Computational acoustics PDF
Preview Computational acoustics
CISM International Centre for Mechanical Sciences 579 Courses and Lectures Manfred Kaltenbacher Editor Computational Acoustics International Centre for Mechanical Sciences CISM International Centre for Mechanical Sciences Courses and Lectures Volume 579 Series editors The Rectors Friedrich Pfeiffer, Munich, Germany Franz G. Rammerstorfer, Vienna, Austria Elisabeth Guazzelli, Marseille, France The Secretary General Bernhard Schrefler, Padua, Italy Executive Editor Paolo Serafini, Udine, Italy Theseriespresentslecturenotes,monographs,editedworksandproceedingsinthe field of Mechanics, Engineering, Computer Science and Applied Mathematics. Purpose of the series is to make known in the international scientific and technical community results obtained in some of the activities organized by CISM, the International Centre for Mechanical Sciences. More information about this series at http://www.springer.com/series/76 Manfred Kaltenbacher Editor Computational Acoustics 123 Editor ManfredKaltenbacher Institute of Mechanicsand Mechatronics Vienna University of Technology Vienna Austria ISSN 0254-1971 ISSN 2309-3706 (electronic) CISMInternational Centre for MechanicalSciences ISBN978-3-319-59037-0 ISBN978-3-319-59038-7 (eBook) DOI 10.1007/978-3-319-59038-7 LibraryofCongressControlNumber:2017941071 ©CISMInternationalCentreforMechanicalSciences2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface ThebookisaresultoftheAdvancedSchoolComputationalAcoustics,whichtook place at the International Centre for Mechanical Sciences (CISM), Udine, Italy, in May 2016. Theaimofthisbookistopresentstate-of-the-artoverviewofnumericalschemes efficientlysolvingtheacousticconservationequations,theacousticwaveequation, and its Fourier-transformed Helmholtz equation. Thereby, the different equations model both vibrational and flow-induced sound generation and its propagation. Chapter “Fundamental Equations of Acoustics” sets the scene by providing the mathematical/physical modeling of acoustic fields. Thereby, the equations of acoustics are based on the general equations of fluid dynamics: conservation of mass, momentum, energy, and closed by the appropriate constitutive equations defining the thermodynamic state. The use of a perturbation ansatz, which decomposes the physical quantities such as density, pressure, and velocity into mean, incompressible fluctuating and compressible fluctuating ones, allows to derive linearized acoustic conservation equations and its state equation. Thereby, we derive acoustic wave equations for both homogeneous and inhomogeneous media. Chapter “Non-conforming Finite Elements for Flexible Discretization with Applications to Aeroacoustics” focuses toward non-conforming finite elements for flexible discretization. Therewith, we allow for each subdomain an optimal grid. The two proposed methods—Mortar and Nitsche-type mortaring—fulfill the physicalconditionsalongthenon-conforminginterfaces.Weexploitthiscapability and apply it to real engineering applications in aeroacoustic. The results demon- strate the superiority of the nonconforming finite elements over standard finite elements concerning preprocessing, mesh generation flexibility, accuracy, and computational time. Chapter “Boundary Element Method for Time–Harmonic Acoustic Problems” presentsthesolutionoftime–harmonicacousticproblemsbytheboundaryelement method (BEM). Specifically, the Helmholtz equation with admittance boundary conditionsissolvedinthree-dimensionalspace.Thechapterstartswithaderivation of the Kirchhoff–Helmholtz integral equation from a residual formulation of the v vi Preface Helmholtz equation. The discretization process with introduction of basis and test functions is described and shown for the collocation and the Galerkin method. Throughout the chapter, numerous different examples are presented, both simple one-dimensional examples having analytical solutions, which may be used for implementation verification, and rather industrial applications such as sedan cabin compartments, diesel engine radiation, and tire noise problems demonstrating the applicability. Chapter “Direct Aeroacoustic Simulations Based on High Order Discontinuous GalerkinSchemes”focusesondirectaeroacoustic simulationsbased on high-order discontinuousGalerkinschemes.Theframeworkpresentedisbasedonaparticular version of the Discontinuous Galerkin method, in which a nodal as well as dis- cretely orthogonal basis is used for computational efficiency. This discretization choice allows arbitrary order in space while also supporting unstructured meshes. Afterdiscussingthedetailsoftheframework,examplesofdirectnoisecomputation arepresented,withaspecialfocusonthenumericalsimulationofacousticfeedback in a complex automotive application. Numerical schemes lead to a system of algebraic equations, which needs effi- cientsolvers.Therefore,Chapter“DirectandIterativeSolvers”presentsacompact introduction to direct and iterative solvers for systems of algebraic equations typ- ically arising from the finite element discretization of partial differential equations. Beside classical iterative solvers, we also consider advanced preconditioning and solving techniques like additive and multiplicative Schwarz methods, generalizing Jacobis and Gauss-Seidel’s ideas to more general subspace correction methods. In particular, we consider multilevel diagonal scaling and multigrid methods. We have pleasure in thanking our colleagues, Gary Cohen, Dan Givoli, Ulrich Langer,SteffenMarburg,Claus-DieterMunz,andMartinNeumüllerforpresenting their lectures, and the students for attending the course and contributing to dis- cussions. Furthermore, we particularly thank the Rectors and officers at CISM for theirenthusiasm,assistance,andhospitality.Finally,wewanttothankSpringerfor their kind assistance, and especially Sooryadeepth Jayakrishnan and his team for their great job in doing the layout. Vienna, Austria Manfred Kaltenbacher Contents Fundamental Equations of Acoustics.... .... .... .... .... ..... .... 1 Manfred Kaltenbacher Non-conforming Finite Elements for Flexible Discretization with Applications to Aeroacoustics . .... .... .... .... .... ..... .... 35 Manfred Kaltenbacher Boundary Element Method for Time-Harmonic Acoustic Problems.. .... .... ..... .... .... .... .... .... ..... .... 69 Steffen Marburg Direct Aeroacoustic Simulations Based on High Order Discontinuous Galerkin Schemes.. .... .... ..... .... 159 Andrea Beck and Claus-Dieter Munz Direct and Iterative Solvers .. ..... .... .... .... .... .... ..... .... 205 Ulrich Langer and Martin Neumüller vii Fundamental Equations of Acoustics ManfredKaltenbacher Abstract The equations of acoustics are based on the general equations of fluid dynamics:conservationofmass,momentum,energyandclosedbytheappropriate constitutive equation defining the thermodynamic state. The use of a perturbation ansatz,whichdecomposesthephysicalquantitiesdensity,pressureandvelocityinto mean,incompressiblefluctuatingandcompressiblefluctuatingones,allowstoderive linearizedacousticconservationequationsanditsstateequation.Thereby,wederive acousticwaveequationsbothforhomogeneousandinhomogeneousmedia,andthe equationsmodelbothvibrational-andflow-inducedsoundgenerationanditsprop- agation. 1 Overview Acousticshasdevelopedintoaninterdisciplinaryfieldencompassingthedisciplines ofphysics,engineering,speech,audiology,music,architecture,psychology,neuro- science,andothers(see,e.g.,Rossing2007).Therewith,thearisingmulti-fieldprob- lems range from classical airborne sound over underwater acoustics (e.g., ocean acoustics) to ultrasound used in medical application. Here, we concentrate on the basicequationsofacousticsdescribingacousticphenomena.Thereby,westartwith themass,momentumandenergyconservationequationsoffluiddynamicsaswell as the constitutive equations. Furthermore, we introduce the Helmholtz decompo- sition to split the overall fluid velocity in a pure solenoidal (incompressible part) andirrotational(compressible)part.Since,wavepropagationneedsacompressible medium, we associate this part to acoustics. Furthermore, we apply a perturbation methodtoderivetheacousticwaveequation,anddiscussthemainphysicalquanti- tiesofacoustics,planeandsphericalwavesolutions.Finally,wefocustowardsthe twomainmechanismofsoundgeneration:aeroacoustics(flowinducedsound)and vibroacoustics(soundgenerationduetomechanicalvibrations). B M.Kaltenbacher( ) InstituteofMechanicsandMechatronics,TUWien,Vienna,Austria e-mail:[email protected] ©CISMInternationalCentreforMechanicalSciences2018 1 M.Kaltenbacher(ed.),ComputationalAcoustics,CISMInternationalCentre forMechanicalSciences579,DOI10.1007/978-3-319-59038-7_1 2 M.Kaltenbacher 2 BasicEquationsofFluidDynamics We consider the motion of fluids in the continuum approximation, so that a body B iscomposedofparticlesRasdisplayedinFig.1.Thereby,aparticleRalready representsamacroscopicelement.Ontheonehandaparticlehastobesmallenough todescribethedeformationaccuratelyandontheotherhandlargeenoughtosatisfy theassumptionsofcontinuumtheory.Thismeansthatthephysicalquantitiesdensity ρ,pressure p,velocityv,andsoonarefunctionsofspaceandtime,andarewritten asdensityρ(x ,t),pressure p(x ,t),velocityv(x ,t),etc.So,thetotalchangeofa i i i scalarquantitylikethedensityρis (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) ∂ρ ∂ρ ∂ρ ∂ρ dρ= dt + dx + dx + dx . (1) 1 2 3 ∂t ∂x ∂x ∂x 1 2 3 Therefore,thetotalderivative(alsocalledsubstantialderivative)computesby (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) dρ ∂ρ ∂ρ dx ∂ρ dx ∂ρ dx = + 1 + 2 + 3 dt ∂t ∂x dt ∂x dt ∂x dt 1 2 3 (cid:2) (cid:3) (cid:2) (cid:3) (cid:4)3 ∂ρ ∂ρ dx ∂ρ ∂ρ dx = + i = + i . (2) ∂t ∂x dt ∂t ∂x dt i=1 i i (cid:5) (cid:6)(cid:7) (cid:8) v i Note that in the last line of (2) we have used the summation rule of Einstein.1 Furthermore,inliteraturethesubstantialderivativeofaphysicalquantityismainly denotedbythecapitalletter DandforanEulerianframeofreferencewritesas D ∂ = +v·∇. (3) Dt ∂t Fluid particleR Fluid body B Fig.1 AbodyBcomposedofparticlesR 1Inthefollowing,wewillusebothvectorandindexnotation;forthemainoperationsseeAppendix.
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