A mapping relationship based near-field acoustic holography with spherical fundamental solutions for Helmholtz equation

Journal of Sound and Vibration 373 (2016) 66–88

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A mapping relationship based near-field acoustic holography with spherical fundamental solutions for Helmholtz equation Haijun Wu a,b,n, Weikang Jiang a,b,n, Haibin Zhang a,b a

State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, China Institute of Vibration, Shock and Noise, Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai Jiao Tong University, Shanghai 200240, China b

a r t i c l e in f o

abstract

Article history: Received 28 August 2015 Received in revised form 14 February 2016 Accepted 9 March 2016 Handling Editor: P. Joseph Available online 29 March 2016

In the procedure of the near-field acoustic holography (NAH) based on the fundamental solutions for Helmholtz equation (FS), the number of FS and the measurement setup to obtain their coefficients are two crucial issues to the successful reconstruction. The current work is motivated to develop a framework for the NAH which supplies a guideline to the determination of the number of FS as well as an optimized measurement setup. A mapping relationship between modes on surfaces of boundary and hologram is analytically derived by adopting the modes as FS in spherical coordinates. Thus, reconstruction is converted to obtain the coefficients of participant modes on holograms. In addition, an integral identity is firstly to be derived for the modes on convex surfaces, which is useful in determining the inefficient or evanescent modes for acoustic radiation in free space. To determine the number of FS adopted in the mapping relationship based NAH (MRS-based NAH), two approaches are proposed to supply reasonable estimations with criteria of point-wise pressure and energy, respectively. A technique to approximate a specific degree of mode on patches by a set of locally orthogonal patterns is explored for three widely used holograms, such as planar, cylindrical and spherical holograms, which results in an automatic determinations of the number and position of experimental setup for a given tolerance. Numerical examples are set up to validate the theory and techniques in the MRS-based NAH. Reconstructions of a cubic model demonstrate the potential of the proposed method for regular models even with corners and shapers. Worse results for the elongated cylinder with two spherical caps reveal the deficiency of the MRSbased NAH for irregular models which is largely due to the adopted modes are FS in spherical coordinates. The NAH framework pursued in the current work provides a new insight to the reconstruction procedure based on the FS in spherical coordinates. & 2016 Elsevier Ltd. All rights reserved.

1. Introduction To locate the position and target the strength of noise for a vibrating structure, near-field acoustic holography (NAH) had been widely adopted as an effective tool. It has a significant influence on the noise diagnostics, which gives a permission to get all desired acoustic quantities, such as pressure, particle velocity, sound power, etc. from a number of discrete field measurement.

n

Corresponding authors. E-mail addresses: [email protected] (H. Wu), [email protected] (W. Jiang).

http://dx.doi.org/10.1016/j.jsv.2016.03.010 0022-460X/& 2016 Elsevier Ltd. All rights reserved.

H. Wu et al. / Journal of Sound and Vibration 373 (2016) 66–88

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It was originally developed by Willams, Manynard, etc. to reconstruct surface velocity of a rectangular plane with Fourier transform technique [1–3]. Initially, the Fourier-based NAH decomposes the field pressure into k-space (wavenumber space) for baffled problems. In other words, the field pressure are expanded into plane waves and the reconstruction procedure is to obtain coefficients of the plane waves based on measured pressure. Although different from the k-space decomposition, concept of Fourier transformation was inherent to the 3D cylindrical and spherical NAH problems as the in-depth discussions in Ref. [4]. Since it was proposed [1], varieties of approaches had been proposed and their superiorities had been proven in various applications, which resulted in several categories according to their underlying theories. Statistical Optimal NAH [5–7] uses the elemental waves to approach the acoustic field, in which the surface-to-surface projection of the sound field is performed by using a transfer matrix defined in such a way that all propagating waves and a weighted set of evanescent waves are projected with optimal average accuracy [6]. Boundary element method (BEM)-based NAH [8–13] are appropriate for arbitrarily shaped model in which a general transformer matrix between the surfaces of structure and hologram is derived from the integral equation. Among the BEM-based NAH, two types of integration equation are adopted: the direct formulation (Helmholtz integral equation) and indirect formulation (single or double layer integral equation). The quantities reconstructed by the NAH derived from direct formulation have clear physical meaning [8–10], while the ones obtained by NAH derived from the indirect formulation are not the real physical quantities [11–13]. The equivalent source method (ESM) [14–19], also named as wave superposition algorithm (WSA) [16,20,21], was proposed by Koopman [22] for solving acoustic radiation problems of closed sources. ESM assumes the field is generated by a series of simple sources such as monopoles and dipoles, and numerical integration is not needed in determining the source strength for a set of prescribed positions. Despite versatility of the ESM and various successful applications, “retreat distance” between the actual source surface and the virtual source cannot be well defined and deserves more attention in the application [23]. The Helmholtz equation least square method (HELS) [24–26] adopted the spherical wave expansion theory to reconstruct acoustic pressure field from a vibrating structure. Coefficients of the spherical wave function, the fundamental solution for the Helmholtz equation (FS), are determined by requiring the assumed form of solution to satisfy the pressure boundary condition at the measurement points. Since the spherical wave functions solve the Helmholtz equation directly, it is immune to the nonuniqueness difficulty inherent in BEM-based NAH [27]. However, HELS works better for spherical or chunky model than elongated model due to the specific basis function [25]. Essentially speaking, NAH is to achieve the desired acoustic quantities by the measured physical quantities such as sound pressure in the field. Most of the methods explicitly require the transfer operator Tðy; xÞ between desired acoustic quantities f ðyÞ and measured physical quantities p ðxÞ. They built a linear system of f ðyÞ ¼ invðTðy; xÞÞp ðxÞ in which invð⋆Þ represents an inverse operator, by either a general numerical method (BEM-based NAH) [8–13], or specific basis spaces such as a general Fourier basis (Fourier-based NAH) [1–4], simplified monopoles, dipoles (ESM, WSA) [14–16,18,20,21], fundamental solutions (HELS) [24–27]. The reconstruction procedure is therefore to solve the linear system to obtain the physical quantities on the boundary, such as pressure or normal velocity in BEM-based NAH, the source strength of equivalent source in ESM, coefficients of basis functions in Fourier-based NAH and HELS, and following by an extrapolation process to achieve desired acoustic quantities. Unfortunately, all the proposed methods are very sensitive to errors which may cause reconstruction to fail. It is primarily due to abundant adoption of basis functions in the transfer operator which amplifies the errors in the inverse process. That is the reason why there have been numerous studies focusing on the development of regularization methods to stabilize this inverse problem, such as truncated singular value decomposition [28] and the Tikhonov regularization [29]. Thus, construction of transfer operator is not a trivial process but is crucial to the feasibility and accuracy of the NAH. Concerning the theory development and practical measurement, it naturally arise a question whether there exists a guideline to determine the number of generalized basis function as well as measurement to obtain their coefficients for a given shape of source surface and prescribed tolerance. To the best knowledge of authors, the number of FS as well as number and position of the microphones array in the measurement are not well studied for the category of NAH based on the FS. Thus, the primary objective of the current work is to build a guideline for the determination of the number of FS and measurement configuration in the FS based NAH by exploring the mapping relationship between the modes in FS between surface and hologram, and investigating approximation of the modes with a set of locally orthogonal patterns. This paper is organized as follows: acoustic modeling with the boundary integral equation (BIE) and its multipole expansion form are briefly introduced in Section 2.1, two types of truncation techniques for the multipole BIE are derived in Section 2.2, which serves as fundamental theory for the determination of necessary degree of modes and also the explanation of the evanescent modes. Procedure of the MRS-based NAH as well as the evanescent waves and comparison with existing methods are described in Section 3. Section 4 is about the measurement configuration for three representative holograms. An algorithm to automatically determine the number and position of microphone array on planar hologram is introduced in part A, a reduced algorithm is developed for cylindrical holograms in part B, part C is about an analytical method for the determination of measurement configuration on spherical hologram. A number of numerical examples are elaborately designed to validate the theory and demonstrate the potential of the proposed MRS-based NAH in Section 5. Section 6 concludes the paper with some discussions and remarks of our method.

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z Ω

y S x

n

Fig. 1. Exterior acoustic problem of a vibrating structure in free space.

2. Acoustic problems modeling 2.1. Boundary integral equation Assume that the fluid is homogenous, inviscid, compressible and only undergoes small translation movement. The time harmonic sound pressure radiated from a vibrating structure into an infinite domain Ω is described by the well-known Helmholtz equation ∇2 pðxÞ þk pðxÞ ¼ 0 for x A Ω 2

(1)

where k is the wave number, relating to the acoustic speed c and angular frequency ω by k ¼ ω=c, and x is a point in the domain. The time component is assumed to be e  iωt . An alternative expression of radiation problems to the Eq. (1) is the boundary integral equation (BIE) [30]  Z  ∂pðyÞ ∂Gðx; yÞ  pðyÞ dSðyÞ (2) Gx; y cðxÞpðxÞ ¼ ∂nðyÞ ∂nðyÞ S in which S is the exterior boundary of the vibrating structure, nðyÞ is the normal direction pointing outward from the domain Ω as shown Fig. 1 for exterior problem, G(x,y) is the free-space Green's function Gðx; yÞ ¼

eik‖x  y‖ 4π ‖x  y‖

(3)

where kxk means the distance of point x to the origin, cðxÞ is a variable depending on the position of and the geometry around the point x. It is one for an exterior point, and one half for when the geometry is continuous around the point x on the surface. If point x is located on a discontinuous surface, such as corner and edge, cðxÞ becomes Ψ=4π where Ψ is the outer solid angle at that point [31]. Since the radiation condition at infinite is automatically satisfied, BIE is widely applied into the analysis of exterior acoustic problems. Instead of using the elementary Green's function in the BIE, the multipole expansion of Green's function is adopted in our analysis, which can be expressed as ∞

Gðx; yÞ ¼ ik ∑

n

m ∑ R−m n ðk; yÞSn ðk; xÞ ¼ lim GN ðx; yÞ;

n ¼ 0 m ¼ −n

n→∞

for

‖y‖o ‖x‖

(4)

m in which GN represents the truncated series with degree being N, Rm n ðyÞ and Sn ðx Þ are two sets of FS of the governing formulation, Eq. (1), for interior and exterior acoustic problems [32], respectively,   m (5) Rm n ðk; yÞ ¼ jn k‖ykÞYn θ ; ϕ

   m Sm n ðk; x Þ ¼ hn k‖xkÞYn θ ; ϕ

(6)

where variables θ and ϕ are the polar angles of a point in the spherical coordinates. jn and hn are the nth spherical Bessel function and spherical Hankel of the firs kind, respectively. Ym n is the normalized spherical harmonic function    imϕ 1 m Ym n θ ; ϕ ¼ pffiffiffiffiffiffiPn cos θ e 2π 

(7)

whose inner product stratifying the Kronecker delta property on an unit spherical surface σ^ , e.g.,  R m t t m ^ Ym n ; Yl ¼ σ^ Yn Yl dσ ¼ δnl δmt , and Pn is the normalized associated Legendre function [33]. Thus, sound pressure at an

H. Wu et al. / Journal of Sound and Vibration 373 (2016) 66–88

z

69

E y

S

x So

n (y )

n (x )

y

x

Fig. 2. A vibrating structure in free space and its equivalent spherical source.

exterior point x can be reformulated by substituting Eq. (4) into Eq. (2) as  Z  1 n X X ∂pðyÞ ∂Rn m ðk; yÞ m  p ð y Þ dSðyÞ Sm ð k; x Þ R ð k; y Þ pðxÞ ¼ ik n n ∂nðyÞ ∂nðyÞ S n ¼ 0 m ¼ n

for J y J o J x J :

It reveals that the coefficients of the FS are subject to the boundary condition of a vibrating structure. Normal gradient of the fundament solution Eq. (6) is   ∂hn ðkkxkÞYm ∂Sm n θ; ϕ n ðx Þ ¼ qm ð x Þ ¼ n ∂nðxÞ ∂nðxÞ

(8)

(9)

which is related to the normal velocity vn by the Euler formulation qðxÞ ¼ ikρcvn ðxÞ

(10)

It should be noted that Eq. (6) and (9) are related as a solution pair for exterior acoustic problems, which means giving one as the boundary condition, the other will be the solution. They form a set of pressure/velocity modes on the boundary of a vibrating structure, which are generally independent on non-spherical surfaces and orthogonal on spherical surfaces. To facilitate derivations, we refer the velocity modes as the normal gradient q instead of the normal velocity vn . Assume that a structure is vibrating in one of its velocity modes Eq. (9), the radiated pressure must be in the form of Eq. (6), which can be derived by making use of the ESM and BIE. The ESM states that radiated field from a vibrating structure can be equivalently reconstructed by an interior spherical source, as shown in Fig. 2. Ref. [34] proved that radiated pressure on the surface of a structure is in the same mode if the interior spherical source is vibrating in the form of Eq.

(9). It was also derived that the radiated sound power of a structure 2 vibrating in one of the mode Eq. (9) is W ¼ 1= 2ρck which is independent of the degree and order of the mode [34]. The forward mapping relation from the interior spherical source to the vibrating structure is one to one and is easily obtained by exploring the orthogonality of the modes on spherical surfaces. To apply it in the inverse process, the backward mapping should also be one to one. It can be proved by the conversation law of radiated sound power. Assume that the structure is vibrating in one of the modes, qm n ðx Þ, and the interior equivalent spherical source is vibrating in a different form, qðx Þ, which is expressed as qðxÞ ¼ qm n ðxÞ þ

1;l a n l;tX ¼ l X l¼0

atl atl ðxÞ

(11)

t ¼ l

In light of the orthogonality and radiated sound power of each mode on the spherical source, the radiated sound power generated by the equivalent spherical source is ! 1;l a n l;lX am X 1 jatl j2 (12) W ¼ 1þ 2 2 ρ ck l ¼ 0 t ¼ l

2 However, the sound power radiated by the structure is assumed to be W ¼ 1= 2ρck . Therefore, all the coefficients atl in Eq. (11) should be zero by the conversation law of radiated sound power. In other words, if the structure is vibrating in one of the modes, an equivalent spherical source must be found which is vibrating in the same mode and producing the same acoustic field as that from the real structure. Thus, a unique mapping relationship between the modes in Eq. (6) or Eq. (9) on surfaces of vibrating structure and its interior equivalent spherical source is formed. Please be noted the mapping relationship is derived under the assumption of radiators with closed convex shapes [34]. It is straightforward to extent the mapping relationship to an exterior convex surface of a vibrating structure, such as holograms in the NAH. Therefore, if a structure is vibrating in one of the modes, the radiated field on an exterior convex surface is also in the same mode. It means that sound pressure radiated from the

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structure vibrating in the velocity pattern qtl ðyÞ is # Z " 1 n X X ∂St ðk; yÞ ∂Rn m ðk; yÞ t  Sl ðk; yÞ dSðyÞ; Sm Rn m ðk; yÞ l Stl ðk; xÞ ¼ ik n ðk; xÞ ∂nðyÞ ∂nðyÞ S n ¼ 0 m ¼ n

for J y J o J x J :

(13)

Since Sm n ðk; x Þ is independent, Eq. (13) implicates that # Z "  m t ∂Stl ðk; yÞ ∂Rn m ðk; yÞ t δ δmt m  Sl ðk; yÞ dSðyÞ ¼ nl (14) Rn ðk; yÞ I Rn ; Sl ¼ ∂nðyÞ ∂nðyÞ ik S  m t t It means Rm n and Sl forms an integral identity I Rn ; Sl on an arbitrarily closed convex surface which is a property similar to Kronecker delta function. Eq. (14) can be derived by a purely mathematical method on a spherical surface which is a very special case of closed convex surfaces, while it is hard to be proved mathematically for a general case. Since, the identity  t I Rm n ; Sl will play a key role in the determination of efficient modes or evanescent modes of a vibrating surface, a mathematical proof on the spherical surface and some numerical validations on several representative surfaces of general case are supplied in the Appendix. 2.2. Truncated multipole BIE The FS based NAH assumes that the radiated field is contributed by fewer most efficient FS or modes. Therefore, Eq. (8) must be truncated. The truncation number, denoted by N, is crucial to the pressure evaluation as well as the number and positions of microphone array required in the NAH. Two types of approaches are introduced for the determination of the necessary truncation number. One is based on a strongly requested criterion of point-wise pressure evaluation. Another is built on a weak criterion of the radiated sound power. Suppose a vibrating structure is contained by a sphere with radius being r and the field point is located exterior to a sphere with radius being r h , as shown in Fig. 3. The relative error for the multipole expansion GN ðx; yÞ can be bounded by (9.1.28) [35] in the low frequency range 0:01 r kr r 10 as ϵT o 

γ −N−1 2 3=2 ; for N 4 ðkr h Þ =2 1−γ −1

(15)

where ratio γ ¼ r h =r. Usually larger ratio σ will lead to larger truncation number N for the same tolerance. Vibrating structures and holograms sometimes have corners and sharpers that will give rise to larger ratio σ . Therefore, the necessary condition in Eq. (15) is sometimes too rigorous which will result in large N and consequently a large number of measurements. To alleviate the influence of some extreme corners and sharpers in determining the truncation number in NAH, equivalent radius for the vibrating structure and holograms are defined. The equivalent radius is the radius of the equivalent spherical source which has the same averaged radiated sound power per unit area as that of the vibrating structure or holograms. Since the radiated sound power from the equivalent source and the vibrating structure as well as the holograms should be same, the requirement of same averaged radiated sound power per unit area makes the equivalent radius satisfy pffiffiffiffiffiffiffiffiffiffiffi (16) r~ ¼ S=4π where S represents area of structure and holograms in the determination of equivalent radius r~ and r~h , respectively. Even though equivalent radiuses are adopted, the necessary truncation number determined by Eq. (15) is also relatively very large in practical point of view. An alternative way to select the efficient number of participant modes is based on an energy criterion which neglects the modes contributing to the radiated sound power very few. As illustrated by Eq. (16) in Ref. [34], the radiated sound power  2 of the modes are same regardless of their degree and order. It is a simple accumulation  for each mode as of the participant coefficients αm n W¼

1 2ρck

2

1 n X X  m 2 a  n

n ¼ 0 m ¼ n

rh

r

Fig. 3. A schematic of vibrating structure contained in an inner sphere and the field point located exterior to an outer sphere.

(17)

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71

Generally, the participant coefficients coefficients αm n are obtained by decomposing the velocity distribution q into the modes on the surface of the vibrating structure as introduced in Section 3.1. To take advantages of the orthogonality of modes in the derivation, an alternative way to achieve those coefficients is to decompose the equivalent normal velocity v~ on surface of the equivalent spherical source as   ~ qm v; n αm ¼ (18) n qm 2 n where qm n are the modes on the surface of equivalent spherical source, the inner product 〈*,*〉 and the norm ||*|| are defined, respectively, as Z   v~ ðxÞ; qm ð x Þ ¼ (19) v~ ðxÞqm n n ðx ÞdΓðxÞ Γ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m qm ‖qm n ðx Þ‖ ¼ n ðx Þ; qn ðx Þ

(20)

and “*” means complex conjugate. By applying Cauchy–Schwarz inequality to αm n , an upper bound of the Eq. (17) is Wr

1 ‖ν~ 2 ‖ X

8πρck

2

ð2n þ 1Þσ n ðkr~ Þ

(21)

n¼0

σ n is the radiation efficiencies of the equivalent spherical source with radius being r~ for a group of modes qm ð n  n rm rnÞ, which has a closed-form expression Eq. (18) in Ref. [34]. Eq. (21) demonstrates that the radiation sound power of a vibrating structure is upper bounded by the product of the superposition of the radiation efficiency and the square normal velocity on its equivalent spherical source. Therefore, determination of the number of most efficient modes is converted to seeking the truncation number of radiation efficiency σ n making the upper bound of the radiated sound power converge to a given tolerance. A relative error of the upper bounded radiated sound power caused by new added modes for degree N with respective to the one produced by existing modes for degree less than N is defined as ð2N þ 1Þσ N ðkr~ Þ

εN ¼ PN  1

(22)

ð2n þ1Þσ n ðkr~ Þ

n¼0

Generally, the relative error is larger than the actual errors since the denominator is less than the real upper bound of the radiated sound power. Fortunately, for a specific dimensionless value kr~ , the radiation efficiency σ N 4 Nc is a strictly decreasing function with respective to the degree N after a certain degree N c which can be obtained by its closed-form expression. Radiation efficiency σ N and the relative error εN for the dimensionless size 0:1 r kr~ r10 and the varying degree from 0 to 7 are presented in Fig. 4(a) and (b), respectively. It shows that σ N is a monotonously decreasing function and clearly distinct from each degree for kr~ o 2:0. There is a plateau on which σ N starts to overlap for kr~ 4 2:0. It means that those degrees of modes contribute to the radiated sound power almost equally and no one can be neglected, which is verified by the relative errors in Fig. 4(b). Therefore, more degrees of modes are needed to make the radiated sound power converge for larger dimensionless value kr~ . The relative error εN presented in Fig. 4(b) can be used as a reference to determine the degree of the most efficient modes for 0:1 r kr~ r10, or in other words the truncation number N for the FS based NAH. However, Eq. (22) only relates to the size of the structure and has nothing to do with the field point or size of hologram. Generally, the truncation number determined by Eq. (22) is less than that by Eq. (15). Those two approaches supply reasonable lower and upper bounds for the estimation of truncation number in the FS based NAH.

0

0

10 N=0

-3

10

N=1

-6

10

N=2

-9

10

N=3

-12

N=2

-6

10

N=3

-9

10

N=4 -12

10

10

N=4 N=5 N=6 N=7

-15

10

N=1

-3

10 Rative error ε N

Radiation efficiency

10

-1

10

0

10 k~r

N=5

-15

1

10

10

-1

10

N=6 N=7 0

10 k~ r

1

10

Fig. 4. (a) Radiation efficiency of a sphere (b) and relative errors of upper bounded radiated sound power, for varying degrees and k̃r .

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H. Wu et al. / Journal of Sound and Vibration 373 (2016) 66–88

3. The mapping relationship based NAH 3.1. The procedure Acoustic holography techniques implicitly make an assumption of a fact that there must be a bijective mapping relationship between the boundary quantities and the radiated field. In other words, a given distribution of the radiated field is uniquely determined by a specific set of boundary quantities, and vice versa. Suppose a set of at least independent velocity modes on the boundary, denoted as vi ðx A SÞ; ði ¼ 1; 2; …Þ, can produce a set of independent pressure modes pi ðx A ΓÞ; ði ¼ 1; 2; …Þ on the measurement surface Γ, correspondingly, and they form a pair of bijective mapping relationship. Generally the pressure patterns pi ðx A ΓÞ are non-orthogonal on the hologram, an orthogonalization process is required, which can be done by the Gram–Schmidt approach as   i1 X pi ðxÞ; uj ðxÞ uj ðxÞ (23) ui ðxÞ ¼ pi ðxÞ  ‖uj ðxÞ‖ j¼1 where the inner product h⋆; ⋆i and the norm k⋆k are defined in Eqs. (19) and (20). After some algebraic operations, the normalized orthogonal modes ei ðxÞ ¼ ui ðxÞ=‖ui ðxÞ‖ can be expressed in the following form E ¼ pR (24)

 where E ¼ ½e1 ðxÞ; e2 ðxÞ; ⋯, p ¼ p1 ðxÞ; p2 ðxÞ; ⋯ and R is an upper triangular square matrix. As indicated in Eq. (24), the normalized orthogonal modes are actually a linear combination of the independent pressure modes pi . It is remarkable that evaluation of the inner produce is performed on holograms which are generally in smooth shapes, such as the three typical holograms in Section 4. Thus, they can be computed on exact geometries which are avoid of discretization errors. Furthermore, orthogonalization of the modes on simple shaped holograms will yield a translation matrix R with good numerical characteristics, such as small condition number. Assume the radiated pressure pðxÞ is decomposed into the normalized orthogonal modes ei ðxÞ on the hologram as pðxÞ ¼

1 X

αi ei ðxÞ

(25)

i¼1

where



αi ¼ pðxÞ; ei ðxÞ

 Γ

(26)

Once those participant coefficients are obtained by the measured pressure, substituting Eq. (24) into Eq. (25) yields pðxÞ ¼

1 X

λ i pi ðx Þ

(27)

Rij αj

(28)

I¼1

where the coefficients are

λi ¼

1 X J¼1

Due to the unique mapping relationship between surfaces of vibrating structure and hologram, reconstruction for the boundary quantities can be performed by multiplying the corresponding modes with the same set of participant coefficients λi on the surface of the vibrating structure. Thus acoustic holography is converted to seek explicit descriptions of the mapping relationship between the modes on the boundary of the vibrating structure and modes on the holograms, and design a proper experimental setup for obtaining the participant coefficients of the modes on the measurement surface. The modes on the boundary are free of restrictions for their form of expression, which could be in any well studied analytical functions or in generally numerical representations. However, it should be expected to have a capacity of fast convergent ratio in the decomposing of boundary quantities, and generate a radiated pressure on the hologram which is easy to be determined by the experiment. In the current work, the FS in spherical coordinates for the Helmholtz equation Eq. (6) and its normal gradient Eq. (9) are chosen as the pressure and velocity modes. Merits of choosing those forms of modes are twofold. First, the radiated modes on the field are also the in the same form; and second, the most effective modes contributing to the field pressure are easy to be determined. The underlying theory of those two merits are investigated in Section 2. Henceforth, the radiated pressure modes in Eq. (27) is chosen as pi ¼ pn2 þ m þ n þ 1 ¼ Sm n in our analysis to facilitate derivations. So do the velocity modes qi . 3.2. Evanescent wave According to the above derivations, mapping relationship puts no restriction of the distance between surfaces of the boundary and holograms. However, holograms cannot be placed without limits in the NAH due to the existence of evanescent wave which decays rapidly away from the vibrating structure and cannot be captured from far distance. For normalized wave modes, evanescent wave can be determined by their coefficients. A power law is provided to qualitatively

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73

describe the degenerative characteristics of the coefficients (Eq. 6.104) [4]. Since the wave mode adopted in the current work are not normalized, it is inappropriate to define the evanescent wave only by coefficient. Thus, it is not sufficient to yield a guideline to distinct the evanescent wave from others only by the investigations of the coefficients. A method is available to make the quantitative analysis of evanescent wave according the theory introduced in Section 2. Suppose the velocity distribution v is decomposed into the velocity modes as v ðy Þ ¼

1 n X 1 X am qm ðk; yÞ; ikρc n ¼ 0 m ¼  n n n

for

yAS

(29)

and Green's function, Eq. (4), is reformulated as Gðx; yÞ  ð1 þ ϵ~ N ÞGN ðx; yÞ

(30)   in which ϵ~ N is a positive real-valued error introduced by the term truncation N and is subject to ϵ~ N j r ϵT which is the  t tolerance in Eq. (15). In light of the integral identity I Rm n ; Sl , Eq. (8) yields radiated sound pressure N

pðxÞ  ð1 þ ϵ~ N Þ ∑

n

m ∑ am n Sn ðk; xÞ

n ¼ 0 m ¼ −n

(31)

with the boundary condition prescribed by Eq. (29). Eq. (31) means a lot. First, the radiated pressure can be assumed to obey the same error as that in the multipole expansion of Green's function. Second, it reveals that the number of most effective modes contributing to the radiated field is determined by the truncation number of Greens’ function, and the radiated pressure produced by the modes with degree larger than N could be viewed as the evanescent wave. Therefore, the most effective modes or evanescent wave can be determined by the error analysis of the multipole expansion of Greens’ function [32,36,37]. Please be aware of that the definition of evanescent wave introduced above is in terms of radiated pressure, another way to determine the evanescent wave based on the radiated sound power is straightforward by following the same process introduced above and the analysis in Section 2. 3.3. Comparison with existing methods FS is extensively studied and widely adopted in various kinds of acoustic problems. Concerning its application in NAH, there are two representative work which are highly relative to our current work. One is the spherical NAH [4], the other is HELS [25]. Both approaches are to obtain the coefficients of their spheroidal functions by measured pressure and reconstruct the acoustic field by evaluating the expansion with obtained coefficients. The spherical functions adopted in the spherical NAH are the same as the ones in the current work. However, the spheroidal functions adopted in HELS is slightly different from that in the spherical NAH as well as the current work, which did not apply Euler equation for the variable ϕ. The spheroidal functions are required to satisfy the Helmholtz equation in both approaches, which is a partial difference equation. Therefore, the mapping relationship of the modes between surfaces of the boundary and hologram is not explicitly explored even it is implicitly assumed in their underlying theories. Due to unawareness of the mapping relationship, hologram of the spherical NAH [4] is restricted to a spherical surface with an aim to employ the orthogonality of spherical harmonics. Moreover, how to effectively and accurately obtain the spherical harmonics was not addressed. While the measurement configurations for three representative holograms are elaborately investigated in the next section, which result in an optimized setups. Instead of obtaining the spectrum on a completely closed space [4], HELS determined the coefficients associated with these independent functions by requiring the assumed form of solution to satisfy the pressure boundary condition at the measurement points [25]. It is free of the restriction for shapes of hologram as that in the spherical NAH [4]. However, it may raise another issue of uniqueness for the reconstructed quantities by only matching the assumed form of solution at some specific points expect the measurement points can represent the radiated modes on hologram properly. In addition, HELS required Gram–Schmidt approach for the independent functions on the surface of vibrating structure, while a Gram– Schmidt is also needed but on the holograms due to the explored analytically mapping relationship in our approach. Since holograms are generally in smooth shapes, such as the three types of holograms studied in following section, it facilities the numerical evaluation of the inner product in both Eqs. (19) and (26). Besides, the transform matrix R generated in the orthogonalization process usually has a good numerical characteristics. Those distinctions make our work different from the spherical NAH and HELS. It provides a new insight to the NAH with FS by exploring the unique mapping relationship between modes on surfaces of structure and holograms.

4. Configuration on holograms Since the modes are distributed on a whole surface, the holograms should form an enveloping surface enclosing the vibrating structure. Otherwise, partially measured pressure cannot represent the modes completely, and consequently cannot be applied to reconstruct the boundary information based on the mapping relationship. The distribution of a specific mode varies on different holograms. Generally, the measurements are subject to the experimental conditions such as microphones and permissible space. How to accurately recognize the field pressure modes

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Fig. 5. Planar holograms.

is one of the crucial factors to MRS-based NAH. In practice, microphones are preferred to be placed on planar, cylindrical or spherical surfaces which are easy to be set up but generally not conformal to the vibrating structure. In this section, configurations for accurate recognition of the pressure modes on those three typical measurement holograms are investigated. 4.1. Configuration on planar holograms Taking surfaces of a hexahedron as the holograms where measurement are performed, as shown in Fig. 5. Therefore, each pressure mode is divided and projected onto six patches, such as Γi ði ¼ 1; ⋯; 6Þ. On each patch, the measured pressure should be able to accurately represent the projected pressure modes. However, once the pressure are discretely sampled, the spectrum or the number of participant modes on that patch is truncated. Suppose the degree of most effective pressure modes on the patch is up to degree N, which is determined by the techniques introduced in Section 2.2. Now, it arises questions of how many microphones are necessarily required and where they should be located to accurately recognize the projected pressure modes. Therefore, the primary task in the measurement is to set up the microphone arrays properly with an aim to approximate all the projected pressure modes on each patch actually. As shown in Fig. 5, on each planar surface, the pressure modes can be expressed by two sets of locally normalized orthogonal polynomials such as polynomial f n ðxÞ and g n ðyÞ of degree n in each direction. Taking the right hologram as an example, which has two independent coordinates x and y, the pressure modes Sm n ðk; xÞ on the patch can be decomposed in the form Sm n ðk; x Þ ¼

1 1 X X

0

am n0 f n0 ðxðx ÞÞg m0 ðyðx ÞÞ

(32)

n0 ¼ 0 m0 ¼ 0

where xðxÞ and yðxÞ represent the two components of point x. By exploring the orthogonality of the polynomials, the coefficients can be obtained by Z   Sm (33) am0 n0 ¼ n ðk; xÞf n0 ðxðx ÞÞ g m0 ðyðx ÞÞ dΓðxÞ Γi

In the application, Eq. (32) is inevitable to be truncated by upper bounds N 0 and M 0 for the polynomials f n and g m in the approximation of the pressure modes, respectively. Hardly will the analytical way be explored to determine the bounds N0 and M 0 . Since the pressure patterns Sm n have analytical expressions, numerical approach can supply an alternative way to the 0 determination of N0 and M 0 . Suppose the point wise truncation errors εM ðxÞ for the upper bound N0 and M0 are N0 0

Sm n ðk; x Þ ¼

0

N M X X n0 ¼ 1 m0 ¼ 1

M0 am0 n0 f n0 ðxðx ÞÞg m0 ðyðx ÞÞ þ ϵN 0 ðxÞ

To determine the tight bounds N0 and M0 , two types of errors ϵ~ p and ϵ~ e are defined as 0

ϵ~ p ¼ max ϵM N 0 ðx Þ o ϵp vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uZ 0

ϵ~ e ¼ u ðxÞϵM ðxÞdΓi o ϵe t ϵM0 N0 N0

(34)

(35)

(36)

Γi

where εp is a point wise tolerance, and εe is the tolerance defined on the patch Γi in the energy form. A numerical process is designed to seek the proper bounds for modes on each patch within the given tolerances. For the pressure mode Sm n of degree n and order m, the process is as follows: Initial: Set N0 ; M0 ¼ 1;

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Step 1: Set N 0 ¼ N0 þ 1, compute the am n0 by Eq. (33);  0 0 ~ p ¼ max ϵM ðxi ÞjÞ; Step 2: Compute εM 0 ðx Þ by Eq. (34) for some specific points xi , set ϵ N N0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R M0 M 0 ðx ÞdΓ numerically on the patch; ð Þ ϵ x ϵ Compute ϵ~ e ¼ i Γi N 0 N0 0

if ϵ~ p o ϵp and ϵ~ e o ϵe , exit; 0 Step 3: if ϵ~ p Z ϵp or ϵ~ e Z ϵe , Set M 0 ¼ M 0 þ 1, compute the am n0 , and perform Step 2; Step 4: if ϵ~ p Z ϵp or ϵ~ e Z ϵe (computed by Step 2), go to Step 1.

It is no necessary to compute the bounds for all pressure modes due to the fact that if the two polynomials with degree up to N 0 and M 0 can accurately approximate the pressure modes with degree being N, the pressure modes with degree less than N can also be accurately approximated. Hence, there are only 2N þ1 modes need to select their bounds for a group of modes with degree being N. Once the upper bounds of the local patterns are determined, pressure distribution pðxÞ on each patch can be approximated by the locally orthogonal patterns as 0

pðxÞ 

0

N M X X

0

am n0 f n0 ðxðx ÞÞg m0 ðyðx ÞÞ; for x A Γi

(37)

n0 ¼ 0 m0 ¼ 0

in which the coefficients are achieved by Z am0 n0 

Γi

pðxÞf n0 ðxðxÞÞ g m0 ðyðxÞÞ dΓðxÞ

(38)

R Kronecker delta property of the orthogonal polynomials, δnn0 δmm0 ¼ Γi f n0 ðxÞg m0 ðyÞf n ðxÞg m ðyÞdΓi ðxÞ are implicitly required  0    in the numerical evaluation. To accurately evaluate Eq. (38), N þ 1 and M 0 þ 1 points Gaussian quadrature for variables x     and y are accurate enough to integrate the polynomials f n0 ðxÞf n ðxÞ ; n0 ; n ¼ 0; ⋯; N0 and g m0 ðyÞg m ðyÞ ; m0 ; m ¼ 0; ⋯; M 0 , which can be expressed as 0

am n0 ¼

0 0 NX þ1 M þ1 X

        wij p xij f n0 x xij g m0 y xij

(39)

i¼1 j¼1

where wij are the weights for abscissas xij in the Gaussian quadrature. Please be noted, the abscissas xij in the quadrature     are also the position where the microphones are placed in the measurement. It means N 0 þ 1  M 0 þ 1 measured pressure signals are required and the positions on which the microphones are placed are the abscissas of Gaussian quadrature of the two independent directions on the patch. Once the participant coefficients of locally orthogonal polynomials are obtained by Eq. (39) on all planar patches, reconstruction is to substitute Eq. (37) patch wisely to Eq. (26) firstly to obtain the participant coefficients of each modes on the holograms, and then apply the mapping relationship back to the boundary surface for desired acoustic quantities. 4.2. Configuration on cylindrical holograms In contrast to the planar holograms which composed of six patches, a closed cylindrical measurement surface Γ as       illustrated in Fig. 6 has three patches, one left circular planar patch Γl x b=2 ¼ x cos θ , one right Γr xb=2 ¼ x cos θ    circular planar patch, and one cylindrical surface Γc xa ¼ x sin θ in the central portion. Similar to the analysis in the previous section, it needs to select a set of locally orthogonal patterns to approximate the pressure modes on each patch. In this case, arguments θ and ϕ in cylindrical coordinates are the two independent variables to define patches. All three patches possess a complete description for the variable ϕ in the range ½0; 2π . In light of the expression for pressure modes, pffiffiffiffiffiffi   Eq. (6), normalized basis g m ϕðxÞ ¼ eimϕðxÞ = 2π is selected as one set of the orthogonal patterns in the ϕ direction. Thus, the determination of the truncation number for another set of local basis function f n0 is simplified to approximate the following function 1 X      F n ðk; xÞ ¼ hn k‖xkÞPm an0 f n0 θðxÞ n cos θ ¼

(40)

n0 ¼ 0

where x is on a line parallel to the axis on the cylindrical surface Γc , and along the radial direction on the two circular patches Γl and Γr , as illustrated in the Fig. 6. Once the truncation number N0 is determined for the basis function f n0 by the algorithm described in the previous section. The pressure distribution on the patch can be decomposed based on the locally orthogonal patterns same as Eq. (37), and the formulation to obtain the local participant coefficients based on the measured signals is same as Eq. (38).         Obviously, integration of polynomials f n0 θðxÞ f n θðxÞ ; n0 ; n ¼ 0; ⋯; N 0 can be accurately evaluated by N0 þ 1 points R 2π iðm  m0 Þϕ Gaussian quadrature. The integration of I1 ¼ 0 e dϕ can be exactly computed by ð2N þ 1Þ point square quadrature [38].

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Fig. 6. Cylindrical holograms.

Therefore, the participant coefficients are numerically evaluated as 0

0

     ∑ wi w j p xij f n0 ðθi Þ g m0 ϕj

N þ 1 2N þ 1

am n0 ¼ ∑

(41)

i¼1 j¼1

~ j ¼ 2N2πþ 1 and ϕj ¼ jw ~ j are weights and the where wi and θi are weights and the abscissas for the Gaussian quadrature, w abscissas of the square quadrature. Please be noted, the abscissas xij in the quadrature are also the position where the   microphones are placed in the measurement. It will result in N0 þ 1  ð2N þ 1Þ measurement on each patch. Once the participant coefficients are obtained on all patches by Eq. (41), the reconstruction process is same as that for planar holograms. 4.3. Configuration on spherical hologram A spherical measurement surface is shown in Fig. 7, which is a conformal patch to the spherical coordinates upon which the FS is obtained. Field modes on the spherical surface are orthogonal. Determination of the field modes on the spherical hologram is actually to identify the spherical harmonic functions based on the measured pressure. The following integral is essentially to be computed by Eq. (26) based on measured pressure Z



^ ^ m0 ^  dΓ δnn0 δmm0 ¼ Γ^ Ym (42) n Γ Yn 0 Γ ^ is a unit spherical surface. Accurate reconstruction now relies on the numerical integration of Eq. (42). The in which Γ   m0   Rπ integration of two spherical harmonics can be separated to the product of I1 and I2 ¼ 0 Pm n cos θ Pn0 cos θ d cos θ , in   which I1 is same as that defined in the previous section. The normalized associated Legendre function Pm n μ is a polynomial pffiffiffiffiffiffiffiffiffiffiffiffiffi   m0   of degree n in μ if m is even and polynomial of degree ðn 1Þ times 1  μ2 if m is odd. If m ¼ m0 , Pm n μ Pn0 μ is thus a 0 0 polynomial of degree ðn þ n Þ in μ. Please be noted that, I1 is equal to zero if m a m . According to analyses in the previous section, the participant coefficients in Eq. (26) can be accurately evaluated by ðN þ1Þ point Gaussian quadrature and ð2N þ 1Þ point square quadrature for variables of θ and ϕ on the spherical surface as 0

am n ¼

N þ 1 2N þ 1    1   ∑ ∑ wi w j p xij Sm n k; x ij 2 jhn ðkrh Þj i ¼ 1 j ¼ 1

(43)

where wi and θi are weights and the corresponding abscissas for the Gaussian quadrature, weights and abscissas of the square quadrature are same as that in Eq. (41). Please be noted, the abscissas xij in the quadrature are also the positions where the microphones are placed in the measurement. Due to conformality of the hologram to the coordinate system of FS, analytical way is available to determine the number and position of the measurement. Thus, it is free of the numerical searching in the determination of the truncation number. In addition, the total number of measurement is the smallest than that requested by the other two holograms. Once the participant coefficients are obtained on the spherical surfaces by Eq. (43), reconstruction will follow the same process of the other two approaches. 5. Numerical examples In this section, numerical examples are set up to investigate the performance of the MRS-based NAH on three types of measurement configurations. First, coefficients reconstruction on the holograms where the radiated field pressure as

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77

Fig. 7. Measurement on spherical holograms.

Table 1 Prescribed participant coefficients for each mode. Order m

Degree n

Real part

Image part

0 1 0 1 2 1 0 1 2

0 1 1 1 2 2 2 2 2

6.8806099118051245E-02 5.3086428069412650E-01 4.0761919704115257E-01 7.1835894320588367E-01 5.3133390656567447E-01 1.0562920332902193E-01 7.7880224182409252E-01 9.0823285787439478E-02 1.5365671759130661E-01

3.1959973518049611E-01 6.5444570775706634E-01 8.1998122278194063E-01 9.6864933023109367E-01 3.2514568182056003E-01 6.1095865874620059E-01 4.2345291896273829E-01 2.6647149077907240E-01 2.8100530253387090E-01

prescribed by superposition of a specified number of modes is studied. Reconstruction for a cubic model is investigated next to demonstrate the accuracy and potential of the proposed NAH for regular model but with corners and shapers. At last, reconstruction for a cylinder with two spherical caps, a representative of irregular model, is examined to investigate the feasibility of the approach for elongated models. In all simulations, Legend function is adopted as the locally orthogonal polynomial. 5.1. Coefficients reconstruction Assume the radiated pressure is generated from a structure with prescribed participant coefficients am n , as listed in Table 1, for modes with degree N up to 2. Therefore, the radiated pressure can be exactly evaluated by Eq. (31) at the specified positions. Reconstructions of the prescribed participant coefficients are examined with the three types of measurement setups, planar holograms in Fig. 5, cylindrical holograms in Fig. 6 and spherical hologram in Fig. 7. Since the modes projected onto each hologram are same regardless of the size, such as a ¼ dx ¼ dy ¼ dz of the planar holograms, ratio a=b of cylindrical holograms, and the radius a of spherical holograms, respectively, the values a and b are set as 1 m, 2 m in simulations. Two frequencies corresponding to ka ¼ π and ka ¼ 9π are analyzed in this case, which is used to validate the approach for lower and higher frequency problems. Numbers and positions of measurement are determined by the algorithm in Section 4.1 for each setup. To investigate the influence of the tolerances εp and εe (εp ¼ εe ) on the final accuracy in the reconstruction of participant coefficients, three tolerances are applied which are denoted as ε1 ¼ 1E-1, ε2 ¼ 1E-2 and ε3 ¼1E-3. Numbers and positions of measurement on the spherical hologram are analytically determined once the degree of the participant modes is prescribed, thus it is no necessary to seek the truncation number on the spherical hologram. However, reconstruction process of the participant coefficients is also performed to validate the theory presented in Section 4.3. Table 2 lists the number of microphones for the three types of holograms determined by the algorithm in Section 4 for different frequencies and tolerances, in which the 1st, 2nd and 3rd coordinates represent the three coordinates in the Cartesian coordinate system, Cylindrical coordinate systems and spherical coordinate system. As indicated in the table, the planar holograms is the worst setup to approximate the modes locally in terms of the number of microphones, which needs totally 6172 for the tolerance ε3 ¼ 1E-3 and frequency ka ¼ 9π . That is because the modes on the planar patch cannot be

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separated but be approximated by two locally orthogonal polynomial basis functions. While azimuth dependence of the modes can be separated from the radial and axial directions in the cylindrical holograms. Therefore, number and position of microphones along azimuth direction can be analytically determined, which are constant value 5 regardless of frequencies and the prescribed tolerances. The largest number of microphones on the cylindrical hologram is obviously for the frequency ka ¼ 9π , which is totally ð25 þ 16Þ5. Because of the spherical hologram is conformal to the coordinates of the modes, analytical approach is available to determine the number and position of microphones, which results in a 35 measurement array regardless of frequency and tolerance along inclination and azimuth directions. It turns out that spherical holograms is the best in terms of number of measurement due to its conformality to the coordinates system of the modes. Relative errors of the reconstructed coefficients as well as the reconstructed radiated sound power based on the number and position determined above are presented in Figs. 8–11. They demonstrate that reconstruction accuracy of the 9 coefficients and the radiated sound power decrease with the tolerance ε, which validates that the proposed reconstruction approach for the three types of holograms is correct. Furthermore, accuracy of the reconstruction with spherical holograms as shown in Figs. 10 and 11 is almost in machine precision. It numerically proves the validity of the analytical way for the determination of number and positions of the microphones. 5.2. A model of regular shape As shown in the above examples, given a set of modes with prescribed coefficients, the number and positions of measurement can be numerically determined, and the participant coefficient of each modes can be accurately reconstructed on the holograms. However, the number of efficient participant modes is hardly to be obtained for a realistic radiator. Therefore, it is inevitable to make an estimation of the most efficient participant modes first and then perform the reconstruction process with the proposed approach. A cube is selected in this case as a radiator of regular shape but with corners and shapers. The radiated pressure is assumed to be equivalently produced by six monopoles locating inside on the three coordinates axes at 7a=4, respectively, as pðyÞ ¼

6 X

Gð x n ; y Þ

(44)

n¼1

Table 2 Number of microphones of the three types of holograms for different frequencies and tolerances. Holograms

Planar Cylindrical Spherical

Frequency

ka ¼ π ka ¼ 9π ka ¼ π ka ¼ 9π ka ¼ π ka ¼ 9π

1st Cord

2nd Cord

3rd Cord

ε1

ε2

ε3

ε1

ε2

ε3

ε1

ε2

ε3

5 10 3 11 — —

9 13 5 14

11 17 7 16

5 10 5 5 3 3

9 13

11 17

5 10 5 16 5 5

9 13 8 21

11 17 11 25

Fig. 8. Relative error of coefficients reconstruction on planar holograms for frequencies (a) ka ¼ π, and (b) ka ¼ 9π.

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Fig. 9. Relative error of coefficients reconstruction on cylindrical holograms for frequencies (a) ka ¼ π, and (b) ka ¼ 9π.

Fig. 10. Relative error of coefficients reconstruction on spherical holograms for frequencies (a) ka ¼ π, and (b) ka ¼ 9π.

Fig. 11. Relative errors of reconstructed sound power for different holograms, tolerances and frequencies (a) ka ¼ π, and (b) ka ¼ 9π:

where xn are the location of monopoles on the axes. The radiated sound power is numerically evaluated by Z 1 i∂p ðyÞ dSðyÞ W ¼ Re pðyÞ 2 kρc∂n S

(45)

in which Euler equation ikρcvn ðyÞ ¼ ∂pðyÞ=∂n is applied, “Re” represents the real part of a complex. In the simulation, size of the cube is set as a ¼ 0:2 m as shown in Fig. 12, analyzing frequency is 340 Hz. The setup of holograms is schematically illustrated in Fig. 12, the planar holograms are away from the surfaces of the cube by Δ, the spherical hologram is away from the equivalent radius r~ , Eq. (16), of the cube model by Δ. As for the cylindrical case, the two

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Fig. 12. Schematic setup for (a) the planar holograms and the front view of cylindrical holograms and (b) top view of cylindrical holograms and spherical holograms.

end circular surfaces are away from the cube by Δ as shown in Fig. 12(a), while the radius of the cylinder on the cylindrical hologram is away from the equivalent radius of the projected square on the topffi view plane by Δ as shown in Fig. 12(b). pffiffiffiffiffiffiffiffiffiffiffi Under this configuration, the equivalent radius of the radiator is r~ ¼ 3=2π a, and the dimensionless vale kr~ ¼ 0:868. According to the relative errors of radiated sound power for different degrees presented in Fig. 4(b), reconstructions are performed by prescribing the largest degree of FS as 2, 3 and 4, respectively. Two types of error for the reconstruction are investigated, one is of the radiated sound power and the other is of infinity norm of pressure, which are defined, respectively, as   εSP ¼ W recon  W =W (46)





εpres ¼ precon  p 1 = pk1

(47)

where p and W are sound pressure and radiated sound power evaluated by Eqs. (44) and (45), precon and W recon are the corresponding reconstructed ones given by the NAH. The influence of the distance Δ to the accuracy is investigated by increasing it from 0.05 m to 0.5 m, which is at most half wavelength. Since the modes are regardless of the size of holograms, the number of measurement positions queried by the algorithm are same for different distance Δ which are listed in Table 3. The mean relative errors of sound power and infinity norm of pressure, which are denoted as ϵSP and ϵPres , are also examined in Table 3. In conjunction with the relative errors visually demonstrated in Figs. 13–15, it could be concluded that all the three types of measurement can deliver the same level of accuracy and the accuracy does increase by prescribing larger degree of FS. Due to the exactly match of the modes on spherical holograms, the results reconstructed by the spherical holograms are a little bit better than the other two approaches. In addition, the spherical hologram has another distinct feature of requiring the smallest number of measurement than the other two approaches. Another phenomenon illustrated in Figs. 13–15 is that relative errors are less influenced by the distance Δ even it is up to half wave length (Δ ¼ 0:5m) in this case. As indicated in Table 3, the number of modes with degree up to 4 is adequate to reproduce the radiated sound power as well as pressure. It means the modes with higher degrees are negligible in this case. Therefore, the accuracy of reconstruction is less affected by the distance Δ except it exceeds a critical value to which the modes with degree less than 4 radiate ineffectively in term of pressure. 5.3. A model of elongated shape Although FS adopted in the approach are in spherical coordinates, elongated shapes are frequently to be encountered in practical problems. To investigate performance of the proposed approaches for elongated shapes, a radiator of cylinder with two spherical caps is setup in this simulation. The radiated pressure is assumed to be equivalently produced by twenty-five h i h i monopoles which are located at xi;j ¼ 0:5a cos θi 0:5a sin θi zj and x4j ¼ 0 0 zj , where θi ¼ iπ =2; ði ¼ 0; 1; 2; 3Þ and zj ¼ jb=3; ðj ¼  2;  1; 0; 1; 2Þ. The radiated sound pressure and sound power are also evaluated by Eqs. (44) and (45), respectively. As shown in Fig. 17, radius of the cylinder is set as a ¼0.1 m, height is varied to make a range of 1:0 rb=2a r 2:0 in the simulation. The distance between surfaces of the radiator and holograms is set fixed as Δ ¼0.1 m for all types of holograms as illustrated in Fig. 17, which can guarantee that r h of the spherical hologram is larger than b=2 by 0.04 m for the largest ratio b=2a ¼ 2:0. Analyzing frequency of this case is set as 1020 Hz, which is chosen to investigate the performance of NAH the for higher frequencies problems.

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Table 3 Numbers of microphones for different degrees and the mean relative errors of Δ for the three types of holograms. Holograms

Planar

Cylindrical

Spherical

Degrees

N ¼2 N ¼3 N ¼4 N ¼2 N ¼3 N ¼4 N ¼2 N ¼3 N ¼4

Number of points

Mean relative errors

1st Cord

2nd Cord

3rd Cord

ϵSP

ϵPres

9 10 11 5 6 7 – – –

9 10 11 5 7 9 3 4 5

9 10 11 8 10 12 5 7 9

5.8959E-05 5.4545E-05 1.0157E-05 1.3352E-04 6.3518E-05 1.0489E-05 1.1112E-05 1.0096E-05 9.8231E-06

3.5698E-2 3.3359E-2 3.3008E-3 3.7531E-2 3.3871E-2 4.3381E-3 2.7835E-2 2.5981E-2 2.7015E-3

Fig. 13. Relative errors of the reconstruction by the NAH with planar holograms for (a) radiated sound power and (b) infinity norm of pressure.

Fig. 14. Relative errors of the reconstruction by the NAH with cylindrical holograms for (a) radiated sound power and (b) infinity norm of pressure.

pffiffiffi Equivalent radius of the model is in the range a r r~ r 2a, therefore the largest non-dimensionless value kr~ ¼ 2:6657. By referring to Fig. 4(b), reconstructions by choosing degrees as 3, 5 and 7 are investigated for each ratio of the model. As demonstrated in Figs. 18–20 for the three types of method, reconstructed sound power are at the same level of accuracy for degrees larger than 3. While accuracies of the reconstructed sound pressure increase as a function of degree for a fixed ratio, and deteriorate along with the increasing ratios for a fixed degree. It implicates that the number and position of measurement determined by the proposed algorithm can capture the radiated sound power accurately. The increased errors of sound pressure along with the ratios illustrate that the larger deformation of the model away from the spherical shape, the worse reconstruction in term of pressure is likely to be yield, even the radiated sound power can be achieved within a

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Fig. 15. Relative errors of the reconstruction by the NAH with spherical holograms for (a) radiated sound power and (b) infinity norm of pressure.

 Fig. 16. Absolute value of hn ðxÞj for different degrees n and x A ½0:6π; 1:2π.

tolerance. The phenomenon is reasonable. Since the modes adopted in the approach is the FS in spherical coordinates, it should be better for models with shape closer to sphere and worse for models with irregular shape away from sphere, such as the elongated studied in this case. It should be noted that reconstructed sound power obtained by the three degrees do show a good convergence for all ratios. However, the reconstructed sound pressure is worse with the degree 7 than by the other smaller degrees since ratio larger than 1.25. It actually reveals a fact that reconstructed sound pressure does not obey monotonous convergence for irregular shapes with respect to degrees in contrast to radiated sound power. In other words, it is not expected to get more accurate pressure reconstruction by setting larger degree for the elongated shapes as demonstrated in figures for the degree  7. It is largely due to the characteristics of spherical Hankel function hn ðxÞ involved in the models. hn ðxÞj is an absolutely increasing function of variable n for fixed parameter x and an absolutely decreasing function of variable x for fixed degree n  [33]. Fig. 16 illustrates hn ðxÞj for different degrees n and variable x ranging from 0:6π to 1:2π which are two extreme distances of the elongated mode to the origin. Therefore, variation of the modes on the two spherical caps and the middle part may be very large, which cannot be balanced. Thus, superposition of the modes for irregular shapes will cause distortion and result in large error. In addition, both the results presented in Figs. 13–15 for the previous example and Figs. 18–20 for this case demonstrate a good consistency for the three types of holograms in terms of errors for radiated sound power as well as sound pressure. Therefore, those three types of holograms can deliver the same level of accuracy for the same number of prescribed degree. However, the spherical hologram is more preferential than the other two methods since it is free of searching the number and positions of measurements and the analytical determined number of measurement points is the smallest one which results in less effort in practical measurement.

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Fig. 17. Schematic setup for (a) the planar holograms and the cylindrical holograms, (b) spherical holograms.

Fig. 18. Relative errors of the reconstruction by NAH with planar holograms for (a) radiated sound power and (b) infinity norm of pressure.

Fig. 19. Relative errors of the reconstruction by NAH with cylindrical holograms for (a) radiated sound power and (b) infinity norm of pressure.

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Fig. 20. Relative errors of the reconstruction by NAH with spherical holograms for (a) radiated sound power and (b) infinity norm of pressure.

6. Conclusions and discussions In this work, a NAH based on the mapping relationship between modes on surfaces of structure and hologram is proposed. The modes adopted in the NAH are FS of the Helmholtz equation in spherical coordinates which are generally independent and not orthogonal except on the spherical surface. The modes on the surface of structure and hologram form a bijective mapping. An analytical way is proposed to prove the mapping relationship by means of the ESM and the boundary integral equation. Uniqueness of the mapping relationship is also verified by the conversation law of radiated sound power. Therefore, the reconstruction procedure can be performed on the holograms and mapped backward for reconstruction. Number of modes prescribed in the MRS-based NAH is crucial to total number of measurement as well as the final reconstruction accuracy. Two approaches are developed to estimate the necessary degree of effective modes. One is based on the multipole expansion of Green's function. It puts a strong request on approximating pressure point wisely within a given tolerance. Generally, a larger value of degree is yield with this method which will result in increased effort in measurement and may cause numerical problems in reconstruction. Besides, corners or sharpers in the model will result in redundant degree. To alleviate the influence of some irregular components of a structure in determining the necessary number of modes, equivalent radius is designed to represent the size of spherical source, which makes the equivalent source have the same averaged radiated sound power per unit area as that for the real model. Another one is built on the energy criteria by exploring the radiation efficiency of the modes on the equivalent spherical source. An upper bounded error is derived for the radiated sound power of a vibrating structure with degree of modes up to a specific value. A relative small value of degree is given by this approach. However, determination of the necessary degree by this approach is only related to the size of equivalent spherical source but nothing to do with the location of measurement. Fortunately, the deficiency can be compensated by the first approach which can judge how many modes can be well captured at a specified position. Therefore, combination of the two approaches can provide a good estimation of the necessary degree in the MRS-based NAH. Once the necessary degree of modes are determined, number and position of measurement, which are also very crucial to the NAH, are investigated. Techniques to approximate the modes on three types of holograms by a set of locally orthogonal patterns are developed. A numerical algorithm is designed to determine the tight bounds for two locally orthogonal patterns on the planar patch. Due to the completeness of polar angles on cylindrical holograms, the algorithm is reduced by one dimension and the number of degree and positions are analytically determined for the local patterns along the polar direction. The number and position of measurement on the spherical hologram are determined by a purely analytical method because of its conformity to the coordinates of modes. At last, numerical examples are elaborately designed to validate the theory of MRS-based NAH. Coefficient reconstructions of a set of prescribed modes demonstrate that the proposed mapping relationship in the NAH is correct. The algorithm to determine the number and position of measurement is also validated by the simulation. Numerical examples show that setting the tolerance εp and εe as 1E-2 in the algorithm are accurate enough for reconstruction, which are the tolerances adopted in the consequent simulations. Simulation of the cubic mode with artificially prescribed boundary condition based on the superposition of monopoles demonstrates that the MRS-based NAH can deliver good results for regular model even with corners and sharpers. However, simulation of the cylinder with two spherical caps shows deficiencies for elongated models. It is because the adopted modes are FS in spherical coordinates and may cause very large variation between different parts of elongated model for large degree. Results obtained for the cubic and elongated models display good consistency for the three types of holograms in terms of errors for radiated sound power as well as sound pressure. It means the three types of holograms can deliver same level of accuracy in the reconstruction. However, spherical hologram is more preferential in practice than the other two methods due to the less effort in measurement.

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The NAH framework pursued in the current work provides a new insight to the reconstruction procedure based on the FS in spherical coordinates. An integral identity of the FS is firstly derived which supplies a useful tool for the analysis of evanescent waves and acoustic radiations. We believe that approaches developed in this paper can supply a guideline to the determination of necessary number of FS, and server to optimize the experiment setup, such as the number and positions of microphone array, for the NAH with FS. The current paper is devoted to the theory development and numerical validation of the MRS-based NAH with FS in spherical coordinates. Experimental study of the approach for engineering practice will conducted in our sequential work, in which error sensitivity analysis and a nose reduction technique are extensively investigated.

Acknowledgments The work is supported by the National Natural Science Foundation of China (Grant no. 11404208).

Appendix  t In this section, brief proofs of the integral identity I Rm n ; Sl , Eq. (14), are presented mathematically for the special case of a spherical surface and numerically for some representative surfaces, as shown in Fig. 21. On the spherical surface, the integral identity will degenerate into Z  

 0 t 2 0 Yn m S^ Ytl S^ dS^ (A1) I Rm n ; Sl ¼ ka jn ðkaÞhl ðkaÞ  hn ðkaÞjl ðkaÞ S^

^ Inner product of normalized spherical harmonic functions on an unit sphere S, δnl δmt . Thus, the proof is converted to seeking the expression of

R

Ym S^ n



^ is obviously equal to S^ Ytl S^ dS,

γ n ¼ j0n ðkaÞhn ðkaÞ  h0n ðkaÞjn ðkaÞ

(A2)

Eq. (A2) is essentially a variant of the cross product of spherical Bessel functions, which was proved in Eq. (13) in the Ref. [34] as

γn ¼

1 iðkaÞ

2

(A3)

 t Therefore, I Rm n ; Sl ¼ δnl δmt =ik proves the integral identity for the case of spherical surface. To the best knowledge of authors, hardly is the analytical approach be obtained on generally non-spherical shapes for the integral identity. Since the identity plays a key role in the determination of efficient modes of a vibrating surface, numerical proofs are provided for the shapes in Fig. 21(b)–(d) on which origins of the coordinates are placed on the geometric centers. In the numerical evaluations, the three shapes are discretized by fine enough triangular elements on which the integrand  t is assumed to have a linear distribution. Thus the numerically evaluated identity I~ Rm n ; Sl is



2 3 J X P

∂Stl k; yjp

∂Rn m k; yjp   X t  m ~I Rm ; St ¼  Sl k; yjp 5 (A4) wjp 4Rn k; yjp l n ∂n yjp ∂n yjp j¼1p¼1 where J denotes the number of triangular elements and P means the number of point in the Gaussian quadrature on each element, wjp and yjp are p-th weight and abscissa of the Gaussian quadrature on the j-th element. Simulations are performed for degrees n and l satisfying 0 r n; l r 14. Relative errors of the simulations are defined as 8    > ~ m t   m t  < I Rn ; Sl 1=ikk; for n ¼ l and m ¼ t (A5) ϵ Rn ; Sl ¼   t > : I~ Rm n ; Sl j ; others

Fig. 21. Models used in the proof of the integral identity: (a) sphere, (b) cube, (c) cylinder, (d) regular tetrahedron.

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Fig. 22. Errors of simultion on the integral identity of the cube for frequencies: (a) k ¼ 10, (b) k ¼ 20.

Fig. 23. Errors of simultion on the integral identity of the cylinder frequencies: (a) k ¼ 10, (b) k ¼ 20.

Fig. 24. Errors of simultion on the integral identity of the tetrahedron for frequencies: (a) k ¼ 10, (b) k ¼ 20.

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Two frequencies of k ¼ 10; 20 are simulated for the three models with the parameter a being one. The errors are plotted in Figs. 22–24, which obviously demonstrates the integral identity are satisfied on the three different convex and closed surfaces. The simulations are numerical validation rather than a complete numerical proof of the integral identity. To be  aware of that special functions jjn ðxÞj and hn ðxÞj go to infinitely small and large along with variable x for fixed degree n, and with degree n for fixed variable x, respectively. Thus, numerical evaluations are prone to give less accurate results, as shown in the left top corners in Figs. 22–24, and even fail to validate the integral identity due to the round off errors. Even though a pure mathematical proof is out of the range authors, it is can be concluded that the integral identity is true for the fundamental solutions with arbitrary degrees and orders on any convex and closed shapes based on the numerical validations and the physical proof in Section 2.1.

References [1] E.G. Williams, J.D. Maynard, E. Skudrzyk, Sound source reconstructions using a microphone array, Journal of the Acoustical Society of America 68 (1980) 340–344. [2] J.D. Maynard, E.G. Williams, Y. Lee, Nearfield acoustic holography: I. Theory of generalized holography and the development of NAH, Journal of the Acoustical Society of America 78 (1985) 1395–1413. [3] W.A. Veronesi, J.D. Maynard, Nearfield acoustic holography (NAH) II. Holographic reconstruction algorithms and computer implementation, Journal of the Acoustical Society of America 81 (1987) 1307–1322. [4] E.G. Williams, Fourier Acoustics Sound Radiation and Nearfield Acoustical Holography, Academic Press, 1998. [5] R. Steiner, J. Hald, Near-field acoustical holography without the errors and limitations caused by the use of spatial DFT, International Journal of Acoustics and Vibration 6 (2001) 83–89. [6] Y.T. Cho, J.S. Bolton, J. Hald, Source visualization by using statistically optimized near-field acoustical holography in cylindrical coordinates, Journal of the Acoustical Society of America 118 (2005) 2355–2364. [7] J. Hald, Basic theory and properties of statistically optimized near-field acoustical holography, Journal of the Acoustical Society of America 125 (2009) 2105–2120. [8] G.T. Kim, B.H. Lee, 3-D sound source reconstruction and field reprediction using the Helmholtz integral equation, Journal of Sound and Vibration 136 (1990) 245–261. [9] M.R. Bai, Application of BEM (boundary element method)-based acoustic holography to radiation analysis of sound sources with arbitrarily shaped geometries, Journal of the Acoustical Society of America 92 (1992) 533–549. [10] W.A. Veronesi, J.D. Maynard, Digital holographic reconstruction of sources with arbitrarily shaped surfaces, Journal of the Acoustical Society of America 85 (1989) 588–598. [11] Z. Zhang, N. Vlahopoulos, S.T. Raveendra, T. Allen, K.Y. Zhang, A computational acoustic field reconstruction process based on an indirect boundary element formulation, Journal of the Acoustical Society of America 108 (2000) 2167–2178. [12] Z. Zhang, N. Vlahopoulos, T. Allen, K.Y. Zhang, Source reconstruction process based on an indirect variational boundary element formulation, Engineering Analysis With Boundary Elements 25 (2001) 93–114. [13] A. Schuhmacher, J. Hald, K.B. Rasmussen, P.C. Hansen, Sound source reconstruction using inverse boundary element calculations, Journal of the Acoustical Society of America 113 (2003) 114–127. [14] M.E. Johnson, S.J. Elliott, K.H. Baek, J. Garcia-Bonito, An equivalent source technique for calculating the sound field inside an enclosure containing scattering objects, Journal of the Acoustical Society of America 104 (1998) 1221–1231. [15] I.Y. Jeon, J.G. Ih, On the holographic reconstruction of vibroacoustic fields using equivalent sources and inverse boundary element method, Journal of the Acoustical Society of America 118 (2005) 3473–3482. [16] A. Sarkissian, Method of superposition applied to patch near-field acoustic holography, Journal of the Acoustical Society of America 118 (2005) 671–678. [17] C.X. Bi, X.Z. Chen, J. Chen, R. Zhou, Nearfield acoustic holography based on the equivalent source method, Science in China, Series E: Technological Sciences 48 (2005) 338–353. [18] C.X. Bi, X.Z. Chen, J. Chen, Sound field separation technique based on equivalent source method and its application in nearfield acoustic holography, Journal of the Acoustical Society of America 123 (2008) 1472–1478. [19] C.X. Bi, J. Stuart Bolton, An equivalent source technique for recovering the free sound field in a noisy environment, Journal of the Acoustical Society of America 131 (2012) 1260–1270. [20] L. Song, G.H. Koopmann, J.B. Fahnline, Numerical errors associated with the method of superposition for computing acoustic fields, Journal of the Acoustical Society of America 89 (1991) 2625–2633. [21] J.B. Fahnline, G.H. Koopmann, A numerical solution for the general radiation problem based on the combined methods of superposition and singularvalue decomposition, Journal of the Acoustical Society of America 90 (1991) 2808–2819. [22] G.H. Koopmann, L. Song, J.B. Fahnline, A method for computing acoustic fields based on the principle of wave superposition, Journal of the Acoustical Society of America 86 (1989) 2433–2438. [23] M.R. Bai, C.C. Chen, J.H. Lin, On optimal retreat distance for the equivalent source method-based nearfield acoustical holography, Journal of the Acoustical Society of America 129 (2011) 1407–1416. [24] S.F. Wu, On reconstruction of acoustic pressure fields using the Helmholtz equation least squares method, Journal of the Acoustical Society of America 107 (2000) 2511–2522. [25] Z. Wang, S.F. Wu, Helmholtz equation-least-squares method for reconstructing the acoustic pressure field, Journal of the Acoustical Society of America 102 (1997) 2020–2032. [26] S.F. Wu, J.Y. Yu, Reconstructing interior acoustic pressure fields via Helmholtz equation least-squares method, Journal of the Acoustical Society of America 104 (1998) 2054–2060. [27] S.F. Wu, Methods for reconstructing acoustic quantities based on acoustic pressure measurements, Journal of the Acoustical Society of America 124 (2008) 2680–2697. [28] A.N. Thite, D.J. Thompson, The quantification of structure-borne transmission paths by inverse methods. Part 1: Improved singular value rejection methods, Journal of Sound and Vibration 264 (2003) 411–431. [29] E.G. Williams, Regularization methods for near-field acoustical holography, Journal of the Acoustical Society of America 110 (2001) 1976–1988. [30] L.G. Copley, Integral Equation Method for Radiation from Vibrating Bodies, Journal of the Acoustical Society of America 41 (1967) 807–816. [31] A.F. Seybert, B. Soenarko, F.J. Rizzo, D.J. Shippy, An advanced computational method for radiation and scattering of acoustic waves in three dimensions, Journal of the Acoustical Society of America 77 (1985) 362–368. [32] M.A. Epton, B. Dembart, Multipole translation theory for the three-dimensional Laplace and Helmholtz equations, SIAM Journal on Scientific Computing 16 (1995) 865–897. [33] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, U.S. Government Printing Office, Washington, 1964.

88

H. Wu et al. / Journal of Sound and Vibration 373 (2016) 66–88

[34] H. Wu, W. Jiang, Y. Zhang, W. Lu, A method to compute the radiated sound power based on mapped acoustic radiation modes, Journal of the Acoustical Society of America 135 (2014) 679–692. [35] N. Gumerov, R. Duraiswami, Fast Multipole Methods for the Helmholtz Equation in Three Dimensions, Elsevier, 2004. [36] J. Rahola, Diagonal forms of the translation operators in the fast multipole algorithm for scattering problems, BIT 36 (1996) 333–358. [37] S. Amini, A. Profit, Analysis of the truncation errors in the fast multipole method for scattering problems, Journal of Computational and Applied Mathematics 115 (2000) 23–33. [38] H.J. Wu, W.K. Jiang, Y.J. Liu, Analysis of numerical integration error for Bessel integral identity in fast multipole method for 2D Helmholtz equation, Journal of Shanghai Jiaotong University 15 (2010) 690–693.