Code_Aster. Finite elements in acoustics. Version 12. Summary:

Transcription

1 Cod_Astr Vrsion 1 Titr : Élémnts finis n acoustiqu Dat : 3/10/015 Pag : 1/14 Rsponsabl : DELMAS Josslin Clé : R Révision : Finit lmnts in acoustics Summary: This documnt dscribs in low frquncy stationary acoustics th quations usd, th variational formulations which rsult from this as wll as th corrsponding translation in finit lmnts, for ach of th two mthods usd in Cod_Astr : classical '' formulation with an unknown factor p (acoustic prssur), and mixd formulation with two unknown factors p, v (prssur and spd acoustics). Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 017 EDF R&D - Licnsd undr th trms of th GNU FDL (

2 Cod_Astr Vrsion 1 Titr : Élémnts finis n acoustiqu Dat : 3/10/015 Pag : /14 Rsponsabl : DELMAS Josslin Clé : R Révision : Contnts 1Introduction... 3 Equations and boundary conditions of th problm Equations and boundary conditions Classical formulation in prssur Mathmatical xprssion of th problm Discrtization by finit lmnts Th matrix of rigidity Th matrix of mass Th matrix of damping Th vctor sourc Mixd formulation prssur-spd Mathmatical xprssion of th problm Local formulation Mixd variational formulation Discrtization by finit lmnts Th matrix of rigidity Th matrix of mass Th matrix of damping Th vctor sourc Ordrs spcific to acoustic modling Conclusion Bibliography Dscription of th vrsions of th documnt Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 017 EDF R&D - Licnsd undr th trms of th GNU FDL (

3 Cod_Astr Vrsion 1 Titr : Élémnts finis n acoustiqu Dat : 3/10/015 Pag : 3/14 Rsponsabl : DELMAS Josslin Clé : R Révision : 1 Introduction Options of modling wr dvlopd in Cod_Astr, allowing to study th low frquncy stationary acoustic propagation, in closd mdium, for filds of propagation to complx topology, i.. to solv thr undr th quotd conditions th quation of Hlmholtz. Th solution by finit lmnts of this quation can b ralizd according to two mthods: a first mthod consists in bing fixd lik unknown factors of th problm, only th nodal complx acoustic prssurs, that is to say 1 dgr of frdom pr nod [bib1]; it is that which on dscribs as formulation to th finit lmnts classical, in th scond mthod, calld to th finit lmnts mixd, on sts lik unknown factors at th sam tim th nodal acoustic prssurs and th 3 componnts nodal vibratory spd, is on th whol 4 dgrs of frdom pr nod [bib5]. To know th ways of propagation of nrgy in th fluid, th acoustics xprt has sizs: activ acoustic intnsity I and ractiv acoustic intnsity J ; ths two sizs ar dfind lik: I = 1 R [ p v* ] and J = 1 Im [ p v* ] éq 1-1 whr v * indicat th combind complx on vibratory spd. Th knowldg of ths sizs brings a vry important furthr information in th rsolution of problms of all kinds, for xampl th masurmnt of th powrs radiatd by th machins, th rcognition and th localization of th sourcs. Th calculation of th acoustic intnsity by th finit lmnt mthod mixd must provid valus mor prcis than th classical mthod; indd in th mixd cas on nsurs th continuity of th drivativ first of th prssur and not simply th continuity of th lattr. Howvr if it is mor prcis, th mixd formulation spnds on th othr hand mor siz mmory and tim CPU, whil kping th advantag of having, with qual numbr of dgrs of frdom pr wavlngth, a rlativ rror incrasingly wakr on th calculation of th acoustic intnsity. Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 017 EDF R&D - Licnsd undr th trms of th GNU FDL (

4 Cod_Astr Vrsion 1 Titr : Élémnts finis n acoustiqu Dat : 3/10/015 Pag : 4/14 Rsponsabl : DELMAS Josslin Clé : R Révision : Equations and boundary conditions of th problm.1 Equations and boundary conditions Th quation to b solvd is th quation of Hlmholtz [bib]: k p=s éq.1-1 k indicat th numbr of wav of with th dalt problm; it can b complx or ral, according to whthr th propagation is carrid out or not in a porous fild [bib6]: k= c éq.1- c indicat th spd of sound, which can b complx in th cas of a propagation in porous nvironmnt. p is a complx siz indicating th acoustic prssur and s, also complx, rprsnts th sourcs trms of th problm. is a rality in all th cass, which indicats th pulsation: = f éq.1-3 f is th work frquncy of th harmonic problm. W rprsnt on th figur [Figur.1-a] th unspcifid confind fild whr applis th quation of Hlmholtz [éq.1-1] and th conditions to th bordrs. is opn limitd R 3 of bordr rgular, partitionné in v and z ; = v z Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 017 EDF R&D - Licnsd undr th trms of th GNU FDL (

5 Cod_Astr Vrsion 1 Titr : Élémnts finis n acoustiqu Dat : 3/10/015 Pag : 5/14 Rsponsabl : DELMAS Josslin Clé : R Révision : n Fluid z Frontièr absorbant d'impédanc localisé Z Sourc vibratoir monochromatiqu d'amplitud normal V n v z x y Figur.1-a: Configuration of th problm Th quation [éq.1-1] is to b solvd in a closd fild. Boundary conditions to tak into account on th bordr fild xprss thmslvs in thir most gnral form lik: / n appoint th oprator of normal drivativ. p p n = éq.1-4,, ar complx oprators, who can b scalars, or intgral oprators according to whthr th bordr of application of th boundary condition is with local raction or nonlocal raction (cas of th intraction fluid-structur). Dvlopmnts currntly carrid out in Cod_Astr concrn only boundary conditions with local raction, for which,, ar scalars; th cass spécifiabls ar th following: =0, 0, 0 who indicats a bordr of th fild at imposd vibratory spd. Indd, thr xists a rlation conncting th acoustic gradint of prssur complxs at th complx particulat vibratory spd. p n = j 0 V n éq indicat th dnsity of th fluid considrd, and on imposs V n, normal vibratory spd with th wall ( V n =v n whr n indicat th unit vctor of th normal xtrnal with th bordr ). 0, 0, 0 rlat to a bordr with acoustic impdanc Z imposd. Acoustic impdanc Z is dfind lik th rport of th prssur at th particulat vibratory spd in th vicinity of th wall with imposd impdanc: Z p éq.1-6 V n Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 017 EDF R&D - Licnsd undr th trms of th GNU FDL (

6 Cod_Astr Vrsion 1 Titr : Élémnts finis n acoustiqu Dat : 3/10/015 Pag : 6/14 Rsponsabl : DELMAS Josslin Clé : R Révision : 0, 0, 0 rprsnt th cas whr th acoustic prssur is imposd p at a bordr (gnrally 0, corrsponding to p 0 ). 3 Classical formulation in prssur 3.1 Mathmatical xprssion of th problm Th standard procdur aiming at posing th problm with th classical finit lmnts is th following on: on supposs th sufficintly rgular solution of th problm, p H. Th quation is multiplid: k p 0 éq by a function tst On intgrats on and th formula of Grn is usd. Th bordr fild, subdivids itslf in zons, a zon at imposd vibratory spd, v and on zon with imposd acoustic impdanc, z. Th quation obtaind can b rwrittn in th form: grad p. grad p. p.. 0 dv j ds j 0 Vn ds 0 c Z z v éq 3.1- dv rprsnt an lmnt of diffrntial volum in and ds rprsnt an lmnt of surfac on. particulat vibratory spd is thn dtrmind by: 3. Discrtization by finit lmnts j p éq v grad In th cas of th classical finit lmnts, th lmntary intgrals ar four K, M, C, U according to th dcomposition indicatd in [éq 3.-3] ( K is th matrix of rigidity, M th matrix of mass, C th matrix of damping and U th vctor sourc). Two of thm com from voluminal intgrals, th two othrs ar th rsult of intgrals rspctivly on a vibrating surfac and a surfac with imposd impdanc. It will b supposd that th total coordinats of an lmnt can b writtn thanks to th data of m lmntary functions of form H i : m OM H i i1 OM i éq 3.-1 On is givn morovr, of th basic functions N i, to dscrib th lmntary prssur. Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 017 EDF R&D - Licnsd undr th trms of th GNU FDL (

7 Cod_Astr Vrsion 1 Titr : Élémnts finis n acoustiqu Dat : 3/10/015 Pag : 7/14 Rsponsabl : DELMAS Josslin Clé : R Révision : Th prssur insid an lmnt will b abl to b writtn: m i p x, y, z N,, P i éq 3.- i1 whr P i is th prssur with th nod i lmnt. In th cas as of isoparamtric finit lmnts, th basic functions N i ar qual to th functions of form H i. On ach lmnt of th fild, th problm with th finit lmnts in prssur is writtn: whr P q 1 K M j C q P j U q q q 1 1 is th matrix column of th nodal valus of th prssur on th lmnt. éq Th matrix of rigidity grad grad Th matrix of rigidity K corrsponds to th calculation of: p. Sh admits lik trm gnral: dv K ij Ni N j dv éq Th matrix of mass Th matrix of mass M corrsponds to th calculation of: Sh admits lik trm gnral: 1 c p. dv M ij 1 c N i N j dv éq Th matrix of damping Th matrix of damping C corrsponds to th calculation of: Sh admits lik trm gnral: 3..4 Th vctor sourc C ij z 0 Z Th vctor sourc U corrsponds to th calculation of: 0 p. ds Z z N i N j ds éq v V ds Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 017 EDF R&D - Licnsd undr th trms of th GNU FDL ( n

8 Cod_Astr Vrsion 1 Titr : Élémnts finis n acoustiqu Dat : 3/10/015 Pag : 8/14 Rsponsabl : DELMAS Josslin Clé : R Révision : H admits lik trm gnral: 4 Mixd formulation prssur-spd 4.1 Mathmatical xprssion of th problm Local formulation U Vn ds 0 N i éq i Th quation of Hlmholtz [éq 1-1] with th boundary conditions [éq.1-3] rsult by way of local quations blow: v i p div v 0 i 0 v grad p 0 1 v. n p Z v. n V n dans dans sur sur z v éq éq éq éq whr 1 / 0 c is th adiabatic cofficint of comprssibility of th fluid. Th mathmatical problm is th following: bing givn functions Z L Z and V n 1 H, to find functions p and v dfind in and with valus in C chcking ths V quations. Thy dscrib, in harmonic mod of pulsation, small fluctuations in prssur p and spd v starting from at-rst stat (c.à.d. acoustic prssur and acoustic particulat spd) of a fluid comprssibl homognous, isotropic, nonviscous, confind in and subjctd to a distribution of normal vlocity V n on V., and c th dnsity, th adiabatic cofficint of comprssibility and th spd of sound rprsnt 0 rspctivly rlating to th fluid, in acoustic absnc of disturbanc; th cofficint 1/ Z is th localisd admittanc of constitunt matrial V with th pulsation considrd. To build a mthod of approximation by finit lmnts of this problm, it is ncssary to put it in a variational form Mixd variational formulation On taks th scalar product of th quation [éq ] in L with an unspcifid function q in H 1 (it is th function-tst). Th formula of Grn and th fact that v chck th boundary conditions [éq ] and [éq ] allow us to lad to: i pq* pq* v. gradq* Vn q* éq z v Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 017 EDF R&D - Licnsd undr th trms of th GNU FDL (

9 Cod_Astr Vrsion 1 Titr : Élémnts finis n acoustiqu Dat : 3/10/015 Pag : 9/14 Rsponsabl : DELMAS Josslin Clé : R Révision : On procds in th sam way with th quation [éq ] by taking his scalar product in L with a function-tst u unspcifid in L 3 on obtains: i 0 v. u * grad p. u* 0 éq Now w multiply [éq ] by j 0 and [éq 4.1.-] by j 0, thn w do it chang of function: j v Thus w obtain th mixd variational formulation [éq ]: To find p, v X M such as: v 0 v. gradq* 1/ c pq* j 0 pq* j 0V n q* q X V Z 0 v. u * 0 grad p. u* 0 u M i L 3 v v i i 1,, 3 ; vi L whr: 1 X H p L ; p / x L i 1,, 3 and: M éq Discrtization by finit lmnts. Th fild and its bordrs V and Z is cut out in lmntary filds and bordrs: on which ar calculatd lmntary intgrals. V Z,, To rprsnt th filds of p and of v insid th lmnt on uss th sam basic functions N i. Insid ach lmnt (comprising m nods) on writs: OM v i1 m m i1 m p N,, i1 N,, OM N,,,, ar th curvilinar coordinats of a thr-dimnsional lmnt; OM i is th vctor position of th nod M i lmnt with m nods; i i i Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 017 EDF R&D - Licnsd undr th trms of th GNU FDL ( p v i i i

10 Cod_Astr Vrsion 1 Titr : Élémnts finis n acoustiqu Dat : 3/10/015 Pag : 10/14 Rsponsabl : DELMAS Josslin Clé : R Révision : N i, i 1, m ar th basic functions on th lmnt ; v i acclration with th nod is th vctor M i lmnt. In this cas th systm of quations [éq ] is writtn matricillmnt for ach lmnt : p * q * q * q v K q p j p j u * v M u * v C u * S u * * éq 4.-1 Whr: t p p p v v v p v v v x y z m v mx my mz v 1, 1, 1, 1,,, is th vctor solution in th lmnt ; 4..1 Th matrix of rigidity K is th matrix of lmntary rigidity, corrsponding to th calculation of th following part of [éq ]: v. grad q* 0 v. u * 0 grad p. u * On can writ it by braking up it into mxm undr matrics K ij dimnsions 4 X 4 lik Ci - blow: 0 K K K j K Ki 1 Kij K K m1 K mj K m im mm pour i, j 1,, m with th following trms for K ij : K ij 4.. Th matrix of mass 0 N N N N j 0 i 0 i j x Ni Ni Ni 0 N j 0 N j 0 N j x y z j 0 i 0 i j y 0 0 N N 0 N N 0 N N N N j 0 i i j z Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 017 EDF R&D - Licnsd undr th trms of th GNU FDL (

11 Cod_Astr Vrsion 1 Titr : Élémnts finis n acoustiqu Dat : 3/10/015 Pag : 11/14 Rsponsabl : DELMAS Josslin Clé : R Révision : is th matrix of lmntary mass, corrsponding with th calculation of: M Its cofficints ar th following: M 1/ c pq* i j r m pour, 1,, 4 3,, 4 3 c 1/ N N avc r = 1,, m ij i j Th othr trms ar worthlss 4..3 Th matrix of damping C is th matrix of lmntary damping, coming from calculation from: Its cofficints ar th following: C 0 pq* V i j r m pour, 1,, 4 3,, N N avc r = 1,, m ij i j V Th othr trms ar worthlss Th vctor sourc S is th vctor lmntary sourc, rprsnting th calculation of th trms: Its componnts ar th following ons: S 0 V n q* Z i j r m pour, 1,, 4 3,, 4 3 V 0 N avc r = 1,, m i n i z Th othr trms ar worthlss. Aftr having obtaind th fild p,v on th fild by rsolution of th quation [éq 4.-1] assmbld, on rturns to th fild p, v by opposit chang of function; on can calculat th dfinit acoustic intnsitis by [éq 1-1] which ar in this cas continuous in all th fild 5 Ordrs spcific to acoustic modling At th tim of a study by modling in acoustic finit lmnts with Cod_Astr on uss gnral ordrs and ordrs which ar spcific to acoustics, or whos kywords and options ar particular with this disciplin; w prsnt th list blow of it. Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 017 EDF R&D - Licnsd undr th trms of th GNU FDL (

12 Cod_Astr Vrsion 1 Titr : Élémnts finis n acoustiqu Dat : 3/10/015 Pag : 1/14 Rsponsabl : DELMAS Josslin Clé : R Révision : Dfinition of th charactristics of th propagation mdiums It is ncssary to giv th dnsity (actual valu) and th clrity of propagation (complx valu); on uss for that th ordr: DEFI_MATERIAU with th following kywords: kyword factor: FLUID kywords: RHO (dnsity 0 ) CELE_C (clrity c ) Exampl: air = DEFI_MATERIAU (FLUIDE=_F (RHO= 1.3, CELE_C: IH )); In this cas 0 =343. j0. Assignmnt of th modl It should obligatorily b spcifid that it is about phnomnon acoustic and to choos on of th 3 modlings possibl of acoustics; th ordr is thus usd: AFFE_MODELE with th following kywords for which on spcifis th possibl valus of assignmnt: Kywords: PHENOMENON = ACOUSTIC MODELING = ' 3D' or PLAN Boundary conditions On must assign valus normal vibratory spd pr fac (or dg into two-dimnsional) to th mshs dfining th bordrs sourcs, and also valus of acoustic impdanc pr fac (dg into twodimnsional) with th mshs dfining th bordrs in imposd impdanc. On uss th ordr spcific to acoustics AFFE_CHAR_ACOU with th following kywords: Kyword: MODEL Kyword factor: Kyword: Kyword factor: Kyword: VITE_FACE MESH GROUP_MA VNOR (normal vibratory spd V n ) IMPE_FACE MESH GROUP_MA IMPE (acoustic impdanc Z ) Kyword factor: PRES_IMPO NODE GROUP_NO NEAR (prssur p imposd on th nods) Calculation of th lmntary matrics Th various lmntary matrics (rigidity, mass and damping) ar calculatd by spcific options. Th ordr is mployd: CALC_MATR_ELEM with th kyword OPTION for which on spcifis th possibl valus of assignmnt: Kywords: OPTION RIGI_ACOU MASS_ACOU Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 017 EDF R&D - Licnsd undr th trms of th GNU FDL (

13 Cod_Astr Vrsion 1 Titr : Élémnts finis n acoustiqu Dat : 3/10/015 Pag : 13/14 Rsponsabl : DELMAS Josslin Clé : R Révision : Not: AMOR_ACOU Th assmbld matrics can b obtaind dirctly with th macro ordr ASSEMBLY and sam options. Calculation of th lmntary vctor sourc Th lmntary vctor is calculatd by on spcific option; th loading should obligatorily b indicatd. Th ordr is mployd: CALC_VECT_ELEM with th kyword OPTION for which on spcifis th only valu of possibl assignmnt: Kywords: OPTION CHAR_ACOU Kywords: LOAD Calculation of th solution Aftr assmbly of th lmntary matrics and vctor th harmonic solution can b calculatd dirctly with th ordr: DYNA_LINE_HARM. Postprocssings Starting from th rsult of th rsolution of th matric transcription of th quations [éq 3.1-] or [éq ], ordrs of postprocssing mak it possibl to obtain th nodal filds of following acoustic sizs: P lvl L p of acoustic prssur P in db : Lp 0 log ral part of th acoustic prssur imaginary part of th acoustic prssur activ acoustic intnsity I 1 v R p * ractiv acoustic intnsity J 1 v Im p * Ths filds ar calculatd by th us of th ordring of postprocssing CALC_CHAMP (th concpt of th rsult is of th typ ACOU_HARMO or MODE_ACOU ): CALC_CHAMP with th kywords RESULT and OPTION for which on spcifis th possibl valus of assignmnt: Kyword: RESULT Kyword: ACOUSTICS PRAC_ELNO (prssur lvl in db ) (ral part of th prssur) (imaginary part of th prssur) INTE_ELNO (intnsity activats) (intnsity ractivats) 6 Conclusion Moduls wr thus intgratd in Cod_Astr, allowing to do calculations of intrior acoustics in low frquncis for complx gomtris by two mthods: finit lmnts acoustic classics and mixd acoustic finit lmnts. Th two formulations wr validatd by comparison with th sam analytical solution; cass tsts ar prsntd in th handbook of V7 validation undr coding AHLV100. As it was nvisagd, th prcision, with idntical grid, is highr in th mixd cas; if account of th obstruction Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 017 EDF R&D - Licnsd undr th trms of th GNU FDL (

14 Cod_Astr Vrsion 1 Titr : Élémnts finis n acoustiqu Dat : 3/10/015 Pag : 14/14 Rsponsabl : DELMAS Josslin Clé : R Révision : mmory is takn this supriority is advantagous only if w want to obtain th fild of intnsity: on should us th mixd finit lmnts only in this cas thr. 7 Bibliography 1) A. BOUIZI: Put in work of a computr cod of finit lmnts for tratmnt of th quation of Hlmholtz in closd spac - Work of nd of studis, E.C.L ) A. BOUIZI: Spctral analysis of th quation of Hlmholtz. - Rport of DEA, Cntral School of Lyon, ) A. BOUIZI: Mixd finit lmnts in stationary linar acoustics: Dvlopmnt of cod AIRMEF - Acoustic Dpartmnt. DER - EDF. HE - 4/ ) A. BOUIZI, MR. COURTADE, D. JEANDEL, E. LUZZATO, A. MIGNOT, C. SURRY. : Conditions of compatibility of Brzzi-Babuska for mthods finit lmnts mixd in conformity in Mchanics and Acoustic AUM, Acts of th 8th Frnch Congrss of Mchanics, Nants, ) A. BOUIZI, MR. COURTADE, D. JEANDEL, E. LUZZATO, C. SURRY: Tratmnt of th quation of Hlmholtz by a cod of mixd finit lmnts in closd spac GAMI, Confrnc Vibrations Shocks, 1988, ECL, ) A. BOUIZI: Rsolution of th quations of linar Acoustics by a mthod finit lmnts mixd. Thsis prsntd in front of th Cntral School of Lyon - Spciality: Mchanics -. Supportd th 3//89. 7) C. HABASQUE: Exprimntal validation of th cod computr of acoustics intrnal, Low frquncy. Rport of intrnship of DEA. ECL 1986 (+ Projct of nd of studis). 8) A.ADOBES: Digital and xprimntal study of th filds of standing wavs Ratio DER/EDF - HE-87. 9) F. STIFKENS, A.ADOBES: Assssmnt of th intgration of th classical finit lmnts in Astr - Ratio DER/EDF - HP-64/ ) F. STIFKENS: Intgration of th mixd acoustic finit lmnts in Astr' Ratio DER/EDF HP-61/ Dscription of th vrsions of th documnt Vrsion Astr Author (S) Organization (S) 3 F.STIFKENS EDF- R&D/AMV Dscription of th modifications Initial txt Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 017 EDF R&D - Licnsd undr th trms of th GNU FDL (