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A novel modelling approach for sound propagationanalysis in a multiple scatterer environment
B. Van Genechten, B. Bergen, B. Pluymers, D. Vandepitte and W. Desmet
K.U.Leuven - Dept. of Mechanical Engineering, Celestijnenlaan 300B - bus 2420, 3001Heverlee, Belgium
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The Wave Based Method (WBM) is an alternative deterministic prediction method for steady-stateacoustic problems, which is based on an indirect Trefftz approach. It uses wave functions, which are exactsolutions of the underlying differential equation, to describe the dynamic field variables. As a result,the WBM does not require a dense element discretization, as opposed to the commonly used elementbased prediction techniques. The relatively smaller system and the absence of pollution errors make theWBM very suitable for the treatment of mid-frequency problems, where the element-based methods areno longer feasible due to the large computational cost associated with the fine discretizations needed toretain accurate results.This paper introduces a new modelling concept for the efficient treatment of radiation and scatteringproblems when multiple distinct objects are involved. Each object is assigned to a separate level, whereit is treated using existing wave based modelling techniques. An adapted weighted residual formulationlinks the multiple levels, yielding a multi-level wave based model describing the entire problem.
1 Introduction
Underwater acoustics, utilizing SONAR technology, isby far the most commonly used technique for detec-tion, assessment, and monitoring of underwater physicaland biological characteristics and objects which may befloating in the water, lying on the seafloor, or buried be-low the sediment. In some applications, such as low fre-quency detection and classification of structurally com-plex objects immersed in a fluid, signal processing tech-niques must be aided by a-priori model based knowl-edge of the target echo. Because objects of interest inpractical applications often consist of a configuration ofmultiple geometrically complex objects with a detailedinternal structure, mathematical techniques capable ofdealing with generic geometries, and capable of couplingdifferent physical domains (i.e. the solid and the fluiddomains) must be used. Moreover, since the frequencyrange of interest is very wide, the development of ef-ficient numerical modelling techniques for the study ofunderwater scattering in the frequency domain are ofgreat interest.
Both the Finite Element Method (FEM) and the Bound-ary Element Method (BEM) are well established deter-ministic CAE tools which are commonly used for theanalysis of real-life structural-acoustic problems. TheFEM (1) discretizes the entire problem domain into alarge but finite number of small elements. Within theseelements, the dynamic response variables are describedin terms of simple, polynomial shape functions. Be-cause the FEM is based on a discretization of the prob-lem domain into small elements, it cannot inherentlyhandle unbounded problems. An artificial boundaryis needed to truncate the unbounded problem into abounded problem. Special techniques are then requiredto reduce spurious reflection of waves at the truncationboundary. Three strategies are applied to this end: ab-sorbing boundary conditions, infinite elements or ab-sorbing layers (2). The BEM (3) is based on a boundaryintegral formulation of the problem. As a result, onlythe boundary of the considered domain has to be dis-cretized. Within the applied boundary elements, someacoustic boundary variables are expressed in terms ofsimple, polynomial shape functions, similar to the FEM.Since the boundary integral formulation inherently sat-isfies the Sommerfeld radiation condition, the BEM isparticulary suited for the treatment of problems in un-bounded domains.
However, since the simple shape functions used in both
the FEM and BEM are no exact solutions of the govern-ing differential equations, a very fine discretization is re-quired to suppress the associated pollution error (4) andto obtain reasonable prediction accuracy. The resultinglarge numerical models limit the practical applicabilityof these methods to low-frequency problems (5; 6), dueto the prohibitively large computational cost.
The Wave Based Method (WBM) (7) is an alternativedeterministic technique for the analysis of vibro-acousticproblems. The method is based on an indirect Trefftzapproach (8), in that the dynamic response variablesare described using wave functions which exactly satisfythe governing differential equation. In this way no ap-proximation error is made inside the domain. However,the wave functions may violate the boundary and conti-nuity conditions. Enforcing the residual boundary andcontinuity errors to zero in a weighted residual schemeyields a small matrix equation. Solution of this ma-trix equation results in the contribution factors of thewave functions used in the expansion of the dynamicfield variables. The WBM has been applied successfullyfor many steady-state structural dynamic problems (9),interior acoustic problems (10) and interior and exteriorvibro-acoustic problems (11). It is shown that, due tothe small model size and the enhanced convergence char-acteristics, the WBM has a superior numerical perfor-mance as compared to the element based methods. Asa result, problems at higher frequencies may be tackled,making it an attractive technique for studying underwa-ter scattering phenomena and sonar applications.
This paper discusses a new modelling concept for acous-tic radiation and scattering problems, particularly suitedfor the treatment of problems involving different dis-tinct objects (scatterers). The main idea is to assigneach object in the problem to a particular model level.In this level, the scattering on this particular object istreated using existing wave based modelling techniquesfor unbounded problems. The general problem is thencomposed by combining the different levels through anadapted weighted residual formulation.
The first part of this paper briefly addresses the generalacoustic problem setting and the WB modelling tech-niques used for the models in each level. A second partis devoted to the discussion of the new multi-level mod-elling concept. Finally, the new method is applied to anunderwater scattering problem, in order to illustrate itspotential and validate the accuracy of the method.
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2 Problem description
X
Y
�
�
��
P
V
Z
�
P
r
Figure 1: A 2D unbounded acoustic problem
Consider a general 2D unbounded acoustic problem asshown in figure 1. The steady-state acoustic pressureinside the problem domain is governed by the inhomo-geneous Helmholtz equation:
∇2p(r) + k2p(r) = −jρ0ωδ(r, rq)q (1)
with ω the circular frequency and k = ω/c the acousticwave number. The acoustic fluid is characterised by thedensity ρ0 and the speed of sound c. The fluid is excitedby a cylindrical acoustic volume velocity source q. Theproblem boundary Γ is constituted of 2 parts: the finitepart of the boundary, Γb, and the boundary at infinity,Γ∞. Based on the three types of commonly appliedacoustic boundary conditions, the finite boundary canbe further divided in three non-overlapping parts: Γb =Γv ∪ Γp ∪ ΓZ . If we define the velocity operator Lv(•)as:
Lv(•) =j
ρ0ω
∂•∂n
, (2)
we can write the boundary condition residuals:
r ∈ Γv : Rv = Lv(p(r))− vn(r) = 0 , (3)r ∈ Γp : Rp = p(r)− p(r) = 0 , (4)
r ∈ ΓZ : RZ = Lv(p(r))− p(r)
Zn(r)= 0 , (5)
where the quantities vn, p and Zn are, respectively, theimposed normal velocity, pressure and normal impedance.At the boundary at infinity Γ∞ the Sommerfeld radia-tion condition for outgoing waves is applied. This con-dition ensures that no acoustic energy is reflected atinfinity and is expressed as
lim|r|→∞
(√r(∂p(r)
∂|r| + jkp(r)))
= 0 . (6)
Solution of the Helmholtz equation (1) together withthe associated boundary conditions (3), (4), (5) and (6)yields a unique acoustic pressure field p(r).
2.1 The wave based method
The Wave Based Method (WBM) (7) is a numericalmodelling method based on an indirect Trefftz approachfor the solution of steady-state acoustic problems inboth bounded and unbounded problem domains. Thefield variables are expressed as an expansion of wavefunctions, which inherently satisfy the governing equa-tion, in casu the Helmholtz equation (1). The degrees offreedom are the weighting factors of the wave functionsin this expansion. Enforcing the boundary and conti-nuity conditions using a weighted residual formulationyields a system of linear equations whose solution vectorcontains the wave function weighting factors.
Partitioning into subdomainsWhen applied for bounded problems, a sufficient condi-tion for the WB approximations to converge towards theexact solution, is convexity of the considered problemdomain (7). In a general acoustic problem, the acousticproblem domain may be non-convex so that a partition-ing into a number of convex subdomains is required.
�� t�
Figure 2: A WB partitioning of the 2D unboundedproblem
When the WBM is applied for unbounded problems,an initial partitioning of the unbounded domain into abounded and an unbounded region precedes the parti-tioning into convex subdomains (10). Figure 2 illus-trates the principle. The unbounded acoustic problemdomain is divided into two non-overlapping regions bya truncation curve Γt. The unbounded region exteriorto Γt is considered as one acoustic subdomain.
Acoustic pressure expansionThe steady-state acoustic pressure field p(α)(r) in anacoustic subdomain Ω(α) (α = 1 . . . NΩ, with NΩ thenumber of subdomains) is approximated by a solutionexpansion p(α)(r):
p(α)(r) =n(α)
w∑w=1
pw(α)Φ(α)
w (r) + p(α)q (r). (7)
The wave function contributions p(α)w are the weighting
factors for each of the selected wave functions Φ(α)w . For
a complete description of the functions Φ(α)w the reader
is referred to Pluymers (10).
Wave based modelThe function expansions used guarantee compliance withthe Helmholtz equation inside the domain and the Som-merfeld radiation condition at infinity. The boundaryconditions and subdomain continuity are enforced bymeans of a weighted residual formulation. This yieldsa square system that is solved for the unknown wavefunction contributions. For a detailed description of thesystem matrices, the reader is referred to Pluymers(10).
3 A multi-level concept in WaveBased Modelling
The WBM has shown to be efficient in modelling 2Dacoustic radiation problems (11). However, when mul-tiple acoustic scatterers are present, the method’s effi-ciency tends to deteriorate. This is due to the fact thatthe circular truncation line Γt, whose interior domain
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is modelled using a set of wave function expansions forbounded subdomains, needs to enclose all the scatterersat once. As a result many unbounded wave functionsneed to be included in the model in order to accuratelycouple the spatial resolution of the expansions in thebounded and unbounded subdomains. Moreover, sinceall the scatterers are included within a single circle, acomplex partitioning of the interior is often needed in or-der to satisfy the requirement of using convex boundedsubdomains.
To remedy this, the concept of multi-level modelling isintroduced. The main idea of the multi-level WBM ap-proach is to consider the multiple objects in the problemas different ’levels’ of the problem. In every level, thescattering of one particular object is studied. The inci-dent field for this problem is the scattered field from theother objects and the external excitations (plane waveor point source). Since this incident field is the resultof a scattering calculation in the other levels, all thecalculations have to be carried out simultaneously. Aweighted residual formulation links the levels, yieldingone system for the coupled problem composed of all thelevels. This system can be solved for all the unknownweighting factors. Using these factors, the scatteringfield can be calculated in each level. The total result-ing pressure field is then composed from the fields of alllevels. This procedure is illustrated in figure 3.
��
+Weighted
Residual Formulation
S1 S2
S2S1
=
�t
��
��
L1
t�L2
Figure 3: Graphic representation of the multi-levelmodelling concept
When looking at the different levels, it is clear that thereare parts of the problem for which the levels geometri-cally overlap. More precisely, this overlap takes placein the unbounded part of the wave models of the dif-ferent levels. Consequently, the pressure field in thisunbounded part of the total problem will be describedas a summation of the fields present in each level:
pub,total =nλ∑i=1
ΦLiw · pLi
w . (8)
where nλ is the number of different levels in the problem.Each of the sets Φ(Li)
w is a complete set for the Neumannproblem on its associated truncation circle ΓLi
t .
With the function set chosen, the wave model for thisdomain can now be constructed by enforcing the bound-ary conditions through the weighted residual formula-tion. The residuals of the boundary conditions are nowevaluated using the new, combined wave function set.Similarly, continuity conditions can be applied using the
same types of residuals. This continuity can be used tocouple the multi-level unbounded wave set with boundeddomains in the appropriate level, in the same way as acoupling would be set up between a conventional un-bounded and bounded wave domain. The bounded do-main in each of the levels can then further be modelledusing the conventional wave based domain division tech-niques and function sets. If the test functions used inthe weighted residual are random functions, then theresiduals on the boundary and continuity conditions areforced to zero in an integral sense, resulting in a nu-merical solution for the physical problem. To obtain anumerical model which can be solved, these test func-tions are written as a randomly weighted sum of certainbasis functions ta:
p(•)(r) =n(•)
a∑a=1
pa(•)t(•)a (r) = t(•)(r)pw
(•), (9)
with pa random weighting factors. The choice of thebasis functions needs to be such that this basis is richenough to be combined to any field on the boundary con-sidered, but may vary for different parts of the bound-ary. When integrating the boundary of a conventional(bounded) domain, an expansion in terms of the samebasis functions as used to describe the acoustic vari-ables can be used (like in de the widely used Galerkinapproach). For the multi-level unbounded domain, analternative selection of test functions is proposed. Sincethe unbounded basis functions ΦLi
w for each of the levelsare chosen such that they can accurately approximateany field on the associated truncation ΓLi
t , this set willsuffice as basis for the test functions on this part of theboundary.
4 Numerical validation example
This section discusses a numerical underwater acousticsexample which illustrates the applicability of the pro-posed concept for complex scattering calculations. Theconfiguration studied is shown in figure 4. To tacklethis problem with the WBM, the problem domain ispartitioned in 10 bounded subdomains. Each object isconsidered in a separate level, yielding in total 5 levelsin the model.
I
�t,L2
�t,L5
�t,L1
�t,L3
�t,L4
II
III
IV
V
VI
VII
VII
I
IXX
qp s
R=1
R=1.5
R=1
R=1
R=0.75
R=0.75
1 2 3 4 5
6
7
8
9
10
Figure 4: Problem definition
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x[m]
y[m
]
−6 −4 −2 0 2 4−5
−4
−3
−2
−1
0
1
2
0 0.2 0.4 0.6 0.8 1
Figure 5: Pressure Amplitude [Pa], 3000Hz
x[m]
y[m
]
−6 −4 −2 0 2 4−5
−4
−3
−2
−1
0
1
2
0 0.2 0.4 0.6 0.8 1
Figure 6: Relative pressure amplitude error [%], 3000Hz
The acoustic fluid is water (c = 1500m/s, ρ0 = 1000kg/m3).The system is excited by a point source in the c-shapewith amplitude q = 1.
Several indirect variational BEM models are built forthis problem using LMS/Sysnoise Rev5.6 . The meshdetails are given in table 1. The number of DOF’s inthe wave model used to solve this problem varies withthe frequency, and ranges between 1225 and 2347 andthe model is constructed such that the calculation timeto calculate a frequency response function (FRF), con-sisting of 900 frequency lines between 3000 and 7500Hzis the same as the time needed by the most coarse BEmodel. All calculations are performed on a 2.66GHzLinux-based Intel Xeon system.
element � calculation εav
size DOF time [s] [dB]2.5mm 27484 540000 /
BEM 5mm 13536 83160 0.74607.5mm 9160 27450 1.938212.5mm 5498 6030 5.2144
WBM / 1225-2347 5760 0.7431
Table 1: Model information
The contours in figure 5 show the amplitude of theacoustic pressure field due to the acoustic point sourceat a frequency of 3000Hz, calculated using the WBM.In figure 6, the relative error of this pressure amplitudewith respect to the results from the most detailed BEMcalculation is plotted. It is observed that the errors re-main well within the range of 0 − 1%, except at thepressure nodal lines, where the error calculation itself isinaccurate due to almost-zero division. The errors arealso higher along the coupling arcs between boundeddomains and the multi-level unbounded domain. Thesecoupling errors do however not influence the predictionaccuracy in the rest of the problem domain.
Figure 7 compares the frequency response function ofthe acoustic pressure amplitude in response point 6 in-dicated in figure 4 for the BEM reference model and theWB model. The good match between the WBM and theBEM reference is evident from this figure. A more pre-cise comparison is made in figure 8 where the prediction
errors (averaged over all the response points in figure 4)for three BEM models and one WBM model are shown.It is clear that the WBM result (bottom figure) is farmore accurate than the one obtained by a BEM modelwith the same calculation time (top figure). As shownin the two middle figures, the BEM mesh needs to berefined twice in order to obtain the same overall aver-age prediction accuracy as the WBM calculation. Thisis also indicated by the frequency averaged predictionerrors εav in table 1. This results in an average compu-tation time of 92.4s per frequency. It can be concludedthat for this validation example the WBM efficiency isbetter by a factor of about 14. This clearly illustratesthe advantageous properties of the proposed modellingconcept.
3000 4000 5000 6000 7000
45
50
55
60
65
70
75
80
85
Frequency [Hz]
Pre
ssur
e [d
B]
BEM 2.5mm Tav
= 660s
WBM Tav
= 6.4s
Figure 7: Pressure amplitude FRF [dB re 2.10−5Pa]comparison: WBM and BEM reference
5 Conclusion
This paper discusses a new modelling concept, particu-lary suited for the treatment of scattering problems in-volving multiple objects. The main idea of the approachis to consider the multiple objects in the problem as dif-ferent ’levels’ of the problem. Each level considers thescattering on one particular object, using existing WBMtechniques for bounded and unbounded problems. A
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0
1
2
3er
ror
[dB
]BEM 12.5mm T
av = 6.7s
0
1
2
3
erro
r [d
B]
BEM 7.5mm Tav
= 30.5s
0
1
2
3
erro
r [d
B]
BEM 5mm Tav
= 92.4s
3000 3500 4000 4500 5000 5500 6000 6500 7000 75000
1
2
3
erro
r [d
B]
Frequency [Hz]
WBM Tav
=6.4s
Figure 8: Pressure amplitude error [dB re 2.10−5Pa] comparison: WBM and BEM models
special compound wave function set for the unboundedpart and an adapted weighted residual formulation linkthe different levels together, yielding a single multi-levelsystem, describing the entire problem.
The new method is validated on a numerical example,indicating both the excellent accuracy and the superiornumerical performance as compared to the BEM. Thisreduction in computational load, combined with the lackof pollution errors, makes the WBM particulary suitedfor the treatment of multiple scatterer problems in anextended frequency range, as compared to the elementbased methods.
Acknowledgements
The research of Bart Bergen is funded by a Ph.D. grantof the Institute for the Promotion of Innovation throughScience and Technology in Flanders (IWT-Vlaanderen)and Bert Van Genechten is a Doctoral Fellow of theFund for Scientific Research - Flanders (F.W.O.), Bel-gium.
References
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[10] B. Pluymers, B. Van Hal, D. Vandepitte, andW. Desmet. “Trefftz-based methods for time-harmonic acoustics”. Archives of Computa-tional Methods in Engineering (ARCME), DOI:10.1007/s11831-007-9010-x:343–381, 2007.
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