Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 44, 4, pp. 849-865, Warsaw 2006 LIMITATIONS IN APPLICATION OF FINITE ELEMENT METHOD IN ACOUSTIC NUMERICAL SIMULATION Tomasz Łodygowski Wojciech Sumelka Faculty of Civil and Environmental Engineering, Poznan University of Technology e-mail: Tomasz.Lodygowski@put.poznan.pl; Wojciech.Sumelka@ikb.poznan.pl In the paper, we introduce information on limitations of the Finite Ele- ment Method in acoustic analysis. Difficulties that appear in acoustic analysis aremainly caused by the form of shape functions and sensitivi- ties to boundaries, so we start with a short description of mathematical background. The propositions how to overcome and simplify those disa- dvantages are summarized and illustrated with a real application. Key words: applied acoustics, finite element method 1. Introduction Although the fundamental part of themathematical descriptionof theacoustic wavewas createdmore thanonehundredyears ago (Strutt, 1945), theproblem of the effective solution to the wave equation in connection with arbitrary domains, boundary conditions and initial conditions is still vivid. In general, the partial differential wave equation can be solved only with using numerical methods. There are several possibilities, and the decision on which type of a numerical tool should be used is strictly determined by the frequency of the acoustic wave. Acoustic problems can be divided into threemain groups according to the wave frequency order (Rabbiolo et al., 2004): Low frequency problems: the response exhibits strongmodal behavior, Medium frequency problems: the response spectra exhibit high irregula- rities, indicating irregularmodaldensity.Boundary conditions, geometry andmaterials play the fundamental role. 850 T. Łodygowski, W. Sumelka High frequency problems: the response spectra are ”smooth”, indicating highmodal ”density”. Boundary conditions, geometry andmaterials are not important. The first and the second type of analysis can be efficiently solved using Boundary Element Method (BEM), and especially Finite Element Method (FEM). The third one, i.e. high frequency problems can be solved on the basis of Statistical Energy Analysis (SEA). 2. Mathematical formulation 2.1. General formulation The FEM equations are obtained from the equilibrium equation for small motions of a compressible adiabatic fluid with velocity-dependent momentum losses in the form [1] ∂p ∂x +γ(x,θi)u̇ f +ρf(x,θi)ü f =0 (2.1) where p is the excess pressure in the fluid (the pressure in excess of any static pressure), x is the spatial position of the fluid particle, u̇f is the fluid particle velocity, üf is the fluid particle acceleration, ρf is the density of the fluid, γ is the ”volumetric drag” (force per unit volume per velocity), and θi are i independent field variables such as temperature, humidity of air, or salinity of water on which ρf and γ may depend. The constitutive behavior of the fluid is assumed to be inviscid, linear and compressible, so p=−Kf(x,θi) ∂ ∂x ·uf (2.2) where Kf is the bulkmodulus of the fluid. The weak form of Eq. (2.1) (the integral one for an arbitrary variational field δp) after using Green’s theorem and substituting the constitutive law, has the final shape for a continuous domain ∫ Vf [ δp ( 1 Kf p̈+ γ ρfKf ṗ ) + 1 ρf ∂ ∂x δp · ∂ ∂x p ] dV − ∫ S δp(T(x)) dS=0 (2.3) where T(x)=−n · ( 1 ρf ∂ ∂x p ) and the vector n represents the inward normal to the acoustic mediumat the boundary. Limitations in application of the FEM... 851 2.2. Boundary conditions The last term in Eq. (2.3) introduces the boundary condition statement for acoustic problems. All of boundary conditions described below can be formu- lated in terms of T(x) – this term has dimensions of acceleration. Dividing the boundary into subregions S, the following conditions can be imposed [1]: — Sfp: where the value of acoustic pressure p is prescribed (δp=0) T(x)=n · ( ü f + γ ρf u̇ f ) (2.4) — Sft: where the inward acceleration of the acoustic medium is prescribed (γ =0) T(x)=n · üf (2.5) — Sfs: the acoustic-structural boundary, where the acoustic and structural media have the same displacement normal to the boundary T(x)=n · ( ü m+ γ ρf u̇ m ) (2.6) u m denotes the displacement of the structure, — Sfr: the reactive acoustic boundary – admittance conditions are applied T(x)=− [ γ ρf 1 c1 p+ ( γ ρf 1 k1 + 1 c1 ) ṗ+ 1 k1 p̈ ) (2.7) where the inverse of 1/k1 and 1/c1 are the spring and dashpot parame- ters, respectively, — Sfrs: the mixed impedance boundary and acoustic-structural boundary T(x)=− [ γ ρf 1 c1 p+ ( γ ρf 1 k1 + 1 c1 ) ṗ+ 1 k1 p̈ ] +n · ( ü m+ γ ρf u̇ m ) (2.8) — Sfi: the ”infinite” boundary conditions, obtained from using infinite ele- ments or from a proper choice of impedance coefficients. For the second proposal, it can be written T(x)=− ( 1 c1 ṗ+ 1 a1 p ) (2.9) — Sff: the boundary between acoustic fluids of possibly different material properties. 852 T. Łodygowski, W. Sumelka 2.3. Formulation for transient response Using Eqs. (2.4)-(2.9) in Eq. (2.3), the final variational statement for the aco- ustic medium is obtained in the following form ∫ Vf [ δp ( 1 Kf p̈+ γ ρfKf ṗ ) + 1 ρf ∂ ∂x δp · ∂ ∂x p ] dV + + ∫ Sfr δp [ γ ρf 1 c1 p+ ( γ ρf 1 k1 + 1 c1 ) ṗ+ 1 k1 p̈ ] dS+ + ∫ Sfi δp ( 1 c1 ṗ+ 1 a1 p ) dS− ∫ Sfs δp(n · üm) dS+ (2.10) + ∫ Sfrs δp [ γ ρf 1 c1 p+ ( γ ρf 1 k1 + 1 c1 ) ṗ+ 1 k1 p̈ ] dS+ − ∫ Sfrs δp(n · üm) dS− ∫ Sft δp(n · üf) dS =0 For the acoustic-structural coupling, assuming that the structural behavior is defined by the virtual work in the form ∫ V δε :σ dV + ∫ V αcρδu m · u̇m dV + ∫ V ρδum · üm dV + (2.11) − ∫ St δum · t dS+ ∫ Sfs pδum ·n dS =0 where σ is the stress at a point in the structure, p is the pressure acting on the fluid-structural interface, n is the outward normal to the structure, ρ is the density of the material, t is the surface traction and δε and δum are strain variation and variational displacement field, respectively. The final FEM equation (after linearization) in a matrix notation is −δp̂P{(M PQ f +M PQ fr )p̈Q+(C PQ f +C PQ fr )ṗQ+ +(K PQ f +K PQ fr +K PQ fi )pQ−SPMfs ü M −PPf }+ (2.12) +δuN{IN +MNMüM +CNM(m) u̇ M +(S QN fs )⊤pQ−PN}=0 where δpP = d2(δp̂P)/dt2. In Eg. (2.12), a semidiscrete approximation is assumed and the pressure and displacement fields are interpolated as follows p=HPpP um =NNuN (2.13) Limitations in application of the FEM... 853 where p=1,2, . . . up to the number of pressure nodes and N =1,2, . . . up to the number of displacement degrees of freedom (for detailed information see Sumelka, 2004). 2.4. Formulation for steady-state response For steady-state analysis, we assume that all degrees of freedom and loads vary harmonically at the angular frequency Ω, so in general for an arbitrary variable field we can write f = f̃ expiΩt (2.14) where f̃ denotes a constant complex amplitude. By analogy to Eq. (2.3) and using the basic assumption (Eq. (2.14)) we obtain − ∫ Vf δp Ω2 Kf p̃ dV + ∫ Vf 1 ρ̃ ∂ ∂x δp · ∂ ∂x p̃ dV − ∫ S δpT̃(x) dS=0 (2.15) where T̃(x)=− 1 ρ̃ ∂ ∂x p̃ ·n and ρ̃≡ ρf + γ iΩ is the complex density. After applying an analogous form of boundary conditions as for the tran- sient one, the final variational statement for steady-state analysis becomes ∫ Vf [ −Ω2δp ( 1 Kf p̃ ) + 1 ρ̃ ∂ ∂x δp · ∂ ∂x p̃ ] dV + ∫ Sft Ω2δpn · ũf dS+ + ∫ Sfr∪Sfi δp (iΩ c1 − Ω2 k1 ) p̃ dS+ ∫ Sfrs δp (iΩ c1 p̃− Ω2 k1 p̃+Ω2n · ũm) dS+(2.16) + ∫ Sfs Ω2δpn · ũm dS =0 ThefinalFEMequation for steady-state problems, as for the transient one, for acoustic-structural couplingwe introduce by comparison to Eq. (2.15) and Eq. (2.11) in the form −δp̂P{[−Ω2(M PQ f +M PQ fr )+ iΩ(C PQ f +C PQ fr )+K PQ f ]∆p̃Q+Ω2SPMfs ∆ũ M+ −∆P̃Pf }+ δu N{[−Ω2MNM +iΩ(CNM(m) +C NM (k) )+K MN]∆ũM + (2.17) +(S QN fs )⊤∆p̃Q−∆P̃N}=0 854 T. Łodygowski, W. Sumelka where δp̃P =−Ω−2δpP ∆p̃Q = [ℜ(p̃Q)+ iℑ(p̃Q)]exp(iΩt) ∆ũM = [ℜ(ũM)+ iℑ(ũM)]exp(iΩt) ∆P̃N = [ℜ(P̃N)+ iℑ(P̃N)]exp(iΩt) ∆P̃Pf = [ℜ(P̃ P f )+ iℑ(P̃ P f )]exp(iΩt) The terms ℜ(p̃Q),ℜ(ũM),ℑ(p̃Q) and ℑ(ũM) are the real and imaginary parts of the amplitudes of the response; ℜ(P̃N) and ℑ(P̃N) are the real and ima- ginary parts of the amplitude of the force applied to the structure; ℜ(P̃Pf ) and ℑ(P̃Pf ) are the real and imaginary parts of the amplitude of the acoustic traction applied to the fluid (see also Sumelka, 2004). 3. Limitations in using FEM 3.1. Choosing proper element size Thecrucial point in thepresentmostpopularfiniteelement applications is that the shape functions are linear or quadratic. Such an approximation, sufficient in most engineering applications, in acoustic analysis, where the frequency of excitations change from several to over a dozen thousand hertzs, determines the delimitation of applicability of the above mentioned method. There is a strong relationship between the wave frequency and finite ele- ment size. The approximation for the pressure field, imposed in Eq. (2.13)1, introduces linear orquadratic shape functions H.Toobtain acceptable results, the following requirements must be fulfilled [1]. For the first order elements (linear shape functions), the element dimensions have to be chosen such that the biggest one is at least six times smaller than the acoustic wavelength. For the second order elements, this requirement is twice smaller. To understand how fast such requirements can cause that hardware possi- bilities are exceeded, let us introduce a simple example. Let imagine a typical room of dimensions 2.5× 2.5× 3.0m. Assuming that computations have to cover the frequency range from 100Hz to 10kHzwith the usingfirst order ele- ments used, the total number of finite elements needed in analysis is as follows. For roomtemperature, in the casewhen frequencyof excitation reaches 100Hz the biggest element dimension should not exceed 0.57m, so ∼ 100 elements need to be used. But for frequency 10kHz, themaximal element dimension is 0.57mm, and the total number of finite elements reaches ∼ 100m. Limitations in application of the FEM... 855 Theproblemofproperdimensions of thedements is evenmore complicated if there is a possibility of wave reflection or interference of waves in themodel. In such situations, we cannot predict a priori the frequency of the resultant wave, so the solution can be obtained in an iterative process only. 3.2. Defining absorptive boundary conditions Results of numerical analysis of wave propagation phenomena are very sensi- tive to boundary conditions. Appropriate conditions have to ensure that all physical phenomena connected with waves like reflection, absorption or trans- mission are correctly defined. Eq. (2.7) introduces one of themost important boundary terms. The ”ab- sorption” coefficients (k1 and c1) enable one to obtain demanded reflection conditions from total absorption to total reflection. Unfortunately, those coef- ficients are not given by supplier of the acoustic material. What is more, the ”absorption” at the boundary is nonlinearly dependent on wave frequency. The factors k1 and c1 should be then functions of frequency, which, in ge- neral, is not implemented in popular finite element systems. The proposition how to manage with the first of the above mentioned disadvantage is shown in Sumelka (2004). Another serious limitation is that the absorptive properties ofmaterials are not only caused by physical properties themselves. The shape of boundaries plays crucial role in the process of acoustic wave energy dissipation, Fig.1. The exact mapping of such complicated boundaries would result in a large number of additional variables, so the process of boundary homogenization is needed (for detailed discussion see Sumelka ans Łodygowski (2004)). Fig. 1. A cross section through a typical acoustic material: (a) pyramid, (b) rectangular prism, (c) other 856 T. Łodygowski, W. Sumelka 4. Example 4.1. Assembly hall ”MAGNA” Assembly hall ”MAGNA” is the central point of the Educational Center (see Fig.2) at Poznan University of Technology (PUT). Fig. 2. A general view of PUTEducational Center Fig. 3. A horizontal cross section of hall ”MAGNA” The main task of ”MAGNA” hall is to create representative place for PUT ceremonies, international symposia and conferences, concertos and even movie shows. During an academic year, the assembly hall is an auditorium for students. Themost interesting thing is that the whole area of ”MAGNA” hall can be divided into three separate areas by using special ”moving” walls, Limitations in application of the FEM... 857 so three independent audiences can be obtained, Fig.3 (thick line indicates borderline). In ahorizontal cross section ”MAGNA”hall is an ellipse (seeFig.3),whose major axis is 33.85m and theminor axis is 29.53mwith total area 691m2. In the top-most point there is nearly 10m (see Fig.4). The total volume of the assembly hall is about 6750m3. Fig. 4. A vertical cross section through the major axis Theelliptic shapeof theassemblyhall isnotoptimal in the senseof acoustic properties because it is similar to a huge mirror which concentrates reflected waves at the center of ”MAGNA”. That is why all surfaces at the rear (the walls and a part of ceiling) have to assure absorptive conditions (absorption coefficient α≈ 0.8)1. In the experimental tests and numerical analysis the possibility of division the area of ”MAGNA” hall into three separated auditoria was omitted due to the fact that the structural integrity of the assembly hall has been a priority. 4.2. Numerical analysis The 3D and 2D numerical models of assembly hall ”MAGNA” are shown in Fig.5 andFig.6, respectively.Because of the abovementioned great sensitivity to the boundary shape in acoustic analysis, all details in geometry such as complexity of ceiling, stairs, seats, stage, etc.weremappedwithhighprecision. Due to the strong relationship between the length of acoustic wave and element dimensions, precisely described in Section 3.1, the analysis of a three dimensional model was impossible. 1The complete information about the problems of the Architectural Acoustic can be found in Sadowski (1971, 1976), Sadowski andWodziński (1959). 858 T. Łodygowski, W. Sumelka Fig. 5. A 3Dmodel of assembly hall ”MAGNA” Fig. 6. A 2Dmodel of assembly hall ”MAGNA” – a cross section through the major axis The total volume of the assembly hall is about 6750m3, as mentioned. Considering the complexity of geometry of ”MAGNA” hall, it can be easily shown that theminimal dimensions of the cubic finite element are about 0.2m, so theminimal number of finite elements required reaches 800k. On the basis of Section 3.1, the abovementionedmeshdensity is suitable up to ∼ 500Hz for elementswith quadratic shape function, and up to ∼ 250Hz for elementswith linear shape function. The relation between frequency and required number of finite elements is inversely proportional to the cube of frequency in 3Dacoustic simulations, so it can be easily computed that for analysis with the highest frequency acceptable by human sense of hearing, i.e. 20kHz, the total number of elements would be nearly 200G. Such high computational requirements caused that a substitutive two- dimensional model of the assembly hall was prepared (Fig.6). The 2D model was a cross section of the 3Dmodel cut through themajor axis with the total area of about 260m2. Themaximumnumber of required finite elements at the Limitations in application of the FEM... 859 whole range of frequency, i.e. from 16Hz-20kHz, was then 10m (for second order elements). Two-dimensional modeling of ”MAGNA” hall has a lot of disadvantages which limit the applicability of such an approach. A three dimensional in- terpretation of the 2D model is an infinitely long ”tube” (Fig.7), so it was impossible to model the reflection from the left and right sides, and in con- sequence, computations of the steady state-response or self response were not feasible. Fig. 7. A 2Dmodel interpretation The only way was to execute the numerical simulations as transient ones. The duration of the analysis step had to guarantee that the wave reflected from the left and right sides would not reach the analyzed cross section. We decided to divide the analysis into two steps. In the first step, the ti- me needed to travel a distance from the source to lateral walls and back to the major axis by an acoustic wave was investigated. In the second step (the principal), the essential computations were made during the time determined in the first step. In each analysis step we assumed that the room tempera- ture is considered, so the following values describing the acoustic medium were established: the density of air ρf = 1.20kg/m 3, and the bulk modulus Kf =141.8kN/m 2. Computationsweremade usingABAQUSExplicit system (based on expli- cit integration method). First step. An additional model was created. The model was a horizontal cross section of the assembly hall at the level of the center of spherical source (Fig.8). The results showed that the time needed to travel a distance from the source to lateral walls and back to themajor axis by an acoustic wave reached 860 T. Łodygowski, W. Sumelka Fig. 8. Acoustic pressure [Pa] – the investigation of the second step time duration ∼ 0.1s. This timewas recognized as a time inwhich the results obtained from the two-dimensional model (from second step) should have satisfied accuracy. Second step. The scheme of themodel withmarkedmeasure points Pi and source S is shown in Fig.6. The length of the model (identical to the major axis) is 33.85m, and in the top-most point the height is nearly 10.00m. The 2Dmodel was meshed by four node linear elements (in ABAQUS no- tation AC2D4R) whose average dimensions were ∼ 0.1m. Suchmesh density (40kdof) enabled one to lead computations for frequencies up to 500Hz. It is important to notice that due to the fact that human sense of hearing accepts the sound pressure from 2 ·10−5Pa (0dB – threshold of hearing) up to 20Pa (120dB – pain threshold), the local results has qualitative meaning in acoustic analysis. The contour maps of acoustic pressure distribution gives a general view of the solution obtained (Fig.9). Fig. 9. A contourmap of acoustic pressure distribution [Pa] for 200Hz The detailed information about analysis is placed in following Section 4.4, where a comparison between the numerical simulations and experimental tests is presented. Limitations in application of the FEM... 861 4.3. Experiment tests To validate the results obtained in computations of the mathematical model, experimental tests were carried out. A precise description of the experiment and test equipment is presented below. The acoustical experiment was pre- pared before the Educational Center was opened, on a holiday, so the results were free of incidental noise. Measurement equipment. Thewave source consisted of the following: wa- ve generator ”Metatronik” G430, frequency counter ”ALFAElectonics” FC-1200, amplifier ”LuxmanAmplifier” LV102 and sphere loudspeaker. The acoustic signal was recorded by amicrophone connected with oscil- loscope ”LeCroy” 9310M and handheld sound and vibration analyser ”SVANTEK” SVAN 912AE. The signal was calibrated by sound level calibrator KA-10. Measurement process. The experimental tests were executed in the follo- wing way. The stationary spherical source S was placed on the major axis, by analogy to numerical analysis. Measurement points Pi were si- tuated on the level of a human head while seated (Fig.6). A sinusoidal signal was generated at a range from 150Hz to 700Hz with the step 50Hz, and the sound level run from 98dB to 107dB dependently on frequency. Fig. 10. Results at the point P3 for frequency 300Hz: (a) transient case, (b) steady-state case At each point and at each frequency of interest two kinds of readings were made. The first one enabled one to observe themoment in which the acoustic wave was reaching themeasure point and the second one enabled observation 862 T. Łodygowski, W. Sumelka of the steady-state response (Fig.10). The individual reading was 0.5s long with the time sampling 1 ·10−5 s. A graphical representation of the output datawas prepared in a free scien- tific software package for numerical computations SciLab 3.0. 4.4. Comparison The comparison between the numerical analysis and experimental tests is shown in Fig.11 and Fig.12. These figures indicate differences in the acoustic pressure [Pa] from 0s to 0.1s. The first appearance of the acoustic wave in the measurement points can be observed. A mathematical idealization of the results in the artificial zero pressure before the acoustic wave came in (black line). Themesh density applied to themodel should gave satisfactory results up to 500Hz, as mentioned. Fig. 11. Comparison for 200Hz: experiment (grey line), numerical solution (black line) In Fig.11, the results of experimental tests and numerical analysis for the excitation frequency equal to 200Hz are compared. The convergence of both theamplitudeandphaseof acoustic pressure is high inallmeasurementpoints. One should notice the number or finite elements per wavelength. The average dimensions of the finite element were ∼ 0.1m, so for 200Hz there were nearly 17 elements per wavelength in the analysis. Limitations in application of the FEM... 863 Fig. 12. Comparison for 300Hz: experiment (grey line), numerical solution (black line) The last sentence has a fundamentalmeaning.Comparisonsmade for other frequencies of interest showed that although the accuracy in phase is high eno- ugh, the convergence in amplitude of traveling waves decreases considerably. In Fig.12 the results of experimental tests and numerical analysis for the exci- tation frequency equal to 300Hz are compared. It is clearly visible that the convergence in amplitudewas lost. By simple computations, we can determine the number of elements per wavelength just as before. Because of the mesh density,whichdidnot changeduring theanalysis, for 300Hz therewere around 11 elements per wavelength in the analysis. For upper frequencies such effects were more andmore sharply outlined. 5. Conclusions Limitations in the application of FEM in acoustic analysis can be divided into three groups. The first group involves problemswith proper choice of element sizes. The second group includes hardware problems, and the last one takes into account boundary difficulties. 864 T. Łodygowski, W. Sumelka A example of assembly hall ”MAGNA” proved all the abovementioned li- mitations. Itwas clearly seen that themeshdensity had fundamentalmeaning. Recommendations in relation to theminimal number of the finite elements per wavelength, introduced in Section 3.1 (based on [1]), are not adequate as the example of hall ”MAGNA” had prowed. The results indicated that the maxi- mal dimensionsof element shouldbeat least 10 times smaller then theacoustic wavelength. A additional limitation was the fact that only a two dimensional model could be analysed. Because of this, the first run of the acoustic wave through the assembly hall could be recognized. But not only the conclusions about the convergence of the numerical and experimental results atmeasurement points could be done. The analysis of the contour maps, showing the rate of change of the acoustic pressure in themodel (Fig.9), had indicated that the shape of the first reflected wave was well matched to the amphitheatrical auditorium, so the uniform loudness should be inside. In spite of the criticism to application of FEM in acoustic analysis, it has to be outlined that great meaning of the method in the field of acoustics is unquestionable, and is still growing along with hardware development. Acknowledgments The authors acknowledge their appreciation to Dr Z. Golec and Dr M. Golec for great help in experimental tests. The authors also acknowledge their appreciation to Prof. M. Fikus, the architect of the Educational Center, for all information about Assembly Hall ”MAGNA”. The work was partially supported by the University Grant BW11-806/06 References 1. ABAQUSTheoryManual, v6.5, 2005 2. Gołaś A., 1995, Metody komputerowe w akustyce wnętrz i środowiska, AGH, Kraków 3. Malecki I., 1969,Physical Fundations of Technical Acoustics, Oxford: Perga- mon Press,Warszawa, PWN 4. Rabbiolo G., Bernhard R.J., Milner F.A., 2004, Definition of a high- frequency threshold for plates and acoustical space, Journal of Sound and Vi- brations, 277, 647-667 5. Reddy J.N., 1976,An Introduction to The Finite Element Method, McGraw- Hill, USA 6. Sadowski J., 1971, Akustyka w urbanistyce, architekturze i budownictwie, Wyd. Arkady,Warszawa Limitations in application of the FEM... 865 7. Sadowski J., 1976,Akustyka architektoniczna, PWN,Warszawa 8. Sadowski J.,Wodziński L., 1959,Akustyka pomieszczeń,WydawnictwoKo- munikacyjne 9. Strutt J.W. (Baron Rayleigh), 1945,The Theory of Sound, Dover Publi- cations, NewYork (originally published in 1877) 10. Sumelka W., 2004, Acoustics in Structural Engineering (in Polish), M.Sc. Thesis, PUT, Poznań 11. SumelkaW., ŁodygowskiT., 2006, Substitute acoustic boundary impedan- ce conditions for boundarieswith periodic geometry in computer simulations of acoustic planar wave traveling, Journal of Mechanical Engineering, submitted Ograniczenia zastosowania Metody Elementów Skończonych w analizie numerycznej pola akustycznego Streszczenie W pracy przedstawiono ograniczenia zastosowania Metody Elementów Skończo- nych w analizie numerycznej pola akustycznego. Wykazano, iż trudności w ocenie rozkładu akustycznych pól ciśnień spowodowane są doborem funkcji kształtu, wrażli- wością nawarunki brzegowe oraz gęstością stosowanych siatekMES.Na bazie porów- nania symulacji numerycznych z przeprowadzonym eksperymentem pokazano, w jaki sposób można niektóre z tych ograniczeń pominąć oraz do jakiego stopnia można uprościć analizę akustyczną, przyjmując modele dwuwymiarowe. Manuscript received February 2, 2006; accepted for print May 5, 2006