three-dimensional phononic band gap calculations using the fdtd...
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148 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 53, no. 1, january 2006
Three-Dimensional Phononic Band GapCalculations Using the FDTD Method and a
PC Cluster SystemPo-Feng Hsieh, Tsung-Tsong Wu, and Jia-Hong Sun
Abstract—This paper aims at studying the band gap phe-nomena of three-dimensional phononic crystals using thefinite difference time domain (FDTD) method and a PCcluster system. In the paper, Bloch’s theorem is applied tothe wave equation and to the boundary conditions of theperiodic structure. We calculate the variations of displace-ments and take discrete Fourier transform to acquire theresonances of the structures. Then, the dispersion relationsof the bulk acoustic wave can be obtained and the band gapsare predicted accordingly. On the other hand, because oflarger data calculation in three-dimensional phononic crys-tals, we introduce the PC cluster system and parallel FDTDprograms written with respect to the architecture of a PCcluster system. Finally, we discuss the numerical calculationof two-dimensional and three-dimensional phononic crystalsconsisting of steel/epoxy and draw conclusions regardingthe band gap phenomena between these phononic crystals.
I. Introduction
There has been research pertaining to artificial crys-tals starting with photonic crystals since the 1980s.
A photonic crystal is a periodic arrangement of dielectricmaterials that an electromagnetic (EM) wave of a spe-cific frequency range is forbidden to pass through—theso-called band gap phenomenon of the photonic crystal.Owing to the analogy of EM waves and acoustic waves,the band gap phenomenon also exists in phononic crystals,the periodic arrangement of elastic materials. The bandgap phenomenon may potentially find many engineeringapplications, such as in acoustic wave filters, the acous-tic waveguide, and acoustic resonators, which enhance thenecessity to develop a fast and efficient wave propagationtheory for the phononic crystal.
Theoretical analyses on the band gaps of phononic crys-tals have been conducted during the last 15 years. For re-search in band gaps of bulk acoustic wave (BAW) modesof phononic structures, the mixed and transverse polar-ization modes have been carried out using the plane waveexpansion (PWE) method [1]–[7]. The PWE method isalso adopted to study level repulsions of bulk acousticwaves in composite materials [8]. The phononic struc-
Manuscript received March 30, 2005; accepted July 6, 2005. Theauthors gratefully acknowledge the financial support of this researchby the National Science Council of ROC (NSC 93-2212-E-002-025).
The authors are with the Institute of Applied Mechan-ics, National Taiwan University, Taipei, Taiwan (e-mail:[email protected]).
tures considered in the aforementioned studies includetwo-dimensional (2D) solid/air, solid/solid, and solid/fluidstructures. It is worth noting that in conducting the acous-tic waves in the solid/fluid structure, a modified PWEmethod was used [4]–[7]. In addition to the band gap anal-yses of bulk modes in 2D phononic structures, there issome literature on the investigation of band gaps of sur-face acoustic waves (SAW) [9]–[11]. More detailed stud-ies of temperature effects of surface wave band gaps andelectromechanical coupling coefficient in a piezoelectricphononic crystal are also reported [12], [13]. Instead of thePWE method, the multiple scattering theory (MST) hasbeen applied to study the band gaps of bulk wave proper-ties in three-dimensional (3D) periodic acoustic compos-ites and the band structure of a phononic crystal consist-ing of complex and frequency-dependent Lame coefficients[14]–[17]. With regard to the numerical simulation of bandgaps in phononic structures, the finite difference time do-main (FDTD) method was applied to interpret the experi-mental data of 2D systems consisting of cylinders of fluids(Hg, air, and oil) inserted periodically in a finite slab ofaluminum host [18]. It was also used to calculate the pe-riodic solid-solid, solid-liquid, and solid-vacuum compos-ites [19]. The coupling characteristics of localized phononsin photonic crystal fibers and analyses of mode couplingin joined parallel phononic crystal waveguides using theFDTD method are reported in [20], [21].
In these studies, most phononic crystals were 2Dstructures; some 3D phononic crystals are reported. The3D structures include the simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC)lattices consisting of water/mercury (liquid-liquid) [22],mercury/aluminum (liquid-solid) [23], and steel/polyester(solid-solid) [24], [25]. In these 3D phononic crystals, theband gaps were evaluated by both the MST and the PWEmethods. However, the MST method has a limitation ofcalculating with overlap scatters, and the PWE methodencounters convergence problems when the phononic crys-tal has a large elastic mismatch. The FDTD method is verysuitable for dealing with different geometric structures andfor handling the convergence problem but requires consid-erable calculation.
The purpose of the present study was to develop a par-allel FDTD method and a PC cluster system to handleand accelerate the calculation of band gap phenomenonof the phononic crystal in 3D cases. By using a heteroge-neous FDTD formulation, the material property and the
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hsieh et al.: three-dimensional phononic band gap calculations 149
geometric shape can be changed easily and the formula-tion can be employed to study the band gap phenomenonin different geometric shapes, material properties, and de-sign patterns. We adopted the message passing interface(MPI) to write the parallel FDTD program in order toaccelerate the whole computational speed in the PC clus-ter system. Owing to the high-speed computation of theFDTD method and the PC cluster system, calculations of3D phononic crystal can be achieved quickly.
II. The Finite Difference Time Domain (FDTD)
Method
Whether the FDTD method or the PWE method isused, the BAW theory of phononic crystal is based on thesame physical concept. In this paper, the periodic bound-ary using Bloch’s theorem is used to describe the peri-odic arrangement of the phononic crystal. The principle ofFDTD is mentioned in this section.
A. Equation of the Wave Propagation
According to the theory of elasticity, all the elastic ma-terials are considered in the linear elastic range. Equationof motion and constitutive law with the anisotropic andinhomogeneous material can be expressed as [26]:
ρui = τij,j + ρ fi, (1)τij = Cijklεkl, (2)
where ρ is the density of materials, ui is the displacement,τij is the stress, fi is body force, Cijkl is elastic constant,and εkl is strain. Eq. (1) and (2) describe the behavior of aninfinitesimal element of an anisotropic material, and theycan apply to the inhomogeneous structure. Therefore, theFDTD method based on (1) and (2) is suitable to analyzethe phononic crystal by arranging the density and elasticconstant periodically.
In this FDTD method, the displacement and stress vari-ables in materials are used as the bases to describe theequation of motion and constitutive law, and further todevelop the difference equations. To cooperate with thestaggered grids adopted in the following difference equa-tion, the anisotropy of elastic material is limited. An or-thotropic material with nine independent elastic constantsis allowed in the program. The matrix form of the elasticityconstant using an abbreviated notation is
CIJ =
⎡⎢⎢⎢⎢⎢⎢⎣
C11 C12 C13 0 0 0C21 C22 C23 0 0 0C31 C32 C33 0 0 00 0 0 C44 0 00 0 0 0 C55 00 0 0 0 0 C66
⎤⎥⎥⎥⎥⎥⎥⎦
, (3)
where the subscripts 1, 2, 3, 4, 5, and 6 of material con-stant C represent xx, yy, zz, yz, xz, and xy, respectively.
From (1), (2), and (3), we can derive the 3D wave equa-tion in which the displacement and stress are regarded asthe variables; thus the nine equations of wave equations inmatrix form are⎡
⎣u1u2u3
⎤⎦ =
1ρ
⎡⎣τ11 τ12 τ13τ21 τ22 τ23τ31 τ32 τ33
⎤⎦
⎡⎣∂1∂2∂3
⎤⎦ +
⎡⎣f1
f2f3
⎤⎦ , (4)
⎡⎢⎢⎢⎢⎢⎢⎣
τ11τ22τ33τ23τ13τ12
⎤⎥⎥⎥⎥⎥⎥⎦
=
⎡⎢⎢⎢⎢⎢⎢⎣
C11 C12 C13 0 0 0C21 C22 C23 0 0 0C31 C32 C33 0 0 00 0 0 C44 0 00 0 0 0 C55 00 0 0 0 0 C66
⎤⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎢⎢⎣
∂1u1∂2u2∂3u3
∂2u3 + ∂3u2∂1u3 + ∂3u1∂1u2 + ∂2u1
⎤⎥⎥⎥⎥⎥⎥⎦
.(5)
In (4) and (5), the subscript numbers 1, 2, and 3 of dis-placement u and stress τ denote the components along thex, y, and z directions in the Cartesian coordinate, respec-tively, and the symbol ∂i represents the partial differentialin direction i.
Then we use the second-order center difference equa-tion and the first-order center difference equation to ap-proximate the second-order differential equation and thefirst-order differential equation, respectively. Therefore wedefine the first-order center difference equation in spacefirst as:
D1 [M(i, j, k, l)] =
1∆x1
[M
(i +
12, j, k, l
)− M
(i − 1
2, j, k, l
)]
D2 [M(i, j, k, l)] =
1∆x2
[M
(i, j +
12, k, l
)− M
(i, j − 1
2, k, l
)]
D3 [M(i, j, k, l)] =
1∆x3
[M
(i, j, k +
12, l
)− M
(i, j, k − 1
2, l
)].
(6)
In (6) and the following formulas, the i, j, and k insidethe bracket represent the position index of the x, y, andz directions, respectively, and the symbol l is the indexof time step. The ∆xi, where i = 1, 2, and 3, representsthe distance of two nearby discrete grids in the x, y, andz directions. By replacing the second-order differential intime with the second-order center difference in time, wehave
∂2
∂t2[M(i, j, k.l)] =
1(∆t)2
[M
(i, j, k, l +
12
)
− 2M(i, j, k, l) + M
(i, j, k, l − 1
2
)]. (7)
After the discrete procedure, we have the difference equa-tion of the displacement and stress (8)–(16) (see next twopages).
In (8)–(16), the symbol Ui and Tij are the discretedisplacement and stress components, respectively, with
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150 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 53, no. 1, january 2006
U1
(i +
12, j, k, l +
12
)= 2U1
(1 +
12, j, k, l
)− U1
(i +
12, j, k, l − 1
2
)
+(∆t)2
ρ
(i +
12, j, k
)
D1
[T11
(i +
12, j, k, l
)]+ D2
[T12
(i +
12, j, k, l
)]
+ D3
[T13
(i +
12, j, k, l
)] + f1
(i +
12, j, k, l +
12
)(8)
U2
(i, j +
12, k, l +
12
)= 2U2
(i, j +
12, k, l
)− U2
(i, j +
12, k, l − 1
2
)
+(∆t)2
ρ
(i, j +
12, k
)
D1
[T21
(i, j +
12, k, l
)]+ D2
[T22
(i, j +
12, k, l
)]
+ D3
[T23
(i, j +
12, k, l
)] + f2
(i, j +
12, k, l +
12
)(9)
U3
(i, j, k +
12, l +
12
)= 2U3
(i, j, k +
12, l
)− U3
(i, j, k +
12, l − 1
2
)
+(∆t)2
ρ
(i, j, k +
12
)
D1
[T31
(i, j, k +
12, l
)]+ D2
[T32
(i, j, k +
12, l
)]
+ D3
[T33
(i, j, k +
12, l
)]+ f3
(i, j, k +
12, l +
12
)(10)
T11(i, j, k, l + 1) = C11(i, j, k) · D1
[U1
(i, j, k, l +
12
)]
+ C12(i, j, k) · D2
[U2
(i, j, k, l +
12
)]+ C13(i, j, k) · D3
[U3
(i, j, k, l +
12
)](11)
T22(i, j, k, l + 1) = C21(i, j, k) · D1
[U1
(i, j, k, l +
12
)]
+ C22(i, j, k) · D2
[U2
(i, j, k, l +
12
)]+ C23(i, j, k) · D3
[U3
(i, j, k, l +
12
)](12)
T33(i, j, k, l + 1) = C31(i, j, k) · D1
[U1
(i, j, k, l +
12
)]
+ C32(i, j, k) · D2
[U2
(i, j, k, l +
12
)]+ C33(i, j, k) · D3
[U3
(i, j, k, l +
12
)](13)
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hsieh et al.: three-dimensional phononic band gap calculations 151
T23
(i, j +
12, k +
12, l + 1
)= C44
(i, j +
12, k +
12
)·
D3
[U2
(i, j +
12, k +
12, l +
12
)]+ D2
[U3
(i, j +
12, k +
12, l +
12
)] (14)
T13
(i +
12, j, k +
12, l + 1
)= C55
(i +
12, j, k +
12
)·
D3
[U1
(i +
12, j, k +
12, l +
12
)]+ D1
[U3
(i +
12, j, k +
12, l +
12
)] (15)
T12
(i +
12, j +
12, k, l + 1
)= C66
(i +
12, j +
12, k
)·
D2
[U1
(i +
12, j +
12, k, l +
12
)]+ D1
[U2
(i +
12, j +
12, k, l +
12
)] (16)
Fig. 1. The unit of the grids in the FDTD method.
i, j = 1 ∼ 3. The positions of the staggered grids are de-fined by these discrete wave equations. Fig. 1 shows theunit of the staggered grids in the calculation of the FDTDmethod, and the relationship between the displacementgrids and the stress grids.
B. Boundary and Initial Conditions
To calculate the dispersion curves of phononic crystals,the periodic boundary condition of FDTD is developed.Because of the periodicity of phononic crystals, we appliedBloch’s theorem to fulfill the boundary condition. The pe-riodic boundary condition satisfying Bloch’s theorem ispromoted by Tanaka et al. in [19] for 2D phononic crys-tal cases. In the present article, we deal with 3D phononiccrystals based on the same principle. According to Bloch’stheorem, the displacement and stress components in theperiodic structure can be expressed in a periodic functionas follows:
Ui(x, t) = eik·xU i(x, t), (17)
Tij(x, t) = eik·xT ij(x, t), (18)
where k = (k1, k2, k3) is a wave vector, and U i and T ij areperiodic functions which satisfy the following relation:
U i(x + a, t) = U i(x, t), (19)
T ij(x + a, t) = T ij(x, t), (20)
where a is a lattice translation vector with the componentsa1, a2, and a3 of each direction.
In [19], the equation of motion (1) and constitutive law(2) are translated to solve the periodic functions U i andT ij and to apply the periodic boundary conditions (19)and (20). Alternatively, we develop the FDTD in Ui andTij , and therefore the periodic boundary conditions arederived as:
Ui(x + a, t) = eik·aUi(x, t), (21)
Tij(x + a, t) = eik·aTij(x, t). (22)
Expanding (21) and (22), we have the discrete forms ofthe surface normal to the X, Y, and Z axes:Normal to the X axis:
U1
(−1
2, j, k, l
)= eik1a1U1
(a1 − 1
2, j, k, l
)
U2
(0, j +
12, k, l
)= eik1a1U2
(a1, j +
12, k, l
)
U3
(0, j, k +
12, l
)= eik1a1U3
(a1, j, k +
12, l
)(23)
T11(0, j, k, l) = eik1a1T11 (a1, j, k, l)
T12
(−1
2, j +
12, k, l
)= eik1a1T12
(a1 − 1
2, j +
12, k, l
)
T13
(−1
2, j, k +
12, l
)= eik1a1T13
(a1 − 1
2, j, k +
12, l
).
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152 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 53, no. 1, january 2006
Normal to the Y axis:
U1
(i +
12, 0, k, l
)= eik2a2U1
(i +
12, a2, k, l
)
U2
(i,−1
2, k, l
)= eik2a2U2
(i, a2 − 1
2, k, l
)
U3
(i, 0, k +
12, l
)= eik2a2U3
(i, a2, k +
12, l
)(24)
T12
(i +
12,−1
2, k, l
)= eik2a2T12
(i +
12, a2 − 1
2, k, l
)T22(i, 0, k, l) = eik2a2T22 (i, a2, k, l)
T23
(i,−1
2, k +
12, l
)= eik2a2T23
(i, a2 − 1
2, k +
12, l
).
Normal to the Z axis:
U1
(i +
12, j, 0, l
)= eik3a3U1
(i +
12, j, a3, l
)
U2
(i, j +
12, 0, l
)= eik3a3U2
(i, j +
12, a3, l
)
U3
(i, j,−1
2, l
)= eik3a3U3
(i, j, a3 − 1
2, l
)(25)
T13
(i +
12, j,−1
2, l
)= eik3a3T13
(i +
12, j, a3 − 1
2, l
)
T23
(i, j +
12,−1
2, l
)= eik3a3T23
(i, j +
12, a3 − 1
2, l
)T33(i, j, 0, l) = eik3a3T33 (i, j, a3, l) .
After setting up the periodic boundaries, we need todefine the initial condition for the calculation. Because theperiodic structure needs to be vibrated in this method, weprovide a small disturbance in a random position of thestructure for the setting of the initial condition, and wehave the initial disturbance which is set in any grid of theunit cell
Ui (x1, x2, x3) = δx1,xδx2,yδx3,z, (26)
where δ is a delta function and (x1, x2, x3) is the randompoint which excites a wide-band signal in the unit cell atthe initial time step l = 0. Thus, all possible wave modesare transported inside the phononic crystal under consider-ation, and then the signal is expanded into a time Fourierseries; this permits the finding of the information aboutpossible types of waves.
Before the calculation is started, the specific wave num-ber in the first Brillouin zone needs to be defined for theboundary condition. During the calculation, the variationof the displacement must be recorded while time increases.After obtaining a sufficiently large number of displacementdata, we Fourier-transformed the variation of the displace-ment into the frequency space. Then the resonant frequen-cies of the periodic structure can be obtained. The reso-nant points are the existing peaks in the frequency spectra;they identified the eigenfrequencies of the phononic crystalfor a given wave vector.
Fig. 2. The hardware architecture of a PC cluster system.
III. Parallel Computing in the PC Cluster
System
The traditional supercomputer no longer satisfies thedemands of large scientific computing in that the hard-ware has met the bottleneck of design and the price of asupercomputer is too expensive to be afforded by generalresearch institutions. So the PC cluster system is becom-ing the most popular high-efficiency computation machinein scientific and technologic fields nowadays. In this study,because of the high-speed computation and low cost of thePC cluster system, the complex research of 3D phononiccrystals can be calculated and discussed by saving muchof the CPU computation time.
As Fig. 2 shows, the PC cluster consists of several PCswith different operating systems and different hardware ar-chitectures. PCs are connected with each other by externalnetwork devices, such as network cards based on Ethernet.Thus the PCs in the system can use the messages passingaround to communicate with each other. Therefore thewhole system of PCs can be regarded as a complete par-allel machine.
The parallel model of the PC cluster is SPMD (singleprogram multiple data). In the model of SPMD, the paral-lelism is controlled by the programs, not the CPU instruc-tions. The controller of the system sends the program toevery computational node involved in the calculation, andcommands every PC to do the task. In the parallel com-puting of the PC cluster system, we adopt MPICH [27],a high-performance portable implementation of the stan-dard MPI, to parallel the program. MPICH offers manykinds of parallel ways in the library so that the program-mer can easily pass the message from one of the computersto the others by simply calling the function offered fromthe library.
In this paper, we accelerate the calculation speed byparalleling the program. Among several kinds of parallel
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hsieh et al.: three-dimensional phononic band gap calculations 153
Fig. 3. Parallel model—working queue.
methods for calculation, such as dividing the calculationtime or space, here we propose the method of dividingthe calculation by wave vectors. Unlike dividing the cal-culation by time or space, which requires numbers of dataexchanging, by calculating different wave vectors aroundthe irreducible part of the first Brillouin zone, there arefewer messages passing in PC cluster system so that thecomputation can achieve higher performance. There is nomessage passing, data exchanging, and synchronization be-tween the calculations with different wave vectors. There-fore, dividing the computation by the wave vector is ex-pected to obtain higher performance in PC cluster system.
In this parallel method, as Fig. 3 shows, the workingqueue is created to store all the wave vectors of the wholecomputation in the server before the calculation. Then theclient sends the request to the server for the jobs asked forin the calculation. If the working queue in the server isnot empty, the new job will be sent to the client. Afterreceiving the reply from the server, the client can do thecalculation with the specific wave vector. Then the clientcan iterate the job requesting process until the workingqueue is empty. Therefore the whole computation is fin-ished completely.
Using this kind of parallel method, the whole computa-tion can save more calculation time by increasing numbersof CPUs. Because the numbers of CPUs in calculation areincreased, jobs dispatched to each node are also reduced.And in this method there is no message passing betweenthe clients and fewer data exchanging between the clientand the server. Thus the total waiting time is spent mostlyin the calculation and just a little in the waiting for mes-sage passing. Therefore this parallel method is very suit-able for use in the FDTD study of phononic crystals.
Fig. 4. (a) A 2D phononic crystal with square lattice arrangement;(b) The irreducible part of the first Brillouin zone.
IV. Examples of Phononic Band GAP
Calculations (2D and 3D)
In this section, the above-mentioned BAW theory of aphononic crystal using the FDTD method is employed todiscuss phononic band gaps with different crystal struc-tures. The results for 2D and 3D phononic crystals arepresented, including 2D cases with square lattice arrange-ment, and 3D cases with SC lattice, BCC lattice, and FCClattice arrangements. In all of the numerical calculationsof phononic crystals, the background and filling materi-als chosen are epoxy and steel. These materials are well-known for the study of the property of both acoustic bandgaps and total band gaps [21], [28]. Thus it is convenientto choose them for demonstrating the results of 2D and3D phononic crystals in this study. The density and theelastic constants C11 and C44 of epoxy are assumed as1180 kg/m3, 7.61 GPa, and 1.59 GPa, respectively, andthose for steel are 7780 kg/m3, 264 GPa, and 81 GPa.To proceed with the calculation of the FDTD method,the phononic crystals are created by assigning the mate-rial constants of corresponding positions according to theshape of inclusions. In this study, although the phononiccrystals are formed with isotropic background and fillingmaterials, the anisotropic lattice structures demonstratethe anisotropic property of wave propagation, as shownbelow.
A. Two-Dimensional Phononic Crystals
In this subsection, we consider a 2D steel/epoxyphononic crystal with square lattice arrangement as shownin Fig. 4(a). The unit cell of the square lattice phononiccrystal can be chosen as either the square with a full in-clusion inside or a square with four quarters of the inclu-sion. In this study, we adopt the former one for simplicity.The first Brillouin zone of the phononic crystal in squarearrangement is shown in Fig. 4(b). Owing to the symme-try of the material and lattice, the irreducible part of thefirst Brillouin zone is an isosceles right triangle. Thereforewe calculate the dispersion relations for a BAW along theboundary of the irreducible part of the first Brillouin zone.The symbols Γ, X, and M in Fig. 4(b) denote the vertexes
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154 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 53, no. 1, january 2006
Fig. 5. Dispersion relation of a circular inclusion phononic crystalwith square lattice arrangement.
of the first Brillouin zone of square-arrayed phononic crys-tals.
First, we discuss the band gap phenomena of a phononiccrystal with circular inclusion. The radii of the cylinder are9 and 15 grids while the lattice constant is 30 and 50 grids.The filling fraction of the 2D phononic crystal is definedas the ratio of the cross-section areas of the inclusion andthe unit cell. Therefore, the filling fraction f is 0.283 forthese cases. To verify the result of the FDTD method, thedispersion relation is also calculated by the PWE method.There are 1681 reciprocal lattice vectors (RLV) for thePWE method while each unit cell is divided into 30 by 30and 50 by 50 grids for the FDTD method. The dispersioncurves are shown in Fig. 5, where the lines with solid trian-gle symbols “” represent the results of the PWE methodand the lines with hollow circle “” and hollow rhombus“” symbols represent those of the FDTD method. Thedispersion relation shows the allowed elastic wave modesinside the phononic crystal. The vertical axis is the nor-malized frequency Ω = ωa/Ct and the horizontal axis isthe reduced wave vector defined as ka/π, where ω is theangular frequency, k the wave vector, a lattice constant,and Ct the transverse wave velocity of the background ma-terial.
As Fig. 5 shows, the results show good agreementbetween the PWE and the FDTD methods. The PWEmethod shows the total band gap from the normalizedfrequency 4.07 to 6.79, and the FDTD method shows theband gap from Ω = 4.08 to 6.67 and 4.05 to 6.67 for 30×30and 50×50 grids per unit cell, respectively. The dispersioncurves match each other in the lower frequency range andshow slight differences in higher frequency range. The pos-sible reason is that the presentation of the inclusion shapeis limited by the orthogonal grids system. Therefore, the
resonance frequencies of higher eigenmodes change slightlywhen the phononic crystal unit cell is defined by moregrids. Basically, the result of the FDTD method with theunit cell divided into 30 × 30 grids shows very high agree-ment with that for 50 × 50 grids, but the calculation timeis less than half that of the latter. It is also worth indicat-ing that the numbers of reciprocal lattice vectors (RLV)affect the accuracy of dispersion curves. The PWE workshows that with the larger number of RLV, the dispersioncurves shift to a lower value. With the results, the FDTDmethod is verified and the division of the unit cell is alsodiscussed for the trade-off of accuracy and efficiency.
Second, Fig. 6 shows the results of 2D phononic crystalswith a square lattice arrangement but the inclusion is asquare rod. Although the orthogonal discrete grids maylimit accuracy of the inclusion shapes, the FDTD methodis very convenient to study the phononic crystals with adifferent inclusion shape. The square inclusions are definedby designating the material constants in the correspondinggrids, and furthermore the square inclusions are orientatedin different directions, such as 0, 15, 30, and 45 degreeswith respect to the X-axis. These cases have an equal fillingfraction, but the apparent band gaps change due to theorientation of square inclusions. Fig. 6(a) shows the resultsof 2D phononic crystals whose filling fraction is 0.27. Thefilling fraction is approximately equal to, but smaller than,that in Fig. 5. The hollow rhombus “” symbols of the 0degree orientation case, for example, represent a total bandgap width of 2.58, with a normalized frequency from 3.97to 6.55. Meanwhile, Fig. 6(b) shows the dispersion curvesof the phononic crystals with filling fraction 0.36, and thusthe length of the side of square rod is 30 grids, equal to thediameter of the circle in Fig. 5. A total band gap of the0 degree orientation case appears to have a normalizedfrequency from 3.73 to 7.67 and the width of the bandgap is 3.94. The other symbols, hollow circle, triangle, andsquare, represent the cases with orientation 15, 30, and 45degrees, respectively, and the band gaps are included inFig. 6(c).
When the filling fractions are equal, we find the totalband gap widths of the phononic crystal with the squareinclusion in the square lattice arrangement approximatelyequal to that with the circular inclusion, but the intervalchanges to a higher frequency range with the increase ofthe orientation angle. In addition, if a general radius isdefined as the vertical distance from the center of the ge-ometric shape to the boundary such as the half length ofa square side and the radius of a circle, then the cases ofequal general radius show that the crystal with the squarerod inclusion exhibits a wider total band gap.
B. Three-Dimensional Phononic Crystals
By replacing the material constant of each grid, theFDTD method is appropriate to study the 3D phononiccrystals. We analyze three types of 3D phononic crystalsin this subsection, including SC lattice, BCC lattice, andFCC lattice arrangements. Due to the considerable com-
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hsieh et al.: three-dimensional phononic band gap calculations 155
(a)
(b)
(c)
Fig. 6. Dispersion relation of a square inclusion phononic crystal withdifferent orientations: (a) The case of filling fraction 0.27; (b) Thecase of filling fraction 0.36; (c) The distribution of total band gaps.
(a)
(b)
Fig. 7. A 3D simple cubic lattice phononic crystal with a sphere in-clusion: (a) Lattice arrangement and the first Brillouin zone; (b) Dis-persion relation.
putation of 3D cases, we introduce the PC cluster systemfor the numerical work. In addition, the unit cell of the 3Dphononic crystal is divided into 30 × 30 × 30 grids. It isbelieved that the accuracy is good enough from the studyof the 2D cases in the previous subsection. The time stepis chosen as 0.002a/Ct to satisfy the convergence conditionof the FDTD method in wave propagation, and the calcu-lation is with the time evolution of 100,000 time steps.
Fig. 7(a) shows the unit cell of the 3D phononic crystalwith a sphere inclusion in the SC lattice arrangement. Thefirst Brillouin zone of an SC lattice is also shown in thefigure. Due to the symmetric property of geometry andmaterial, the irreducible part is a tetrahedron with cornerslabeled Γ, X, M, and R. The radius of the sphere is 9 grids,and thus the filling fraction of this SC phononic crystalis 0.113. The dispersion relation for a BAW in the firstBrillouin zone is represented in Fig. 7(b) and a total bandgap is observed from a normalized frequency of 4.68 to4.90 with a width of 0.22.
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156 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 53, no. 1, january 2006
(a)
(b)
Fig. 8. A 3D simple cubic lattice phononic crystal with a cube inclu-sion: (a) Lattice arrangement; (b) Dispersion relation.
Next, we replace the sphere inclusion in the SC latticewith a cube inclusion, as shown in Fig. 8(a). The lengthof the cube side equals 16 grids and the filling fraction is0.152. The dispersion relation is analyzed along the irre-ducible volume again and a band gap from 4.48 to 5.38is obtained. The width of the band gap is 0.9, which iswider than that of the sphere inclusion. The band gap be-comes larger when the side length of cube inclusion equal18 grids. Therefore a cube inclusion can create a largerband gap than a sphere one for an equal general radius.
Further, the 3D phononic crystal with a BCC latticearrangement and its first Brillouin zone are shown inFig. 9(a). The tips of the irreducible part are labeled Γ, H,N, and P. The radius of the sphere is 9 grids, the same asin the SC case. Since each cubic unit cell in the BCC lat-tice arrangement contains two spheres, the filling fractionis 0.226. The dispersion curves for a BAW are calculatedalong the selected direction and shown in Fig. 9(b). A to-tal band gap appears from a normalized frequency of 5.37to 8.07 and the width is 2.7.
(a)
(b)
Fig. 9. A 3D body-centered cubic lattice phononic crystal with sphereinclusions: (a) Lattice arrangement and the first Brillouin zone;(b) Dispersion relation.
The final 3D phononic crystal case is the FCC lattice ar-rangement in Fig. 10(a). The tips of the irreducible part offirst Brillouin zone are labeled Γ, X, W, K, L, and U. Withthe same dimension of the lattice constant and sphere, eachcubic unit cell contains four spheres and the filling fractionis 0.452. There is a noticeable total band gap from 6.79 to14.60 with a width of 7.81.
Thus we find that the 3D phononic crystal with thecube inclusion has a larger band gap than that with thespherical inclusion; this is also true in the 2D case. It isapparent that in the 3D phononic crystal the band gapphenomenon is affected by the geometric shape of the in-clusion. Furthermore, the 3D phononic crystals, includingSC, BCC, and FCC lattice arrangements, show that theband gap width in the FCC case is larger than those ofthe BCC and SC cases and that the band gap width ofthe BCC case is larger than that of the SC case. We canalso inspect the phononic band gaps in terms of relativeband width ∆ω/ω0, defined as ratio of the band gap to thecentral frequency of gap. The relative band width valuesof the SC, BCC, and FCC lattice cases of spherical inclu-sions are 0.046, 0.40, and 0.73, respectively. So the bandgap enlarges as the filling fraction increases, and a high-density package arrangement has a larger band gap in the
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hsieh et al.: three-dimensional phononic band gap calculations 157
(a)
(b)
Fig. 10. A 3D face-centered cubic lattice phononic crystal with sphereinclusions: (a) Lattice arrangement and the first Brillouin zone;(b) Dispersion relation.
3D phononic crystal. The FCC lattice has a considerabletotal band gap of its BAW and thus can potentially beapplied to the design of 3D waveguides and filters.
V. Conclusions
This paper proposes a FDTD theory of the BAW in 3Dphononic crystals consisting of materials with orthotropicsymmetry. The FDTD method is very suitable for deal-ing with various periodic phononic structures and easilysatisfies the convergence condition of materials with largeelastic mismatch. With the periodic boundary conditionderived from Bloch’s theorem, the FDTD method can beused to analyze the dispersion relations of 2D and 3Dphononic crystals efficiently. We also develop and executeparallel FDTD programs on a PC cluster system to mini-mize the calculating time for the 3D structures. The paral-lel programs are realized by distributing jobs with specificwave vectors, and they have increased our work efficiencymany times.
In the discussion of phononic crystals, we generalize therelation between the band gap phenomenon and the geo-metric shape by extensive study of phononic crystals ofdifferent geometric designs. Some band gap characteris-tics of both 2D and 3D phononic crystals drawn from theaforementioned study are as follows:
1. In the 2D cases of square lattice arrangement,phononic crystals with the square inclusion arrange-ment have a larger band gap. The range of total bandgaps varies with the orientation angle of the squareinclusion while the widths do not show obvious differ-ences.
2. The geometric shapes of inclusions in 3D phononiccrystals also change the appearance of total band gaps.A cube inclusion creates a larger band gap than asphere inclusion of equal general radius.
3. The band gaps of 3D phononic crystals with a constantsphere size are calculated for SC, BCC, and FCC lat-tice arrangements. The largest band gap is obtainedin the FCC case and the next largest for the BCCcase. This can be understood by the increase of thefilling fraction. These analyses are the fundamentalsteps toward development of further applications. Forexample, the BCC lattice phononic crystal with realdimensions of the 10-mm lattice constant and the 3-mm spherical radius supports a total band gap from99.2 kHz to 149.1 kHz. This property can be of usefor an acoustic wave filter and further to accordinglydesign a 3D acoustic waveguide.
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Po-Feng Hsieh received his B.S. degree fromthe Department of Mechanical Engineering,National Chiao Tung University, and his M.S.degree from the Institute of Applied Mechan-ics, National Taiwan University, in 2002 and2004, respectively. Currently, he is a researchengineer in the Behavior Design Corpora-tion, Science-Based Industrial Park, Hsinchu,Taiwan, R.O.C. His research work is mainlyon nature language processing and machinetranslation.
Tsung-Tsong Wu received his doctorate intheoretical and applied mechanics from Cor-nell University, Ithaca, NY, in 1987. Then hejoined the National Taiwan University facultyand is now a professor in the Institute of Ap-plied Mechanics. He was awarded the distin-guished research prizes of the National Sci-ence Council (NSC) three times for six yearsfrom 1995 to 2001 and is currently a distin-guished research fellow of the NSC. He is theexecutive board director of the Taiwanese So-ciety of Nondestructive Testing and the board
director of the Quartz Industry Association of Taiwan. He is a mem-ber of the American Society of Mechanical Engineers.
Jia-Hong Sun received his B.S. degree fromthe Department of Aeronautics and Astronau-tics, National Cheng Kung University, in 1995and his M.S. degree from the Institute of Ap-plied Mechanics, National Taiwan University,in 1997. Currently, he is a Ph.D. candidatein the Institute of Applied Mechanics, Na-tional Taiwan University. His research workis mainly on phononic crystals and the calcu-lation of elastic wave propagation using theFDTD method.
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