Hybrid smoothed finite element method for acoustic problems

Accepted Manuscript Hybrid smoothed finite element method for acoustic problems Eric Li, Z.C. He, X. Xu, G.R. Liu PII: DOI: Reference:

S0045-7825(14)00344-2 http://dx.doi.org/10.1016/j.cma.2014.09.021 CMA 10377

To appear in:

Comput. Methods Appl. Mech. Engrg.

Received date: 21 October 2013 Revised date: 14 September 2014 Accepted date: 17 September 2014 Please cite this article as: E. Li, Z.C. He, X. Xu, G.R. Liu, Hybrid smoothed finite element method for acoustic problems, Comput. Methods Appl. Mech. Engrg. (2014), http://dx.doi.org/10.1016/j.cma.2014.09.021 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Hybrid smoothed finite element method for acoustic problems Eric Li1, ZC He1*, X Xu2, GR Liu3 1

State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082 P. R. China 2

College of Mathematics, Jilin University, Changchun 130012, China

3

School of Aerospace Systems, University of Cincinnati, Cincinnati, OH 45221-0070, USA

Abstract It is well known that “overly-stiff” finite element (FE) model fails to provide accurate results to the Helmholtz equation with large wave numbers due to the well-known “pollution error” caused by the numerical dispersion. In this paper, the hybrid smoothed finite element method (HS-FEM) using triangular (2D) and tetrahedron (3D) elements that can be generated automatically for any complicated domain is formulated to solve acoustic problems in order to overcome the shortcomings of standard finite element method (FEM). In the formulation, a parameter α is equipped into HS-FEM, and the strain field is further assumed to be the weighted average between compatible strains from standard FEM and smoothed strains from node-based smoothed finite element method (NS-FEM). The smoothing technique can provide a softening effect to the model and have a very close-to-exact stiffness of the continuous system and hence significantly improve the accuracy of solution in both structural and acoustic analyses. Intensive numerical studies have been conducted here to demonstrate the effectiveness of the HS-FEM.

Key words: acoustic, finite element method (FEM), hybrid smoothed finite element method (HS-FEM), numerical method, discretization error * Corresponding author. E-mail address:[email protected] (ZC He)

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1. Introduction With the increasing requirements on the sound quality of enclosed cavities, such as the automobile passenger compartments and aircraft cabins, careful considerations are needed in designing these sophisticated structures. As the analytical solution is only available for very simple cases, numerical methods for solving acoustic and structural-acoustic problems are becoming more and more prevalent. Over the past several decades, there are many numerical methods proposed for approximate solutions of acoustic, aero-acoustic and structural-acoustic problems [1-6]. Currently, the standard finite element method (FEM) is the most widely used numerical technique for acoustic and structural-acoustic problems. However, the standard FEM solution of acoustic problem has serious issues on accuracy and reliability, due to the pollution errors rooted on the FEM discretization of the differential operator of the governing equation [7]. At higher frequencies, the FEM solutions with linear interpolation become worse and worse mainly due to the increased pollution error, in addition to the usual interpolation error. Although the use of more elements per wavelength can improve the accuracy of acoustic problem, the use of fine mesh will lead to the dramatic increase in computational cost, especially for large scale 3D acoustic problems. In order to overcome the pollution error, many numerical methods have been developed such as Galerkin Least Squares Finite Element Method (GLS) [8], discontinuous enrichment method (DEM) [9], Element-free Galerkin method (EFGM) [10] and the residual-free finite element method (RFFEM) [11] with enhanced

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computational efficiency. Another approach is the development of multi-scale techniques [12-13] with the application to Helmholtz problems [14]. However, the pollution error in general two- and three-dimensional acoustic problems still cannot be eliminated properly with these methods. Our recent studies [15-18] revealed that the “compounded” effects of differences in “stiffness” between the exact continuous system and the approximate discrete model is the main reason to cause dispersion error. The discretized FEM model based on the standard Galerkin weak form behaves

stiffer than the corresponding system,

leading to the speed of sound propagation in the numerical model faster than the real speed of sound in the original continuous system, and thus the wave length of numerical model is longer than the exact one [15-18]. This also implies that the wave number in the finite element (FE) model is smaller than the actual one, leading to the so-called numerical dispersive error. In order to soften the discretized system, the strain smoothing method was proposed by Chen et al. in [19], and later a generalized gradient smoothing (GGS) technique for discontinuous functions, namely G space theory, and a notion of weakened weak (W2) formulation have been established by Liu[20-23].Using the generalized gradient smoothing technique, Liu and coworkers have developed a series of novel numerical techniques, which allow effectively softening the stiffness of the model and hence make these methods possess a number of attractive features [24-28]. Using the node-based strain smoothing technique, a node-based smoothed point interpolation method (NS-PIM or LC-PIM) [29-31] and node-based finite element

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method (NS-FEM) [32] have been formulated. It has been found the NS-PIM and NS-FEM can provide an upper bound solution to the exact one in terms of energy norm for elasticity problems with homogeneous essential boundary conditions [27-32]. This property implies that the NS-PIM and NS-FEM model are much softer than the exact model. However, the NS-PIM and NS-FEM models behave “overly-soft” leading to the so-called temporal instability observed as spurious non-zero energy modes in vibration analysis [29]. For this reason, a strain-constructed hybrid smoothed finite element method (HS-FEM) is proposed using the existing FEM and NS-FEM techniques [33]. In HS-FEM, a parameter α is equipped into the FEM and NS-FEM formulation. By regulating this parameter, both the upper and lower bound solutions of exact solution in terms of strain energy for solid mechanics problems can be obtained. Furthermore, a super accurate solution that is very close to the exact solution can be obtained for a proper parameter α [33]. It is expected that the HS-FEM will overcome the issue of pollution error in acoustic problems, because the “close-to-exact” stiffness can be generated by a proper HS-FEM formulation.

The paper is organized as follows: Section 2 briefly

describes the fundamental of gradient smoothed method. Section 3 introduces the detailed formulation of pure acoustic problems and acoustic-structural interaction problems using HS-FEM. The issue of pollution error is illustrated in Section 4. In Section 5, a number of examples are presented in detail. Finally, the conclusion from the numerical results is drawn in Section 6.

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2. Fundamental of gradient smoothed method For the sake of simplicity, a two dimensional problem is considered here to explain the gradient smoothed method. As shown in Fig. 1, the smoothing operation on the gradient of a field  for a point x k in a smoothing domain  ks can be expressed as follows

  xk    s   x W  x, x  x k  d  k

(1)

W  x, x  x k  is a smoothing function that generally satisfies the following properties [19]:



ks

W  x, x  x k   0

(2)

W  x, x  xk d   1

(3)

For simplicity, the smoothing function in current study is set to be piecewise constant over the smoothing domain as follows: 1/ Vi w( x  xi )   0

x  i

(4)

x  i

where Vi stands for the area or volume of the smoothing domain  i . Apply divergence theorem, Eq. (1) can be written as follow:

  xk  

1 Vks



ks

  x nds

(5)

where  ks represents the boundary of the gradient smoothing domain, and n denotes the outward-pointing unit normal vector on  ks , as shown in Fig. 1.

3. Mathematical model of acoustic problem 3.1 Pure acoustic domain and hybrid discretized system equations Assumed the acoustic fluid is homogeneous, inviscid, compressible and only s5

undergoes small translational movement, the acoustic field variable p (acoustic pressure) is governed by the following wave equation in a finite domain Ω:

p 

1  2 p 0 c 2 t 2

in Ω

(6)

where c and t represent the speed of sound traveling in the fluid and time, and ∆ is the Laplace operator. Assumed that the acoustic pressure p is a small harmonic perturbation around the steady state, and it can be expressed as follows:

p  pe jt

(7)

where ω is the angular frequency, j= 1 , and p is the spatial distribution of complex sound pressure and satisfies the following Helmholtz Equation:

p  k 2 p  0

(8)

where k is the wave-number defined by k



(9) c The acoustic particle velocity v in ideal fluid is linked to the gradient of acoustic pressure p by the equation of harmonic motion which can be written:

p  j v  0

(10)

where  is the gradient operator. Three sets of following boundary conditions: the Dirichlet, Neumann and admittance (Robin) boundary conditions on  D ,  N and  A are prescribed:

p  pD v  vn or p  n   j vn v  An p p  n   j  An p

D

Dirichlet condition

(11)

N

Neumann condition

(12)

A

Robin condition

(13)

or

where vn is the normal velocity on the boundary ΓN, ρ and An denote the density of s6

medium and the admittance coefficient on boundary ΓA, respectively. The weak form of Helmholtz equation is obtained using the method of weighted residuals, and the standard weak form for acoustic problems can be expressed as follows

 w 

T

pd  k 2  wpd  j   wvn d  j   wpAn d  0 

N

A

(14)

The field variable of acoustic pressure can be expressed in the following approximate form: m

p   Ni pi  Np

(15)

i 1

where Ni are the shape functions and pi are unknown nodal pressure. In the smoothed Galerkin weak form, the “smoothed” gradient obtained in Eq.(1) using the generalized gradient smoothing operation over the smoothed domains is used, and the smoothed Garlerkin weak formulation of the acoustic problems can be obtained as follows[8]:

 N 

T



NPd  k 2  NT NPd  j   N T vn d  j   N T NAn Pd  0 

N

A

(16)

The discretized system equations can be finally written in the following matrix form:

[Κ f  k 2 Μ f  j C f ]Ρ   j  F

(17)

where

 

T

K f   N Nd   B Bd

The acoustic stiffness matrix

(18)

Μ f   NT Nd

The acoustic mass matrix

(19)

C f   NT NAn d

The acoustic damping matrix

(20)





A

Ff   NT vn d N

T



The vector of nodal acoustic forces

(21)

P is the nodal acoustic pressure vector in the domain. Compared to the standard FEM, it is noted that the stiffness matrix in Eq. (18) has been modified into a hybrid form of stiffness which combines FEM with NS-FEM.

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3.2 HS-FEM for Acoustic-structural interaction problem 3.2.1

Plates problem and hybrid discretized system equations

The influence of the acoustic field on the vibrating structure cannot be neglected in an elastic structure sometimes, especially in a thin-walled structure. Such structure is commonly modeled using shell or plate elements. In this paper, lower-order Reissner-Mindlin plate elements are adopted due to its simplicity and efficiency. In the formulation, it is assumed that the normals to the neutral surface of the plate remain straight, but they are not necessarily normal to the deformed mid-surface. Based on the assumption, the displacements in Cartesian coordinate system can be written as follows: u  x, y, z   z x  x, y  v  x, y, z   z y  x, y 

(22)

w( x, y, z )  w0  x, y 

where w0 denotes the displacements of the mid-plane of the plate,  x and  y are the rotations of the transverse normals with respect to the undeformed mid-plane, in the xz and yz planes, respectively. The bending strain ε b and shear strain vector ε s can be expressed in terms of the mid-plane deformations of Eq. (22), which gives   x    x  w     xx  x             xz   x  y  yy    zε b   z   ,    ε s   w  y      yz   y   xy   y   y        x  y    x 

(23)

Substituting the strains shown in Eq. (23) into the principle of virtual work, the s8

variation equation can be obtained as follows:



 ε b D b ε b dV    ε s D s ε s dV  T

s

T

s



s

(24)

 u MudV    u f dV  0 T

T

s

where  s is the structure volume, f is the load on the structure , the u denotes

  , x

, w0  and bending stiffness constitutive coefficients Db , transverse shear T

y

stiffness constitutive coefficients D s and M are defined as 0  1  Et 3  Db   1 0  2   12 1    0 0 1  2 

,

1 0  D s   tG   0 1 

(25)

t 3 /12 0 0   3 M 0 t /12 0   0 0 t  

(26)

in which G is shear modulus,   5/ 6 is the shear correction factor, E is Young‟s modulus and  is Poisson ratio. In the hybrid smoothed finite element formulation for the discrete Reissner-Mindlin plate, the generalized field variable u   x , y , w in the element T

is approximated by the nodal displacement, and the field variable u can be express as: n

n

u   Ni  x  ui  N s u e ,

 u   N i  x  u i  N s u e

i 1

(27)

i 1

where ui  xi , xi , wi  and Ni are the nodal displacement and shape function at node T

i, respectively. Ns are the generalized FEM shape functions and ue is the vector of unknown displacement of element for the structure. Substitute Eq. (27) into Eq. (24), leads to the following matrix form:

K   M u  F 2

(28)

where

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K   Bb Db Bb d   B s Ds B s d T

T





(29)

Μ    NT Nd

(30)

F   NT fd

(31)



2

where B b and B s are the smoothed bending strain and shear strain matrices, respectively. It is noted that shear locking cannot be avoided when the thickness of the plate becomes small. This is because the transverse shear strains cannot vanish under the pure bending condition. In order to alleviate this effect, the „„discrete shear gap‟‟ (DSG) method [34] is adopted here to avoid the shear locking phenomenon. Due to the length limit of paper, the detailed formulated is not presented here [18]. 3.2.2

Coupling of HS-FEM for structural-acoustic analysis

Consider the structural domain  s coupled with the fluid domain  f at the interface  sf as shown in Fig. 2. The boundary conditions of the fluid represent  f , while  u and  t in the structural domain are the essential boundary and natural boundary as shown in Fig. 2. During the interaction between the plate and acoustic fluid, the structure and fluid particle move together in the normal direction of the interface and the boundary condition can be expressed as follows: us n  u f n

on sf

(32)

where n is the normal vector along the fluid boundary, u s and u f are the displacement of plate and fluid acting on the plate. At the interface, the continuity between the structural stress and acoustic pressure can be expressed as follows:

 s n =-p where  s

n

on sf

(33)

is the projection of  along the normal of plate. Note that at the interface, s10

the normal on the plate n s is opposite the normal on the boundary of fluid n f , that is n  n f  n s . The fluid force acting on the structure can be computed as:

Fs  

sf

NTs ns s d   

sf

NTs n f pd     NTs n f N f d   p  sf 

(34)

The force from the structure acting on the acoustic fluid can be obtained as: Ff    

sf

NTf u f d     

sf

NTf us d       NTf n f N s d   u s  sf 

(35)

By introducing the spatial coupling matrix:

H

 sf

N s n f N f dS

(36)

The coupling force in Eq. (34) and Eq. (35) can be written as

Fs  Hp,

Ff    HT u s

(37)

Based on Eqs. (36) and (37), the structural-acoustic problem can then be described by an unsymmetrical system of equations:  M   HT 

0  u s   K  M f   p   0 

H  u   Fs       K f  p  Ff 

(38)

Assume that the pressure and displacement are both time harmonic. So the force frequency response analysis equation for structural-acoustic problems can be given by: K   2M  u   Fs  H       2 T K f   2M f  p  F f    H

(39)

3.3 Numerical integration using hybrid smoothed finite element method In the two dimensional hybrid smoothed finite element method (HS-FEM), the shape functions are constructed with the background of triangular elements in order to ensure efficiency and reliability. The triangular elements can be generated by many

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types of commercial software automatically. Based on the triangular mesh, the problem domain Ω is further divided into N smoothing domains associated with nodes of the tetrahedron such that Ω1  Ω2  … ΩN = Ω and Ωi∩Ωj=Ø, i≠j, where N is the total number of nodes. As outlined in Fig. 3(a), the smoothing domain Ωk for node k is created by connecting sequentially the mid-edge-point to the centroids of the surrounding triangular of the node of interest. Extending the smoothing domain Ωk in 3D problems, the domains discretization is the same as that of standard FEM using tetrahedral elements and the smoothing domain is formed associated with the nodes of tetrahedrons. As shown in Fig. 3(b), the smoothing domain Ωk for node k inside cell I are formed by connecting sequentially the mid-edge-points, the centroids of the surface triangles, and the centroids of cell I. In the formulation of HS-FEM, the smoothed velocity v which is linked to the gradient of acoustic pressure and smoothed strain any points within any sub-domain

ε in the displacement domain at

k ,i for k =1,2,..., N , i =1,2,...,M is the linear

combination of smoothed strain from NS-FEM and compatible strain from FEM:

vk , j   vk , j  1    vk

Acoustic domain

(40)

εk , j   εk , j  1    εk

Displacement domain

(41)

where 0 ≤α ≤1, vk , j and ε k , j are the compatible velocity in acoustic domain and strain in displacement domain,

vk 

1 Vk



vk , j d 

Acoustic domain

(42)

εk 

1 Vk



k, jd

Displacement domain

(43)

k

k

where v k and ε k are the smoothed velocity in acoustic domain and strain in

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displacement domain computed by NS-FEM [23], and Vk is the area or volume of smoothed domain Ωk . Based on Eqs. (40) and (41), all the compatible velocities in the acoustic domain and strain in displacement domain for FEM formulation are replaced by smoothed velocity and strain. The implementation of boundary condition, assembly of stiffness matrix, and computation of mass matrix are exactly the same as FEM. From the above analysis, it is noted that the accuracy of numerical results is controlled by α value. When α=1, the HS-FEM is equivalent to standard FEM model which behaves “overly-stiff” property. When α=0, the HS-FEM is the same as the NS-FEM model which behaves “overly-soft” property. In order to achieve a close to exact stiffness, the selection of α value can be very important to control the accuracy of numerical solutions. Nevertheless, the optimal α value depends on both nature of problems and size of mesh. In a real problem, it is impossible to discretize the model with constant mesh size especially for complicated geometry. Hence, the selection of α value is cumbersome in the simulation of acoustic problem. We are still searching for a rigorous algorithm to find an optimal α value, so as to remove the worries of many. This will not be easy, but we believe that it is possible. Based on our research experience, α2 ∈[0.5, 0,7] will always give a better result compared with those overly-stiff FEM model and overly-softy NS-FEM model in 2D and 3D acoustic problems.

4. Numerical error in acoustic problems The critical issue in the numerical solution of mid-frequency of acoustic s13

problems is to control the pollution error. There is a so called “the rule of thumb kh=1” which describes a certain number of linear elements are needed per wavelength to obtain a reliable solution to the Helmholtz equation. However, the criterion is not reliable with the increase of wave number even if the rule of thumb is followed. In the computational acoustic problem, the gradient of acoustic pressure p is usually adopted as the global error indicator. Based on the relation between the pressure and velocity described in Eq. (11), the numerical error indicator in terms of velocity can be written as:



en   v 

exact

 vh

 v T

exact



 v h d

(44)

where v is complex conjugate of the velocity v, the superscript exact represents the exact solutions and h is the numerical solutions from numerical methods. Ihlenburg et al. showed that the error can be estimated and the relative error for a uniform hp-mesh of finite element method is bounded by [35]: e  n  ee

  v  v   v  v  d  C '  kh     p  v d h

T

h



2



1

p

 kh   C2 ' k    p

2p

(45)

where C1‟, C2‟ are constant parameters that are independent of the parameters k and h, and p here is the degree of polynomial approximation used in the numerical methods. There are two terms in the relative error: the interpolation error that defines the difference between the exact and the interpolation; the pollution error that defines the difference between the interpolation exact wave and the numerical solution of acoustic. For linear interpolation (p=1) discussed here, it is shown in Ref [36, 37] that if kh<1, the relative error for acoustic problems can be expressed by:

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  C1kh  C2 k 3h 2

(46)

From Eq. (46), it is observed that wave number k and mesh size h are very important to control the error.

5. Numerical Examples 5.1 2D problem with Neumann boundary condition The first example is a 2D tube filled with water as shown in Fig. 4. The dimension of this tube with length l=1m and width b=0.5m is considered. The right of the tube is excited by the harmonic motion with normal velocity vn=0.01, the left end of the tube is rigid wall and the normal velocity v=0m/s. The density of water ρ is 1000 kg/m3 and the speed of sound in the water is 1500 m/s. The analytical solutions for this problem can be easily derived and the pressure and velocity are given by p   j  cvn

v

5.1.1



cos  1    sin( )

vn sin  1     sin( )



(47)

(48)

Convergence study

This section studies the convergence by employing four models with 68, 236, 874 and 3352 uniformly distributed nodes. In this 2D example, α2=0.6. Fig. 5 illustrates the convergence rate in terms of global error against the average mesh size h at frequency of 2500Hz and 4000Hz using HS-FEM and FEM respectively. In order to test the performance of HS-FEM, the numerical results from FEM using Galrekin least square (GLS) [38] using the same node are also presented. From Fig. 5(a), it is

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easily noticed that the present HS-FEM and GLS gives very similar accuracy, but better result than FEM when the frequency is equal to 2500Hz. As the frequency becomes 6500Hz, the HS-FEM stands out clearly, and the standard FEM gives very poor solution. The computational efficiency for different numerical methods is presented in Fig. 6, in which it is found that with the same set of mesh, the computational time for HS-FEM is larger than FEM and GLS. There are two factors associated with computational cost: (1) the „„overhead‟‟ cost for all operations until the stiffness matrix is formed. In the HS-FEM model, the overhead cost is apparently increased due to the additional effort of creation of smoothing domain which leads to relatively complicated data storage structures. (2) The solution time to compute the resultant system equations. In the FEM model with triangular element, each integration of stiffness matrix only needs three nodes to assemble the global stiffness matrix. However, the HS-FEM requires more information from all the nodes belonging to the elements that include the current node to assemble. Based on these two factors, the computational cost for HS-FEM is a little higher for solving the resultant system equations. However, in term of computational efficiency (CPU time for the same accuracy in terms of time), the HS-FEM is much better than FEM with consideration of numerical accuracy and time. 5.1.2

Accuracy of acoustic field

The numerical solutions of acoustic pressure using HS-FEM, GLS and FEM along the x-axis at frequency of 2500Hz and 6500Hz are presented in Fig. 7. All the models s16

are solved with 236 nodes.

It can be clearly observed from Fig. 7 that the present

HS-FEM and GLS solutions agree very well with exact solutions. Comparatively, there is a large deviation between the FEM and the analytical solutions due to the dispersion error. Figure 8 plots the global error as a function of wave number k with 236 nodes, in which the global error tends to infinity where the numerical wave numbers are close to the exact eigenvalues. With the increase of wave number, the FEM numerical solutions are becoming worse due to the dispersion of the numerical wave; while the GLS and HS-FEM can still provide very accurate solution even when kh > 1, which is known as “the rule of thumb”. However, the accuracy for GLS model is slightly less than HS-FEM. This is because the gradient smoothing operation has been conducted in the HS-FEM which can partially soften the structure and provide the “right-stiffness” to the acoustic model. 5.1.3

Effects of nodal irregularity

In order to study the influence of the mesh irregularities on the accuracy, the numerical example about irregular mesh is analyzed in this section. The irregularly distributed nodes are generated based on nodal irregularity degree defined in the following expression:

x  x  x  rc  ir y  y  y  rc  ir where x and y are original regular coordinates, x and the irregular mesh, x and

(49)

y  are the coordinates of

y are the initial regular nodal spacing in x- and s17

y-directions, , and βir is a prescribed irregularity degree, rc is a computer-generated random number between -1.0 and 1.0. The larger value of βir leads to more irregular nodes distribution to be used in the nodal irregularity study. Fig. 9(a) and 10(a) presents the results of two mesh models of different nodal irregularity. The study is performed at frequency of 5000Hz. The acoustic pressure distributions computed using the FEM, HS-FEM and GLS along the x-axis are plotted in the Fig. 9(b) and Fig. 10(b) with different types of mesh together with the exact solution. The numerical error indicator in terms of velocity using different irregular mesh is outlined in Fig. 9(c) and 10(c). The pictures clearly indicate that the GLS and FEM results will become worse as the irregular meshes are employed. However, the numerical solutions from HS-FEM still agree very well with analytical solution even irregular mesh is applied. In real engineering problems, it is impossible to discretized the domain into regular node, which imply that the present HS-FEM is potentially to extend in many engineering problems.

5.2 Acoustic eigenfrequencies analysis The modal quantities such as eigenfrequencies and eigenmodes are very important to investigate the acoustic performance in the design stage. It is known that FEM is overly-stiff model resulting in a higher predicted eigenfrequencies in acoustic analysis. Comparatively, the NS-FEM model behaves softer than the continuum counterpart [17]. Hence, the NS-FEM predicts a lower predicted eigenfrequencies. It is natural to expect that the HS-FEM with the right softened effect is able to provide more accurate solutions in terms of eigenfrequencies. The computational model is

s18

shown in Fig. 11. The density of water ρ is 1000kg/m3, and the speed of sound c in the water is 1500m/s. The analytical solutions of eigenfrequencies are available: 2

c m n f      2  l  b

2

m=0,1,2

, n=0,1,2

(49)

where f is the eigenfrequencies of this problem, n and m cannot be zero simultaneously. The tube is discretized with average mesh size of 0.025m which satisfies the “the rule of thumb”, guaranteeing a frequency limit of 9554Hz. Table 1 summarizes the first twenty-four natural eigenfrequencies obtained from the HS-FEM, FEM and NS-FEM with the same mesh. The analytical solutions are also listed in this table for comparison. As shown in Table 1, the numerical solutions are bound by FEM and NS-FEM. In the lower eigenfrequencies, all the numerical models match the analytical solution very well. However, the errors obtained from NS-FEM and FEM become larger with increase of eigenfrequencies; while the solutions of HS-FEM are still very close to the analytical solutions even in the high eigenfrequencies. This numerical example demonstrates that HS-FEM provides soft effect to the model and has close to the exact stiffness of the continuous system again. Figure 12(a) shows the convergence rate of eigenfrequencies for mode shape 24. Although convergence rates for both HS-FEM and FEM are quite similar, the HS-FEM produces more accurate results than that of FEM. In order to analyze the influence of α on the accuracy of eigenfrequnecy for mode shape 24, the numerical results obtained from different α value using 306 nodes is presented in Fig. 12(b) with analytical solution. With adjustment of α value, the upper and lower bound solutions of eigenfrequency are obtained. From Fig. 12(b), it is clearly indicated that α2=0.64 gives s19

the exact solution. Fig. 13 compares the computational efficiency between FEM and HS-FEM. It is noticed that with the same set of nodes, the computational time for HS-FEM model is slightly larger than FEM model.

However, in terms of

computational efficiency, the HS-FEM model performs much better than FEM model.

5.3 2D car acoustic problem In this section, acoustic pressure distribution in a car passenger compartment is analyzed using HS-FEM and FEM. As shown in Fig. 14, it is assumed that the front panel of the passenger compartment has the velocity of 0.01 m/s due to the engine vibration. The density of air ρ in the car passenger compartment is 1.225 kg/m3 and the speed of sound in the air is 340 m/s. The roof of the passenger compartment is fixed with absorbing material with admittance of 0.00144m3/(Pa·s). This domain is discretized with 361 nodes, which gives a frequency limit of 721Hz based on “the rule of thumb”. The acoustic pressure at different frequency values of 600Hz and 700Hz will be investigated using the HS-FEM here. Figs. 15(a) and (b) show the acoustic pressure distributed in the passenger compartment along the defined path ab in Fig. 14 at 600Hz and 700Hz obtained from the HS-FEM, FEM and GLS. Because the analytical solution is unavailable for this problem, a reference configuration using FEM with a very fine mesh (22524 nodes) is employed. It is observed that the numerical solutions obtained from HS-FEM have a good agreement with reference one. At lower frequency (600Hz), the results from GLS are better than FEM but less than HS-FEM. However, a large deviation in the FEM and GLS model is observed compared with reference one when

s20

frequency is equal to 700Hz as outlined in Fig. 15(b). The direct frequency response analysis is studied using present HS-FEM and FEM. At each frequency, the system equations is set up and solved to obtain the pressure distribution. A full range frequency sweep is tested from 1Hz to 750Hz at intervals of 1.0Hz, and the response (sound pressure level) at the driver‟s ear point shown in the Fig. 14 is computed. The results using HS-FEM and FEM are plotted in Fig. 16. As the analytical solution is unavailable, the reference solution using FEM with 22524 nodes is also provided. From Fig. 16, it is noticed that the HS-FEM is able to provide much better result than FEM in the full frequency range. Even the frequency is beyond 721 Hz which is the limit of the FEM based on the rule of thumb, the HS-FEM solution also matches the reference solution very well. From this numerical example, it has proven again that HS-FEM possesses a close to exact stiffness due to a more appropriately softened effect and provides much better numerical results even at high frequencies.

5.4 3D acoustic analysis of Mini car occupant compartment Due to the excellent features of HS-FEM validated in the above mentioned 2D problems, the 3D formulation of HS-FEM is extended to analyze the acoustic pressure distribution in three dimensional car passenger compartment. As shown in Fig. 17 (a), the front panel of the passenger compartment is subjected to the vibration from the engine with the velocity of 0.01m/s. The roof of the passenger compartment is fixed with absorbing material with admittance of 0.00144 m/(Pa·s). In this numerical example, α2=0.65. s21

The acoustic pressure distributions with a frequency of 300 and 400Hz using HS-FEM and FEM with 5364 nodes (the discretized information shown in Fig. 17 (b)) are plotted in Figs. 18 and 19. As the analytical solution is unavailable for this problem, the reference solution using FEM with a very fine mesh (221885 tetrahedron meshes with 44278 nodes) is obtained. As shown in Fig. 18 and 19, the contour lines obtained from HS-FEM model are much smoother than that of FEM result due to right softened effect. In addition, the numerical solution using HS-FEM is also closer to the reference solution, especially in the high gradient domain as shown in Figs.18 and 19. In order to compare the numerical results quantitatively, the real part of pressure obtained from the HS-FEM and FEM along the defined path

ab shown in Fig. 17(a)

are plotted in Fig. 20. In order to show the performance of the proposed HS-FEM, the numerical results obtained from 3D edge-based smoothed finite element method (ES-FEM) using 5364 nodes [39] are also presented herein. The 3D ES-FEM has been demonstrated more accurate and efficient than the standard FEM and modified integration rule (MIR) method [40] in acoustic problems. Again, it is found that there is a large deviation between FEM model and reference one. On the other hand, the numerical solution obtained from HS-FEM model is very close to the reference solution, and even better than 3D ES-FEM. This 3D numerical example has clearly demonstrated that the present HS-FEM is an excellent candidate to solve three dimensional mid-frequency acoustic problems.

s22

5.5 Structure-acoustic coupling problem Noise and vibration problems generated by structural-acoustic interaction are quite common in industry such as airplane cabin and vehicle passenger compartment. As the coupling between the flexible structure and an acoustic cavity has a strong effect on the interior noise level of such enclosure, the coupling effect of the passenger compartment system is analyzed to investigate the coupling performance of HS-FEM/HS-FEM and FEM/FEM. As shown in Fig. 21, a unit load is applied at the center of front panel excited noise in the passenger compartment. The front panel can be modeled as clamped plate with  x =0,  y =0 and w=0 at all the edges. In this example, the flexible plate is made of aluminum (ρ=2700kg/m3, v=0.3 and E=71GPa), and the air property is taken as ρ=1.21kg/m3 and speed c=343m/s. As the forced frequency response analysis is a good way to provide information of eigenfrequencies, where the peaks of the acoustic pressure occur. The forced frequency response for this coupled passenger compartment is studied in this section. The responses at the driver‟s ear point in acoustic fluid of the passenger compartment and the center of the plate in the structure domain are computed. Because the frequencies ranging from 1 Hz to 280 Hz is of great importance in designing a vehicle, the analysis for this range of frequency is studied. The numerical solutions from both FEM/FEM and HS-FEM/HS-FEM (260 nodes for plates, 4761 nodes for acoustic domain) at these two different points obtained are plotted in Figs. 22 and 23, respectively. For the purpose of comparison, the reference results obtained using the coupled FEM/FEM in SYSNOISE with a very fine mesh (1421 nodes for

s23

plates, 65301 nodes for the acoustic domain) are presented together. From Figs. 22 and 23, it is easily observed that the numerical solutions from both the FEM and HS-FEM models are very close to reference solution when the frequency is less than 150Hz. However, the deviation between the FEM model and reference one is becoming larger and larger with increase of frequency due to the dispersion error. As HS-FEM is able to provide a close to exact stiffness, the accuracy obtained from HS-FEM is still in good agreement with the reference one in the whole frequency range.

This numerical example has also validated the proposed method can be

applied in the practical structural-acoustic problems and predicts much more accurate solutions than FEM.

6. Conclusion and Discussion In this work, the hybrid smoothed finite element (HS-FEM) is further formulated for solving acoustic problems. The smoothed Galerkin weak form is adopted to formulate the discretized system equations and the numerical integration is performed based on the smoothed strain which combines NS-FEM and FEM. A number of acoustic problems are investigated in detail to study the accuracy, convergence and efficiency of the HS-FEM. Within the limitation of the studies, the following conclusions can be drawn: 

The implementation of HS-FEM using triangular elements in 2D and tetrahedral elements in 3D is very simple; no additional parameters or degrees of freedoms are needed.

s24



The HS-FEM has an appropriately softened effect and close-to-exact stiffness due to gradient smoothing operation with the optimal alpha used in the model, which dramatically reduces the dispersion error.



The HS-FEM is less sensitive to the distortion of mesh compared with FEM and GLS.



For the practical acoustic problems with complicated geometry, the HS-FEM solutions are much more accurate than the FEM with the same nodes.



The eigenfrequencies of acoustic problem can be bound by HS-FEM with adjustment of alpha value.



The couple HS-FEM provides much better result in structural-acoustic problem compared with FEM.

Acknowledgements The project is supported by the Science Fund of State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body no.31315002 and 51375001. The authors wish to thank the support of National Natural Science Foundation of China (Grant No. 11202074)

s25

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Figure

y  ks

xi  ks

n

x

Figure 1: smoothing domain for a point at xi

f Interface

HS-FEM nf

f

HS-FEM

s s Figure 2: structural solid domain  s couple to acoustical fluid domain  f

s29

ni Centroid of triangle

k k

Field node k

Mid - edge - point

(a) background triangular cells and nodal smoothing domains for node k in 2D

Cell I

n

Node k

1

Node k

n p

Field node

n 2

Centroid of the surface triangle

Centroid of the tetrahedron

Mid  edge point

(b) background tetrahedral cells and nodal smoothing domains for node k in 3D

Figure 3: Schematic of one partition of the node-based smoothing domain for node k as one vertex of four-node tetrahedral cell I (k-k2-k3-k6)

s30

Figure 4：Two dimensional tube filled with water

s31

-5 FEM with T3 HS-FEM with T3 GLS with T3

-5.5

log10(en)

-6

-6.5

-7

-7.5

-8

-8.5 -2

-1.9

-1.8

-1.7

-1.6

-1.5 -1.4 log10(h)

-1.3

-1.2

-1.1

-1

(a) 2500Hz -3.5 FEM with T3 HS-FEM with T3 GLS with T3

-4 -4.5

log10(en)

-5 -5.5 -6 -6.5 -7 -7.5 -8 -2

-1.9

-1.8

-1.7

-1.6

-1.5 log10(h)

-1.4

-1.3

-1.2

-1.1

-1

(b) 4000Hz

Figure 5： Convergence rate

s32

-3.5 FEM with T3 HS-FEM with T3 GLS with T3

-4

log10(en)

-4.5

-5

-5.5

-6

-6.5

-7 0

0.2

0.4

0.6 0.8 computational time (s)

1

1.2

1.4

Figure 6： Comparsion of computational efficency

s33

4

2

x 10

FEM with T3 Analytical solution HS-FEM with T3 GLS with T3

1.5 1

Pressure

0.5 0 -0.5 -1 -1.5 -2 0

0.1

0.2

0.3

0.4

0.5 x

0.6

0.7

0.8

0.9

1

0.7

0.8

0.9

1

frequency=2500 4

6

x 10

FEM with T3 Analytical solution HS-FEM with T3 GLS with T3

4

Pressure

2

0

-2

-4

-6 0

0.1

0.2

0.3

0.4

0.5 x

0.6

frequency=6500 Figure 7：Pressure distribution along x axis at different frequencies

s34

0 -1 -2 -3

log10(en)

-4 -5 -6 -7 -8

FEM with T3 HS-FEM with T3 GLS with T3

-9 -10 0

5

10



15

20

25

Figure 8: Error for different wave number

s35

(a) Irregular mesh a 4

3

x 10

FEM with T3 Analytical solution HS-FEM with T3 GLS with T3

2.5 2 1.5

Pressure

1 0.5 0 -0.5 -1 -1.5 -2 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

(b) Acoustic pressure distribution -5

1

x 10

0.9 0.8 0.7

en

0.6 0.5 0.4 0.3 0.2 0.1 0

GLS

FEM

HS-FEM

(c) Comparison of error

Figure 9: Effect of irregular mesh a on accuracy s36

(a) Irregular mesh 4

4

x 10

FEM with T3 Analytical solution HS-FEM with T3 GLS with T3

3 2

Pressure

1 0 -1 -2 -3 -4 -5 0

0.1

0.2

0.3

0.4

0.5 x

0.6

0.7

0.8

0.9

1

(b) Acoustic pressure distribution -5

6

x 10

5

en

4

3

2

1

0

GLS

FEM

HS-FEM

(c) Comparison of error

Figure 10: Effect of irregular mesh b on accuracy

s37

Figure 11：Two dimensional tube filled with water

s38

Error for natural eigenfrequency of mode shape 24

5 HS-FEM with T3 FEM with T3

4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0

500

1000

1500

2000 2500 DOF

3000

3500

4000

4500

(a) Convergence rate for mode shape 24

Natural eigenfrequency for mode shape 24

11000 10500 HS-FEM with T3 Analytical solution

10000 9500 9000 8500 8000 7500 7000 0

0.1

0.2

0.3

0.4

0.5 2

0.6

0.7

0.8

0.9

1

(b) Influence of accuracy for different α value Figure 12: Comparison or accuracy from different numerical methods

s39

Error for natural eigenfrequency of mode shape 24

5 HS-FEM with T3 FEM with T3

4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0

20

40 60 80 Computational time (Seconds)

100

120

Figure 13： Comparsion of computational efficency

s40

Driver ear's position

absorbing material

Vn =0.01m/s

a

the defined path ab

b

Figure 14: Two dimensional acoustic problem in the car

s41

4 FEM with T3 HS-FEM with T3 Reference solution GLS with T3

3 2

Pressure

1 0 -1 -2 -3 -4 -5 -6 0

0.5

1

1.5

X

a) 600 Hz 10

5

Pressure

0

-5

FEM with T3 HS-FEM with T3 Reference solution GLS with T3

-10

-15 0

0.5

1

1.5

X

b)700 Hz Figure 15: Acoustic pressure distribution along the path ab at different frequency values

s42

Figure 16: Acoustic frequency response at driver‟s ear point for 2D car problem

s43

a) Model of 3D car

b) Mesh Figure 17: Three dimensional acoustic problem in the car

s44

Y

X

Pressure

Y

9 8 7 6 5 4 Z 3 2 1

a) Reference solution X

Pressure 9 8 7 6 5 4 3 2 1 Y

Z

b) HS-FEM

X

Pressure 9 8 7 6 5 4 3 2 1

c) FEM

Figure 18: Contour of pressure for 300Hz

s45

Z

Y

X

Pressure 12 11 10 9 8 7 6 5 4 3 2Z 1

a) Reference solution

Y

X

Pressure 12 11 10 9 8 7 6 5 4 Z3 2 1

Y

b) HS-FEM

X

Pressure 12 11 10 9 8 7 6 5 4 3 2 1

c) FEM

Figure 19: Contour of pressure for 400Hz

s46

4 Reference FEM with T4 HS-FEM with T4 ES-FEM with T4

3 2

Pressure

1 0 -1 -2 -3 -4 -5 -6 1

1.5

2

2.5

3

3.5

4

4.5

X

a) 300Hz 10 Reference FEM with T4 HS-FEM with T4 ES-FEM with T4

8 6

Pressure

4 2 0 -2 -4 -6 1

1.5

2

2.5

3

3.5

4

4.5

X

b) 400Hz Figure 20: acoustic pressure distribution along the path ab at different frequencies

s47

Figure 21: Structural solid domain Ωs couple to acoustical fluid domain Ωf

s48

150

FEM Reference HS-FEM

140

Response(dB,ref=2e-5)

130 120 110 100 90 80 70 60 50 50

100

150 Frequency

200

250

Figure 22: Driver's ear point response

FEM Reference HS-FEM

220

Response(dB,ref=1e-12)

200

180

160

140

120

100 50

100

150 Frequency

200

250

Figure 23: Fender response at the center of the plate

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Table 1: 2D tube natural eigenfrequencies calculated by FEM, HS-FEM and NS-FEM Error Eigenvalue

Exact (Hz)

FEM (Hz)

of

HS-FEM

FEM

(Hz)

(%) 1

750.00

2

1500.00

3

2250.00

4

3000.00

5

3750.00

6

4500.00

7

5250.00

8

6000.00

9

6750.00

10

7500.00

11

7500.00

12

7537.40

13

7648.50

14

7830.20

15

8077.70

16

8250.00

17

8385.30

18

8746.40

19

9000.00

20

9154.90

21

9604.70

22

9750.00

23

10090.00

24

10500.00

750.1228 1500.981 2253.309 3007.831 3765.264 4526.311 5291.658 6061.972 6837.894 7620.027 7623.387 7664.556 7788.394 7990.989 8267.168 8408.99 8610.867 9015.405 9205.076 9474.413 9977.122 10013.57 10530.26 10822.73

0.016 0.065 0.147 0.261 0.407 0.585 0.793 1.033 1.302 1.600 1.645 1.687 1.829 2.053 2.346 1.927 2.690 3.076 2.279 3.490 3.877 2.703 4.363 3.074

749.983 1499.864 2249.543 2998.921 3747.907 4496.411 5244.353 5991.66 6738.265 7484.113 7522.428 7560.947 7675.171 7861.868 8116.048 8229.159 8431.634 8802.045 8973.35 9220.71 9681.2 9716.869 10178.36 10459.17

Error of HS-FEM (%)

0.002 0.009 0.020 0.036 0.056 0.080 0.108 0.139 0.174 0.212 0.299 0.312 0.349 0.404 0.475 0.253 0.553 0.636 0.296 0.719 0.796 0.340 0.876 0.389

NS-FEM (Hz)

749.7625 1498.098 2243.572 2984.728 3720.079 4448.073 4705.364 4710.15 4921.927 4940.112 5167.061 5236.328 5613.614 5874.649 6033.852 6399.418 6572.932 6665.595 6787.903 6853.497 6931.248 7080.113 7110.274 7219.418

Error of NS-FEM (%)

0.032 0.127 0.286 0.509 0.798 1.154 10.374 21.498 27.083 34.132 31.106 30.529 26.605 24.974 25.302 22.431 21.614 23.790 24.579 25.138 27.835 27.383 29.531 31.244

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