A new pressure-dominant approximation model for acoustic–structure interaction

Applied Acoustics 105 (2016) 116–128

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A new pressure-dominant approximation model for acoustic–structure interaction Heekyu Woo, Young S. Shin ⇑ Division of Ocean Systems Engineering, KAIST, Daejeon, Republic of Korea

a r t i c l e

i n f o

Article history: Received 23 April 2015 Received in revised form 25 November 2015 Accepted 7 December 2015 Available online 21 December 2015 Keywords: External acoustics Acoustic–structure interaction Doubly asymptotic approximation

a b s t r a c t A high accuracy approximation modeling approach for the acoustic–structure interaction problem with a shell structure is presented in this paper. The new approximation model aims to accurately reveal the relationship between pressure and velocity in the acoustic field. The main idea of this model is to separate the velocity terms into a combination of velocity and pressure by using a weighting parameter. The modal analysis was performed to find an appropriate weighting parameter for the new model for the spherical case. The stability range of the model is limited during this process. An approximation model was coupled with the equation of motion of a spherical shell to check the performance of the model. Responses of a spherical shell excited by a plane step wave and cosine-type incident pressure from the new model were compared to the exact solution and solutions from former approximation models such as Doubly Asymptotic Approximations (DAAs). The new proposed model can approximate high accuracy responses in both early and late time. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Acoustic–structure interaction has been studied by many researchers during and after World War II. The vibration of an elastic surface of a structure in compressible fluid can generate a pressure wave around the structure itself. If the density of fluid is low enough, it is possible to neglect the fluctuating pressure. However, if the fluid density is high, interaction between acoustic pressure and structure has to be considered. In this case, the structure can be affected by a radiating and scattering acoustic wave. In underwater explosion situations, the surrounding medium is water, which has sufficiently high mass density. This is why researchers became interested in this topic during and after the war. Several studies were done to obtain an analytic solution to this acoustic–structure interaction problem. In spite of this, only a handful of analytic solutions for only a few limited cases have been found due to the complexity of the interaction. The response of an elastic cylindrical shell to a shockwave was studied first [1–3]. In addition to the response of the cylindrical shell, that of the spherical shell, which is another special case, was also studied [4]. Huang provided the first exact shell response of a hollow spherical shell caused by the special properties of cylindrical and spherical shapes. In those cases, the response of interaction could be ⇑ Corresponding author. E-mail address: [email protected] (Y.S. Shin). http://dx.doi.org/10.1016/j.apacoust.2015.12.002 0003-682X/Ó 2015 Elsevier Ltd. All rights reserved.

expanded in trigonometric function or Legendre polynomial function. After studying special cases, some approximations were developed to deal with complex general geometries. Since it is an acoustic–structure coupled problem, the boundary element method (BEM) is widely used. BEM is efficient for this kind of problem because the boundary integral decreases the dimension from volume to surface integral. Approximations for general geometries can be derived from Kirchhoffs retarded potential formula [5]. Kirchhoff’s formula is obtained from Kirchhoff–Helmholtz integral equation, which is the solution to a wave equation in integral form [5]. Approximations can be categorized into three classes: Earlytime approximations (ETA), Late-time approximations (LTA), and Doubly Asymptotic Approximations (DAAs) [6]. ETA and LTA are the simplest approximation models that can be developed for two limited cases – early time and late time. ETA can be used for a high frequency wave. On the other hand, LTA can be used for a low frequency wave. Among the approximation models, DAA approaches the exact solution in both early-time and late-time limits [7]. The first- and second-order DAAs were developed in the 1970s by matching their trial equations to first- and secondð1Þ

order ETA and LTA. DAA1 and DAA2 denote these first- and second-order DAAs in this paper. DAA methodology was modified ð2Þ

after its initial development to improve the accuracy. DAA2 which is a modified version of the second-order DAA was introduced in 1994 [8–10]. Even though a great deal of DAA research was done for the acoustic–structure interaction problem, some research

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has applied DAA to solve problems besides acoustic–structure interaction [11–14]. Since DAA is the methodology for approximation of the boundary integral, this methodology can be applied to other open boundary problems such as elastodynamics [11,13], bubble dynamics [15], and dam-reservoir interaction problems [12,14]. However, several shortcomings of DAAs have been observed. As mentioned before, DAAs were originally formulated with trial equations, and those trial equations were matched to exact solution for special cases such as a sphere and an infinite cylinder. Thus, it is hard to find a relation between matrix form and modal form equations for special cases. There are also other ð1Þ

performance issues. DAA1 and DAA2 cannot represent the impulse impedance. In some cases, they cannot follow the impulse response very early on because the dominant acoustic scattering ð2Þ

pressure modes are not often represented. DAA2 is hard to apply for general geometry because the equation contains curvature correction terms. The third-order DAA (DAA3) is also introduced in 2000 [16]. The result of the DAA3 is accurate, however, it is complicate to apply. Also, the stability of the model should be considered before application. Lee et al. added the advanced potential in Kirchhoffs retarded potential, and employed the weighted combination of the retarded potential and the advanced potential to improve the performance of the approximation model [17]. In that way, Lee et al.s model was consistent from the early time. However, this model shows a large damping effect in the late time, and the magnitude of the response become smaller than those of other DAAs. In this study, the main shortcoming of Lee et al.’s approximation model, the large numerical damping in the late-time, is improved. To improve the approximation model, some proportion of the velocity term is changed to the pressure term. A first-order approximation model is used for the transformation. The values of a new weighting parameter are determined by a modal study. By using new models, it is possible to get more accurate early-time and late-time responses than with existing approximation models. Next, the approximate model in matrix form is applied to the discretized spherical shell structure. This can prove that the new approximation model is applicable for a general geometry. In Section 2, there is a brief review of existing approximation models including DAAs. Those models are derived from Kirchhoff’s retarded potential formula [8–10,17]. DAAs can be obtained by finding unknown parameters in trial equations with matching to ETA and LTA models. In Section 3, the new approximation model with a new weighting parameter is proposed. A second-order approximation model was obtained by dividing the velocity term into both velocity and pressure terms. In Section 4, an appropriate weighting parameter for the proposed approximation model is chosen through a modal study. The performance of approximation model is compared to existing models. The responses of a spherical shell excited by two different types of pressure are calculated with approximation models. Result such as impedance, resistance, normalized velocity, and pressures are compared with those of existing models. Also, the new approximation model is applied for a discretized spherical shell.

2. Approximation derived from Kirchhoff’s retarded potential formula The Kirchhoff–Helmholtz integral equation is an exact solution of a wave equation in integral form for a spherical wave. It can be derived from the Helmholtz equation and, by applying Green’s theorem equation, can be expressed in the following form [5]: 4p
/ðP; tÞ ¼ 

Z ( S

)

1 @/ðQ; tr Þ 1 @RPQ 1 @RPQ _ þ 2 /ðQ ;tr Þ þ /ðQ ;tr Þ dS RPQ @n cRPQ @n RPQ @n

ð1Þ

where S denotes a bounding surface in a volume of acoustic medium that is assumed to be smooth; P and Q are the location of an integration and the position of the observation on the surface S, respectively; t is the time; c is the speed of sound; RPQ is the distance from point P to point Q ;
is a solid angle that takes on (1, 0.5, 0) depending on whether point P is within the fluid, on surface S, or inside surface S, respectively; / is the velocity potential; tr is R

; and @=@n denotes partial difthe retarded time which is t r ¼ t  PQ c ferentiation along the outward normal to surface S. The velocity potential occupied at time t r and point Q is as follows:

  RPQ ½/r ¼ / Q ; t  ¼ /eikRPQ c

ð2Þ

where ½/r is the retarded potential. It is possible to write the Kirchhoff–Helmholtz integral equation using this retarded potential as follows:

4p
/ðP; tÞ ¼

   ) Z ( @RPQ 1 @/ @RPQ 1 1 @/ dS ½/r   @t r @n cRPQ RPQ @n r @n R2PQ S ð3Þ

Eq. (3) is called Kirchhoff’s retarded potential formula [5]. As the velocity potential can be written in terms of pressure p and the fluid particle velocity in outward normal direction u, it is possible to find out the relationship between pressure and velocity. This formula can be expressed as follows:

) Z ( @RPQ 1 @RPQ 1 q 4p
pðP; tÞ ¼ ½pr  ½p_ r þ ½u_  dS @n R2PQ @n cRPQ RPQ r S

ð4Þ

where q is the fluid mass density. ½ r denotes it is retarded value. This formula has been used to derive an approximation model that can be applied to BEM. 2.1. Doubly asymptotic approximation Solving Kirchhoff’s formula directly requires all time histories at all integral surfaces, and this requirement affects computational efficiency. Singly asymptotic approximations such as ETAs and LTAs have been developed for this reason. However, as computer performance has improved over the years, other approximations methods have been developed. Among the approximations, DAAs have been found efficient and accurate in many interaction problems [7]. DAA asymptotically approaches the exact solution in early time (high frequency range) and late time (low frequency range); however, it cannot cover all time ranges. DAAs are modeled using trial equations with unknown parameters [7–9]. Unknown parameters can be determined by matching ETA and LTA. The original matrix form of the first- and second-order DAAs are written as follows [7]:

M f p_ þ qcAp ¼ qcM f u_

DAA1 : ð1Þ DAA2

:



€ þ qcAp_ þ qcXf Ap ¼ qc M f u € þ Xf M f u_ Mf p



ð5Þ ð6Þ

where M f is the fluid mass matrix [18], A is the area of the surface matrix, and Xf is the frequency matrix. p and u are the vector of nodal surface pressures and the vector of normal fluid particle velocities, respectively. After the first DAA model was introduced, several modified DAA models were developed. A modified secondorder model was introduced to improve the accuracy of DAA [8,9]. To improve accuracy, curved wave approximation was added ð2Þ

ð2Þ

to DAA2 , but DAA2 is hard to implement to general geometry because of mean curvature j and Gaussian curvature s terms. The curvature term has to be a known value at every node in complex ð2Þ

geometry. The matrix form of DAA2

contains complex integral

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operations in their respective formulas, and the matrix formulation as follows: ð2Þ

€ þ cðX 0 þ K Þp_ þ c2 X 0 B1 Cp ¼ qcðu € þ cX 0 u_ Þ DAA2 : p

ð7Þ

where

h ih i1 X 0 ¼ B1 C  K I  B1 AB1 C Z K ¼ J 1 wðX ÞjðX Þv T ðX ÞdS A ¼ J 1 B ¼ J 1

Z

S

Z wðX ÞdS 

Z

S

Z wðX Þ

S

C ¼ J 1

S

Z

Z

wðX Þ S

S

S

ð8Þ ð9Þ

v ðX ÞdS

ð10Þ

 0 jX 0  Xj1 v T X 0 dS dS

ð11Þ

   0 jX 0  Xj3 X 0  X  n X 0 v T X 0 dS dS

ð12Þ

Here, X denotes a point on the surface of the shell, v is the column vector of shape functions, w is the column vector of weight R functions, and J is S wðX Þv T ðX ÞdS. n is the unit normal to the surface-outward direction, and r is the surface Laplacian. The modal form of all DAA models were also introduced. In this paper, these modal forms are used to compare the result to that of the proposed model. These modal forms of DAAs are not derived directly from their matrix forms. Trial equations of each model are matched to the exact modal transfer function of the spherical case [9]. The following equations are modal DAA models in the Laplace domain for the spherical case: 2 s

n þ ð1 þ nÞp  n ¼ su n DAA1 : sp ð1Þ DAA2 ð2Þ DAA2

2

ð13Þ

n ¼ s un þ ð1 þ nÞsu n þ ð1 þ nÞ p n : s pn þ ð1 þ nÞsp 2

2

2

 n þ nsu n þ nð1 þ nÞp n ¼ s2 u n : s pn þ ð1 þ nÞsp

ð14Þ ð15Þ

The bar over p and u means that terms are in the Laplace domain, and n denotes the mode number. Applying the initial value theorem to these models, neither DAA1 nor DAA2 satisfy the specific acoustic impedances in low mode. The specific acoustic impedance, pn =un , of DAAs does not match ðpn =un Þexact while t ! 0 [17,19].

 2p ðP; sÞ þ c2 B2 p ðP; sÞ þ ð1 þ vÞcA1 sp ðP; sÞ vAs 2 2   ¼ vqcAs uðP; sÞ þ qc A1 suðP; sÞ

ð18Þ

 A1 , and B2 are the boundary integral operators as where the A; follows:

 Z 1 Z 1 1 @RPQ ðQ; sÞdS 2 q dS dS S RPQ S RPQ @n S Z 1 ðP; sÞ ¼ ðQ ; sÞdS q A1 q R PQ S Z 1 @RPQ ðQ ; sÞdS þ 4p
dðP  QÞ ðP; sÞ ¼ q B2 q 2 S RPQ @n

q ðP; sÞ ¼ A

Z

ð19Þ ð20Þ ð21Þ

The d in the boundary operator is the Kronecker delta. This approximation model shows an exact early-time response, although its late-time response shows larger error than DAA2. The response of this model is also included in the results in a later section. 3. Proposed model with new weighting parameters In this section, a new approach is introduced to remedy the shortcomings of Lee et al.’s model, and a new approximation model is proposed. The main shortcoming of Lee et al.’s model is the damping error. Their model has larger error in the damping ratio than other models, as shown in Fig. 8, and its effect is shown in Figs. 10–20. To reduce the damping effect of the approximate model, other handling is applied to the model. The main idea of the proposed model is changing some part of the fluid particle velocity term into the pressure term by using a first-order approximation model. By doing this, the goal of reducing the damping effect is achieved. It is already known, by potential theory, that there is a relationship between velocity and pressure. Thus, it is possible to change one term to another. Here, the simplest relationship that can be derived from Kirchhoff’s retarded potential formula is used. The simplest case, v ¼ 0, has been used for the conversion. If v ¼ 0, Eq. (18) will be reduced to a first-order equation, which is the same as with DAA1. This v ¼ 0 means that retarded and advanced potential has same ratio in modified potential.

2.2. Approximation model adopting modified velocity potential

 þ cA1   sp 1 B2 p ¼ sqc u

To solve the early-time inconsistency problem and match the pn =un to ðpn =un Þexact , Lee et al. introduced a modified velocity potential that is a combined form of retarded and advanced potentials [17,19]. The definition of the modified velocity potential is as follows:

To apply this relation to the approximation model, first, the right-hand side of Eq. (18) was split into two parts using new weighting parameter a.

1 1 ð1  vÞ½/r þ ð1 þ vÞ½/a 2 2

r ¼ /eikRPQ ½/a ¼ / Q ; t þ c

/mod ¼

ð16Þ ð17Þ

where ½/a is the advanced potential which has the meaning of an out-going wave. Modified velocity potential is expressed as a combination of these two potentials using weighting parameter v. By changing weighting parameter v, it is possible to change the ratio between the retarded and advanced potentials. Following the same process in obtaining the retarded potential formula shown in the previous section, it is possible to get Kirchhoff’s formula for the modified velocity potential. An approximation model with a weighting parameter v can be derived by taking the Laplace transform of the modified potential, and expanding the exponential terms in terms of the Taylor expansion series. After adopting the plane wave approximation, @R ! 1, an approximation model can @n be arranged as follows [17]:

 2p ðP; sÞ þ c2 B2 p ðP; sÞ þ ð1 þ vÞcA1 sp ðP; sÞ vAs    þ ð1  aÞðvsA þ cA1 Þsqcu  ¼ aðvsA þ cA1 Þsqcu

ð22Þ

ð23Þ

 term in Eq. (23), After substituting Eq. (22) into the second sqcu the equation can be written as follows:

i  1 B2 sp   þ c2 B2 p ðv þ aÞcA1  ð1  aÞvcAA 1 a    ¼ vsA þ cA1 sqcu

vs2 A p þ

1h

ð24Þ

This is a new approximation model. Eq. (24) shows that only the second term of the proposed model is different from Lee et al.’s model. Thus, this model can reduce the amplitude error by controlling the dominance between the velocity and pressure parts with a value of weighting parameter a. 4. Modal analysis of the proposed models In this section, modal analysis was performed to find out the appropriate weighting parameter and accuracy of a new approxi-

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mation model. This study was done for the spherical shell case, which is one of the special cases that has an exact solution. Thus, the results from an approximation model can be compared with its exact solution. A weighting parameter was chosen from acoustic impedance comparison. The performance of the approximation model was checked by case studies with two types of incident pressure. 4.1. Modal equations of approximation models There are two special cases that have the exact solutions. One is the cylindrical shape case, and the other is the spherical shape case. In the case of a sphere-shaped structure, pressure and displacement on the surface of shell can be expanded as in the Legendre polynomial series [20], and the model can be changed into modal form in the Laplace domain. Through the modal analysis, it is possible to understand the characteristics of interaction phenomena and to find the meaningful relationships. For simplification, terms are non-dimensionalized via

t ¼ tc ; a

W w¼ ; a

v

V ¼ ; a

R r¼ ; a

P p¼ 2 qc

ð25Þ

where V; W and P are meridional displacement, radial displacement, and pressure on the surface of the shell, respectively. There is a well-known relationship between velocity potential and pressure. It is possible to rearrange the wave equation and get the relationship among the incident wave pressure pi , the scattering wave pressure ps , and the fluid particle velocity u.

_ tÞ pðr; tÞ ¼ /ðr; @/  uðr; tÞ ¼  ðr; tÞ @r @p  @pi @ps ð1; t Þ ¼ ð1; tÞ þ ð1; tÞ ¼ u_ @r @r @r

ð28Þ

where jn is the modified Bessel function of the third kind as follows [22]:

2

j0n ðzÞ ¼ Cnm ¼

ð30Þ

m¼0

djn dz

ð31Þ

ðn þ mÞ! 2 m!ðn  mÞ!

ð32Þ

m

PQ

@R

related to RPQ into r; a, and h. @nPQ is the same to cosw, and RPQ itself can be express as Eq. (34) by trigonometric identity.

@RPQ r cos h  a ¼ cos w ¼  1=2 2 @n ðr þ a2  2ar cos hÞ

ð33Þ

R2PQ ¼ r 2 þ a2  2ar cos h

ð34Þ

terms into the Legendre

1 X 1 1 r n ¼ Pn ðcos hÞ RPQ a a n¼0

ð35Þ

1 @RPQ cos w r cos h  a ¼ 2 ¼ 3=2 R2PQ @n RPQ ðr 2 þ a2  2ar cos hÞ @ 1 @a ðr2 þ a2  2ar cos hÞ1=2 1 X ðn þ 1Þrn ¼ Pn ðcos hÞ anþ2 n¼0 ¼

ð36Þ

Now, we can get the modal form for the spherical shell case by  is substituting Eqs. (35) and (36) into Eqs. (19)–(21), respectively. p Legendre expansion, and the orthogonality of Legendre polynomials is used to obtain the following equations.

Z

1 @RPQ ðQ; sÞdS p R2PQ @n 1 X 4pa2 ðn þ 1Þ r n n ðr; sÞ; Bn2 ¼ Bn2 p ¼ 2n þ 1 anþ2 n¼0 Z 1  ¼ ðQ ; sÞdS A1 p p S RPQ 1 X 4pa2 rn n ðr; sÞ; An1 ¼ An1 p ¼ 2n þ 1 anþ1 n¼0 Z

1 Z Z 1 1 @RPQ p ðQ ; sÞdS 2  ¼ dSQ p A dSQ S RPQ S RPQ @n S 1 2 n X  n ¼ 4pa r np n ðr; sÞ; A ¼ A 2n þ 1 an n¼0  ¼ B2 p

S

Z

4.1.1. Modal equations of approximation models Modal equations of approximation models can be obtained by using the geometrical property of the structure because this is a spherical shell case. From Fig. 1, it is possible to change all terms

@RPQ @n

polynomial series [23].

ð27Þ

ð29Þ

n p X jn ðzÞ ¼ ez Cnm zðmþ1Þ

Deploying the known properties of the spherical coordinate system, it is possible to change R1PQ and R12

ð26Þ

With these terms, the relationship between the scattering pressure and the fluid particle velocity on the surface of the spherical shell can be derived from the wave equation in a spherical coordinate. The exact mode-by-mode solutions for the spherical wave equation is expressed as follows [21]:

j0n ðsÞ s  n ðsÞ p ðsÞ ¼ u jn ðsÞ n

Fig. 1. Spherical shell surrounded by acoustic medium.

(

1

1

Pn ðxÞPm ðxÞdx ¼

0

for m – n

2 2nþ1

for m ¼ n

ð37Þ

ð38Þ

ð39Þ

ð40Þ

After non-dimensionalization, finally modal forms of approximation models are obtained.

1 1 1  þ ð1 þ nÞp n ¼ s2 u  ð41Þ n þ ð1 þ Þsp  n þ su Lee etal: : s2 p vn n vn vn n    v 1 1  þ ð1 þ nÞp n n þ 1  n n þ 1 þ sp Proposed : s2 p fn vn n vn 1 n  n þ su ¼ s2 u ð42Þ

vn

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H. Woo, Y.S. Shin / Applied Acoustics 105 (2016) 116–128

where fn ¼ an vn . This fn is introduced to simplify the term in the approximation model. Again, this weighting parameter fn is only included in the second term on the left-hand side. Thus, it is possible to control the damping effect of the approximation model by choosing an appropriate value of the weighting parameter. The new weighting parameter fn is formed up with vn . vn has a matrix form for a discretized BEM model. Thus, it is expected that the new weighing parameter fn is adoptable for the discrete model case. Finding the value of the new weighting parameter, the exact mode-by-mode wave solution’s roots for a sphere were used. Those roots can be obtained from Eq. (29). Fig. 2 shows mode-by-mode poles of a pressure characteristic equation. At the mode numbers ð2Þ DAA2

n ¼ 0 and n ¼ 1, the roots of are located away from the exact roots. The exactness in the low mode is the strong point of Lee et al.’s approximation model. To keep this advantage, the value of the new weighting parameter has to be fn ¼ 1 at the n ¼ 0 and n ¼ 1. The characteristic roots of the proposed model have to be

moved to the right-side in the real axis after n ¼ 2. In that sense, the new weighting parameter was chosen as follows:

v1 n ¼ n þ dð0Þ

ð43Þ

f1 n

ð44Þ

i 1h ¼ 1 þ 2ðn þ dð0ÞÞ  ðn þ dð0ÞÞ1 2

where dð0Þ has a value of 1 if n ¼ 0, and has value 0 if n > 0. With these parameters, the approximation model can be reduced as follows:

  1 1 1 n þ ð1 þ nÞp n ¼ s2 u  n ð45Þ n þ 1 þ  n þ su sp Lee etal: : s2 p

vn

n þ ¼ s2 u

n=0 Huang (exact) DAA2 Lee et al. Present

Imag

Imag

0 −2

−4

−4

−3.5

−3

−2.5

−2

−1.5

−1

−6 −4

−0.5

−3.5

−3

Real

4

4

2

2

Imag

Imag

6

0

−2

−4

−4

−2.5

−2

−1.5

−1

−6 −4

−0.5

−3.5

−3

4

4

2

2

Imag

6

0

−2

−4

−4

−2

Real

−2.5

−2

−1.5

−1

−0.5

−1.5

−1

−0.5

0

−2

−2.5

−0.5

n=5

n=4 6

−3

−1

Real

Real

−3.5

−1.5

0

−2

−3

−2

n=3

6

−3.5

−2.5

Real

n=2

Imag

ð46Þ

2

−2

−6 −4

n su

4

0

−6 −4

1

vn

6

2

−6 −4

vn

n=1

6 4

vn

  1 1 1 n þ vn  1 vn n þ 1 þ spn þ ð1 þ nÞpn Proposed : s2 p 2 vn vn

−1.5

−1

−0.5

−6 −4

−3.5

−3

−2.5

−2

Real

Fig. 2. Mode-by-mode characteristic roots of the pressure equation for a spherical shell.

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H. Woo, Y.S. Shin / Applied Acoustics 105 (2016) 116–128

n=1

n=1

1.5

1.5

1

Reactance

Resistance

Huang (exact) DAA2 Lee et al. Proposed

0.5

0 −1 10

1

0.5

10

0

0 −1 10

1

10

0

1

10

λst/λac

10

λst/λac

Fig. 3. Specific acoustic impedance of approximation models for a sphere, n ¼ 1.

Fig. 2 shows mode-by-mode poles of a pressure characteristic ð2Þ

equation. At n ¼ 0 and n ¼ 1, only roots of DAA2 are different from ð2Þ

exact roots. This mismatch of the roots of DAA2 affects the response. After n ¼ 2, the characteristic roots of the proposed model are located close to the dominant exact roots. It is possible to locate the roots of the proposed model closer to the exact soluf1 n ,

tion by changing the weighting parameter but doing so affects the stability of the structure-pressure coupled equation. 4.1.2. Specific acoustic impedance of approximation models Acoustic impedance shows how much pressure can be given by the particle velocity in an acoustic medium at a given frequency. The specific acoustic impedance Z is the ratio of pressure and velocity. Each approximation model can give the relationship between pressure and particle velocity. Specific acoustic impedance is defined as

Z ð xÞ ¼

pðxÞ ¼ RðxÞ þ iX ðxÞ uðxÞ

ð47Þ

where RðxÞ is the resistive part and X ðxÞ is the reactive part of the impedance. Resistance represents the energy transfer, pressure is in-phase with motion, and reactance represents pressure that is out of phase with the motion when there is no net energy transfer. To find out the characteristics of each models, specific acoustic impedances of models are calculated. Figs. 3–5 show the resistance and the reactance of models at mode n ¼ 1 to mode n ¼ 3, respectively. The structural wavelength for the surface motion is given by kst ¼ 2pa=ðn þ 1Þ, and the acoustic wavelength is kac ¼ 2pc=x [24]. In Fig. 3, it is shown that the resistance and reactance of the new proposed models are exactly the same as the exact acoustic impedance at mode n ¼ 1. At n > 1, the resistance and reactance of the proposed model are almost the same and are always larger than in Lee et al.’s model. The reactances of the proposed model follow exact values until n ¼ 2; however, all approximation models have large error at higher modes. The resistance and reactance of ð2Þ

DAA2 are not related to the mode number and cannot follow exact impedance in any mode. 4.2. Modal acoustic–structure interaction equation

the surface of an elastic spherical shell can be expanded as in the Legendre polynomial series [8,20]. Thus, coupling with the modal wave equation is possible. Using this expansion, the modal equations of motion for an elastic spherical shell can be expressed as a matrix equation [20]:

v ðh; tÞ ¼

1 X

v n ðtÞ

n¼0

wðh; tÞ ¼

1 X

d Pn ðcos hÞ dh

ð48Þ

wn ðtÞPn ðcos hÞ

ð49Þ

n¼0

"

kn s2 þ Avv n Avn w

Avn w 2 s þ Aww n

8 9 #> v n >

= 0 0 < n ¼ w i n > s > lp l : n ; p

ð50Þ

Avv n ¼ kn ð1 þ bÞnn c0 Anv w ¼ kn ð1 þ m þ bnn Þc0 Aww ¼ ½2ð1 þ mÞ þ kn bnn c0 n

ð51Þ 2

where l ¼ ðq=qs Þða=hÞ, c0 ¼ c20 =c2 , b ¼ ðh=aÞ =12, kn ¼ nðn þ 1Þ, and nn ¼ kn  1 þ m. q; qs ; a; h; c0 ; c, and m are the density of the acoustic medium, density of the shell, plate velocity, sound velocity and Poisson’s ratio, respectively. To satisfy geometric compatibility on the surface of the spherical shell, the shell velocity on the surface has to be the same as the fluid particle velocity. Therefore, modal pressure equations can be coupled with equations of the motion of a spherical shell.

_ n ðtÞ ¼ uin ðtÞ þ usn ðtÞ w

ð52Þ

Exact :

 n þ jn u  in j0n psn ¼ jn sw

DAA1 :

sn ¼ s3 w  n  s2 u  in ½s þ ðn þ 1Þp h

ð1Þ

DAA2 :

ð53Þ ð54Þ

i

sn s2 þ ðn þ 1Þs þ ðn þ 1Þ2 p

 n  ½s þ ðn þ 1Þsu  in ¼ ½s þ ðn þ 1Þs2 w     1 1 sn s þ ðn þ 1Þ p s2 þ 1 þ vn vn     1 2 1  in n  s þ s w su ¼ sþ

ð55Þ

Lee et al: : 4.2.1. Modal acoustic–structure interaction equation Next, a modal acoustic–structure interaction equation is constructed to check the performance of each approximation model. In a spherical coordinate, radial and meridional displacements on

vn

vn

ð56Þ

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n=2

n=2

1.5

1.5 Huang (exact) DAA2 Lee et al. Proposed

Reactance

Resistance

1

0.5

1

0.5

0 −1 10

10

0

0 −1 10

1

10

10

λst/λac

0

1

10

λst/λac

Fig. 4. Specific acoustic impedance of approximation models for a sphere, n ¼ 2.

n=3

n=3

1.5

1.5 Huang (exact) DAA2 Lee et al. Proposed

Reactance

Resistance

1

0.5

1

0.5

0 −1 10

10

0

1

10

0 −1 10

10

λst/λac

0

1

10

λst/λac

Fig. 5. Specific acoustic impedance of approximation models for a sphere, n ¼ 3.

 

1 1 1 vn  1 vn n þ 1 þ s þ ð1 þ nÞ psn 2 vn vn     1 2 1  in n  s þ s w su ¼ sþ

Proposed :

s2 þ

vn

vn

ð57Þ Now, we can combine the modal equations of motion for an elastic spherical shell, Eq. (51), and the modal pressure equation and obtain a coupled modal equation.

2 6 4

kn s2 þ Avv n

Anv w

Anv w

s2 þ Aww n

0

sQ n ðsÞ

9 8 9 38 0 > > < v n > = > < = 7  l 5> wn > ¼ > lpin > : s ; :  in ; Q n ðsÞu pn Rn ðsÞ 0

Exact :

Q n ðsÞ ¼ jðsÞ; Rn ðsÞ ¼ j0 ðsÞ

DAA1 :

Q n ðsÞ ¼ s; Rn ðsÞ ¼ s þ ðn þ 1Þ

ð1Þ

ð2Þ

ð58Þ

Q n ðsÞ ¼ sðs þ ðn þ 1ÞÞ; Rn ðsÞ ¼ s2 þ ð1 þ nÞs þ ð1 þ nÞ2   Lee etal: : Q n ðsÞ ¼ s vn s þ 1 ; Rn ðsÞ ¼ vn s2 þ 1 þ vn s þ ðn þ 1Þ  Proposed : Q n ðsÞ ¼ s vn s þ 1 ; h

i  Rn ðsÞ ¼ vn s2 þ 1  vfnn vn n þ vn þ 1 s þ ðn þ 1Þ DAA2 :

To solve the above equation, it is possible to treat Eq. (58) itself as a transfer function. The characteristic roots of the modal acoustic–structure interaction equation for a spherical shell are shown in Figs. 6 and 7. The real part of the roots is related to damping, and the imaginary part is related to phase. Fig. 7 shows a magnified view of dominant roots. In the n ¼ 2 case, a root of the proposed approximation model is the closest to the exact root. After n ¼ 2, the real part of a root of the proposed approximation model is located closer to the exact solution than the roots of any other approximation models. On the other hand, the imaginary part of DAA2 is located closer to the exact solution than roots of other approximation models after n ¼ 2. We can now expect that ð2Þ

DAA2 will perform better in low frequency approximation and that the proposed approximation model will be better in high freð2Þ

quency approximation; moreover, DAA2 will show phase difference in the late-time. Fig. 8 shows the damping ratio of this coupled system. The exact model has a very low damping ratio in every mode. Lee et al.’s model shows the largest error in every mode. The proposed approximation model has larger error than ð2Þ

that of the DAA2 model only in the second mode and has smaller error in all other modes.

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n=2

n=2 15 10

Exact solution DAA2 Lee et al. Proposed

1.22 1.21

Imag

Imag

5 0

1.2 1.19

−5

1.18

−10

1.17

−15 −15

−10

−5

0

−0.06

5

−0.05

−0.04

−0.03

−0.02

Real

Real

Fig. 6. Mode-by-mode free vibration characteristic roots of coupled acoustic–structure interaction equation for a spherical shell submerged in water for n ¼ 2.

n=2

Imag

1.21

Exact solution DAA2 Lee et al. Proposed

1.89 1.88

Imag

1.22

n=5

1.2

1.87

1.19 1.86 1.18 1.85 1.17 −0.06

−0.05

−0.04

−0.03

−0.02

−0.03

−0.02

−0.01

Real

Real

n=8

n = 11

0

0.01

2.255

2.595

2.25

2.59

2.245

Imag

Imag

2.26

2.24

2.585 2.58

2.235

2.575

2.23 2.225

2.57 −15

−10

−5

Real

0

5 x 10

−3

−10

−8

−6

−4

Real

−2

0

2 x 10

−3

Fig. 7. Mode-by-mode free vibration characteristic roots of coupled acoustic–structure interaction equation for a spherical shell submerged in water for n ¼ 2 through n ¼ 11.

4.2.2. Stability check for proposed approximation models DAAs and Lee et al.’s model are unconditionally stable. However, the proposed approximation model is a conditionally stable model. It was found that the value of a affects the value stable range of the a=h. To find the stable range, the Routh–Hurwitz stability criterion was used for a stability check. To apply the proposed model to a wide range of a=h; a has to approach to 1, which means the proposed model is similar to Lee et al.’s model. However, in this study, the limitation of stable range has been set to increase the accuracy of the approximation model. This means that the proposed model is only applicable for special cases. Finally, the values of weighting parameters are decided as Eq. (44).

1 The proposed approximation model, which uses f1 n ¼ 2 ½1þ

2ðn þ dð0ÞÞ  ðn þ dð0ÞÞ1  is stable for aspect ratio a=h < 220. This range of a=h is reasonable for the underwater structure. 4.2.3. Cesaro summation Cesaro summation has been used to achieve the solution for transient responses in this paper. This superposition technique is effective in reducing oscillations and can improve convergence [25]. For this kind of problem, the modal solutions for n ¼ 1 and n ¼ 2 cases are important. Standard Cesaro summation gives weights of less than 1 for n ¼ 1 and n ¼ 2, so the procedure has been modified as follows [8,10]:

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Damping ratio Huang (exact) DAA2 Lee et al. Proposed

0.045 0.04

Damping ratio

0.035 0.03 0.025 0.02 0.015 0.01 0.005 Fig. 9. A submerged spherical shell excited by plane incident wave.

1

2

3

4

5

6

7

8

9

10

11

12

Mode number Fig. 8. Mode-by-mode damping ratio of coupled acoustic–structure interaction equation for a spherical shell submerged in water for n ¼ 1 through n ¼ 12.

x¼

N X C n xn

ð59Þ

n¼0

where

( Cn ¼

1

for 0 6 n 6 2

Nþ1þn N1

for 3 6 n 6 N

In modified Cesaro summation, the weights for n ¼ 0; 1; 2 are all the same, and the convergence of the solution can be improved with this modification [10]. In the case study sections, Huang’s solution using this modified Cesaro summation is considered as the exact solution [4]. 5. Shock analysis

ð2Þ

This is the case in which a submerged elastic spherical shell is excited by a plane step wave in the direction of h ¼ p as in Fig. 9. In this study, the following parameters are used: a=h ¼ 100; qs =q ¼ 7:7 and c0 =c ¼ 3:7. The following equations are the incident wave pressure and incident wave velocity for the plane step wave [10].

Z 2n þ 1 p Hðt  cos h  1ÞPn ðcos hÞ sin hdh 2 Z0 p 2n þ 1 uIn ¼ cos hHðt  cos h  1ÞP n ðcos hÞ sin hdh 2 0

pIn ¼

2n þ 1 2

uIn ¼

2n þ 1 2

c aT1

1 Z acT1 1

In the previous subsection, a spherical shell was excited by a plane step wave. In that case, the high frequency term disappears

Plane step wave case, front side (θ = π) 0.5

ð61Þ

Pn ðzÞdz

ð62Þ

zP n ðzÞdz

ð63Þ

Figs. 10 and 11 show a normalized radial velocity response on the surface of the spherical shell at h ¼ p, where the incident plane step wave originates. All approximation models show very good performance in very early-time. However, error increases as time goes on. Fig. 11 shows the late-time response on the front side, ð2Þ

5.2. Case study 2: cosine-type impulsive pressure

ð60Þ

Hð Þ is the Heaviside function. It is possible to change Eqs. (60) and (61) into time-related functions, and doing so is more convenient for computation.

Z

ð2Þ

side, h ¼ 0. It is not difficult to find that the period of DAA2 is shorter than the period of Huang’s solution in late-time. The proposed approximation model also shows better response than the others at this point. For an error measure, Geers’ Magnitude, Phase, and Comprehensive Error Factors, Zilliacus’ Error Index, and RootMean-Square deviation are adopted [26–28]. By these factors, it is easy to check the error of each response. M, P, C, Z, and RMSD in Tables 1–4 are Geers’ Magnitude, Phase, and Comprehensive Error Factors, Zilliacus’ Error Index, and Root-Mean-Square deviation, respectively. The Error Factors for the response of the front side are in Table 1. In Table 1, M, P, C, Z, and RMSD of the proposed model are 46%, 44%, 45%, 65%, and 67% of Lee et al.’s model, respectively. All error factors imply that the error of the proposed model is much smaller than the error of the Lee et al.’s model. The Error Factors for the response of the back side are also in Table 2. In Table 2, M, P, C, Z, and RMSD of the proposed model are 51%, 52%, 51%, 71%, and 72% of the Lee et al.’s model, respectively. However, Error factors for DAA2 and the proposed model show conflicting results. It is hard to say which one has a smaller error.

5.1. Case study 1: plane step wave

pIn ¼

approximation model outperforms the others in this time duration. Figs. 12 and 13 show normalized radial velocity at the opposite

h ¼ p. As we expected, DAA2 shows phase error in late-time, and Lee et al.’s model shows large amplitude error. The proposed

Huang (exact) DAA2 Lee et al. Proposed

0 −0.5

Normalized velocity

0

−1 −1.5 −2 −2.5 −3 −3.5

0

2

4

6

8

10

Normalized time (tc/a) Fig. 10. Normalized radial velocity of submerged spherical shell excited by plane incident wave at h ¼ p using modal equations (n ¼ 0  7).

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Plane step wave case, front side (θ = π)

Table 1 Error factors for the normalized radial velocity of submerged spherical shell excited by plane incident wave at h ¼ p.

0.5 Huang (exact) DAA2 Lee et al. Proposed

0

Normalized velocity

−0.5 −1

DAA2

Lee et al.

Proposed

M P C Z RMSD

0.0025 0.0054 0.0060 0.0867 0.2158

0.0145 0.0059 0.0157 0.0875 0.2247

0.0066 0.0026 0.0071 0.0570 0.1502

−1.5 −2

Table 2 Error factors for the normalized radial velocity of submerged spherical shell excited by plane incident wave at h ¼ 0.

−2.5 −3 −3.5 10

12

14

16

18

20

Normalized time (tc/a) Fig. 11. Normalized radial velocity of submerged spherical shell excited by plane incident wave in late-time at h ¼ p using modal equations (n ¼ 0  7).

Error factor

DAA2

Lee et al.

Proposed

M P C Z RMSD

0.0029 0.0043 0.0052 0.0786 0.1954

0.0157 0.0046 0.0164 0.0805 0.2030

0.0080 0.0024 0.0083 0.0568 0.1453

Table 3 Error factors for the normalized radial velocity of submerged spherical shell excited by cosine impulsive pressure at h ¼ p.

Plane step wave case, front side (θ = 0) 4 3.5 3

Normalized velocity

Error factor

2.5

Error factor

DAA2

Lee et al.

Proposed

M P C Z RMSD

0.0137 0.0111 0.0177 0.1412 0.1225

0.0363 0.0124 0.0384 0.1507 0.1307

0.0177 0.0091 0.0199 0.1259 0.1109

2 1.5

Table 4 Error factors for the normalized radial velocity of submerged spherical shell excited by cosine impulsive pressure at h ¼ 0.

1 Huang (exact) DAA2 Lee et al. Proposed

0.5 0 −0.5 0

2

4

6

8

10

Normalized time (tc/a) Fig. 12. Normalized radial velocity of submerged spherical shell excited by plane incident wave at h ¼ 0 using modal equations (n ¼ 0  7).

Plane step wave case, front side (θ = 0) 4

Error factor

DAA2

Lee et al.

Proposed

M P C Z RMSD

0.0107 0.0133 0.0171 0.1417 0.1234

0.0475 0.0145 0.0497 0.1505 0.1307

0.0265 0.0106 0.0286 0.1260 0.1109

fast and low frequency vibration is the main response. To observe the effect of high-frequency response, the incident pressure is changed from a plane step wave to cosine-type impulsive pressure. Cosine-type impulsive pressure excites the spherical shell as shown in Fig. 14 at time tc=a ¼ 0. The equation of this impulsive pressure can be expressed as follows:

3.5

pIn ¼ 

Normalized velocity

3

2 1.5 1 Huang (exact) DAA2 Lee et al. Proposed

0 −0.5 10

12

14

16

18

Z p

cos hdðtÞU n ðcos hÞPn ðcos hÞ sin hdh

ð64Þ

0

This kind of impulsive pressure can be a very tough case in which to calculate the response of the structure. In this case, the existing DAA models cannot predict the response accurately. Figs. 15–17 show a normalized radial velocity response on the surface of as spherical shell at h ¼ p. These figures show that these results contain more high-frequency terms than the plane step incident wave case in early-time. That means that higher modes

2.5

0.5

2n þ 1 2

ð2Þ

are used to calculate the response. In Fig. 16, DAA2 shows large ð2Þ DAA2

20

Normalized time (tc/a) Fig. 13. Normalized radial velocity of submerged spherical shell excited by plane incident wave in late-time at h ¼ 0 using modal equations (n ¼ 0  7).

error the peak value of is over 50% larger than the exact model at the first peak, because of the early-time inconstancy and inexactness of low mode pressure impedance. Fig. 17 is a graph of late-time responses. The response of Lee et al.’s model has large damping error, just as it did in the previous section. In Table 3, M, P, C, Z, and RMSD of the proposed model are 49%,

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Cosine impulse case, front side, (θ = π) 0.8 Huang (exact) DAA2 Lee et al. Proposed

0.6

Normalized velocity

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

Fig. 14. A submerged spherical shell excited by a cosine-type impulsive pressure.

−1.2

73%, 52%, 84%, and 67% of the Lee et al.’s model, respectively. Figs. 18–20 show normalized radial velocity on the opposite side,

0

0.5

ð2Þ

h ¼ 0. In Fig. 19, only DAA2 shows 19% error, just like the response on the front side. In Table 4, M, P, C, Z, and RMSD of the proposed model are 56%, 73%, 58%, 84%, and 72% of the Lee et al.’s model, respectively.

ð65Þ

p

Huang’s exact solution was also used for a reference for this case. From Figs. 21–24, the results of each approximation model are compared to this exact solution. The results of each approximation model contain more fluctuation than the analytic solutions in the previous sections. In Figs. 21 and 22, the normalized velocity of the surface of the structure at the front side, h ¼ p, is shown. In the early-time, all approximation models do not follow the exact solution at the first peak. In Figs. 23 and 24, the normalized velocity of the surface of the structure at the back side, h ¼ 0, is shown. In

Cosine impulse case, front side, (θ = π)

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1.2 10

12

14

16

18

20

Normalized time (tc/a) Fig. 17. Normalized radial velocity of submerged spherical shell excited by cosine impulsive pressure in late-time at h ¼ p using modal solution (n ¼ 0  12).

Cosine impulse case, front side, (θ = π)

Cosine impulse case, front side, (θ = 0) 1.5

Huang (exact) DAA2 Lee et al. Proposed

0.6 0.4

1

0.2

Normalized velocity

Normalized velocity

Huang (exact) DAA2 Lee et al. Proposed

0.6

0.8

0 −0.2 −0.4 −0.6 −0.8

0.5

0

Huang (exact) DAA2 Lee et al. Proposed

−1 −1.2 0

2

0.8

Normalized velocity

h i 5 2 pi ðt; hÞ ¼ cos h pffiffiffiffi exp 52 ðt  0:5Þ Hð cos hÞ

1.5

Fig. 16. Normalized radial velocity of submerged spherical shell excited by cosine impulsive pressure in early-time at h ¼ p using modal solution (n ¼ 0  12).

5.3. Case study 3: discretized spherical shell excited by cosine-type impulsive pressure A discretized spherical shell with 386 nodes was constructed for this case. The purpose of this case study was showing the possibility of application of the proposed model. A cosine-type impulsive pressure was applied to the structure, as in Fig. 14. To avoid the discontinuity during calculation, the following Gaussian function was applied for an input function instead of the Delta function.

1

Normalized time (tc/a)

2

4

6

8

10

Normalized time (tc/a) Fig. 15. Normalized radial velocity of submerged spherical shell excited by cosine impulsive pressure at h ¼ p using modal solution (n ¼ 0  12).

−0.5 0

2

4

6

8

10

Normalized time (tc/a) Fig. 18. Normalized radial velocity of submerged spherical shell excited by cosine impulsive pressure at h ¼ 0 using modal solution (n ¼ 0  12).

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Cosine impulse case, front side, (θ = 0)

Cosine impulse case, front side, (θ = π )

0.5

1.5

1

Normalized velocity

Normalized velocity

0

0.5

0

Huang (exact) DAA2 Lee et al. Proposed

−0.5 0

0.5

1

1.5

-0.5 -1 -1.5 -2 -2.5

Huang (exact) DAA2 Lee et al. Present(2nd)

-3 -3.5 0

2

2

4

6

8

10

Normalized time (tc/a)

Normalized time (tc/a) Fig. 19. Normalized radial velocity of submerged spherical shell excited by cosine impulsive pressure in early-time at h ¼ 0 using modal solution (n ¼ 0  12).

Fig. 21. Normalized radial velocity on a spherical shell excited by Gaussian impulsive pressure at h ¼ p (tc=a ¼ 0  10).

Cosine impulse case, front side, (θ = 0)

0.5

1.5

Cosine impulse case, front side, (θ = π )

-0.5

1

Normalized velocity

Normalized velocity

0

0.5

0

−0.5 10

Huang (exact) DAA2 Lee et al. Proposed 12

14

16

18

-1 -1.5 -2 -2.5

Huang (exact) DAA2 Lee et al. Present(2nd)

-3 -3.5 10

20

12

late-time, non of the approximation models can follow the exact solution, and they show large phase error as time goes on. In late-time, the magnitude of the response is more important than the accuracy of the phase. The proposed approximation model shows smaller magnitude error than other models; however, it is not as obvious as the improved modal analytic response in case studies 1 and 2. Not only this paper, but also other many papers use the spherical and cylindrical shell for examples. Those cases are selected because of the existence of exact solutions, as mentioned before. Those are very useful to validate the performance of the new approximation model. However, there are structures which the approximation model cannot be applied because of its shape. This problem is caused by unfortunate restriction of the methodology of DAA which cannot be applied to non-convex problem. This problem can be overcome with a FE modeling technique [29,30]. The key-point is the creation of new surface for approximation model. The new surface which is the convex geometry needs to be generated apart from the structure. Next, the buffer elements have to be filled between the structure and the new surface as an acoustic medium. The material for these buffer elements is the water for the underwater explosion situation. The equation

16

18

20

Fig. 22. Normalized radial velocity on a spherical shell excited by Gaussian impulsive pressure at h ¼ p (tc=a ¼ 10  20).

Cosine impulse case, front side, (θ = 0)

1.6 1.4

Normalized velocity

Fig. 20. Normalized radial velocity of submerged spherical shell excited by cosine impulsive pressure in late-time at h ¼ 0 using modal solution (n ¼ 0  12).

14

Normalized time (tc/a)

Normalized time (tc/a)

1.2 1 0.8 0.6 0.4 Huang (exact) DAA2 Lee et al. Present(2nd)

0.2 0 -0.2 0

2

4

6

8

10

Normalized time (tc/a) Fig. 23. Normalized radial velocity on a spherical shell excited by Gaussian impulsive pressure at h ¼ 0 (tc=a ¼ 0  10).

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H. Woo, Y.S. Shin / Applied Acoustics 105 (2016) 116–128

Cosine impulse case, front side, (θ = 0)

1.6 1.4

Normalized velocity

1.2

was supported by WCU (World Class University) program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (R31-2008-000-10045-0).

Huang (exact) DAA2 Lee et al. Present(2nd)

References

1 0.8 0.6 0.4 0.2 0 -0.2 10

12

14

16

18

20

Normalized time (tc/a) Fig. 24. Normalized radial velocity on a spherical shell excited by Gaussian impulsive pressure at h ¼ 0 (tc=a ¼ 10  20).

of state has to be set for the acoustic medium to propagate a shockwave properly. 6. Conclusions This paper has presented a new approximation model for the external acoustic–structure interaction problem that can be applied for the shell problems. The approximation model is fully derived from Kirchhoff’s retarded potential formula instead of using a trial equation. New weighting parameter a is introduced to control the dominance of the pressure term over the velocity term in the model. This weighting parameter can reduce the dominance of the pressure in the model and can decrease the damping error of the model. The accuracy of the model can be increased by the reduced damping error, however, a limitation of a stable range had set to apply the model. A value of the new weighting parameter is determined to follow the exact solution from modal analysis. The performance of the proposed approximation model was estimated by applying approximation models to spherical shell problems for two different types of pressure condition. The proposed approximation model also can follow the exact response in earlyð2Þ

time, while DAA2 shows large error at the initial impulsive response. The new model can relieve the shortcoming of the large damping effect of previous approximation model. Thus, the proposed model can also follow late-time response better than the former approximation model while using weighting parameter a. In a discretized spherical shell, the response of the proposed model has large magnitude in late-time, which means it has a smaller damping effect in it. In conclusion, it is considered that the effect of particle velocity is excessive to predict the apposite response in existing approximation models. Acknowledgements We would like to express our appreciation to Professor K.C. Park for his patient guidance and advice of this research. This research

[1] Baron M. The response of a cylindrical shell to a transverse shock wave. In: Proceedings of second US national congress of applied mechanics. Michigan: University of Michigan; 1954. p. 210–2. [2] Huang H. An exact analysis of the transient interaction of acoustic plane waves with a cylindrical elastic shell. J Appl Mech 1970;37(4):1091–9. [3] Mindlin R, Bleich H. Response of an elastic cylindrical shell to a transverse step shock wave. J Appl Mech 1953;20:189–95. [4] Huang H. Transient interaction of plane acoustic wave with a spherical elastic shell. J Acoust Soc Am 1969;45(3):661–70. [5] Baker B, Copson E. The mathematical theory of Huygens principle. 3rd ed. New York: AMS Chelsea; 1987. [6] Felippa C. A family of early time approximations for fluid structure interaction. J Appl Mech 1980;47(4):703–8. [7] Geers T. Doubly asymptotic approximation for transient motion of submerged structure. J Acoust Soc Am 1978;64(5):1500–8. [8] Geers T, Zhang P. Doubly asymptotic approximations for submerged structures with internal fluid volumes: formulation. J Appl Mech 1994;61(4):893–9. [9] Geers T, Zhang P. Doubly asymptotic approximations for submerged structures with internal fluid volumes: evaluation. J Appl Mech 1994;61(4):900–6. [10] Zhang P, Geers T. Excitation of a fluid-filled, submerged spherical shell by a transient acoustic wave. J Acoust Soc Am 1993;93(2):696–705. [11] Geers T, Lewis B. Doubly asymptotic approximations of transient elastodynamics. Int J Solids Struct 1997;34(11):1293–305. [12] Prempramote S, Song C, Tin-Loi F, Lin G. High-order doubly asymptotic open boundaries for scalar wave equation. Int J Numer Meth Eng 2009;79:340–74. [13] Underwood P, Geers T. Doubly asymptotic, boundary-element analysis of dynamic soil-structure interaction. Int J Solids Struct 1981;17:687–97. [14] Wang X, Jin F, Prempramote S, Song C. Time-domain analysis of gravity damreservoir interaction using high-order doubly asymptotic open boundary. Comput Struct 2011;89:668–80. [15] Geers T, Hunter K. An integrated wave-effects model for an underwater explosion bubble. J Acoust Soc Am 2002;111(4):1584–601. [16] Geers T, Toothaker B. Third-order doubly asymptotic approximations for computational acoustics. J Comput Acoust 2000;8(1):101–20. [17] Lee M. A study on approximation models for acoustic–structure interactions in exterior problems. Ph.D. thesis. Korea Advanced Institute of Science and Technology; 2009. [18] DeRuntz J, Geers T. Added mass computation by the boundary integral method. Int J Numer Meth Eng 1978;12(3):531–49. [19] Lee M, Park Y, Park Y, Park K. New approximations of external acoustic– structural interactions: derivation and evaluation. Comput Methods Appl Mech Eng 2009;198(15–16):1368–88. [20] Junger M, Feit D. Sound, structures, and their interaction. 1st ed. Massachusetts: MIT Press; 1972. [21] Morse P, Ingard K. Theoretical acoustics. New York: McGraw-Hill; 1968. [22] Abramowitz M, Stegun I. Handbook of mathematical functions with formulas, graphs, and mathematical tables. New York: Wiley; 1972. [23] Milne-Thomson L. Theoretical hydrodynamics. New York: Macmillan; 1968. [24] Geers T. Transient response analysis of submerged structures. In: Belytschko T, editor. Finite element analysis of transient nonlinear structural behavior. AMD, vol. 26. New York: ASME; 1975. [25] Apostol T. Mathematical analysis: a modern approach to advanced calculus. Massachusetts: Addison-Wesley; 1957. [26] Geers T. An objective error measure for the comparison of calculated and measured transient response histories. Shock Vib Bull 1984;54(Part 2):99–107. [27] Whang B, Gilbert W, Zilliacus S. Two visually meaningful correlation measures for comparing calculated and measured response histories. Tech rep. Bethesda (MD): Carderock Division, Naval Surface Warfare Center; September 1993. [28] Russell D. Error measures for comparing transient data. In: The proceedings of the 68th shock and vibration symposium. Hunt Valley (MD): David Taylor Research Center; 1997. p. 175–98. [29] Schneider N. Prediction of surface ship response to severe underwater explosions using a virtual underwater shock environment. Master’s thesis. Monterey (CA): Naval Postgraduate School; June 2003. [30] Shin Y, Schneider N. Ship shock trial simulation of USS Winston S. Churchill (DDG 81); Part 1: modeling and simulation strategy, Part 2: surrounding fluid volume effect. In: Proceedings of 74th shock and vibration symposium, SAVIAC, San Diego, California; 2003.