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Arch. Comput. Meth. Engng.
Vol . 9, 4, 291-370 (2002) Archives of Computational
Methods in Engineering
State of the art reviews
Developments in Structural–Acoustic Optimization
for Passive Noise Control
Steffen Marburg
Institut f¨ur Festk¨orpermechanik
Technische Universit¨at
Mommsenstr. 13, 01062 Dresden, Germany
marburg@ifkm.mw.tu-dresden.de
Summary
Low noise constructions receive more and more attention in highly industrialized countries. Consequently,
decrease of noise radiation challenges a growing community of engineers. One of the most efficient techniques
for finding quiet structures consists in numerical optimization. Herein, we consider structural–acoustic opti-
mization understood as an (iterative) minimum search of a specified objective (or cost) function by modifying
certain design variables. Obviously, a coupled problem must be solved to evaluate the objective function.
In this paper, we will start with a review of structural and acoustic analysis techniques using numerical
methods like the finite– and/or the boundary–element method. This is followed by a survey of techniques
for structural–acoustic coupling. We will then discuss objective functions. Often, the average sound pres-
sure at one or a few points in a frequency interval accounts for the objective function for interior problems,
whereas the average sound power is mostly used for external problems. The analysis part will be completed
by review of sensitivity analysis and special techniques. We will then discuss applications of structural–
acoustic optimization. Starting with a review of related work in pure structural optimization and in pure
acoustic optimization, we will categorize the problems of optimization in structural acoustics. A suitable
distinction consists in academic and more applied examples. Academic examples include simple structures
like beams, rectangular or circular plates and boxes; real industrial applications consider problems like that
of a fuselage, bells, loudspeaker diaphragms and components of vehicle structures. Various different types of
variables are used as design parameters. Quite often, locally defined plate or shell thickness or discrete point
masses are chosen. Furthermore, all kinds of structural material parameters, beam cross sections, spring
characteristics and shell geometry account for suitable design modifications. This is followed by a listing of
constraints that have been applied. After that, we will discuss strategies of optimization. Starting with a
formulation of the optimization problem we review aspects of multiobjective optimization, approximation
concepts and optimization methods in general. In a final chapter, results are categorized and discussed.
Very often, quite large decreases of noise radiation have been reported. However, even small gains should
be highly appreciated in some cases of certain support conditions, complexity of simulation model and large
frequency ranges. Optimization outcomes are categorized with respect to objective functions, optimization
methods, variables and groups of problems, the latter with particular focus on industrial applications. More
specifically, a close–up look at vehicle panel shell geometry optimization is presented. Review of results is
completed with a section on experimental validation of optimization gains. The conclusions bring together
a number of open problems in the field.
1INTRODUCTION
The decrease of noise emission of machines, vehicles, domestic appliances etc. will likely
become one of the major challenges of engineering in the 21st century. Research and
development departments are directed to come up with the right design the first time. A
recent book entitled “Designing for Quietness”, cf. Munjal [238], comprises eighteen papers
covering various aspects of the book’s title.
Development of computers has enabled a wide range of applications of numerical meth-
ods, such as the finite–element method and the boundary–element method during the past
three decades. Further and further improvement of hardware systems and simulation meth-
ods has encouraged researchers to employ methods of numerical optimization for structural–
acoustic optimization over the last 15 years.
In this paper, structural–acoustic optimization is understood as the improvement of
certain acoustic characteristics of a structure that emits sound or noise by mathematically
c
2002 by CIMNE, Barcelona (Spain). ISSN: 1134–3060 Received: May 2002
292 Steffen Marburg
controlled modification of the structure. To accomplish that, a multifield problem has to
be solved, hence the subject is also considered as a problem of multidisciplinary optimiza-
tion. Solution of multifield problems usually appears as a procedure that is computationally
expensive. If structural–acoustic simulation covers a wide frequency range, the multifield
problem has to be solved for a certain number of discrete frequencies. An outer loop of
optimization encompasses the entire solution of the multifrequency multifield problem in
a single step. This, however, considers analysis only. A number of additional technical
problems are encountered when actually implementing this type of multidisciplinary opti-
mization. These difficulties may explain why comparatively few publications are known in
the field of structural–acoustic optimization up to now. In fact, we have not yet been able
to find any paper on this subject for time–domain problems. For that reason, a harmonic
time dependence of eiωt is presumed throughout this article.
As the acoustic characteristics to be influenced, we can name the sound power, sound
pressure level, directivity patterns or other measures. In practical applications, a number
of very different objectives must be considered in the design process. Cases of multicriteria
optimization are familiar because it often happens that two or even more conflicting design
criteria should be taken into account simultaneously. Then, we have to make a trade–off
among them.
Structural modifications involve almost all possible changes of an existing construction.
To identify a few of them, modifications include plate and shell thicknesses, material data,
size and location of added masses, damping characteristics and shell curvature. In general,
modifications must satisfy design and functionality requirements. Any further variation is
subject to the engineers’ experience and imagination. Many types of modifications necessi-
tate certain constraints. These constraints typically comprise mass conditions but others,
like specific cost functions or differently defined conditions, can be regarded too. Often, the
mass is minimized, whereas acoustic properties account for side constraints.
In some cases, it is important to find the least change of design variables required
to meet specific properties. Then, we formulate the optimization problem in such a way
that the acoustic characteristics are taken as design constraints and the total change of
all parameters, from the initial design to the final design, as the objective function. This
procedure is called inverse optimization.
Optimization involves a number of calculations of the objective function. Depending
on the number of variables, condition of the optimization problem, optimization method
and other features of the particular problem, this calculation is repeated between 10 and
106times. The upper limit is an estimate and may be exceeded in some cases. This,
however, demands efficient analysis techniques for calculation of the objective function. In
most applications, structural models are analyzed using the finite–element method. The
variety is greater in noise emission evaluation. So, methods based on finite elements and
on boundary elements are both used. Additionally, several methods based on the Rayleigh
integral have been reported.
The optimization method will essentially influence the success of the entire procedure.
Since there is a high degree of non–linearity of the objective function in terms of design
parameters, only a numerical treatment like an iterative minimum search meets the needs of
this problem. Deterministic and stochastic algorithms are appropriate for these purposes.
A number of the above discussed problems has been addressed in the books by Koop-
mann and Fahnline [180] and Kollmann [177]. One problem that received very little focus
in these volumes is the technical completion for complex structural models. This includes
mesh coupling between structure and fluid, parametrization, and data handling. For ap-
preciation of optimization result, this point should not be neglected since it often requires
a major “manual” effort. Software tools exist for applications in structural–acoustic opti-
mization; in most cases, however, one needs to combine several different tools or develop
Developments in Structural–Acoustic Optimization for Passive Noise Control 293
a particular code, because general purpose applications do not fit together with specific
requirements of the particular application.
It should be mentioned that the title of papers often promise structural and/or acoustic
optimization. A closer look, however, shows that only a number of different variants are
compared, while an optimization in the sense of an (autonomous) iterative minimum search
of an objective function (or cost function) is not reported, cf. references [26,92,243,315,321].
Moreover, the field of structural–acoustic optimization is also relevant to problems in active
noise control. This subject is excluded from our reflections for this paper.
A number of papers discuss vibrational or acoustic optimization neglecting either the
structural or the acoustic field problem. These will also be briefly discussed in this paper.
The paper is structured as follows: We start with a review of structural and acoustic
analysis including fluid–structure interaction, objective functions and sensitivity analysis.
In the third section, we list related problems in either structural or acoustic optimization as
well as in structural–acoustic optimization. Parameters and constraints that were used by
other authors account for the subject of the fourth section. In section five, we review aspects
of multidisciplinary and multiobjective optimization including approximation concepts and
optimization algorithms. Results of optimization in structural acoustics will be categorized
and discussed in the sixth section. A particular focus is directed to results in the design of
sedan body panel geometry. The article will be completed by conclusions and a summary
of open problems in the field.
2 STRUCTURAL–ACOUSTIC SIMULATION AND OBJECTIVE FUNCTION
2.1 Structural Analysis
Some simple examples have been reported where structural behaviour was investigated in
part analytically, see for example Sorokin [300] or la Civita and Sestieri [184].
In most of the other applications, finite–element models were investigated for simula-
tion of the structure. In general, the finite–element method provides the linear system of
equations like (see for example [23,344])
As(ω)u(ω)= fs(ω)(1)
where uand fsdescribe the column matrices containing the nodal displacement vectors
and the nodal excitation force vectors. Parameter ωdenotes the circular frequency. Asis
the global system matrix more commonly known as the dynamic stiffness matrix given by
As(ω)=Ks+iωBs−ω2Ms.(2)
In equation (2), the matrices Ks,Bs,andMsrepresent the (static) stiffness matrix, the
damping matrix and, the mass matrix. Further, iis the imaginary unit. Formal inversion
of the dynamic stiffness matrix provides the nodal displacement vectors, formally written
as
u(ω)= A−1
s(ω)fs(ω).(3)
Time derivation of the displacement vector of node k(represented
uk) leads to the velocity
vector
vkas
vk(ω)= iω
uk(ω).(4)
In order to evaluate the surface particle velocity of the structure vsk, the normal component
of the nodal velocity vectors is extracted. Obviously, this is only relevant for nodes located
at the interface between structure and fluid. This extraction is achieved by
vsk(ω)=
nk·
vk(ω)= iω
nk·
uk(ω).(5)
294 Steffen Marburg
Assembling the components of all normal vectors
niinto a matrix Nand filling it with
zeros for all nodes not belonging to the structure–fluid interface, the column matrix of the
structural particle velocity is written in terms of the nodal displacements as
vs(ω)= iωNu(ω)(6)
and in terms of the force excitation and the system properties as
vs(ω)= iωNA
−1
s(ω)fs(ω).(7)
Equation (7) represents a formulation to evaluate the structural particle velocity at the
fluid interface in terms of a force excitation. In practice, however, explicit inversion of As
is computationally too costly if this is required for many different frequencies. Hence, a
modal analysis of the (non–damped) structure prior to the harmonic analysis is a must for
efficient simulation. The eigenvalue problem is expressed in the form
(Ks−λMs)φ=0.(8)
In equation (8), λdenotes the eigenvalue (being the square root of the eigenfrequency) and
φrepresents the corresponding eigenvector. Assembling the eigenvectors that have been
determined in a reduced modal matrix Φ, we can substitute the inversion of the dynamic
stiffness matrix by
A−1
s=Φ(Λ+Θ−Ω)−1ΦT.(9)
The frequency–dependent part is reduced to inversion of a diagonal matrix, being the sum
of three simple diagonal matrices. Λcontains the eigenvalues λk=ω2
k,whereωkis the
equivalent circular eigenfrequency of the system. Ωis given by Ω=ω2Iwhere ωis the
current circular frequency and Idenotes the identity matrix. Elements of Θare evaluated
as the product 2iωkωϑkwhere ϑkis the damping ratio of the corresponding natural modes.
Finally, normalization of the eigenvectors must be defined. Equation (9) requires that the
condition
ΦTMsΦ=I(10)
be fulfilled. A major drawback in this formulation is the damping formulation. For well–
damped complex structures like trimmed sedan body models, a spatially distributed damp-
ing can be necessary. In such a case, the damping matrix must be considered in the modal
analysis. The algebraic eigenvalue problem for a second order polynomial matrix is to be
solved. It should be mentioned that mass, stiffness, and damping matrices may be complex.
An application of spatially distributed damping of a circular plate was given in the paper
on structural–acoustic optimization by Wodtke and Lamancusa [330]. In their application,
a velocity–proportional damping matrix is omitted. Concerning treatment of algebraic
eigenvalue problems of polynomial matrices, we refer to Wilkinson [328] for further details.
In the paper of Christensen et al. [70], it was mentioned that, from a formal mathemat-
ical viewpoint, the expansion of structural response in terms of eigenvectors is the optimal
general way to compute the structural response. The authors point out that computa-
tional efficiency can be increased if some presumptions apply. For frequencies less than
the first eigenfrequency of the system and light fluid, structural response can be calculated
by expansion of static Ritz vectors. This technique might be useful for large–scale finite–
element models. Then, the extraction of eigenvectors maybe much more time–consuming
than construction of Ritz vectors. For special techniques and more detailed discussion even
for increasing frequencies, we refer to [70,158, 289, 329, 333].
Developments in Structural–Acoustic Optimization for Passive Noise Control 295
2.2 Acoustic Analysis
The variety of different types of analysis for the fluid is more distinctive than in struc-
tural analysis. Finite– and boundary–element methods seem to be most popular for closed
and open domains. Furthermore, there is a substantial count of papers indicating that
sound power evaluation using the Rayleigh integral and clever derivations of it is a valuable
alternative to the above mentioned methods.
Closed domains can be easily analyzed by using standard finite elements or boundary
elements. The advantage of finite–element methods is that they usually lead to a system
of equations similar to equations (1) and (2) as
Af(ω)p(ω)= ff(ω)= Cv
f(ω).(11)
Here, pand ffrepresent, respectively, the sound pressure and a vector of equivalent nodal
forces that is derived from the fluid particle velocity vf.Cis a sparse symmetric matrix
equivalent to a mass matrix. Afis the dynamic stiffness matrix of the fluid given by
A(ω)=K−ω2M.(12)
Using boundary–element models for closed domains, a system of equations as
H(ω)p(ω)= G(ω)vf(ω) (13)
is set up. One serious drawback of boundary–element methods is the implicit dependence
on the frequency of both system matrices Gand H. A number of approaches are re-
ported to overcome this problem. Likely the most promising method is the dual–reciprocity
method [241] which gives the same results as particular–integral method [4] for our purposes,
cf. Polyzos et al. [259]. Both methods were originally developed for problems in structural
dynamics but have also been applied to formulate the algebraic eigenvalue problem and the
system response for acoustics [5,20,46,77, 209]. The full theory of these methods is beyond
the scope of this paper; but in general, the formulation leads to a system of equations like
A(ω)p(ω)= iωGLvf(ω) (14)
where
A(ω)= HL+ω2M.(15)
Now, GLand HLdenote real–valued matrices corresponding to the Laplace equation,
i. e. the static part of the Helmholtz equation. Mass matrix Mresults from a number
of algebraic transformations and contains real elements too. These three matrices are
independent of the frequency. Unfortunately, this method encounters some difficulties. A
critical review was provided by Ali et al. [5]. A meshless collocation method was developed
by Kang and Lee [166] for two–dimensional cavities and by Chen et al. [61].
Acoustic damping is usually incorporated by an admittance or impedance boundary
condition. An analogous formulation to the velocity–proportional type in structural anal-
ysis is required for high–frequency acoustic problems only. For low frequencies, material
damping in air or water is usually negligible compared to absorption at the fluid surface.
Therefore, a local boundary condition is introduced as
vf−vs=Yp , (16)
where Yis the complex boundary admittance. In the technically important case of hard
reflecting surfaces, i. e. Y= 0, the structural particle velocity is equal to that of the fluid.
In realistic problems, the boundary admittance depends on the frequency. Chen et al. [64]
296 Steffen Marburg
developed a technique to evaluate acoustic eigenfrequencies in the presence of frequency–
dependent boundary admittance conditions extending the formulation of equations (14)
and (15). This technique can also be applied to the finite–element formulation.
Finite– and infinite–element methods for acoustic problems in open domains were sur-
veyed in Givoli [120], Bettess [39], Ihlenburg [153], and Kollmann [177]. Harari et al. [143]
presented a comprehensive state–of–the–art review of the finite–element method using
Dirichlet–to–Neumann maps to model absorbing boundary conditions. Remarkable progress
has been reported in this field over the past decade. Concerning infinite elements that are
applied to fulfill the Sommerfeld radiation condition at infinity, one distinguishes between
the so–called unconjugated Burnett elements, cf. Burnett and Holford [50–54] and the
so–called conjugated Astley–Leis elements, cf. Astley et al. [11, 12, 14, 15]. Gerdes [117]
investigated and compared these types of elements from a mathematical point of view and
emphasized that both methods may have advantages for single–frequency solutions. He
showed that a near–field solution is efficiently obtained by using the unconjugated Burnett
formulation, and that for solution a far–field solution one should employ the conjugated
Astley–Leis formulation.
Evaluation of the emitted sound power requires knowledge of the sound pressure and
the fluid particle velocity either on the surface of the vibrating structure or on an artificial
boundary circumscribing the structure. Hence, determination of the surface sound pressure
is sufficient. This could favor the unconjugated Burnett formulation. However, it has not
been mentioned yet, that conjugated Astley–Leis elements supply a system of equations,
cf. equation (11) where the explicit frequency–dependent matrix Afhas the form
Af(ω)=Kf+iωBf−ω2Mf.(17)
Obviously, the (non–Hermitian) system matrix Afcan be evaluated as the sum of (frequen-
cy–weighted) stiffness, damping, and mass matrix. Consequently, it may be factorized to
simplify the repeated inversion of matrix Affor many different values of ω. Unconjugated
Burnett elements still contain implicit frequency–dependent terms and it will not be possible
to factorize them. Thus, each frequency requires a separate solution of the Hermitian sparse
system of equations.
A rather new approach in infinite–element analysis employs harmonic basis functions.
Originally proposed by Melenk and Babuˇska [228], it was applied to some two–dimensional
problems by Laghrouche and Bettess [187]. Use of harmonic basis functions can decrease
computational costs significantly, especially for high–frequency radiation and scattering
problems. Application to three–dimensional problems is yet to be developed.
The alternative approach in finite–element analysis of exterior domains that does not
involve infinite elements is the use of an artificial circumscribing boundary where non–
reflecting boundary conditions are used. This type of boundary conditions is called the Di-
richlet–to–Neumann (DtN) condition and it represents a spatially distributed, frequency–
dependent admittance boundary condition. The DtN map was originally developed by
Keller and Givoli [169]. Interesting reviews were presented by Harari et al. [143] and,
more recently, Givoli [121]. Pinsky and co–workers [206, 244–246] published numerous pa-
pers on efficiency improvements of finite–element analysis with DtN boundary conditions.
Malhotra and Pinsky [207] presented a Pad´e approximation technique for efficient sound
field computations over a large frequency range that is to be used for efficient multifre-
quency shape design sensitivity analysis and optimization in acoustics [105] and in coupled
structure–fluid–interaction problems [258].
A number of books have been published on boundary–element methods in acoustics.
Collections were edited by Ciskowski and Brebbia [72], by Wu [332] and by von Estorff [322],
giving reviews of the state of the art at the time of their edition. A monograph was written
by Kirkup [175].
Developments in Structural–Acoustic Optimization for Passive Noise Control 297
The efficiency and accuracy of boundary–element methods in acoustic applications have
been investigated in the author’s papers [212] and, together with Schneider, [220]. These
investigations focused at interior noise problems. They showed that boundary–element
collocation methods perform most efficiently with discontinuous elements and nodes at the
zeros of Legendre polynomials. This effect of superconvergence was discovered earlier by
Chandler [57] and Chatelin and Lebbar [58]. It was further investigated for boundary–
element collocation in general by Atkinson [17] and for acoustic resonances in particular by
Tadeu and Antonio [307].
A particular problem of the boundary–element method for exterior domains arises at
certain frequencies, so–called irregular frequencies. Several approaches have been reported
to overcome this disadvantage. To mention some of them: There are the so–called combined
Helmholtz integral equation formulation (CHIEF) by Schenck [278], combined approaches
as those by Panich [254] or Burton and Miller [55] and some that use an additional series
in the fundamental solution by Ursell [316] and Jones [163]. An excellent review of the first
three formulations, including a number of additional references, can be found in do Rego
Silva [265]. An advanced Galerkin boundary–element formulation was presented by Chen et
al. [65]. This method provides Hermitian matrices similar to the one that was described in
the paper by Gaul et al. [114,115]. However, both formulations also encounter the problem
of irregular frequencies and will therefore require application of one of the above mentioned
formulations.
A recent paper by Cremers et al. [83] proposed a multi–domain boundary–element
technique utilizing infinite boundary elements. In that approach, it was reported that
the irregular frequencies had not been observed. Currently, this method still suffers from
inefficient integration techniques for the infinite elements. It can be presumed that further
development of this technique will likely overcome this disadvantage.
A serious drawback with respect to computational efficiency is the implicit dependence
of the system matrices on the frequency. Application of ordinary boundary–element for-
mulation demands matrix evaluations in every frequency step. Benthien and Schenck [30]
proposed a frequency interpolation of the matrix elements in boundary–element formula-
tions. Solving the linear system of equations in every frequency step, as required, makes
the method still costly for real applications. An alternative technique using Pad´e approxi-
mation was proposed by Coyette [78]. This method will significantly accelerate the analysis
process but it requires the storage of a number of global matrices equal to the order of Pad´e
approximation. This reduces practicability of that scheme. Attempts to construct explicit
frequency–dependent formulations for boundary elements using the dual–reciprocity or the
particular–integral method have not yet been successful. Other accelerations of boundary–
element formulations are based on special element types like axisymmetric boundary ele-
ments [182,298]. Substantial speed–ups were gained for certain problems and also utilized
for structural–acoustic optimization, cf. Kessels [170].
Finally, it should be mentioned that we can observe activities of mathematicians which
make boundary–element solutions more competitive even for large scale problems. Particu-
larly, three methods can be named: besides wavelet algorithms, cf. Lage and Schwab [186],
and panel clustering techniques, cf. Sauter [275], which were applied to Galerkin bounda-
ry–element methods and lead to sparse symmetric matrices, a third technique, known as
the fast multipole method [125, 247], may be used to accelerate boundary–element solu-
tions significantly. These recent methods were surveyed, tested and compared in Lage and
Schwab [185].
The above mentioned fast methods perform best if very high accuracy is desired. This,
however, implies that the element size should be one or more orders lower than the wave-
length. For technical application, this would be considered inefficient because acoustic prob-
lems usually do not require such a high accuracy. Other alternative concepts involve system-
298 Steffen Marburg
atic combination of these methods using a concept of hierarchical multilevel boxes [118] (for
Laplace equation only) or fast Fourier transformation techniques [38] (for Maxwell equa-
tion). Both methods, the so–called multilevel fast multipole analysis (MLFMA) and the
regular grid method (RGM), have been applied to acoustic problems by Schneider [279], in
particular to the so–called direct boundary–element method [72,322, 332] for internal prob-
lems and to the Burton and Miller formulation [55, 175, 265, 332] for the external acoustic
problem.
These methods require usage of iterative solvers for the arising system of equations.
Although successfully applied and documented in several papers [6,7, 164,176, 204], iterative
solvers are not yet considered standard for boundary–element methods in acoustics. Fast
boundary–element methods are based on kernel approximation such that the matrix–vector
product v=Au needed by the iterative solver is split into a near field part Anearuand a
far field part vfar =Afaru
v=Au =Anearu+Afaru=Anear u+vfar .(18)
It is well known that Arepresents a fully populated and, for most boundary–element
collocation methods, a non–Hermitian matrix. Anear is evaluated by integration in the
usual matter in the vicinity of the source point, i.e. in the near field. This part is a
sparse and non–Hermitian matrix. The remaining part Afar is not computed. The matrix–
vector product vfar is approximated either by the RGM or the MLFMA. Performance of
iterative solvers for standard boundary–element methods, for RGM, and for MLFMA has
been investigated by Marburg and Schneider [221]. It became obvious that real structures
with many sharp vertices, i.e. the power train of a car or a wheel box with a tire on
a half–plane, require a suitable preconditioner if the hypersingular formulation of Burton
and Miller is used. A well performing preconditioner is the ilut(τ, p) by [270] applied to the
near field matrix Anear by Schneider and Marburg [280].
An alternative to the solution of the acoustic boundary value problem using neither
finite nor boundary elements consists in the sound power evaluation by using the Rayleigh
integral. Originally defined to calculate the sound power that is emitted from a baffled
plate, it can also be reliably used for more complicated structures like a sedan oil pan [287]
or a diesel engine [286]. The Rayleigh integral omits the solution of a boundary–value
problem in the classical sense. It is a consequence of the surface integral representation
formula
p(
y)= Γ
[−iωρ G(
x,
y)vf(
x)−H(
x,
y)p(
x)] dΓ(
x) (19)
where Γ denotes the surface of the radiating structure, i, ρ,andωbeing the imaginary unit,
fluid density, and circular frequency, respectively. Gand Hcorrespond to the fundamental
solution of the Helmholtz equation and its normal derivative (in three–dimensional full
space) as
G(
x,
y)= e−ikr
4πr and H(
x,
y)= ∂G(
x,
y)
∂n(
x)where r=|
x−
y|.(20)
Clearly, contribution of the surface sound pressure vanishes for a plane surface like a plate
because H= 0. Consequently, the sound pressure can be written as
p(
y)= Γ
[−iωρ G(
x,
y)vf(
x)] dΓ(
x).(21)
Integration and matrix assemblage of G, cf. equation (13), should be performed, though.
However, its explicit storage is not required. Seybert et al. [286, 287] pointed out that
Developments in Structural–Acoustic Optimization for Passive Noise Control 299
the Rayleigh integral appears to be a reasonable alternative to predict the sound power
for bodies of arbitrary shape. It is not that well suited for calculation of sound pressure
distribution. Thus, it will not be an alternative for solution of the boundary–value problem
in general. Later in this paper we will discuss the use of the emitted sound power to compare
and judge different designs.
Further simplification that can be applied for sound power estimation based on the
Rayleigh integral was proposed by Fahnline and Koopmann [103,104, 180] and by H¨ubner
et al. [150–152]. These techniques mainly use simplified representations of the particle
velocity distribution on the body surface.
Substitution of the piecewise polynomially approximated particle velocity by a constant
value over a boundary element allows analytic integration. Both methods involve different
assumptions. However, they provide identical solutions, cf. Fritze et al. [112], but were
given very different names. Fahnline and Koopmann called their technique a ”lumped
parameter model for calculating the acoustic power output of a vibrating structure” while
H¨ubner used the name ”direct finite–element method” which seems somewhat difficult to
understand.
Subsequent to Borgiotti [43] and Photiadis [257] on singular value decomposition of
the radiation operator and Sarkissian [274] on eigenvalue analysis of the real part of the
impedance matrix, i. e. in general being A−1Cin equation (11) and H−1Gin equa-
tion (13), Cunefare and Currey [85, 87, 88, 90] developed a radiation mode technique that
contains a number of parallels to the ordinary modal analysis. There, one has to solve the
eigenvalue problem of a real symmetric matrix. Many analogies can be formulated. So, the
Rayleigh quotient provides the radiation efficiency, and the eigenvectors form an orthogonal
basis that is called “radiation modes”. The radiation modes can be treated similarly to
structural or interior modes. However, it is pointed out that they still depend on a partic-
ular frequency. Their storage to reconstruct the impedance matrix H−1G, cf. equation 13,
can be advantageous if only few radiation modes per frequency are found to be sufficient
for this purpose. Otherwise, one could store the frequency–dependent impedance matrix
instead. Obviously, considerable disc space will be required to store this matrix for many
frequency steps. An early application of radiation modes in optimization has been offered
by Naghshineh et al. [240]; a more recent one was published by Kessels [170].
The attempt to compare performance and usability of different methods is always dan-
gerous. A number of papers present comparisons between finite and boundary–element
methods for sound radiation, i.e. [50, 144, 286]. Seybert et al. [286] concluded for their ap-
plication of sound radiation from a diesel engine that the finite–element model could not be
managed for high frequencies since the number of degrees of freedom exploded. A highest
frequency of 500 Hz (in air) was reported for the finite–element mesh while the boundary–
element solution was evaluated and compared with experimental data and with the Rayleigh
integral results up to 2 kHz. A very large finite–element model of about 105or more degrees
of freedom is difficult to mesh and to handle. However, solution is usually achieved in a very
short time since the matrices are banded and mostly symmetric. In contradiction to these
results, Burnett [50] published some graphs comparing CPU time over degree of freedom for
finite– and boundary–element method applications for sound radiation. This comparison
seems to be specific for the problem. Furthermore, a boundary–element model will usually
give lower errors for the same degree of freedom. This is simply due to the fact that its
surface discretization is one dimension smaller than the domain discretization using finite
elements. Another comparison of finite–element method and boundary–element method
was published by Harari and Hughes [144]. Their approach compares both methods with
respect to different dimensionality. They showed convincingly that, based on computation
resources, conventional boundary–element techniques are not competitive to finite–element
methods for large–scale systems. The situation has likely changed with the development of
300 Steffen Marburg
fast boundary–element methods as discussed above.
All of these methods that have been discussed encounter some particular problems.
Finite– and infinite–element methods involve the so–called pollution error at high frequen-
cies, cf. Ihlenburg [153]. This has not been observed for boundary–element methods yet, cf.
Marburg [212, 220]. However, the irregular eigenfrequencies make the boundary–element
method more complicated, because each of the methods that are used to overcome this
problem encounter new difficulties. A problem like the irregular frequencies has not been
reported for finite–element solutions. Sound power computation by the Rayleigh integral
is likely restricted to comparatively simply shaped, i.e. convex, structures where little
reflection and diffraction occurs.
Apparently, there are many methods that may be suited for acoustic analysis. It is
always a challenge for the user to decide which method might be best suited for a particular
purpose.
2.3 Structure–Fluid Interaction
In this subsection, we will discuss some ideas of coupling both analysis parts. In general,
two approaches are distinguished, the uncoupled and the coupled structure–fluid interac-
tion. The uncoupled structure–fluid interaction is not actually an approach that omits all
coupling effects. It assumes that the structural vibrations in vacuo are calculated first.
Their results serve as boundary conditions for the acoustic analysis part. This corresponds
to a physical meaning such that the structural vibrations influence and excite those of the
fluid but the fluid vibrations have no effect on the structure. Reaction of the fluid on the
structure is included in the coupled structure–fluid interaction.
The uncoupled version is the most popular formulation for structural–acoustic optimiza-
tion. This is likely due to its simple formulation because two smaller models are computed
one after the other, instead of one large model. Examples of the application of uncoupled
structure–fluid interaction in structural–acoustic optimization were reported for interior
acoustics by Cunefare and Engelstad et al. [82, 86, 94] and by Marburg and Hardtke et
al. [210, 213–215,218], and for exterior problems by Koopmann and Fahnline [102,180], by
Koopmann and other co–workers [28, 74, 89, 240, 301], by Tinnsten et al. [309, 311, 312], by
Hall [138], by Fisher [106], and by Milsted et al. [232].
The uncoupled formulation can be used if a heavy structure (steel) and a light fluid
(air) are considered. Hence, an engine or a gear box are usually analyzed uncoupled. In
contradiction, light structures in air like a loudspeaker or a microphone membrane and
heavy structures in a heavy fluid like ships in water require coupled simulation. In a
number of cases, both coupled and uncoupled simulation were performed. In this context,
we mention sedan body models with coupled analysis, cf. [67,68,133,145,199], and uncoupled
analysis, cf. [155,214–216, 304].
Coupled structure–fluid interaction was formulated even in discretized form about thirty
years ago by Craggs [80]. This paper and most of those that have followed focused on closed
domains. They used finite–element models for both the structure and the fluid. These
techniques were reviewed by Nefske et al. [242], by Morand and Ohayon [236] and, more
recently, by Astley [10].
It is common practice that authors who perform coupled analysis using finite elements
for structural and fluid meshes start with Craggs’s formulation that can be written in
frequency domain as
Kss −ω2Mss Ksf
−ω2Mfs Kff −ω2Mff us
pf=fs
ff(22)
Developments in Structural–Acoustic Optimization for Passive Noise Control 301
where indices sand fcorrespond to the structure and the fluid, respectively. Square ma-
trices Mand Kand column matrices u,pand fhave similar meanings as explained with
equations (1) and (11). Matrices have now two indices, as they denote blocks in global
system matrices. Mixed indices indicate coupling matrices. Choi et al. [68] discussed com-
putational efficiency of solution methods for equation (22). The straightforward approach
of solving this equation directly for each single frequency requires enormous computational
costs if it is to be solved for a large frequency range. An alternative is the modal–frequency
finite–element analysis proposed by Flanigan and Borders [107]. Their approach is based on
solution for the algebraic eigenvalue problem of the asymmetric system matrix. A number
of methods were proposed and/or applied to end up with a symmetric system matrix, cf.
Bretl [47], Yamazaki and Inoue [337], Luo and Gea [198], and Sielaff et al. [293]. Other
useful references on this are [42, 251, 272, 273].
Craggs’s formulation [80] is often referred to as the pressure formulation. This as-
sumes that structural behavior is described by displacement variables, whereas pressure
variables are used for the fluid. Alternative formulations utilize displacement variables for
the structure, as well. However, they can be distinguished by the fluid variables, e.g. the
velocity potential [99], a combination of sound pressure and potential [235], displacements
for the fluid [172], a combination of displacements and pressure [326] etc. All these alter-
native formulations result in symmetric, positive–definite matrices, but may encounter new
problems. Kiefling and Feng [172] observed spurious modes when using the displacement
formulation together with Lagrangian elements. A solution to circumvent them was pre-
sented by Hamdi et al. [141]. Moreover, these spurious modes vanish if special interface
elements, Raviart–Thomas elements [264], cf. Bermudez and Rodriguez [32, 33], are used.
Wang and Bathe [326] suggested a combined displacement/pressure formulation to avoid
these spurious modes. In general, the displacement formulations appear interesting for re-
search because they allow an easy handling of the interface condition and give rise to sparse
symmetric matrices. Furthermore, they can easily be extended to nonlinear problems, cf.
Bathe et al. [24]. In addition to the spurious modes, it is a disadvantage that the displace-
ment vector field is larger than the scalar pressure field. A more detailed investigation and
comparison of the pressure and the displacement formulation are found in the recent paper
by Bermudez et al. [31].
Fully coupled structural–acoustic interaction using boundary elements for the fluid
appears less popular than using finite elements for both models. Nevertheless, a num-
ber of papers are dedicated to coupling a finite–element mesh for the structure and a
boundary–element mesh for the fluid. In this context, we mention the paper by Benthien
and Schenck [30]. The straightforward approach leads to asymmetric matrices, which are
inefficient for computation of frequency ranges. Thus the authors recommend modal decom-
position of the structure and employment of the coupling procedure based on that. Asym-
metric matrices also occurred in the methods proposed by Everstine and Henderson [100]
and Rajakumar et al. [262]. Chen et al. [66] proposed an alternative technique using finite
elements for the structure and Galerkin BEM for the fluid. Other symmetric coupling for-
mulations were published by Mariem and Hamdi [222], by Jeans and Mathews [159] and,
recently, by Gaul and Wenzel [116]. The latter formulation includes frequency interpolation
as explained in the previous subsection in connection with Benthien and Schenck [30], see
above. Cabos and Ihlenburg [56] applied the so–called added mass effect that uses the
assumption of an incompressible fluid vibration analysis of container vessels.
An interesting semi–analytical approach was presented by Slepyan and Sorokin [294].
They involved an analytic representation of Green functions for segments of the structure
that directly interact with the fluid. It is an advantage of this idea that the resulting
matrices are much smaller than those in other formulations. The method should be com-
bined with ordinary finite–element representations, though. For further discussion, we refer
302 Steffen Marburg
to [70, 300].
In addition to finite– and boundary– element techniques for structure fluid interaction,
radiation modes may be used to formulate a modal–based interaction. This feature was
reported by Chen and Ginsberg [63]. They extended the real matrix that is necessary for
sound power evaluation by its imaginary part to accomplish coupling at fixed frequencies.
More applications were presented in the paper by Chen [62].
Modeling of acoustic damping by absorption is another important question in structural–
acoustic modeling. Condition (16) is usually used for uncoupled problems. A summary of
boundary conditions for the uncoupled problem was supplied by Suzuki [303, 304]. Simi-
larly, Chen et al. [66] were using the boundary admittance for acoustic damping in their
fully coupled finite– and boundary–element formulation. Freymann et al. [110] studied the
physics of acoustic damping. Their simple model consisted of a duct with two rigid plates
representing simple spring–mass dashpot systems. Some parameter studies were performed
on force excitation, on spring stiffness, dashpot existence and on the location of damping
pads either at one end or in the middle of the tube. The authors derived and solved a
system of equations for the coupled vibrations. Seybert et al. [288] formulated and studied
sound propagation in a duct containing bulk absorbing material, i.e. foam, at its surface.
Although material data of bulk–absorbing media is complex, the same integral equation
method as for a non–damping fluid could be used. A multi–domain formulation became
necessary. Astley and Cummings [13] developed a finite–element scheme for considering
sound propagation in air and in porous media using one formulation for both. In another
paper, Astley [9] compared this formulation with that using the admittance boundary con-
dition, cf. equations (16).
Obviously, a simulation model should be as exact as necessary. For optimization pur-
poses, it is crucial to make use of as much simplification as possible. Clearly, this must not
falsify vibration properties and acoustic behaviour. Full structure–fluid interaction is not
always necessary where it is applied. In the case of vehicle body acoustics, many authors
have reported fully coupled formulation and simulation. The uncertainties of modeling such
a complex structure for acoustic purposes, however, are likely to be several orders higher
than the numeric error that is involved by simplification from coupled to uncoupled systems.
Few papers are known for optimization of realistic and fully coupled models. Christensen
et al. [69–71] reviewed formulations of structural–acoustic simulation and optimization.
They applied this to some cases of shape optimization of a loudspeaker diaphragm using a
fully coupled finite– and boundary–element model. While optimizing a coupled model they
were able to accelerate their simulation at the point of modeling by applying axisymmetric
analysis for structure and fluid.
2.4 Objective Functions
Approaches to formulate an objective function to judge the desired properties can be cat-
egorized into three different groups, the first one being the sound pressure level at one or
more specified points, primarily utilized for closed domains [82,86, 94,181,184,210, 213,215,
216,218,252, 253, 293,299]. For general–purpose noise control in open domains, the emitted
sound power accounted for the objective function in a number of papers [28, 70, 74, 75, 102,
106, 138, 160, 161, 180, 189, 190, 232, 240, 301, 330]. The remaining third category compre-
hends all other proposals of objective functions [69, 71, 139, 146, 191, 205, 239, 263, 309, 312].
A few papers discuss concepts to substitute pure structural measures for sound field eval-
uations [160,161,214,263,330].
Looking at the first category, it becomes obvious that judging a structure in terms
of the sound pressure at just one discrete point might result in a very low value at this
particular point and high values at other points. The question is often discussed but it was
not possible for the author to find any clarification in the literature. In what follows, we
Developments in Structural–Acoustic Optimization for Passive Noise Control 303
will supply a brief discussion that is – so far – only based on assumptions and heuristic
experiences for a certain objective function.
An objective function Fthat was introduced and discussed by Marburg et al. [210,214–
216] is written as
F=˜
F1
n(23)
where ˜
Fis given by
˜
F=1
ωmax −ωmin
ωmax
ωmin
Φ{pL(ω)}dω . (24)
The operator Φ{} applied to pLrepresents a kind of a weighting function. A reasonable
weighting function is
Φ{pL}=(pL−pRef)nfor pL>p
Ref
0forpL≤pRef .(25)
So, the objective function simply appears as the frequency–averaged sound pressure level,
and the exponent ncontrols the type of average. For n= 1, equations (23)–(25) lead to
the mean value, where only values higher than a certain reference level pRef are taken into
account. Similarly, this form provides the root mean square value for n=2. Themajor
advantage of the square mean value is that high–level peaks are valued higher than low–level
parts of the function. This helps to reduce these high–level peaks during an optimization
procedure and avoids deep valleys as compensation of high peaks. As ntends to infinity,
the objective function approaches the maximum value of the noise transfer function.
The objective function given above should be used for low–frequency problems only,
e.g. sedan body interior noise. In these problems, a linear frequency spacing is still applica-
ble. For higher–frequency problems in wide frequency ranges, a logarithmic spacing of the
frequency is recommended. For that, Hambric [139] suggested computation of a one–third
octave band level. His recommendation included mean square sound pressure calculation
which is then transformed into a band level. Alternative adjustments may be reasonable,
e.g. the band width could be decreased.
With respect to the question introduced above, of using only one specific point for
judging noise in a cavity, we assume that the objective function indeed depends very little
on the location of the specified point provided
•frequency range is large enough,
•exponent n>1, and
•boundary admittance Y>0.
Apparently, the frequency range plays the most important role since vibration nodes usually
occur at discrete resonance frequencies, normally at different locations for different reso-
nances. The simple average should not be used as the objective function, except possibly
in combination with large absorption and high reference level pRef. It is clear that fre-
quency step size must be greater than a certain minimum set very small for low damping.
Otherwise, narrow peaks in the noise transfer function might not be detected.
Thus, results of the examples of minimization of the sound pressure level at one discrete
point for just one (or few) discrete frequencies [82,181,252,253] could possibly encounter the
problem that optimization favors a design with vibration nodes at these particular points.
In Crane et al. [82], application of another, much finer model indicates that the authors
are aware of this objection since they were using up to 188 data recovery points for that.
304 Steffen Marburg
Sielaff et al. [293] recommend using the root mean square value of the sound pressure level
at different points and for different load cases. Finally, an averaged noise transfer function
is to be judged using a criterion like the objective function in equations (23)–(25).
Koopmann and Fahnline [180] suggested using the mean square sound pressure pav in
the enclosure:
pav =1
V
V
[p∗(
x)p(
x)] dV (
x) (26)
where p∗represents the conjugate complex of the sound pressure p. This value is propor-
tional to the potential energy due to compression of the fluid particles. Another idea was
the use of the overall acoustic energy including the potential energy and an additional term
for the kinetic energy.
For radiation into open domains, the emitted sound power Pmostly accounts for the
objective function. It is calculated by
P(ω)= 1
2
A
p(
x,ω)v∗
f(
x,ω)dA(
x).(27)
Hence, it is the integral over a chosen surface Aof the real part of the sound pressure,
multiplied by the complex conjugate normal component of the fluid–particle–velocity vector.
The sound power corresponds to the arbitrarily chosen surface A. Usually, the entire surface
of the structure under consideration, or an artificial (spherical) surface circumscribing the
emitting structure, is considered for sound power evaluation. However, more sophisticated
investigations could estimate the sound power emitted to certain (selected) directions.
Averaging of the sound power in a specified frequency range has been proposed in
different ways. Similar to calculation of a mean value of the sound pressure level in equa-
tions (23)–(25), Lamancusa [189] and together with Eschenauer [190] proposed a simple
averaging of the sound power. It should be emphasized that the sound power and not the
sound power level was taken. A similar technique with simplified sound power evaluation
has been proposed by Milsted et al. [232]. In a number of examples, authors suggest that
the sound power level should be calculated based on the sum of sound power values at
resonance frequencies assuming that structural eigenfrequencies and acoustic resonances
coincide [28, 146, 177, 180]. This approach is limited to weakly damped structures and
avoids computationally costly evaluations at many frequencies, cf. Kollmann [177]. While
this seems to be a reasonable technique for weakly damped structures with rather few res-
onances, we recommend the root mean square sound power level if a large frequency range
is analyzed. Again, a logarithmic frequency spacing might be useful.
In his papers, Jog [160,161] compared the emitted sound power with a particular defini-
tion of the dynamic compliance, in order to find an equivalent measure that avoids having
to solve the acoustic equations. For that formulation, sensitivity analysis becomes rather
simple and efficient.
In a number of papers, alternative objective functions have been proposed. Esping [97]
reported the improvement of the noise emission characteristics by maximizing the lowest
structural eigenfrequency of a sedan floor panel. A similar strategy was applied by Marburg
and Hardtke [214]. This procedure was then justified by comparison of the noise transfer
functions. We will return to this example later in this paper since this objective function is
not always advantageous. Hibinger [146] and, based on his investigations, Kollmann [177]
minimized the sum of the average squared particle velocity of the structure. Since this
measure gives an upper limit of the sound power estimation, their work has already been
mentioned in the context of sound power minimization. This remark is also valid for the
approach of Jog [160,161]. Christensen and Olhoff et al. [69,71] prescribed a certain desired
Developments in Structural–Acoustic Optimization for Passive Noise Control 305
directivity pattern. By formulating the objective function as the squared deviation from
a prescribed curve, they tried to adjust the actual directivity pattern in an optimization
process for one and three frequencies.
Hambric [139] formulated a general objective function that could be used for very dif-
ferent purposes. The original objective function was extended by a constraint condition
considering the sound pressure above a target level. This type of objective function is ex-
cellently suited for minimization of acoustic measures like the sound pressure or the sound
power level. As a reasonable extension of this objective function, many authors have intro-
duced a mass condition to avoid or, at least, to define aprioritheincreaseofmasswhen
decreasing the emission of noise [28,69,82,86,94,106,138,139,146,177,180,189,190,232,301].
Usually, this mass condition is formulated in terms of an inequality relation:
1−minitial
mnew
≤0.(28)
This condition is fulfilled if the weight of the new design does not exceed the initial mass.
Instead of minimizing sound pressure or sound power with a mass constraint several authors
emphasize suitability of minimizing mass while trying to fulfill certain noise constraints, for
instance to decrease sound power b elow a certain level [82, 86, 146, 177, 189, 190, 325]. One
advantage of this formulation consists in a linear dependence of the objective function in
terms of design variables, provided that shell thickness, layer thickness, or size of discrete
masses accounts for the optimization variables. Other authors use weighted sums of two or
more objective functions like mass and noise level, cf. [82,86,170]
Koopmann et al. [180, 240, 301] coined the terminus of “weak radiator”. Such a weak
radiator is a design of minimum sound radiation at a specified mode. Obviously, this
approach is very robust with respect to different load cases. However, originally developed
for one mode shape, it is valid for weakly damped structures only. For practical purposes,
it could hardly be applied for more than five to ten modes.
Finally, we want to discuss some alternative objectives that have been applied in
structural–acoustic optimization.
The first one consists in the maximization of transmission loss through structures. To
our surprise, this subject has seldom been addressed by researchers up to now. Dym et
al. [191, 205] focus on the optimization of sandwich panels. Besides the maximization of
transmission loss itself, a weight constraint was required to get meaningful results. Obvi-
ously, a thick and heavy panel would guarantee worst transmission. Ratle and Berry [263]
considered such a problem in the acoustic optimization section of their paper on optimiza-
tion of plates. They minimized far–field sound pressure on one side due to plane–wave
excitation on the other side of the baffled plate.
The second alternative for objectives in the context of structural and acoustic optimiza-
tion appears in the design of bells [267,281, 317]. Roozen–Kroon [267] and van Houten [317]
formulated the objective function as
F=wDfD
fDopt
−12
+
Nf
i=1
wf,i¯
fi−¯
fopt,i2+
Nf
i=1
wa,iηai−ηaopt,i2.(29)
In this function, wD,w
f,iand wa,idenote weighting factors for each single term. Nfis the
number of structural eigenfrequencies being considered, ¯
fiand ηairepresent the eigenfre-
quencies and corresponding acoustic damping values of the current design, while ¯
fopt,iand
ηaopt,i account for the target values. Acoustic damping has been defined as a frequency–
scaled relation between the radiated sound power and the kinetic energy of the structure.
The values of fDand fDopt correspond to a relation between the diameter of the bell and the
306 Steffen Marburg
shell thickness. The measure fDwas thus defined as Dlf1, the product of the lip diameter
and the first eigenfrequency. Excluding this first term in equation (29), this objective func-
tion could also utilized for autonomous model updating, where experimental and simulated
data of a modal analysis are compared.
Tinnsten et al. [309, 311, 312] used the sound intensity in a point to have an objective
function for radiation problems.
Nagaya and Li [239] wrote their objective function as
F=ω0
0Aw1p2
i+w2vs2
kdA dω . (30)
Squared values of the sound pressure at a particular point in space, and the particle velocity
on the structure, were weighted and integrated over the structural surface and the frequency
range.
Pal and Hagiwara [181, 252, 253] formulated an objective function for inverse optimiza-
tion. The intent here is to achieve minimum modification of design variables with con-
straints on sound pressure level and mass. Their objective function consisted in the sum of
squared modification of design variables.
2.5 Sensitivity Analysis
Barthelemy and Haftka [21] and van Houten [317] reviewed and discussed approximation
concepts. Basically, they distinguished between local, global, and mid–range approxima-
tion of the objective function. These approximation concepts will be further discussed with
optimization methods. Especially local approximation is often based on polynomial approx-
imation (linear or quadratic polynomials) in the vicinity of a certain reference point. This
truncated Taylor series requires sensitivities represented by the gradients of the objective
function with respect to every single parameter.
The simplest way to evaluate sensitivities is to compute global finite differences. As-
suming the objective function Fin terms of a parameter set ϑ=(ϑ1,ϑ
2,...ϑ
n)wecan
write the sensitivity as
∂F(ϑk)
∂ϑk
=F(ϑk+∆ϑk)−F(ϑk)
∆ϑk
+O∆ϑk
∂2F
∂ϑ2
k.(31)
The error term O∆ϑk∂2F
∂ϑ2
kis omitted for numeric calculations. Often, this is called a
forward finite difference scheme. A more sophisticated sensitivity analysis is achieved by
using a central finite difference scheme like
∂F(ϑk)
∂ϑk
=F(ϑk+∆ϑk)−F(ϑk−∆ϑk)
2∆ϑk
+O∆ϑ2
k
∂3F
∂ϑ3
k.(32)
Clearly, this formulation will be more reliable if the perturbation ∆ϑkis sufficiently small
and the objective function sufficiently smooth. It is an advantage of central finite difference
schemes that they give access to second–order sensitivities.
Global finite difference schemes are widely used in structural–acoustic optimization
since they are easy to implement. This technique was applied and discussed in papers
by Lamancusa [189, 190, 330], by Hambric [140], by Christensen et al. [69–71], and by
Marburg et al. [214, 218]. Cunefare et al. [82, 86, 94] separated structural and acoustic
sensitivities. Acoustic sensitivities were provided by a commercial boundary–element code.
These sensitivities are well discussed in the paper by Coyette [79]. Structural sensitivities
are provided by global finite differences.
Developments in Structural–Acoustic Optimization for Passive Noise Control 307
It is well known that the global finite difference scheme performs inefficiently especially
if many variables are involved. Often, reliability of the resulting sensitivity depends strongly
on step size ∆θk. Doubtlessly, a large step size would supply erroneous sensitivities due to
the non–linearity of most objective functions with respect to design variables. If the step
size is chosen too small, numeric errors may occur due to a finite number of digits for floating
point operations in a computer. These disadvantages and computational phenomena can be
avoided by analytic or semi–analytic sensitivity analysis. A similar formulation is outlined
in what follows.
We start with differentiation of the objective function in equation (23)
∂F(ϑk)
∂ϑk
=1
n˜
F
1−n
n(ϑk)∂˜
F(ϑk)
∂ϑk
(33)
Assume that ˜
F(ϑk) is evaluated as the frequency average of a function similar to equa-
tions (24) and (25)
˜
F(ϑk)= 1
ωmax −ωmin
ωmax
ωmin
[Π(ϑk,ω)−ΠRef]ndω . (34)
Here, Π represents the specific measure to be minimized, i.e. sound pressure por sound
pressure level pLfor internal problems, and sound power Por sound power level PLfor
external problems. ΠRef denotes a suitable reference value. Differentiation of Fwith respect
to ϑkyields
∂˜
F(ϑk)
∂ϑk
=n
ωmax −ωmin
ωmax
ωmin
[Π(ϑk,ω)−ΠRef]n−1∂Π(ϑk,ω)
∂ϑk
dω . (35)
This reduces sensitivity analysis to its computation at each single frequency, the logarithmic
level operator being
pL(
y)= 20log
10 |p(θk,ω,
y)|
p0(36)
with p0=2·10−5Pa for the sound pressure level and
PL=10log
10 |P(θk,ω)|
P0(37)
with P0=10
−12W for the sound power level. For sensitivity analysis, it is useful to
substitute the natural logarithm for the common logarithm. It can be presumed for this
purpose that P(θk,ω)≥0. Therefore, equation (37) transforms to
PL=10
ln 10 ln P(ϑk,ω)
P0.(38)
It is useful to square the sound pressure value in equation (36). This yields (temporarily
omitting
ydependency)
pL=10
ln 10 ln |p(ϑk,ω)|2
p0=10
ln 10 ln 2p(ϑk,ω)+2p(ϑk,ω)
p0(39)
308 Steffen Marburg
where and represent the real and imaginary parts, respectively. Differentiating sound
pressure level gives (temporarily omitting dependencies on ϑkand ω)
∂pL
∂ϑk
=20
ln 10
p∂p
∂ϑk
+p∂p
∂ϑk
2p+2p.(40)
As the sensitivities of the real and the imaginary part of the sound pressure are equal
to the real and the imaginary part of the sensitivities of the sound pressure, respectively,
equation (40) is rewritten as
∂pL
∂ϑk
=20
ln 10
p∂p
∂ϑk+p∂p
∂ϑk
2p+2p.(41)
Sound power level sensitivity is simply found by differentiation of equation (38)
∂PL
∂ϑk
=10
ln 10
1
P
∂P
∂ϑk
.(42)
Hence, the sensitivity of the sound power level is represented in terms of the sensitivity of
the sound power. The latter may be formulated by differentiating equation (27). Assuming
that the surface of integration remains independent of model variables, this leads to
∂P
∂ϑk
=1
2
A
∂p
∂ϑk
v∗
f+p∂v∗
f
∂ϑkdA . (43)
Thus, sensitivity of the sound power is evaluated in terms of sensitivities of sound pressure
and fluid particle velocity.
Examples of analytic or semi–analytic sensitivities for full structure–fluid interaction
problems have been presented for the sound pressure or sound pressure level [67, 68, 134,
199, 324, 325] and for sensitivities of modal measures like eigenvectors and natural fre-
quencies [198, 200, 276, 277]. Furthermore, analytic sensitivities have been used for spe-
cific measures like total energy level by Bitsie and Bernhard [41], dynamic compliance by
Jog [160, 161] and the sum of the average squared particle velocity of the structure by
Hibinger [146] and Kollmann [177].
Returning to examples of analytic or semi–analytic sensitivities in fully coupled structure–
fluid interaction problems for sound pressure, it is worth mentioning that, basically, three
approaches have been used. Ma and Hagiwara [199] presented sensitivity analysis on behalf
of an adjoint variable method. Similarly, Choi et al. [67, 68] and Wang et al. [324, 325] used
the adjoint variable method for sound pressure sensitivities based on Craggs’s formulation
in equation (22). One major advantage of the adjoint variable formulation is that the sound
pressure sensitivity is efficiently obtained for an extremely large number of design variables.
Choi et al. [68] reported sensitivity analysis and optimization for 36,000 variables. Pertur-
bation techniques were used for sensitivity analysis of coupled structural–acoustic systems
by Hahn and Ferri [134]. However, their paper was rather dedicated to local approximation
of a frequency response function in terms of a certain design parameter. Their method
avoided time–consuming reanalysis of the coupled system.
For further considerations on analytic and semi–analytic sensitivity analysis, we con-
centrate on uncoupled problems, i.e. it is assumed that the structure vibrates in vacuo
Developments in Structural–Acoustic Optimization for Passive Noise Control 309
whereas its vibration patterns are used as input data for fluid analysis, cf. section 2.3. We
proceed with equation (43). Incorporating the boundary condition of equation (16) and
assuming the boundary admittance independent of design variable ϑkgives
∂P
∂ϑk
=1
2
A
∂p
∂ϑk
v∗
s+∂p
∂ϑk
Y∗p∗+p∂v∗
s
∂ϑk
+pY∗∂p∗
∂ϑkdA . (44)
This expression is reordered and simplified as follows
∂P
∂ϑk
=1
2
A
∂p
∂ϑk
v∗
s+p∂v∗
s
∂ϑk
+2Y∂p
∂ϑkp+∂p
∂ϑkpdA. (45)
As can be seen, sound power sensitivity reduces to sensitivities of the sound pressure and
the structural particle velocity. If the sound power is evaluated by integration over the
structural surface, the vector of sound pressure at nodal points is evaluated by
p=Zv
s(46)
where Zdenotes the impedance matrix. Consequently, surface sound pressure sensitivity
is simply written as
∂p
∂ϑk
=∂Z
∂ϑk
vs+Z∂vs
∂ϑk
.(47)
Discretization of the representation formula (19) and incorporation of boundary condi-
tion (16) allows us to evaluate the sound pressure at a field point p(
y) in the fluid as
p(
y)= gT(
y)vs−hT(
y)−gT(
y)Yp.(48)
Substitution of pas given in equation (46) yields
p(
y)= gT(
y)−hT(
y)−gT(
y)YZvs(49)
being equivalent to
p(
y)= bT(
y)vs.(50)
Sensitivity of p(
y) is determined as
∂p(
y)
∂ϑk
=∂bT(
y)
∂ϑk
vs+bT(
y)∂vs
∂ϑk
,(51)
where brepresents certain influence coefficients, cf. [3,79, 155,208,210]. Recently, they were
called acoustic transfer vectors [196]. It is easy to realize that the influence coefficients
indicate the sound pressure’s sensitivity with respect to structural particle velocity.
Equation (45) supplied sensitivity of sound power in terms of sensitivities of surface
sound pressure, equation (47), and structural particle velocity. Sensitivities of surface sound
pressure, equation (47), and field point sound pressure, equation (51), are given in terms
of sensitivities of structural particle velocity and another matrix containing information
representing the fluid material, fluid geometry etc. There is an abundance of literature on
sensitivity analysis of the fluid, in particular of the impedance matrix and the vector of
influence co efficients, see for example [34, 36, 105, 122, 165, 178, 179, 224, 229, 295].
A number of technical applications require modification of the structure only. Sensitivity
of the impedance matrix (or the influence coefficients, respectively) vanishes if modifications
310 Steffen Marburg
of the fluid remain small compared to the fluid wavelength [210]. Obviously, for that kind of
application, sensitivity analysis requires evaluation of structural sensitivities only [89,271].
According to equation (6), sensitivity of particle velocity is formulated in terms of
displacement sensitivity as (indicating dependence on ϑkbut omitting ωdependence)
∂vs(ϑk)
∂ϑk
=iωN∂u(ϑk)
∂ϑk
.(52)
This presumes that the normal vectors are independent of design variables. This is likely if
the assumption given above holds, i.e. impedance matrix is independent of design variables.
Displacement sensitivity is calculated by implicitly differentiating equation (1), see also
[130],
∂A(ϑk)
∂ϑk
u(ϑk)+A(ϑk)∂u(ϑk)
∂ϑk
=∂f(ϑk)
∂ϑk
.(53)
Rearrangement leads to
∂u(ϑk)
∂ϑk
=A−1(ϑk)∂f(ϑk)
∂ϑk
−∂A(ϑk)
∂ϑk
u(ϑk).(54)
Note that the original system matrix Ais to be inverted. Suitable factorization by using
modal basis or Ritz vectors allows fast evaluation of sensitivities, provided that matrix
derivatives are available. These matrix derivatives can be calculated numerically by finite
differences or analytically by further differentiating element matrices. Sensitivities of modal
basis or Ritz vectors are not required.
An alternative to conventional sensitivity analysis is based on adjoint operators [2, 27,
130]. This is briefly outlined for the sound pressure at a field point as the product of
influence coefficients and structural particle velocity, cf. equation (50). Writing this in
terms of structural displacement requires us to incorporate equation (6) and yields
p(
y,ϑ
k)= iωbT(
y)Nu(ϑk)=cH(
y)u(ϑk) (55)
where the substitute cis given by
c(
y)= iωbT(
y)NH.(56)
Superscript Hdenotes the transposed conjugate complex of a vector or matrix, i.e. Hermi-
tian. The derivative of p(
y)iswrittenas(furtherassumingcindependent of ϑk)
∂p(
y,ϑ
k)
∂ϑk
=cH(
y)∂u(ϑk)
∂ϑk
(57)
which is – according to equation (54) – obviously equivalent to
cH∂u(ϑk)
∂ϑk
=cHA−1(ϑk)∂f(ϑk)
∂ϑk
−∂A(ϑk)
∂ϑk
u(ϑk)(58)
=yH(ϑk)∂f(ϑk)
∂ϑk
−∂A(ϑk)
∂ϑk
u(ϑk).(59)
It is easy to realize that yis the solution of the system of equations (omitting ϑkdependence)
AHy=c.(60)
Developments in Structural–Acoustic Optimization for Passive Noise Control 311
This system need be solved only once in a sensitivity analysis for an arbitrary number of
design variables. Major computational efforts are required for evaluation of sensitivity of
A. A similar approach is possible for sound power sensitivity. A discretized version of
the sound power, equation (27), is written as (cf. equivalent formulation in Salagame et
al. [271])
P=1
2uTCu∗.(61)
Adjoint variable sensitivity analysis is formulated analogously to that for the sound pressure.
Obviously, this is just a small extract of sensitivity analysis on structural–acoustic ob-
jective functions. Other objectives require modified sensitivity analysis, cf. Tinnsten et
al. [312] and Jog [160, 161].
2.6 Special Techniques
It became clear in the previous subsections that structural–acoustic analysis as a category
of coupled field analysis appears computationally expensive. As the optimization process
requires sequential (or parallel) solution of single coupled problems, it seems reasonable to
use special techniques to evaluate the objective function more efficiently. In what follows,
we will briefly discuss several of these techniques.
Hibinger [146], Kollmann [177] and Jog [160,161] substituted the use of structural mea-
sures as objective function for solution of the acoustic boundary value problem. While
Hibinger [146] and Kollmann [177] minimized the mean square velocity of their structures,
Jog [160, 161] defined the dynamic compliance to be minimized. Furthermore, Jog [160]
compared minimization of this dynamic compliance with sound power optimization. Basi-
cally, the results indicated that, for radiation from simple structures like plates or a box,
objectives based only on the structure may work well. Their applicability for more complex
structures has not likely been tested and reported yet. It is assumed that pure structural
objectives account for stronger measures than structural–acoustic ones. The latter allow
creation of vibration loops that erase each other. Doubtlessly, creation of these dipole struc-
tures requires very accurate simulation models, but they seem to have a larger potential for
optimization.
Another aspect that may be utilized for efficient evaluation of acoustic data is based
on the circumstance that many structural modifications do not affect the fluid system
properties. It is clear that thickness modifications, material data adjustment or even small
modifications – small with respect to the fluid’s wavelength [210, 218] – will hardly affect
system data of the fluid. Unfortunately, by now, this feature could be utilized reasonably
for internal problems only. Evaluation of emitted sound power would require too much data
storage. Although many applications are known in structural acoustic optimization, most
authors could have done without repeatedly solving the acoustic problem. Modifications of
the fluid domain that necessitated repeated analysis of fluid vibration have been reported
Roozen–Kroon [267], van Houten [317], Christensen and Olhoff [69], Scarpa and Curti [277],
and Soize and Michelucci [299]. The remaining applications did not require or would not
have required acoustic reanalysis in each optimization step if appropriate methods had been
available.
To omit repeated fluid analysis, three concepts have been followed: Pal and Hagi-
wara [252,253] reported that they calculated fluid modes first and used them repeatedly in
each optimization step for fully coupled structural–acoustic analysis. Kessels [170] stored
radiation modes at 50 frequencies. As the radiation modes decay rapidly, only few of them
were necessary at each frequency. Therefore, data storage remained moderate for that
axisymmetric case. The impedance matrix was reconstructed by using radiation modes.
Marburg and Hardtke et al. [210, 213, 215, 216, 218] proposed single evaluation of acoustic
312 Steffen Marburg
influence coefficients, cf. equation (50). Similar to the above mentioned modes, acoustic
influence coefficients are evaluated in the first step of optimization and they are repeatedly
used in the entire optimization process. This procedure demands that modification of the
fluid domain remain small with respect to the fluid wavelength. This presupposition holds
for most applications in structural–acoustic optimization. Variations as mentioned above
(thickness, material properties, other stiffness, mass or damping parameters) and small
variations of shell curvature strongly influence structural vibrations whereas the properties
of the fluid remain unchanged. Recently, a comprehensive package on evaluation and use of
influence coefficients as acoustic transfer functions and their further development in acous-
tic transfer matrices as well as efficient interpolation in a large frequency range has been
the subject of a patent application by Cremers et al. [84].
Application of this concept for radiation into infinite domains encounters the problem
that the integral of the product of sound pressure and particle velocity over the radiating
surface or over a surrounding surface must be calculated. This, however, requires evaluation
of the sound pressure at many positions, usually at every node of the surface. In the case of
cavity analysis, influence coefficients represent the solution for the sound pressure at a single
position or at few positions. Normal modes allow fast approximation of the impedance
matrix to compute the sound pressure for modified distributions of the surface particle
velocity. To the author’s knowledge, literature does not yet discuss the approximation of
the impedance matrix by frequency–independent normal modes or similar techniques. Some
approaches in radiation analysis may have the potential of such fast technique. In most
cases, however, they contain a flaw, such as the requirement for too much data storage,
cf. papers on Pad´e approximation by Coyette et al. [78] on the boundary–element method,
and Malhotra and Pinsky [207] on the finite–element method and DtN boundary conditions.
It would be most desirable to use explicit frequency–dependent matrices with discretizing
numeric methods. Conjugated infinite elements based on the Astley–Leis formulation may
be utilized for this application.
Another special technique was developed for optimization of large–scale models. In the
particular case of large–scale application, modifications of the structure are often limited
locally. Then, a component model appears as a suitable alternative in the optimization
process. This component model consists of at least two parts. On the one hand, there is
a finely modeled part of finite elements where all optimization variables act. On the other
hand, the remaining part of the structure is reduced in size and a substructure technique
is applied, cf. Hermans and Brughmans [145] and Marburg and Hardtke [215, 216]. Static
condensation or Guyan reduction, see for example [344, Chapter 20] was applied in [215]
whereas the advanced technique of Component Mode Synthesis [81] was utilized in [145,216].
Finally, it should be mentioned that optimization problems are often well suited for par-
allel processing. Local and global approximations of the objective function often allow many
steps to be done simultaneously. Usually, the underlying structural and acoustic analysis is
also excellently suited for parallel processing. Surprisingly, only one application of parallel
processing in the content of structural–acoustic optimization could be detected [297]. This
field might be an additional area of research to improve performance of optimization in the
future.
3 PROBLEMS IN STRUCTURAL–ACOUSTIC OPTIMIZATION
3.1 Related Work in Structural Optimization
The topics of structural dynamics and acoustics are closely connected. In some cases,
an engineer will be able to estimate major acoustic properties by analyzing structural
dynamic behaviour only. Several approaches of this type are reported in the literature.
In this subsection, we will briefly discuss related work in optimization of modal spectra
Developments in Structural–Acoustic Optimization for Passive Noise Control 313
and dynamic response. Some references on structural shape optimization will be supplied
because its extension to the field of structural–acoustic shape optimization is likely to receive
particular attention in the future. Doubtlessly, we can discuss only little part of the related
work in structural optimization.
In his early review papers, Olhoff [249,250] surveyed methods and techniques to opti-
mize vibrating structures. Numerical methods and computers allowed researchers at that
time to investigate vibrating rods, beams and plates. Further, plates with prescribed spec-
tra and minimized weight were discussed. In an earlier paper, Olhoff [248] maximized the
fundamental eigenfrequency of a rectangular plate. Other related work on the optimiza-
tion of structural modal spectra was provided by Hinton et al. [147] and by Watts and
Starkey [327]. Wang [323] developed an analytic formulation for eigenvalue sensitivity of
structures. Simple examples of a rod carrying discrete masses were presented. McMillan
and Keane [225,226] added discrete masses to a plate structure to force a frequency gap in
the mode spectrum. This gap was planned in the range of the fortieth eigenfrequency of
the plate (1000 kg). At first the authors utilizing up to five masses of 10 kg each, cf. [225],
but did not succeed until forced larger gaps or at least very low responses in the focused
frequency band [226].
Grandhi’s [124] general paper on the optimization of structures including certain fre-
quency constraints, and the comprehensive review of static and dynamic sensitivity analysis
by Haftka and Adelmann [130], gave excellent overviews on some important techniques in
this field. These two papers contained a number of valuable references about this subject.
Structural optimization of musical instruments is closely linked to acoustic properties.
The paper of Tinnsten and Carlsson [310] addressed control of spectral properties, in partic-
ular the first three eigenfrequencies of a violin top plate. Both varying thickness distribution
and arch height (geometry of the shallow shell) allowed these eigenfrequencies to be adapted
very well to prescribed values. Background of their research was to show whether (natu-
rally occurring) small variations in material parameters can be compensated by thickness or
arch height modification. It can be expected that further research in structural vibrations
of violins and other musical instruments will also involve structural–acoustic optimization.
Optimization of eigenfrequency spectra is not only popular in optimization for vibra-
tional comfort, radiation control or other tuning techniques for constructions. This subject
is closely linked to the field of modal model updating as mentioned in connection with the
objective function, equation (29). For further information in this field, we refer to the work
by Frishwell and Mottershead [111,237] and the paper by Imregun and Visser [154].
The wide field of optimization of dynamic response function was also addressed in many
papers in the past 30 years. An early article was published by Kwak et al. [183] on the
optimal design of isolators. Keane [168] investigated minimization of dynamic response of a
two–dimensional truss. It is remarkable that a mid–range broad frequency band was chosen
for optimization. Substantial gains of the dynamic response level were reported.
A more recent and advanced paper was written by Yim and Lee [340]. These authors
addressed a vibration problem of sedan bodies. It is well known that idle shake vibrations
and their joint stiffness crucially influence the vibration behaviour of the entire body struc-
ture. Therefore, after careful preparation of a simulation model of these components, their
shape will substantially influence and improve global vibration modes of the body. This
technique was developed to estimate a design in the early stage of vehicle development.
Consequently, rather simplified modeling techniques were applied. The simulation model
should be further refined and extended by details, after the optimization process has been
successfully completed. A paper that dealt with optimization of vibrational comfort was
published by H¨anle and Sielaff [142]. It surveyed some general techniques of vehicle body
noise, vibration, and harshness problems and also considered optimization. In particular,
these authors minimized the maximum amplitude of a vibrating panel.
314 Steffen Marburg
Another interesting paper by Ballinger et al. [19] discussed shape optimization tech-
niques of a head expander component of an electrodynamic exciter that was used for vibra-
tion tests. The authors actually addressed modification of eigenfrequencies but considered
dynamic response functions in the optimization process. Similar to the above described
methods, they shifted modes into higher frequency ranges.
Koopmann and Fahnline [180] distinguished between the categories of – first – fre-
quency and mode shape control and – second – shape optimization. With respect to the
first category, some remarks have been made above. Additionally, we will discuss more
issues in the structural–acoustic subsection. With respect to the second category, two re-
view papers should be mentioned. They include the papers by Haftka and Grandhi [131]
and by Hsu [149]. These articles did not yet explicitly contain references to dynamic re-
sponse problems. The work by Gates and Accorsi [113] covered the field of shell curvature
modification and optimization but also left out the dynamic problem. Shape optimiza-
tion problems appear most challenging because many technical problems must be solved,
including parametric description of the geometry or autonomous remeshing procedures.
Finally, we refer to a review paper by Esping [97]. There, the author explained a
number of techniques, problems and applications of design and/or shape optimization of
structures. Beside frequency optimization and discussion of acoustic response, stresses
of contact problems, design safety, manufacturing costs, and topology optimization were
discussed inter alia.
A considerable number of papers were published in the field of topology optimization
with respect to dynamic properties, as well. Among others, cf. [29,162], the extensive work
of Steven and co–workers, see for example [195, 261, 334, 343] and the book by Xie and
Steven [335] should be mentioned. For detailed review of topological optimization we refer
to Rozvany [269] and Eschenauer and Olhoff [96].
3.2 Related Work in Acoustic Optimization
Obviously, acoustic optimization is closely linked to the field of structural–acoustic opti-
mization. Often, pure acoustic analysis will still require structural vibrations either incorpo-
rated as sound sources or as velocity boundary conditions. In this subsection, we distinguish
between related work on classes of practical problems and on numerical methods.
Starting with the practical problems, we refer to Lamancusa [188], who optimized ten
shape variables of an air intake system. While the objective function was computed an-
alytically, the exit sound pressure was minimized over a finite frequency range. 10 to
20 dB decrease were gained in that process. A similar system was optimized by Hacken-
broich [128], who applied finite–element analysis. In that paper, the author proposed to
add volumes. This actually decreased the sound pressure amplitudes although the num-
ber of modes increased. These papers were reflected in van Houten’s dissertation [317].
The example served for testing optimization methods. The shape of reactive mufflers was
optimized by Bernhard [35]. He utilized a transmission matrix technique and developed
a semi–analytic sensitivity analysis scheme. A recent paper by Mechel [227] was focused
on optimization of porous and other absorbers. The objective function, i.e. the absorp-
tion coefficient in a frequency range, was evaluated analytically whereas optimization was
performed numerically.
A closed rectangular cavity was investigated by Papadopoulos [255]. The author modi-
fied this room by adding little portions of rectangular cavities to the outer surface. Clearly,
the modified volume had a modified spectrum of eigenfrequencies and modes. Modes were
optimally controlled, i.e. they were redistributed. It was stated that for a quieter cavity,
evenly distributed modes should be favored since sound energy is more evenly distributed
in the frequency range and, therefore, the effect of certain maxima and minima of sound
energy reduced.
Developments in Structural–Acoustic Optimization for Passive Noise Control 315
A number of papers were published on the optimization of absorbing material. In
general, this class of problems belongs to the optimization of boundary conditions of acoustic
problems. Surface excitation by specified particle velocity distribution was addressed by
Miccoli [231]. This very simple model of an earth–moving machine was optimized to perform
a minimum sound pressure level at the driver’s ear. Further, the boundary admittance was
optimized. The study of Yang et al. [339] addressed optimal placing of acoustic materials
on the boundary of closed domains to minimize noise. Simple rectangular two–dimensional
problems and three–dimensional sedan cabin compartments were examined. Locating the
proper positions by optimization techniques was formulated as one of the problems that
receive major practical interest. Values of acoustic impedance or admittance were optimized
at fixed positions by Bernhard and Takeo [37] and by L¨uet al. [197].
B¨uhrmann [49] managed to optimize the (periodic) shape of a traffic noise barrier. It
was shown that the flat barrier should periodically vary in its height. Compared to a unit
height barrier of the same area, 0.9 dB decrease were gained for the root mean square sound
pressure.
Noise emission by fan and cooling circuits were investigated and even optimized by
Cleon and Willaime [73]. Their concise paper presented a number of actions suitable to
improve fan design. Unfortunately, details about the optimization procedure including the
particular method, objective function etc. are missing.
A study of optimal design from a mathematical point of view was published by Hab-
bal [127]. As one of the results of this paper, it turned out that the field sound pressure is
not very sensitive with respect to shape modification for (fixed) low frequencies. It becomes
more sensitive for higher frequencies. In that frequency range, the optimal shape differs
for the objective function, being either the sound pressure maximum or the root mean
square sound pressure. It was emphasized that the optimal shapes strongly depend on the
frequency. Hence, a frequency–range solution would be useful for practical applications.
Another application of shape identification and optimization in acoustics arises in inverse
scattering. Among others, we refer to a recent paper by Givoli and Demchenko [122]. There,
a technique was presented to identify the geometry of an obstacle by shape optimization
techniques.
Optimization in inverse acoustics also considers identification of material data. Two
recent examples on the reconstruction of the speed of sound in terms of spatial coordinates
are known by Mastryukov [223] for time–harmonic problems and by Gustavsson and He [126]
for problems in time domain.
With respect to numerical methods, there are papers on optimization and sensitivity
analysis on finite–element [34, 79] and boundary–element methods [3, 36, 79, 165, 179, 224,
229,295].
Surprisingly, very little work is found on pure acoustic optimization or sensitivity analy-
sis using finite elements. An early approach, supplied by Bernhard [34], discussed paramet-
ric finite–element matrices to have a simple tool for shape modification and adjustment.
Coyette et al. [79] surveyed sensitivity analysis including two sections on finite–element
formulation. Concerning direct frequency response, they recommended the semi–analytic
approach, cf. [130,171]. In their following section, a formulation for sensitivity analysis of
modal frequency response was proposed. In general, Coyette et al. [79] provided a man-
ual for sensitivity analysis using certain commercial software packages. In a recent paper,
Feij o o et al. [105] presented an adjoint operator approach for shape–design sensitivities in
sound radiation using finite elements and absorbing (DtN) boundary conditions. Other
work that has been mentioned previously [122, 127, 128] utilized the finite–element method
but the method itself was not the primary point of these papers.
As mentioned above, there are a number of papers on pure acoustic optimization and
sensitivity analysis on boundary–element formulations. This could be due to several rea-
316 Steffen Marburg
sons. On the one hand, formulation of the boundary–element method is widely considered
more complicated than finite–element formulations. For that reason, publications focus on
method developments. On the other hand, a surface discretization suffices for boundary–
element formulations. Shape modification usually does not demand exhaustive remeshing
procedure, as is often required in finite–element approaches. As mentioned in the previous
section, finite–element research for external problems in acoustics has made remarkable
progress in the past decade. If this trend continues, we can certainly expect more papers
on acoustic optimization in the near future.
As already shown, optimization techniques for acoustic problems based on boundary–
element formulations received considerable attention over the last 15 years. So, sensitivity
analysis using boundary–element methods was presented in some papers. Kane et al. [165]
proposed a semi–analytic approach for shape sensitivities similar to the well–known tech-
nique based on finite elements. They estimated sensitivities of the surface and field pressure
and particle velocity. Similar to the above mentioned part on finite–element acoustic sen-
sitivity analysis, Coyette et al. [79] supplied formulations for the sound power sensitivity,
for the sensitivity with respect to structural variables and for field pressure sensitivities
with respect to structural vibrations. The latter was also discussed in papers by Ishiyama
et al. [155], Adey et al. [3] and Marburg [208]. Matsumoto et al. [224] supplied extensive
formulation of sensitivities in general. These include sensitivities with respect to shape,
frequency, fluid density and boundary admittance values. Another survey of sensitivity
analysis using boundary–element methods was provided by Bernhard and Smith [36, 295].
Alternative approaches on sensitivity analysis and optimization were presented in the
papers by Meric [229], Koo [178] and Koo et al. [179]. Meric proposed the use of an adjoint
variable method for sensitivity analysis. Four two–dimensional examples demonstrating
shape optimization were presented in that paper. Unfortunately, the method demanded
evaluation of a domain integral that required internal cells. This might turn out to be a
handicap for more complicated three–dimensional domains. In these papers, Meric [229] and
Koo et al. [178,179] used material derivatives originally employed in continuum mechanics.
A brief review of the material derivative concept and justification was supplied in the
article [179] that uses direct analytic differentiation of the boundary integral equation.
As an example of an irregular domain, the shape of a two–dimensional sedan cavity was
optimized with respect to the sound pressure at the driver’s ear.
3.3 Applications in Structural–Acoustic Optimization
Papers that focus on problems in structural–acoustic optimization are expected to solve
structural and acoustic problems. The relative small number of contributions to this field
can be categorized into academic examples and more realistic applications. An advantage
of simple examples is that one can experience and investigate many effects of a method.
In some cases, however, particular adjustments become necessary if problems and models
become more complex.
We start with the academic examples. These can be grouped into problems of beams,
plates, shells, ducts and boxes.
Few papers considered baffled beams, cf. Naghshineh et al. [240], Jog [160, 161], and
Koopmann and Fahnline [180]. The beam structure was likely chosen for simplicity. Some
interesting effects could be demonstrated by investigating this example. Moreover, beam
structures are suitable to test an optimization method.
Plates were investigated and optimized quite often [28,69,71,74,102, 112, 146, 160, 161,
180, 189, 190, 239, 263, 271, 301, 309, 311, 330]. However, the category of academic examples
seems to regard either rectangular or circular ones. It was possible to find one article
presenting the example of an engine cover plate [28] that will be discussed with the re-
alistic applications. Choice of circular or rectangular plates could be critical since these
Developments in Structural–Acoustic Optimization for Passive Noise Control 317
structures contain symmetry properties that may have unfavourable effects on sound power
radiation. Sound radiation problems were investigated in all plate examples. Usually,
the sound power was to be minimized. Few examples on circular plates considered dif-
ferent targets. Fahnline and Koopmann [102] sought a circular plate of maximal power–
conversion efficiency. Christensen et al. [69,71] optimized the directivity pattern of sound
radiation from a circular plate. Tinnsten [309, 311] chose a circular plate to construct an
example for experimental verification of structural acoustic optimization. He measured
the sound intensity above the plate. Wodtke and Lamancusa [330] optimized a circu-
lar baffled plate with an unconstrained damping layer for minimum sound power emis-
sion. In most of these examples of circular plates, the structure was excited by a point
force [69,71,309, 311,330] or a ring load [330]. One example was based on modal properties
and, therefore, independent of the excitation [102]. Rectangular plates were excited by
point loads [28,74, 160, 161,180,189, 239,263,271,301] or by distributed loads [112,190,263].
Lamancusa [189,190] optimized modal properties to achieve an optimal solution indepen-
dent of a particular excitation. Plates were mostly clamped. Few examples of different
boundary conditions were reported. Cases of clamped and free boundary conditions were
compared by Wodtke and Lamancusa [330] and by Tinnsten [309]. Additionally, opti-
mization of an elasticly supported circular plate was investigated in [330]. The example
of Nagaya and Li [239] comprised a plate that was clamped at two edges while the other
two edges were free. Ratle and Berry [263] and Fritze et al. [112] investigated a simply
supported rectangular plate. In some of these examples, the authors reported and com-
pared both broad–band and single–frequency excitation [28, 271, 301]. Only one driving
frequency or mode was considered (for the plate example) in [69, 71, 74]. Two or more
frequencies or modes were included in [239, 263, 309, 330]. Broad–band sound radiation
including many modes was minimized by Lamancusa and Eschenauer [189, 190], by Hi-
binger [146] and by Fritze et al. [112]. Plates seem to be well suited for experimental
verification as supplied in [146, 180, 239, 301, 309, 311]. In addition to the above discussed
axisymmetric examples, symmetry of vibration modes was presumed in some of the exam-
ples, cf. [160,161,180,189,190,271,301].
Structural–acoustic optimization of shell structures without a direct application was
addressed in papers [71, 74, 75, 139, 299]. Hambric’s [139, 140] example was a cylindrical
shell with hemispherical end caps. In one test case the shell was simply supported, in
another one these supports were omitted. The structure was excited by a ring load. Single–
frequency and broad–band excitation cases were investigated for optimization. For this shell
structure submerged in water, a full structure–fluid interaction became necessary as well
as for the thin conical membrane that was considered by Christensen et al. [71]. This
axisymmetric structure clamped at its edge was excited by a point load in the center. The
directivity pattern was optimized for a single frequency. Constans et al. [74, 75] chose the
example of a half–cylindrical shell clamped along the two lower straight edges. The emitted
sound power due to a driving force was minimized for the lowest five modes. Although
structural and loading configuration were symmetric the optimal solution was asymmetric.
An experimental verification was provided [75]. Soize and Michelucci [299] investigated the
aspect ratio of an elastic dome filled with water. Their axisymmetric model was excited by
flow induced pressure from outside while the average sound pressure was minimized.
A simple duct example was investigated by Shi et al. [291, 292]. These authors stated
that the one–dimensional duct system with a coupled spring–mass system should represent
the rear window of a vehicle, but complexity of vehicle structures is much higher than in
their example. Sound pressure at a single point was calculated for a narrow frequency band.
Boxes were addressed for both internal [181, 213, 252, 253] and external [146, 177, 312]
noise problems. Starting with the internal problem, a simple box model was investigated
by Pal et al. [181,252,253]. Fully coupled structure–fluid interaction was considered. The
318 Steffen Marburg
box was excited by a point load and analyzed for two discrete frequencies. The paper
by Marburg et al. [213] addressed modeling techniques of a steel box structure that was
reinforced by two frames. Although a frequency range containing about sixty modes was
chosen, a reliable structural model was created for a range of about twenty modes. The root
mean square sound pressure level over the frequency range was minimized for one test point.
The paper of Tinnsten et al. [312] addressed sound radiation. A point load was applied
for one or two discrete frequencies. The authors reported an uncoupled analysis. The box
example of Hibinger [146,177] was primarily focused on radiation problem. Sound radiation
of this box was not computed, though. Minimization of the sum of mean square velocities
of the box surface accounted for the optimization target. A point load was applied. The
frequency range contained the first 15 modes.
The more realistic applications of structural–acoustic optimization can be grouped into
problems of sandwich panels, loudspeaker diaphragms, bells, a cylinder representing an
aircraft fuselage and vehicle noise problems. For the latter, we can further distinguish
between interior noise problems and others.
Dym et al. [191, 205] investigated optimization of sandwich panels. Their target was
to maximize transmission loss of these plates. A large frequency range was analyzed and
several, at that time new findings were reported.
Christensen et al. [69] optimized distribution of ring masses and geometry of the ax-
isymmetric shell contour of a loudspeaker diaphragm. Driven by a point load in the centre
and clamped at the edge, they found an optimal ring mass distribution and a geometry
to match prescribed directivity patterns for a certain frequency. In another example, the
diaphragm was put into a soft surround (mount) to achieve a uniform pattern even at low
frequencies. For that example, optimization considered three discrete frequencies.
A comprehensive research project on the optimization of carillon bells, in particular
design of a major third bell, was initiated and kept up for about two decades by Lehr
and Schoofs. An early three–part paper gave an excellent and thorough survey of that
problem [193, 284, 318]. Structural–acoustic optimization was incorporated by Roozen–
Kroon [267]. Although still a modal approach, cf. equation (29), she had added modal
damping coefficients to the formulation. Another important step in bell design was com-
pleted by the dissertation of van Houten [317]. Schoofs and van Campen summarized these
and other activities on bell design and optimization in a longer paper [285]. Knowledge
of axisymmetric shell analysis was then applied to optimization of scanners for magnetic
resonance imaging, cf. Kessels [170].
A cylinder clamped at both ends accounted for a design problem similar to an aircraft
fuselage in several papers by Cunefare, Engelstad et al. [82,86,94]. The structure was excited
by a monopole source outside at a single frequency that coincided with an eigenfrequency of
the original model. Optimization included thickness of shell rings [82], cross section [86,94]
and location [94] of stiffening elements such as stringers and frames. The sound pressure
was evaluated at certain data recovery points inside the cylinder.
There are few papers on application to vehicle acoustics which do not primarily consider
body vibrations. All of them address design and support of the engine or components. An
engine cover plate accounted for the practical application in Belegundu et al. [28]. The
plate was clamped with respect to rotations and supported on springs of a high degree
of stiffness. Excitation forces of different phase angles were applied. In the optimization
process, sound radiation was minimized with respect to the first three structural modes.
The paper by Milstedt et al. [232] discussed design and composition of an engine that
consisted of several single substructures. Optimization was accomplished for the frequency
range of 0–5000 Hz. Their model, however, was known as a concept model. Therefore, the
objective of that study was to show the potential of structural optimization and to illustrate
the concept. Similar problems were studied by Hall [138] and by Fisher [106]. Real engine
Developments in Structural–Acoustic Optimization for Passive Noise Control 319
analysis requires finer models, as the authors stated. La Civita and Sestieri [184] described
design of an optimal engine mounting system by using a very simple analytic model for
computation. They computed the sound pressure level at the driver’s ear in the frequency
range up to 120 Hz. The paper regarded very different possibilities for constructing engine
supports, i.e. position and orientation, non–linear spring characteristics, and, at the same
time, they imposed different technical and geometrical constraints.
Most of the applied papers in structural–acoustic optimization addressed sedan body
construction. These examples are categorized in terms of design variables. One category
consists of seeking an optimal distribution of piecewise constant shell thickness [68, 133,
145, 199, 325]. These applications usually require many variables. The paper of Choi et
al. [68] indicates use of 36,000 parameters. Hermans and Brughmans [145] treated the
vehicle body as an assembly of several components being investigated independently in a
first design stage. The third category utilizes modification of shell geometry, i.e. shell
curvature was optimized [214–218]. Several sedan components were investigated, e.g. a
roof [218], a dashboard [215], a hat–shelf [214], a floor structure [216] and a spare–wheel
well [217].
All of these examples of sedan interior noise aimed at pure reduction. Future work
will likely design sound in cars as suggested by Freymann [109]. The importance and
necessity of optimization in the body design process to achieve a “design right at first
time” was emphasized in a paper on optimization of the noise, vibration and harshness
engineering process by Roesems [266]. Ginsberg [119] listed numerous limitations of large–
scale simulations for automotive applications including (among others) noise, vibration
and harshness modeling, as well as multidisciplinary optimization of noise and crash. He
briefly discussed another recent paper on this subject by Sobieszczanski–Sobieski et al. [297]
that was directly dedicated to large–scale combined optimization of noise, vibration and
harshness problems, and crash. However, crash analysis was computationally much more
demanding than the acoustic optimization which just considered eigenfrequency of the first
torsional mode of the car.
A number of papers that did not actually address structural–acoustic optimization but
were focused on structural–acoustic sensitivity analysis. Again, we start with the academic
examples. Hahn and Ferri [134] demonstrated sensitivity analysis for a cylindrical shell
submerged in water. Shepard and Cunefare [290] used a semi–infinite plate. Both papers
mainly addressed modeling techniques, including the question of how coarse or fine certain
details should be modeled. A number of papers can be found on interior noise of simple
boxes [41, 67, 198–201, 276, 277] and just one on radiation of a box [89]. The latter also
presented an example of a cylindrical shell. While most of the papers that consider interior
noise of boxes concentrate on sensitivities of coupled modes, the contribution of Bitsie and
Bernhard [41] is devoted to an unusual topic in the field, namely high–frequency noise
problems. Applying energy finite–element methods, the authors studied the reduction of
the energy level of structure and fluid due to an excitation of either the structure or the
fluid. Sensitivity analysis of vehicle body structures was accomplished in [199,200,324,325].
Some of the papers that consist of two parts discussed theoretical aspects in part one.
The review paper of Christensen et al. [70] supplied detailed description of efficient analysis
in structural acoustics. Marburg [210] described theoretical aspects of the optimization of
vehicle noise transfer functions. In another recent paper by the author [211], concepts of
parametric description of shell structures were developed. Geometry–based models were
compared with a newly developed modification concept that enabled a user to directly
modify the shell geometry of a finite–element mesh.
Finally, it should be mentioned in this context that a review of the book by Koopmann
and Fahnline [180] was published by Elliott [93].
320 Steffen Marburg
4 DESIGN VARIABLES
4.1 Choice of Parameters
When looking at design variables one can easily recognize that most authors favored opti-
mization of plate thickness, shell thickness, or thickness of individual layers, any or all of
these being piecewise constant or linear. Other authors used variables for material data,
cross section data of beams, rib size and location, size of discrete masses and location,
stiffness, and damping of absorbers, and geometrical data of shell structures and cavities.
In the following pages, these categories will be described in more detail.
Starting with plate and shell thickness as design parameters, we categorize contributions
where each element accounts for its own single variable [28,67,68, 106,133,138, 146,160,161,
177,181,189,190,198–200, 232, 252, 253, 271,312]. This approach is straightforward but will
usually fail for large–scale structures, since it requires too many variables to be optimized.
Even sensitivity analysis becomes more challenging. An adjoint operator approach appears
useful for sensitivity analysis. This was demonstrated by Choi et al. [68] who optimized
shell thickness of as many as 36,000 elements. However, it seems useful to create groups of
elements of the same thickness. Thus, the number of variables may be kept at a reasonable
level. Apparently, shell domains of constant thickness are easier to produce than those of
elementwise constant thickness. A realistic application with predefined element groups of
piecewise constant plate thickness was presented by Tinnsten et al. [309, 311]. His circular
plate consisted of several rings of constant thickness. It was first optimized and then
produced for experimental verification. Other applications of this category were given
in [69,82,86,133,134,139,140,145,199,200,276,277,297,324,325]. To mention some examples,
Ma and Hagiwara et al. [133, 199, 200] optimized plate thickness of the walls of a system
similar to a sedan–cavity. Cunefare and Engelstat et al. [82,86] generated strips of constant
thickness in two directions of a cylinder, circumferential strips and strips parallel to the axis
of revolution. Scarpa et al. [276, 277] investigated sensitivity and dependence of different
acoustic properties in terms of thickness of the radiating plate. Christensen and Olhoff [69]
combined thickness parameters with geometric variables to optimize them simultaneously.
Milsted et al. [232] applied a so–called effective thickness that considered the stiffening
effect of ribs. Hall [138] gave an extension when the relative distance between nodes was
parameterized. This, however, was nothing more than varying thickness of stiffening ele-
ments. Similar variables were used by Fisher [106]. Wodtke and Lamancusa [330] varied
the thickness of the damping layer of an axisymmetric circular plate. Piecewise linear dis-
tribution of this layer was assumed. Dym et al. [191, 205] optimized layer thickness and
layer properties of sandwich plates. Kessels’s [170] investigation of scanners for magnet res-
onance imaging tested optimization of thickness of individual layers, including viscoelastic
glue layers.
Beam properties include properties of the cross section, in particular area and area
moment of inertia. This corresponds to typical input parameters of commercial programs
as applied in the paper by Hermans and Brughmans [145]. They used values of the pbar
card of MSC/Nastran [203] combined with shell thickness. Based on parameter choice, a
similar paper was published by Wang [324]. However, this was essentially dedicated to
analytic sensitivity analysis. Naghshineh et al. [240] varied thickness of thin–walled profiles
that may be used to calculate cross–sectional area and area moment of inertia. Similarly,
Jog [161] optimized thickness of a long plate that might also be considered as a beam.
Cunefare and Engelstad et al. [86, 94] investigated p erformance of stiffening stringers and
frames on an aircraft skin. Cross–sectional size accounted for their variables, also combined
with shell thickness [86]. Marburg et al. [213] sought optimal positions and lengths of
reinforcing profiles of a test box.
Few authors reported optimization of ribbed structures. Hambric [139,140] described a
Developments in Structural–Acoustic Optimization for Passive Noise Control 321
search for the optimal location of four ribs. A fundamental problem is encountered when
optimizing discrete positions of stiffeners in a discretized simulation model. Consequently,
rib positions should coincide with element interfaces of the finite–element mesh. This
remark is also valid for the above mentioned optimization of the reinforced box [213].
Hibinger [146] predefined ribs at all element interfaces and had them vanish during the
optimization process.
Material data were optimized in [134, 190, 191, 205, 240, 312]. Dym et al. [191, 205]
combined layer thickness and layer density optimization. Naghshineh et al. [240] presented
results for distribution of density and Young’s modulus. Lamancusa and Eschenauer [190]
investigated possibilities to find an optimal distribution of the volume fraction between fiber
and matrix of a fiber reinforced composite sandwich panel. While the fiber angle remained
constant in that case, it was optimized by Tinnsten et al. [312]. Hahn and Ferri [134]
studied dependence of acoustic measures in terms of Young’s modulus and plate thickness.
A number of papers discuss optimization of discrete absorbers, i.e. springs, dampers,
masses and joint occurrences. A single spring stiffness accounted for the only design variable
in the test case of Shii et al. [291, 292]. In a parameter study, Milsted et al. [232] discussed
importance of damping to smooth the objective function because it was dominated by res-
onances and contained numerous maxima and minima. Hambric [139] combined thickness
optimization and rib positioning with optimization of loss factors. Loss factor sensitivities
were used for better understanding of structural–acoustic systems in medium–frequency
range. Size of discrete added masses was optimized by Koopmann et al. [180, 301] whereas
Christensen and Olhoff et al. [69, 71] used ring masses in the axisymmetric case. Alterna-
tively, the positions of one or five discrete masses were controlled by Ratle and Berry [263].
Constans et al. [74, 75] and also Ratle and Berry [263] combined sizing and positioning
of discrete masses. As previously discussed with stiffener location, positioning of discrete
masses also encounters the problem of the discretized structure. Masses coincided with
element vertices. Full combination of absorbers was proposed in [184, 239]. The analytic
model of la Civita and Sestieri et al. [184] allowed continuous variation of all parameters
including location and orientation of mounts. It is remarkable that these authors used non-
linear engine mount characteristics. Nagaya and Li [239] combined five variables for each
of three absorbers, i.e. two variables for position, and one each for mass, damping, and
stiffness. Their final position did not show the absorbers at element vertices.
Optimization of carillon bells [267,284,317] required radii and shell thickness as design
variables. Roozen–Kroon [267] proposed the use of seven radii and 13 thickness values.
Sensitivity studies of Scarpa and Curti [277] comprised examples of variable cavity size.
An axisymmetric dome was varied in size, i.e. the aspect ratio of length and radius of the
dome. In that paper by Soize and Michelucci [299], only one variable was used. This variable
did not only modify the cavity but strongly influenced shell geometry and curvature.
The geometry of a shell, i.e. curvature, was optimized for a loudspeaker diaphragm
by Christensen and Olhoff [69]. Since their finite–element mesh was much finer than the
geometric approximation necessary for representation of the loudspeaker diaphragm, a ge-
ometry model based on spline interpolation was used. Vertical components of eight of nine
keypoint positions accounted for design variables.
Similarly, Marburg et al. [210, 215, 218] utilized geometry–based models of sedan body
panels. In doing that, keypoint positions were parametrically defined. These keypoints
were connected by straight or curved lines that spanned areas to be meshed by finite shell
elements. This concept performed well for large and smooth panels like a sedan roof [218].
Even for the more complicated structure of a dashboard, the geometry–based model worked
reasonably [215] although problems occurred if the remeshing procedure created a new mesh.
A geometry–based model of the more complicated structure of a sedan hat–shelf would
have become too complex for an optimization procedure [214]. Therefore, the concept of
322 Steffen Marburg
direct modification of the finite–element mesh was developed by Marburg [211]. To do
that, we distinguish between global and local modification in a user–defined modification
domain. Global modification means modification on the entire modification domain using
a complicated function. A local modification function is a simple function that lives on
part of the modification domain. It can be freely moved within the modification domain.
Quadratic polynomials as global functions and bead shapes as local functions were tested
for sedan body panels by Marburg and Hardtke [216,217]. Local modification functions
were further investigated by Fritze et al. [112]. The actual choice of variables depends on
the choice of the modification functions [211].
Finally, we summarize that very different variables have been used in the field of
structural–acoustic optimization. The number of design parameters ranged from only
one [299] to approximately 36,000 [68]. Most authors try to keep the number of variables
very low, i.e. between 2 and 50. The number increases rapidly, if the method demands
equipping every element or every node with its own variable. For complex models, this re-
quires grouping the elements affected by the same variable or using a modification concept
similar to those proposed in [211].
4.2 Constraints
In most cases, design variables require certain constraints. Even if restrictions are not
explicitly mentioned, they are necessary to limit the feasible domain of the new design.
Optimization will usually demand lower and upper bounds of variables. These bounds
include minimum shell thickness values or maximum feasible modifications of geometry. A
suitable list of reasonable restrictions in general was supplied by Lamancusa et al. [189,190].
In the previous subsection, we have discussed a modification concept that was called
direct modification of the finite–element mesh. One disadvantage of this concept is that
maximum geometric modification can hardly be predicted and controlled by an a priori
bounded interval of the design variable [112,216, 217]. The same problem occurs with the
position of the rectangular function domain of a bead function [112,211].
In the context of thickness optimization, there is some work on limiting thickness jumps,
i.e. the thickness difference between two neighboring elements must not exceed a certain
value. This restriction was applied by Hibinger [146,177] for a plate and for a box structure.
Roozen–Kron [267] and van Houten [317] used such a condition for bell design.
The choice of some variables affects the overall mass of a structure. Usually, the op-
timizer will favor a stiff and heavy construction. Therefore, the overall mass must be
controlled in some way. It appears straightforward to define a mass constraint as de-
scribed in [69, 71, 82, 86, 94, 106, 138, 146, 177, 180, 189, 232, 301, 309,311, 312]. Other au-
thors substituted the mass constraint by a volume constraint [160, 161, 330], which is fully
equivalent. Instead of minimizing an acoustic property and introducing a constraint for
the mass, several authors minimized the mass and set a constraint for the acoustic prop-
erty [68,82,86,146,189,190,325]. Few of them compared both variants, cf. Lamancusa [189],
Crane et al. [82], Cunefare et al. [86] and Hibinger [146]. Lamancusa [189,190] proposed an
alternative to the traditional inequality constraint for the mass in thickness optimization.
Based on sensitivity analysis, the optimizer should find two regions, one of positive and the
other one of negative sensitivity with respect to two different parameters. The additional
mass in one region should be taken from the other part of the structure. The author found
that optimization with an inequality mass constraint converged much slower than one with
the equality constraint, or did not yield such an improvement of the acoustic property at
all.
Other restrictions necessitate full analysis of the structure or even of the structural–
acoustic system. La Civita and Sestieri [184] required maximum static stiffness for non–
Developments in Structural–Acoustic Optimization for Passive Noise Control 323
linear mount characteristics and maximum displacement of mounts for certain driving ma-
neuvers.
Roozen–Kron [267] and van Houten [317] used constraints as an additional controller
for parts of the objective function, equation (29). So, they defined that the frequency could
vary by about ±1%. Moreover, ±20% were allowed for the damping value and ±5% for the
lip radius.
Mass minimization by Sobieszczanski–Sobieski et al. [297] considered numerous con-
straints. These constraints included static stiffness, the roof crush as a particular crash
case, the lowest torsional mode as the particular noise, vibration and harshness measure
and others. In that case, the mass was minimized.
Inverse optimization, cf. Pal et al. [181, 252, 253], that tries to minimize modification
of variables, requires introduction of noise constraints. For certain parameter choices, e.g.
shell thickness, an additional mass condition might be useful.
Although seldom applied, constraints may be omitted by introduction of penalty func-
tions. Only few applications of penalty functions were found. Acoustic properties as the
constraints in a mass–and–cost minimization were transformed into penalty functions to
apply an unconstrained optimization algorithm, cf. Hambric [139]. Tinnsten [311] had
to omit constraints to use a stochastic optimizer. Fritze et al. [112] introduced penalty
functions to stabilize a commercial finite–element code (and its optimizer) that could not
handle warped shell elements or bead functions which left the feasible modification domain.
5 STRATEGY OF OPTIMIZATION
5.1 Formulation of Optimization Problem
In general, the optimization problem is formulated to minimize an objective function (or
cost function) as, cf. Lamancusa [189]
F(ϑ)−→ min (62)
while the design variables ϑiremain in a prescribed interval of lower and upper bounds
ϑil≤ϑi≤ϑiu.(63)
Many applications require the fulfilling of certain constraints as discussed in the previous
section. These include equality conditions as
Gj(ϑ) = 0 (64)
and inequalities as
Hk(ϑ)≥0.(65)
Examples for objective functions and for restrictions have been discussed in previous sec-
tions. Now, we are focusing on optimization strategies.
Different objectives, different applications and different variables necessitate different
techniques to minimize the objective function as autonomously as possible. Acoustic prop-
erties covering a large frequency range will likely construct an objective function with
numerous minima and maxima. Dependence of overall mass in terms of discrete masses or
in terms of shell thickness is linear. However, nonlinear constraints must be considered in
that case.
Nonlinearity and complexity of problems in structural–acoustic optimization rarely al-
low analytic solution. Usually, they demand numeric treatment. When dealing with real-
istic applications in passive noise control, the engineer will hardly be able to decide that
a certain solution is really a global optimum. Mostly, the engineer is more interested in
substantially decreasing the objective function by controlling parameters, than in finding
the global optimum after a long and exhausting search.
324 Steffen Marburg
5.2 Multiobjective Optimization
In many technical applications, it happens that more than one objective function should
be minimized. There are several ways to handle this problem of multicriteria optimization.
The mathematical formulation assumes that a vector of objectives Fmis to be minimized.
Usually, there is no general set of design variables that minimizes each single objective
globally. Since the improvement of one objective might result in worsening of another
objective one has to find a trade–off between these different criteria. Some more details
and problems on multiobjective optimization were discussed in Eschenauer et al. [95] and
in numerous recent papers, cf. [16, 45, 167, 194].
One of the possibilities for solving a multiobjective optimization problem is based on
defining a single (major) objective and formulating constraints on the remaining objec-
tives. Examples of this practice, such as minimization of mass with constraints on acoustic
properties, have already been discussed in the previous section.
Other methods require a weighting of different objectives. Assuming nobjectives, we
introduce weighting factors wmwith
n
m=1
wm= 1 (66)
The most commonly applied methods for actual multiobjective optimization are the wei-
ghted sum method and the goal attainment method. A good survey and comparison of both
methods and their application to structural–acoustic problems was given by Kessels [170].
They are briefly sketched in what follows.
The weighted sum method is likely to be the straightforward approach for multiobjec-
tive optimization. Formulations of this method were presented by Lamancusa [189], by
Hambric [139], by Crane et al. [82], by Cunefare et al. [86], and by Kessels [170]. We as-
sume that ndifferent objectives Fmshall be minimized. Every objective is equipped with
a weighting factor wm. A superobjective Fis then formulated by
F(ϑ)=
n
m=1
wmFm(ϑ) (67)
The optimization method is applied to the superobjective function F.Theweightedsum
method was tested in papers by Crane et al. [82] and Cunefare et al. [86]. These authors tried
to minimize sound pressure and mass simultaneously. They reported much worse results
than those yielded by the above outlined constraint optimization with a single (major)
objective function. The authors emphasized that the weighted sum method is not suited
for multiobjective optimization if conflicting objectives are to be minimized simultaneously.
Kessels [170] and Statnikov and Matusov [302] explained the reasons why the weighted sum
method failed in these cases.
An alternative choice is the goal attainment method. This method was used by la Civita
and Sestieri [184]. In that paper, however, the goal attainment method was applied in the
sense of a general optimization scheme. Therefore, the only reported application in the
sense of multicriteria optimization is that by Kessels [170]. For this method, we start with
defining a goal F∗
mfor each objective. It is useful to choose values that cannot be reached.
It is tried to approach this goal as close as possible but keeping the design feasible.
With fixed F∗
m, the actual single objective function Fmis given by the linear function
Fm=F∗
m+αmγ. (68)
Developments in Structural–Acoustic Optimization for Passive Noise Control 325
The values of αmrepresent coefficients related to the weighting factors as
αm=F∗
m(1 −wm) (69)
whereas γis calculated as
γ=max
mFm−F∗
m
αm.(70)
The original multiobjective optimization problem is transformed to a standard problem
with the objective function γ. Minimization of γwithin feasible design space performed
excellently for simultaneous minimization of acoustic properties and mass, cf. Kessels [170].
5.3 Approximation Concepts and Approximate Optimization
Many optimization methods require particular types of functions which are usually simpler
than the actual objective function. If both the objective function and all constraints depend
linearly (linear programming problem) or quadratically (quadratic programming problem)
upon the design variables, the optimization problem can be solved by standard algorithms,
see for example Haftka and G¨urdal [129]. However, this is usually not the case. Therefore,
the objective function may be approximated to create a linear or higher–order subproblem
that can be dealt with by using standard techniques. Different approximation schemes are
categorized in terms of validity in the design space [21,317]. These categories comprehend
local and global approximation, whereas the gap between both is filled by mid–range ap-
proximation. Local approximation assumes that the approximation is only valid in the
vicinity of one point in the design space. A global approximation is valid in the entire
design space or, at least, in a large region of it. Mid–range approximation tries to utilize
strengths of local and global approximation. Usually, it covers a larger range of the design
space than local methods. More information on structural optimization with emphasis on
approximation concepts is found in the paper by Vanderplaats et al. [320].
Local function approximation techniques are often based on the function itself and its
first derivatives at a single design point being ϑ0. The simplest approximation is the Taylor
series that approximates the objective function F(ϑ) (and usually all constraints) in the
vicinity of ϑ0as
F(ϑ)=F(ϑ0)+
n
i=1
(ϑi−ϑ0i)∂F(ϑ0)
∂ϑi
.(71)
As this approximation is inaccurate for many applications, modified first–order and second–
order approximations may be used. Second–order Taylor approximation formulates as
F(ϑ)=F(ϑ0)+
n
i=1
(ϑi−ϑ0i)∂F(ϑ0)
∂ϑi
+1
2
n
i=1
n
j=1
(ϑi−ϑ0i)(ϑj−ϑ0j)∂2F(ϑ0)
∂ϑi∂ϑj
.(72)
Linear approximation is commonly used in connection with sequential linear programming,
whereas quadratic approximation results in sequential quadratic programming techniques.
Suitable approximation schemes should be substituted for the computation of second or-
der sensitivities (Hessian) in equation (72), see for example Powell [260] and Jawson and
Morris [156, 157]. Other authors suggest using of half–quadratic approximation schemes,
cf. [233], considering diagonal terms of the Hessian only. An application of half–quadratic
approximation in structural acoustics was reported by Hambric [139].
Sequential linear programming was used for minimization of the objective function
in [68, 69, 71, 146, 240, 267, 325]. Roozen–Kroon [267] applied both sequential linear and se-
quential quadratic programming for optimization in structural acoustics. In that case, these
326 Steffen Marburg
techniques performed well for the analytically known global approximation that will be dis-
cussed later. Sequential quadratic programming was also used by Sobieszczanski–Sobieski
et al. [297] and by Lamacusa and Eschenauer [190]. The latter utilized an optimization
code that was also used by Wodtke and Lamancusa [330]. Both methods are connected
with a generalized reduced–gradient line search. The flexible tolerance method that was
accomplished by Makris et al. [205] fits also into this category.
Choosing the reasonable step size amounts to one of the most crucial questions for
optimization methods. Sequential linear and sequential quadratic programming require
limits to the allowable changes of the design variables, so–called move limits. These move
limits should be chosen carefully. If chosen too small, the optimization process converges
very slowly; if chosen too large, the approximation error rises and is no more acceptable.
This will likely lead to wrong results or, at least, slow convergence. A method that works
without move limits and that performed very well in comparison with sequential quadratic
programming was proposed by Snyman and Stander [296]. Its application did not require
any explicit evaluation and storage of the Hessian matrix. The authors showed several
examples indicating better convergence than sequential quadratic programming. However,
no indication could be found that this method has been tested in the context of structural–
acoustic optimization up to now.
An improved linear approximation was proposed by Svanberg [305]. This method was
called the method of moving asymptotes. It was used for structural–acoustic purposes
by Tinnsten et al. [309, 312], see also Tinnsten et al. [311] and Esping [97]. The method
of moving asymptotes performs approximation of the objective function in the vicinity of
ϑ0=(ϑ01,ϑ
02,...,ϑ
0n)Tby
F(ϑ)=r(ϑ0)+
n
i=1 pi
Ui−ϑi
+qi
ϑi−Li(73)
where piand qiare calculated as
pi=
(Ui−ϑ0i)2∂F(ϑ0)
∂ϑi
for ∂F(ϑ0)
∂ϑi
>0
0for
∂F(ϑ0)
∂ϑi
≤0
(74)
and
qi=
0for
∂F(ϑ0)
∂ϑi
≥0
−(ϑ0i−Lj)2∂F(ϑ0)
∂ϑi
for ∂F(ϑ0)
∂ϑi
<0.
(75)
The value of r(ϑ0) is computed as
r(ϑ0)=F(ϑ0)−
n
i=1 pi
Ui−ϑ0i
+qi
ϑ0i−Li.(76)
The optimization subproblem is then solved by a suitable method, such as the gradient
or the conjugate gradient method, cf. Svanberg [305]. A recent review of the method of
moving asymptotes was given by Bruyneel et al. [48].
Developments in Structural–Acoustic Optimization for Passive Noise Control 327
Where the method of moving asymptotes had been applied, the users reported excellent
performance for different examples, cf. Esping [97, 312]. As the approximation concept is
local, optimization may be easily trapped in a local minimum, as could be seen in [311].
Another popular gradient–based method is called the method of feasible directions. This
method was accommodated in the computer program Conmin and described by Vander-
plaats [319]. A method description is also found in the book by Haftka and G¨urdal [129].
Basically, the method consists of a line–search algorithm where the updated vector of design
variables ϑk+1 is evaluated as
ϑk+1 =ϑk+αd.(77)
In this equation, αand drepresent step size and gradient, respectively. Once the gradient is
known, the new parameter set is sought in the direction of the steepest descent by variation
of step size α. The method was tested in numerous applications in structural acoustics,
see for example Lamancusa et al. [189,190], Cunefare and Engelstadt et al. [82, 86,94], and
Koopmann et al. [180, 301]. Comparison with other methods was reported by Lamancusa
and Eschenauer [190].
A method that works similarly to the method of feasible directions is the first–order
optimization method of the commercial computer code Ansys [306]. This optimizer was
used in the author’s papers [112,214–218]. Convergence and handling of this tool requires
further improvement, since it converged very slowly and, in some cases, did not perform
stably.
Linear local approximation of the objective function and one constraint is required
by the modified optimality criterion method that was proposed by Ma et al. [202] and
successfully applied by Jog [160, 161].
There are some papers in structural–acoustic optimization where an explicit description
of the approximation concept or the optimization method was not found. Though not
clarified in detail it is assumed that linear local approximation and a first–order optimization
method was used in references [28,145,181,240,252,253].
Global approximation accounts for the complementary approximation concept. For this,
the approximation should be valid for the entire design space, or at least for large regions
of it. Similar to local approximation, the idea is to use a comparatively simple function, for
example a polynomial, to substitute the complicated and usually implicit objective function.
In some methods, global approximation concepts require other optimization methods.
The review paper of Barthelemy and Haftka [21] started with the response surface
technique for global approximation. This is likely the most common and straightforward
approach. It uses function values of different design sets and creates a globally valid poly-
nomial. According to Barthelemy and Haftka [21], there are only a limited number of
applications of response surface models in the literature, and applications are small in size.
Global schemes can work very efficiently for these problems, as presented in [282]. They
are also applied to create substitute models, cf. Papila and Haftka [256], and to handle
discrete variables, cf. Hajela [136].
For their application in structural acoustics, Milsted et al. [232] and Hall [138] used a
polynomial of order mthat can be written as
F(ϑ)=α0+
n
i=1
αiϑi+
n−1
i=1
n
j=i+1
αij ϑiϑi+
n−2
i=1
n−1
j=i+1
n
k=j+1
αijk ϑiϑjϑk+
(78)
+...+
1
i=1
2
j=2
3
k=3
...
n
m=n
αijk...m ϑiϑjϑk...ϑ
m.
328 Steffen Marburg
The coefficients αcan be computed after 2nevaluations of the objective function provided
that m=n. These authors minimized a multimodal objective function. It is hard to
imagine that the response surface represented the behavior of the objective function well,
but the final result was convincing.
Another technique for global approximation by a response surface is known as design
of experiments. There are a number of specifications of this method. It was well discussed
by Montgomery [234] and van Houten [317]. Schoofs et al. [284] applied design of exper-
iments to eigenfrequency optimization of carillon bells. This strategy was motivated by
the observation that, essentially, eigenfrequencies depend linearly on the shell thickness.
Roozen–Kroon [267] included damping considerations but approximation by the global
scheme was rather poor.
Another category of global schemes consists in the approximation by the concept of
neural networks. As the neural network attempts to imitate a neurobiological process that
processes input and generates output, it can be trained by pairs of input and output data.
During this training period, weight factors are adjusted in the connections between certain
nodes of the network so that the generated output matches expected output data. Once
trained, the network is used as a simple function that substitutes for an original one. A
technique for general subsequent usage of neural networks was proposed by Arslan and
Hajela [8].
Few applications of neural networks as approximation scheme are known for structural
acoustics. Nagaya and Li [239] applied a three–layered neural network system for optimiza-
tion of fifteen variables. According to the authors’ remark, the convergence could be very
rapid, although the matter was not discussed in much detail. Another example of neural
networks utilized holographic neural network, cf. Shi et al. [291, 292]. The method required
very few evaluations of the objective function but only one variable was optimized.
The above mentioned examples of neural networks used in structural–acoustic optimiza-
tion are hardly suited for any clear estimation of performance or for deciding whether they
are appropriate for applications in structural–acoustic optimization.
The third global approximation scheme is very simple: If only one or two variables are
in use for optimization, the graph can be viewed. This technique was used by Soize and
Michelucci [299]. The global minimum was easily found since they only considered one
variable.
In the paper of Marburg et al. [213], it was originally expected to optimize eight vari-
ables. Four could be easily fixed. The remaining four were optimized by viewing the
objective function for two of them, and then for the other two. This scan – actually per-
formed as a random search – of the objective function became necessary since reinforcing
beams could not be continuously moved on the finite–element mesh. So their position was
constrained discretely along the nodes. Essentially, the solution appeared reasonable. In
another paper by Marburg and Hardtke [214], graphs were used to find initial values for
optimization. A coarse discretization was used for this purpose.
The gap between local and global approximation is filled with mid–range methods. This
category assembles all methods that utilize more than only one location for formulation of
an optimization subproblem but do not attempt to approximate the whole problem at once.
They are still sequentially working methods. According to Kessels [170], mid–range methods
possess most of the advantages of local and global approximation schemes. However, highly
sophisticated move–limit strategies are required for their successful application. As many
decisions must be made, many parameters contribute to the performance of mid–range
methods. In mid–range methods one can categorize single–point paths, multi–point paths
and local–global approximations. An example for the latter scheme was given by Free et
al. [108] who applied design of experiments as local approximations in the vicinity of a
current design point. They solved the approximate optimization problem and modified
Developments in Structural–Acoustic Optimization for Passive Noise Control 329
the valid range of the approximation by contracting or expanding and shifting the current
reference point as appropriate.
Most singlepoint path methods use function values and their sensitivity information.
Essential contributions came from Haftka et al. [132] and from Fadel et al. [101]. These
methods utilize point information and take optimization history into account as well. This
concept, however, leads to the problem that solution of the optimization subproblem may
become complicated.
Toropov [313] proposed a multipoint approximation by response surface function of
successively contracting size. Development of this strategy was continued by Toropov et
al. [314]. This resulted in a multipoint path method for mid–range optimization with an
appropriate move–limit strategy. This move–limit strategy was improved and described
by Etman [98] and van Houten [317]. In a recent paper on multipoint approximation,
Abspoel et al. [1] discussed applicability of mid–range methods for discrete design variables
(or objectives) and objective functions or constraints being stochastic functions.
Mid–range optimization was reported for structural–acoustic optimization by van Hou-
ten [317] and Kessels [170]. Performance of mid–range methods appeared convincing re-
gardless of whether bells were designed or magnetic resonance imaging scanners were op-
timized over a large frequency range. These methods are a reasonable alternative to local
approximation schemes that are widely used in structural–acoustic optimization.
5.4 Optimization Methods
Having discussed approximation concepts, we will briefly survey optimization methods in
general now. In most cases, an adequate choice of the optimization method appears crucial
for successful minimization of the objective function. Seldom do arbitrarily chosen methods
perform efficiently. Such an example was published by Mechel [227], who utilized the find
minimum function of a commercially distributed mathematical software system. Though
appearing trivial, simple evaluation of the objective function and its (likely) smooth be-
havior helped to guarantee successful optimization and reasonable performance. Since this
is usually not the case, efficient optimization methods account for one major basis in the
entire process.
Subsequent solution of local approximate optimization problems appears as the most
popular method for applications in structural acoustics. This has already been mentioned
in the previous subsection. Again, it is mentioned that sequential linear programming
and sequential quadratic programming, the method of moving asymptotes, the method
of feasible directions and other gradient–based methods were successfully applied in the
field. This is surprising insofar as, in most cases, multimodal objective functions containing
numerous local minima and maxima were considered. It is well–known that local approx-
imation schemes tend to be guided into the nearest local minimum. Tinnsten et al. [311]
published an interesting example comparing results of local approximate optimization us-
ing the method of moving asymptotes and the global stochastic method called “simulated
annealing”. Similarly, subsequent solution of relatively simple problems is employed by
mid–range approximate optimization methods.
Direct optimization algorithms were briefly reviewed and clearly discussed by Hajela [136].
Essentially based on that paper, direct search methods will be categorized as zero–order
local search methods, global search methods, heuristic methods, and global function ap-
proximation. The latter was discussed previously in this paper and will not be considered
again in this part.
Usually, it is hardly possible to decide if the solution is actually the global optimum.
The only method generally supplying the global optimum (if there is any) is known as
exhaustive search or exhaustive enumeration. This method, however, is limited to small–
scale problems with very few variables, because the number of required evaluations of the
330 Steffen Marburg
objective function grows exponentially with the number of variables. It was discussed
previously in the context of global approximation that certain problems can be solved by
viewing the graph of the objective function. This technique may also be understood as
exhaustive search. It was applied to optimization problems in structural acoustics by Soize
and Michelucci [299] and, combined with random iteration, by Marburg et al. [213]. If
applicable, exhaustive search of simplified models may be used to find an initial parameter
set, cf. Marburg and Hardtke [214].
A number of deterministic zero–order local search methods were developed in the early
sixties. To name three of them, we mention the Rosenbrock method [268], the method of
Box [44], and the pattern search method by Hooke and Jeeves [148]. The latter was used
for structural–acoustic optimization by Lang and Dym [191] and, basically, performed well
in that example. This algorithm is easy to survey. Starting at an initial set of variables, the
first variable is varied until a minimum is found in the direction of this variable. This value
is used as starting point for the next search varying the second parameter. Subsequent
search in the direction of all variables provides us with a new parameter set. Then, the
minimum along the line between the initial parameter set and the new parameter set is
determined. The resulting point accounts for the new reference value. It was mentioned
in [191] that the method is less flexible than the method of Box [44] but better suited to
handle bounds of variables.
Hajela [136] gave a brief survey of a random local search method, i.e. random walk.
The method is easy to implement but likely to perform inefficiently if evaluation of the
objective function is expensive. The paper further surveyed deterministic global search
methods, which comprehend subsequently performed searches as known from mid–range
methods, interval methods, and clustering methods [136]. The latter may be of interest in
connection with subsequent local search methods as it attempts to avoid identifying the
same local minimum repeatedly.
One of the most popular optimization methods is known as simulated annealing. This
method belongs to the category of stochastic global search methods. It is equivalent to
the annealing process, cf. Metropolis et al. [230] and Kirkpatrick et al. [174]. Starting
from a user–defined parameter set ϑ0the algorithm moves to a random direction. The new
parameter set ϑ1is accepted as a new reference set if it decreases the objective function.
However, it may also be accepted if the move was in the uphill direction. Then, the new
design has a certain probability of being accepted. This probability is formulated as
P=e−F(ϑ1)−F(ϑ0)
T.(79)
According to their work [230], this rule is called Metropolis criterion [76,123]. Trepresents
the temperature. It starts at a high level and decreases during the optimization process. As
the method also accepts uphill moves, it seems well suited for problems with multiple max-
ima and minima that are expected for multimodal objective functions in structural–acoustic
optimization. Bilbro and Snyder [40] claimed that simulated annealing was developed and
performed best for discrete optimization. They proposed an alternative for continuous
variables and called this method tree annealing.
Simulated annealing was successfully used for optimization in structural acoustics by
Constans et al. [74, 75] and by Tinnsten et al. [311]. Constans et al. [74, 75] optimized
positions of discrete masses on a half–cylinder. Since this was carried out with a discretized
cylinder, the positions of masses were constrained to discrete positions of nodes. Tinnsten
et al. [311] compared results of simulated annealing and those yielded by the method of
moving asymptotes. Hambric [139] favored simulated annealing for solution of the local
approximation problem.
Developments in Structural–Acoustic Optimization for Passive Noise Control 331
Genetic algorithms as global search methods received much attention over the last three
decades. There is still a large community developing these algorithms, cf. [22, 59, 137, 192,
331,336,342]. The method possesses a combinatorial and stochastic basis. It works similar
to development of species in nature. To employ the method, we define a population of
individuals. Each individual possesses a genetic code (parameter set) that gives a certain
fitness value (the objective function). Different types of operators are applied to that basis
population. Based on the fitness value, a selection operator chooses a pair of parents
providing a descendant. A crossover operator arbitrarily creates the genetic code of the
descendant, and a mutation operator forces random modifications to avoid remaining with
the initial genetic permutations. Apparently, a large number of variations of this method are
possible. Bessaou and Siarry [22] compared performance of a derived genetic algorithm with
that of several other methods including simulated annealing. Their focus was directed to
different multimodal functions. Several enhancements of genetic algorithms were discussed
in Hajela’s paper [136]. They included constraint handling, see also [137], applications
in multiobjective optimization, see also [135, 192, 341, 342], parallel implementations, see
also [192], combination with other methods e.g. with Hooke–Jeeves pattern search [336],
and immune network modeling, cf. [341,342].
Most likely, genetic algorithms are ideally suited for optimization of multimodal func-
tions as occurring in structural acoustics, especially if the objective function consists of an
average over a large frequency region. The particular case of multimodal functions was
addressed and tested by Siarry et al. [22, 59]. These developments have not been tested
for structural–acoustic applications yet. La Civita and Sestieri [184] used a commercially
available code for their optimization application. As their model allowed an analytic rep-
resentation of the objective function, a large number of function evaluations would have
hardly been noticed. Fisher [106] tested performance of genetic algorithms and varied pa-
rameters of the optimization strategy, e.g. size of population. A maximum number of 100
evaluations of the objective function was prescribed. Ratle and Berry [263] tested their
algorithms in several examples positioning discrete masses on a plate. They estimated the
number of almost 1016 discrete cases to distribute five point masses on a mesh of 64 ×64
nodes. To do that, they chose a population of 100 individuals and 200 generations. Find-
ing the optimum required about 2 ·105function evaluations. Even with respect to the
impressive results, the method seems to perform inefficiently for large–scale problems. The
above mentioned genetic algorithms [22,59] may be suitable for optimization in structural
acoustics. Further testing of suitable genetic algorithms remains one of the key problems
for efficient structural–acoustic optimization.
As another stochastic global search method that looks suitable for application in the
focus of this paper we mention evolutionary strategies. Recent papers in the field of evo-
lutionary algorithms have been published by Back [18], Tan et al. [308], and Kincaid et
al. [173]. Among other items, the latter paper considers Levy’s function as a multimodal
test function. According to the experience of users, performance of evolutionary strategies
is competitive for our applications especially if it is modified to escape local minima [173].
While the evolution–based methods given above involve numerical strategies, the term
“evolutionary optimization” can also be related to a physical strategy, as it assumes to
remove material where it is not required. In the field of simultaneous shape and topology
optimization Xie and Steven [334] considered the field of structural dynamics while Zhao et
al. [343] maximized the difference between two eigenfrequencies. This strategy was called
evolutionary structural optimization, cf. [335]. Based on the technique that is utilized,
Rozvany [269] suggested using the name of sequential element rejection and admission.
Finally, Hajela [136] reviewed heuristic methods including tabu (or: taboo) search,
design classifier systems, and a hybrid method that combines an expert system and numer-
ical optimization. Similar to the above considered evolutionary strategies, an application
332 Steffen Marburg
of these methods in structural acoustics is not yet known. Tabu search was applied to
multimodal test functions [25, 60, 91, 338].
Doubtlessly, this collection of optimization algorithms cannot be expected to be a com-
plete one. However, this selection is regarded as a choice of some methods having been used
or may prove to possess the potential to perform well in the context of structural acoustics.
In some cases combination of two methods was reported. Combination of two methods
became necessary if a sequentially applied, locally based approximate optimization trapped
into a local minimum. A random search in the vicinity of the current design point provided
an escape from the trap, cf. Marburg and Hardtke [216] and Fritze et al. [112]. Authors of
the latter paper suggested controlling parameters of the objective function, equation (24)–
(26) in order to make progress when stagnation occurs.
Generally, it appears useful to test different optimization strategies. Normally, choice
of an appropriate strategy requires careful observation of its performance. Different models
may be solved best by using different methods. Obviously, a problem with few optimization
variables cannot necessarily be optimized using the same strategy as for a problem with
many variables even if identical simulation models and objective functions are considered.
As known from many other applications in non–linear mechanics, the experienced user will
usually choose a more efficient optimization strategy than the beginner.
6 SURVEY OF RESULTS
6.1 Categories and General Results
Review of results in structural–acoustic optimization clarifies that successful applications
require a profound understanding of the underlying physics, of the numerical analysis for
evaluation of the objective function, and of the optimization algorithm. It is rarely possible
to consider the entire problem as a black box to decrease only one single value. This is
only possible if the analysis is extremely fast, cf. [184, 227]. Additionally, it is necessary
to distinguish between different categories since we can expect successful optimization in
some of them and less successful ones in others. Furthermore, small gains of the objective
functions can be appreciated in some categories much more than large improvements in
other cases. These categories encompass examples of control of a noise transfer function
only, and exclude certain applications, e.g. bell design [267,317].
Applications can be categorized with respect to frequency ranges. We distinguish four
categories. There are a number of papers on optimization of acoustic measures at a single
frequency [69, 71, 82, 86, 94, 160, 161, 309, 311, 312] or at few discrete frequencies [69, 160,
161, 181, 252, 253]. Narrow frequency bands (the second group) were considered in the
papers [68, 133, 145, 263, 325]. Numerous papers are known where authors investigated
problems of wide frequency ranges usually with many modes involved, at least in the initial
design, cf. [138–140, 146, 184, 189, 190, 213–216, 218, 232, 239, 299, 330]. As an example,
Wodtke and Lamancusa [330] stated that optimization gains are higher the fewer modes
are involved. The fourth category collects contributions where authors optimized sound
radiation of certain mode shapes [28, 74, 180, 240, 301, 312]. The latter category can be
understood as an extra one since optimization of mode shapes may affect a large frequency
range that is influenced by this particular eigenvector. In this regard, Naghshineh et al. [240]
coined the term “weak radiator”, a structure that is optimized with respect to a certain
mode shape. Radiation is particularly low for these mode shapes. A weak radiator for
a certain mode shape can, however, perform as a poor radiator for another frequency or
mode shape. Obviously, it is more difficult to control a noise transfer function over a large
frequency range. Basically, the problem of noise radiation may be shifted off the frequency
range under consideration. The strategy may be reasonable if the excitation is really limited
to such a band. However, wide frequency ranges are excited in many technical applications,
Developments in Structural–Acoustic Optimization for Passive Noise Control 333
e.g. combustion engines and aircraft noise. For these problems, it is better to cover the
range from infranoise to a certain upper limit. Often, higher frequencies are better damped
than lower ones. The argumentation, though often being correct, is convenient but should
be carefully examined for each particular application.
Focus of the second collection of categories is directed to the complexity of the simulation
system. The first category covers simple simulation models including analytic models,
beams, plates and simply formed shell structures [28, 71, 74, 75, 112, 160, 161, 180, 184, 189,
190,239,240,263,301,309,311,330]. Boxes, freely formed shells, simply formed but reinforced
plates and shell structures are assembled in the second group [69, 82, 86, 94, 133, 139, 140,
146, 181, 218, 252, 253, 299, 325]. This category comprises simulation systems of medium
complexity. Highly complex simulation models, e.g. large–scale and/or realistic models of
vehicle engines [138, 232], sedan body structures [68,145,214–216,297], magnetic resonance
imaging scanners [170], and a reinforced box structure [213] account for the third category.
Often, it is easier to optimize simple structures. More complex structures are usually
influenced by a large number of modes that produce multiple local minima and maxima. In
these cases, optimization convergence slows down and an appropriate strategy is required.
Noise transfer functions as the basis for the objective function will mostly act globally,
whereas most design variables in complex structures have local effects only. This is another
reason why it is more difficult to gain large improvement of the objective function for
complex structures by locally acting variables.
We categorize members of the third group according to support conditions. We distin-
guish based on eigenfrequencies of rigid body modes. Most of the examples in the literature
clearly focused on problems with fixed support conditions, i.e. there are no rigid body modes
at all [28, 69, 71, 74, 75, 82, 86, 94, 146, 160, 161, 180, 189, 190, 214, 218, 239, 240, 263, 301, 312].
Then, there are applications involving elastic supports. In these examples, rigid body modes
are coupled with elastic modes. This happens for actual elastic supports [330] and for sub-
structures embedded in larger structures [215, 216]. Few papers are known with free–free
boundary conditions, i.e. rigid body modes occur at frequency of 0 Hz or, at least, at much
lower frequencies than elastic modes [69,309, 311,330]. There is a fourth category of (usually
more complex) examples that do not fit into the first three groups. The freely supported
box structure in [213] fits rather into the first category than into the third. This is due
to the stiff edges of a box. Panels of the box behave similar to clamped plates or shallow
shells. Wodtke and Lamancusa [330] and Tinnsten et al. [309, 311] showed comparisons
between clamped or simply supported plates and free plates. It could be clarified in these
examples that passive noise control by structural optimization is much more challenging if
rigid body modes of the noise–emitting panels occur at 0 Hz or, at least, in the very low
frequency range. These properties were not observed in the case of the free box [213] since
rigid body modes radiated noise neither into the cavity (as considered there) nor to the ex-
terior. Wodtke and Lamancusa [330] lowered the radiated A–weighted average sound power
by 13.3 dB for the clamped plate by using damping material and only 0.4 dB were gained
for the free plate that was excited identically. Tinnsten et al. [309, 311] gained 18.7 dB for
the intensity calculated at one or two points above a clamped circular plate, whereas this
intensity could be decreased by 4.0 dB only for the free plate. Soft support or virtually free
boundary conditions have been utilized to achieve better radiation at low frequencies for a
loudspeaker diaphragm [69]. This corresponds to the discussion above.
Some papers clearly indicate that the position of excitation is particularly sensitive with
respect to design modifications in order to yield a quiet structure. Stiffening in the vicinity
of excitation as a general concept for quiet designs resulted from optimization in the work
of Belegundu et al. [28], Hibinger [146], Tinnsten et al. [311] and Fritze et al. [112]. The
situation appears less clear if discrete masses are added and optimized. It was mentioned
that the intuition of an engineer would favor this added mass close to the excitation, cf. [263],
334 Steffen Marburg
but for minimum radiated sound power, optimization supplied different solutions as shown
in papers by Constans et al. [74,75] and by Ratle and Berry [263]. Similarly, positions and
sizes of three absorbers, i.e. spring–mass dashpot systems, on a plate were optimized [239].
This resulted in a distribution of these absorbers that is not easily understood at first
glance. Christensen and Olhoff [69] showed that the target of a uniform directivity pattern
of radiation favors a single mass, i.e. a ring mass in the case of an axisymmetric plate.
There are numerous papers considering just symmetric halves or even only quarters of
structures. Usually, these applications cover flat rectangular plates [160, 161, 180, 189, 190,
301] but some vehicle panel shell structures have also been investigated [215, 218]. It is not
reported how good these optimization results were in comparison to consideration of the
whole structure. However, other authors found that solutions were asymmetric even for a
priori symmetric systems, cf. Ratle and Berry [263] and Constans et al. [74,75].
Several authors stated that optimization results are specifically valid for the excitation
case that is used in the analysis, cf. [170,330]. This disadvantage may be overcome in the
case of low modal density. As mentioned earlier in this section, creation of a so–called
weak radiator [180, 240] leads to a structure of minimum noise–emitting mode shape(s). It
is likely, that some (few) mode shapes can be tuned to perform as a weak radiator. The
concept will likely fail if the modal density increases.
In the absence of rigid body modes in the frequency range of investigation, it can be
useful to drastically increase the lowest eigenfrequency. This technique was successfully
applied in the optimization of sedan body structures [297] and panels of sedan body struc-
tures, cf. Esping [97]. Doing so, Marburg et al. [218, 219] minimized the average sound
pressure level inside a sedan cavity (driver’s ear position) due to harmonic pressure acting
on the roof from above. Provided there was enough design freedom, the optimizer favored
a structure of substantially increased first eigenfrequency. Furthermore, the lowest two
mode shapes occurred virtually at the same frequency, but they did not cause any peak
in the noise transfer function. It is likely that these modes neutralized each other. The
new roof was 30 cm higher than the initial one. The lowest eigenfrequency was increased
from 88 Hz to 209 Hz, the number of modes up to 250 Hz decreased from 68 to 2. The
relatively flat roof was designed more spherically as shown in Figure 1. Experiences from
the roof investigation could be utilized in the optimization of a sedan hat–shelf, cf. Mar-
burg and Hardtke [214]. For this panel, the lowest eigenfrequency was maximized at first.
In a second step, three cases of excitation, two frequency ranges, two cases of boundary
admittance distribution, and two variants of the objective function, equations (23)–(25),
were tested. The newly designed panel proved to possess better acoustic properties, i.e.
depending on the particular case (excitation, objective function, acoustic damping), the
objective function was lowered between 4.4 and 13.9 dB.
The strategy of maximizing the lowest eigenfrequency may fail if radiation to external
space is considered. Increasing radiation efficiency at higher frequencies is identified as the
reason for this failure. Application of this strategy appears reasonable if the excitation
spectrum decays for higher frequencies. For example, a four–stroke, four–piston engine
excites a structure mainly with the second engine order, i.e. 6000 rotations per minute
correspond to an excitation frequency of 200 Hz. Therefore, it can make sense to limit
optimization to the frequency range up to about 200 Hz.
As the final point in this subsection, we briefly discuss the problem of a robust optimum.
This issue has seldom been addressed. The engineer is usually interested in both an optimal
solution for the simulation model and a real construction. There is no reason to find an
optimal solution that loses optimality if certain margins of acceptable tolerance are reached.
The optimum must be robust with respect to small parameter changes. Sensitivity of the
optimal design was investigated and discussed in the paper of Christensen et al. [71]. Among
others, they compared their uniform directivity patterns yielded for a specified frequency
Developments in Structural–Acoustic Optimization for Passive Noise Control 335
Figure 1. Before (upper subfigure) and after (lower subfigure) optimization:
Sedan roof, simply supported in six points, optimized with respect
to sound pressure level at driver’s ear due to harmonic pressure from
above, horizontal view from the left
at the different levels of 64 and 80 dB. It became evident that the solution found for the
lower level behaved insensitively with respect to frequency modification compared to the
optimum for the higher level. For practical usage of structural optimization it appears
necessary to check robustness of the solution.
6.2 Results with Respect to Objectives and Optimization Algorithms
In this section, we will review experience and results for objective functions and for opti-
mization algorithms. In few papers, different choices of objective functions or optimization
methods have been compared. Most papers were focused on a particular problem like so-
lution of an engineering task, development of analysis for optimization, or application of
particular algorithms.
Suitability of different objectives was first tested by Lamancusa [189]. He discussed
numerous possible objective functions, e.g. frequency average of the radiated sound power
with and without mass constraint, frequency average of mean square velocity, frequency
average of radiation efficiency, modal radiated sound power, matching of a weak radiator
mode shape, maximized lowest eigenfrequency and mass. Clearly, combinations of these
objectives resulting in a multiobjective optimization problem could also occur. It was
reported that frequency average sound power and frequency average of mean square velocity
performed very well in the optimization process. If a mass constraint was required this
should be rather formulated as an equality constraint than as an inequality constraint.
Optimization of modal radiation efficiency achieved remarkable decreases of this measure,
but a decrease in radiated sound power was not observed. Radiation efficiency combined
with mean square velocity performed similarly to the mean square velocity optimization.
Modal power resulted in satisfying improvements but was difficult to control if modes were
moved outside the frequency range under consideration. The highly nonlinear constraint of
radiated power did not allow a mass optimization to produce any acceptable improvement.
It was suggested that other optimization methods than that of feasible directions would
give better solutions. Minimized and maximized lowest eigenfrequencies were tested and
compared. Noise transfer function of the latter case was more favorable than for the former.
It was also possible to match a weak radiator shape at a specified frequency. This, however,
did not result in a remarkably lower sound power even at that particular frequency.
As a consequence of these investigations, Wodtke and Lamancusa [330] compared opti-
336 Steffen Marburg
mization of radiated sound power and mean square velocity. Basically, they found for a cir-
cular plate that both cases performed similarly and equivalently if fixed support conditions
were applied, e.g. a clamped plate. For the free plate, decrease of the frequency–averaged
mean square velocity by 8.4 dB led to an increase of the radiated sound power by 4.8 dB.
Similarly but at lower level, optimization of radiated sound power obtained 0.4 dB gain
whereas the mean square velocity increased by 0.8 dB. The contradiction is not as strong
for the elastic supports but a clear tendency of substantial differences between both objec-
tives was observed. Usually, the optimizer utilized whatever was possible. Consequently,
the highest gain was always achieved for the objective function. The remaining measure, i.e.
mean square velocity or radiated sound power, respectively, was usually much less affected.
Ratle and Berry [263] compared optimization of mean square velocity and far–field sound
pressure at a specified point. The average over a frequency band accounted for the objective
functions, one of both was optimized whereas the other one was evaluated and compared
afterwards. Basically, the results were similar to those of Wodtke and Lamancusa [330] but
only a simply supported plate was investigated by Ratle and Berry [263]. They gained up to
70 dB for the frequency–averaged mean square velocity by shifting all eigenfrequencies off
a certain band containing the fourth and the fifth mode of the original plate. Optimization
favored shifting all modes off the specified frequency band. In this context, it should be
mentioned that a graph of the objective function with respect to two design variables was
shown. They clearly indicated that the objective function contained many minima and
maxima, a result of numerous eigenfrequencies in the frequency range under investigation.
Jog [160,161] substituted minimization of a newly defined dynamic compliance for sound
power minimization. He compared minimization of this compliance function with that of
sound power minimization [160]. In test cases, excellent correlation was observed between
both objectives. Test cases involved excitation at one or two discrete frequencies, and fixed
supports. It has not yet been clarified whether minimization of dynamic compliance is
equivalent to sound power minimization if more complex models and/or more compliant
support conditions apply.
Crane et al. [82] investigated practicability of different objectives for gradient–based
optimization. Excitation at one discrete frequency was considered. In a first test case, the
authors minimized the average sound pressure at up to 188 data recovery points distributed
inside a cylindrical cavity. The second test case involved an additional mass constraint. In
their third investigation, they turned the problem and minimized the mass with average
sound pressure as constraints, and they finally tested simultaneous minimization of average
sound pressure and mass. The latter formulation was based on weighted sum method.
All test cases guaranteed design variables in a specified interval. The authors attested
reliable and robust behavior if the average sound pressure had been minimized with and
without mass constraint. Mass minimization was the fastest strategy but suffered from some
numerical instabilities. Simultaneous minimization of sound pressure and mass turned out
to perform slowly and inconsistently. According to the authors, this behavior is due to
the fact that sound pressure and weight represent conflicting objectives. A nearly linear
dependence between noise level gain and optimal weight was found. Similar results on the
formulation of these objectives were reported by Cunefare et al. [86].
Excellent performance in simultaneous minimization of noise radiation and mass was
reported by Kessels [170] by using the goal attainment method. After few iteration steps,
the radiated sound power of a magnetic resonance imaging scanner was lowered by 10 to
15 dB, whereas the mass was decreased when that was given. It is likely that the excellent
performance of his optimization was due to using the goal attainment method.
The objective function given by equations (23)–(25) was applied in several papers of
Marburg and Hardtke [214–216] and, with sound pressure level substituted by sound power
level, in the paper of Fritze et al. [112]. Most applications performed better if the root
Developments in Structural–Acoustic Optimization for Passive Noise Control 337
mean square value was optimized instead of the mean value. Fritze et al. [112] used the
mean value but manually controlled the reference level to exclusively consider high level
peaks. The reference level was slowly decreased during the optimization process. The
difference between mean value and root mean square value becomes evident in the presence
of many resonance peaks [215]. If we consider relatively smooth noise transfer functions
with few eigenfrequencies in the frequency range under consideration, then differences in
the objective function are negligible, cf. [214].
Hambric [139] compared two approximation schemes, i.e. first–order Taylor series and
half–quadratic approximation, with two optimization formulations, i.e. the method of fea-
sible directions and the Broydon–Fletcher–Goldfarb–Shanno (BFGS) method. The approx-
imate problems were then exhaustively searched by utilizing a simulated annealing algo-
rithm. His objective function comprised structural mass, manufacturing costs and acoustic
properties. Similar to applications of Crane et al. [82] and Cunefare et al. [86], the method
of feasible directions performed inefficiently and unstably. However, Hambric [139] reported
good performance of first–order Taylor series approximation and even higher efficiency of
half–quadratic approximation.
Lamancusa and Eschenauer [190] compared performance of the method of feasible direc-
tions with sequential quadratic programming combined with a generalized reduced–gradient
line search. Using the latter method, the average radiated sound power was decreased by
13.2 dB whereas the method of feasible directions stopped after gaining 2.1 dB. Therefore
it is assumed that the method of feasible directions is less suited for multimodal objec-
tive functions such as applications in structural acoustics. Furthermore, Lamancusa and
Eschenauer [190] pointed out that sufficient design freedom can influence the optimization
results significantly. Fixing bounds as ratio between thickest and thinnest element at 4, a
gain of 3.4 dB was obtained. A ratio of 10 allowed 13.2 dB improvement.
As mentioned above, it is difficult to compare performance of approximation concepts
and optimization algorithms based on different examples. When looking at local approx-
imation schemes, sequential quadratic programming, cf. [190,330] and method of moving
asymptotes, cf. [309, 312], seem to perform very well. Simplicity and availability may be
arguments for other gradient–based methods but most comparisons with other methods
showed evidence of limitations. Van Houten [317] and Kessels [170] reported excellent con-
vergence of mid–range approximation and optimization methods. While optimization of
bells is likely a problem of a rather smooth objective function, cf. [317] and also Roozen–
Kroon [267], that could also be investigated by using response surface techniques [267,284],
a mid–range method seems much better suited for optimization of a multimodal objec-
tive function [170]. Examples of utilizing neural networks hardly allow any decision about
practicability of this approach [239, 291, 292].
Results of the method of moving asymptotes as a local–approximation–based optimiza-
tion method were compared with simulated annealing as a stochastic algorithm by Tinnsten
et al. [309,311]. It was clarified that the method of moving asymptotes returned a local min-
imum. Simulated annealing found another minimum gaining an additional improvement of
the objective function, i.e. the sound intensity at a specified point above the circular plate.
Simulated annealing was also used by Constans et al. [74, 75]. Their parameters, b eing
positions of additional discrete masses, had to be considered as discrete variables since the
positions were constrained to nodes of the finite–element mesh. The optimum was found
after 160 evaluations of the objective function. This corresponded to five percent of the
evaluations of the total design space of these four variables. Basically, finding the optimal
solution after 160 evaluations is not bad for four variables but searching five percent of
the total design space may become very expensive if more variables and more values per
variable are possible.
Three papers are known where authors utilized genetic algorithms [106,184,263]. How-
338 Steffen Marburg
ever, only the paper by Ratle and Berry [263] allows review of results with respect to
performance of the optimization method. In the case of optimization of five masses, a rea-
sonable optimum was found after 2×105evaluations of the objective function. A scan of the
total design space would have required virtually 1016 evaluations of the objective function.
Although many times faster than an exhaustive search, 2 ×105evaluations appear to be
an inefficient performance for realistic problems.
6.3 Results with Respect to Variables and Specified Problems
It could be seen in Section 4 that a large variety of design variables has been tested for
structural–acoustic optimization. These include piecewise polynomial, i.e. constant or lin-
ear, plate thickness and shell thickness, addition of discrete masses or reinforcing structural
elements, optimization of damping layers, positions of supports and/or absorbers, control
of material data, and geometric properties like shell curvature. Almost all of them have
in common that optimization of these variables leads to substantial gains of the objective
function. Therefore, comparison is rather based on practicability for actual use of the
application in real life than on performance of variables.
Piecewise polynomial plate thickness and shell thickness account for the most popular
variables in the field. Numerous authors parameterized the thickness of single plate or shell
elements, cf. [28,67,68,146,160,161,189,190,312]. This is a suitable procedure to understand
the effects of optimization. It seems, however, hardly applicable for real constructions.
For practical use, it is better to define regions of constant thickness as proposed in [82,
86, 309]. This has the advantage that the number of variables can be reduced for finely
meshed structures. Use of 36,000 variables as proposed by Choi et al. [68] should remain
an exception, even with the knowledge that their sensitivity analysis performed extremely
efficiently for that many variables.
Optimization of layered materials seems to possess a great potential for practical appli-
cations. This was shown in papers about maximizing transmission loss of plates [191,205],
about optimization of a damping layer on a circular plate [330] and in the thesis of
Kessels [170] for the sound radiation of a magnetic resonance image scanner.
Engelstad et al. [94] investigated influence of reinforcing structures like stringers and
frames on acoustic properties. It was shown that these thin–walled beam structures could
be easily thinned out whereas the general shape of the stiffeners strongly influences the
noise level inside the investigated structure, i.e. a clamped cylinder basically representing
an aircraft fuselage. In an earlier paper, cf. Cunefare et al. [86], these authors compared
optimization of circumferential rings of constant shell thickness and stiffeners. Again, they
found a tremendous potential for optimization. It was stated that longitudinal variability
yielded a somewhat greater impact on interior noise levels than circumferential variability.
Shell thickness dominated the process if both, shell thickness and stiffener parameters, were
optimized at once.
It is well–known that shell curvature strongly influences shell stiffness. Therefore, opti-
mization of shell geometry appears as a useful alternative to thickness optimization. Based
on practicability for production of the optimized structure, it is useful to design for a
large series but difficult to build for test purposes. With respect to practical applications,
shell geometry was optimized for design of carillon bells [267, 284, 317], of a loudspeaker
diaphragm [69], and of sedan body panels [214–216, 218]. Optimization of a bell or a
loudspeaker diaphragm presupposed an axisymmetric design. Although three–dimensional
radiation was considered, a two–dimensional finite–element mesh was sufficient. Moder-
ate geometric modifications of the contour always reproduced the same mesh topology.
For free–form shells, e.g. sedan body panels, we distinguish between the geometry–based
model and the finite–element model, whereas the latter is created by meshing the geometry–
based model. At first glance, parametrization is suitably applied to the geometry–based
Developments in Structural–Acoustic Optimization for Passive Noise Control 339
model that is remeshed after design changes. However, if the topology changes during the
optimization process, it is likely that the objective function does not remain continuous.
Therefore, a concept of direct modification of a finite–element mesh was developed by the
author [211] and applied to different problems and structures [112, 216, 217]. It is still
necessary to reformulate the specific functions of global and local modification in order
to achieve more flexibility to reasonably control the variables. This particularly regards
higher–order polynomials as global modification functions since parameters of different size
and physical meaning are in use. Work is in progress to substitute piecewise–formulated,
spline–interpolated lines and surfaces for these polynomials. Then, all variables describe a
length. As the modifying surface represents just a projected surface, there is no need to
completely adjust it to every detail of the model. It can even be applied to complicated
shell structures.
Results with respect to problems are again categorized into academic and more real-
istic problems. Starting with plates as a more academic application, we can state that
tremendous gains are possible if a stiffly supported structure with locally acting excitation
force is considered. As mentioned before, free or elastically supported plates have much
less potential if minimization of radiated noise is desired. If uniform radiation for all fre-
quencies is aimed, free boundary conditions support a better radiation at low frequencies
where the radiation efficiency is low. A locally acting load helps to achieve better im-
provements. This became clear in papers by Lamancusa and Eschenauer [190] and Ratle
and Berry [263]. They compared point–load excitation and uniform–pressure excitation.
In [263], point–load excitation allowed the decrease of the objective function by 40–70 dB
in a frequency band of the fourth and fifth mode of the original plate. Only 4 dB were
gained with uniform–pressure excitation. Most applications of plates and shells considered
locally acting loads. Often, the optimizer favored stiffening in the vicinity of the excitation.
This was shown in a parameter study by Fritze et al. [112]. Similar results were reported
in [146, 177, 189, 311]. Wo dtke and Lamancusa [330] suggested putting damping material
into regions where large deflections were observed. Since their example considered an ax-
isymmetric plate with large deflections occurring where large bending moments around the
circumferential axis were observed, it is assumed that damping material is optimally placed
in regions of large bending energy.
Box structures behave like plates. It is likely that most boxes were chosen to achieve
the closed surface of a radiator, either emitting noise to the cavity or to open space. Op-
timization of boxes produced similar results as that of plates, likely because boxes consist
of plates and/or shallow shells. In Marburg et al. [213], position of stiffeners proved to be
optimal when certain mode shapes could be extinguished. One of four reinforcing beams
could not be reasonably used. Therefore, the optimizer decided to place that beam close to
an edge where it hardly affected any structural mode.
Similar to plates and boxes, optimization of simple shell structures showed the large
potential of optimization. Test cases allowed manipulations of 10–20 dB. Investigation of the
aspect ratio of structural geometry of a dome made evident that there was a clear optimum,
cf. Soize and Michelucci [299]. This optimum held for different frequencies. Optimization
of a symmetric half–cylindrical shell under symmetric point load by Constans et al. [74,75]
ledtoanasymmetricdesign.
Bitsie and Bernhard [41] investigated energy–level reduction in a 1.152m3air–filled
box with a 0.64m2elastic wall in the frequency range between 130 Hz and 10 kHz. The
authors categorized their test cases into excitation and observation of structure and fluid.
They chose structural and acoustic loss factors, surface absorption and radiation efficiency
as variables. For excitation of the structure, an increase of structural loss factor caused
substantial energy reduction of structure and fluid, whereas increase of radiation efficiency
caused substantial energy reduction of structure and magnification in the fluid. Structural
340 Steffen Marburg
vibrations due to excitation of the structure were negligibly influenced by the acoustic loss
factor and surface absorption. At low frequencies, fluid energy level due to excitation of
the structure was hardly affected by acoustic loss factor, whereas sensitivity due to surface
absorption was more significant. For higher frequencies, acoustic loss factor became more
important and surface absorption lost significance. Looking at structural vibrations in
the case of acoustic excitation, the situation was very similar to fluid energy levels due to
structural vibration. Looking at fluid energy levels due to fluid excitation, it was reported
that sensitivities of structural loss factors and radiation efficiency were negligible for energy–
level reduction whereas, again, acoustic loss factor gave high impacts at high frequencies.
Surface absorption significantly affects fluid energy–level for low frequencies.
In the category of more realistic applications, we start with reflection of optimization
results for transmission loss maximization of layered plates [191, 205]. Basically, this was
accomplished by substantially increasing the coincidence frequency. Two strategies were
mentioned. At first, the plate could be designed more massively and less stiffly, which
cannot be considered a practical solution. Second strategy was therefore the only sensible
one for practical applications. It favored a stiff core of low density.
Optimization of the directivity pattern of a loudspeaker diaphragm by Christensen
and Olhoff [69] confirmed that different types of design parameters could be used. These
parameters consisted of rings of additional mass, distribution of shell thickness and shell
geometry. Obviously, variables could be combined. The authors showed that uniform
directivity patterns were accomplished for the entire audio frequency range. Optimization
created a point source in the center of the diaphragm. When optimizing at a single (and
relatively high) frequency, uniformity of the patterns was also yielded for the remaining
frequency range, however, at lower levels. Uniform pattern over a wide frequency range was
achieved for simultaneous optimization at three frequencies, in a first case at 500 Hz, 10 kHz
and 15 kHz and in a second case at 500 Hz, 7 kHz and 10.5 kHz. Soft (or free) boundary
conditions supported a better radiation at low frequencies. In a test case of simultaneous
optimization of thickness distribution and shell geometry, thickness distribution hardly
changed, whereas the geometry was modified and adjusted.
Schoofs, van Campen and many co–workers [267, 283, 284, 317] investigated and devel-
oped shapes of bells. This development of bell optimization over the last twenty years
was surveyed in their paper [285]. The project was aimed at design of a major third bell
based on a minor third bell. In a first step, Schoofs [284] created a minor third bell of the
targeted eigenspectrum. In addition to spectral properties, Roozen–Kroon [267] extended
the optimization by consideration of acoustic damping. She found a bell that possessed a
significant but unwanted bulge in its contour. This bulge vanished after improved analysis
and optimization methods could be used. Van Houten’s dissertation [317] presented a minor
third bell without that bulge.
A clamped cylinder as a substitute for an aircraft fuselage was investigated and opti-
mized in [82, 86, 94]. Performance of optimization strategies with respect to the objective
function was checked. Furthermore, it was investigated which type of parameters were in-
vestigated to determine which type would have the highest impact on the transmission loss
between an engine source outside the fuselage and a receiver inside. The general statement
of these papers leads the reader to believe that the potential of optimization is quite sub-
stantial. We have discussed choice of variable earlier in this section. It will be interesting
to experience optimization results for the free cylinder and for frequency ranges.
Milsted et al. [232] argued that the simplified model of an engine gave basic insight into
the optimization process of its structure. They demonstrated how damping and thickness
affected the objective function and showed that it smoothed for higher damping values.
Low damping emphasized resonance peaks and led to numerous minima and maxima. For
thickness variables, the authors presented a sensitivity estimate, showing how the emitted
Developments in Structural–Acoustic Optimization for Passive Noise Control 341
sound power varies per millimeter thickness and per kilogram weight. This also implied that
the (simplified estimated) A–weighted sound power as objective function depends linearly on
the variables, an assumption that is indeed surprising. In a similar case, Milsted’s co–author
Hall [138] demonstrated that a (relatively) simple response surface sufficed for simplified
representation of the objective function in terms of the design variables. In both papers,
it was argued that a trade–off between mass and radiated sound power became necessary.
For constant mass, Milsted et al. [232] reported 3 dB(A) gain, whereas the worst design for
the same mass returned a 5 dB higher value in the objective function. Hall [138] concluded
the case study with a trade–off of a 5% weight reduction with 1.4 dB(A) noise reduction.
Higher gains in either direction were possible. Another engine structure was investigated by
Fisher [106]. That work was focused on the use of genetic algorithms, although the simplified
structure and variables looked very similar to those of the two preceding examples. Only
100 evaluations of the objective function were sufficient to gain 1.7 dB(A) for a structure of
the same mass, and 2.3 dB(A) for a 5% heavier structure. Comparing this with the papers of
Milsted et al. [232] and Hall [138], it seems likely that better improvements would have been
possible. Milsted et al. [232] and Hall [138] reported that their techniques required between
45 and 150 evaluations and led to better designs. However, more complicated structures and
other variables may be better optimized by the use of a stochastic algorithm. La Civita and
Sestieri [184] succeeded using a genetic algorithm. Their analytic model was investigated
and heuristically optimized by other experienced engineers. Optimization, however, allowed
them another 5 dB average decrease of the noise transfer function. With the exception of
the latter paper, engine applications dealt with – in comparison to most other applications
– complex and rather compact structures. This is assumed to be the reason for relatively
small noise reductions in these examples.
The final group of applications comprises sedan bodies and sedan body panels. So-
bieszczanski–Sobieski et al. [297] combined optimization of crash and noise, vibration, and
harshness (NVH) problems. In more detail, mass reduction was used as the objective
function and numerous constraints like static stiffness and roof crush were fulfilled. As a
constraint for NVH purposes, the lowest torsional eigenfrequency was controlled between
26.65 and 29.32 Hz. Finally, the mass was decreased by little more than one percent
without violating any of the constraints. The torsional mode was then found at a frequency
of 29.32 Hz, equivalent to a 10% increase compared to the original value. Choi et al. [68]
minimized mass with constraints on the sound pressure at a certain point for two narrow
frequency bands. Control of 36,000 thickness variables resulted in 6 and 8 dB decrease of
the noise transfer in these frequency bands, respectively. Wang and Lee [325] lowered a
particular peak in their noise transfer function by about 15 dB. Primarily, they minimized
the mass by optimally sized and distributed shell thickness. Mass was reduced by 2%.
Hermans and Brughmans [145] reported 2–10 dB gain of the sound pressure level at the
driver’s ear in two frequency bands by adjustment of shell thicknesses and beam cross–
section properties. More applications that stem from the present author’s work will be
discussed in the following section.
6.4 Optimal Geometry of Sedan Body Panels
Geometric optimization of sedan body panels was performed and discussed in the author’s
papers [214–219]. These applications comprised a roof [218, 219], a dashboard [215], a
hat–shelf [214], a floor panel [216] and a spare–wheel well [217]. According to the local
approximation scheme and in view of the complicated objective function, it is likely that
the global optimum was not found. In most cases, we considered significant gains after
a finite number of function evaluations as more important than the long and exhaustive
search for a global minimum.
The first example, a vehicle roof, dealt with a model that was not realistic [218,219]. It
342 Steffen Marburg
was assumed that the roof was simply supported at six points. Excitation being a harmonic
pressure from above was applied. As already discussed before, these two properties made
it very easy to achieve large improvements of the noise transfer function, being the sound
pressure level at the driver’s ear. The mean sound pressure level in a frequency range
between 0 and 200 or 300 Hz, respectively, accounted for the objective function. For the
freely modified shape, approximately 50 dB were gained in the frequency range between
0 and 300 Hz, see also Figure 1 for the new shape. Besides the unrealistic simulation
model of the roof, the optimized design would have been unacceptable for vehicle designers.
Therefore, the optimization was then constrained to a maximum modification of only 1 cm
in normal direction. Even this small modification led to an improvement of virtually 8 dB.
Alteration of damping showed little impact on optimization performance. The optimization
process was quickly guided to large gains. Less than 200 evaluations of the objective function
succeeded for six variables, even using the first–order optimizer of Ansys [306], which was
difficult to handle in other applications. From an engineering point of view, two strategies
were followed. If possible, the optimizer tried to shift the eigenfrequencies off the frequency
range under consideration. Usually, modes tended to appear as dipoles or higher–order
multipoles at these higher frequencies, although the original structure contained numerous
monopole mode shapes. Once the modes were shifted to much higher frequencies the
optimizer started to control mode shapes directly. If the eigenfrequencies could not be
substantially increased, the optimizer controlled mode shapes immediately and, at the same
time, thinned out the eigenspectrum. In the case of only 1 cm modification, the number
of modes up to 200 Hz was decreased from 39 to 24. The first eigenfrequency decreased
from 88 to 64 Hz, though. In [219], optimization results were compared with different cases
of conventional stiffening of the roof. In most cases, the lowest eigenfrequency could be
slightly enlarged. This led to a little decrease of the noise transfer function up to this first
eigenfrequency. However, beyond the first mode, noise transfer function increased so that
the objective function was hardly affected by that type of stiffening. About 1 dB was gained
in the best case.
Experience with roof optimization could be utilized for optimization of a sedan hat–
shelf [214]. Similar to the roof model, the structure was assumed to be simply supported
at numerous positions. Three cases of excitation were considered. These three excitations,
i.e. discrete forces acting at the belt retractor, at the subwoofer, and at the rear window–
cross member, were located in the hat–shelf structure itself. Clearly, this model could be
considered as a much more realistic simulation model than the roof. For this type of ex-
citation, simply supported conditions seemed reasonable. Requests of three different load
cases, and the experience with the roof optimization guided us to the strategy of maximiz-
ing the lowest eigenfrequency. This eigenfrequency was shifted from approximately 32 to
101 Hz. Noise transfer functions were compared for both structures. The objective function
was evaluated 24 times for different configurations. Consideration comprised three cases
of excitation, analysis with and without consideration of acoustic damping by absorption,
objective function as mean sound pressure level and root mean square sound pressure level
at driver’s ear, and lower limit of frequency range being 0 and 20 Hz, whereas the upper
limit of 100 Hz remained unchanged. Consequently, 24 values of the objective function
were compared for the original and the optimized structure. This comparison confirmed
that maximization of the lowest eigenfrequency accounted for a successful strategy to lower
the sound pressure level at the driver’s ear. Between 4.4 and 13.9 dB were gained. Again,
it should be mentioned that a simplified model was successfully used for finding initial pa-
rameter sets. The new design was essentially higher than the original structure, i.e. the
entire design freedom of 6 cm vertical modification was utilized.
A dashboard embedded in the entire body structure (symmetry assumed), cf. Figu-
re 2, was optimized in another application [215]. Here, a superelement represented the
Developments in Structural–Acoustic Optimization for Passive Noise Control 343
dashboard
Figure 2. Sedan dashboard embedded in entire body structure
vibrational behavior of the remaining sedan body whereas the dashboard was considered
as a finite–element shell structure. This component model implied that only radiation
by structural vibrations from the dashboard was taken into account. Obviously, reflec-
tion and absorption by all other panels were considered. The sound pressure level at the
driver’s ear was decreased by 3.8 dB in the root mean square value as objective function,
see also Figure 3. Evaluation of the objective function, cf. equations (23–25), assumed
frequency bounds of 24 Hz and 200 Hz and a reference level pRef (= 76 dB). Clearly, the
high–level frequency ranges between 80 and 150 Hz were substantially lowered, while other
regions were used to compensate. A comparative parameter study of the Young’s modulus
of the dashboard showed that stiffening would have had a similar effect but returned a
completely different noise transfer function. Similar gains in the objective function were
realized for a structure ten times as stiff. Further stiffening led to further reductions in
the noise transfer function. However, considerable gains have also been observed for a
substantially more compliant structure. This effect is likely due to more local vibration
modes in the higher frequency range. The eigenspectrum was marginally affected during
the optimization process. Optimization altered mode shapes to establish cancellation of
their effect in the cabin. In the resulting design of that application, well–known elements of
construction were observed, cf. Figure 4. Major modifications occurred in the upper part.
A horizontal bead was directed to the cavity whereas, below, two bulges pointed to the
engine. Little modification was required in the lower part of the dashboard. The optimizer
utilized the entire design freedom of 10 mm modification in normal direction. This value
was exceeded for modeling reasons since optimization was carried out on a geometry–based
model. A finite–element mesh model was created for each evaluation of the objective func-
tion. Optimization performed slowly, and sometimes erroneously, since modification of the
finite–element discretization produced discontinuities in the objective function.
For the smaller floor panel, cf. [216], creation of a geometry–based model was bypassed.
Instead, the finite–element mesh was modified directly. The general concept of this direct
modification was explained in detail in the author’s paper [211]. Similar to the dashboard
example, the floor panel behind the driver’s seat (left–hand drive) was embedded in a su-
perelement environment that is visualized in Figure 5. Lines were drawn to visualize the
body structure. Excited at two engines, the noise transfer function as the sound pressure
level at the driver’s ear was evaluated and minimized, cf. Figure 6. The objective func-
tion, i.e. the root mean square value in the frequency range between 0 and 100 Hz with
344 Steffen Marburg
Frequency in Hz
520 35 50 65 80 95 110 125 140 155 170 185 200
70
75
85
95
105
80
90
100
24 Hz
76 dB
Original model
Optimized model
Sound pressure level at driver’s ear in dB
Figure 3. Noise transfer functions for original and optimized dashboard
pRef = 0 dB, was decreased by approximately 2 dB. In Figure 6, we can clearly identify sub-
stantial gain in the upper frequency range and also improvements between 30 and 50 Hz.
Compensation in the domain between 50 and 80 Hz is observed since two additional peaks
were created. In practical applications, the user should decide about this trade–off. As
shown in the more recent paper [112], variation of the reference level pRef (starting at high
values and successively decreasing the reference level) can help to smooth down the objec-
tive function and to avoid such a large increase of the noise transfer function in certain
regions. In this example of a floor panel, only the bottom plate was modified. In that
region, a quadratic polynomial accounted for the locally invariant global modification func-
tion. Local modification was realized by four simple but locally variable bead functions.
The new shape is shown in Figure 7. We will discuss the problem of gains as small as 2 dB
or even smaller, later in this section.
A spare–wheel well embedded in a superelement was investigated in another applica-
tion [217]. Excited at two engine supports, vibrations were transferred through the entire
body to the trunk bottom. It was assumed that the trunk partition wall did not exist.
Although not realistic, this simplification was introduced to circumvent analysis of sound
transmission through the trunk partition wall. It was clear that the simplified model dif-
fered from the actual one. However, the results appeared reasonable independent of the
actual structure. The root mean square value of the sound pressure level was lowered by
only 1.2 dB in the frequency range between 40 and 100 Hz. Prior to optimization, a contri-
Developments in Structural–Acoustic Optimization for Passive Noise Control 345
MX
0.00
1.11
2.24
3.36
4.48
5.60
6.72
7.84
8.96
10.09
Figure 4. Sedan dashboard: New shape as modification of the original geometry
Figure 5. Sedan floor panel embedded in superelement environment representing
the remaining body structure, arrows indicating excitation forces
346 Steffen Marburg
Frequency in Hz
Sound pressure level at driver’s ear in dB
10 20 30 40 50 60 70 80 90 100
10
20
30
40
50 Original
Optimum
Figure 6. Noise transfer functions for original and optimized floor panel
bution analysis over the entire frequency range identified a zone of particular mobility. A
remarkable vibration loop was observed at the bottom of the wheel well. That loop affected
the entire frequency range. Increasing Young’s modulus by a factor of 106enabled investi-
gation of stiffening. Stiffening provided substantial gains in selected frequency ranges, i.e.
between 40 and 50 Hz and between 75 and 100 Hz. Hardly any decrease was observed below
40 Hz and between 50 and 75 Hz. Modification was prepared to destroy the vibration loop
at the bottom of the spare–wheel well. The success of this strategy became evident as the
contribution analysis of the original and the optimized structures were compared. However,
only 1.2 dB were gained.
There are some reasons that only little improvements of the noise transfer function were
reported for sedan body panels like the floor panel [216] or the spare–wheel well [217]. One
of them is that the excitation far away from the particular component has some charac-
teristics of excitation by prescribed displacements. This is due to the comparatively great
inertia of the superelement environment. Therefore, we cannot consider just local modes
for optimization. For most of the eigenvectors, component and superelement vibrations are
strongly coupled, and excitation of the component is introduced at every coupling point.
We considered about 100 eigenfrequencies up to 100 Hz. When investigating a panel like
the spare–wheel well, a rigid body motion due to a global vibration mode, e.g. global
bending, will hardly cause noise at the driver’s ear. Mathematically, the contribution of
the wheel well is compensated by the vibration of the trunk lid. It is further assumed that
optimization of a panel close to the excitation has more potential for a successful process
because structural modifications can better control stiffness of the excitation region which
seems to have the most essential impact for a quieter construction.
As most papers in structural–acoustic optimization consider more simple constructions
like plates, simple shells and boxes, tremendous gains could have been reported. Mostly
force excitation was assumed in these constructions. As can be seen in more realistic ap-
plications of engines or sedan body panels, improvements become much smaller if these
assumptions are violated. Reasonable results indicate that the strategy to optimize com-
ponents is a useful technique for quieter constructions.
Developments in Structural–Acoustic Optimization for Passive Noise Control 347
MN
MX
-9.90
-8.42
-6.94
-5.46
-4.47
-2.99
-1.51
-0.52
0.96
2.44
3.43
4.91
6.39
7.37
8.86
10.34
11.82
Figure 7. Sedan floor panel: New shape as modification of the original geometry
6.5 Experimental Validation
Few papers have been published on experimental verification of results in structural–
acoustic optimization. Basically, the argumentation assumes that a verified simulation
model can be arbitrarily modified. It should still be valid after modification. In general,
this appears as reasonable argumentation, and all parameter studies with simulation models
are based on this assumption. Obviously, the modified simulation models perform on the
same mathematical basis as the original structure. However, surprisingly large gains gave
rise to the question whether these results can also be verified in the experiment.
St. Pierre and Koopmann [301] investigated a square plate. They distributed nine
discrete masses on the plate and minimized the radiated sound power due to harmonic
force excitation in the plate’s center. Minimization was limited to resonance peaks. One
case of optimization was then experimentally investigated. The case of minimization of
a single resonance peak allowed 30 dB gain in the simulation. The experiment confirmed
22 dB reduction. Although a gap of several decibels remained, the substantial gain could be
validated. St. Pierre and Koopmann argued that the experimental setup lacked non–ideal
boundary conditions and rotary inertia contribution of the added masses.
Koopmann and Fahnline [180] considered a very similar example of lesser complexity.
The authors compared experimentally determined sound power level with those of the op-
timized simulation models. Investigation focused on performance of one variable only, i.e.
size of an added lumped mass. Parameter study showed a distinctive minimum where the
sound power level could be decreased by approximately 40 dB in the second doubly sym-
metric mode. Simulation results in the 1/3 octave spectra agreed well with the experimental
data, especially in the vicinity of the original second resonance peak.
A half–cylindrical shell was investigated in the paper by Constans et al. [74]. Positions of
two small discrete masses were optimized to achieve a minimum sound power radiation in the
resonance peaks. Five resonances were considered. The second and the third mode radiated
noise most significantly. In a first step, the sound power was experimentally measured for
348 Steffen Marburg
these resonances. It was, however, stated that the fourth and the fifth mode could not
be uniquely separated. Therefore, these modes were omitted in further comparisons of
experimental and simulated data. Compared to second and third mode, the fourth and
fifth mode contributed marginally to sound radiation of the half–cylinder. Optimization
of the sum of sound power values for the first five structural resonances supplied a gain of
9.5 dB. Essentially, this improvement resulted from the second mode. The experimental
investigation returned 8.25 dB decrease of the objective function, being the sum of the sound
power values of first three resonance peaks. This is considered as an excellent agreement of
simulated and experimental data.
Nagaya and Li [239] optimized a rectangular plate and searched for optimal positions
of absorbers. These optimal absorber parameters were found. Presentation of results,
however, lacked clarity. It did not become clear in every detail how the experimental data
correlate with simulation data. A direct comparison between both would have been useful
and more convincing.
Two plates perpendicular to each other accounted for one example in the dissertation of
Hibinger [146]. He minimized mean square velocity instead of sound power. Similar to the
above mentioned applications, only resonances were considered. In contradiction to other
papers, this work considered 12–15 modes instead of only 1–5 in other papers. Comparison
of simulation and experimental data showed good agreement for highly radiating modes
and larger deviations for modes that were less radiating. Apparently, this can be regarded
as a principal ill–conditioning. Basically, the experiment confirmed simulation results. Ribs
were used as design variables.
A circular plate was optimized and experimentally investigated by Tinnsten et al. [309,
311]. In two cases, the plate was simulated as free and as clamped structure [309]. The
intensity for a specified frequency at a specified point above the plate accounted for the
objective function. By using the method of moving asymptotes, simulation achieved gains
of 4 dB for the free plate and 14.7 dB for the clamped structure by controlling the thickness
of six rings. The intensity was measured in three points of equal radius from the axis of
revolution and then averaged. Numeric prediction of 4 dB was not achieved for the free
case. Only 2.3 dB of improvement were reported. 19.5 dB gain were observed in the ex-
periment whereas the simulation predicted only 14.7 dB. A number of possible reasons for
deviations were collected. They included a substantial influence of damping, the statement
that idealized boundary conditions were used in simulation, and a remark that the simu-
lation assumed structural vibrations of the plate only, whereas a more complex structure
was excited in the experiment. In a further study [311], it was clarified that the original
optimum for the clamped plate showed up as a local minimum. Usage of a stochastic
optimization method, i.e. simulated annealing, resulted in another thickness distribution.
For that, 18.7 dB were gained. Experimental investigation showed 24.1 dB, even a larger
difference than for the sub–optimum.
A much more complex model was investigated by Marburg et al. [213]. A steel box rein-
forced by an external and an internal beam frame, cf. Figure 8, was produced with similar
technologies as sedan bodies. In the first and most demanding step, a simulation model
was prepared. This simulation model was validated by adding a steel block of specified
size. For that particular model, the frequency range for reliable prediction was found to be
between 10 and 50...60 Hz. The noise transfer function represented the sound pressure level
at a specified point inside the box due to a unit force acting at the external beam frame.
Optimization considered the frequency range between 0 and 100 Hz, but hardly any con-
tribution in the frequency range below 10 Hz and beyond 60 Hz was obtained. It was one
idea of that project to provide evidence that a design optimization of a complex structure
can actually lead to an improved physical construction. Such evidence presupposes non–
destructive modification of the model since destruction would change vibration behaviour
Developments in Structural–Acoustic Optimization for Passive Noise Control 349
1500
590
370
1000
650
1100
200
700
Figure 8. left: photo of the box, two beams on the front panel show positions of
optimized stiffeners, right: drawing of the structural model (measures
in mm), view from the left (upper subfigure) and from above (lower
subfigure)
significantly. For that reason, only two panels were selected for modification. Two beams
were distributed on each of these panels, see for example two added stiffeners in the upper
and lower parts of the front panel in Figure 8. These beams were mounted while washers
were used to bridge the local distances between straight beams and distorted metal sheets.
This technique ensured that the beams were coupled tightly but did not modify the uneven
geometry of the metal plate. Welding of the beams would have likely changed the geometry
and, consequently, the modal properties of the panel. Moreover, the chance of dismantling
would have been spoiled. The optimizer favored a construction that did not contain certain
modes. More specifically, three modes were more or less destroyed by optimization. The
fourth beam, placed at the lower panel, proved to be unnecessary and was, therefore, shifted
to one edge. This type of selective stiffening hardly affected the overall modal density. The
generally predicted trend of simulation was reproduced by the experiment. Correlation of
results improved the more reliable the simulation model was. As an example, 6.3 dB of gain
were predicted by simulation, whereas the experiment returned 6.7 dB. Consideration of
additional frequency ranges would either reduce gain or enlarge the gap between prediction
and comparison with experiment. The major message of the paper was the statement that
experimental confirmation requires a reliable simulation model.
In general, experimental validation showed that even large gains can be confirmed by
measurements. It is assumed that this requires that model modification not essentially
alter the simulation model in the optimization process. However, it is not yet clear which
properties can and which properties must not be modified.
350 Steffen Marburg
7OPENPROBLEMS
The study of contributions in the field of structural–acoustic optimization confirmed that
this field contains large potential for designing quiet structures and, as a consequence, for
quietening a noisy world. In particular, tremendous gains seem to be possible for thin
structures like shells and for layered structures. However, numerous problems still remain
unsolved or, at least, insufficiently solved. In this final section, we will point out several
directions to encourage further research. These directions cover the fields of harmonic
analysis, objectives, extension of applications, search for general design rules and optimiza-
tion algorithms. Besides development of new methods, extensive testing and collection of
experiences account for key questions in the field of structural–acoustic optimization.
Koopmann and Fahnline [180] emphasized that efficient analysis techniques account
for the basis of optimization. Though regarding the entire analysis, fast solutions of the
fluid problem or even of the coupled problem involving radiation into open space and
considering frequency ranges are strongly desired. There are a couple of ideas to perform
efficient analysis over a large frequency range.
In finite–element analysis, infinite elements based on Astley–Leis formulation, cf. Ast-
ley et al. [15], return a system matrix that quadratically depends on the frequency, cf.
equation (17). Transformation of this problem into state space allows reconstruction of the
inverse system matrix based on eigenvalues and eigenvectors. Evaluation of eigenvectors
requires considerable computational resources. However, assuming that structural modi-
fication does not or only negligibly alters fluid surface, spectral data remain constant in
the entire optimization process. The inverse system matrix could be easily and efficiently
reconstructed. Another promising technique consists in Pad´e approximation schemes as
proposed by Malhotra and Pinsky [207]. It will be interesting to experience performance
and practicability for real three–dimensional structures.
Neither Pad´e approximation nor frequency interpolation schemes have yet supplied suf-
ficient efficiency gains for multifrequency boundary–element analysis, since they demand
extremely large storage or require solution of a linear system of equations, respectively.
Since sound power evaluation necessitates availability of both sound pressure and particle
velocity on the structural or at an artificial surface, the storage of influence coefficients
needs resources equal to the storage of the entire impedance matrix. Formulation of an ex-
plicit frequency–dependent system matrix similar to equation (17) by the dual–reciprocity
method would likely require a particular solution that fulfills the Sommerfeld radiation
condition. Hardly any paper on the application of the dual–reciprocity boundary–element
method for external problems is known. A reasonable trade–off for fast solution of multi-
frequency problems could consist in storage of a few eigenvectors of the system matrix (for
each frequency) and their use as a spectral preconditioner for the iterative solution of the
system of equations. Application of fast boundary–element methods ensures that matrix
set–up time is small compared to solution time.
A similar concept consists in the use of radiation modes to reconstruct the impedance
matrix. The few applications hardly allow conclusions on general performance of opti-
mization that utilizes radiation modes. Though promising, this concept must be further
developed and tested.
Improvement of other methods, e.g. the lumped parameter model, can also lead to
a substantial impact on the computational efficiency even though applicability of these
rough approximation methods is limited. Only one contribution on parallel processing is
known up to now in the field of structural–acoustic optimization. Also the question should
be investigated whether substituting minimization of the dynamic compliance for sound
power optimization holds as a reasonable strategy for more complex structures.
Besides computational efficiency, analysis should clarify a number of other problems.
Developments in Structural–Acoustic Optimization for Passive Noise Control 351
Although we have proposed three conditions that should be fulfilled to substitute the sound
pressure at a particular point for the potential energy in a cavity, this assumption is still
to be tested, extended, and proved. Further questions comprise consideration of different
load cases and modification of finite–element mesh topology.
Comparatively few applications of structural–acoustic optimization could be collected.
Most papers considered more academic examples. Practical examples covered sedan power
train and sedan body, carillon bells, loudspeaker design, aircraft fuselage, magnetic res-
onance imaging scanners and sandwich plates. Other applications could encompass tire
noise, concert halls, hearing aids, aircraft engines, and others. More test examples are
required for better understanding of the optimization processes.
Optimization should play an important role in collecting rules for quiet designs in gen-
eral. Most likely, this challenging problem requires many test examples. However, research
has already presented several rules. One of these rules consists in stiffening in the vicinity
of excitation.
Further, there is a need to gain experience in multidisciplinary optimization combining
structural–acoustic demands and other objectives which are more significant for the design
process, e.g. durability, fatigue and safety. As mentioned at the beginning of the paper,
not any application in time domain was found yet. Tests and examples in time–domain
acoustics will be interesting to acquire a new field of applications.
Experimental validation might be further required to dispel doubts in the results of
optimization. However, experimental data should not be mistaken for an analytic solution.
Comparison of simulation data and experimental measurements will likely help to further
introduce optimization methods in industrial development of prototypes.
Similar to analysis in structural acoustics, optimization algorithms contain a large po-
tential for future development and tests. In the field of approximate optimization, sequential
quadratic programming as local approximation and multipoint approximation as a mid–
range method seem to possess the highest potential for optimization in the context of this
paper. Their performance should be compared with that of stochastic methods like simu-
lated annealing, genetic algorithms, tabu search and, possibly, other methods. Extensive
tests and comparisons will be necessary to clarify which method performs best for certain
applications. This problem should be illuminated from different directions. It will be in-
teresting to experience which method finds a reasonable improvement very soon and, also,
which method finds the best solution. Obviously, a robust method is more desired than one
that performs best in some cases, but is not stable. There are many parameters to control
optimization methods. Use of these optimization methods requires experienced choice of
these control parameters. Although presented for some analytic test functions, application
to realistic multimodal objective functions in structural–acoustic optimization could require
further tuning and modifications.
With respect to numerous research demands and other requirements in noise control, the
field of structural–acoustic optimization will likely attract the attention of more and more
researchers and engineers. It is to be hoped that this will contribute to the achievement of
a quieter world in the future.
ACKNOWLEDGEMENT
We wish to acknowledge that most of the simulation was run on the SGI Origin 2000 at
the Zentrum f¨ur Hochleistungsrechnen of the Technische Universit¨at Dresden. Finally, the
author thanks Gilbert Adams for patiently checking and explaining the English language.
352 Steffen Marburg
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