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JMST Advances

, Volume 1, Issue 1–2, pp 153–159 | Cite as

Topology optimization for phononic band gap maximization considering a target driving frequency

  • Guilian Yi
  • Yong Chang Shin
  • Heonjun Yoon
  • Soo-Ho Jo
  • Byeng D. YounEmail author
Letter
  • 204 Downloads

Abstract

A phononic crystal (PnC) is an artificially engineered periodic structure that exhibits extraordinary phenomena, such as a phononic band gap. The phononic band gap refers to a certain range of frequencies within which mechanical waves cannot propagate through the PnC. The main purpose of this paper is to propose a topology optimization formulation for phononic band gap maximization that simultaneously takes into account a target driving frequency. In the proposed topology optimization formulation, a relative band gap is considered as an objective function to be maximized. In addition, an equality constraint is imposed on the central frequency of the band gap. The topology optimization problem is solved using the globally convergent method of moving asymptotes, which is a gradient-based optimization algorithm. Numerical examples are computed to demonstrate the effectiveness of the proposed topology optimization formulation.

Keywords

Phononic band gap maximization Equality constraint Gradient-based topology optimization Target driving frequency 

1 Introduction

A phononic band gap refers to a frequency regime within which mechanical waves (i.e., elastic and acoustic waves) cannot propagate through a phononic crystal (PnC). Many engineering applications of the PnC take advantage of the extraordinary features of the phononic band gap to manipulate the propagation of mechanical waves.

For a given lattice constant and component materials, the phononic band gap of PnCs can be tailored using different topological shapes of a unit cell. Topology optimization has been used to find the best topological shape of a unit cell for phononic band gap maximization. Many researchers have attempted to conduct phononic band gap maximization using topology optimization techniques, such as non-gradient-based methods [1] and gradient-based methods [2, 3]. Barbarosie and Neves [4] designed a 2D PnC with a maximized band gap for a frequency filtering application. Li et al. [2] conducted band gap maximization of 2D cellular PnCs whose stiffness requirements were satisfied through constraints on the static effective bulk or shear modulus. Zhang et al. [5] proposed optimal designs of 1D infinite PnCs that yielded minimum boundary frequencies for band gaps of longitudinal waves and minimum pass band ranges in the target frequency area. Yi and Youn [6] provided a comprehensive survey on topology optimization of PnCs; they also thoroughly investigated the topology optimization techniques used for the maximization of phononic band gaps.

Mechanical waves in real applications have physical uncertainty such as frequency fluctuations. It is thus required that the driving frequency is better to be close or equal to the central frequency of the band gap, for promising effective applications of band gaps in practical engineering problems, such as wave focusing, wave localization, and wave filtering. Previous methods are limited in that the central frequency of the phononic band gap changes in each iteration. This implies that the previous methods for topology optimization do not guarantee whether the driving frequency is located at or close to the center frequency of the phononic band gap. This study proposes a new topology optimization formulation that maximizes the phononic band gap while considering a target driving frequency. In the topology optimization formulation, a relative band gap is considered as an objective function to be maximized. An equality constraint is imposed on the central frequency of the band gap. The topology optimization problem is solved using the globally convergent method of moving asymptotes (GCMMA).

The rest of this paper is organized as follows. Section 2 describes the novel topology optimization design formulation for a unit cell of 2D PnCs for a maximal band gap around a target driving frequency. Section 3 discusses topology optimization results of 2D PnCs with a maximized band gap for given materials. The conclusions of this research are given in Sect. 4.

2 Topology optimization of 2D PnCs for a maximal band gap around a target driving frequency

To conduct the topology optimization of a 2D PnC to maximize the phononic band gap, a band structure of the 2D PnC should be calculated first to examine the existence of phononic band gaps. In this research, the finite element (FE) method, first used by Sigmund and Jensen [7], was implemented to compute the band structure of the elastic waves propagating within the 2D PnC. The main idea of the FE method is introduced in Sect. 2.1. Topology optimization for band gap maximization around a target driving frequency using the FE method combined with a gradient-based optimization method is described in Sect. 2.2.

2.1 Finite element method for band structure calculation

For a 2D PnC, it is assumed that the waves propagate only in the xy plane [1]. In other words, the in-plane wave modes are considered for the band structure calculation in this study. The wave propagation is governed by
$$- \rho u\omega^{2} \left( {\mathbf{k}} \right) = \left( {\lambda + 2\mu } \right)\frac{{\partial^{2} u}}{{\partial x^{2} }} + \lambda \frac{{\partial^{2} v}}{\partial x\partial y} + \mu \left( {\frac{{\partial^{2} u}}{{\partial y^{2} }} + \frac{{\partial^{2} v}}{\partial x\partial y}} \right),$$
(1)
$$- \rho v\omega^{2} \left( {\mathbf{k}} \right) = \mu \left( {\frac{{\partial^{2} v}}{{\partial x^{2} }} + \frac{{\partial^{2} u}}{\partial x\partial y}} \right) + \left( {\lambda + 2\mu } \right)\frac{{\partial^{2} v}}{{\partial y^{2} }} + \lambda \frac{{\partial^{2} u}}{\partial x\partial y},$$
(2)
where λ and μ are Lame’s coefficients, ρ the material density, and (u, v) is the displacement vector of the elastic wave propagating within the 2D PnC. k = (kx, ky) is the wave vector and ω is the angular frequency of the wave. Equations (1) and (2) are coupled with each other.
The main idea of using the FE method for the band structure calculation is to apply the periodic boundary condition on the boundary edges of the unit cell of a 2D PnC through nodal displacements. Based on the Floquet–Bloch theory, the displacement field of the waves within the 2D PnC can be expressed as
$${\mathbf{u}}_{{\mathbf{k}}} \left( {{\mathbf{r}},t} \right) = {\mathbf{u}}_{{\mathbf{k}}} \left( {\mathbf{r}} \right)e^{{ - i\omega \left( {\mathbf{k}} \right)t}} ,$$
(3)
$${\mathbf{u}}_{{\mathbf{k}}} \left( {{\mathbf{r}} + {\mathbf{R}},t} \right) = {\mathbf{u}}_{{\mathbf{k}}} \left( {{\mathbf{r}},t} \right)e^{{i{\mathbf{k}} \bullet {\mathbf{R}}}} ,$$
(4)
where r and t denote the position vector and time, respectively; \({\mathbf{R}}\) is the lattice periodicity of the PnC unit cell. ω is the angular frequency. uk(r) = up(r)eik∙r is the spatial part of the displacement vector and up(r) is a periodic displacement field of r with the same periodicity as the structure. By inserting into Eqs. (1) and (2), an eigenvalue problem can be obtained as
$$\left( {\left[ {\mathbf{K}} \right]\left( {\mathbf{k}} \right) - \omega^{2} \left( {\mathbf{k}} \right)\left[ {\mathbf{M}} \right]} \right)\left[ {\mathbf{u}} \right] = 0,$$
(5)
where u = uk(r) and ω2 are, respectively, the eigenvectors and eigenvalues of the eigenvalue problem. [K] and [M] are the global stiffness and mass matrices, respectively. By sweeping the wave vector \({\mathbf{k}}\) along the edges of the irreducible Brillouin zone, shown by triangle \({\varvec{\Gamma}} \to {\mathbf{X}} \to {\mathbf{M}} \to {\varvec{\Gamma}}\) in Fig. 1, all dispersion relations (also called a band structure) of the Bloch waves propagating within a 2D PnC can be obtained.
Fig. 1

Wave number vector and irreducible Brillouin zone for a square unit cell with the lattice constant a

2.2 Topology optimization formulation

In this research, a method is proposed to maximize the phononic band gap while considering a target driving frequency. This new method addresses the optimal topology shapes of the unit cell of the PnC. A gradient-based topology optimization algorithm is used together with external calls to the FE code to evaluate the optimized topology shapes. To maximize the phononic band gap while considering a target driving frequency, the topology optimization problem is formulated with an objective of maximizing the relative band gap width and a constraint on the central frequency of the band gap equal to the target driving frequency. The objective function to be maximized is expressed as
$$f\left( \varSigma \right) = \frac{{\min_{{\mathbf{k}}} :{\mkern 1mu} \omega_{n + 1} \left( {\varSigma ,{\mathbf{k}}} \right) - \max_{{\mathbf{k}}} :{\mkern 1mu} \omega_{n} \left( {\varSigma ,{\mathbf{k}}} \right)}}{{\left( {\min_{{\mathbf{k}}} :{\mkern 1mu} \omega_{n + 1} \left( {\varSigma ,{\mathbf{k}}} \right) + \max_{{\mathbf{k}}} :{\mkern 1mu} \omega_{n} \left( {\varSigma ,{\mathbf{k}}} \right)} \right)/2}},$$
(5)
where Σ denotes the topological distribution within the unit cell of the PnCs, f(Σ) the relative band gap width, and n is the total number of wave vectors on the irreducible Brillouin zone edges. The relative band gap width described above is used to enable evaluation of the width of a phononic band gap regardless of the scale of the lattice constant. Because the lattice constant of the PnC, also known as the size of the unit cell, scales with the wavelength of the interfering waves, the order of magnitude of the phononic band gap frequencies is inversely proportional to the lattice constant, which is called the scaling property. The magnitude of the central frequency of the phononic band gap depends on the lattice constant of the PnC. That is why the relative band gap is calculated as the absolute band gap divided by the central frequency. The constraint function is expressed as
$$g\left( \varSigma \right) = \frac{{{{\left( {\min_{{\mathbf{k}}} :{\mkern 1mu} \omega_{n + 1} \left( {\varSigma ,{\mathbf{k}}} \right) + \max_{\text{k}} :{\mkern 1mu} \omega_{n} \left( {\varSigma ,{\mathbf{k}}} \right)} \right)} \mathord{\left/ {\vphantom {{\left( {\min_{{\mathbf{k}}} :{\mkern 1mu} \omega_{n + 1} \left( {\varSigma ,{\mathbf{k}}} \right) + \max_{\text{k}} :{\mkern 1mu} \omega_{n} \left( {\varSigma ,{\mathbf{k}}} \right)} \right)} 2}} \right. \kern-0pt} 2}}}{{\omega_{\text{t}} }} - 1,$$
(6)
where ωt represents the target driving frequency. The constraint function g(Σ) is the relative difference between the central frequency of the phononic band gap and the target driving frequency. Note that the objective and constraint functions are both normalized into the same scale for improving the optimization efficiency. For the equality constraint, Eq. (6) is set as g(Σ) = 0.
Because of the periodicity of a fully symmetric unit cell, only one-eighth of the unit cell area (as shown in Fig. 2a) is considered as the actual design domain of the topology optimization problem. However, since it is impossible to exactly take one of eighth of the unit cell due to the element-based discretization, more than one-eighth of all elements is selected (as shown in Fig. 2b). Therefore, for the original whole design domain with a meshing size of \(m \times m\), the number of design variables for the actual design domain becomes N = (1 + m/2) × m/4. In this paper, bilinear rectangle elements (Q4) are used for meshing the unit cell. Within the actual design domain, element-wise design variables are introduced for each element as
$$0 \le x_{i} \le 1,\,i = 1, \ldots ,\,N,$$
(7)
where the subscript i denotes the element number and represents the design variable of the ith element. A material interpolation scheme is employed to evaluate the material properties of the ith element in a formulation as
$$A_{i} \left( {x_{i} } \right) = \left( {1 - x_{i} } \right)A_{1} + x_{i} A_{2} ,$$
(8)
where Ai represents the material density (ρ) and elastic properties (i.e., Lame’s coefficients λ and μ) and the subscripts 1 and 2 denote the material phases 1 and 2, respectively. In general, the denser the inclusion and the lighter the matrix, the easier it is for the band gap to form. Therefore, the material phase 1 (matrix) is assumed to be lighter than the material phase 2 (inclusion). If the design variable is zero, the element has a pure material phase 1; if the design variable is one, the element has a pure material phase 2. The physical meaning of Eq. (8) lies in approximating that the material distribution of the element can be defined as a linear mixture of the two materials throughout the domain with a continuous parameter (design variable). The material interpolation scheme expressed herein yields 0–1 solutions naturally when maximizing the phononic band gaps. There is no need to penalize based on elastic properties of each element [8, 9].
Fig. 2

A fully symmetric unit cell: a design domain with on eighth of the unit cell and b the actual design domain

To eliminate mesh dependency and checker-board issues, a density filtering method is used to evaluate the actual design variables in the full design domain of the unit cell [10]. The filtered design variables are further used to evaluate the actual material properties of the elements for the band structure calculation.

2.3 Sensitivity calculation

The derivatives of eigenfrequencies with respect to the design variables can be obtained as
$$\frac{{\partial \omega_{n} \left( {\varSigma ,{\mathbf{k}}} \right)}}{{\partial x_{i} }} = \frac{1}{{2\omega_{n} }}\left( {{\mathbf{u}}_{n} } \right)^{\text{T}} \left( {\frac{{\partial \left[ {\mathbf{K}} \right]\left( {\mathbf{k}} \right)}}{{\partial x_{i} }} - \omega_{n}^{2} \frac{{\partial \left[ {\mathbf{M}} \right]}}{{\partial x_{i} }}} \right){\mathbf{u}}_{n} ,$$
(9)
where un is the displacement vector (eigenvector) corresponding to the nth eigenfrequency (eigenvalue). ∂[K]/∂xi and ∂[M]/∂xi are the derivatives of the global stiffness and mass matrices, respectively. Note that the eigenvalue sensitivity calculation shown above in Eq. (9) is only applicable for an eigenvalue problem with a distinct eigenfrequency. Sensitivities for repeated eigenvalues are considered as well with the technique described in literature [3]. Using the derivatives of eigenfrequencies, the sensitivities of the objective and constraint functions can be calculated by
$$\begin{aligned} \frac{\partial f}{{\partial x_{i} }}\, & = - 4\frac{{\max_{\text{k}} :{\mkern 1mu} \omega_{n} \left( {\varSigma ,{\mathbf{k}}} \right)\frac{{\partial \min_{{\mathbf{k}}} :{\mkern 1mu} \omega_{n + 1} \left( {\varSigma ,{\mathbf{k}}} \right)}}{{\partial x_{i} }}}}{{\left( {\min_{{\mathbf{k}}} :{\mkern 1mu} \omega_{n + 1} \left( {\varSigma ,{\mathbf{k}}} \right) + \max_{\text{k}} :{\mkern 1mu} \omega_{n} \left( {\varSigma ,{\mathbf{k}}} \right)} \right)^{2} }} \\ & \quad + 4\frac{{\min_{{\mathbf{k}}} :{\mkern 1mu} \omega_{n + 1} \left( {\varSigma ,{\mathbf{k}}} \right)\frac{{\partial \max_{\text{k}} :{\mkern 1mu} \omega_{n} \left( {\varSigma ,{\mathbf{k}}} \right)}}{{\partial x_{i} }}}}{{\left( {\min_{{\mathbf{k}}} :{\mkern 1mu} \omega_{n + 1} \left( {\varSigma ,{\mathbf{k}}} \right) + \max_{\text{k}} :{\mkern 1mu} \omega_{n} \left( {\varSigma ,{\mathbf{k}}} \right)} \right)^{2} }}, \\ \end{aligned}$$
(10)
$$\frac{\partial g}{{\partial x_{i} }} = \frac{1}{{2\omega_{t} }}\left( {\frac{{\partial \min_{{\mathbf{k}}} :{\mkern 1mu} \omega_{n + 1} \left( {\varSigma ,{\mathbf{k}}} \right)}}{{\partial x_{i} }} + \frac{{\partial \max_{\text{k}} :{\mkern 1mu} \omega_{n} \left( {\varSigma ,{\mathbf{k}}} \right)}}{{\partial x_{i} }}} \right).$$
(11)

In this research, two topology optimization problems, (1) with an equality constraint and (2) without an equality constraint, are described through formulations in Eqs. (5) and (6). The globally convergent method of moving asymptotes (GCMMA) [11] is used as the gradient-based optimization algorithm for addressing the above optimization problems. The maximization problem is converted to a minimization problem in GCMMA by multiplying the objective function with − 1. To solve the optimization problem with an equality constraint in GCMMA, which was proposed for a general constraint format of g(Σ) ≤ 0, the equality constraint is separated into two constraints: Constr. 1—g(Σ) ≤ 0, and Constr. 2—g(Σ) ≥ 0. The sensitivities corresponding to these two constraints are ∂g/∂xi and − ∂g/∂xi in Eq. (9), respectively.

3 Results and discussion

Aluminum and tungsten were selected for the two material components to compose the 2D PnC; their properties are shown in Table 1. The Lame’s coefficients of the two materials can be calculated in terms of Young’s modulus and Poisson’s ratio. The lattice constant of the unit cell is \(a = 20\,{\text{mm}}\) and the meshing size on the unit cell is 40 × 40. 46 wave vectors along \({\varvec{\Gamma}} \to {\mathbf{X}} \to {\mathbf{M}} \to {\varvec{\Gamma}}\), which indicates 46 eigenvalue problems, are considered to calculate one band in the band structure. The filtering radius is set as 1.4 times the element length in this research. A frequency of 120 kHz is considered as the target driving frequency. To verify the effectiveness of the proposed topology optimization formulation with the equality constraint, the results of the proposed method are compared to those obtained from the unconstraint topology optimization formulation, which has the same objective function as in Eq. (5).
Table 1

Material properties

 

Aluminum

Tungsten

Density

2700 kg/m3

19,300 kg/m3

Young’s modulus

70 GPa

411 GPa

Poisson’s ratio

0.33

0.28

Li et al. [3] reported that different baseline designs of a unit cell with given materials and lattice constant may produce different topology shapes and band gaps in the final results of topology optimization of 2D PnCs. In this study, therefore, three different baseline designs are considered as the initial designs of the unit cell, including an X-cross-shaped baseline design (D1), a multi-cross-shaped baseline design (D2), and a square-shaped baseline design (D3), as shown in the first row of Fig. 3. The green and blue materials in each baseline design represent aluminum and tungsten, respectively. The band structures of three baseline designs are shown in the second row of Fig. 3, which are normalized with respect to the lattice constant and wave speed:
$$\omega_{\text{normalized}} = \frac{\omega a}{{2\pi c_{\text{t}} }},$$
(12)
where ω is the angular frequency, a the lattice constant, and ct is the wave speed.
Fig. 3

Three different baseline designs of the unit cell: a an X-cross shape (D1), b a multi-cross shape (D2), and c a square shape (D3), d the band structure of D1, e the band structure of D2, and f the band structure of D3

The baseline design (D3) has a relative band gap width of 0.26 between the 3rd and 4th bands and a normalized central frequency of 0.75; whereas, D1 and D2 exhibit no band gap in the band structures. Since the normalized central frequency of 0.77 represents the target driving frequency of 120 kHz, which can be possibly centered between the 3rd and 4th bands, this study examined the band gap between the 3rd and 4th bands, and used topology optimization to find the largest band gap.

The results of the topology optimization for maximizing the relative band gap are summarized in Table 2. The first, second, and third columns exhibit the optimized topology shapes and corresponding band structures computed using the three baseline designs (D1), (D2), and (D3), respectively. For a clear observation, the band structures with normalized frequencies are converted to regular ones without normalization. The gray shading areas indicate the absolute phononic band gaps, of which the edges are denoted as ‘upper’ and ‘lower’. The central frequencies, band gaps, and relative band gaps obtained from two optimization formulations with respect to three baseline designs are summarized in Table 3.
Table 2

Results of topology optimization with respect to the three baseline designs

Table 3

Central frequencies, band gaps, and relative band gaps from three baseline designs

Constraints

Baseline

Relative band gap

Central frequency (kHz)

Absolute band gap (kHz)

Unconstraint

D1

0.4363

114.55

49.98

D2

0.4311

112.11

48.33

D3

0.4432

115.72

51.29

Equality constraint

D1

0.4290

120.00

51.48

D2

0.4254

120.00

51.05

D3

0.4287

120.00

51.44

For a clear comparison between the locations of the target and central frequencies, Fig. 4 shows the band gap widths of the unconstraint cases in blue solid lines and those of the equality-constraint cases in red solid lines. The black dash line indicates the target frequency of 120 kHz. The solutions of the equality-constraint cases satisfy the requirement of the target frequency, exactly sitting at the central frequency of the phononic band gap. This implies that the equality-constraint cases would be more robust than the unconstraint cases against the driving frequency fluctuation.
Fig. 4

The locations of the phononic band gaps of the three optimized designs with respect to the target frequency

The optimization histories of the objective are listed in Table 4, where the red curves denote the history of the relative band gap, and green and black curves represent the histories of the constraints, respectively. For the unconstraint case, the denominator of Eq. (5), which is the central frequency of the phononic band gap, changed in each iteration, and the optimization problems converged rapidly when the design with the largest relative band gap was found. Due to the random location of the central frequency during the optimization, the unconstraint band gap maximization problem easily yielded at local optimums. Thus, the unconstraint band gap maximization problem has a tendency to be dependent on the baseline design by producing different optimized shapes with different baseline designs. This phenomenon of dependency on the baseline design has been previously described in the literature [3], but the reason of it has not been discussed in earlier literature. For the equality-constraint case, the central frequency matches the target frequency to satisfy the equality constraint at first, and the largest band gap is found after several iterations.
Table 4

Optimization histories for objectives (read curves) and constraints (green and black curves)

In addition, the equality-constraint cases, the much similar circle-like topological shape was obtained from the different baseline designs. This implies that the baseline design has little influence on the results of the topological shapes of the unit cell.

4 Conclusions

This paper proposed a topology optimization formulation for phononic band gap maximization that allows a target driving frequency to be simultaneously taken into account. In the topology optimization formulation, a relative band gap is considered as the objective function to be maximized, and the equality constraint is imposed on the central frequency of the relative band gap.

The results of the numerical example demonstrate the stability and validity of the proposed optimization formulation in two perspectives: (1) the influence of the baseline design of a PnC unit cell on the topology optimization results is very minor, and (2) the proposed optimization formulation yields a larger band gap and more stable topology shapes than the without-constraint optimization formulation. Future research to build upon this work will examine topology optimization of PnCs made of a single-phase elastic medium for the easier fabrication and integration to possible applications of PnCs.

Notes

Acknowledgements

This research was supported by the National Research Council of Science & Technology (NST) grant by the Korea Government (MSIT) (No. CAP-17-04-KRISS). This research was also supported by a Grant from the Institute of Advanced Machinery and Design at Seoul National University (SNU-IAMD).

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Copyright information

© The Korean Society of Mechanical Engineers 2019

Authors and Affiliations

  • Guilian Yi
    • 1
  • Yong Chang Shin
    • 1
  • Heonjun Yoon
    • 1
  • Soo-Ho Jo
    • 1
  • Byeng D. Youn
    • 1
    • 2
    • 3
    Email author
  1. 1.Department of Mechanical and Aerospace EngineeringSeoul National UniversitySeoulRepublic of Korea
  2. 2.Institute of Advanced Machines and DesignSeoul National UniversitySeoulRepublic of Korea
  3. 3.OnePredict InC.SeoulRepublic of Korea

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