Efficient boundary element analysis of periodic sound scatterers

Transcription

1 Boundary Element and Meshless Methods in Acoustics and Vibrations: Paper ICA Efficient boundary element analysis of periodic sound scatterers M. Karimi, P. Croaker, N. Kessissoglou 1 School of Mechanical and Manufacturing Engineering, UNSW Australia, Sydney, Australia, m.karimi@unsw.edu.au Abstract A boundary element technique is used to formulate exterior acoustic problems comprising of periodic arrangements of sound scatterers. The matrix equation formulated by the boundary element method for this acoustic scattering problem is a block Toeplitz matrix. The discrete Fourier transform is then employed in an iterative algorithm to solve the block Toeplitz system. Solving a periodic acoustic problem using the block Toeplitz system significantly reduces computational time and storage requirements. Solid cylindrical scatterers in a periodic square lattice arrangement are examined. Result for the insertion loss of the sonic crystal barrier is presented. Directivity and contour plots of the total acoustic field at selected discrete frequencies are also presented and compared with those obtained by the finite element method. Keywords: Boundary element method; Periodic scatterers; Block Toeplitz matrix; Sonic crystal noise barrier

2 Efficient boundary element analysis of periodic sound scatterers 1 Introduction Sonic crystals are periodic arrangements of acoustic scatterers embedded in a fluid medium [1]. An interesting property of these structures is the occurrence of ranges of frequencies where no propagating modes are supported by the periodic structure, known as band gaps. The physical mechanism that explains this phenomenon is the destructive Bragg interference. Vasseur et al. [2] examined the sound attenuation by a square array of parallel copper cylinders in air both theoretically and experimentally. Their results revealed that rigid hollow and solid cylinders produce similar sound transmission. The performance of a two-dimensional arrangement of rigid hollow cylinders in air was investigated by Sanchez-Perez et al. [3]. Koussa et al. [4] combined two geometries of sonic crystal scatterers aligned parallel to the ground with a rigid straight noise barrier. To further extend the barrier insertion loss, the scatterers were modelled as either rigid cylinders or resonant cavities, where the cavities were either rigid or lined with an absorbent material. The outdoor experiment on periodic arrays of scatterers showed that they can be exploited as a suitable device to reduce noise in free-field conditions. A Toeplitz system is a linear system of equations in which the coefficient matrix has constant elements along each diagonal [5]. Levinson [6] proposed an algorithm which solves n Toeplitz equations in O(n 2 ) operations. The solution of block Toeplitz matrices with Toeplitz entries was studied by Kalouptsidis et al. [7]. Gohberg et al. [8] described a unified approach to obtain the solution of a linear system of equations with a recursive structure which includes Toeplitz, Hilbert and Vandermonde matrices. A fast stable solver was derived using a modified QR algorithm for non-symmetric Toeplitz-like matrices [9]. Saad [10, 11] showed that the minimum residual method and generalized minimal residual method (GMRES) can be used to solve indefinite and non-symmetric Toeplitz systems, respectively. Czuprynski et al. [12] showed that for acoustic radiation problems with rotationally symmetric boundary surfaces, the coefficient matrix is a block circulant matrix. They also demonstrated that a distributed parallel algorithm can speed up the solution of block circulant linear systems. The authors recently showed that formulating an acoustic problem involving a unidirectional periodic structure leads to a block Toeplitz system [13]. A small segment of a structure was selected as a unit cell and the boundary element method was implemented to model the problem. In this work, sonic crystal noise barriers with periodicity in three directions are examined. Consequently, a 3-level block Toeplitz matrix is formed using the boundary element method. Due to the translational invariance of the free-space Green s function, applying the boundary element method to the multi-directional periodic system leads to the formation of a multilevel block Toeplitz matrix. The GMRES algorithm is then used to solve the multilevel block Toeplitz system. Rigid cylindrical scatterers in a square lattice arrangement are examined. Results for the insertion loss, directivity and contour plots of the sound pressure at receiver locations in the shadow zone of the sonic crystal barrier are presented and compared with those obtain using the finite element method. 2

3 2 Numerical Methodology Assuming a time harmonic dependence of the form e iωt, the Helmholtz equation is given by p(x) + k 2 f p(x) = F (1) where p(x) is the acoustic pressure at field point x, is the Laplacian operator, k f = ω/c f is the acoustic wave number, ω is the angular frequency, c f is the speed of sound and F is the source. Equation (1) can be written in a weak formulation after integrating by parts twice as follows [14] G(x,y) c(x)p(x) + n(y) p(y)dγ(y) = iωρ f G(x,y)v f (y)dγ(y) + p inc (x) (2) Γ where ρ f is the fluid density, v f (y) is the fluid particle velocity and i = 1. The vector n(y) represents the outward normal vector at the source point position y on the boundary Γ. / n(y) is the normal derivative. In equation (2), c(x) is a free-term coefficient and equals one in the domain interior and 0.5 on a smooth boundary. G(x, y) is the free-space Green s function for the Helmholtz equation given by G(x,y) = eik f r 4πr Γ where r = x y (3) p inc is the incident acoustic pressure as a result of the acoustic source. In this work, a monopole source was considered. Assuming the monopole source to be a pulsating sphere with strength Q, the incident pressure is given by [15] p inc (x) = iωρ f Q eik f r (4) 4πr The incident acoustic field radiated by the monopole sound source is applied as a load in the boundary integral equation (2). The sonic crystal barrier examined in this work is considered as a rigid structure. Hence the fluid particle velocity at the scatterer surface is zero, that is, v f (y) = 0, y Γ. The boundary element method (BEM) formulation then becomes a linear system of equations which can be expressed as follows T a = b (5) where T is the BEM coefficient matrix and a, b represent the sound pressure and incident pressure at nodal points, respectively. For an acoustic scattering problem which involves periodic structures, the matrix equation formulated by the BEM is a block Toeplitz matrix. A block Toeplitz matrix has constant blocks along each diagonal and has the form: T 0 T 1 T 1 m T 1 T 0 T 1 T 2 m T =.. T (6) T 1 T m 1 T m 2 T 1 T 0 3

4 where each T i is an n n matrix if the structure is periodic in only one direction. Inspection of equation (6) reveals that the block Toeplitz matrix T can be specified by its first block row and its first block column. If the structure has periodicity in multiple directions, T is a multilevel block Toeplitz matrix. As such, the number of levels in T corresponds to the number of periodic directions. 3 Results and Discussion The acoustic performance of a periodic array of acoustically rigid cylinders in free space is examined. The surface of the 3D model was discretised by linear discontinuous quadrilateral elements. In the numerical model, the cylinder exterior diameter is 0.2 m, the cylinder height is 1 m and the lattice constant is 0.3 m, where the lattice constant is defined as the distance between the centres of adjacent scatterers. Figure 1 shows a schematic diagram of the acoustic domain and relative position of the source and the receiver points with respect to the barriers. A monopole source is placed in front of the barrier and the receiver is located in the barrier shadow zone, as shown in Figure 1. Sound attenuation by a noise barrier is expressed in terms of insertion loss (IL) as follows [16] IL = SPL without barrier SPL with barrier (7) where SPL without barrier and SPL with barrier respectively correspond to the sound pressure levels at the same receiver position without and with the presence of the barrier. Source Receiver 0.95m 3m Figure 1: Configuration of a 4 5 square lattice sonic crystal barrier showing the point source and receiver locations Figure 2 presents the insertion loss as a function of frequency for a 4 5 square lattice sonic crystal barrier using solid cylinder scatterers under monopole source excitation. Results obtained using the periodic BEM (PBEM) technique and from a similar model of the sonic crystal barrier using the finite element method (FEM) are presented. In the periodic BEM model, a small segment of a single scatterer was chosen as a unit cell and the sonic crystal barrier is 4

5 represented by periodicity in three directions. The FEM model was developed as a reference solution using the commercial software COMSOL Multiphysics (v5.0). The acoustic domain in the FEM model was discretised using tetrahedral elements. A non-reflective boundary condition was applied on the boundary of the acoustic domain to allow the outgoing acoustic waves to leave the domain with minimal reflections. Very good agreement in the results from the periodic BEM technique and FEM can be observed. High insertion loss is achieved between 500 Hz and 700 Hz. The centre frequency predicted by Bragg s law for a lattice constant of 0.3 m is 572 Hz which is nearly midway between the lower and upper frequencies of the high insertion loss. Since the sonic crystal barrier in this study is in free space, the insertion loss results include diffraction around the top and bottom of the scatterers as well as around the edges of the square lattice array. Insertion loss (db) PBEM FEM Frequency (Hz) Figure 2: Insertion loss obtained using the 3D periodic BEM (PBEM) technique and from a 3D FEM model for a 4 5 square lattice sonic crystal barrier comprising of solid cylinder scatterers The directivity of the sound pressure in the barrier shadow zone at two distinct frequencies of 500 Hz and 850 Hz is presented in Figure 3. The direct incident pressure denoted by Pinc is also shown. The source was located in front of the barrier at 270. The periodic BEM technique was used to calculate the scattered sound pressure at data recovery points located on a circle with radius of 3 m from the centre of the sonic crystal barrier in the barrier shadow zone. The directivity of the sound pressure field predicted by the periodic BEM technique is in very good agreement with results obtained from FEM. At 500 Hz, a low pressure region is generated in the barrier shadow zone due to destructive interference between scattered waves within the periodic arrangement of the sound scatterers. In contrast, at 850 Hz, constructive interference between scattered waves amplifies the sound pressure in the barrier shadow zone at 90. 5

6 150 Frequency=800 Hz Frequency=850 Hz PBEM FEM Pinc (a) 500 Hz PBEM FEM Pinc (b) 850 Hz Figure 3: Directivity of the total sound pressure (in Pascals) for the 4 5 square lattice sonic crystal barrier Figure 4 presents contour plots of the sound pressure level for the sonic crystal barrier at 500 and 850 Hz, obtained from both the 3D FEM model and using the periodic BEM technique by computing the sound pressure on a rectangular data recovery mesh consisting of almost 83,000 points located on a horizontal plane at the bottom of cylinders. Very close agreement in the contour plots of the sound pressure level predicted using FEM and the periodic BEM technique is observed. At 500 Hz in Figure 4(a), the sound waves within the sonic crystal barrier destructively interfere resulting in significant attenuation in the barrier shadow zone. 4 Summary A periodic array of sound scatterers was modelled by exploiting the block Toeplitz structure in the boundary element formulation. The block Toeplitz system for this acoustic scattering problem was solved using the GMRES algorithm. Periodicity of the structures in three directions was taken into account. Results for the insertion loss, directivity and contour plots of the sound pressure at receiver locations in the shadow zone of the sonic crystal barrier were presented. Very good agreement was observed from the comparison between the results obtained by the periodic BEM and those obtain using the finite element method, showing that in addition to significant reduction in computational time and storage requirements, the current approach produces accurate results. 6

7 (a) 500 Hz (b) 850 Hz Figure 4: Sound pressure level (db) for the 4 5 square lattice sonic crystal barrier using FEM (left) and periodic BEM (right) at (a) 500 Hz and (b) 850 Hz 7

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