Error estimation in vibroacoustic problems solved by means of finite elements

1 E o es ima ion in ib oacous ic p oblems sol ed by means o ini e elemen s Poble -Puig, J.; Rod íguez-Fe an, A. A ilia ion: {Labo a o i de Càlcul Numè ic, E.T.S. d'Enginye s de Camins, Canals i Po s de Ba celona, Uni e si a Poli ècnica de Ca alunya} e-mail: {jo di.poble @upc.edu; an onio. od iguez- e [email p o ec ed]u} Abs ac The ib oacous ic equa ions can be sol ed by means o he ini e elemen me hod. A disc e isa ion o he s uc u e and he acous ic domains is equi ed and highly in luences he quali y o he nume ical solu ion. The e exis meshing c i e ia (a p io i e o es ima o s) o he case o he Helmhol z equa ion bu hese s udies ha e no ocused hei a en ion in he case o he ib oacous ic p oblem. The luid s uc u e in e ac ion ep esen s a new sou ce o nume ical e o s and meshes in he in e ac ion zone should be designed by no only aking in o accoun he physical p ope ies o he acous ic medium bu also he mechanical p ope ies o he s uc u e. The goal o he wo k is o ob ain an a p io i e o es ima ion c i e ion o he ib oacous ic p oblem and Illus a e i s e iciency by means o nume ical expe imen s. Keywo ds: ini e elemen , bounda y elemen , e o es ima ion. 1 In oduc ion The solu ion o physical p oblems whe e he expec ed solu ion is a wa e can be s ill conside ed an open p oblem o he nume ical echniques such as he ini e elemen me hod (FEM) o he bounda y elemen me hod (BEM) [17]. I he wa e leng h o he solu ion ield is oo sho , he disc e isa ion o he physical domain mus be ine enough in o de o desc ibe his wa e and he compu a ional cos s o he p oblem inc ease ill hey become una o dable. Acous ics and s uc u al ib a ion a e among hose physical p oblems whose solu ion is a wa e. They a e go e ned by he ib oacous ic equa ions. Se e al o mula ions a e a ailable [5,14]. Howe e , he discussion and he examples shown he e a e based on he s eady ha monic p essu e-displacemen equa ions: INTERNOISE 2010 │ JUNE 13-16 │ LISBON │ PORTUGAL 2 (1) Whe e p is he phaso o acous ic p essu e, k is he ai wa e numbe , he acous ic o ce e m is due o poin sou ces, n is an imposed eloci y, ρ 0 is he ai densi y, n a e ou wa d no mals and linea elas ici y is assumed o he solid pa o he p oblem. The nume ical o mula ion o his p oblem is e iewed in [1]. Examples o he nume ical esolu ion o hese equa ions in o de o model physical si ua ions o indus ial in e es can be ound in he li e a u e. The sound ansmission h ough walls has been s udied by means o he FEM in [11] o BEM in [3]. S udies on he nume ical e o o he uncoupled (acous ic o s uc u al) p oblem ha e been done. In he ield o nume ical acous ics, a usual ule o humb is ha six linea ini e elemen s pe wa e leng h a e enough in o de o ob ain accu a e esul s. Mo e de ailed analyses [9,10]) p edic ha he in e pola ion e o o linea elemen s can be es ima ed as (2) whe e and a e dimensionless wa e numbe and elemen size, is a cha ac e is ic leng h o he p oblem, C local is a cons an ha depends on each pa icula si ua ion bu i is independen o and , p is he exac solu ion and p I is an in e pola ion o p. The six-elemen s-pe -wa e-leng h ule o humb only akes in o accoun his local in e pola ion e o . Ne e heless, wo addi ional phenomena make his c i e ion insu icien : he dispe sion e ec (k-singula i y) and he exis ence o eigen equencies o he equa ion (λ- singula i y) [2]. The wa e numbe o he nume ical solu ion has been p o ed o be di e en om he exac wa e numbe in he Helmhol z equa ion ( o a linea one-dimensional ini e elemen solu ion o Helmhol z equa ion we ha e . This causes an inc easing phase shi o he disc e e solu ion. The e o a ec s all he domain. Due o dispe sion, a nume ical solu ion ob ained by keeping cons an would ha e mo e e o o high INTERNOISE 2010 │ JUNE 13-16 │ LISBON │ PORTUGAL 3 equencies. The inc ease o e o due o he inc ease o dimensionless wa e numbe is known as pollu ion e ec [4]. In oducing he nume ical wa e numbe (a ec ed by dispe sion e o ) in he analysis, an exp ession o he o al e o can be ob ained as (3) p h is he nume ical solu ion ob ained by FEM. C 1 and C 2 a e cons an s ha should be calcula ed o each pa icula si ua ion (bu hey a e independen o and ) and q he deg ee o polynomial in e pola ion. This is an impo an d awback because hese a p io i e o es ima es gi e a endency bu he mesh should be co ec ly designed o each p oblem. No e ha in equa ion (3) we can associa e he i s e m (mul iplied by C 1 ) wi h he in e pola ion o local e o (e I = (p-p I /p)) and he second e m (mul iplied by C 2 ) wi h he pollu ion e o (e POL = (p I -p h )/p h ). Nume ical expe imen s ha show he e olu ion o nume ical e o s in he BEM o he Helmhol z equa ion can be ound in [12,13]. In [16] an a pos e io i e o es ima o o he case o s uc u al dynamics in he ime-domain has been de eloped. The wo k in he p esen con ibu ion is ocused in he analysis o possible sou ces o nume ical e o due o he luid-s uc u e coupling in he ib oacous ic p oblem. In all he p e ious e e ences, he s udy is done in uncoupled acous ic o solid p oblems and uni o m domains. Thus, he wa es o he p oblem (p essu e o displacemen wa es) depends only on he physical p ope ies o he medium and he go e ning equa ion. The wa eleng h o he solu ion ields is p o ided by he dispe sion ela ion. In a bounded p oblem, he solu ion ends o be simila o he eigenmode whose eigen equency is close o he p oblem pulsa ion. Howe e , he solu ion o he p oblem can also be composed o o ced wa es. They appea when he solu ion is equi ed o i i egula i ies a he bounda y o he sys em, o sa is y some imposed o ce. In gene al, hey can be di e en o any cha ac e is ic mode o oscilla ion o he sys em. This possibili y is no conside ed in uncoupled sys em e o analysis because o usual o ces ac ing on he p oblem (poin o ces, uni o m loads,...) o ced wa es a e no gene a ed. This is no he case o coupled ib oacous ic p oblems whe e an spa ial oscilla o y p essu e is o en he exci a ion o ce on a s uc u e o a sinusoidal eloci y is imposed on an acous ic con ou . I he disc e isa ion o a medium is designed aking as e e ence he wa e leng h o esonan wa es and he solu ion is composed o o ced wa es o sho e wa e leng h he nume ical e o s would be la ge han expec ed. A simila idea is ound when modal analysis is used o sound ansmission p oblems. In [6] bo h esonan and o ced modes mus be conside ed in o de o accu a ely desc ibe he p essu e ields in he ooms. 2 S uc u al p oblem The dynamic esponse o he wo s uc u al sys ems in Figu e 1 is analysed. The i s one, Figu e 1(a), is a simply suppo ed beam wi h a poin o ce placed a om he suppo ( is he leng h o he beam). I is a e e ence case whe e he solu ion will be only composed o INTERNOISE 2010 │ JUNE 13-16 │ LISBON │ PORTUGAL 4 esonan wa es. On he con a y, in he second case (Figu e 1(b)) a sinusoidal p essu e is imposed ( ). The ib a ion ield can be composed o esonan and o ced wa es, depending on he ela ionship be ween he geome ic and mechanical p ope ies o he beam, he pulsa ion o he p oblem and he o ce pa ame e s K and θ . Figu e 1 – Ske ches o he wo s uc u al p oblems analysed: (a)Beam loaded wi h a poin o ce a he posi ion (b)Beam loaded wi h a sinusoidal load . In all he examples shown in he con ibu ion, he coupling and o cing e ms a e calcula ed wi h enough accu acy. As shown in [8] o s a ic beams loaded wi h sinusoidal o ces, he inco ec in eg a ion o he o ce e m can be a sou ce o e o , especially i he o ce is in eg a ed using he same accu acy ha has been used o he mass ma ices ( ypically linea o quad a ic polynomials o s uc u al analysis FEM). In any case, his possible sou ce o e o in ib oacous ic coupled p oblems will no be analysed in he con ibu ion. The e o measu e in his s uc u al p oblem is (4) u h is he phaso o he nume ical alue o he s uc u al displacemen o a mesh o size h, while u is he e e ence alue. I is calcula ed by means o he modal analysis solu ion. The ma e ial p ope ies o he s uc u e a e shown in Table 1. The calcula ed nume ical e o s can be seen in Figu es 2 and 3. The poin o ce example should be unde s ood as he e e ence o he con e gence beha iou . The in e pola ion o he no mal displacemen in an Eule Beam elemen is o deg ee q = 3 and he slope o he log 10 (||e||) - log 10 (h) cu e is q +1 = 4. The e o is clea ly ela ed wi h he s uc u al wa e numbe and inc eases wi h equency as shown in Figu e 1. The exp ession o he s uc u al wa e numbe o an Eule beam is (5) Table 1 – Geome ical and mechanical p ope ies o he s uc u e. Meaning Symbol Value Thickness 5 mm Young’s modulus 2,061·10 11 N/m 2 Densi y 7500 kg/m 3 Damping 2 % Leng h 4 m INTERNOISE 2010 │ JUNE 13-16 │ LISBON │ PORTUGAL 5 Figu e 2 – E o analysis o a poin loaded beam: on he le , dependence on he numbe o elemen s pe wa e leng h; on he igh dependence on he equency. In he sinusoidal load case o Figu e 3 he alue o he ` o ce wa e numbe ' K = 100 π / , has been chosen in o de o pe ec ly ma ch he exci a ion o ce wi h he mode ha has 50 wa e leng hs inside he beam. In ha case he exac solu ion is composed by only one o ced wa e. Figu e 3 – E o analysis o a beam loaded by means o an oscilla o y o ce whe e K = 100 π / (exac ly 50 load wa es pe beam leng h) : (a) ela i e e o depending on he numbe o nodes pe s uc u al wa e leng h; (b) ela i e e o depending on he numbe o nodes pe p essu e load wa e leng h. In Figu e 3 (a) he e o has been plo ed depending on he s uc u al wa e numbe and in Figu e 3 (b) depending on he o ce wa e numbe K (numbe o s uc u al elemen s pe imposed p essu e wa e leng h). The esul s ake sense in he second case because he INTERNOISE 2010 │ JUNE 13-16 │ LISBON │ PORTUGAL 6 displacemen ield wa e leng h is K . This means ha he mesh mus be designed by conside ing he exci a ion wa e numbe K ins ead o he s uc u al wa e numbe . 3 Acous ic p oblem The e ec o spa ial oscilla o y exci a ions can be also ound in he acous ic p oblem. The acous ic adia ion in a semi-in ini e semi-space has been calcula ed by means o bounda y elemen s (a simila si ua ion has been analysed in [15]). An imposed eloci y n = K y = 2 π n/L y has been imposed along a 4 m leng h ib a ing bounda y. The e o measu e in his acous ic p oblem is (6) whe e p h is he phaso o he nume ical alue o he acous ic p essu e o a mesh o size h, while p e is he e e ence alue. This alue is now aken as he p essu e in a mesh wi h smalle elemen s. The in eg a ion o he ou pu s is pe o med along he 4 m leng h con ou wi h bounda y elemen s ( adia ing pa o he con ou ). Figu e 4 – Con e gence o he ela i e e o in acous ic p essu e o he acous ic adia ion o sound inside a semi-in ini e space. Cons an equency o he p oblem (250 Hz), wi h inc easing alue o he spa ial wa e numbe o he imposed eloci y a he con ou . The ob ained esul s a e shown in Figu e 4. We can see how he con e gence o he nume ical e o depends again on he wa e numbe o he imposed eloci y. Fo he case o n = 15 wa es inside he compu a ional domain he expec ed con e gence slope is ob ained. INTERNOISE 2010 │ JUNE 13-16 │ LISBON │ PORTUGAL 7 Howe e , o la ge numbe o wa es inside he compu a ional domain (n = 30 and n = 50), he con e gence wi h he expec ed slope s a s o a smalle elemen size. A less, mo e han one elemen pe exci a ion wa e leng h is equi ed in o de o each he con e gence o o de 2. In hese cases he exci a ion wa e leng h is sho e han he leng h o p opaga ion wa es in he ai . The wa e numbe o he exci a ion oscilla ion is smalle han he wa e numbe o ai (k = ω /c). In his si ua ion he p essu e ield is mainly composed o e anescen wa es. This wa es dec ease wi h dis ance and anish away om he sou ce loca ion ( ib a ing con ou in his case). 4 Conclusions I has been shown ha o bo h s uc u al and acous ic p oblems he con e gence o nume ical me hods such as FEM o BEM can depend on he exci a ion ac ion. In hese me hods, he spa ial disc e isa ion o he physical domain mus be designed acco ding o he expec ed wa es in he solu ion. In gene al he leng h o hese wa es can be p edic ed by conside ing only he physical p ope ies o he medium ( esonan o p opaga ing wa es). Howe e o some exci a ion ac ions o ced wa es can be induced. This causes he inal solu ion o ha e con ibu ions ela ed o di e en equencies and consequen ly modi ies he expec ed pe o mance o he conside ed nume ical me hods. This beha iou in uncoupled p oblems should be s udied o ib oacous ic p oblems in he ollowing cases: • Acous ic domains ha a e in con ac wi h s uc u es ib a ing wi h a wa eleng h smalle han he leng h o he wa es gene a ed in he ai . I is he case o acous ic domains coupled wi h ligh weigh s uc u es and equencies below he c i ical equency. The disc e isa ion in he acous ic pa o he p oblem should be done by aking in o accoun he s uc u al wa e leng h. • S uc u es su ounded by acous ic luids and equencies abo e he c i ical equency. The disc e isa ion in he s uc u al pa o he p oblem should be done by aking in o accoun he acous ic wa e leng h. Acco ding o he p esen ed p elimina y esul s, i can be concluded ha he elemen size should be decided no only by aking in o accoun he medium p ope ies bu also he exci a ion o ces o he p oblem. In addi ion, i will be checked i o coupled ib oacous ic p oblems his e ec is also ele an in he coupling in e ace and he sho es wa e leng h o any o he media in ol ed in he p oblem should be conside ed. Acknowledgmen s The inancial suppo o he Minis e io de Educación y Ciencia (BIA2007-66965, DPI2007- 62395) and he Facul a de Ma emà iques i Es adís ica de la UPC (FME, CERMET) is g a e ully acknowledged. F ee so wa e has been used [7]. INTERNOISE 2010 │ JUNE 13-16 │ LISBON │ PORTUGAL 8 Re e ences [1] A alla,N.; Be nha d, R.J. Re iew o nume ical solu ions o low- equency s uc u al- acous ic p oblems, Appl. Acous ., Vol 43, 1994, pp 271-294. [2] Bouilla d,Ph.; Ihlenbu g,F. E o es ima ion and adap i i y o he ini e elemen me hod in acous ics: 2D and 3D applica ions, Compu . Me hods Appl. Mech. Eng g., Vol 176(1-4), 1999, pp 147-163. [3] Coye e, J.P. The use o ini e-elemen and bounda y-elemen models o p e-dic ing he ib o-acous ic beha iou o laye ed s uc u es, Ad . Eng. So w., Vol 30, 1999, pp 133- 139. [4] De aemaeke , A.; Babuska,I.; Bouilla d,Ph. Dispe sion and pollu ion o he FEM solu ion o he Helmhol z equa ion in one, wo and h ee dimensions, In .J. Nume . Me h. Engng., Vol 46, 1999, pp 471-499. [5] E e s ine, G.C. Fini e elemen o mula ions o s uc u al acous ics p oblems, Compu . S uc ., Vol 65(3), 1997, pp 307-321. [6] Gaglia dini, L.; Roland, J.; Guyade ,J.L. The use o a unc ional basis o calcula e acous ic ansmission be ween ooms, J. Sound Vib ., Vol 145(3), 1991, pp 457-478. [7] Geuzaine, C.; Remacle, J.-F. Gmsh: a h ee-dimensional ini e elemen mesh gene a o wi h buil -in p e- and pos -p ocessing acili ies, In . J. Nume . Me h. Engng., Vol 11(79), 2009, pp 1309-1331. [8] Gudla, P.K.; Ganguli,R. 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