SOUND SCATTERING FROM ACOUSTICALLY TREATED BODIES USING A DOMAIN DECOMPOSITION TECHNIQUE

Transcription

1 th AIAA/CEAS Aeroacoustics Conference (th AIAA Aeroacoustics Conference) - Ma 9, Miami, Florida th AIAA/CEAS Aeroacoustics Conference (th AIAA Aeroacoustics Conference) AIAA 9-9 AIAA-9-9 SOUND SCATTERING FROM ACOUSTICALLY TREATED BODIES USING A DOMAIN DECOMPOSITION TECHNIQUE Mohamed A. Saed Department of Mechatronics, German Universit in Cairo, Cairo, Egpt and Basman Elhadidi Department of Mechanical and Aerospace Engineering, Sracuse Universit, Sracuse, NY, Airframe noise is a significant component of the aircraft signature. In this paper the impact of coating airfoils with an acoustic absorbing material is considered. The finite element method (FEM) is implemented to model the homogenous uniform air, the non-homogenous flow and the isotropic thin absorbing laer. Each domain has different governing equation which requires the use of a domain decomposition (DD) approach using appropriate coupling conditions. Appropriate causalit conditions are implemented at the far field to ensure no reflections of waves in the near field. The numerical technique was verified using the method of manufactured solution (MMS) and validated with some of the results published in literature. Results show that the unstead lift depends on the airfoil camber/chord ratio, reduced frequenc of incident acoustic wave and direction, and absorbent impedance. The results show that a thin isotropic absorber coating an airfoil modifies both the acoustic pressure lobes in the far field, the unstead lift and attenuates the propagating sound power. The sound power attenuated for the NACA coated with a thin absorber with acoustic permittivit ε = -.i, subject to an incident plane wave with reduced frequenc k=, is 7. db. Nomenclature Latin Greek Superscripts a clinder radius α Angle of attack R Vortical velocit A The comple admittance ε Acoustic permittivit ' Perturbation c Speed of sound ϕ Velocit potential i Imaginar unit ( ) Γ Artificial boundar Subscripts k Wave number L Differential operator n Normal direction M Mach number Ω Natural frequenc o Stead flow quantit n Normal direction ρ Densit abs Absorber p Pressure ζ, η Quadrilateral coordinates inc Incident field R Radius w µ Acoustic permeabilit air Outer air domain Galerkin weighting function τ MMS test function Position vector I. Introduction HE number of operational aircrafts will dramaticall increase in the net decade. This increase is associated Twith the larger travel capacit and larger trade demand. Such an increase will necessaril increase aircraft traffic volume at currentl congested airports which will increase the perceived sound levels and affect the communities living close to the airports. The challenge of increasing aircraft numbers while pertaining (and reducing) the current Assistant Lecturer, mohammed.saed@guc.edu.eg Visiting Assistant Professor (on sabbatical leave from Aerospace Engineering, Cairo Universit, Giza, Egpt), belhadid@eng.cu.edu.eg Copright 9 b the, Inc. All rights reserved.

2 th AIAA/CEAS Aeroacoustics Conference (th AIAA Aeroacoustics Conference) AIAA-9-9 noise levels has led the industr and research communit on a quest to reduce the aircraft noise b db b. Figure summarizes NASA s long-term goal to reduce aircraft noise b. Such aggressive noise reduction requires the investigation of new sound reducing technologies, new engine concepts, reduction of airframe noise, and new landing and take-off strategies to reduce airport footprints. The objective of this paper is to simulate the scattered acoustic waves propagating from acousticall treated bodies. Different effects will be eamined in this paper, such as the treatment material, thickness and frequenc of the incident acoustic waves impinging on the D bodies. Figure. Net generation technolog aim to reduce noise b - db b [] In aeroacoustics, the convected wave equation is the basic equation governing the propagation and scattering of time-harmonic sound in uniform flow []. The acoustic propagation inside the absorber is governed b the comple Helmholtz's equation. This equation is derived b Baumeister and Kreider [8] for general non isotropic material. In their stud the eamined the benefit of using soft absorbing bodies and compared the results to hard surfaces. In this paper the benefit of using thin absorbing material will be eamined to attenuate the scattered sound field. The standard Galerkin FEM has recentl become the most popular method to solve the Helmholtz's equation since the do not have the restrictions of BEM's, however; the need special treatment for unbounded problems. The numerical advantage of the FEM is that the lead to sparse matrices which significantl speeds up computations and reduces memor requirements. The solution of the acoustic wave equation requires two tpes of boundar conditions which are: (i) the impermeabilit boundar condition which is imposed on the solid surface of the bod and (ii) the non-reflecting boundar condition. Resiga and Atassi [] studied the effect of the different far field non-reflecting boundar conditions for the Helmholtz's equation with FEM implementation on the solution accurac and compared the simulations to the analtical eact solution. Different tpes of non-reflecting boundar conditions used in the acoustic problems are summarized b Thompson []. In this paper the simplest condition known as the Sommerfeld condition is applied since the emphasis is on modeling and eamining the effect of the absorbing materials, which performs quite well for large frequencies values []. To ensure the validit of the calculations, the numerical model is compared to the results presented b Baumeister and Kreider [8] whom solved the Helmholtz's equation. To verif the numerical solver the Method of Manufactured Solution (MMS) technique proposed b Roache [9] was applied, since the method can easil trace errors in the absence of eact solution. In summar, in this paper the standard Galerkin FEM is implemented to model the propagation of acoustic waves in air and the isotropic thin absorbing laer. Each domain has different governing equation which requires the use of the Domain Decomposition (DD) approach. Appropriate coupling causalit conditions are implemented at the far field to ensure no reflection of waves in the near field. The numerical technique is verified using the Method of Manufactured Solution (MMS) and validated with some of the results published in literature.

3 th AIAA/CEAS Aeroacoustics Conference (th AIAA Aeroacoustics Conference) AIAA-9-9 II. Problem Formulation A. Governing Equations In the present work a Domain Decomposition Method (DDM) for an eternal scattering problem in an infinite domain is developed. The DDM splits the domain into two subdomains; the absorber and the homogenous air domain because the have different governing equations as shown in figure. Farfield boundar Г Isotropic absorbent Rigid bod surface +ve α w Homogeneous uniform mean flow domain Absorbing bod surface Incident plane wave Figure. Schematic for the different domains in the model problem. In the homogenous domain the governing equations for inviscid flow and non-heat conducting ideal gas with constant specific heat coefficient reduce to the Euler equations. For D irrotational flows, the disturbance velocit, u, can be further separated to two distinct disturbances, u= ϕ+u R, where ϕ represents the acoustic potential and u R, represents the vortical disturbances [,]. The vortical component is independent of the pressure term, while the acoustic part is governed b an inhomogeneous convected wave equation. The governing equations, for no mean stead flow, reduce to the unstead wave equation for acoustic disturbances with u R =, and is given b, ϕ ρo ( ρo ϕ ) = () t co t The unstead pressure perturbation p' is defined b, ϕ p = ρo () t ) ) i t In the frequenc domain the solution is assumed in the form, { p, ϕ} = { p, ϕ } e ω, where ω, represents the disturbance frequenc. Substituting this into equations () and (), we get the classical Helmholtz equation, (dropping the hat), ϕ + k ϕ = () p = iωρoϕ where k = ω co. Absorbing materials are used to absorb and attenuate the incident acoustic waves and scattered waves from the solid surface. Inside the absorber the material wave equation is used to solve for unknown values of the pressure perturbations given absorbent acoustic impedance properties permittivit ε, and permeabilit µ [8]. For time harmonic pressure propagation in a heterogeneous bulk material, Baumeister and Dahl (987) [7] used the continuit and momentum equations to derive the dimensionless wave equation (). p p + + ω µ p = () ε ε Assuming the interior domain to be isotropic (ε and µ are constant) this reduces to, + k p = () ( abs ) The reduced frequenc used in this equation is defined b, kabs = ω εµ. B. Boundar Conditions and Incident Wave

4 th AIAA/CEAS Aeroacoustics Conference (th AIAA Aeroacoustics Conference) AIAA-9-9 For the solution of the acoustic problem with an absorbent material covering the solid surface we solve equation (). The unknown values for this equation are the acoustic pressure p. From the momentum equation, the condition for zero normal velocit is, p n b = (6) where, n b is the normal to the bod. The coupling conditions at the absorber-air interface are the continuit for both pressure and the normal velocit so, pabs = pair + pinc. ( + ) n (7) uabs uair uinc = Where p abs, p air, and p inc, are the absorber, air and incident pressure respectivel and u abs, u air and u inc are the absorber, air and incident velocit respectivel, and n is the normal direction to the absorber-air bod. The acoustic velocit for disturbances in air is given b, uair n = ϕ n (8) The normal velocit in the sound absorbent is related to the pressure inside the absorbing material [8], b the relation, A uabs n = pabs n (9) ikabs Where the comple admittance A, is zero for the case of a hard wall. For a soft absorbing wall the admittance is given b, A = ( ρocoε). The incident acoustic wave is represented b a plane wave, i( k ωt) ϕ = e. () inc The reflections caused b the boundar condition on Γ will contaminate the solution in the interior homogenous domain so to avoid the reflection of the outgoing waves on the boundar of the computational domain will be either infinite domain which is not applicable or b using an eact boundar condition at specified finite domain to prevent all reflections at the boundar as indicated b the name non-reflecting boundar conditions []. The Sommerfeld condition is used because it is eas to appl, least epensive, and gives reasonable results compared to the others non-reflecting boundar conditions for high frequenc []. The boundar condition for the D Helmholtz's equation is given b, ϕ lim r ikϕ =. () r r III. Finite Element Implementation A. FEM for Homogeneous Domain The governing equations for both homogenous domain and the isotropic domain are discretized b the Finite i t Element Method (FEM). Appling the the Galerkin weighted residual, w( ) = we ˆ ω, the convected wave equation () in weak form reduces to, (dropping hat for convenience), k wϕ + w ϕ dv w ϕ ds = n V Appling the FEM we approimate ϕ, as n ϕ = Niϕi i= S Where ϕ are the nodal values and the basis functions, N i for two dimensional iso-parametric elements are given b, i N ( ζη, ) = ( + ζ)( η), N ( ζη, ) = ( + ζ)( + η) () N ( ζη, ) = ( ζ)( η), N ( ζη, ) = ( ζ)( + η). The standard Galerkin FEM equations reduce to, () ()

5 th AIAA/CEAS Aeroacoustics Conference (th AIAA Aeroacoustics Conference) AIAA-9-9 ϕi ( k Ni N j + N j Ni ) dω = N j dγ () i Ω Γ ϕ n The use of numerical integration is essential for practical evaluation of integrals over iso-parametric element domains. The standard practice has been to use Gauss integration because such rules use a minimal number of sample points to achieve a desired level of accurac. B. FEM for Isotropic Absorber Domain For the isotropic domain the Helmholtz's equation is used to solve the unknown values of the pressure perturbations in the absorber. The finite element model in this domain reduces to, p pi ( kabs Ni N j + N j Ni ) dω = N j dγ (6) n i Ω Γ C. Dumm Element Method The method of dumm element is used to appl the boundar conditions at the interface between the different domains, in this method the dumm elements (zero length in the normal direction) are to be used as shown in figure. Solid Absorber Dumm Laer Air Domain Figure : Dumm Elements Line. The pressure and velocit potential continuit conditions are applied over this line element, so from equations (6) and () we get, p = ρ iω ϕ + ϕ (7) ( inc ), o,. ϕ A p ϕ N j dγ = N j dγ n, ikabs n, n (8) Γ Γ inc Using the partitioned stiffness matri is helpful in this case because the unknown nodal values will be in terms of acoustic velocit potential and acoustic pressure and the dumm elements will couple the solution for whole domain. Figure shows the stiffness matri of the domains coupling between the absorber and air, between the unknown nodal values of pressure perturbation and velocit potential.

6 th AIAA/CEAS Aeroacoustics Conference (th AIAA Aeroacoustics Conference) AIAA-9-9 Figure : The stiffness matri representation indicated the coupling at (a) the air-absorber interface for DD D. Validation and Verification of the Numerical Model The Method of Manufactured Solutions (MMS) [6, 9] is a ver simple process and is recommended as a general procedure for verification of codes. A manufactured solution is an eact solution to the PDE or set of PDE s that has been constructed b solving the problem backwards. The technique can be used to estimate the accurac of the numerical method and the order of the numerical scheme. To appl the procedure an eact solution is assumed that does not satisf equation the PDE in general. This then reduces to, l ( ϕ ) = R, (9) where l represents the operator of equation () and R is the resulting residual. The finite element method is then applied to solve equation (9), and the solution is compared to the assumed solution and error is computed. This method can be used to verif the numerical scheme accurac as well as validation of the numerical code. Figure presents the convergence of the numerical code as a function of the number of elements using a test functionτ = cos cos. - NOE = NOE = 6 Numerical error - - NOE = NOE = - No. of Elements (NOE) Figure : The results of MMS to check the all derivatives using test function τ = cos cos. Figure 6 compares the numerical solution of acoustic scattering of a plane wave incident on a clinder with the eact solution published in [8]. The total acoustic pressure (scattered and incident) directivit is shown for the case of a single domain. The results compare well at two radii r=7.9a, and r=.9a, where a is the radius of the clinder. The reduced frequenc, k =k=, and the incident wave is moving parallel to the -ais. 6

7 th AIAA/CEAS Aeroacoustics Conference (th AIAA Aeroacoustics Conference) AIAA R=7.9a R=.9a Figure 6: The results of the single domain at k = for (a) r = 7.9 and (b) r =.9 E. Validation of the Domain Decomposition To eamine the accurac of the DD method used here, a test case was performed in which the inner domain material properties was selected ε =, µ = such that absorbent material can be modeled as solid bod. The solution is compared to a single domain solution and it is shown that the coupling interface between the absorber and the air is performing well with no reflections in the outer domain. Figure 7 compares the far field directivit at R=.8a, for those two simulations and figure 8 compares the imaginar component of the total acoustic pressure in the domain. The results are a good indication that the domain decomposition methods works efficientl in the numerical model. The Directivit at the Far field Single domain -Domains The contour plot of the imaginar part (single domain) The contour plot of the Imaginar part (-domains) Figure 7: DD validation, Polar plot of the total acoustic pressure around a solid clinder subjected to a plane wave. The results of the single domain compared to the DD results at k =, ε =, µ =, and at r = (a) (b) Figure 8: DD validation, The contour plot of the imaginar part of the total acoustic pressure for (a) single domain, (b) -domains and The results of the single domain compared to the DD results at k =, ε =, µ =, and at r =

8 th AIAA/CEAS Aeroacoustics Conference (th AIAA Aeroacoustics Conference) AIAA-9-9 IV. Stud Results One of the aircraft noise sources is that generated b incident acoustic waves on the tail unit. This arises from scattered wave produced from the engines, structural noise generated b control surfaces and landing gears. The cross section of the vertical tail unit is that of a smmetric airfoil, whereas the horizontal stabilizer unit can be that of a cambered airfoil. Both smmetric and cambered airfoil will be eamined in this section. For all simulations it is assumed that the far field boundar is at a radius R ma = c, where c is the airfoil chord. Several simulations were performed with R ma > c, to verif that no reflections eist in the smaller domain and that the far field boundar is placed far enough. For all simulations the flow velocit is zero. A. Sound Scattering For Different Reduced Frequencies Figure 9 compares the real part for the total acoustic pressure for an incident acoustic wave with k =k, and k =. As the reduced frequenc increases the number of lobes increase and there is stronger directivit in the aft lobes compared to the forward lobes as shown in figure. The intensit of the acoustic pressure increases as the frequenc increases as well. The acoustic pressure here is normalize b ρ o ω. This suggests that the scattering from the airfoil surface after the point of maimum thickness is sensitive to the airfoil contour and camber (as will be shown). The contour plot of the real part.6 The contour plot of the real part (i) k = (ii) k = The contour plot of the real part The contour plot of the real part (iii) k = (iv) k = Figure 9: Contour plots of the total acoustic pressure of incident plane wave k =k, k = for NACA

9 th AIAA/CEAS Aeroacoustics Conference (th AIAA Aeroacoustics Conference) AIAA k =. k = k = k = k =. 8 7 Figure : Directivit plot of the acoustic pressure of incident plane wave k =k, k = for NACA. B. The Effect of Blade Thickness Tpicall blade thickness is important for structural considerations. Smmetric airfoils are alwas used in the vertical fins of the tail unit. Here we eamine the effect of blade thickness on the acoustic sound propagation. Figures a shows the directivit plot of the far field scattered acoustic pressure for different smmetric NACA airfoils for k =k =, k =. The results show that the increase in airfoil thickness increases the magnitude of the scattered acoustic wave as the reduced frequenc increases. The results suggest that the increase is a linear function of the airfoil thickness; however the increase in magnitude is ver small which is epected in this case, because the airfoil is smmetric and the incident wave has zero angle, so there is noise cancellation. Figure b shows the same set of airfoils with k=, and the incident wave angle is o with respect to the horizontal. The effect of thickness here is small, and appears in the lower lobes onl, however there is more significant noise radiation since the upper and lower surfaces have different pressure distributions and there is no noise cancellation NACA 6 NACA 9 NACA NACA NACA (i) k =k =, k = (ii) k =, α= o Figure : Directivit plot of the scattered acoustic pressure of incident plane wave k =, k = for different blade thickness C. The Effect of Incident Wave Angle To eamine the effect of the wave incident angle, consider the case of NACA airfoil with different reduced frequencies k = and different incidence angle α inc =, and 9. Figures compares the far field pressure directivit for the scattered acoustic pressure for this case, and the results show that the pressure intensit depends on the incidence angle of the acoustic wave and can change considerabl. The maimum value of the intensit is the case of a wave incident normal to the airfoil chord. To compute the unstead lift, the normalized unstead pressure difference is computed, which is defined b, plower pupper p =, () ρω o 9

10 th AIAA/CEAS Aeroacoustics Conference (th AIAA Aeroacoustics Conference) AIAA-9-9 As shown in figure the unstead pressure difference is zero at all reduced frequencies at zero wave angle, which is epected due to smmetr of the airfoil. As the wave angle increase the unstead pressure difference increase due to the difference between the upper and lower surface and also increase with the reduced frequenc increases without shape distribution changes. The unstead pressure difference distribution α w = α w = α w = 9 Real p' Figure : Directivit plot of the scattered acoustic pressure of incident plane wave for different wave angle and at k =, k = for NACA. - α w = -. α w = α w = /c Figure : The real plot of the unstead pressure distribution of incident plane wave for different wave angle for NACA. D. The Effect of Blade Camber This stud compares the case of non-lifting bodies with lifting bodies without an absorbing material. To see the effect of loading the case of cambered airfoils at the same thickness to chord ratio are eamined. Figure compares the scattered real component of the pressure propagation from the airfoil surface for a NACA and NACA airfoil sections with k =k =, k =. The results suggest that as the camber increases the sound propagation from the upper surface increase while decrease from the lower surface due to the camber of the airfoil. The contour plot of the real part NACA. The contour plot of the real part NACA Figure : The contour plot of the real part of the acoustic pressure of incident plane wave for k =, k = for different airfoil sections. Figure compares the far field pressure for different cambered airfoils with k =k =, k =k =, k = and k =k =, k =. For low reduced frequenc k, the effect of the cross section is negligible in the far field. However as k increases the camber increases the pressure intensities on the upper surface and reduces the intensities on the lower surface. Also for ver large k (k = ) a front lobes is created which is the effect of the leading edge scattering which was not apparent for low k. -

11 th AIAA/CEAS Aeroacoustics Conference (th AIAA Aeroacoustics Conference) AIAA NACA NACA NACA NACA NACA NACA NACA NACA NACA NACA NACA NACA NACA NACA NACA (i) k = (ii) k = (iii) k = Figure : Directivit plot of the acoustic pressure of incident plane wave for k =k, k = different airfoil sections. 7 E. Sound Scattering From a Coated Loaded Airfoil Here we consider a NACA, coated with a thin absorbent, chord c =, R ma = c, and an incident acoustic wave k =k = and. The airfoil aerodnamic characteristics are ver critical in the aircraft design; hence the coated material thickness should be within certain limit such that it doesn't change the aerodnamic performance. For this limitation the thickness of the absorber will be.% of the chord which has minimal effect on the aerodnamic shape. Figure 6 compares the far field directivit for different absorber materials suggested in this stud from the report b Baumeister and Kreider [8]. From this figure it is clear that most tested materials attenuate the upper lobes, more than the frontal lobes. solid ε = -.i ε = -i ε = -.i ε = -i Solid ε = -.i ε = -i ε = -.i ε = -i (i) k = (ii) k = Figure 6: Directivit plot of the acoustic pressure of incident plane wave at zero angle and R ma = for different absorber material properties ε and µ at constant absorber thickness. % for NACA. 7 To assess the effect of the absorber, the sound pressure level (SPL) in decibels is defined b, p ( ) log rms SPL db =. where, p = RMS p. rms ( far ) () () Table compares the values of the unstead lift and the reduction in SPL. From this we can see that the absorber material increase the unstead lift which ma here causes some structural impact. Also we find that the material with acoustic permittivit ε = -.i which gives the largest decrease in the SPL b 7. db.

12 th AIAA/CEAS Aeroacoustics Conference (th AIAA Aeroacoustics Conference) AIAA-9-9 Table : The unstead lift coefficient and the SPL for different absorbent material for NACA at zero wave angle. ε c l k = k = SPL (db) SPL decrease in db c l SPL (db) SPL decrease in db Solid i i i i i i V. Conclusion A finite element method has been successfull implemented to model the sound propagation scattered from an incident wave on an acousticall treated two dimensional bod. To account for the different governing equations for the propagation inside the absorbing material and in the air, a domain decomposition technique has been implemented with appropriate matching conditions on the interface. This technique results in a partitioned matri with coupling terms which can be parallelized in the future easil to solve for higher reduced frequencies. The numerical model was validated with published data, and verified b using the method of manufactured solutions technique. Numerical results suggest that the amplitude and number of acoustic lobes in the far field downstream of the airfoils increase as the reduced frequenc of the incident acoustic waves increases. Furthermore, the amplitude of the scattered pressure waves increase linearl with an increase in airfoil thickness. The results also show that the airfoil camber tends to increase the intensit of the upper right lobes and reduces the lower right lobes as suggested b the camber profile for all frequencies. The results eamine the effectiveness of appling a thin absorbent laer on a NACA solid airfoil to attenuate the unstead lift and far field scattered sound intensit. The results show some promising results with sound power level attenuation of up to 7 db for some of the eamined materials. The results of the paper are encouraging and for future work, it is proposed to search for a material with optimal sound attenuation capabilit and to work closel with material science eperts to eamine how it is possible to manufacture such suggested materials. The model has been developed to include uniform mean flow which is subject of a paper in the net AIAA meeting. Acknowledgments Mohamed Ahmed would like to epress his gratitude to the German Universit in Cairo for supporting this work. The authors would like to epress gratitude to Professor M. Madbouli, Professor of Aerodnamics, Facult of Engineering, Cairo Universit for his comments and help. Basman Elhadidi would like to acknowledge the support of the AFSOR, DR. John Schmisseur, Program manager for funding portions of this stud. References Kreszig. E., Advanced Engineering Mathematics, 9 th ed., Wile,. Appelö. D, Non-reflecting Boundar Conditions for Wave Propagation Problems, Licenciate s Thesis, Roal Institute of Technolog, Department of Numerical Analsis and Computer Science, Stockholm. Givoli, D., and Patlashenko, I., Finite-Element Solution of Nonlinear Time-Dependent Eterior Wave Problems, Journal of Computational Phsics, Vol., 998, pp. 8. Atassi, H.M, Unstead aerodnamics of vortical flows: Earl and recent developments, Smposium on Aerodnamics and Aeroacoustics, Feb. 8 March, 99. Harari, I., and Slavutin, M., Analtical and numerical studies of a finite element PML for the Helmholtz equation, Journal of Computational Acoustics, Vol. 8, No.,, pp. 7.

13 th AIAA/CEAS Aeroacoustics Conference (th AIAA Aeroacoustics Conference) AIAA Salan, K. and Knupp, P., Code Verification b the Method of Manufactured Solutions, SAND- Unlimited Release, June. 7 Baumeister, J. and Dahl, M., A Finite Element Model for Wave Propagation in an Inhomogeneous Material Including Eperimental Validation, NASA-TM-9, Sunnvale, California, October 9-, Baumeister, J., and Kreider, K., Modal Element Method for Scattering of Sound b Absorbing Bodies, NASA-TM- 69, Anaheim, California, November 8-, Roache. P., Verification and Validation in Computational Science and Engineering, st ed., Hermosa Publishers, 998. Susan-Resiga, R., and Atassi, H., A Domain Decomposition Method for the Eterior Helmholtz Problem, Journal of Computational Phsics, Vol. 7, 998, pp. 88. Susan-Resiga, R., and Atassi, H., Finite element implementation of nonreflecting far-field conditions for unstead aerodnamics and aeroacoustics, 9 th AIAA Aeroacoustics/CEAS Conference,, AIAA--9. Thompson, L., Finite element methods for acoustics, Journal Acoustic Societ America,.