Dynamic analysis of a coupled structural-acoustic problem

Finite Elements in Analysis and Design 14 (1993) 225-234

225

Elsevier FINEL 305

Dynamic analysis of a coupled structural-acoustic problem Simultaneous multi-modal reduction of vehicle interior noise level by combined optimization Chinmoy Pal and Ichiro Hagiwara Vehicle Research Laboratory, Nissan Motor Co., Yokosuka shi, Kanagawa ken, Japan Abstract. In the present study, a finite element sensitivity analysis is performed for a higher- and lower-order

truncated modal coupled structure-acoustic problem, to find out the rate of change of vibration responses of the body panels and sound pressure levels at the position of the ear of a vehicle passenger/driver. An inverse multi-objective optimization problem, based on a structural synthesis method and a pseudo-inverse method, is solved to determine the required minimum changes of the design variables from the baseline design parameters, in order to attain the selected design response by the shortest route. It is found that for a highly nonlinear dynamic response optimization problem, the structural synthesis method and the pseudo-inverse method converge to the same solution when the step size at each iteration level of the optimization process is kept small.

Introduction

With the advancement of computational capability and speed in recent years, the finite element method (FEM) has become a powerful tool for structural analysts and structural designers solving practical problems having very many degrees of freedom. But for general dynamic problems a large amount of computational time is usually consumed by time integration schemes for time domain analysis. However, such problems can be tackled by modal analysis, which basically assumes linear superposition of the first few important lower-order modes relevant to the designer's region of interest. However, such an efficient and economical computational technique becomes computationally expensive for calculating the acoustic response of a coupled structural-acoustic problem, particularly where the lower-order modes of the sound field correspond to the higher-order modes of the structural field. Further, for an optimization problem such as the reduction of the sound pressure level at a certain frequency band, the above difficulty becomes severe even when one applies existing modal superposition techniques which are capable of truncating only the higher-order modes, due to the repeated estimation of unwanted lower-order modes of the structural field at each stage of the design cycle of a modified structure. From the sales point of view, accurate estimation and reduction of the internal noise level by a cost-efficient numerical simulation technique plays an important role in designing high-quality vehicles. A p r o g r a m m e of research based on a new modal superposition technique, with truncated higher- a n d / o r lower-order modes, and the corresponding sensitivity analysis of a coupled structure-acoustic Correspondence to." Dr. Chinmoy Pal, Vehicle Research Laboratory, Nissan Motor Co., T-237, 1 Natsushima cho,

Yokosuka shi, Kanagawa ken, Japan. 0168-874X/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

226

C. Pal, I. Hagiwara / Dynamic analysis of a coupled structural - acoustic problem

system for modal frequency response analysis, was discussed in detail in the previous research works of Ma and Hagiwara [1,2] with application to a car cabin model. It was shown that this new method has a higher order of computational accuracy and better convergence characteristics than those of the other well known existing methods, such as the modal displacement method, modal acceleration method for modal superposition analysis and the Fox-Kapoor modal method, Wang's improved modal method and Nelson's method for modal sensitivity analysis. Moreover, the present lower- and higher-order truncated modal superposition and sensitivity analysis is one of the most general methods, in the sense that it can be reduced to the other above-mentioned existing methods by selecting different values of a shift parameter. The same method was further extended to the transient response analysis by Ichikawa et al. [31. However, this paper deals with the application of the above-mentioned sensitivity analysis to optimize the structural response and the sound pressure level within a box. In fact, the present optimization technique is based on the inverse problem, which determines the required minimum change in design variables to satisfy or achieve a certain level of response. Comparisons are made between two different algorithms, (a) the structural synthesis method and (b) the pseudo-inverse method, which are both very suitable for finding optimum solutions when the number of design variables exceeds of the number constraints. At present, different commercial optimization packages such as VMCON, ADS, and CONMIN[4] based on mathematical programming are quite often used for industrial structural optimization problems. However, most of them are not very efficient and sometimes they fail to converge when the number of variables and constraints is large, say a few hundred or thousand. Further, it is often a requirement to reach a feasible optimum design from an initial design which may or may not be feasible. Again, for a complex problem like a car, often the designer has to satisfy, or to make a trade-off among, a number of mutually conflicting criteria, the target values of which are set by the designer to suit his/her constraints of the design environment. Hence in this paper we have extended the concept of the structural synthesis method to multi-criteria compliant optimization, and explained it with the help of a numerical example.

Coupled structural-acoustic analysis The coupled structural-acoustic FEM equation can be written as follows: (1)

Mii + Ku = f ,

where M =

Ma s

.7

/,/T

01

Ma a ,

g =

ST= { f ? ,

ga aj,

(la,b)

(lc-e) and u s, u a, fs and fa are the displacement vector of the structure, the sound pressure vector, and the exciting forces acting on the structure and the sound source in the acoustic field, respectively. Mss and Kss are the mass and stiffness matrix of the structure, Maa and Kaa are the mass and stiffness matrix of the acoustic field and Mas and Ksa are the coupling matrices. The subscripts 's' and 'a' denote the structural system and the acoustic system respectively; .... indicates the second time derivative. Orthogonality conditions, which generally do not hold good for a coupled system, can be obtained by introducing right and left eigenvectors as described in Ref. [1]. Using the following relation between the left eigenvector ~i and the right eigenvector tbi, =

~/=

/

4~s~, 'C-4~.i

fT},

for ,~i ~ 0,

Ks a = _ M a s ,

(2)

C. Pal, L Hagiwara / Dynamic analysis of a coupled structural-acoustic problem

227

where A i is the eigenvalue of the coupled system and (~si and t~a i represent the eigenvectors of the structural system and acoustic system respectively, a modal response analysis by truncating lower- and higher-order modes and corresponding sensitivity analysis for a coupled structural-acoustic system are derived in detail in Ref. [1,2]. The relation between the present approach and the other existing methods are given in the Appendix.

Multi-criteria optimum design Structural synthesis method In ordinary optimization analysis, we usually find optimum values of the design variables which minimize a certain objective function subject to some equality and non-equality constraints, as denoted by the following equation [4,5]: Min F ( x ) subject to

g ( x ) <~0 h(x) = 0

objective function, inequality constraint, equality constraint.

(3)

However, in some cases, it is important to find the least change of design variables required to meet a certain design response level. For such cases (i.e. an inverse problem) we have to formulate the optimization problem in such a way that the response level will be taken as the design constraints and the total change of design variables, from the initial design to the final design, as the objective function. Further, when we choose two or more than two conflicting design criteria simultaneously, we have to make a trade-off among the individual optimum solutions. In the multi-criteria inverse optimization problem presented here, the objective function is the sum of the changes of design variables (say, the total change of plate thickness from the initial/starting design values) and the constraints are the desired performance Characteristics or responses (such as the sound pressure level at some point in the acoustic field at one or more frequency levels) set up by the designer. Assume an eigenvalue problem defined by: ([R'] - A[~t]){~} = 0,

(4)

where we denote the stiffness and mass matrices by K and M, and the eigenvalue and eigenvector by A and ~b respectively. The upper bar denotes the values at any arbitrary design level. Modifying the selected design variables (say, the thickness of a plate or the cross-sectional area of a bar, etc.) by a percent (see eqn. (4a)) from the present design values, we can assume a linear change in the design response z (say, the eigenvalues, eigenvector, sound pressure level, etc.) by the same percentage. Hence, using the Taylor series expansion, we can write the following set of equations for the perturbed design space. The primed quantities indicate the sensitivity matrices and vectors of the corresponding quantities with respect to design parameters a: N

N

[x] = [~] + Y'~ x',,a,,,

[ K ] = [ K ] + ~ K'a,,,

n=l N

[M]=[.~t]+

~' M'an ' n=l N

[~b] = [~] + Z ~b~,a,, n=l

(4a,b)

n--1 N

[A]=[A]+

Y'~ A'na n,

(4c,d)

n=l N

[z] = [~'] + Z z'a,,. n~l

(4e,f)

C. Pal, L Hagiwara / Dynamicanalysisof a coupledstructural-acousticproblem

228

Extending the structural synthesis formulation [6], for more than one (say, J) conflicting design response criteria, we can form a functional /7 of the following form

J /7=

N

J(

E E (O~n--Og:J) 2 + E j=l n=l j=l

N )

t3~

~j-z? +

2

N

, 2 E Zjn~n + E an, n=l

(5a)

n=l

subject to

N ~j+ E Zjna ' n -- Zj* < o ,

(Sb)

n=l

where a new perturbed design value is defined by eqn. (Sa). The first term is for multi-criteria optimum design where a n*] is the ideal solution vector of the design variable for the jth performance criterion, obtained by minimizing the individual optimal criteria subject to a set of prescribed design constraints corresponding to N design variables a n denoting the fractional changes for the design parameters x n. The upper bar indicates quantities corresponding to the baseline design variables xn, the asterisk the allowable response limit or target response values, and primes the response sensitivity with respect to the design parameters calculated by the above truncated modal sensitivity analysis. For the ordinary single-criterion optimum design problem, differentiating eqn. (5a) with respect to Lagrange parameters ~j and a fractional change a n of design variables, we have MI --=0

i = 1,...,N,

(6a)

a/7 --=0 a~j

j = l . . . . . J,

(6b)

and for reducing eqns. (6a, b) to a set of linear simultaneous equations in terms of a i and fli, we consider the following linear approximation with respect to a~, in place of the second term of eqn. (5a):

z*) E Zj'nan "

/3j ~ - z/*)z- 2 ( ~ -

(6C)

n=l

The final values of the design parameters of the system can be determined by solving a set of simultaneous equations by iteration, until a i and /zj become zero, to satisfy the equality condition of eqn. (5b):

B•

¢~1 '

0

•

0

'¢1

=

(z,*-~)

2 '

(7)

where A is an (N x N) square diagonal matrix, whose elements are Ann=2,

Anm=O ( n ~ m )

(7a)

and matrices B and C consist of the following elements:

B.j = - 2 ( z ?

- ~)~j'.,

C,~ = o

(7b,c)

C Pal, I. Hagiwara / Dynamic analysis of a coupled structural-acoustic problem

229

To find out the individual optimum values of a,*J, eqn. (5a) should be minimized by omitting its first term. For multi-criteria optimization the zero terms of the right-hand side vector of eqn. (7) corresponding to a i should be replaced by 2a*, and the diagonal terms A , , of [A] are to be replaced by 4.

Pseudo-inverse method The above inverse optimization problem can also be solved with the help of the pseudo-inverse method [5,6]. Assuming the following relation

exists, we have to find those values of change in design variable Ax that minimize the following functional ~r, which is a function of Lagrange multipliers K and Ax. ~ ( a x , K) = ½{AxIT{Ax} - ( [ z ' ] { A x t - {AzI)T{K}.

(9)

From the stationary condition O,

(10)

where Bnj = Zj'n,

(10a)

{Ax} = [z']W{K}.

(10b)

we get

Using condition (10b), and eqn. (8), we have [Z'][z']T{K} = {Az},

(11)

and assuming the inverse of [z'][z'] T exists, then from eqn. (11) we have {r} = ([ z'][ z']T)-I{Az},

(12)

and substituting eqn. (12) in eqn. (10b) {Ax} = { X - - ~ } = [ z ' I T ( [ z ' I [ z ' I T ) - ' ( Z ~ - - Z * ) = B ( B T B ) - ' ( ~ - - Z * ) .

(13)

Depending on the degree of nonlinearity of the design constraints, the percentage change of design variables a , or Ax after each iteration has to be restricted to a certain value for stable convergence.

Numerical simulation and discussion

A car cabin is modelled as a box made of thin plates, as shown in Fig. 1. There are 125 nodal points and 96 quadrilateral plate elements in the structural field, and the enclosed acoustic field is divided by 64 hexahedral elements. The overall flow chart of the present calculation is shown in Fig. 2. Since the present analysis is based on structural modification, the acoustic analysis is carried out first, and the corresponding eigenvalues and the eigenvectors are repeatedly used in a coupled structure-acoustic modal response analysis and in the corresponding coupled structure-acoustic sensitivity analysis in each design cycle of the

C. Pal, L Hagiwara / Dynamic analysis of a coupled structural - acoustic

230

problem

d Region of Design Variables

0

y

.i~ --r

~00c~ .....wl t = 4mm

PANEL

26

4

4

27

A

A A

2_~

~ 'outpJ

A

~a

a

A

LNEL 3

inn~

PANEL 1 Fig. 1. Finite e l e m e n t m o d e l of a car cabin.

optimization process. For simplicity, we take the thickness of the bottom plate (the shaded region) as design variable and the sound pressure level (SPL) at the point B as constraint, the values of which are to be reduced to a certain level at different frequencies, as shown in Fig. 3. The point A is excited by a sinusoidal force of unit magnitude in the y-direction. The sound pressure levels at frequencies of 134 Hz and 147 Hz are to be reduced to a 50 dB level. The results are plotted in Figs. 4 and 5. Table 1 shows the tabulated values of the above results, and that of the combined optimization when the SPLs at two different frequencies are to be controlled simultaneously. It is clear from Fig. 4(a, b) that both the structural synthesis method and the pseudo-inverse method almost converge to the same results, i.e. the total change of plate thickness is almost identical when the sound pressure levels at 134 Hz and 147

I ACOUSTIC ANALYSIS[

t STRUCTURALAALYSIS [

't

I COUPLEDSTRUCTURE'-ACOUSTIC[ SENSITIVITYANALYSIS ] ,l

OPTIMIZATIONANALYSIS] ANDEVALUATION Fig. 2. Flow chart of the optimization procedure.

C. Pal, L Hagiwara / D y n a m i c analysis o f a coupled s t r u c t u r a l - acoustic p r o b l e m

231

Sound Pressure Level ( at node 33) 100

S,

---"-initial ~

_

80 60 40

"~

20

I

-target

~evel

-

01

0

140

150 Fig. 3. Initial r e s p o n s e c u r v e f o r s o u n d p r e s s u r e level,

Frequency(Hz)

(8) 856 1 - - _ -_ ~. . -. .~. . ~. . 4

Initial t(synthesis 134Hz) t(pseudo inverse 134Hz)

"~

.... 3 2

..x=,__

"

' 4

0

~

' ' " ' " ' 8 1 2 1 6

P l a t e No,

t b) 5 " ' ~ / _ : ~ _ .... _ ~ _ - - ~ . . . . . . ,, .... ~ - - i[ . . . . . . . . . . . . . . . . . . . . .

4

•

3

,

0

.

4

,

.

,

8

.

1

2

Initial t ( s y n t h e s i s 147Hz) t ( p s e u d o i n v e r s e .147Hz)

,

1 6

P l a t e No. r

° ' ~ o-'~ _

__ -

~ ..... ~

...........

Combined Opt.(struc. sys)

. _ , . ~ , . ~ ~ , . , j ~ _ _ _ :_-~,_ _ - ~,~ m . . . .

t ( s t r u c , sys. 1 4 7 H z . ) initial thickness

20

4

8

1 2

1

6

Plate No.

Fig. 4. R e s u l t s f o r p l a t e t h i c k n e s s .

100

"~ ~

90 o

8O

~

134

lOO

Iz(m udo) l Iz(s,f , t h ¢ ~

I

I '/

70 _m

1214]

~l

"~ o

80

Z

60

I

14

¥"l

'60

--t,-

5O

I

40 0

4 8 12 16 Iteration Number (a)

"~ 40

(

2o

0

2

4

6

Iteration

8

Number

(b)

Fig. 5. C o n v e r g e n c e o f r e s p o n s e : (a) a t 134 H z ; (b) at 147 H z .

10

232

C. Pal, 1. Hagiwara / Dynamic analysis of a coupled structural-acoustic problem

Table 1 Required total change of design variable Case

Pseudoinverse method

Structural synthesis method

134 Hz 147 Hz

1019% 211%

1039% 217%

Combined optimization

500%

Hz are taken as individual target levels. As the response (the sound pressure level) is a highly nonlinear function of the design variables, the upper bound of maximum change of plate thickness is taken as 5% for each iteration. Figure 5 indicates that the structural synthesis method converges faster than the pseudo-inverse method. For combined optimization, the total change of plate thickness lies between the above two results, as shown in Table 1. These results for combined optimization are also shown in Fig. 4(c). Although in the present numerical analysis the base plate is taken as the design region, it would be practical from the designer's point of view to take each panel or part of a panel of an actual car cabin as one design variable. Further, the present structural modification technique has some limitations and it is effective only in modifying those acoustic response levels which are affected by the structural modes. In other words, it is valid only for a structurally resonant acoustic response. In our future research work, investigations will be carried out on active vibration control with structural optimization in the coupled structural-acoustic noise reduction problem.

References [1] Z.D. MA and I. HAGIWARA,"Development of a new mode-superposition technique for modal frequency response analysis of coupled acoustic-structural systems", Finite Elements in Analysis and Design 14, pp. 209-223, t993. 12] I. HAGIWARA,Z.D. MA, A. ARM and K. NAGABUCH1,"Technical development of eigenmode sensitivity analysis for coupled acoustic-structural system", Trans. Jpn. Soc. Mech. Eng. 56 (527), pp. 1704-1711, 1990 (in Japanese). [3] T. ICHIKAWA,I. HAGIWARAand Z. MA, Tram. Jpn. Soc. Mech. Eng. 58 (545), p. 92, 1992 (in Japanese). [4] G.N. VANDERPLAATS,"Numerical optimization techniques", in: Computer-Aided Optimal Design Structural and Mechanical System edited by C.A. MOTA SOARES,Springer-Verlag, Berlin, 1987. [5] R.H. GALLAGHARand O.C. ZIENKIEWICZ(eds.), Optimum Structural Design--Theory and Applications, Wiley, New York, 1973. [6] S. NAKAGIRIand K. StJzura, Trans. Jpn. Soc. Mech. Eng. 53, p. 2439, 1987 (in Japanese). [7] YANAI and TAKEUCHI, Projection Matrix, Generalized Inverse Matrix and Particular Value Decomposition, Tokyo University Press, 1983 (in Japanese). [8] M. OKUMA, M. NANPE, S. PARK and A. NAGAMATSU,"Technique of structural dynamic optimization--I st Report: Use of mathematical pseudo-inverse method and substructure synthesis method", Tram. Jpn. Soc. Mech. Eng. 54 (504), pp. 1753-1760, 1988.

Appendix

Modal response analysis by truncating lower- and/or higher-ordermodes The displacement response U can be written as [1]

U=(K+jo%C-oj2M)-IF + ~ ¢~iQd,

(A.I)

i=m

Q~=

a i - ol2 ..[_ 2j~ioJitOc

_

2 = ziai, tOc

(A.2)

C. Pal, L Hagiwara / Dynamic analysis of a coupled structural-acoustic problem

233

where M, C and K are the mass, damping and stiffness matrices respectively and 0)c is the given shift frequency, and z i and Qi are defined as follows: ~"~2 -- 0)c--2 2j~i0)i(~('~ -- 0)c)

(A.3)

0)2 + 2j~i0)i0)c -- 0)c2

Zi

~F (A.4)

2"

Qi = 0)2 + 2J~i0)i0)c _ tOc

Sensitivity analysis for coupled structural-acoustic system If we consider the following eigenvalue problem ( K - A,M) ~, = 0,

(A.5)

where Ai and ~ i a r e the corresponding eigenvalues (with no repeated roots) and eigenvectors respectively, and differentiate it with respect to design parameter a , , we have the following formulations for the sensitivities of eigenvalue and eigenvector: Ascb: = b/,

A./= K - A:M,

b: = ( A : M - K ' + as-M')q5 s,

(A.6)

t~= ejj, Ess =

( K' - ajM')4a/.

Since A/ is singular, different approaches are proposed to obtain a solution. We get three formulations for our new sensitivity analysis method, as follows:

~;= E ~i CO',

(A.7)

i=1

with -1 --Ei/

C °•

fori4=j,

A i -- Aj

E 0 = ~ii(K'-

~j = K-lbj

a/M')cb/,

"~iPi ° =

L ~t)_iclij,

- - I2~"~'i - - T*'* M ' ~"t"i,"

(A.8)

i=1

with C:s = As 1 h i Xj - h~iEi:

c:i

=

for i ~ j ,

c,°; n

ok; = ( K - I x M ) - l b s + Y'. qSiCis,

i=m

(A9)

234

C. Pal, 1. Hagiwara / Dynamic analysis of a coupled structural - acoustic problem

with Aj - I-~ Cij

1

h i - ~ hj--h i

Eiy

for i :~j,

"~'i' Ci i = _ ~ii M X g _ l2"q'i'" ~--T~,,h

Xi = ( K - / x M )

-1 bi"

They correspond to Fox's modal method (when joJ c ~ oo), Nelson's method (when toc = o~/) and Wang's modified modal method (when to c = 0). However, they are not general, and have some limitations. The method proposed by Hagiwara et al. [2], is one of the simplest and most general methods, which can be transformed to any of the above three methods by selecting different values of a shift p a r a m e t e r to c.