Calculation of sound radiation in infinite domain using a meshless method

PROCEEDINGS of the 22 nd International Congress on Acoustics Structural Acoustics and Vibration: Paper ICA2016-41 Calculation of sound radiation in infinite domain using a meshless method Shaowei Wu (a), Yang Xiang (b) (a) School of Energy and Power Engineering,Wuhan University of Technology,China,thinkwsw@qq.com (b) School of Energy and Power Engineering,Wuhan University of Technology,China,yxiang@whut.edu.cn Abstract A meshless method coupling with a variable order infinite acoustic wave envelope element for sound radiation calculation in infinite domain is presented with the aim of accurately calculating the sound radiation and improving computational efficiency. It is based on using the elementfree Galerkin method in the inner region enclosing the radiator and a variable order infinite acoustic wave envelope element in the outer region for the proper modeling of the pressure amplitude decay to satisfy the Somerfield condition explicitly. The details are provided for the derivation and implementation of this method. The factors of influencing the performance of the method, which include the shape function constructing, the weight functions, and the support domain, are discussed. The suitable radius of the influence domain for the acoustic field calculation in free space is also determined by use of numerical experiments. An infinitely long cylinder is designed for simulation to validate the method. The results illustrate the accuracy, applicability and effectiveness of this method. Keywords: meshless method; acoustic wave envelope element; infinite domain;

1 Introduction For sound pressure prediction in infinite domain within low-frequency range, the boundary element method (BEM) is established as a well-known numerical tool. However, nonuniqueness problem occurs at certain critical frequencies and that various orders of singular integrals of Green s function have to be numerically described when a field point is on the structure surface. Different methods to overcome these problems have been suggested [1-3]. From a practical point of view, a disadvantage is that full and complex system matrices are yielded which can lead to inefficient in computation. With this complexity in mind,koopmann et al. [4] proposed wave superposition method based on the Kirchhoff Helmholtz integral equation, in which the singular integrals of Green s function are eliminated by placing the equivalent sources inside the radiator. However, the calculation accuracy depends heavily on the positions of the equivalent sources [5-9]. Astley et al. [10-13] developed a variable order infinite wave envelope element (WEE). In this method, the shape function consisting of a reciprocal decaying eave-like variation, which has the asymptotic behavior and satisfies the Somerfield condition explicitly, is established by a finite to infinite geometry mapping. The nonuniqueness problems occurring at critical frequencies are removed in the infinite wave envelope scheme. Furthermore, an advantage is that the calculation speed is improved in comparison with BEM, due to sparsity of the system matrices. To ensure uniqueness of the infinite geometry mapping, the radial infinite edges must be divergent. As a result, the infinite wave envelope element should be combined with conventional finite element to calculate sound pressure for radiator of complex geometric shape. Meshfree methods are relatively new in the field of numerical calculation compared with FEM. Meshfree methos have some advantages over the classical mesh-based methods [14-16]. Mesh for constructing shape functions is no need in these methods. Shape functions are calculated based on arbitrary field point distribution in domain. Consequently, it is possible to obtain high order shape functions of arbitrary continuity and add some specific terms into basis function. The Meshfree methods have been extensively used in the field of sound pressure prediction for bounded domains and given accurate results for interior acoustic problems [17-20]. The element free Galerkin method originally developed by Belyschko et al. [21] is one of meshfree methods. It is based on the moving least square approximation (MLS).To improve the accuracy for the prediction of infinite acoustical fields, a new coupling method, which is based on using meshfree in inner region enclosing the radiator and a variable order infinite acoustic wave envelope element (WEE) in the outer region for the proper modeling of the pressure amplitude decay to satisfy the Somerfield condition explicitly, is presented. 2 Theoretical background First, a discrete trial function p h (x) is chosen for the acoustic pressure amplitude p. This takes the general form M h p ( x, ) N ( x, ) p (1) j 1 j j 2

where N j (x,ω) (j=1,,n) are known shape functions. Application of the weighted residual formulation to the Helmholtz equation, the boundary conditions of prescribed velocity profile and the Sommerfeld radiation yields [10-12] 2 K k M P F (2) where, and ( ). It should be noted that S b represents the boundary surface of the radiator and v(x) is vibration velocity on the surface S b. To predict the sound pressure, V is subdivided into an inner region V inner and outer unbounded subregion V outer and they are separated by an interface Γ. Element Free Galerkin method (EFG) in the near-field region and a variable order infinite acoustic wave envelope element in the outer region are used to discrete the domains, respectively. 3 Application of the coupling method to sound pressure prediction 3.1 Inner region 3.1.1 Moving least square approximation In case of meshfree methods, a series of discrete points is need for representation of the considered domain Ω compared with mesh-based methods. These points can be arbitrarily distributed in the domain and its boundary. The shape functions of EFG are determined by the moving least square approximation (MLSA) which is now used widely for constructing meshless shape functions. The MLSA is defined on a set of n nodes. Consider the sound pressure p(x) in V inner. The MLS approximation of sound pressure p(x) is defined at x as m h T p ( x, x) p ( x) a ( x) p ( x) a( x ) (3) i 1 i where represents coordinates of these points within the neighborhood domain Ω x of the calculated point x, m is the number of terms in the basis ( ). In general, a complete polynomial basis of order M is taken the form of ( ) for twodimensional case. It should be noted that the coefficients a(x) are function of x. The coefficients are determined by minimizing the following weighted discrete L 2 norm: n m 2 T I i I i I I 1 i 1 i J w ( x)[ p ( x ) a ( x) ˆ p ] ( Pa pˆ ) W( x )( Pa p ˆ ) (4) with T ˆ ( ˆ p, ˆ p,, ˆ p ) 1 2 N p (5) T P [ p( x ), p( x ),, p( x )] (6) 1 2 n 3

W( x) diag( w( x x ), w( x x ),, w( x x )) (7) 1 2 where n is the number of nodes in the support domain Ω x of x for which w I (x) 0, is the nodal parameter of p at x=x I and w I (x) is a weighting function of representing the influence of the node x I at a given point x (as shown in Figure 1 (a)), which is equal to unity at the node, decreases with the distance to the node increasing and is equal to zero outside the domain of the node influence. The weighting function w I (x) are often chosen as the cubic spline, the quartic spline or or the exponential function [22]. n J/ a=0 yields Figure 1 (a) Influence domains of field pint x I (b) Background cell for integral Substituting Eq. (8) into Eq. (3) yields 1 T T ˆ a( x) ( P w( x) P) P w( x) p (8) 1 T T T ˆ ˆ h p ( x) p ( x)( P w( x) P) P w( x) p N( x) p (9) It has to be noted that the shape function N I (x J ) δ IJ, namely, p h (x I ). p h (x I ) should be determined by Eq. (3) after being obtained. In order to calculate the acoustic stiffness and mass matrices in inner region, background cells, which practically don t exist in the model and merely assist in locating the integration points for integral are need, as shown in figure 1 (b). Numerical forms of K ij and M ij are given as N N J J J (10) nc ns nv nc ns nv lhk i i 1 T 1 j j T (, )[ ] [ ](, ) ij h k ij h k l l l l 1 h 1 k 1 l 1 h 1 k 1 s v s v K w w K w w N N nc ns nv nc ns nv lhk ij h k ij h k i j l l 1 h 1 k 1 l 1 h 1 k 1 M w w K w w N N J (11) 4

where n c is number of background cells, n s and n v the number of Gauss integral points in each background cell, w represent the weighting factors and J l represents the Jacobian matrix. For Gauss integral point in each background cell, and are equal to zero if node i and node j both don t fall in the Ω x of the Gauss integral point. Numerical forms of F i is given as nct ns 2 2 x (12) i i Qh l 1 h 1 v 1 F ik c N v ( ) ( x / s ) ( y / s ) where x Qh represents the Gauss integral point. 3.1.2 Influence Domain A circular domain is characterized by influence radius r w and rectangular domain is represented by r wx and r wy in x and y directions, respectively, for each node x I. For simplicity, r wx = r wy is often used. For two-dimensional case, ( ). α s is dimensionless size and ( ) is average nodal spacing in the vicinity of this point with A s being the area of the estimated support domain and n As the number of nodes covered by the estimated domain with the area of A s, α s has great influence on accuracy of numerical result and should be determined by numerical test for sound pressure prediction. 3.2 Outer region To imitate the acoustic radiation characteristics in free field space, variable order infinite wave envelope element (WEE) is used to satisfy the Sommerfeld radiation condition. The element maps from square of side 2 in the parent t-s space. Figure 2 shows both the mapped and parent elements. The infinite wave envelope element takes the form of elongated quadrilaterals or blocks that extend to infinity in the radial direction. Only two-dimensional linear elements are considered in the current analysis. The construction of The shape function and the choice of the weighting function are presented in detail in reference [10-13]. Figure 2 2-D geometry mapping 5

4 Numerical investigations In what follows, acoustic radiations from an infinitely long cylinder of radius R=1 with analytical solutions listed in references [10-12] are investigated. The domain of 1 r<2 is defined as inner region. The acoustic medium is air with speed of sound c=343 m/s and density ρ=1.21 kg/m 3. In this paper, the velocity amplitude is taken as 0.001 m/s.the calculation frequency satisfies kr=π in order to give an acoustical wavelength equal to the diameter of the cylinder. As indicated in [12], first order WEE can be used to model correctly the radial behavior of monopole, second order WEE for dipole and third order WEE to model qaudrupole. 4.1 Effect of influence size In this paper, ε 1 and ε 2 represent the relative errors of the amplitude and phase of the sound pressure, respectively. The maxima of the relative errors for sound pressure on nodes of the boundary surface versus α s with α s =0.1 are plotted in figures 3-5 for different weighting functions, using 420 nodes combined with third order WEE. Figure 3 Relative errors for monopole case Figure 4 Relative errors for dipole case 6

Figure 5 Relative errors for quadrupole case These results depicted in Figure 3-5 show that the size of domains of influence has a large influence on the prediction error for sound pressure. The prediction accuracy becomes unaccepted for small values of α s <2. The reason is that the number of nodes used to constructing shape function is not enough, which leads to the smoothness of shape function decreasing. At large values of α s 4, the shape function is too smooth to represent local properties of field function for sound pressure, which leads to the prediction error for sound pressure increasing. The choice of 2 α s 3 can assure good accuracy for sound pressure prediction, which is consistent with the findings for EFG in reference [22]. 4.2 Convergence analysis In this section, convergence of the method is numerically studied using uniformly distributed nodes with linear basis and α s =2.4. Figure 6 gives average errors at k=π for different weighting functions. These curves show that the prediction accuracy for the sound pressure is improved as d being smaller. Figure 6 Convergence with respect to d (a) cubic spline (b) quartic spline (c) exponential 7

4.3 Comparisons between MFree+WEE and FEM+WEE Both approaches are compared on qaudrupole case with quartic spline as weighting function. For each approach, the same configuration is used, as shown in figure 7. Figure 7 (a) MFree+WEE (b) FEM+WEE The prediction accuracies are shown in figures 8. Table 1 gives the values of the errors for each coupling approach. In contrast with the FEM coupled with WEE, the devised method converges faster and can obtain higher prediction accuracy except the errors of the pressure amplitude in the cases of 80 and 120 nodes. Figure 8 Maximum error for MFree+WEE and FEM+WEE approach Table 1 Relative error for MFree+WEE and FEM+WEE NDof Maximum of ε 1 Maximum of ε 2 NDof Maximum of ε 1 Maximum of ε 2 MFree FEM MFree FEM MFree FEM MFree FEM 80 11.94 4.06 58.36 66.31 280 1.25 3.30 9.14 10.64 120 6.08 5.74 29.48 38.74 352 0.76 2.80 4.19 7.9 180 3.33 4.95 14.53 21.04 450 0.20 2.23 1.20 5.71 204 2.11 4.23 15.72 16.62 8

5 Conclusion Details of the meshfree method coupled with mapped wave envelope element for sound radiation in infinite domain have been presented. Numerical results indicate that the choice of dimensionless size 2 α s 3 can assure good accuracy for sound pressure prediction. This devised method has good convergence. It also shows that the devised method has better convergence rate and the prediction error for sound pressure can be reduced significantly in comparison with the classical FEM coupled with WEE. Acknowledgments This work was supported by the National Natural Science Foundation of China ( Research on structure sound radiation matrix deriving, computing and optimizing Grant No.51079118, Research on computational auditory scene analysis based internal combustion engine noise sources identification Grant No.51279148). References [1] Visser, R. A boundary element approach to acoustic radiation and source identification, Ph.D. thesis, University of Twente, Enschede, The Netherlands, 2004. [2] Davey, K.; and Farooq, A. Evaluation of free terms in hypersingular boundary integral equations, Engineering Analysis with Boundary Elements, vol 35 (9), 2011, pp 1060-1074. [3] D Amico, R.; Neher, J.; Wender, B.; Pierini, M. On the improvement of the solution accuracy for exterior acoustic problems with BEM and FMBEM, Engineering Analysis with Boundary Elements, vol 36 (7), 2012, pp 1104-1115. [4] Koopmann, G.H.;Song, L.; Fahnline, J. B. A method for computing acoustic fields based on the principle of wave superposition, Journal of the Acoustical Society of America, vol 86 (6), 1989, pp 2433-2438. [5] Fahnline, J. B.; Koopmann, G. H. A numerical solution for the general radiation problem based on the combined methods of superposition and singular-value decomposition, Journal of the Acoustical Society of America, vol 90 (5), 1991, pp 2808-1819. [6] Hwang, Jyh-Y.; Chang, S. C. A retracted boundary integral equation for exterior acoustic problem with unique solution for all wave numbers, Journal of the Acoustical Society of America, vol 90 (2), 1990, pp 1167-1180. [7] Song, L.;Koopmann, G. H.;Fahnline, J. B. Numerical errors associated with the method of superposition for computing acoustic fields, Journal of the Acoustical Society of America, vol 89 (6), 1991, pp 2625-2633. [8] Gounot, Y. J. R.; Musafir, R. E. On appropriate equivalent monopole sets for rigid body scattering problems, Journal of the Acoustical Society of America, vol 122 (6), 2007, pp 3195-3205. 9

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