Analysis of thin acoustic barriers over an

Transcription

1 Analysis of thin acoustic barriers over an impedance plane using a dual BEM formulation L.A. de Lacerda*\ L.C. Wrobel*, WJ. Mansur^ "Wessex Institute of Technology, Ashurst, Southampton, UK ^COPPE/URRJ, Programa de Engenharia Civil, Cx.P , Rio de Janeiro, Brazil ^Brunei University, Department ofmechanical Engineering, Uxbridge, Middlesex UBS 3PH, UK Abstract In this paper the Dual Boundary Element method is formulated and applied to study outdoor sound propagation where acoustic barriers are modelled as nonthickness bodies which may have different impedance conditions at each side. Standard and hypersingular integral equations are applied on each side of the barrier. The infinite impedance plane is incorporated into the Green's function in both equations avoiding the discretization of the ground. A numerical result is presented validating the formulation and the use of the non-thickness barrier approximation. 1 Introduction The conventional boundary integral equation method has been used to study outdoor sound propagation in a neutral quiescent atmosphere and its abatement using acoustic barriers for more than 20 years. Seznec [l]firstused this formulation to study acoustic barriers over a plane rigid ground. Discretization of the ground was avoided employing the method of images. For the analysis of thin barriers his results were compared with those from geometrical acoustic approaches; good agreement was achieved, and it was concluded that the thickness of the barrier has no effect at all on the excess attenuation, providing it stays bellow one wavelength. If the plane ground is not rigid the method of images is no longer valid and a Green's function that incorporates the plane absorbing ground is necessary for the efficient application of the boundary element method. This function was shown to be the solution for a rigid ground with a correction term which is basically an infinite integral with an oscillatory integrand. Many different authors studied the evaluation of this integral for the problem of a point source over a plane impedance boundary and suggested similar asymptotic solutions for the far field [2, 3, 4].

2 44 Boundary Elements Chandler-Wilde and Hothersall [5] developed a similar asymptotic expression for the two-dimensional case using a modified steepest descent method, based on the work of Kawai et al. [4] and Brekhovskikh [6], and proposed the use of Gaussian quadrature for the evaluation of the correction term at the near field. Later on, using the boundary element method, they studied the performance of different barrier profiles over a homogeneous impedance ground [7]. Based on the observation of Seznec [l] on the influence of the barrier thickness (also confirmed in [12]), an attempt is made in this paper to consider thin barriers as infinitely thin boundaries. This assumption makes the modelling easier, reduces computer time and memory requirements and avoids near-singular integration in some cases. The conventional boundary integral equation previously used to study acoustic barriers [1, 5, 8] cannot be used alone in non-thickness boundaries because it produces a degenerate system of equations [10]. Therefore, a hypersingular boundary integral equation is also used and the second derivative of the Green's funtion incorporating the plane impedance boundary is obtained. 2 Mathematical Formulation and Green's Functions A sketch of the problem of outdoor sound propagation in a homogeneous quiescent atmosphere and its attenuation using thin barriers is shown in figure 1. Figure 1: Two-dimensional model of outdoor sound propagation The plane ground is locally reacting and has a homogeneous normalized surface admittance j3. The admittance of the barrier surfaces is also locally reacting and may be different on each side depending on the treatment applied. Linear sound propagation in this medium is described by the Helmholtz equation (er * time-dependence) VV (x) + k^cf) (x) = h (x) (1) where 0 is the velocity potential, k is the wave number, h(x.) is the known source in the domain H and V^ the Laplacian operator. The boundary conditions are represented by

3 Boundary Elements 45 for x on the ground Fg or in the barrier F&, n the unit outward normal vector at x and /3 the admittance (inverse of the specific acoustic impedance) which may be J3 = 0 for a rigid surface or Re(/3) > 0 for an absorbing one. The velocity potential must also satisfy the Sommerfeld radiation condition at infinity. In order to avoid the discretization of the flat ground, it is possible to use a Green's function G which satisfies equations (1) with /i(x) replaced by a Dirac delta function, (2) and the radiation condition. Generally /? ^ 0 and the Green's function of interest is the sum of GO(, x) (Green's function for a rigid ground obtained by using the method of images) and a correction term G^((,x), (2) G(,x) = Go(,x) + G0($,x) (3) The development of an expression for G^(,x) suitable for numerical evaluation has been an object of study by many authors [2, 3, 4, 5] and the one used here follows the work of Chandler- Wilde and Hothersall [5] which is a two-dimensional version of what has been proposed by Kawai [4] in a three-dimensional field. By substituting (3) into (2) for x on the ground and applying direct and inverse Fourier transformation, as well as changing the variable of integration, the following integral representation is obtained [9] with Re(\/l g%), Im(%/l g%) > 0. The numerical evaluation of this integral is efficiently performed by using a modified steepest descent method, but nevertheless, it is still very expensive in computational terms. For application of the boundary element method, it is also necessary to evaluate the normal derivative of the correction term with n\ and n^ the components of the unit normal vector n at x. The second term on the right side of equation (5) can be expressed as a function of G^((, x) and a zero-order Hankel function of thefirstkind, while thefirstcan be analyzed like G0(,x), by using the same method. Once the Green's function correction term G@(, x) and its normal derivative c%20((, x)/<9n(x) are established, the boundary element method can be employed to study sound attenuation using acoustic barriers very efficiently [5, 7, 8]. From previous analysis it could be seen that the thickness of the barrier does not affect significantly the sound pressure field behind the barrier, particularly at points far from it. Based on this observation, an attempt of considering the barrier as a non-thickness structure is made. This assumption certainly facilitates the mesh generation of the problem (especially in 3-D cases) as well as saves computer time and memory. In this work the dual boundary element method [13] is applied over the non-thickness barrier, which means that a hyper singular integral equation is

4 46 Boundary Elements employed and an extra derivation of the Green's function and its derivative has to be done, with respect to the normal direction m at the collocation point. In the hypersingular formulation the following functions appear as will be seen in next Section and _^_ f\t, -/ vui{$) uiii{$) a /^ 2z: -{- ^ The obtention of expressions for the first term on the right side of equations (6), (7) and their numerical evaluation is straightforward. The last term in these equations can be expanded to and with mi and 7712 the direction cosines of the unit normal m( ). The first and second derivatives of G^((,x) on the right side of equations (8) and (9) were developed in [14] and they are all expressed in terms of Gp(, x), its ^-derivative and Hankel functions of thefirstkind. 3 Dual BEM formulation for the non-thickness barrier Applying Green's second identity to the boundary value problem stated in the previous Section, the following integral equation is obtained for collocation points inside the domain SI (10) where 0/( ) is the known incident wave contribution coming from monofrequency concentrated sources. Taking the collocation point ( to the boundary F via a limiting process and recalling that the Green's function G((, x) was chosen such that the term between brackets is equal to zero at Pg, a standard boundary integral equation (SBIE) is found to be :

5 Boundary Elements 47 (n) where the Green's function and its derivative have only a weak singularity. The integral equation (11) has been employed by Chandler- Wilde and Hothersall [5] and more recently by Park and Eversman [8] for the study of acoustic barriers over impedance planes. In order to use the BEM, the contour F& is discretized into Nb elements over which the discretized form of equation (11) is applied to generate a system of Nb equations and Nb unknowns (assuming that simple constant elements are used). Consider now a barrier with its thickness tending to zero as shown in figure 3. The application of the integral equation (11) over the boundaries F^~ and F^, 0 ground Figure 2: Thin barrier geometry. which comprise the barrier vertical face and one-half of the top horizontal face, generates the equations : J dn+(x) 0n+(x and

6 48 Boundary Elements Equations (12) and (13) can be applied to solve problems with thin barriers but fine discretizations (comparable to the thickness of the structure) must always be used, even for low frequencies, to avoid the near-degenerate system of equations as well as the near-singular integrations. The first problem arises when the thickness is very small as the equation produced for + is nearly the same as the one for ~ at the same height in the barrier. The second is evident when the element to be integrated is at one side and the collocation point is just opposite to it at the other. Depending on the problem to be analyzed this refinement may be very costly in terms of computer time and memory, especially in 3-D cases. In the limit when e > 0, F^ and F^ occupy the same position F&, n+(x) = n(x), n~(x) = n(x) and the boundary integral equation for a collocation point on any side of the barrier will read [10] 9G(S,i where $+(x) = 00+(x)/0n(x) = -<?+(x) and <T(x) = 0</r(x)/0n(x) = g~(x). Equation (14) alone cannot be used to solve a non-thickness problem because it forms a degenerate system of equations. This means that the integral equation is the same for both sides and can be rewritten using the notation :, x where I#(f) = <^(f) + <T (f), A#x) = 9^N - ^ W and 0~(x). Applying equation (15) over N& elements produces twice more unknowns than equations. In order to overcome this difference a hypersingular boundary integral equation (HBIE) is also applied along the barrier. The HBIE for non-thickness problems is obtained by differentiating equation (10) with respect to m( ), applying the limit when e» 0 andfinallytaking to the boundary via a limiting process, thus - f where E%) = q*~(x) 4-?"(x) and the sign in the second integral indicates its interpretation as a Hadamardfinitepart integral. It has been shown in (14) that this new integral equation, like the SBIE, behaves in such a way that the integration over the plane impedance ground Fg does not need to be evaluated. The dual boundary element method basically consists on the application of equations (15) and (16), one on each side of the non-thickness barrier, in order to generate a well-conditioned system of equations.

7 Boundary Elements 49 After assembling the system of equations and applying the boundary conditions g^(x) = -2k/)+(x)(^(x) (17) ^(x) = 2&/T(x)(Mx) (18) the system can be solved and values of potential before and after the barrier obtained. The variables /3+(x) and /3~(x) are the surface admittances at the front and back of the barrier, respectively. In this paper, they are calculated according to the Delany-Bazley model [11] where each surface consists of a hardbacked layer of finite thickness. The correction terms depend on the value of the ground normalized admittance J3 which is also calculated with this model considering the ground as an infinitely thick porous layer. 4 Numerical aspects In two-dimensional problems like this, computation of the matrix of coefficients of the final system of equations is usually more time consuming than the solution of the system. In this computation the evaluation of the correction terms G0(,x) and dgp(, x)/<9x(x), which has to be performed for every integration point, is very expensive. In a thick barrier, each integration point belongs to one single element and G^(f,x) and its ^-derivative have to be calculated at each of them. On the other hand, in a non-thickness barrier, each integration point geometrically belongs to two elements (one on each side), so the evaluation of the same terms in each point is performed once for every two elements reducing the required computational time for this task in almost 50%. In problems where the barrier is rigid at both sides, the dual BE method has an extra advantage in that the system of equations to be solved is half the size of the one obtained from the conventional method. The hypersingular equation (16) can initially be solved to obtain A0(x) and afterwards, the values of 0(x) at each side of the barrier are directly computed with the standard equation (15). The computation of potential values atfieldpoints is carried out with equation (11) where the coefficient of 0( ) is now 1 instead of 1/2 and the integration is carried over both sides of IV For the cases where the barrier has zero admittance at both sides this computation is faster because, for the same reason above, it can be made in terms of the jump A0(x). In problems where higher frequencies are involved the discretization meshes must be finer. As a result, all the advantages mentioned above become more evident, confirming the efficiency of the proposed dual method for this type of analysis. 5 Numerical test A comparison of results from the proposed dual formulation (6 = 0.0m) with the conventional method for two different barrier thicknesses (8 = 0.05m and m) is shown where eight constant elements per wavelength were used in

8 50 Boundary Elements all analyses. Consider a plane vertical barrier with height h 5.0m located over a plane boundary. A unit monofrequency source is located 10.0m from the barrier and 1.0m above the ground which has a normalized admittance characterized by a flow resistivity a = Nsm~~* (grassland) and the frequency of the source. Values of the velocity potential modulus at the vertical barrier are computed when the source has a frequency of looffz and the barrier is considered rigid on both sides. Figure 3 shows excellent agreement between the SBIE and the proposed dual formulation T boundary elements Figure 3: Velocity potential modulus at a rigid vertical barrier for a frequency of loohz. 6 Conclusions This paper has presented a dual boundary element formulation for studying sound propagation around acoustic barriers over a homogeneous impedance plane. The dual BE formulation involves the application of two integral equations (the standard Helmholtz integral representation and its hypersingular counterpart) over a non-thickness barrier. The Green's function employed directly incorporates the plane absorbing ground, so the only necessary discretization is that over the barrier axis. Comparison with results obtained with barriers of thickness 5cm and 20cm confirmed convergence of the numerical results to the non-thickness case, attesting the correctness of the theoretical formulation and its related numerical algorithms. It is concluded that the dual BE formulation is very effective and accurate for the analysis of thin barriers. The present research is now being extended to

9 Boundary Elements 51 three-dimensional problems, in which case substantial savings in computer time and memory should be expected. 7 Acknowledgements We thank Dr. H. Power of the Wessex Institute of Technology for many helpful discussions. The first author would like to acknowledge the financial support provided by CAPES, Ministry of Education, Brazil. References [1] R. SEZNEC 1980 Journal of Sound and Vibration 73, Diffraction of sound around barriers: use of the boundary elements technique. [2] C. F. CHIEN and W. W. SOROKA 1975 Journal of Sound and Vibration 43, Sound propagation along an impedance plane. [3] S. I. THOMASSON 1976 Journal of the Acoustical Society of America 59, Reflection of waves from a point source by an impedance boundary. [4] T. KAWAI, T. HIDAKA and T. NAKAJIMA 1982 Journal of Sound and Vibration 83, Sound propagation above an impedance boundary. [5] S. N. CHANDLER-WILDE and D. C. HOTHERSALL 1985 Journal of Sound and Vibration 98, Sound propagation above an inhomogeneous impedance plane. [6] L. M. BREKHOVSKIKH 1960 Waves in Layered Media. New York: Academic Press. [7] D. C. HOTHERSALL, S. N. CHANDLER-WILDE and N. M. HAJMIRZAE 1991 Journal of Sound and Vibration 146, The efficiency of single noise barriers. [8] J. M. PARK and W. EVERSMAN 1994 Journal of Sound and Vibration 175, A boundary element method for propagation over absorbing boundaries. [9] S. N. CHANDLER-WILDE and D. C. HOTHERSALL 1995 Journal of Sound and Vibration 180, Efficient calculation of the Green function for acoustic propagation above a homogeneous impedance plane. [10] G. KRISHNASAMY, F. J. RIZZO and Y. LIU 1994 International Journal for Numerical Methods in Engineering 37, Boundary integral equations for thin bodies. [11] M. E. DELANY and E. N. BAZLEY 1970 Applied Acoustics 3, Acoustical properties of fibrous absorbent materials. [12] D. C. HOTHERSALL, D. H. CROMBIE and S. N. CHANDLER-WILDE 1991 Applied Acoustics 32, The performance of T-profile and associated noise barriers. [13] A. PORTELA, M. H. ALIABADI and D. P. ROOKE 1992 International Journal for Numerical Methods in Engineering 33, The dual boundary element method : effective implementation for crack problems. [14] L. A. DE LACERDA, L. C. WROBEL and W. J. MANSUR Dual boundary element formulation for sound propagation around barriers over an impedance plane, submitted to Journal of Sound and Vibration.