Field reconstruction by acoustic holography technique based on BEM Y.A. He, Z.Y. He Acoustic Engineering Institute, Harbin Engineering University, Harbin, Heilongjiang 150001, China Abstract This paper presents a technique of surface field reconstruction for nonseparable geometry based on the boundary element method and the singular value decomposition. The sound sources are axisymmetric bodies with arbitrary boundary conditions. The hologram data are got from a cylindrical surface that encircles the source. By the use of axisymmetric property of the body, the two-dimensional integral formulation is reduced to one-dimensional integral along the generator of the body, as the pressure and normal velocity on the surface is expanded in a Fourier series with regard to the angle of revolution. The Fourier coefficients can be evaluated by using elliptic integrals and Gaussian quadrature formula. An improved SVD method is given to get the generalized inverse of the complex matrix. The acoustical quantity on the source surface can be calculated by inverse transformation. The method is tested on numerically generated data. 1 Introduction The inverse problems in some different fields of modern physics are importance to science and engineering. As one of inverse problem of acoustic, the near field acoustic holography (NAH/ or the spatial transformation of sound fields (STSF) is a method for the research of sound radiation. It provides the capability of projecting an image in a three-dimensional space, based on the sound data on a two-dimensional hologram surface, by use of FFT algorithm. It can be applied to location of the position of sound noise sources, identification of the sound propagation path, determination of the major noise sources,
34 Boundary Elements and investigation of the vibrating characteristics on the surface of transducers of on sonar arrays, et. ' Since the FFT-based algorithm has been limited to separable coordinate systems with some simple geometries. Veronesi and Maynard extended the NAH to non-separable coordinate systems by applying boundary element method/ In the past, some papers dealt mainly with conformal surfacefieldtransformation and the singular value decomposition (SVD) is used to process the generalized inverse of a complex matrix with ill-posed nature. ' ' The SVD is extremely time consuming, especially at high frequencies. The reconstruction technique of non-conformal surfacefieldbased on BEM and SVD is developed in this study. The sources are axisymmetric bodies with arbitrary boundary conditions. The hologram is a cylindrical surface that encircles the sound source surface. It is more convenient for pressure measuring on the cylindrical hologram than on a closed hologram. The pressure and normal velocity on the source surface are related to complex pressure on hologram by Helmholtz integral equation. By expanding the velocity potential function on the boundary into a Fourier series with regard to the angle of revolution, the angular dependence of the boundary condition is eliminated. The Fourier coefficients can be evaluated by elliptic integrals and a Gaussian quadrature formula where the singularity has been removed. The integration along the generator is carried out by dividing the generator into segments using quadratic isoparametric line elements. Fewer elements are required for the field reconstruction when compared to triangular elements or quadrilateral elements as described in ref 7. Then the pressure and normal velocity on the source surface can be calculated by the inverse transformation based on an improved SVD. One of the advantage of this technique is that it don't required large matrices, and can handle the any arbitrary boundary conditions for axisymmetric problems. The CHIEF interior points are used to insure uniqueness of the solution at the characteristic frequencies. The field reconstruction from cylindrical hologram to axisymmetric source surface are studied. As test case, an oscillating sphere with radial velocity is used to numerical simulate. The error curves show that the relation between the reconstruction precision and the parameters of geometry. 2 Fundamental relations 2.1 The Helmholtz integral formulation The classical Helmholtz integral formula for radiating problem is ^ = f f«fi) CD & \ This formula is valid in an infinite homogeneous medium outside a closed body B with a surface S. In the medium, the scalar velocity potential <j> satisfies
Boundary Elements 35 V^ 4- k* = 0 for the time harmonic waves. Q is a point on the surfaces, and P is point inside, on the surface of, or outside the body B. The G(P,Q) = Qxp^-ikR(P,Q)]/ R(P,Q) is the Green function in free space; R(P,Q) = \P-Q\ is the distance between P and <g; k = ( >/c is the wave number; / is the imaginary unit and n is the unit normal to the surface S at the point Q directed away from the body. The C(P) has the value 0 for P inside B and 47i for P outside B. In the case of P on the surface S, it is 2 71 for a smooth surface, and it equals the solid angle measured from the medium when P is on an edge or a corner. Here, the value of C(P) is given by C(P) = 4n + ds(q) (2) Equation (1) and (2) form the boundary integral equation (BIE) formulation for acoustic radiation from any closed arbitrary body B. In common case, only 4 or 5<j) / dn is not known on the surface S. By placing P on S, the unknown part of the pair (,dfy I dn ) is determined by use of Eq(l). Once and 3<j) / dn are known on the surface, the <j> value at any desired exterior field point P can be calculated using Eq(l). In the inverse problem of acoustic. The and d<j> / dn on the surface of body B are all unknown and need to be determined by using complex pressure on hologram. 2.2 Axisymmetric formulation When the Helmholtz integral equation is applied to a axisymmetric body B shown in Fig.l, equation(l) then becomes ^ In this case, it is convenient to use a cylindrical co-ordinate system (p,0,z), as defined in Fig.l, i.e., R(P,Q)= R(PQ, 90, Zg, p,, 9,, Z,) (4)
36 Boundary Elements where p(jg) is the radial co-ordinate of an arbitrary point Q on the body B ; d6 is the differential angle of revolution, and Jl(6) is the differential length of the generator, the other symbols are as in equation (1). For any body of axisymmetric shape, the <K2)' 4>(/0 &%d d l dn can be expanded in a Fourier series with respect to the angle of revolution with a period of 2n. Similar, the exp[-ikr(p,q)]/r(p,q) and its normal derivative can be also expanded in a Fourier series. For examples: B Figure. 1 An axisymmetric body sin(w0 p ) + (j) ^ (P) cos(w6 p ) (5) I (Q) sin(«0 cos(«6 Q (6) (7) n=l Substituting all expanding formulations into Eq(3) and make use of the orthogonality of sine and cosine function, it is given by where n=0,l,2..., and a = 5- or c indicate coefficient term of sine or cosine function, respectively. H» and H^ are the integral functions about angle 0 of revolution. When point P outside or inside B, // and HJ are nonsingular. The integral can be evaluated using a Gaussian quadrature formula. When P is moved on surface of body B, they include the singular integral. We can divide them into two terms, the first ones is nonsingular, and can be evaluated by a standard Gaussian quadrature formula. The remaining part of the integral is the singularities and evaluate it analytically by using elliptic integrals. 3 Discretization of boundary integral equations To reconstruct the pressure and normal velocity on body B, the BEM is employed for solving the acoustic holography problem. The boundary elements are used for approximating the integral equations. Here, the interior HIE is used to insure the unique solution of the line system at the characteristic frequencies. We choose a cylindrical as the hologram for measuring convenience. The num-
Boundary Elements 37 bers of discrete points on the hologram are IQ x l^. By expanding the pressure in Fourier series, the independent nodes are ^. The generator L of the body B is discretized by 3 -node quadrilateral elements, and nodes are N. The coordinates p and z of any point Q on L are given by * p(o = imopp 4) = ixcozf. (9) P=I P=I here, N^ (^\ are the shape functions given below where is the local coordinate parameter. Similarly, the boundary variables <(>"(0), < >"'(6) (a =,s, c, H = 0,1...) are also described in Eq(10). Thus, on element m: P=l where \j/,( ) denotes any of the mentioned boundary variables on element m, and the xj; ^ are the corresponding nodal values of the variables. Substituting Eq(9)-(ll) into Eq(8), the exterior, surface and interior HIE can be all expressed in the following matrix, respectively. (12) where a = s or c, denote the sine and cosine components, respectively. [A*], [ " ] are l^ x N complex coefficient matrix; [C" ], [D* ] are TV x N complex matrixes; [ "], [F"] are ^ x N complex matrixes. The Eq(13),(14) may be rewritten in the form of a single matrix equation, as where two complex coefficient matrix are all ( TV + /<, ) x N matrix. ^ is interior point numbers. Apparently, the pressure and normal velocity on radiating body B surface can be determined from Eq(12) and Eq(15). The inverse transformation technique is given in the following discussion. 4 The SVD-based acoustic holography algorithms To get the steady solution for inverse problem, singular value decomposition (S VD) is incorporated into the algorithms in order to alleviate and deal with the
38 Boundary Elements ill-posed nature frequently encountered in the backward reconstruction of acoustic field. By utilizing the SVD. Eq (12) and (15) can be rewritten as where [G] is M x N (M > TV, M = /,) complex matrix as here, [ ]~* denote the generalized inverse by the use of SVD. To finishing inverse transformation, the generalized inverse of [G] matrix is also given by SVD. According to the theory of Line Algebra, the M x N matrix [G] (M > N ) can be given by where [U] and [V] are M x M and N x N unitary matrix, respectively. E 1 ~ dtog (? i 2 (?r 0...0) is N x N real diagonal matrix and M > N > r. a, = ^/X~ are the singular values of [G] matrix, and a i ><J2 >...><?,. >0. T indicates the conjugate transpose. Thus, the [G] M' = [F] g ' O][t/f (19) where ^ ' = diagjjs? a''... a;' 0...0). In some Linepacks, the SVD calculating is inefficient. For example, we use the EIGCH in the IMSL library at the VAX780 computer. When M=200, the CPU time is over 2 hours. As M=700, consume the CPU time over 10 hours, but have not result. Thus, a kind of efficient SVD is needed. In the light of matrix theory^, the column vectors of unitary matrix [U] and the row vectors of unitary matrix [V] are same as eigenvectors of the Hermitian matrix [C/][G] and [^T[G], respectively. There are also same eigenvalues between [C?][G]^ and [G]^[G] matrix, e.g. X, > A^ >...> A,,. > 0. The singular value decomposition of M x N matrix [G] can be divided into two steps. First, calculate the eigenvalues of [G][G]^ and [G]^[G], and corresponding eigenvectors [ /'] and [F 1. Second, calculate the unitary matrix [U] and [K]. First, assume the [G][G]^ and [G]^[G] can be decomposed as [GM = MWI. [G]'[G] = [V][X][V]' (20) where [k] and [X] are M x M and N x N real diagonal matrix, and A,. = Xj > 0 (i, j = 1,2,... r); 1. =0 (i=r+l,..ju), ^ = 0 (j=r+l,...7v). The Eq(20) can be regarded as the eigenvalues decomposition of a / x / complex as
Boundary Elements 39 Hermitian matrix [H] (1= M or N). The / - / times elementary unitary similar transformations are applied to [//]. The transforming matrix of number / is given by [/>]= [7] - P; M}{W,y (21) where [/] is Ixl unit matrix, p. is complex coefficient, and {w.} is Ix 1 column vector. The selecting rule for p. and {%/.} is that make the absolute value of the elements of transformed matrix largeness as far as possible. After carried out / -1 times similar transformation, we can transfer the [//] into a triangular matrix of real symmetry S] as =1 - /=! Then, the eigenvalues of S] are got by use of QL method as [S] = [X] [la] [X]-' (23) where [X] is a Ixl orthogonal matrix. Due to the decomposition for [H] matrix doesn't change the eigenvalues of its, Thus [* *] = [^D] (24) Note: the CPU time and memory space can be reduced by fully utilizing symmetry of matrix. Second, to calculating the [U] and [F],changes Eq(18) into [E/f [Gl = ( 1 [yf (25) L J L J [_ 0 J Then, replaced the [[/] and [V] with [f/'l and \V'] matrixes, respectively. By use of the property for matrix equality, the ratio values for the unknown complex coefficients of every double eigenvectors corresponding singular values a, ( >0 ) can be determined alone. The complex values can be chosen arbitrarily, and don't effect the results. Now, we have finished the singular values decomposition for matrix [G]. Thought numerical test, the CPU time is consumed near 1 hour for a 700 x 700 complex matrix at VAX780 computer. 5 Verification of the reconstructing method By means of this reconstructing method just proposed, the numerical calculation is conducted to investigate the performance of this method. Consider the problem of radiation from an oscillating submerged sphere (radius is a) with radial velocity UQ cos(0), the exact analytical solution for the pressure and normal velocity are given by Respectively. p(r) = A e~* cos(0) «, (r) = A " -* cos(0) (26)
40 Boundary Elements where 8 = 0 is motion direction of sphere, e.g. z axis direction, and p - /cope)), u^ = -<))'. Figure 2 show the contrasting results between the theoretical values and reconstructing values from hologram. g 11 i 0.8 0.6 1 1 0.4 < 0.2 0 _n 9 \ \ \ \\ /\ /,,\ / / /\ I / /,,sf ^ 1 2 3 4 5 6 7 8 9 (a) node number 0.5 \ \ -0.5 1 2 3 4 5 6 7 8 9 urn r Figure.2 Reconstruction result versus exact solutions for the sound field radiated by an oscillating sphere. Here, ka=2, r/a=2 (r is radial of the cylindrical), L/a=3 (L is the length of the hologram), AL/X=l/8 (AL is the space of measuring points on the hologram), and M=7V=9. It is can be seen that the reconstructing errors for the magnitude and phase of the normal velocity on the sphere are larger than ones of the sound pressure. The error values at the near two polar points of the sphere are larger than other points. The most values of error for the pressure and normal velocity magnitude are 0.35 and 0.98dB, respectively. The most values for their phase error are 0.84 and 1.73 degree, respectively. The errors formulation of magnitude and phase for the pressure and normal velocity as e,(x) = (trans) (27) where X indicates the pressure or normal velocity on surface of source. The effect of different spacing between the sphere surface and the cylindrical hologram are shown in Figure 3. The results show that the mean errors of
Boundary Elements 41 reconstruction for the normal velocity are larger than ones of the pressure obviously. But the mean errors of the magnitude and phase for the pressure and normal velocity can all be controlled effectively by choosing the suitable parameters, e.g. ^ < 1.5d!B, ^ < 2 as r/a<3, L/a<3, AL/A,=l/8, M=N=9. Here, the formulas of the mean error as (28) - \X(trans))l\X(exact)\t (29) 1 J 10 Jr 11 11 11 / 1 1 1 I ^ ^* 0 1.2 1.6 2 2.4 2.8 3.2 3.6 4 0-2 1.2 1.6 2 2.4 2.8 3.2 3.6 4 6 Conclusion Figure.3 Effect of different spacing between source surface and hologram.( L/a=3, AL/^=l/8, M=N=9.) (a): # ^ * e. (b): + ^(p) e+(u) (R is the radial of the cylindrical.) We have presented a technique that reconstructs the pressure and normal velocity of an axisymmetric body surface with arbitrary boundary conditions by util-
42 Boundary Elements izing the BEM and an improved SVD. Satisfactory agreement between the reconstructing results and the exact solution exhibits the effectiveness of this method. There are two advantages : (1) Requires less memory space since twodimensional problem is reduced to one-dimensional; (2) The numerical calculating efficiency can be increased remarkably due to improved singular values decomposition. 7 references 1. Maynard, J.D., Williams, E.G., and Lee. Y. Nearfield acoustic holography: I. Theory of generalized holography and development of NAH, J. Acoust. Soc. Am, 1985, Vol.78, 1395-1413. 2. Hald, J. STSF-a unique technique for scan based near-field acoustic holography without restrictions on coherence, Technical Review, 1989, 1,1-49. 3. Adin, J., Mann, III., Pascal, J.C. Locating noise sources on an industrial air compressor using broadband acoustical holography from intensity, Noise Control Engineering Journal, 1992,Vol.29,No.l, 3-12. 4. Zhang, G.X., He, Z.Y., and Zhang, B. Nearfield holography (NAH) on underwater sound sources, P8-4, Proceeding of the 14th Int. Conf. on Acoust. in Beijing, 1992. 5. Veronesi, W.A., Maynard, J.D. Digital holographic reconstruction of sources with arbitrarily shaped surfaces, J. Acoust. Soc. Am, 1989, Vol.85, 588-598. 6. Borgiotti, G.V., Sarkissian, A., Williams, E.G., and Schuetz, L. Conformal generalized near-field acoustic holography for axisymmetric geometries. J. Acoust. Soc. Am, 1990, Vol.88, 199-209. 7. Bai, M.R. Application of BEM (boundary element method)- based acoustic holography to radiation analysis of sound sources with arbitrarily shaped geometries, J. Acoust. Soc. Am, 1992, Vol.92, 533-549. 8. He, Y.A, Zhang, J.D, and Zhang, W.Q. Velocity reconstruction of vibrating sources with arbitrarily shaped surfaces, Journal of Harbin Shipbuilding Engineering Institute, 1993, Vol.14, 64-73. 9. Banerjee, P.K and Butterfield, R. Boundary Element Methods in Engineering Science, Mcgraw-Hill, New York, 1981. 10. Golay, A.R. Numerical method of matrix eigenvalues problem, (Translated by Tang, H.Z.), Shanghai Scientfic & Technique Publisher, 1980.