0.15. Along with the increase of the porosity, more ABGs are opened at 3rd–4th, 4th–5th, 7th–8th, and 8th–9th bands. The central frequencies of the ABGs are decreased with wider band widths except for the 4th–5th band. Its central frequency is a little increased along with the increase of the porosity from f ¼0.3 to f¼ 0.34. Comparison between Figs. 7a and 8a indicates that wider ABGs could be opened in Sierpinski equilateral triangle FPPC whilst Sierpinski right-angled isosceles FPPC is easier to form a band gap at lower porosity and lower band. When n¼ 1, (Figs. 1b and 8b), more ABGs are opened at lower order bands. It is seen that the 2nd–3rd band is opened at f¼ 0.23. Along with the increase of the porosity, the central frequencies are decreased with wider bandwidth. It is noticed that the 5th–6th and 4th–5th bands are closed at f ¼0.4 and f ¼0.35, respectively. Comparison between Fig. 8a and b indicates that the central frequencies of the ABGs are decreased. When n¼ 2 (Fig. 1b and 8c), an ABG is firstly opened at the 4th– 5th band at f ¼0.25, which is larger than f ¼0.22 when n ¼1. Along with the increase of the porosity, no ABGs are opened. The 2nd– 3rd, 5th–6th and 8th–9th ABGs are opened at f¼ 0.3, f ¼0.35, and f¼ 0.34, respectively, which are larger than the corresponding
Fig. 8. Variation of the ABGs with respect to the porosity in the Sierpinski right-angled isosceles triangle FPPCs when (a) n ¼0; (b) n¼1; (c) n ¼2; and (d) n¼ 3.
K. Wang et al. / Physica B 498 (2016) 33–42
values at n ¼ 1, see, f¼ 0.23, f ¼0.23 and f ¼0.30 respectively. When n ¼3 (Figs. 1d, and 8d), the 5th–6th band is opened at f ¼0.42 and keeps open, whilst the 4th–5th band is still closed at f ¼0.45. Moreover, it is noticed that when n ¼3, along with the increase of the porosity, the bandwidth is decreased. Comparison among Fig. 8a–d indicates that in Sierpinski right-angled isosceles triangle FPPCs, fractal hierarchy will not affect the structures of ABGs. But lower fractal hierarchical is easier to form wider ABGs. Comparison Figs. 8–7 shows that it is easier to form low band ABGs in Sierpinski right-angled isosceles triangle FPPCs. In order to see clearly this variation due to the fractal hierarchy, Fig. 9 displays the variation of the band edges of the lowest band (see, 4th–5th band for Sierpinski equilateral triangle FPPC, and 2nd–3rd band for Sierpinski right-angled isosceles triangle FPPC) with respect to the fractal hierarchy at different porosities. Seen as Fig. 9, the bandwidth of ABGs in different fractal hierarchy and porosities are marked out by using rectangles with different colors. When n ¼0, that is, when the fractal program has not been implemented, the ABGs are with wider bands and higher central frequencies. Along with the increase of the porosity, the central frequencies of ABGs are decreased. No matter Sierpinski right-angled isosceles triangle FPPCs or Sierpinski equilateral triangle FPPCs, the first order fractal (n ¼1) corresponds to lower
a
0.60 f =0.3
0.55
f =0.4
f =0.5
2nd-3rd Band
ωa/2πVt
0.50 0.45 0.40 0.35
41
central frequencies and wider bandwidth. When the fractal hierarchy is increased, see, n ¼2 or n¼ 3, the central frequencies are increased. But for Sierpinski right-angled isosceles triangle FPPCs, the effect of the fractal hierarchy is not so distinct. And along with the increase of the porosity, the sensitivity to the fractal hierarchy is also weakened.
4. Summary In this paper, the band gap structures in two-dimensional Sierpinski triangle fractal porous phononic crystals are investigated by using finite element (FE) simulation. Summarizing the results above we can conclude that: (1) Compared to the normal structure (n ¼0), Sierpinski triangle fractal increases the critical porosity for the opening of the ABG; whilst the central frequencies of the ABGs are decreased; (2) Along with the increase of the porosity, the central frequencies of the ABGs in different hierarchies display a unified downward trend. Increase of the hierarchy increases the sensitivity of the central frequency to the porosity; (3) Along with the increase of the hierarchy, the bands of the ABGs are not changed, but the central frequencies are increased. When n Z2, further increase of the fractal hierarchy has limited effects on the band structures in Sierpinski triangle FPPCs; (4) On the same hierarchy, wider ABGs could be opened in Sierpinski equilateral triangle FPPC; whilst in Sierpinski rightangled isosceles FPPC, it is easier to form a band gap at lower porosity and lower band. (5) The results above to some extend disclose the intrinsic relation of the ABGs with the fractal hierarchy and topological parameters in Sierpinski triangle FPPCs. It would provide useful guidance in tuning of the band structures in fractal porous phononic crystals.
0.30 Acknowledgments
0.25 Sierpinski right-angled isosceles triangle FPPC
0.20 0
1
2
3
n
The second author thanks the support from the Fundamental Research Funds for the Central Universities of China (No. 2014JBZ014). Supports from the National Natural Science Foundation of China, China (11272046), and National Basic Research Program of China (973Program) (2015CB057800), are also acknowledged.
References
Fig. 9. Variation of the upper and lower edges of the first ABGs with respect to the fractal hierarchy. (a) Sierpinski right angled isosceles triangle FPPCs; (b) Sierpinski equilateral triangle FPPCs.
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Physica B journal homepage: www.elsevier.com/locate/physb
Band structures in Sierpinski triangle fractal porous phononic crystals Kai Wang, Ying Liu n, Tianshu Liang Department of Mechanics, School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 8 December 2015 Received in revised form 31 May 2016 Accepted 19 June 2016 Available online 23 June 2016
In this paper, the band structures in Sierpinski triangle fractal porous phononic crystals (FPPCs) are studied with the aim to clarify the effect of fractal hierarchy on the band structures. Firstly, one kind of FPPCs based on Sierpinski triangle routine is proposed. Then the influence of the porosity on the elastic wave dispersion in Sierpinski triangle FPPCs is investigated. The sensitivity of the band structures to the fractal hierarchy is discussed in detail. The results show that the increase of the hierarchy increases the sensitivity of ABG (Absolute band gap) central frequency to the porosity. But further increase of the fractal hierarchy weakens this sensitivity. On the same hierarchy, wider ABGs could be opened in Sierpinski equilateral triangle FPPC; whilst, a lower ABG could be opened at lower porosity in Sierpinski right-angled isosceles FPPCs. These results will provide a meaningful guidance in tuning band structures in porous phononic crystals by fractal design. & 2016 Elsevier B.V. All rights reserved.
Keywords: A. Sierpinski triangle fractal B. Porous phononic crystal C. Band structure D. FEM
1. Introduction Since the proposing of the concept [1], phononic crystals (PCs) have become a prominent research topic due to their unique physical properties and many promising applications [1–7]. One of the distinctive characteristics of PCs is the existence of absolute band gaps (ABGs) that the elastic/acoustic wave at a certain frequency range could not propagate, which leads to a variety of important applications, such as wave guides, sound filters, etc. [8– 12]. Lately, Liu et al. [13] proposed the concept of ‘porous phononic crystals’ (PPCs), and since then more and more attention [14–21] has been attracted due to the advantages of the lower density and potential ABGs tuning through the pore topology design. Up to now, a lot of efforts have been paid to explore the band structures in different PPCs with the aim to find an optimized strategy in pore topology design for optimized ABGs. Liu et al. [22] discussed the influence of the lattice structures and pore shapes, as well as the lattice transformation, on the band structures in PPCs. Wang et al. [23] proposed a grading strategy in pore design and studied the influence of grading on the band structures. Liu et al. [24] analyzed the wave localization in Kagome honeycombs, which is constituted by hexagonal component and triangular cells. For the three-dimensional lattice, Liu et al. [25] studied acoustic wave propagation in a novel three-dimensional porous phononic crystal-Kagome lattice. Yang et al. [26] investigated the ABGs in a nano-scale three-dimensional (3D) Silicon PC with spherical pores. n
Corresponding author. E-mail address: [email protected] (Y. Liu).
http://dx.doi.org/10.1016/j.physb.2016.06.018 0921-4526/& 2016 Elsevier B.V. All rights reserved.
These works shed some lights on the possibility of the band tuning through the pore topology design. However, more efforts have to be paid to find more optimized pore design strategy. As an optimized structure of natural evolution, fractal structures, which have self-similarity topology, wisely exist in the nature, such as mountains, coastlines, trees, and seashells. It has been a hot topic to use fractals to describe or model the universe since Mandelbrot put forward the mathematical concept [27]. The fractal structures have never-ending patterns, which are created by repeating a simple process in an ongoing feedback loop, and treated as a very clever solution to the problems of the nature. There has been ongoing interest in developing strategies based on fractal structures to obtain optimized ABGs. Kuo and Piazza [28] obtained ultra high frequency bandgaps by the design of fractal phononic crystals in aluminum nitride. Yan et al. [29] proposed layered phononic crystals with different fractal super-lattices and studied elastic wave localization in them. Liu et al. [30] presented a T-square fractal two-dimensional phononic crystal and studied the local resonant and Bragg scattering in it. Gao and Wu et al. [31,32] investigated the band gaps of two-dimensional (2D) PCs which is constituted by self-similarity shape cells. Although these efforts can broaden the ABGs to a certain degree, the relationship between geometrical parameter of fractal structure and ABGs has still not been clarified. Serving as an example of a more general route, the fractal structures based on Sierpinski triangle have attracted growing attention in many fields [33–35]. In the present discussion, one kind of fractal PPCs (FPPCs) based on Sierpinski triangle routine is firstly proposed. Through controlling the pore size, the variation of
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K. Wang et al. / Physica B 498 (2016) 33–42
ABGs in Sierpinski triangle FPPCs is investigated. The influence of the fractal hierarchy on the formation of the ABGs is clarified with the aim to provide some guidance on the optimizing design of FPPCs.
2. Theory A fractal is a mathematical set or a natural phenomenon that exhibits a repeating pattern at every level. If the replication is exactly the same at each level, it is called a self-similar pattern. The way in which they scale makes fractals be different from each other. So fractal dimension is an important indicator in describing the fractal patterns with various degrees of self-similarity. According to Hausdorff-Becikovich [34], we have
D = ln N (ε)/ln (1/ε) ,
(1)
where D is the dimension of the structure, ε is the size of the basic unit, N(ε) is the number of units we obtained based on ε. In the present study, we introduce Sierpinski triangle, a typical fractal pattern, following which the porous phononic crystals are established. Let us start with a triangular plate, as shown in Fig. 1, n ¼0, where n is the fractal hierarchy. Connecting the midpoint of each side of the triangle, we can get a new small triangle in the graphic center, and three entity triangles. At this time, we have the first order fractal, n ¼ 1. Repeat this process for each entity triangle, we can get Sierpinski triangle structure with n ¼ 2 and n ¼3, respectively, as shown in Fig. 1. It is seen that the triangle side is reduced by half. The new triangle can be seen as a repeat of the unit in next level of fractal. If we measure the structure by the triangle with the side length 1/2n, the amount of the units is 3n. Then the Hausdorff-Becikovich dimension of the Sierpinski triangle fractal is n
D = ln 3n /ln( 1/2) = 1.585.
(2)
Fig. 2 is a diagrammatic sketch of the FPPCs following Sierpinski triangle strategy. Here equilateral and right angled isosceles triangle structures are considered, which are shown in Fig. 2a and b, respectively. In the figures, the white part represents the pores. The pores are treated as vacuum. Porosity is the ratio between the pore area and the surface, which is an important factor in the description of PPCs. The porosities of these two types of Sierpinski
Fig. 2. Diagrammatic sketch of 2D fractal PPCs. (a) Sierpinski equilateral triangle structure; (b) Sierpinski right angled isosceles triangle structure. The red line marks out the unit cell used in the calculation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 1. Sierpinski triangle fractal up to three hierarchies. (a) Sierpinski equilateral triangle structure; (b) Sierpinski right angled isosceles triangle structure.
K. Wang et al. / Physica B 498 (2016) 33–42
where r¼ (x, y) denotes the position vector, ρ is the mass density, ω is the angular frequency, μ and λ are the shear modulus and Lamé constant. u¼ (ux, uy) is the displacement vector in the transverse plane, and is the 2D vector differential operator. Based on the Bloch theorem, for the periodic structure, the displacement field can be expressed as
triangle PPCs are calculated as
f = A pore /A unit ⎛ 3 3 2 3 2 3 2⎞ =⎜ b2 + ic + 3 jd + 9 ke ⎟ 4 4 4 4 ⎝ ⎠ =
b2 ⎛ 1 1 3 9 ⎞ ⎜ + i+ j+ k⎟. 8 32 128 ⎠ a2 ⎝ 2
⎛ 3 ⎞ ⎜ a2⎟ ⎝ 2 ⎠ (3)
In this manuscript, we have nmax ¼3. According to Eq. (3), when n ¼0, i¼j¼ k¼ 0; when n ¼1, i ¼1, j¼ 0, k ¼0; when n ¼2, i ¼j¼1, k ¼0. As shown in Fig. 2, the z-coordinate is set perpendicular to the vacuum holes. The elastic waves can be decoupled into the mixed anti-plane shear mode and the in-plane mode since they propagate in the transverse plane (x0y plane), which means that the displacement vectors are independent of the z-coordinate. Accordingly, the in-plane frequency-domain wave equations are expressed as
⎛ ∂ ⎞ −ρ( r)ω2ui = ∇⋅ μ( r)∇ui + ∇⋅⎜ μ( r) u⎟ ∂xi ⎠ ⎝ ∂ + λ( r)∇⋅u , (i = x, y) ∂ xi
(
(
)
35
)
(4)
u( r) = ei( k ⋅ r)uk ( r) ,
(5)
where uk(r) is a periodical vector function with the same periodicity as the crystal lattice and k ¼(kx, ky) is the wave vector within the first Brillouin zone of the reciprocal lattice. COMSOL Multiphysics is utilized to solve the governing equations under the 2D plane strain application mode (ACPN). In the unit cell, the eigenvalue equations in the discrete form can be written as
( K − ω M)U = 0 , 2
(6)
where K and M are the stiffness and mass matrices of the unit cell and U is the displacement at the nodes. On the surface of the pore, the free boundary condition is applied. The Bloch theorem should be applied on the two opposite boundaries of the unit cell, yielding
Fig. 3. Variation of the band structures of Sierpinski equilateral triangle FPPCs with respect to the porosity in the first fractal level. (a) f¼ 0.2; (b) f¼ 0.4; (c) f¼ 0.5; (d) f ¼0.6.
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K. Wang et al. / Physica B 498 (2016) 33–42
U( r + a) = ei( k ⋅ a)U( r) ,
(7)
where a is the lattice basis vector associated with the PPCs and r is located at the boundary nodes. The unit cell is meshed by using the triangular Lagrange quadratic elements provided by COMSOL. The direct SPOOLES is chosen as the linear system solver and the eigen-frequency analysis is selected as the solver mode. In addition, parameter settings in symmetry direction in the advanced solver and the Hermitian transpose of constraint matrix are required. The model built in CAD is imported into COMSOL and it is saved as a MATLAB-compatible ‘.m’ file. This ‘.m’ file is programmed to let the wave vector k sweep the edges of the first irreducible Brillouin zone to obtain the whole dispersion relations. Since the present results have compared with the ones given in Ref. [13], and well coincidence has been obtained, we do not give the detailed convergence analysis of the calculation here anymore.
3. Numerical examples and discussion Seen as Fig. 2, the geometrical parameters of the Sierpinski triangle FPPCs could be changed under certain fractal hierarchy or porosity. In this section, based on the structure given in Fig. 2, we will discuss the effects of the porosity and fractal hierarchy on the band structures in FPPCs, respectively. In the calculation, the
matrix material is Aluminum, and the pore is vacuum. The elastic parameters of the matrix are taken as (in units of GPa): λ ¼68.3, μ ¼28.3, and ρ ¼2700 kg/m3. The lattice constant a ¼0.02 m. As shown in Fig. 2, fixing the central positions, we change the pore areas simultaneously to obtain different porosities by considering the geometrical limitation of the lattice. Fig. 3 shows the band structures in Sierpinski equilateral triangle FPPCs with different porosities in the first fractal level, that is, n ¼1. In the figures, the orange rectangle marks out the possible ABGs. The dimensionless frequency ω a /2πVt is used in the y axis, with Vt ¼3110 m/s, the velocity of the elastic transverse wave in Al matrix. Seen as Fig. 3a, when f¼ 0.2, one ABG is opened between the 10th and the 11st bands (Fig. 3a). The central frequency of the ABG is 1.160 with the bandwidth 1.183 1.136 ¼0.047. Along with the increase of the porosity, the central frequency of the ABG is lowered down and more ABGs are opened. As shown in Fig. 3b, when f ¼0.4, four more ABGs are opened at 4th–5th, 12nd–13rd, 18th–19th and 20th–21th bands, respectively. The 18th–19th band has the widest bandwidth, that is, 1.52 1.31 ¼0.21. Along with the further increase of the porosity, the central frequencies of the ABGs are decreased with wider gap width. When f ¼0.6, it is noticed that the central frequency of the 18th–19th ABG is dropped to 1.226 with the largest bandwidth 1.660–0.793 ¼0.867. In the same way, the ABGs in the Sierpinski right-angled isosceles triangle FPPCs are also investigated and shown in Fig. 4.
Fig. 4. Variation of the band structures of Sierpinski right angled isosceles triangle FPPCs with respect to the porosity. (a) f ¼0.25; (b) f ¼0.3; (c) f¼ 0.35; (d) f ¼0.4.
K. Wang et al. / Physica B 498 (2016) 33–42
Seen as Fig. 4a, when f¼ 0.25, three ABGs are opened at the 2nd– 3rd, 4th–5th, and 5th–6th bands. The central frequency of the first ABG is 0.47 with the gap width 0.478 0.462 ¼0.016. The bandwidth of the 4th–5th band is 0.031 with the central frequency 0.762. The central frequency of the 5th–6th band is 0.923 with the bandwidth 0.936 0.908 ¼0.028. Along with the increase of the porosity, the central frequency of the ABG is lowered down and the 5th–6th bandwidth is increased. When f¼ 0.35 (Fig. 4c), a new ABG is opened at 8th–9th band and the 4th–5th band is closed. The central frequency of the 8th–9th band is 1.216 with the bandwidth 1.252 1.179 ¼0.073. Along with the further increase of the porosity, see f ¼0.4 (Fig. 4d), the 5th–6th band is also closed.
37
The variation of ABGs in FPPCs with different hierarchies is also discussed. The detailed variation of the ABGs with respect to the porosity is plotted in Figs. 6 and 7, respectively. In order to verify the accuracy of the numerical calculation, the transmission spectra through the Sierpinski triangle FPPCs composed by a finite array of 8 1 units are calculated and showed in Fig. 5. A line wave source is located at the leftmost boundary of the structure and the response at the rightmost boundary is computed. For the in-plane mode, only the translational (displacement) wave sources are considered. The red lines represent the x-polarized translational (displacement) wave sources with unit amplitude meanwhile the black lines represent the y-polarized direction.
Fig. 5. (a) The band structure and (b) transmission spectra of the Sierpinski equilateral triangle structure; (c) the band structure and (d) transmission spectra of Sierpinski right angled isosceles triangle structure. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
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K. Wang et al. / Physica B 498 (2016) 33–42
Fig. 6. Eigenmode shapes and displacement vector fields of Sierpinski equilateral triangle FPPC structure at (a) the upper band edge at n ¼1, f¼ 0.4; (b) the lower band edge at n¼1, f¼ 0.4; (c) the upper band at n ¼1, f¼ 0.2; and (d) the lower band at n¼ 1, f ¼0.2.
K. Wang et al. / Physica B 498 (2016) 33–42
The band structure and the transmission spectra along ΓX direction for the Sierpinski equilateral triangle lattice at n ¼1, f ¼0.6 are given in Fig. 5a and b. A very obvious attenuation appears between 0.793 and 1.660. Correspondingly, the band structure and the transmission spectra along ΓX direction for the Sierpinski right angled isosceles triangle structure at n ¼1, and f ¼0.3 are also given in Fig. 5c and d. It can be noticed that attenuations are conspicuously nearby 1.35 and 0.55 which gets a good match with the band structure. The results indicate the good agreement in the directional band structure and the transmission spectra. Fig. 6 gives the vibration eigen-modes of unit cell at the upper and lower edges of the 12nd–13rd ABG when n ¼1, f ¼0.4 (Fig. 3 Au and Ab). Color contour represents the magnitude of displacement. It can be found that for the upper-edge mode (Fig. 6a), a slight torsion occurred in the central of the matrix. The unit cell vibrates centrally symmetrically to the three red dash dot lines as plotted; whilst for the lower-edge mode (Fig. 6b), there is a slightly longitudinal deformation in the central of the matrix and the unit cell vibrates symmetrically to the vertical dash dot line. In order to understand the mechanism of the bandgap generation, the vibration eigenmodes of unit cell at n ¼1, f ¼0.2 with the same wave vector (Fig. 3, marked out by Bu and Bb) are given comparatively in Fig. 6c and d. At this porosity, the 12nd–13rd ABG is not opened. It is seen that although the unit cell makes the same symmetrically
39
vibration at the 12nd band as that when f¼ 0.4, its vibration mode at 13rd band is not symmetry. That is, the unit cell makes the harmonic resonance at the adjacent bands simultaneously, the complete bandgaps will be opened. Fig. 7 shows the variation of the ABGs with respect to the porosity at different hierarchies in Sierpinski equilateral triangle FPPCs. Seen as Fig. 7a, for a normal one with n ¼0 (Fig. 7a and Fig. 8a), an ABG is firstly opened at the 6th–7th band at f ¼0.18. Along with the increase of the porosity, two more ABGs are opened at 10th–11th and 4th–5th bands. When f¼ 0.34, a higher order ABG is opened at the 14th–15th band. Along with the increase of the porosity, the bandwidth is gradually increased, especially for the 6th–7th band. The bandwidth is increased from 0.016 at f ¼0.2 to 0.4128 at f ¼0.45. Conversely, the central frequencies of the ABGs are dropped. When n ¼1 (Fig. 7b), an ABG is firstly opened at the 10th–11st band at f ¼0.2. Along with the increase of the porosity, two ABGs are opened at 18th–19th and 20th–21st bands. When f¼ 0.4, except for an ABG at the 12nd–13rd band, a lower band ABG is also opened at the 4th–5th band with lower central frequency. It is seen that the increase of the porosity leads the wideness of the bandwidth, especially for the 18th–19th band. The bandwidth is increased from 0.025 at f ¼0.3 to 0.867 at f ¼0.6. It is noticed that the central frequencies of the ABGs are dropped sharply at higher
Fig. 7. Variation of the ABGs with respect to the porosity in the Sierpinski equilateral triangle FPPC when (a) n ¼0; (b) n¼1; (c) n ¼2; and (d) n¼ 3.
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K. Wang et al. / Physica B 498 (2016) 33–42
porosity, except for the higher order ABG, see, the 20th–21st band. Its central frequency is a little increased along with the increase of the porosity. Comparison between Fig. 7a and b shows that the critical porosity for the opening of the first ABG is increased, see, from 0.18 at n ¼0 to 0.2 at n¼ 1. Fractal rises up the bands of ABGs and the central frequency is more sensitive to the variation of the porosity. When n ¼ 2 (Figs. 1a and 7c), an ABG is firstly opened at 10th– 11st bands at f ¼0.28, which is larger than f ¼0.2 when n ¼ 1. The higher band, see, the 20th–21st ABG, is not opened again. Along with the increase of the porosity, three ABGs are opened at 18th– 19th, 12nd–13rd, and 4th–5th bands at the porosities f ¼0.32, f ¼0.43, and f ¼0.45, which are larger than the corresponding values at n ¼1, see, f ¼0.3 and 0.4, respectively. Along with the further increase of the fractal hierarchy, see, n ¼3, the critical porosities for the opening of ABGs at different bands are all postponed, especially for the 12nd–13rd band, extended to f ¼0.53. The bandwidth of higher ABGs tends to be narrower, and the central frequencies are decreased. Comparing Fig. 7c and d to Fig. 7b indicates that lower fractal hierarchy is easier to form wider ABG. When n ¼1, the bandwidth of the 18th– 19th ABG is 0.867 at f ¼0.6, which is far wider than the one when n ¼2 at the same porosity, see 0.089, but when n Z2, further increase of the fractal hierarchy has limited effects on the band structures in FPPCs.
Correspondingly, Fig. 8 shows the variation of the ABGs with respect to the porosity in different hierarchies in Sierpinski rightangled isosceles triangle FPPCs. It is seen that when n ¼0 (Fig. 1b and Fig. 8a), a 3rd–4th ABG is firstly opened at f ¼0.15. Along with the increase of the porosity, more ABGs are opened at 3rd–4th, 4th–5th, 7th–8th, and 8th–9th bands. The central frequencies of the ABGs are decreased with wider band widths except for the 4th–5th band. Its central frequency is a little increased along with the increase of the porosity from f ¼0.3 to f¼ 0.34. Comparison between Figs. 7a and 8a indicates that wider ABGs could be opened in Sierpinski equilateral triangle FPPC whilst Sierpinski right-angled isosceles FPPC is easier to form a band gap at lower porosity and lower band. When n¼ 1, (Figs. 1b and 8b), more ABGs are opened at lower order bands. It is seen that the 2nd–3rd band is opened at f¼ 0.23. Along with the increase of the porosity, the central frequencies are decreased with wider bandwidth. It is noticed that the 5th–6th and 4th–5th bands are closed at f ¼0.4 and f ¼0.35, respectively. Comparison between Fig. 8a and b indicates that the central frequencies of the ABGs are decreased. When n¼ 2 (Fig. 1b and 8c), an ABG is firstly opened at the 4th– 5th band at f ¼0.25, which is larger than f ¼0.22 when n ¼1. Along with the increase of the porosity, no ABGs are opened. The 2nd– 3rd, 5th–6th and 8th–9th ABGs are opened at f¼ 0.3, f ¼0.35, and f¼ 0.34, respectively, which are larger than the corresponding
Fig. 8. Variation of the ABGs with respect to the porosity in the Sierpinski right-angled isosceles triangle FPPCs when (a) n ¼0; (b) n¼1; (c) n ¼2; and (d) n¼ 3.
K. Wang et al. / Physica B 498 (2016) 33–42
values at n ¼ 1, see, f¼ 0.23, f ¼0.23 and f ¼0.30 respectively. When n ¼3 (Figs. 1d, and 8d), the 5th–6th band is opened at f ¼0.42 and keeps open, whilst the 4th–5th band is still closed at f ¼0.45. Moreover, it is noticed that when n ¼3, along with the increase of the porosity, the bandwidth is decreased. Comparison among Fig. 8a–d indicates that in Sierpinski right-angled isosceles triangle FPPCs, fractal hierarchy will not affect the structures of ABGs. But lower fractal hierarchical is easier to form wider ABGs. Comparison Figs. 8–7 shows that it is easier to form low band ABGs in Sierpinski right-angled isosceles triangle FPPCs. In order to see clearly this variation due to the fractal hierarchy, Fig. 9 displays the variation of the band edges of the lowest band (see, 4th–5th band for Sierpinski equilateral triangle FPPC, and 2nd–3rd band for Sierpinski right-angled isosceles triangle FPPC) with respect to the fractal hierarchy at different porosities. Seen as Fig. 9, the bandwidth of ABGs in different fractal hierarchy and porosities are marked out by using rectangles with different colors. When n ¼0, that is, when the fractal program has not been implemented, the ABGs are with wider bands and higher central frequencies. Along with the increase of the porosity, the central frequencies of ABGs are decreased. No matter Sierpinski right-angled isosceles triangle FPPCs or Sierpinski equilateral triangle FPPCs, the first order fractal (n ¼1) corresponds to lower
a
0.60 f =0.3
0.55
f =0.4
f =0.5
2nd-3rd Band
ωa/2πVt
0.50 0.45 0.40 0.35
41
central frequencies and wider bandwidth. When the fractal hierarchy is increased, see, n ¼2 or n¼ 3, the central frequencies are increased. But for Sierpinski right-angled isosceles triangle FPPCs, the effect of the fractal hierarchy is not so distinct. And along with the increase of the porosity, the sensitivity to the fractal hierarchy is also weakened.
4. Summary In this paper, the band gap structures in two-dimensional Sierpinski triangle fractal porous phononic crystals are investigated by using finite element (FE) simulation. Summarizing the results above we can conclude that: (1) Compared to the normal structure (n ¼0), Sierpinski triangle fractal increases the critical porosity for the opening of the ABG; whilst the central frequencies of the ABGs are decreased; (2) Along with the increase of the porosity, the central frequencies of the ABGs in different hierarchies display a unified downward trend. Increase of the hierarchy increases the sensitivity of the central frequency to the porosity; (3) Along with the increase of the hierarchy, the bands of the ABGs are not changed, but the central frequencies are increased. When n Z2, further increase of the fractal hierarchy has limited effects on the band structures in Sierpinski triangle FPPCs; (4) On the same hierarchy, wider ABGs could be opened in Sierpinski equilateral triangle FPPC; whilst in Sierpinski rightangled isosceles FPPC, it is easier to form a band gap at lower porosity and lower band. (5) The results above to some extend disclose the intrinsic relation of the ABGs with the fractal hierarchy and topological parameters in Sierpinski triangle FPPCs. It would provide useful guidance in tuning of the band structures in fractal porous phononic crystals.
0.30 Acknowledgments
0.25 Sierpinski right-angled isosceles triangle FPPC
0.20 0
1
2
3
n
The second author thanks the support from the Fundamental Research Funds for the Central Universities of China (No. 2014JBZ014). Supports from the National Natural Science Foundation of China, China (11272046), and National Basic Research Program of China (973Program) (2015CB057800), are also acknowledged.
References
Fig. 9. Variation of the upper and lower edges of the first ABGs with respect to the fractal hierarchy. (a) Sierpinski right angled isosceles triangle FPPCs; (b) Sierpinski equilateral triangle FPPCs.
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