numerical optimization techniques for structural-acoustic design of rectangular panels

NUMERICAL OPTIMIZATION TECHNIQUES FOR STRUCTURAL-ACOUSTIC DESIGN OF

RECTANGULAR PANELS

J. S. LAMANCUSA Department of Mechanical Engineering, 157 Hammond Building, The Pennsylvania State University,

University Park, PA 16802, U.S.A.

(Received 1 April 1992)

Ahatraet-Numerical methods are commonly used to calculate the noise radiation from vibrating structures in order to evaluate the acoustic performance of a particular design. Improvements are then often achieved through trial and error by an experienced designer. While automated numerical optimi~tion techniques are extensively utilized in many fields of structural design, their application to acoustic problems has been limited. In this paper, methods for the design optimi~tion of ‘quiet’ structures using mathematical programming techniques are investigated. The objective of this fundamental study is to develop effective, general methods for sizing optimization of acoustically radiating structures. The specific design example addressed here is the optimal thickness distribution of isotropic, lightly damped, rectangular, flat plates with clamped edges. Acoustic response to applied point forces is minimized either at a single frequency, or over a wide frequency bandwidth. Radiated sound power is calculated using a discretized Rayleigh integral formulation, in conjunction with a finite element solver (SAPIV) for the solution of the structurai problem. The CONMIN opti~~tion program is used.

It is shown that the selection of appropriate objective functions and constraints is critical to the success of the optimization process for this multi-disciplinary, computational expensive problem. Potential candidates for objective functions and/or constraints include: total weight, placement of structural natural frequencies, mode shape, mean square velocity, radiation efficiency or total radiated sound power. Several alternative formulations are presented and their effectiveness is compared. It is demonstrated that significant reductions in radiated sound power can be achieved by optimal thickness variation.

1. BACKGROUND

1.1. Introduction

In our daily lives, we are constantly bombarded with noise radiated from vibrating panels, partitions or walls. Many of these structures can be classified as some type of rectangular plate and examples can readily be found in every automobile, aircraft, and residential or commercial building.

Recent advances in acoustic analysis, materials science, and computing technology are gradually changing the process of designing quiet structures. Where designers were once forced to rely heavily on past experience, art, and sometimes a little luck; a more organized, scientific approach is now possible. New structural and acoustic analysis techniques now make it possible to predict the acoustic performance of complicated mechanical structures prior to fabri- cation. Recent advances in computing resources and optimization procedures have made numerical optimization feasible for complex structures with many design variables. However, the design of structures for minimal sound radiation is a multi~i~plin~ problem which involves large numbers of design variables, complex objective functions, and computationally expensive calculations. These qualities can pose severe obstacles to the successful use of numerical optimization techniques.

The objective of this paper is to examine the fundamental issues, problems and potential benefits of applying numerical optimization techniques to the design of acoustically radiating structures. Potential structures that can benefit from this optimal design procedure for noise reduction include: lightweight noise barriers, automobile engine cover panels, and aircraft fuselage panels. With very slight modifi- cations, the same procedure can be used to maximize the performance of systems where acoustic radiation is desirable, such as audio loudspeakers and acoustic drivers.

1.2. Literature review

Shape, sizing and topology optimization techniques are becoming well-known tools in many indus- tries, particularly in the aircraft industry. The typical aircraft problem which is addressed [l-3] is the minimization of mass, with specified constraints on static stresses or deflections, aerodynamic loads, dimensions, etc. With the advent of more powerful personal computers it is now possible to apply some of these same successful techniques to problems which were previously neglected due to their computational intensity or multi-disciplinary nature, such as the problem of structural-acoustic design.

The design of quiet plate and shell structures requires both strnctural vibration and acoustic analy-

661

662 J. S. LAMANCUSA

ses. Several references in the literature have addressed the vibrational response optimization of structures. These include a minimum cost design of a rectangular plate under acoustic and blast loading [4]. The system loss factor of a plate was optimized by optimum thickness distribution of an unconstrained visco- elastic damping laminate in [5]. References [6] and [7] employ a technique which endeavors to remove natural frequencies from undesirable frequency bands of lightly damped structures. The material distribution of flat plates or beams for controlling the frequency of the first structural mode is addressed in [8,9]. Placement of nodal lines as a design objective by optimal mass distribution is addressed by [lo].

Shape optimization techniques have also been re- cently extended to coupled structural-acoustic problems. The transmission loss of a sandwich panel was optimized in [l 1, 121. Several objective functions are evaluated and the authors point out that the choice of objective functions is crucial to the success of the optimization. Finite element analysis and a Rayleigh integral formulation have been used without optimization to evaluate the noise radiation from engine rocker covers [13]. Shape optimization has also been applied to the acoustic design of automobile intake and exhaust systems [14, 151.

2. METHOD

Shell elements are a basic component of many noise producing structures and are therefore an important element in any strategy for the design of quiet structures. In this study, the optimal thickness distribution of flat plates is investigated, which minimizes the total radiated sound power. The case which is addressed in this paper is that of a flat, rectangular plate having built-in edges and excited at its center by a concentrated force. However, any structure whose dynamic response can be accurately analyzed by finite element analysis and whose acoustic radiation can be predicted using the Rayleigh integral, can be optimized with the Structure QUIet Design (SQUID) package described in this paper.

In this study, the thickness variation of the plate is determined which minimizes the radiation sound power of the structure. Optimal designs are determined for single frequency excitation and for excitation over a wide frequency bandwidth. While the present example is limited to the thickness variation, the same methods can be applied to find the optimal size and placement of lumped masses, or surface damping treatments. It is hoped that the knowledge gained from this study on efficient optimization methods for acoustic systems will also be applicable to design modification alternatives other than thickness variation.

Several measures of sound radiation, radiation efficiency and structural vibration efficiency will be employed as measures of the success of the design. The required equations for calculation of the struc-

tural and acoustic responses, as well as a brief description of acoustic radiation phenomena, are presented in the following sections.

2.1. Structural analysis

Finite element analysis is used to analyze the vibratory motion of the structure. The dynamic equations of motion for the forced harmonic response are assembled into the form

[M](n) + [C]{i} + [K]{x} = {F} e’“‘, (1)

where [Ml, [C] and [K] denote the mass, damping and stiffness matrices of the discretized system. The eigenvalue analysis of the plate is performed using the SAPIV program [16]. This program was chosen over some newer, commercial alternatives because of the availability of its source code, and thereby the ability to easily merge it with optimization algorithms within a larger program structure. Thin shell elements, with four nodes and six degrees of freedom per node, were used. The eigenvector matrix [@)I defines the relation between the physical coordinates of each node, x, and the principal or modal coordinates qi

Ix>= PIhI. (2)

Substituting into (1) yields

P7PflPl~B~ + P7[clPl~4~

+ PwaPl{q~ = [47If-) = &?I. (3)

Assuming proportional damping (a modal damping ratio of t), the uncoupled equations of motion for each of the i modes simplify to

qi + 2tiwiqi + ojqi = Q, (4)

e, (li = w2 + 25iwo, + wf ’

where wi is the natural frequency, w is the forcing frequency, and & is the damping ratio for the ith mode. The normal velocities at each of the i node points in the structure are then computed by superposition of the normal modes

{I?} = jw{x> = jw[@]{q}. (6)

The normal mean square velocity of the plate is obtained by averaging the surface normal velocity over all n node points of the structure

(7)

2.2. Acoustic analysis

The acoustic radiation from a planar surface in an infinite baffle may be calculated once the surface

Numerical optimization techniques for structural-acoustic design 663

normal velocity is known from eqn (6), by use of the Rayleigh integral formulation. The sound pressure at any point r is evaluated by [ 171

p(r) =‘z . s B(rs) e-jkR dS

R ’ (8) s

where p(r) is the complex sound pressure at r, r is the position vector of the observation point, r, is the position vector of the elemental surface SS, R = lr - r,l, fi(r,) is the normal velocity of SS, k is the acoustic wave number (o/c), w is the frequency (rad/sec), p is the mean density of fluid, c is the sound velocity of fluid and j = ,/- 1.

The pressure is evaluated at the centroid of the 111 th element having area A, by summing the contributions due to the average normal velocity fi, (the average of the four nodal point velocities for the Ith element) of all n elements using the discretized form of eqn (8)

pm = ‘2 $ fin, e-JkR’mA, (for ,,, _+ I)

I- 1 R I,m

fi e-‘kRm~

+” Rlll

m . 1

(9)

The second term in the equation represents the pressure at the center of the element due to its own velocity. This term is an approximation which avoids the singularity which occurs (when R = 0) by sub- dividing the element into four equal parts and using the distance from the element center to the center of each sub-element as the distance R,. This discretized formulation gives accurate results provided that the size of each element is less than approximately one quarter of an acoustic wavelength.

The acoustic intensity I is evaluated on the surface of the plate by

where Re denotes the real part of a complex quantity and the asterisk the complex conjugate.

The total radiated sound power W is obtained by integrating the intensity over the surface S. Inte- gration over the surface is more computationally efficient, except for low values of ka where direction- ality is negligible. Surface integration requires fewer discrete field points, than integration over a hemi- sphere in the far-field

W = I(r,) dS. (11)

In discretized form for numerical computation, the power is summed for each surface element by eqn (12)

W = i Re[il A,,#,,#:]. (12)

A commonly used measure of the sound radiation characteristics is the radiation efficiency Q, which is defined as the ratio of power radiated by the structure in question, to that of a reference structure

W fJ=-Z

W’ (13)

where W is the power which would be radiated by a uniformly vibrating baffled circular piston of the same area S and with velocity equal to the average mean square velocity of the structure in question (ka % 1)

w= ;pcS(B;). (14)

Combining eqns (13) and (14) yields

(15)

The so-called ‘cut-on’, coincidence, or critical frequency w,, is the frequency at which each individual mode begins to radiate efficiently (a z 1) and is defined as the frequency at which the structural wavelength equals the acoustic wavelength

w==c/m, (16)

where c is the sound velocity in air, p, q is the mode index (i.e. the number of half sine waves in each direction), a is the plate length in the same direction as p, and b is the plate length in the direction of q.

From the preceding equations, a number of quali- tative observations can be made which are useful to keep in mind while formulating the optimization problem, and while reviewing the optimization results. These include

Minimizing the mean square velocity of the plate will decrease the radiation power of a structure independent of the frequency. To decrease the radiation efficiency of a given mode shape, its natural frequency should be as far below its coincidence frequency as possible. Above critical frequency, radiated power is proportional to the mean square velocity. At any frequency below coincidence, a modal pattern with as many zero-crossings as possible is most desirable for minimum radiation efficiency.

3. MATHEMATICAL PROGRAMMING PROBLEM FORMULATION

3.1. Problem dejnition

The minimization of radiation sound power from a vibrating structure is a multi-disciplinary, computationally intensive problem having the potential for large numbers of design variables and constraints.

664 J. S. LAMANCUSA

Mathematical programming techniques were chosen will be seen, the choice of an appropriate objective for this study due to their flexibility in formulation of function is itself a compromise between speed and objective functions and constraints and the existence accuracy. The most direct measure of sound radiation of several proven optimization algorithms. for this problem is the radiated acoustic power,

Following standard notation, the general form of evaluated over the entire frequency band of interest. the nonlinear programming problem may be stated However, for wide frequency bands, this becomes

[181 difficult to efficiently calculate, hence the need for a

Find a vector of design variables x = (x,, x2, . , x,,) which minimizes the cost or objective function f(x),

subject to equality constraints hi(x) = 0, i = 1, . . , p and inequality constraints g,(x) < 0, j = 1, . , m.

Upper and lower bounds on the design variable also may be imposed such that x:Twer < xi < .x~P~~. Math- ematically and numerically, these are usually treated like inequality constraints. The choice of design variables, constraints, and optimizer specific to this problem are described below.

3.2. Design variables

The dynamic performance of an isotropic structure depends on the local mass, stiffness and damping. These properties can be altered by addition of lumped masses, layered damping materials, and changes in geometry such as thickness. The actual choice of design variables is dictated by the specifics of the problem to be solved. While stepped thicknesses (as utilized in this study) are acceptable for machined metal or molded plastic parts, other manufacturing processes may dictate a different choice of design variables. Continuous thickness variation may be more appropriate in some cases. With stamped sheet metal parts, the placement and size of formed bends or stamped-in stiffening ridges would be a natural choice for design variables. Weldments offer the ability to add stiffening ribs or lumped masses. The location and size of surface mounted constrained- layer damping treatments is also a viable choice in some instances.

A complete treatment of all possible choices of design variables is obviously beyond the scope of this study. Thickness variation was chosen due to the relative simplicity of the required analysis model, in order to gain an initial understanding of the complex- ities and requirements for successful acoustic optimization. While not certain, it is hoped that the results of this fundamental study, comparing the effectiveness of several alternative optimization formulations, will be a useful starting point for other choices of design variables.

3.3. Objective function formulations

Numerous possibilities exist for formulating a performance measure suitable for use as an objective function, f(x). A number of different measures were selected for use in this study, in order to compare their effectiveness and computational efficiency. As

more efficient, although approximate method. Poten- tial candidates for objective functions, including several approximation alternatives, are listed below. Unless otherwise indicated, lower and upper bound constraints are placed on all thickness variables.

1. Radiated sound power (frequency average). The average sound power over a frequency bandwidth is calculated by evaluating eqn (12) at nlrcq closely spaced frequency increments between f,,, and f,,,

(17)

In this study, either single frequency excitation (and single frequency response), or multiple frequency excitation (and banded frequency response) were considered. Several frequencies were examined for the single frequency case. For the banded frequency case, 150 frequency increments were used with f,,, = 100 Hz and f,,, = 1000 Hz. No constraint was placed on the total weight of the structure.

2. Radiated sound power with constant mass (frequency average). One physically intuitive approach to varying the plate geometry, while maintaining a constant mass, is that any material which is added to one plate element, must be taken from another element on the plate. The receiving element will have a positive sensitivity (power decreases with increasing thickness), while the ‘donor’ element must have a less positive (or negative) sensitivity in order for a net power reduction to occur. A new design variable a, is used which describes the material sharing between a pair of elements

xf = aixO and XI = (2 - a,)x,,,

i=l,... , n,P, (18)

where xh is the thickness of the more sensitive element, x’ is the thickness of the less sensitive element, n& is the original number of design variables (the number of finite elements) and x0 is the initial thickness of the elements (and also the average thickness).


The two elements in each pair are chosen after an initial sensitivity analysis on all the thickness variables. The variables are then rank ordered (rank = r) and paired off, x’ to x1+““/* for r = 1, . . , ,n,/2. A new sensitivity analysis is then performed on the reduced design variables ai and the optimization then commences. Besides guaranteeing constant mass, this formulation also has the advantage of halving the number of design variables. This simplifies the optimization process and halves the time spent performing sensitivity analyses. Upper and lower bounds are placed on the sharing coefficients a, to maintain the actual thicknesses within specified limits. This formulation implicitly results in thickness limits which are symmetric about the initial thickness value.

The objective function in this case is also the radiated acoustic power, calculated as in eqn (17)

f2 = Pm = ; y W(q), (19) hqi=l

where F?m signifies the total band power with constant mass achieved by design variable pairing.

3. Upper bound mass constraint. As will be seen shortly, most of the optimizations for minimum power result in plates with increased mass. The mass is typically concentrated near the center of the plate. The predominant effects of this are: to decrease the velocity response of the plate by making it less receptive to the applied central force; and to decrease the frequency of the first mode, thereby lowering its radiation efficiency. In many instances, it may be desirable to specify, in addition to upper and lower bounds on the thickness variables, a maximum plate mass. This is accomplished using an inequality constraint of the form

g3(x)= -1+*, mm,,

where mplats(x) is th e calculated mass for the present design and m,, is the specified mass limit. This constraint is linear with respect to the design variables and is considered to be violated when its value becomes positive. While any of the objective functions can be used with this mass constraint, only results using the radiated power fi [eqn (17)] will be presented.

4. Mean square velocity (frequency average). Since acoustic power is determined by the surface velocity, one alternative and less computationally expensive objective function is to consider only the vibrational efficiency of the structure as expressed by the mean square normal velocity. The average mean square velocity over a frequency band is calculated by evaluating eqn (6) at nrral frequency increments

Even though evaluated at small increments over the frequency band, this calculation eliminates the computational expense of the Rayleigh integral calculation [eqn (S)] and is therefore quite rapid. Lower and upper bound constraints are placed on the design variables (the plate thickness variables) to ensure that the plate stays within reasonable limits.

5. Radiation eficiency (frequency average). Radi- ation efficiency is a commonly used measure of the acoustic performance of a structure and is therefore a candidate for an objective function. The frequency average radiation efficiency can be evaluated over the bandwidth in a similar fashion as the acoustic power

f,=6 =;“~a(wJ. (22) frq,=l

However, as an objective function, this provides no advantage over the direct calculation of radiated sound power, since it involves exactly the same calculations, and does not include the vibrational efficiency of the structure. Therefore, this will not be used as an objective function in this study. Radiation efficiency may still be a valid measure, but only if it offers advantages in computational speed.

6. Modal radiation eficiency. Physically, most of the radiated sound energy occurs at the resonant peaks of the vibrating structure. Making use of this phenomenon, considerable computational savings over eqn (22) can be achieved if the radiation efficiency is calculated only at the natural frequencies of the structure. The sum of the individual radiation efficiencies (the modal radiation efficiency a’) of the first nmode normal modes of the system is defined as

fs = 8’ = “jy a(w,). (23) i=,

7. Modal radiated sound power. A similar technique for streamlining the acoustic power calculation is to calculate and sum the power at the peaks of the response spectrum. The power is evaluated at the first nmode peaks in the response, the location of which is approximated as the peak frequency for a single degree of freedom system

w~~=c+/~, i=l,..., nmode. (24)

The modal radiated sound power is then

hncdr

f,=W’= C R(oFk), i=l,..., nmode. (25) i= 1

8. Matching of a pre-determined ‘weak radiator’ mode shape. This objective attempts to force the nth mode of the structure to match a prescribed mode shape and frequency. This desired mode vector corre- sponds to a vibrational pattern which is known to

CAS 48,4-H

666 J. S. LAMANCUSA

inefficiently or weakly radiate sound. This shape is determined by a prior analysis such as in [22] or by an iterative search to find a surface velocity distribution which minimizes the radiation efficiency. The similarity of the actual mode 9, to the desired mode Q,, is measured by using the orthogonality criteria

fs = i = 11 - @,‘Gd]. (26)

If the actual and desired modes are identical, this quantity is zero. An initial analysis is performed to determine which of the original mode shapes is numerically the most orthogonal to the desired shape. That mode is then chosen for subsequent optimization. This method is fundamentally a single frequency optimization technique since the desired mode shape is only an inefficient radiator at one frequency. The computational advantages of this method are significant, since it completely negates the necessity for acoustic power calculations during the optimal search.

9. Multiple objective functions. Various combi- nations of any of the previously mentioned performance measures are possible. In particular, a weighted sum of modal radiation efficiency and mean square velocity will be employed

f9= C,a’+ C*G. (27)

The values for the weighting factors are typically chosen to impart equal importance to each of the equation elements, in this case C, = C, = 1.0.

10. Fundamental frequency. Historically, some of the first efforts to optimize dynamic performance did so by maximizing the fundamental frequency of the structure, in order to force the lowest resonance to be above an objectionable frequency range. This can be formulated as a conventional minimization problem by using the inverse of the fundamental frequency as the objective function

fill= l/r%. (28)

Conversely, the fundamental frequency may also be minimized, in which case the objective function becomes

_flo=w,. (29)

11. Muss with upper bound power constraint. All the objective functions mentioned up to this point are highly nonlinear with respect to the design variables, however, the mass constraint is a linear function. The use of mass as an objective function, with power included as a constraint may potentially result in a more well-formed problem of some optimization algorithms. The objective function in this instance is the total mass m of the structure

fi,=m=fx,A,p i-l

and the inequality constraint which limits the maximum power is

I@ g,,(x)= -1+-_,

W (31)

max

where p is the mass density; A, the element area; n,,, the total number of elements; and F@,,,,=, the maximum allowable sound power.

While the objective function is now linear with respect to the design variables, the inequality constraint is highly nonlinear. It will be seen sub- sequently that this formulation does not result in an improved design, at least with the optimizer used in this study.

3.3. I. Computational eficiency. A critical difference between these objective function alternatives is the time necessary for calculation of each design iteration. The eigenvalue extraction time (required for all analyses) as well as the actual calculation times per design iteration on a 486 PC are listed in Table 1. Clearly, direct calculation of power over a wide frequency band is the most time consuming. The eigenvalue extraction time is a significant contributor to the total analysis time.

3.4. Choice of constraints

The numerical values of the constraints (lower and upper bounds) depend on the actual problem to be solved. In practice, the physical bounds on thicknesses are a function of the material and the manufacturing process to be used. In addition, other factors, such as maximum allowable stress, may determine the minimum thickness. Limits on element aspect ratio, or thickness variation from element to element may also be necessary to ensure modeling accuracy, de- pending on the boundary conditions and the type of finite element used in the analysis. The optimization process is always more efficient with fewer constraints. Upper and lower bounds on design variables should be as generous as possible to give the optimizer as much design freedom as possible.

3.5. Choice of optimizer

The CONMIN program [19] was chosen for this problem due to its flexibility and because of previous personal successes in its application. CONMIN uses the method of feasible directions. In this method, each design cycle consists of determining a new vector

Table I. Computation time comparison for various objective functions for 56 design variables

Calculation Time/iteration (set)

Mesh generation <O.Ol Eigenvalue extraction (10 modes) 28.8 & (single frequency) 0.63 ct (150 frequencies) 94.9 t; (150 frequencies) 29.8 B (10 modes) 35.3 4 <O.Ol

Numerical optimization techniques for structural-acoustic design

t Initial Design -1 CDNMIN OPTIMIZER I___c Optimal Design

I Design Variables x

+

FEA Mesh Parameters

**“^“‘“_

+i Wi

661

Fig. 1. SQUID acoustic optimization program.

of design variables xk+ , from the current design vector x, [20]

Xk+,=Xk+Udk. (32)

The step size a is chosen so that the new design is feasible and has a smaller cost function. dk is the vector of ‘usable feasible’ redesign variables. Once dk is determined, a line search is performed to determine the size of a. The CONMIN algorithm is composed of two basic elements: search direction determination, and step size determination. The direction is determined by linearizing at the current feasible design, while maintaining feasibility with respect to the linearized constraints. This technique is normally applied to problems with inequality constraints. The major drawbacks of this technique are that a feasible starting point is needed, and that equality constraints are difficult to impose. CONMIN requires the gradients of the objective function with respect to the design variables, as well as the gradients of all

constraints. These quantities may be. supplied by the user, which is most efficient when analytical information is available, or can be calculated by CONMIN using divided difference.

3.6. Program implementation and verification

The structural and acoustic equations were im- plemented in a Fortran program using the Lahey F77L-EM/32 compiler for execution on a 33 MHz 486 personal computer (PC). The validity of the coupled vibration-acoustic calculation was verified by comparison to the closed form solution for a simply supported square plate by Wallace [21].

The completed acoustic optimization package SQUID is described by the flowchart shown in Fig. 1. All option selections and initial dimensions are provided by user-supplied files. The communication of the mode vectors and frequencies from the SAPIV program is accomplished using a data file. The program can be run on a VAX VMS platform, or on a

104 100 200 300 400 600 600

4 700 600 900 1000

FREQUENCY - Hz

Fig. 2. Effect of mesh refinement on sound power frequency response for a 3 mm thick clamped steel plate with dimensions 546 x 682 mm.

J. S. LAMANCUSA

1.2

1000 z ’ f5

ii B 100 0.6

$

E w

0.6 5 9 10 I=

0.4 % 1 d

0.2 - SOUND POWER - SIGMA

0.1 2 4 6 6 10 12 14 16 16 20’ PLATE THICKNESS _ mm

Fig. 3. Effect of varying the thickness of a uniform plate on the single frequency response at 500 Hz.

386 or higher PC. A speed comparison between a batch job on a VAX 8800 and real-time job on a 486 PC (33 MHz) resulted in CPU times of 7 hr 02 min and 6 hr 54min, respectively, for a single frequency optimization of radiated power with 56 design variables (58 iterations, 952 objective function evaluations). Besides being slightly faster in actual time spent in computation, the PC was superior in terms of the total elapsed time from job entry to job completion by more than 400%. Clearly, the availability of relatively inexpensive and powerful desktop workstations with power rivalling that of a main- frame computer has dramatically increased the applicability of numerical optimization methods to previously ignored applications, such as structural- acoustic design.

4. APPLICATION AND NUMERICAL RESULTS

4.1. Problem description

A flat rectangular plate with a 45 aspect ratio, and having dimensions 546 mm (21.492 in.) x 682 mm (26.865 in.) was used as the standard structure for optimization. The thickness of any element on the plate was constrained to remain above the physically realistic limit of 2mm. Several values for the upper limit up to 20mm were investigated. In general, the higher the allowable upper limit, the more successful is the optimization, since there is more freedom in the

design and more material with which to work. The chosen frequency range for study was 100-1000 Hz corresponding to values of acoustic wavenumber ka = l-10, based on the shorter plate dimension (hence the unusual units for the plate width). A modal damping ratio of ci = 0.02 was found from experiments to be representative of a flat plate with clamped edges.

Symmetry was employed in the finite element analysis to reduce the number of elements. One quarter of the plate was modeled by 56 elements (7 x 8 mesh), providing a total of 56 possible thickness variables. Each element was described by a constant thickness. A zero slope boundary condition was enforced along the two edges of the model which correspond to the centerline of the plate. As a result of the assumed symmetry, only the modes which were symmetric about the centerlines of the plate are calculated. In acoustic terminology, these are called the nonsymrnetric or odd modes because they have an odd number of half sinewaves in their modeshapes. These modes are the most significant contributors to acoustic radiation because they have the least amount of inherent self-cancellation at frequencies below coincidence [17]. The 7 x 8 mesh was chosen as a compromise between the mutually exclusive goals of modeling accuracy and minimizing the number of elements (and hence computation costs and the number of potential design variables). The effect of mesh

- CONSTANT THICKNESS + OPTlMlZED

0.1 2 4 6 6 10 12 14 16 16 20

MEAN PLATE THICKNESS - mm

Fig. 4. Results of optimization at SOOHz compared to uniform thickness plate performance.


Fig. 5. Thickness distribution of plate optimized at 500 Hz for initial thickness values of 5.2 mm: (a) to scale, (b) exaggerated vertical scale.

refinement and the sufficiency of the 56 element mesh can be seen in the calculated sound power as a function of frequency shown in Fig. 2. There is no significant difference in the results for the 7 x 8 element mesh versus a 10 x 12 mesh, therefore the coarser mesh will be used.

4.2. Design space visualization for single variable problem

A prior knowledge of the behavior of the objective function is always helpful in the selection of an appropriate objective function, optimizer algorithm, and in the formulation of constraints. While it is not feasible to directly search the design region in all the n variables, it is reasonable and instructive to look at the problem behavior for one or a few variables. The variation in performance as a function of plate thickness is shown in Fig. 3 for a single frequency excitation at 500 Hz. The complex dynamics of the structure result in several peaks in the power and the mean square velocity (not shown). These peaks occur at thickness values where one of the normal modes is excited by the 500 Hz input. The mode indices (p, q) for the excited modes are indicated. This single variable behavior leads one to expect that any multi- variable optimization which uses power or mean square velocity as an objective function, will be

highly dependent on initial values of the design variables.

4.3. Single frequency optimization results

The results of optimization to minimize radiated sound power at 500 Hz are shown in Fig. 4. The performance of the original constant thickness plate is shown as well as the optimized performance for a number of different initial design points. All cases resulted in substantial reductions of radiated power. Nearly all the optimal designs tend to increase weight during the course of the optimization. The choice of starting point determines the position of the forcing frequency in relation to the modal frequencies of the structure, and essentially dictates a design regime which will be searched by the optimizer. For example, if in the original structure, the forcing frequency falls between the second and third modes (such as the 5.2 mm thick case), so will the optimal design. The optimizer cannot ‘escape into the next valley’ since to do so would require it to increase the objective function, if only temporarily. Different starting points within the same ‘valley’ converge toward the same design, as seen with the 7, 11, and 15 mm initial designs of Fig. 4.

The optimized thickness distribution for a typical case (initial thickness 5.2 mm) is shown in Fig. 5. It

1CQOQO - ORIGINAL - OPTIMIZED

0.14’ 0 200 400 600 600 loo0 1200 1400 1600 1600 2 FREQUENCY-Hz

Fig. 6. Frequency response of 500 Hz optimal plate (5.2 mm initial thickness) compared to uniform plate of same total mass.

670 J. S. LAMANCUSA

\ 0.8-

0.6-

O-i 0 10 20 30 40 50 60 70 80 90 ’

ITERATION NUMBER 0

Fig. 7. Iteration history for optimization process at 500 Hz (for 5.2 mm initial thickness).

can be seen that the increased mass collects near the center of the plate. This decreases the mean square velocity of the plate by increasing the driving point impedance. The optimal design of Fig. 5 has a frequency response as shown in Fig. 6. Not only has the response been significantly improved at 500 Hz (15.4 dB improvement with respect to the initial design) but the broad band performance has also been improved by 8.2dB from ~~-1~OHz (with respect to a plate having the same mass as the optimized design). Since most of the improvement in these designs is achieved by decreasing the mean square velocity, it is to be expected that these same improvements would extend over a wider frequency range. Similar results are obtained for the other cases shown in Fig. 4. The progress and rapid convergence of the optimization is illustrated in Fig, 7.

4.4. Wide bandwidth optimization results

Optimizations were performed using the objective function alternatives described in Sec. 3.3. The plate was excited by a center force over a frequency band of 100-1000 Hz. The starting points for the optimizations were plates having constant thickness values of 3, 7 and 11 mm. The thickness variables were limited by lower and upper bounds to 2-20mm, except for the constant mass formulation where the limits were

2-4,2-12 and 2-20 mm, respectively for the 3, 7 and 11 mm starting thickness pIates. A summary of the results frequency average objective function results is shown in Fig. 8 in comparison to the performance of a uniform thickness plate. The results for the approximate (i.e. not true frequency average) objective functions are shown in Fig. 9.

4.4.1. Frequency average objective fictions (@, %, and C). Both the power fi and mean square velocity i, objectives tended to increase the plate mass. The power objective provided the greatest reductions when compared to plates of the same mass as the optimized designs. When the same thickness limits are applied, the constant mass objective (@,,,) provides the best results, without the complication of increased mass. The best results are obtained for the thinner plates and especially when the original design has its fundamental frequency below the excited frequency range, such as was the case for the 3 mm initia1 thickness plate. The first mode shape for lightly damped structures is always a relatively efficient radiator, since its surface motion is all of the same phase. An interesting effect of the optimization was that, when the initial design excluded the first mode from the frequency range, so did the optimal design. However, initial designs which included the first

0.1 i ’ i ij

1 8 10 12 14 16 18 20

MEAN PUlTE THICKNESS - mm

--w -SIGMA - <V!-

+ osJ& x OBJ&m F OBJ- 3

Fig. 8. Broad band optimization results (1004000Hz) using objective functions: (a) total power #‘, (b) total power with constant mass rk,, and (c) mean square velocity 8.

Numerical optimization techniques for structural-acoustic design

0.14 d’ I

2 4 6 8 10 12 14 16 18 MEAN PLATE THICKNESS - mm

2

671

0

--w - SIGMA - <VL + OW-SIGMA X OW=SIGMA+ + )* oElJ=w

Fig. 9. Broad band optimization results (100-1000 Hz) using objective functions: (a) resonant radiation efficiency 6’, (b) resonant power @‘, and (c) 6’ + v^.

mode in the frequency range results in optimal designs which also included that mode. This phenomenon leads to a useful technique for choosing the initial values of the design variables namely, when considering wide frequency bandwidth excitation, the initial design should be chosen such that the fundamental natural frequency is below the excitation frequency band.

A typical frequency response result using frequency average sound power as the design objective is shown in Fig. 10 compared to the original uniform thickness design (7 mm). The optimized design exhibits significant reductions over most of the frequency range.

4.4.2. Modal iadiation eficiency (a’). The modal radiation efficiencies before and after optimization using eqn (23) as the objective function for a 7 mm initial thickness plate are shown in Fig. 11. Modes which show the most improvement are those which are initially near to coincidence frequency. After optimization, all modes had radiation efficiencies of less than 0.1. However, even though this formulation did achieve significant decreases in modal radiation efficiency, it did not result in broad band reductions in radiated power, as can be seen in the frequency

response shown in Fig. 12. The major reason for the poor performance was that these designs had dra atically increased mean square velocity. The other cases (3 and 11 mm initial thickness) also showed no improvement in radiated sound power as seen in Fig. 9.

4.4.3. Radiation ejiciency plus mean square velocity (a’ + ir). The combined objective function of modal radiation efficiency and frequency average mean square velocity [eqn (27)] was investigated. While reductions in power were achieved as seen in Fig. 9, no advantage was gained over using only mean square velocity as the objective function.

4.4.4. Modal power (W’). Optimization using W’ as the objective function [eqn (25)] produced improvements in modal sound power as seen in Fig. 9. However these improvements were outweighed (both literally and figuratively speaking) by the resulting increase in plate mass. A problem with this formulation is that a fixed number of modes must be tracked, and there is no incentive to move a mode outside of the excited frequency band. All designs tended to a similar result regardless of starting point.

100000 - ORIGINAL - OPTIMIZED

1 oooor B 1

P 1000 ,/

L

0 200 400 600 600 loo0 1200 1400 1600 1800 ; FREQUENCY _ Hz

00

Fig. 10. Frequency response of plate optimized for minimum radiated power from 100-1000 Hz compared to original constant thickness plate (7 mm).

672 J. S. LA~~ANCLJSA

2 .

1.6 .

> 1.6-

Y 9 '.4-

0 ii 1.2-

l-

6 F 0.6-

3 0.6-

2 0.4-

0.2-1

.

. .

. I

I

. ORlGtNAL Cl OPTIMIZED

o “+- r-x-7 =co 0 500 1000 1500 2000 2

FREQUENCY - Hz 30

Fig. 11. Optimization results (7 mm initial thickness) using modal radiation efficiency as the objective.

4.45. Minimum mass (m). The traditional optimization approach of mass minimization, using maximum radiated sound power as a constraint [eqns (30) and (31)] did not produce any mass reduction. The highly nonlinear inequality constraint on radiated power prevented the optimizer from finding a good solution. Perhaps an optimizer based on a different search algo~thm may have more success with this approach.

4.4.6. Fundamentalfrequency (CO,). The fundamental frequency of plates with initial thicknesses of 3, 7 and 11 mm were maximized, and minimized in separate analyses. Using the 11 mm case as an example, with maximization, the first natural frequency was raised from 300 to 545 Hz. In the process, the mass also increased (but by a smaller factor) from 32 to 53 kg. The frequency minimization case lowered the frequency to 51 Hz with a mass of 10.4 kg. The optimal thickness dist~butions for both cases are shown in Fig. 13. The frequency response of the original and optimized thickness plates are shown in Fig. 14 for frequency maximization and Fig. 15 for frequency minimization. The primary benefit of this technique would be to force the fundamental frequency out of an excited frequency band. The specifics of the problem at hand would dictate whether

it is best to position the fundamental frequency above or below that band.

4.4.7. Prescribed ‘weak radiator’ shape (Cp). A ‘weak radiator’ surface velocity distribution at 1500 Hz for this size plate was determined by a separate optimization procedure. This pattern was then used as the desired mode shape to be achieved by the structure through optimal thickness distribution using @ [eqn (2611 as the objective function. The desired mode shape and the mode shape (for the eighth mode) of the actual structure after optimization are compared in Fig. 16. The radiation efficiencies of the desired shape, and the actual shape after optimization were 0.00012 and 0.15, respectively. The optimization process was successfully able to match this quite complicated mode shape, even with this relatively coarse finite element mesh. The complexity of the required mode shape will obviously dictate the required finite element model ~finement and the number of design variables, and a finer mesh is required to achieve a better match. The frequency response of the optimized structure is shown in Fig. 17 compared to the original structure before optimization. Of interest is the decreased radiation efficiency at 1.500 Hz, but an increase in the radiated sound power. This is due to the resonant amplification at this frequency, and the fact that

~4 - ORIGINAL - OPTIMIZED

FREQUENCY. Hz

Fig. 12. Frequency response for plate optimization using modal radiation efficiency as the objective (7 mm initial thickness).


Fig. 13. Thickness distribution of plates optimized for (a) maximum or (b) minimum fundamental frequency.

Fig. 14. Frequency response of plate optimized for maximum fundamental frequency compared to initial design (11 mm thick).

the radiation efficiency still possessed a significant be worsened. A different input force location value. will produce a different optimal design. The thick-

4.5. Sensitivity to input force location ness distribution of a plate is optimized for a force applied at the center node, uniformly over all

A disadvantage of several of the objective nodes, at the node closest to comer and at approxi- functions is that a specific loading location must mately one third the distance from the comer to be considered and analyzed. While the optimal plate center. The dramatic differences in thick- design will improve the performance for that con- ness distribution for these cases are illustrated in dition, its performance for other load cases may Fig. lg.

- ORIGINAL - OPllMlZED

FREQUENCY - Hz

Fig. 15. Frequency response of plate optimized for minimum fundamental frequency compared to initial design (11 mm thick).

674 J. g. LAMANCLJSA

desired mode shape

Fig. 16. Desired mode shape and actual mode shape after optimization.

4.6. Effect of mass constraint

Many of the unconstrained optimizations resulted in significant increases in total structure mass. In some cases it may be desirable to limit this mass increase in accordance with a design requirement. The use of a mass constraint as in eqn (20) may seem like a good idea, however care must be taken not to restrict the optimization process unnecessarily. This mass constraint may prevent the optimizer from

finding a good solution. An example of this can be seen by comparing: the results for the constant mass objective function without constraints [eqn (19)]; to the results obtained by imposing a mass constraint [using eqn (20)] equal to the initial mass of the plate. While the constant mass case achieved an improvement of 6.7 dB in radiated power, the second case only managed a 0.8 dB gain. Both plates had the same initial constant thickness of 11 mm and the same final mass. It should be pointed out that

loo00

5 - ORIGINAL - OPTIMIZED

0.014 I 0 ml 1000 1500 2GQa

FREQUENCY - Hz

Fig. 17. Frequency response of plate optimized to match a ‘weak radiator’ shape at 1500 Hz compared to initial constant thickness plate.

Fig. 18. Comparison of thickness distributions of plates optimized for single frequency input (500 Hz) applied at: (a) center, (b) uniform, (c) corner element and (d) one-third from the edge.


optimal thickness distribution of the constant mass case is also a geometrically admissible solution for the other case. However, the optimizer could not ‘find’ that solution due to the restrictions posed by the inequality constraint.

5. CONCLUSIONS

It has been demonstrated that optimization of thickness sizing variables can effectively be used to:

minimize the radiated sound power from rectangular plates at a single frequency or over a wide frequency bandwidth; maximize or minimize the fundamental frequency of a structure; force a structure to vibrate in a specified modal pattern; and minimize the radiation efficiency of the individual modes of a structure.

Several important lessons were learned which re- inforced the opinions that optimization cannot be used simply as a ‘black box’ to solve all problems, and a fast computer does not eliminate the need to understand the underlying physics. The choice of an appropriate objective function is critical to the success of numerical optimization. Power is the most direct measure of acoustic performance and when used as the objective function, produces the most consistently improved designs. Disadvantages of using radiated power are that it is costly to compute for many frequencies, and requires a specific load case. The constant mass formulation is effective, cuts design variables in half, negates the need for inequality constraints, and is the most effective way of controlling mass. Of the other objective functions used, only mean square velocity consistently improved the plate performance. Upper and lower bounds must be wide enough to allow the optimizer proper freedom to find a good solution. Inequality constraints can restrict the optimizer from finding good optimal solutions even when those solutions are known to be in the feasible design region. Single frequency optimization for acoustic power is effective and produces very good broad band results. While the radiation efficiencies of individual modes can be significantly decreased, this does not guarantee decreases in radiated power due to modal superposition effects and to potential increases in mean square velocity.

Future extensions of this work will include the use of composite materials, other design variables such as mass or damping distribution, the use of higher order finite elements, parametric or polynomial descrip- tions to decrease the number of design variables, and investigation of other optimizers.

Acknowledgement-This work was supported by the Alex- ander von Humboldt Foundation while on sabbatical leave at the Research Centre for Multi-disciplinary Analysis and

Applied Structural Optimization, University of Siegen, Germany.

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REFERENCES

H. A. Eschenauer, G. Schuhmacher and W. Hartzheim, Optimization of fiber composite aircraft structures by means of the procedure Lagrange considering multidisciplinary criteria. Comput. Struct., to be published. V. B. Venkayya, New software systems for interdisci- plinary design. Proceedings of COMETT Seminar, Thurhau, Germany, 18-22 June (1990). J. Sobieszczanski-Sobieski, Multidisciplinary optimization for engineering systems: achievements and potential. In Optimization: Methodr and Applications, Possibilities and Limitations, pp. 42-62. Springer (1989). S. S. Rao. Structural optimization under combined blast and acoustic loading. AIAA Jnl 14. 276-278 (1975). A. Yildiz and K. Stevens, Optimum thickness distribution of unconstrained visco-elastic damping layer treatments for plates. J. Sound Vibr. 103, 183-199 (1985). J. M. Starkey and J. E. Bernhard, A constraint function technique for improved structural dynamics. ASME J. Vibrations, Acoustics and Reliability in Design 108, 101-106 (1986). D. Watt and J. Starkey, Design optimization of response amplitudes in viscously damped structures. ASME J. Vibr. Acoustics 112, 275-280 (1990). N. Olhoff, Optimal design of vibrating rectangular plates. Int. J. Solid Struct. 10. 93-109 (19741. D. Thambiratnam and V. ‘Thevendran, ‘Optimum vibrating shapes of beams and circular plates. J. Sound Vibr. 121, 13-23 (1988). J. I. Pritchard, H. M. Adelman and R. T. Haftka, Sensitivity analysis and optimization of nodal point placement for vibration reduction. J. Sound Vibr. 119, 277-289 (1987). M. A. Lang and C. L. Dym, Optimal acoustic design of sandwich panels. J. Acoust. Sot. Am. 57, 1481-1487 (1974). S. Makris, C. L. Dym and J. Smith, Transmission loss optimization in acoustic sandwich panels. J. Acoust. Sot. Am. 79, 1833-1843 (1986). J. Sivakumar, S. H. Shung and D. J. Nefske, Noise reduction of engine component covers using the finite ~~ element method. ASME Struct. Vibr. Acoustics 24, 133-137 (1991).

14. R. J. Bernhard, A finite element method for synthesis of acoustical shapes. J. Sound Vibr. 98, 55-65 (1985).

15. J. S. Lamancusa, Geometric optimization of internal combustion engine induction systems for minimum noise transmission. J. Sound Vibr. 127, 3033318 (1988).

16. K. J. Bathe, E. Wilson and F. Peterson, SAPIV, a structural analysis program for static and dynamic response of linear systems. UC. Berkeley Report EERC 73-11, Revised (1974).

17. F. Fahy. Sound and Structural Vibration: Radiation, Transmission and Response. Academic Press, London (1985).

18. J. S. Arora, Introduction to Optimum Design. McGraw- Hill, New York (1989).

19. G. N. Vanderplaats, CONMIN-a FORTRAN program for constrained function minimization. NASA Ames Research Center, NASA TM X-62.282 (1973).

20. R. L. Fox and M. P. Kapoor, Structural optimization in the dynamics response regime: a computational approach. AIAA Jnl8, 1798-1804 (1970).

21. C. E. Wallace, Radiation resistance of a rectangular panel. J. Acoust. Sot. Am. 51, 946-952 (1972).

22. K. Cunefare, The design sensitivity and control of acoustic power radiated by three-dimensional structures. Ph.D. thesis, The Pennsylvania State University (1990).

numerical optimization techniques for structural-acoustic design of rectangular panels

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