INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids (2016) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/fld.4244 Structural–acoustic sensitivity analysis of radiated sound power using a finite element/ discontinuous fast multipole boundary element scheme Leilei Chen1,*,† , Haibo Chen2 , Changjun Zheng3 and Steffen Marburg4 1 College of Civil Engineering, Xinyang Normal University, Xinyang 464000, Henan, China Key Laboratory of Mechanical Behavior and Design of Materials, Department of Modern Mechanics, University of Science and Technology of China, Hefei 230026, Anhui, China 3 Institute of Sound and Vibration Research, Hefei University of Technology, Hefei 230009, Anhui, China 4 Institute of Vibroacoustics of Vehicles and Machines, Faculty of Mechanical Engineering, Technical University of Munich, 85748 Garching bei München, Germany 2 CAS SUMMARY The complete interaction between the structural domain and the acoustic domain needs to be considered in many engineering problems, especially for the acoustic analysis concerning thin structures immersed in water. This study employs the finite element method to model the structural parts and the fast multipole boundary element method to model the exterior acoustic domain. Discontinuous higher-order boundary elements are developed for the acoustic domain to achieve higher accuracy in the coupling analysis. Structural–acoustic design sensitivity analysis can provide insights into the effects of design variables on radiated acoustic performance and thus is important to the structural–acoustic design and optimization processes. This study is the first to formulate equations for sound power sensitivity on structural surfaces based on an adjoint operator approach and equations for sound power sensitivity on arbitrary closed surfaces around the radiator based on the direct differentiation approach. The design variables include fluid density, structural density, Poisson’s ratio, Young’s modulus, and structural shape/size. A numerical example is presented to demonstrate the accuracy and validity of the proposed algorithm. Different types of coupled continuous and discontinuous boundary elements with finite elements are used for the numerical solution, and the performances of the different types of finite element/continuous and discontinuous boundary element coupling are presented and compared in detail. Copyright © 2016 John Wiley & Sons, Ltd. Received 11 June 2015; Revised 13 March 2016; Accepted 10 April 2016 KEY WORDS: acoustic–structure interaction; FEM/FMBEM coupling; design sensitivity analysis; discontinuous boundary element; radiated sound power 1. INTRODUCTION Analysis of the acoustic radiation or scattering from elastic structures in heavy fluid is a classical problem of underwater acoustics. Analytical solutions to structural–acoustic interaction problems are available only for simple geometric structures with simple boundary conditions [1]. Analytical solutions to practical problems with complicated geometries remain unavailable. Thus, efficient numerical methods need to be developed. The finite element method (FEM) is often used to model the structural parts of the problem because of its high flexibility and applicability to large-scale practical models. Meanwhile, the boundary element method (BEM) is used to model the sound field to avoid the need to mesh the *Correspondence to: Leilei Chen, College of Civil Engineering, Xinyang Normal University, Xinyang 464000, Henan, China. † E-mail: chenllei@mail.ustc.edu.cn Copyright © 2016 John Wiley & Sons, Ltd. L. CHEN ET AL. acoustic domain that is often infinite or semi-infinite. For the analysis of structural–acoustic interaction problems, researchers have paid much attention on the FEM/BEM coupling approaches [2–10], where FEM is used to discretize the structure parts and BEM is used to model the acoustic area. For BEM, the use of continuous linear or quadratic elements is often applied, and alternatives to discontinuous elements with high accuracy have been previously investigated [11–13]. Mostly, discontinuous boundary elements were applied when a hypersingular integral was discretized, because, in that case, C 1 -continuity of the surface at the collocation point is required. Applications are known from crack analysis [14], and from solution of the Navier–Stokes equation [15]. Atkinson [16] reviewed the effect of superconvergence for error dependence on the size of discontinuous boundary elements when collocation points are located at the zeros of orthogonal functions for the standard interval. Error dependence in terms of frequency, element size, and location of nodes on discontinuous elements is presented in [13]. This study found that discontinuous boundary elements perform more efficiently than continuous elements. The performance of discontinuous boundary elements for the acoustic analysis with a rigid structure has also been thoroughly investigated. However, so far, there is no report that describes the performance of discontinuous boundary elements coupled with FEM when the interaction between the structure and the sound field is taken into account. A known disadvantage of conventional BEM is that it produces a dense and non-symmetrical coefficient matrix that induces O.N 3 / arithmetic operations to directly solve the system of equations, such as by using the Gauss elimination method. The fast multipole method (FMM) [17–21] has been applied to accelerate the solution of the integral equation. To solve large-scale practical problems, iterative solvers have been well suited [22, 23]. Thus, the coupling algorithm based on FEM and fast multipole BEM (FMBEM) can be effectively applied to solve large-scale acoustic–structure interaction problems [9, 10]. The FMM used to solve the Helmholtz equation has two forms: one is the original FMM and the other is the diagonal form. Both forms fail in some way outside their preferred frequency ranges. However, the wideband FMM formed by combining the original FMM and the diagonal form FMM can overcome the aforementioned problems [24–27]. Mostly, FMM is used for numerical analysis based on constant boundary element discretization. Few papers apply the FMM for numerical analysis based on discontinuous high-order boundary element discretization because of the complexity of the computing procedure. In this paper, different types of continuous and discontinuous boundary elements are used for the numerical solution, and FMM is applied to accelerate the solution of the integral equation. Finally, the coupling algorithm FEM/discontinuous wideband FMBEM is proposed to solve the large-scale acoustic–structure interaction problems. Acoustic design sensitivity analysis can provide insights into the effects of geometric changes on the acoustic performance of the given structure, and thus, this analysis is important to the acoustic design and optimization processes [28]. Marburg [8] presented an overview on developments in structural–acoustic optimization for passive noise control. However, a sound power sensitivity analysis for acoustic–structure interaction problems with respect to design variables is time-consuming in gradient-based optimization. According to Lamancusa [29], Hambric [30], and Marburg et al. [31], the global finite difference method (FDM) is widely applied to structural–acoustic optimization because it is easy to implement. However, FDM performs inefficiently especially when many design variables are concurrently considered. Aside from global finite differences, semi-analytic and analytic sensitivity analyses have also been distinguished. These categorizations have been reviewed and discussed by Haftka and Adelmann [32] (see also [8, 33]). Analytic and semi-analytic sensitivity analyses are considerably more accurate than global finite differences. The former also require less computational costs than the latter. An analytic sensitivity analysis, which has appeared as the direct differentiation method in recent years, has been applied to coupled structural acoustic problems in [34–36]. Another acceleration in computational time, especially for problems with many design variables, becomes possible using the adjoint operator approach that has been applied to structural acoustic problems [37, 38]. This study is the first to formulate equations for sound power sensitivity with respect to design variables including fluid density, structural density, Poisson’s ratio, Young’s modulus, and structural shape/size. The derivative formulation of the vectors of nodal displacement and sound pressure on the interaction surface with respect to design variables are obtained by directly differentiating the Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2016) DOI: 10.1002/fld RADIATED SOUND POWER SENSITIVITY coupled boundary element equation with respect to design variables, where the coupled boundary element equation is obtained by substituting the structural equation into the acoustic equation. The derivative formulation of the radiated sound power on the structural surface is obtained using the adjoint operator approach, and the derivative formulation of the radiated sound power on an arbitrary closed surface around the radiator is obtained using the direct differentiation approach. For different design variables, the corresponding formulations used to solve the derivative of the radiated sound power are presented in detail. The sound power sensitivity is calculated using the FEM/BEM coupling schemes, including the FEM/discontinuous FMBEM scheme proposed in this paper. A numerical example is presented to demonstrate the accuracy and validity of the present algorithm. Several types of coupling elements are used for the numerical solution, and the performances of different types of FE/BE coupling elements are presented and compared. 2. STRUCTURAL–ACOUSTIC ANALYSIS 2.1. FEM/BEM for acoustic–structure interaction The fluid and structural sub-domains are modeled with the BEM and the FEM, respectively. The discretized boundary integral representation of the solution to the Helmholtz equation in fluid can be expressed as the following formula [39]: Hp D Gq C pi ; (1) where the matrices H and G are the frequency-dependent BEM influence matrices. p and q, respectively, are the vectors with the nodal values for pressure and its normal derivative. pi denotes the nodal pressure from the incident wave. The approach of continuous linear and quadratic elements is often applied, and alternatives to discontinuous elements with high accuracy have been previously investigated [11–13]. Discontinuous boundary elements perform more efficiently than continuous elements [13]. For discontinuous boundary elements, interpolation nodes are located inside the element, and the expressions of the interpolation functions are dependent on the position of the node inside the element. Thus, the numerical solution with different computation accuracies can be obtained by setting different positions of interpolation nodes. In this paper, discontinuous and continuous elements are both used in the numerical calculation. When a harmonic load with a transitory function e i !t is applied to the structure, the steadystate response of the structure can be calculated from the frequency-response analysis. When the acoustic–structure interaction is considered, the linear system of equations to compute the nodal displacements u is derived by [34] Au D fs C Csf p; (2) where A D K C i!C  ! 2 M and the matrices K, C, and M are the stiffness, damping, and mass matrices of the structure, respectively. The vector fs represents the nodal structural forces, and Csf p represents the acoustic loads. The coupling matrix Csf can be expressed as [34] Z NTs nNf d €; (3) Csf D € where € denotes the interaction surface, Ns and Nf are the global interpolation functions for the structure and fluid domains, respectively, and n is the surface normal vector. Equations (1) and (2) are linked via the continuity condition q D i!vf across the interaction surface. The normal velocity vf can be expressed as a function of the displacement u [34] as follows: vf D i!S1 Cfs u; (4) where S D € NTf Nf d € and Cfs D CTsf . The fully coupled system of equations of an elastic structure submerged in a heavy fluid can be obtained by substituting Eq. (4) into Eq. (1) and combining Eqs (2) and (1), and is given by R Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2016) DOI: 10.1002/fld L. CHEN ET AL. " A Csf ! 2 GS1 Cfs H #" # u p DB " # u p D " fs pi # : (5) The use of an iterative solver (e.g., generalized minimal residual method (GMRES)) on system Eq. (5) shows poor convergence. A suitable method is substituting the finite element formulation into the boundary element equation to generate a reduced system equation [34] as follows: Hp  GWCsf p D GWfs C pi ; (6) where W D ! 2 S1 Cfs A1 . Solving directly A1 in Eq. (6) is time-consuming. In fact, A1 need not be directly solved. The term A1 fs in Eq. (6) represents the solution x of the linear system of equations Ax D fs . This symmetric and frequency-dependent system is easily solved by using a sparse direct solver. Directly solving the term A1 Csf on the left side of Eq. (6) is also unnecessary. In this study, the iterative solver GMRES [22, 23] is applied to accelerate the calculation of the solution to the coupled boundary element system equation. The current iterative solution is assumed as pk . The matrix–vector product of Csf pk and a new vector y are obtained, where y D Csf pk . Then, the solution of A1 y can be efficiently obtained when a sparse direct solver is used to solve the symmetric and frequency-dependent system of the linear equation Ax D y. After solving Eq. (6) and substituting the solution of the vector p into Eq. (2), we can obtain the solution of vector u. A well-known disadvantage for directly solving the previous equation is that the coefficient matrices H and G are dense and non-symmetrical, which induce O.N 2 / arithmetic operations when dealing with a problem with N unknowns. The FMM accelerates the solution of the conventional boundary element system of equations and decreases the memory requirement. The main idea of FMM is to approximate the fundamental solution for BEM in terms of spherical Hankel functions, Legendre polynomials, and plane waves. The coefficient matrices consist of two parts. One is the near-field part, which is evaluated by integration in the usual way in the vicinity of the source point. The other is the far-field part, which is not directly computed. Applying FMM on a hierarchy of clusters reduces the complexity of BEM from O.N 2 / to O.N log N /. FMM has two forms. One is the original FMM (low-frequency method) based on a series expansion formula of the fundamental solution; the other is the diagonal form FMM (high-frequency method) based on a plane wave expansion formula of the fundamental solution. The original FMM is inefficient for high-frequency problems, and the diagonal form FMM has instability problems for the solution of low-frequency Helmholtz equations. However, the wideband FMM obtained by combining the original FMM and the diagonal form FMM can be used to overcome these difficulties [24–27]. Detailed information about the wideband FMM algorithm can be found in [26, 27]. 2.2. Radiated sound power formulation For radiation into open domains, the emitted sound power P on an arbitrary closed surface around the radiator can be expressed as Z ± ° 1 PA .!/ D < p.y; !/vf .y; !/ dA.y/; (7) 2 where p is the sound pressure, vf represents the conjugate complex of the particle velocity vf , and < ¹º denotes the real part. The real part of the complex sound power is radiated into the acoustic far field, whereas the imaginary part only contributes to the evanescent near field. Discretization of Eq. (7) using BEM leads to a matrix expression for the sound power that is given by PA .!/ D ¯ 1 ® T < pA SA vA ; 2 (8) R where SA D NTf Nf dA.y/, pA is the nodal sound pressure vector on surface A, and vA is the particle velocity vector on surface A. A is an arbitrary closed surface around the radiator. When the Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2016) DOI: 10.1002/fld RADIATED SOUND POWER SENSITIVITY sound power on the structural surface needs to be calculated, we can conveniently use the structural surface € as a substitute to A for sound power evaluation. Thus, Eq. (8) is rewritten as P .!/ D 1 ® T ¯ < p Svf : 2 (9) The vectors p, u, and vf can be successively obtained by solving Eqs (6), (2), and (4), respectively. Finally, the radiated sound power P on the structural surface can be obtained by solving Eq. (9). When the surface A is not the structural surface, the sound power on surface A is calculated by using Eq. (8). pA .y/ and vA .y/ are the pressure and the particle velocity at field point y on the surface A, respectively. By solving pA .y/ and vA .y/ at all the nodes on surface A, we can obtain the vectors pA and vA . The boundary integral equation defined on the interaction surface € to evaluate the sound pressure pA .y/ at a field point can be expressed as Z Z G.x; y/q.x/d €.x/  F .x; y/p.x/d €.x/; (10) pA .y/ D € € where G.x; y/ D e i kr ; 4 r (11) and q.y/ and F .x; y/ are the normal derivatives of p.y/ and G.x; y/, respectively. Differentiating Eq. (10) with respect to n.y/ and using the continuity condition qA .y/ D i!vA .y/ (q and v have the same direction), we can obtain the following formula: Z Z i @pA .y/ @G.x; y/ @F .x; y/ i i vA .y/ D D q.x/d €.x/  p.x/d €.x/ (12) ! @n.y/ ! € @n.y/ ! € @n.y/ After the discretization of Eqs (10) and (12), we can obtain the following formulas: pA .y/ D gT .y/q  hT .y/p; (13) and vA .y/ D i T i T g1 .y/q  h .y/p; ! ! 1 (14) where g, h, g1 , and h1 are coefficient vectors. Using Eqs (13) and (14), we can obtain pA .y/ and vA .y/ at all the nodes on the surface A. Thus, the nodal sound pressure vector pA and the particle velocity vector vA on surface A are solved. Finally, the radiated sound power PA .!/ on the surface A can be solved by using Eq. (8). 3. SOUND POWER SENSITIVITY FORMULATION 3.1. Sound power sensitivity formulation on the structural surface An implicit differentiation of Eq. (5) with respect to the design variable # and isolating the resulting sensitivities of structural displacement and sound pressure leads to 3 2 @u 4 @# 5 D B1 r; (15) @p @# where rD Copyright © 2016 John Wiley & Sons, Ltd. " r1 r2 # 3 " # @fs u @B @# 5 ; D4 @pi @# p @# 2 (16) Int. J. Numer. Meth. Fluids (2016) DOI: 10.1002/fld L. CHEN ET AL. 3 @Csf @A  @B @# 7 6  @#  D4 5: @ ! 2 GS1 Cfs @# @H  @# @# 2 (17) For different design variables, different expressions of @B are obtained in what follows. When the @# fluid density  is chosen as the design variable, @B is derived by @   @B 0 0 ; (18) D ! 2 GS1 Cfs 0 @ and the vector r is derived by rD " r1 r2 # D " 0 ! 2 GS1 Cfs u # : (19) When the structural parameter is chosen as the design variable, such as structural density s , Poisson’s ratio v, Young’s modulus E, and thickness of the spherical shell h presented in the following numerical example, @B is derived by @# " # @A 0 @B D @# ; (20) @# 0 0 and the vector r is derived by rD " r1 r2 # D " #  @A u @# : 0 (21) When the parameter determining the structural nodal coordinate is set as the design variable, such as the radius of the spherical shell r presented in the following numerical example, @B is derived by @# 3 2 @A  @Csf @B @# 1 @# 5 : (22) D4 @H 2 @.GS Cfs / @# !  @# @# The vector r is derived by rD " r1 r2 # 2 3 @fs  @A u C @Csf p @# @# @# 6 7 D4 5: 1 @Pi C ! 2  @.GS Cfs / u  @H p @# @# @# (23) It is difficult to obtain the derivative of A, Csf , Cfs , S1 , H, and G by using the direct differentiation method for complex structures. However, the semi-analytical derivative method, through which variations of the coefficient matrices can be calculated by using the FDM, can be applied to conquer this difficulty. For example, @C can be calculated by using a small perturbation  , as follows: @# C.# C  /  C.#/ @C D : @#  (24) In this work, a step size =# D 103 is used. Differentiating Eq. (9) with respect to design variable #, we can express the sound power sensitivity on the structural surface as Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2016) DOI: 10.1002/fld RADIATED SOUND POWER SENSITIVITY 1 @P D < @# 2 ´ @p @# T @Cfs  w1 C i!pT u C wT2 @#  @u @#  μ ; where w1 D i!Cfs u and wT2 D i!pT Cfs . In what follows, the expression ³ ² ²   ³ @u @u D < wH < wT2 2 @# @# (25) (26) will be used. The superscript ./H denotes the conjugate complex transposed. Applying this to Eq. (25), the sound power sensitivity is yielded as ³ ² @P 1 @p @u @Cfs  : (27) D < wT1 C i!pT u C wH 2 @# 2 @# @# @# The sum of the first and third terms in the right side of Eq. (27) can be rewritten as 3 2 @u @p @u wT1 C wH D dT 4 @# 5 D dT B1 r; 2 @p @# @# @# (28) where dD " w2 w1 # : (29) Substituting Eq. (28) into Eq. (27), we can express the sound power sensitivity on the structural surface as ³ ² 1 @P T 1 T @Cfs  (30) D < d B r C i!p u : @# 2 @# The sound power sensitivity on the structural surface consists of two terms. The first term on the right side of Eq. (30) can be solved in two ways. One is solving the linear system of equations BNz D r and then solving dT zN . The other is solving the linear system of equations zB D dT and then solving zr. According to Eq. (16), r depends on the derivatives of certain terms with respect to the design variable #, whereas d does not. Consequently, m design variables #j with j D 1; 2; :::; m will produce m right-hand sides r. When using the first way, the linear system of equations BNz D r needs to be solved m times, which is time-consuming. However, when using the second way, the linear system of equations zB D dT only needs to be solved once for different design variables. The use of an iterative solver on the adjoint equation zB D dT , for example, GMRES, shows poor convergence. The adjoint equation can be rewritten as   zB D zs zf B D dT ; (31) where s denotes the structural degree of freedom and f is the fluid degree of freedom. A comfortable method that is separating the adjoint equation into two reduced coupled sensitivity equations is applied as follows: zs A  ! 2 zf GS1 Cfs D wH 2: (32) zf H  zs Csf D wT1 : (33) Substituting Eq. (32) into Eq. (33) and eliminating the vector zs lead to 1 zf H  zf GWCsf D wT1 C wH 2 A Csf : (34) The unknown fluid vector zf can be obtained by solving the previous reduced coupled sensitivity equation that is the same as solving Eq. (6). Then, Eq. (32) can be used to obtain the unknown Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2016) DOI: 10.1002/fld L. CHEN ET AL. structural vector zs . When the fluid density , structural density s , Poisson’s ratio v, Young’s modulus E, or structural thickness h is set as the design variable, the second term in Eq. (30) vanishes because of @Cfs D 0. However, when the structural shape parameter is set as the design variable, @# such as the radius of the spherical shell r, the term @Cfs does not vanish. An exact analytical rep@# resentation of the sensitivity of Cfs is possible but challenging. However, a simple and applicable method is the use of finite differences to overcome this difficulty. 3.2. Sound power sensitivity formulation on an arbitrary closed surface around the radiator Differentiating Eq. (8) with respect to the design variable #, we can obtain the following formula: @PA .!/ 1 D < @# 2 ´ @pA @# T SA vA C @SA  pTA vA @# C pTA SA  @vA @#  μ : (35) @v .y/ @pA .y/ and A , we can obtain the solution of all elements in By using the expressions of @# @# @pA vectors and @vA . Differentiating Eqs (10) and (12), we can obtain the following formulas: @# @# Z Z Z @q.x/ @d €.x/ @G.x; y/ @pA .y/ G.x; y/ G.x; y/q.x/ D q.x/d €.x/ C d €.x/ C @# @# @# @# €Z €Z € Z ; @p.x/ @d €.x/ @F .x; y/  F .x; y/ F .x; y/p.x/ p.x/d €.x/  d €.x/  @# @# @# € € € (36) and Z @2 G.x; y/ @G.x; y/ @q.x/ i q.x/d €.x/ C d €.x/ ! € @n.y/ @# € @n.y/@# Z Z 2 i @G.x; y/ @ F .x; y/ @d €.x/ i C q.x/  p.x/d €.x/ ! € @n.y/ @# ! € @n.y/@# ; Z Z i @F .x; y/ @p.x/ @F .x; y/ i @d €.x/  d €.x/  p.x/ ! € @n.y/ @# ! € @n.y/ @# Z Z @G.x; y/ @F .x; y/ i @1 i @1 C q.x/d €.x/  p.x/d €.x/ ! @# € @n.y/ ! @# € @n.y/ @vA .y/ i D @# ! Z (37) where @r @G.x; y/ e i kr D .1  ikr/ @# 4 r 2 @xi  @yi @xi  @# @#  (38)     @r @r @F .x; y/ e i kr  @yi @xi 2 2 3  3ikr  k r  .1  ikr/ni .x/ D  @# 4 r 3 @n.x/ @xi @# @# e i kr @r @ni .x/  .1  ikr/ 4 r 2 @xi @# (39)    @r @r e i kr @xi @yi @2 G.x; y/ 2 2 .3  3ikr  k r /  .1  ikr/ni .y/ D  @n.y/@# 4 r 3 @n.y/ @yi @# @# e i kr @r @ni .y/  .1  ikr/ 4 r 2 @yi @# (40) Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2016) DOI: 10.1002/fld RADIATED SOUND POWER SENSITIVITY   @r @2 F .x; y/ @r @r e i kr  15  15ikr  6k 2 r 2 C ik 3 r 3 D 4 @n.y/@# 4 r @n.x/ @n.y/ @yk      @r @r @r @yk @xk 2 2  33ikr  k r nk .y/  nk .x/  nj .x/nj .y/  @n.x/ @n.y/ @yk @# @# ;     e i kr  @r @n .x/ @n .y/ @r @r @r i j 2 2 3  3ikr  k r C C 4 r 3 @n.y/ @xi @# @n.x/ @yj @#   @nj .y/ @nj .x/ C .1  ikr/ nj .y/ C nj .x/ @# @# (41) and # " @d € @2 x i @2 x i  ni .x/nj .x/ d €; (42) D @# @#@xi @#@xj where @xi will be evaluated when the boundary of the analyzed domain is fully parameterized with @# the design variable. Discretizing Eqs (36) and (37), we can obtain the following equations: @q @p @pA .y/ D gT2 .y; #/q C gT .y/  hT2 .y; #/p  hT .y/ ; @# @# @# (43) @vA .y/ i T @q i T @p i T i T D g .y; #/q C g .y/  h .y; #/p  h .y/ @# ! 3 ! 1 @# ! 3 ! 1 @#   ; i @1 T g1 .y/q  hT1 .y/p C ! @# (44) where g2 , g3 , h2 , and h3 are coefficient vectors. When the fluid density  is set as the design variable, g2 and h2 in Eq (43) and g3 and h3 in Eq (44) vanish. Thus, Eqs (43) and (44) can be rewritten as @q @p @pA .y/ D gT .y/  hT .y/ ; @ @ @ (45)   @q @p @vA .y/ i T i T i gT1 .y/q  hT1 .y/p : D g1 .y/  h1 .y/  2 @ ! @ ! @  ! (46) When the structural parameter is selected as the design variable, such as structural density s , Poisson’s ratio v, Young’s modulus E, and thickness of the spherical shell h presented in the following numerical example g2 and h2 in Eq. (43) and g3 and h3 in Eq. (44) vanish. Thus, Eqs (43) and (44) can be rewritten as @q @p @pA .y/ D gT .y/  hT .y/ ; @# @# @# (47) i T @q i T @p @vA .y/ D g .y/  h .y/ : @# ! 1 @# ! 1 @# (48) When the structural shape parameter is set as the design variable, such as the radius of the spherical shell, g2 , g3 , h2 , and h3 do not vanish. Therefore, Eqs (43) and (44) can be rewritten as @pA .y/ @q @p D gT2 .y; #/q C gT .y/  hT2 .y; #/p  hT .y/ ; @# @# @# Copyright © 2016 John Wiley & Sons, Ltd. (49) Int. J. Numer. Meth. Fluids (2016) DOI: 10.1002/fld L. CHEN ET AL. i T i T @q i T i T @p @vA .y/ D g3 .y; #/q C g1 .y/  h3 .y; #/p  h1 .y/ : @# ! ! @# ! ! @# (50) Equations (43) and (44) show that the derivatives of pA .y/ and vA .y/ are both determined by p, q, and their derivatives. The vectors p, u, and vf can be successively obtained by using Eqs (6), (2), and (4), respectively. Then, the vector q can be obtained by using the continuity condition across the interaction surface. However, now, we still need to obtain the solution of the unknown vectors @q @p and . @# @# Directly solving Eq. (15) is inefficient because the system matrix is large for realistic problems. The system Eq. (15) can be decomposed into two equations: @p @u  Csf D r1 @# @# (51) @p @u  ! 2 GS1 Cfs D r2 : @# @# (52) A and H Substituting Eq. (51) into Eq. (52), we can obtain the following formula: H @p @p  GWCsf D GWr1 C r2 : @# @# (53) Equation (53) is highly similar to Eq. (6); thus, the same solving method is implemented. By solving @p the previous equation, the sensitivity of the nodal sound pressure on the structural surface can @# @u can be solved by using Eq. (51). Differentiating Eq. (4) be obtained, and the unknown vector @# with respect to the design variable and using the continuity condition across the interaction surface lead to   @u @q @.S1 Cfs / @ : (54) D i! vf C ! 2  u C S1 Cfs @# @# @# @# @q Using the previous equation, we can obtain . The derivatives of pA .y/ and vA .y/ at all the nodes @# on the surface A can be obtained using Eqs (43) and (44). The derivative of SA on the right side of Eq. (35) can be expressed as Z @SA @dA.y/ NTf Nf D : (55) @# @# A The second term on the right side of Eq. (35) vanishes because of @SA D 0 when the computing @# @p surface does not vary with design variable changes. After obtaining the solution of A and @vA , @# @# the derivative of the radiated sound power on the surface A can be obtained using Eq. (35). 4. DEFINITION OF CONTINUOUS AND DISCONTINUOUS ELEMENTS The approach of continuous linear or quadratic boundary elements is often applied. Alternatives to discontinuous boundary elements have been previously investigated [11–13]. For discontinuous elements, the interpolation nodes are located inside the element, and the expressions of the interpolation functions are dependent on the position of the node inside the element (see Figure 1 for quadrilateral elements and Figure 2 for triangular elements). In the two figures, ‘BEmn’ and ’FEmn’ denote boundary element and finite shell element with ‘m’ geometry nodes and ‘n’ interpolation nodes, respectively. For example, ‘BE41’ in Figure 1 is the constant boundary element with four geometrical nodes denoting the presence of linear shape functions, ‘BE44’ is the discontinuous linear boundary element with four geometrical nodes, ‘BE91’ is the constant boundary element with nine geometrical nodes denoting the presence of quadratic shape functions, ‘BE94’ Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2016) DOI: 10.1002/fld RADIATED SOUND POWER SENSITIVITY Figure 1. Distribution of geometric nodes and interpolation nodes in quadrilateral element. Figure 2. Distribution of geometric nodes and interpolation nodes in triangular element. Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2016) DOI: 10.1002/fld L. CHEN ET AL. is the discontinuous linear boundary element with nine geometrical nodes, ‘FE44’ is the isoparametric linear finite element or the continuous linear boundary element (CBE44), and ‘FE88’ is the eight-node isoparametric quadratic finite element. For constant boundary elements, the interpolation node is defined at the centroid of the element. For discontinuous linear boundary elements, ai and bi decide the positions of these interpolation nodes referred to [11]; in this study the program adopts a1 D a2 D a3 D a4 D 1=2 for linear quadrilateral discontinuous elements and a1 D a2 D a3 D a4 D 1=3 for quadratic quadrilateral discontinuous elements. For discontinuous triangular elements, bi is assumed to be b1 D b2 D b3 D 1=4 for linear elements and b1 D b2 D b3 D 1=6 for quadratic elements. The structure is discretized using Shell63 finite element in ANSYS for linear geometric approximation with three or four geometric nodes, such as elements FE33 and FE44, and Shell281 finite element in ANSYS for quadratic geometric approximation with six or eight geometric nodes, such as FE66 and FE88. Continuous and discontinuous boundary elements are used to discretize the acoustic domain. Every structural shell element is set as a fluid boundary element, and it denotes that the identical mesh for structure and fluid at the interface is used for the numerical solution. The coupled element ‘FEmn/BEpq’ means that ‘FEmn’ is used for the structure discretization and ‘BEpq’ for the acoustic domain discretization. For different ‘FEmn/BEpq’ coupled elements, different calculations of coupling matrix Csf are implemented. For the ‘FE44/BE41’ coupled element, the element coupling matrix is calculated using four structural interpolation nodes and one fluid interpolation node, and the surface integration is implemented through linear geometric approximation with four nodes. For the ‘FE88/BE94’ coupled element, the element coupling matrix is calculated using eight structural interpolation nodes and four fluid interpolation nodes, and the fluid surface integration is calculated through quadratic geometric approximation with nine geometric nodes. 5. NUMERICAL EXAMPLE: AN ELASTIC SPHERICAL SHELL EXCITED BY A UNIT FORCE Excitation in the form of a concentrated force F applied at point A ( D 0) is considered, as shown in Figure 3.  denotes the central angle between the calculated and excitation points, and the sound pressure p./ and displacement u./ at the surface of the sphere can be obtained in [1, 40]. Given the symmetry along the x axis, we can express the sound power on an arbitrary closed surface around the radiator as PA D R 2 Z  0 ® ¯  < pA ./vA ./ sin./d; (56) Figure 3. Spherical shell excited by a single force at point A. Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2016) DOI: 10.1002/fld RADIATED SOUND POWER SENSITIVITY Table I. Material and geometrical data for a submerged spherical shell. Density (water) Speed of sound Density (steel) Young’s modulus Poisson’s ratio Radius of sphere Thickness of shell  c s E v r h 1000 1482 7860 210 0.3 5 0.05 kg=m3 m=s kg=m3 GPa — m m Table II. Number of quadrilateral element and triangular element for different mesh sizes. Mesh size d (m) 1.33 1.00 0.67 0.57 0.50 0.35 Number of quadrilateral element 216 384 864 1176 1536 3456 Number of triangular element 432 768 1728 2352 3072 6912 where R is the radius of the surface A and pA ./ and vA ./ are the sound pressure and the particle velocity on the surface A, respectively. By differentiating Eq. (56) with respect to the design variable, the sound power sensitivity on the surface A is derived by ³ Z  ² @v  ./ @pA ./  @PA sin./d < D R2 vA ./ C pA ./ A @# @# @# 0 ³ ² (57) n  .m / @vA  2 R2 X @pA .m /   sin.m /; < vA .m / C pA .m / n mD1 @# @# where m D m=n and n is sufficiently large. Not considering damping, the material data for structure and fluid and the geometric data are listed in Table I. These data are the same as those presented by Peters et al. [40]. Different meshes for the quadrilateral and triangular elements are considered. Detailed information about the mesh size and number of elements is given in Table II. By assembling the element matrices output from ANSYS, the global structural stiffness and mass matrices can be obtained. Because of the symmetry of the structural global matrices, only the nonzero elements in the upper triangular matrix need to be saved, and every global matrix has three arrays. The first one is used to save the number of nonzero elements per line, the second one saves the column number of every nonzero elements per line, and the third one saves the value of every nonzero elements per line. There are six nodal DOFs at a structural node i, which consist of three displacement DOFs and three rotational DOFs and are given as ui D Œuix ; uiy ; ui´ ; ix ; iy ; i´ . Figure 4 presents the analytical and numerical solutions for the sound pressure at point .2r; 0; 0/ in terms of frequencies. In Figure 4, ‘FE88/FMBE94’ denotes that the coupled element FE88/BE94 is used for the numerical calculation, the wideband FMM algorithm is applied to accelerate the product of the matrix–vector in the coupled boundary element Eq. (6). The analytical solution is evaluated every 0.01 Hz up to 100 Hz. The numerical solution is evaluated every 2 Hz, and extra points for evaluation are inserted around the resonant peaks. The element size d D 0:67 m is used to discretize the spherical surface. This figure shows that the numerical solution with FE88/FMBE94 element and FE88/BE94 element agrees well with the analytical solution at extrinsic frequencies; it denotes that the use of the wideband FMM algorithm maintains the high accuracy of conventional BE method. Figure 5 presents the analytical and numerical solutions for the radiated sound power on the spherical surface in terms of frequencies. The FE44/FMBE44 element with linear shape function and Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2016) DOI: 10.1002/fld L. CHEN ET AL. Figure 4. Sound pressure at point .2r; 0; 0/. Figure 5. Radiated sound power on the spherical surface. the FE88/FMBE94 element with quadratic shape function are used to calculate the numerical solution. A mesh size d D 0:67 m is used to discretize the spherical surface. The figures show that the solution obtained using the FE88/FMBE94 element is in agreement with the analytical solution at extrinsic frequencies, but the deviation of the solution with FE44/FMBE44 from the analytical solution around the eigenfrequencies is larger. This result indicates that an accurate numerical solution can be obtained using a discontinuous linear boundary element with quadratic shape approximation in a moderate mesh dense. Figure 6 shows the relative error for the radiated sound power on the spherical surface with different quadrilateral elements and mesh discretization at 50 Hz. FMM is used to accelerate the solution of FE/BE method. This figure shows that the linear continuous coupled element FE44/CBE44 performs most inefficiently. Coupled elements FE44/BE41, FE88/BE91, and FE44/BE44 perform similarly. The coupled quadratic continuous element FE88/CBE88 presents a larger error compared with coupled quadratic discontinuous boundary element FE88/BE98 or FE88/BE99 at refined mesh. The FE88/BE94 element with quadratic shape function approximation performs more efficiently than the FE44/BE44 element with linear shape function approximation. Although elements FE88/BE98 and FE88/BE99 have a similar performance as FE88/BE94, the quadratic discontinuous element generates more interpolation nodes and needs more computing time and memory storage compared with the linear discontinuous element at the same mesh size. In summary, the coupled element FE88/BE94 performs most efficiently in problems with curved surfaces. Figure 7 shows the relative error for the radiated sound power on the spherical surface with different triangular elements and mesh discretization at 50 Hz. This figure shows that the linear Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2016) DOI: 10.1002/fld RADIATED SOUND POWER SENSITIVITY Figure 6. Relative error in terms of element size for different types of quadrilateral elements at 50 Hz. Figure 7. Relative error in terms of element size for different types of triangular elements at 50 Hz. continuous element FE33/CBE33 performs the most inefficiently. Obviously, the coupled element FE66/BE63 performs the most efficiently in problems with curved surfaces. Figure 8 compares the accuracy of the different discontinuous quadrilateral elements and linear continuous quadrilateral elements. The surface error in Euclidean norm is defined by jje A jj2 ; jjpeA jj2 e2A D (58) where jje A jj2 and jjpeA jj2 are defined by n A jje jj2 D 1X A jje .xi /jj2 n i D1 !1=2 ; (59) !1=2 ; (60) and n jjpeA jj2 D 1X jjpe .xi /jj2 n i D1 Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2016) DOI: 10.1002/fld L. CHEN ET AL. Figure 8. Comparison for the surface error in Euclidean norm for radiated sound pressure. where xi is the discretized node on the surface A, n is the number of the discretized nodes, and the error jje A .xi /jj at point xi is defined by jje A .xi /jj D jjpe .xi /  pn .xi /jj; x 2 A; (61) where pe represents the exact solution for the sound pressure, and pn is the numerical solution. In Figure 8, the surface errors for the radiated sound pressure at a distance R D 2r from the origin point are calculated according to Eq. (58). For calculating the surface error e2A , building and discretizing a closed surface are not necessary because of the symmetry along the x axis. The discrete points only need to be distributed on the half circles between  D 0 and  D 180 on the xy plane. We can obtain 91 computing points when the step size is set as 2 degrees. In Figure 8, similar interpolation nodes are set for different coupled elements. For the FE44/BE41 and FE88/BE91 elements, d D 0:35 m generates 3456 interpolation nodes (the BE collocation points); for the FE88/BE94 and FE44/BE44 elements, d D 0:67 m also generates 3456 interpolation nodes; for the FE44/CBE44 element, d D 0:35 m generates 3458 interpolation nodes; for the FE88/CBE88 element, d D 0:57 m generates 3530 interpolation nodes; and for the FE88/BE98 and FE88/BE99 elements, d D 1:0 m generates 3072 and 3456 interpolation nodes, respectively. Comparison of the result of ‘FE88/BE94’ with that of ‘FE44/CBE44’ shows that the linear discontinuous boundary element with quadratic shape function has a higher accuracy than the linear continuous boundary element with linear shape function. Meanwhile, comparison of the result of ‘FE88/BE94’ with that of ‘FE88/BE91’ shows that the linear discontinuous boundary element with quadratic shape function also has a higher accuracy compared with the constant boundary element. The quadratic discontinuous boundary element with nine interpolation nodes performs more efficiently than that with eight interpolation nodes at a high frequency. The surface error with the FE88/BE98 or FE88/BE99 element is large because of the coarse mesh discretization. In summary, the linear continuous element FE44/CBE44 performs most inefficiently. The FE44/BE44 element with linear shape function performs very inefficiently for the practical problem with curved surfaces. However, the coupled element FE88/BE94 performs most efficiently in problems with curved surfaces. Figures 9 and 10 present the analytical and numerical solutions for the amplitudes of the radiated sound pressure sensitivity with respect to fluid and structural densities, respectively, at 50 Hz. The analytical solution for the sound pressure can be obtained in [1, 40]. The analytical solution for the sound pressure sensitivity can be obtained using FDM. The numerical solutions are computed with the FE44/FMBE44 and FE88/FMBE94 elements with d D 1:0 m mesh discretization. The figures show that the numerical solution obtained using the FE88/FMBE94 element with quadratic shape approximation is in agreement with the analytical solution, but the deviation of the solution with the FE44/FMBE44 element from the analytical solution is large. This result denotes that a more accurate solution can be obtained using discontinuous linear boundary element with quadratic shape approximation. Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2016) DOI: 10.1002/fld RADIATED SOUND POWER SENSITIVITY Figure 9. Radiated sound pressure sensitivity with respect to fluid density at distance R D 2r. Figure 10. Radiated sound pressure sensitivity with respect to structural density at distance R D 2r. Figure 11. Radiated sound power sensitivity (magnitude) with respect to the radius r of the sphere. Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2016) DOI: 10.1002/fld L. CHEN ET AL. Figure 11 shows the radiated sound power sensitivity on the structural surface with respect to the radius of the sphere. In this figure, ‘Analytical’ and ‘FE88/BE94’ denote the analytical solution and the numerical solution for the radiated sound power sensitivity on the structural surface, respectively. ‘First term’ and ‘Second term’ denote the solution of the first term and the second term on the righthand side of the radiated sound power sensitivity Eq. (30), respectively. A step size of 0.2 Hz up to 100 Hz is used for evaluation of the analytical solution. The numerical solution is evaluated in steps of 1 Hz. The figure shows that the sound power sensitivity is small in the low-frequency range and Figure 12. Normalized sensitivities of radiated sound power on the surface A with respect to different design variables. (a) Fluid density , (b) structural density s , (c) Poisson’s ratio v, (d) Young’s modulus E, (e) thickness h, and (f) radius r. Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2016) DOI: 10.1002/fld RADIATED SOUND POWER SENSITIVITY then goes up in the vicinity of resonance peaks. Conversely, in the low-frequency range (f < 40Hz), the solution of the first term is contradictory, and its amplitude is smaller than that of the second term. This result denotes that both terms cancel each other. When f > 40Hz, the amplitude of the first term is significantly greater than that of the second term. Figure 12 shows the analytical and the FE88/FMBE94 solutions for the normalized sensitivities of the radiated sound power on the surface A with radius R D 2r with respect to different design variables, where the design variable # is set as the fluid density , structural density s , Poisson’s ratio v, Young’s modulus E, thickness h, and radius r, respectively. The sensitivity results of sound power on the surface A have large differences in the order of magnitude for different design variables. The reason for these differences is certainly that the design variables have different orders of magnitude themselves. Therefore, it appears useful to normalize these sensitivities. The normalized sensitivity of sound power is defined by ˇ ˇ # ˇˇ @PA ˇˇ N : P;# D PA ˇ @# ˇ (62) A mesh size d D 0:5 m is used for the numerical solution. These figures show that the normalized sensitivity of the radiated sound power remains rather small in the low-frequency range and then goes up in the vicinity of resonance peaks. The normalized sensitivity of the radiated sound power rapidly decreases at some frequencies. This result denotes that the algebraic signs for radiated sound power sensitivity change in the vicinity of these frequencies. The numerical solution agrees with the analytical solution, indicating the validity and correct implementation of the proposed algorithm. 6. CONCLUSIONS A coupling algorithm based on FEM and BEM is presented to simulate acoustic–structure interaction and structural acoustic sensitivity analyses. The FEM is used to model the structural parts of the problem. To avoid the need to mesh the acoustic domain, the BEM is used to discretize the acoustic domain boundary, which is also the boundary of the considered structure. The FMM is used to accelerate the matrix–vector products in the boundary element analysis. This study is the first to formulate equations for the radiated sound power sensitivity of fully coupled structural–acoustic systems. An adjoint operator approach is developed to calculate the sensitivity of the radiated sound power on the structural surface. The direct differentiation approach is used to calculate the sensitivity of the radiated sound power on an arbitrary closed surface around the radiator. The design variables include fluid density, structural density, Poisson’s ratio, Young’s modulus, and structural shape/size. For different design variables, the corresponding formulations used to solve the derivative of the radiated sound power are presented. A numerical example is presented to demonstrate the accuracy and validity of the proposed algorithm. Different types of coupled elements are used for the numerical solution, and the performances of different types of finite element/boundary element coupling are presented and compared. It is found that the discontinuous linear boundary element with quadratic shape approximation performs most efficiently. The proposed algorithm can be used to numerically predict the effects of different design variables on the sound field for large-scale practical problems. Future work includes applying the structural–acoustic design sensitivity analysis to optimization problems and extend the developed algorithm to practical engineering problems. ACKNOWLEDGEMENTS This study is financially supported by the China Scholarship Council (CSC), National Natural Science Foundation of China (NSFC) under grant nos. 11172291, 11402071, and U1504505; Research Fund for the Doctoral Program of Higher Education of China under grant no. 20133402110036; and USTC under grant no. WK2090000007. 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