University of Pisa, Department of Civil and Industrial Engineering, Italy

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1 Strojniši vstni - Journal of Mchanical Enginring 60(2014)5, Rcivd for rviw: Journal of Mchanical Enginring. All rights rsrvd. Rcivd rvisd form: DOI: /sv-jm Spcial Issu, Original Scintific Papr Accptd for publication: Finit Elmnt Formulations Applid to Outr Ear Modling Volandri, G. Carmignani, C. Di Puccio, F. Fort, P. Gaia Volandri* Costantino Carmignani Francsca Di Puccio Paola Fort Univrsity of Pisa, Dpartmnt of Civil and Industrial Enginring, Italy Th wor dscribd in this papr is part of a broadr rsarch activity on th dvlopmnt of a virtual ar. Th prsnt study focuss on th tympanic mmbran and auditory canal modling, which ar important componnts in sound transmission. Th standard finit lmnt mthod (FEM) and an altrnativ mthod (th gnralizd FEM), suitabl for modling sound propagation at high frquncis, wr applid. Two domains (fluid and structural) for th auditory canal and th tympanic mmbran, rspctivly, wr considrd in ordr to valuat th coupling of th diffrnt mthods and to apply a fluid-structur intraction formulation. ANSYS softwar was usd for solving FEM analyss, whil GFEM simulations wr obtaind by implmnting th mthod in Wolfram Mathmatica. Simulation rsults includ modal rspons, prssur distribution in th auditory canal and displacmnt distribution in th tympanic mmbran. Th idntifid modal frquncis of th auditory canal agr with publishd data rportd in th litratur. Th validation of such mthod with standard FEM simulation at incrasing msh dnsity shows that FEM is mor suitabl for simulations of th human ar in th audibl frquncy rang, although th gnralizd formulation could b convnint if an ar modl including th whol had or th ultrasound frquncy rang wr invstigatd. Kywords: finit lmnt mthod, auditory canal, simulation, sound transmission 0 INTRODUCTION Th wor dscribd in this papr is part of a broadr rsarch activity on th dvlopmnt of a modl of th human haring prcption, a ind of virtual ar. It dals with th analysis and simulation of th vibratory bhavior of th auditory apparatus in th convntionally considrd audibl frquncy rang, 20 Hz to 20 Hz. In particular, th prsnt study is focusd on th auditory canal (AC) including th tympanic mmbran (TM) that rprsnts a fundamntal portion of th normal acoustic path formd by th outr, middl and innr ar. In th last dcads, many bionginring mthods hav bn applid to simulat th dynamic bhavior of som parts of th ar, both distributd or lumpd paramtr mthods, such as thos basd on lctromchanical analogy or multi-body dynamics. Howvr, for th simulation at low frquncis (<10 Hz) of sound propagation in th AC and for th mchanical-acoustic transmission through th TM, th finit lmnt mthod (FEM) is th most frquntly usd approach [1] to [7]. Th first finit lmnt (FE) modls of th TM appard in th 70s [1]; sinc thn, th FEM has bn widly mployd to modl th ar structurs du to its rmarabl capability for analyzing complx gomtris and th mchanical proprtis of anisotropic and inhomognous matrials. Modls including also, at last, th middl ar wr dvlopd by Klly t al. [2], by Koi t al. [3], by Gan t al. [4] to [6] and L an Chn [7]. In th last yars, hybrid FE and multi-body modls of th TM and middl ar wr also proposd by th prsnt authors [8] and [9]. For simulating wav propagation with a standard FE modl, i.. with polynomial lmnt shap functions, th msh should rspct th rul of thumb, commonly accptd for many wav problms, that thr should b at last 10 nods pr wavlngth λ [10]. For sound transmission in air, λ rangs from 15.6 mm (20 Hz) to 15.6 m (20 Hz), approximatly. This mans that at high frquncis, convntional FEM modling can hav a high computational cost, as it rquirs a vry dns discrtization of th domain. Excpt for som attmpts to apply domain dcomposition with paralll procssing, altrnativ mthods or gnralizations of convntional FEM [11] hav bn proposd in th litratur for low wavlngth acoustic problms. Ths mthods hav high accuracy and a minor nd of msh rfinmnt with rspct to standard FEM. Among ths mthods thr ar th spctral mthods, as th spctral lmnt mthod (SEM) or spctral finit lmnts (SFE), basd on th fast Fourir transform (FFT) [12] or on orthogonal polynomials [13], and th wav lmnt mthods (WEM), as th ultra-wa variational formulation mthod (UWVF) [14], th partition of unity mthod (PU(FE)M) [15] and th gnralizd finit lmnt mthod (GFEM) [16] and [17]. In this papr a convntional FEM analysis, carrid out on an approximatd modl of th systm formd by th AC and th TM, is compard with th rsults of an altrnativ mthod: th GFEM was slctd as suitabl for furthr xtnsion of modling and simulation of thr-dimnsional sound propagation problms to highr frquncis or highr dimnsions of problm domain. This mthod was implmntd in a commrcial cod (Wolfram Mathmatica ) and applid prliminarily to simplifid gomtris *Corr. Author s Addrss: Dpartmnt of Civil and Industrial Enginring, Largo Lazzarino 56122, Pisa, Italy, gaia.volandri@ing.unipi.it 363

2 Strojniši vstni - Journal of Mchanical Enginring 60(2014)5, that approximat th anatomy of th outr ar for a comparison with standard FEM in ANSYS. 1 OVERVIEW OF COMPARED FE METHODS Th prsnt invstigation is focusd on diffrnt FE formulations compard in trms both of accuracy and computational tim. In this sction, th thory of th adoptd standard and gnralizd formulations is rportd in brif. 1.1 Thortical Bacground of th Finit Elmnt Mthods As it is wll-nown, th basic ida of th FEM is to divid th continuous domain of a problm in a discrt st of lmntary subdomains (lmnts) whr th fild function (.g. displacmnt) can b approximatd by mans of simpl basis functions [11]. Such basis functions ar typically continuous, with picwis continuous drivativs, and morovr can b asily intgratd. In ordr to guarant th global continuity of th displacmnt fild, adjacnt lmnts should hav sam valus along thir boundaris, thus th basis functions also support an intrpolatory solution. Accordingly, th displacmnt u at a point x within an lmnt can b writtn as: u ( x) = N ( x) c, (1) whr N (x) ar th basis functions and c th intrpolating cofficints. Diffrncs btwn th FE mthods slctd for comparison can b attributd mainly to th basis functions adoptd in lmnt formulations. Th pculiar charactristics of standard FEM, though wll nown, ar hr rportd to as th comparison with th othr mthod Standard Finit Elmnt Mthod In standard FEM, intrpolating cofficints ar th lmnt nodal displacmnts a ( is th nod numbrs), so that Eq. (1) bcoms: u ( x) = N ( x) a, (2) and th shap functions ar typically low (first or scond) ordr polynomials. A mor compact xprssion can b obtaind introducing th matrix form of th basis functions N(x): u ( x, t) = Nx ( ) a ( t), (3) which introducs also th gnralization to timdpndnt problms. It can b worth noting that in cas of isoparamtric lmnts th sam basis functions ar usd for mapping th lmnts from a rfrnc domain into th physical domain. Accordingly, th lmnt mass and stiffnss matrics ar obtaind from th following intgrals on th lmnt domain Ω : T M = N ρ NdΩ, K = B DBdΩ, (4) Ω Ω whr ρ is th matrial dnsity and B = SN, S bing a diffrntial oprator for calculating strain (ε = Su = SNa = Ba ), and D an lasticity matrix [11]. FE cods calculat such intgrals numrically, i.. Ω fx ( ) dω = fx ( ), (5) in particular by mans of Gaussian quadratur [11], which for polynomials givs th xact solution. Various procdurs xist for th rfinmnt of finit lmnt solutions: th local approximation can b improvd by polynomials of incrasingly highr dgr (p vrsion), or, having fixd th polynomial dgr p (typically p 2), by dcrasing th msh siz h (h vrsion) or incrasing th msh siz h and th dgr p of polynomials (hp-vrsion) [11]. h h w h Gnralizd Finit Elmnt Mthod Th gnralizd finit lmnt mthod, GFEM, is a combination of th standard FEM and th partition of unity mthod (PU(FE)M), aimd at introducing additional trms in th approximating function which nhanc th global bhaviour of th solution, also rflcting th nown information about th boundary valu problm [16]. Thus th standard polynomial FE solution is nrichd with spcial functions u ( x) (or handboo functions) usually non-polynomials, that mans: ux (, t) = Nxa ( ) ( t) + u( x). (6) Howvr such spcial functions must rspct th global rgularity constraints, and thus ar obtaind by som spcific (solution or problm-dpndnt) nrichmnt or handboo function hf(x), multiplid by th partition of unity (PU) N PU (x), i.. T 364 Volandri, G. Carmignani, C. Di Puccio, F. Fort, P.

3 Strojniši vstni - Journal of Mchanical Enginring 60(2014)5, ux (, t) = Nxa ( ) ( t) + N PU ( xhf(x)a ) ( t), (7) whr a * ar nodal unnown paramtrs that adjust th nrichmnt. Sinc this concpt is at th basis of th xtndd mthod, it is worth rminding that a partition of unity in a domain Ω is a st of functions ψ i (x) such that: ( x)= 1, x Ω. (8) i ψ i Consquntly any function f(x) can b rproducd by its product with th functions ψ i (x). Frquntly, but not ncssarily, th N PU functions corrspond to th standard FE shap functions N, oftn linar or bilinar, with a hat shap, dfind on patchs (much widr than lmnts) and zro vrywhr ls. A 2D rprsntation of patchs and lmnts is shown in Fig. 1. Th gnralization to th 3D ttrahdral cas is dirct. Fig. 1. Elmnts and patchs * π with θl = l and l = 1, 2,..., p, and similarly p 2π φm = m, m= 12,,..., q, and mployd as q handboo functions hf(x) in Eq. (7). Such a dirct gnralization of th 2D cas provids a distribution of dirctions which ar concntratd around th pols of an imaginary sphr. As statd in [18], it is impossibl to obtain an qually spacd distribution of dirctions. Howvr, th choic of th logic of distribution of dirctions rprsnts an important issu sinc a diffrnt logic of slction of propagation dirctions, basd on a priori nowldg about th solution, can allow rducing th numbr of dirctions rquird to obtain a givn lvl of accuracy. A main issu in th implmntation of GFEM is th possibl linar (or almost linar) rlation btwn th addd handboo functions and th standard basis FE ons with consqunt ill-conditioning problms [16]. As rportd by many authors, an incrasing numbr of wav dirctions involvs a highr rsult accuracy with th drawbac of introducing illconditiond systm matrics and rquiring ddicatd intgration tchniqus. Thus, it is oftn ncssary to introduc som chcs in th implmntation on th condition numbr of th matrics and, if ndd, to updat th dirction numbr assignd to ach nod. 1.2 Fluid-Structur Intraction Formulation Th local approximability is incrasd du to th handboo functions, whil maintaining th xisting infrastructurs of th FE cods. In fact, th ssntial boundary conditions can b imposd in GFEM xactly as in standard FEM. In [17], GFEM is applid to th solution of th problm of Hlmholtz with th valuation of spcial plan wav and wav band functions and functions of Vua as handboo functions and th conclusion is that th us of plan wav functions involvs a lowr computational burdn without substantial variations in th asymptotic accuracy, compard to mor complx functions. In th 2D cas th plan wavs can b xprssd as shown in [17]. For 3D problms, as in th simulation of th fluid domain corrsponding to th auditory canal, an xtnsion is rquird and th plan wavs can b xprssd as a dirct gnralization of th 2D cas, dtaild in [18]: () W i i ( x sin l cos m y sin l sin m z cos = l ), j θ φ + θ φ + θ (9) Finit Elmnt Formulations Applid to Outr Ear Modling Th coupling btwn partial domains of th whol modl rprsnts a significant aspct of modling that includs th dbatd issu of th fluid structur intraction (FSI) formulation. Th fluid structur intraction at th domain intrfac implis that th acoustic prssur xrts a load on th structur and that th structural motion producs an ffctiv load on th fluid. Th introduction, in th systm dynamics govrning quations, of a coupling matrix accounting for th ffctiv surfac ara and th normal to th intrfac ara, rprsnts a possibl FSI formulation [19]. In th prsnt study, th FSI coupling formulation dscribd in [20] was adoptd at th intrfac btwn fluid and structural domains. Such a formulation involvs th building of an asymmtric lmnt matrix for th intrfac lmnts (typically fluid lmnts), which hav th prssur and thr translational DoFs. For th intrfac fluid lmnt, th structural mass, damping and stiffnss matrics, as wll as th fluid mass, damping and stiffnss matrics, 365

4 Strojniši vstni - Journal of Mchanical Enginring 60(2014)5, assum th typical FE form. Mass (M FS ) and stiffnss (K FS ) coupling matrics ar dfind according to th following quations: M = ρ A T 0, K = A, (10) FS whr ρ 0 is th fluid dnsity, and th coupling matrix A is obtaind by th following intgration on th intrfac surfac S: T A= Nn N T ds, (11) S whr N and N rprsnt th fluid and structural shap function matrics, rspctivly, and th normal to th intrfac is indicatd with n. 2.1 Anatomy FS 2 OUTER EAR MODELS Th auditory canal blongs to th outr portion of th ar, th tympanic mmbran is instad gnrally includd in what is calld th middl ar, bing at th intrfac of th outr and middl ar (Fig. 2). Fig. 2. Schmatic drawing of th human ar Th AC convys th vibratory wavs propagating in air to th middl ar. Th morphology of th AC, though oftn approximatd with a cylindrical gomtry, typically prsnts two curvs [21]; this shap (rfrrd to as S ) not only facilitats th channling of th wav but introducs variations (typically amplifications) at som rsonanc frquncis. Although intr-subjct biological diffrncs xist, thr is a gnral agrmnt on th valu of adult AC lngth of 25 to 32 mm. Th crosssctional ara rangs from to mm 2 at th TM to to mm 2 at th canal ntranc [4] and [7]. Th undrstanding of th pculiaritis of th AC that most influnc th transmission of th signal and its coupling to th middl ar ar important in th dsign of prosthss and in rconstructiv surgry. Th TM, locatd at th nd sid of th ar canal, forms an angl of about 140 with th uppr and lowr walls of th channl; such an orintation givs a usful ara of about 85 mm 2, gratr than th orthogonal cross-sction of th AC. Th TM has a typical conical shap with an opning angl of 132 to 137 and a hight of th con of about 1.42 to 2 mm. Th lliptical bas of th con has a vrtical axis lngth ranging from 8.5 to 10 mm and a horizontal axis lngth ranging from 8 to 9 mm, with th apx (namd umbo and assumd as rfrnc point) facing th mdial sid [22]. Th thicnss of th TM is a critical paramtr for modling, sinc to dat accurat dtaild xprimntal masurmnts of th thicnss distribution ar not availabl and sinc it prsnts a high intr-subjct variability in trms of absolut valus. Although thr is a gnral agrmnt in stimating a nonuniform thicnss of th mmbran, in modling an approximatd singl thicnss valu, ranging from 30 to 150 µm (with an avrag valu of 74 µm) for th human TM is usually adoptd [4], [7] and [22]. Th tissu of th TM is mad of a multilayr structur with fibrs orintd mainly in th radial and circumfrntial dirctions. Two main rgions ar usually distinguishd a Pars Tnsa (PT) and a Pars Flaccida (PF), having diffrnt siz and mchanical proprtis. Th TM is connctd on th mdial sid to th ossicular chain, whil it is anchord along its circumfrnc to th wall of th tympanic cavity by mans of a fibro-cartilaginous ring (annular ligamnt). 2.2 Simplifid Modls of th Auditory Canal and Tympanic Mmbran Gomtry and Matrial Proprtis of th AC Two simplifid modls of th auditory canal wr usd that approximat th anatomy of an ar canal. A 22 mm long cylindrical and a psudo-anatomical gomtris (shown in Fig. 3) of a human auditory canal wr adoptd and importd into ANSYS nvironmnt for th FEM analysis. Th psudoanatomical gomtry (including an air portion in th auricl) was xtractd, through a smi-automatic sgmntation algorithm, from computd tomography data providd by th Dpartmnt of Otolaryngology II of Cisanllo Hospital in Pisa. Siz and morphology of th AC rconstructd gomtry wr in th litratur rang for adult halthy subjcts (lngth of about 28 mm, avrag diamtr of about 9 mm) [4]. 366 Volandri, G. Carmignani, C. Di Puccio, F. Fort, P.

5 Strojniši vstni - Journal of Mchanical Enginring 60(2014)5, As rgards th matrial proprtis, for th fluid containd in th AC th air mdium was assumd as a comprssibl, inviscid fluid with uniform man prssur and dnsity; a g/m 3 dnsity valu and a 340 m/s spd of sound valu wr, thus, adoptd. Th damping was not considrd in this valuation study. Th AC bony wall was simulatd with clampd boundary conditions and a distributd uniform harmonic sound prssur load of 90 db SPL (corrsponding to P was applid at th inlt of th AC whil th outlt was not constraind Msh Dfinition for th AC Modl Ttrahdral lmnts wr mployd for th cylindrical and th psudo-anatomical modls (Fig. 3), as it is typically adoptd for complx biological gomtris. Th siz of th lmnts was st by mans of a convrgnc critrion basd on th valu of th first natural frquncis up to 20 Hz; th prcnt rlativ rror vs. th logarithm of numbr of DoFs was stimatd (variation lss than 1%) [11]. An isotropic matrial modl with Young s modulus of 20 MPa was considrd [1]. Th Poisson s ratio was st qual to 0.3; dnsity was assumd qual to g/m 3. Triangular shll lmnts with thr nods wr usd for th TM, whn includd. As concrns th boundary conditions, th TM was fully clampd at th priphry, i.. th annular ligamnt was not considrd, as wll as th ossicular chain. 2.3 Implmntd Mthods Th implmntation and comparison of th mthods was carrid out in Mathmatica nvironmnt on th AC approximatd gomtris. Firstly, th GFEM implmntation rquird th dfinition of lmnts and patchs and th stting of th wavnumbr. Th partition of unity φ i, and th shap functions N wr chosn as th linar shap functions for th bric lmnt with ight nods of th standard FEM. In this papr spcial plan wav functions, W () i j, wr implmntd, whos mor gnral xprssion is: () i i W = r, (12) j Fig. 3. Cylindrical and anatomical gomtris and msh of a human auditory canal Tympanic Mmbran Modl Th TM was includd in th modl with th aim of accounting for th fluid-structur intraction. Although th TM has a pculiar shap (S Sction 2.1), a flat circular gomtry was adoptd with th aim of facilitating th dfinition of th fluid-structur intrfac. A mm thicnss was adoptd, dducd from [4]. whr r is th position vctor and th wav vctor. Th unit vctors of th propagation dirctions of th plan-wav functions, in th 3D problms, wr slctd basd on a priori nowldg of th solution or following th optimizd sphrical covring logic, borrowd from anothr mthod (.g. ultra wa variational formulation, UWVF) [14]. Th optimizd sphrical covring logic idntifis n points on an imaginary sphr, cntrd in ach patch nod, so that th maximum distanc of ach nod blonging to th sphr from th narst of th n considrd points is minimizd. Such a critrion allows obtaining a dirction distribution as homognous as possibl. In th prsnt study, n was chosn in th 4 to 124 rang du to rquirmnts of computational cost and th sam numbr of dirctions was assignd to ach vrtx of th patchs, although th mthod rquirs nithr an qually spacd distribution of dirctions nor an qual numbr of DoFs at ach nod. Th GFEM mthod was implmntd for th simulation of th fluid domain whil th FEM mthod was implmntd, in Mathmatica as wll, for th simulation of th structural domain (limitd to th TM), whn includd. Th ANSYS FLUID30 and SHELL63 (Discrt Kirchoff lmnt, DKT) formulations wr adoptd Finit Elmnt Formulations Applid to Outr Ear Modling 367

6 Strojniši vstni - Journal of Mchanical Enginring 60(2014)5, and implmntd for th fluid and structural lmnts, rspctivly. Th ANSYS FLUID30 lmnt formulation was slctd as suitabl for fluid/structur intraction problms and sound wav propagation applications. Th lmnt has a bric shap with ight cornr nods and four DoFs pr nod: thr displacmnts (only at nods on th intrfac) and prssur. Th lmnt adopts linar shap functions without xtra shap functions, and a standard Gaussian quadratur for brics (2 2 2 points). For th comparison and validation of th gnralizd mthod, th standard FEM solution at diffrnt lvls of msh rfinmnt obtaind with th commrcial cod ANSYS, was usd as a rfrnc. 2.4 Comparison Tsts Modal and Harmonic Analyss on th AC Modls Numrical fr-fr or constraind modal and harmonic analyss wr carrid out on th modls, intnding as constraind and fr-fr conditions whn sound prssur load is applid or not, rspctivly, at th AC inlt. Prssur distribution in th AC was valuatd as main rsult of th harmonic analysis. In addition to a chc on ignfrquncis in a spcific frquncy rang ([20 Hz to 20 Hz]), th modal shap corrlation was quantifid by th modal assuranc critrion (MAC) [23]. MAC rsults ar usually rprsntd as a squar matrix, corrlating th rfrnc mods (FEM, tst, thory tc.) to vrification mods. It assums a narly zro valu in th prsnc of incompatibl mod shaps, a unitary valu in cas of prfct corrlation and intrmdiat valus in cas of partial corrlation FSI of Auditory Canal and Tympanic Mmbran Harmonic analyss wr also carrid out on th cylindrical gomtry with circular mmbran accounting for th fluid-structur intraction. Th cylindrical gomtry was prfrrd sinc it allows a simpl dfinition of intrfacs for th fluid-structur formulation. Th matrial proprtis and th boundary conditions of th AC, xcpt for th outlt, wr st as prviously dscribd. Th harmonic analysis on th implmntd modls provids th distribution of prssur and normal displacmnt in th fluid and structural domain, rspctivly (coupld by FSI). 3 RESULTS 3.1 Comparison of Mthods on th AC Modl In ordr to compar th two finit lmnt formulations, fr-fr and constraind modal analyss wr prformd on th simplifid gomtris (Fig. 3) of th fluid domain. Eignfrquncis in th 20 Hz to 20 Hz frquncy rang (typically th first thr in th psudo-anatomical cas) wr compard and mod shaps corrlatd, using th MAC matrix. Th modal frquncis obtaind in th fr-fr and constraind modal analysis with FEM and GFEM in th cylindrical and psudo-anatomical gomtris ar rportd and compard in Tabl 1. Tabl 1. First thr modal frquncis; th rsults ar shown for, cylindrical gomtris; and for c), d) psudo-anatomical gomtris;, c) for th fr-fr modal analysis;, d) for th constraind modal analysis f [Hz] FEM GFEM rror [%] FEM GFEM rror [%] c) FEM GFEM rror [%] d) FEM GFEM rror [%] It is worth noting that th modal frquncis obtaind with th psudo-anatomical gomtry ar in agrmnt as ordr of magnitud with th thr rsonanc frquncis in th audibl rang (3176, 9528 and Hz) inhrnt to th xtrnal auditory canal, dpnding on its morphology and lngth, rportd in [24]. Morovr, in th litratur th first natural frquncy inhrnt to th auditory canal is oftn idntifid in th 3 to 4 Hz rang [2] and [24] confirming th stimatd valu. Finally, th rsults obtaind by th two numrical approachs appar to diffr majorly at low frquncy whr standard FEM is considrd rliabl and convnint whil thy ar comparabl at highr frquncis whr GFEM should bcom mor advantagous. Th comparison of th mod shaps was carrid out in Mathmatica nvironmnt by importing th rsults obtaind in th diffrnt softwar nvironmnts for th proposd mthodologis. Fig. 4 shows, for th cylindrical (a, and for th psudo-anatomical 368 Volandri, G. Carmignani, C. Di Puccio, F. Fort, P.

7 Strojniši vstni - Journal of Mchanical Enginring 60(2014)5, gomtry (c, d), for th fr-fr (a, c) and constraind (b, d) modal analysis rspctivly, th matrix rprsntations of th modal shap corrlation of th two mthods, for an qual numbr of nods. Th maximum/minimum valus on th MAC matrix diagonal as wll as th out of diagonal maximum valus ar shown in Fig. 4 for th four abov mntiond cass. th advantag of ping th standard FEM msh, vn with ttrahdral lmnts, with nods blonging to th lmnt boundary. c) d) Fig. 4. Modal shap corrlation by MAC; th rsults ar shown for th, cylindrical and c), d) psudo-anatomical gomtris, for th fr-fr, c) and constraind, d) modal analysis All cass show consistncy in th first thr natural frquncis obtaind by th various mthodologis, lss in th mod shaps. As an xampl of th harmonic analysis rsults, th prssur distribution at 10 Hz in th psudoanatomical modl of th AC is shown in Fig. 5. Th GFEM rsults obtaind with a coars lmnt msh (144 nods, 288 DoFs) ar compard with th ANSYS FEM rsults at qual and rfind (3216 nods, 3216 DoFs) msh. Th TM sid is indicatd in th figurs. Th GFEM and FEM harmonic rsults, in th audibl frquncy rang agr within tight tolrancs. Th GFEM is mor xpnsiv from th computational point of viw, sinc GFEM prsnts th difficulty of calculation in th complx fild and ntails problms of ill-conditioning and linar dpndnc in th phas of construction of th lmnt matrics, for which it oftn rquirs havy intgration tchniqus. Howvr, with rspct to altrnativ mthods of th litratur, GFEM prsnts c) Fig. 5. Harmonic analysis rsults at 10 Hz: prssur distribution in th psudo-anatomical gomtry with GFEM (coars msh) and coars msh, and c) FEM rfind msh In conclusion, FEM is mor suitabl for outr ar simulations in th 20 Hz to 20 Hz frquncy rang. Morovr it is availabl in commrcial cods. Howvr, furthr invstigations lucidat that th GFEM can b mor suitabl for othr applications involving highr frquncis (.g. a 22 mm long cylindrical modl at 100 Hz) or highr charactristic dimnsions of th problm (.g. a scald/homothtic 22 cm long cylindrical modl, at 10 Hz). Th rsults of ths furthr simulations ar shown in Figs. 6 and 7, rspctivly, for diffrnt msh dnsity. Th longitudinal sction of th cylindr, instad of th cylindrical surfac, is plottd in Figs. 6a and 7a. As on can s, th rsults of Fig. 6a ar comparabl to thos of Fig. 6c which rprsnts th Finit Elmnt Formulations Applid to Outr Ear Modling 369

8 Strojniši vstni - Journal of Mchanical Enginring 60(2014)5, c) Fig. 6. Harmonic analysis in a cylindrical modl (22 mm long) at 100 Hz with: GFEM (longitudinal sction) and (b, c) FEM varying th msh dnsity: 137 nods and 166 DoFs, 137 nods and DoFs, c) nods and DoFs FEM rfrnc solution, obtaind with a considrably highr numbr of dgrs of frdom. Thn, if Fig. 6a is compard with Fig. 6b, obtaind with an qual numbr of dgrs of frdom, th diffrncs and th nhancmnt ar vidnt. Th sam obsrvations hold for Fig. 7. Th us of th implmntd gnralizd lmnt formulations can b convnint, spcially as frquncy incrass (and thrfor th wav numbr) bcaus it allows to achiv, with a coars msh (which dos not satisfy th rul of thumb of tn nods pr wavlngth convntionally accptd for standard FEM), an accuracy comparabl to that obtaind with a fin msh in standard FEM formulations. Ths first indications suggst that th intgration of ths advancd tchniqus in a FE modl of th ar could b usful if, for xampl, th whol had wr includd or if th ultra-sound fild wr invstigatd. c) Fig. 7. Harmonic analysis in a cylindrical modl (scald 22 cm long cylindr) at 10 Hz with: GFEM (longitudinal sction) and, c) FEM varying th msh dnsity: 137 nods and 166 DoFs, 137 nods and DoFs, c) nods and DoFs 3.2 Comparison of Mthods on th FSI Problm Concrning th simulation of th fluid- structur problm by th combination of tchniqus GFEM and FEM for th fluid and structural domains, rspctivly, th distributions of prssur insid th AC and TM displacmnt at 200 Hz ar compard with th rsults obtaind with standard FEM in ANSYS with a coars and rfind msh, in Figs. 8 and 9, rspctivly. As highlightd in th prvious sction, th rsults indicat that GFEM and FEM rsults, in th invstigatd frquncy rang, ar in agrmnt within accptabl tolrancs. 4 CONCLUSIONS In this wor som modling aspcts of th human ar canal and tympanic mmbran wr xamind also 370 Volandri, G. Carmignani, C. Di Puccio, F. Fort, P.

9 Strojniši vstni - Journal of Mchanical Enginring 60(2014)5, c) Fig. 8. Harmonic analysis rsults at 200 Hz in th FSI problm: prssur distribution in th auditory canal with: GFEM (coars msh, longitudinal sction) and FEM (coars msh), c) FEM (rfind msh c) Fig. 9. Harmonic analysis rsults at 200 Hz in th FSI problm: normal displacmnt distribution in th tympanic mmbran with: GFEM and, FEM (coars msh), c) FEM (rfind msh) considring th fluid-structural coupling that occurs btwn ths two componnts of th acoustic path. Standard and gnralizd finit lmnt modls wr implmntd, tstd and compard. GFEM and FEM modal and harmonic rsults in th 20 Hz to 20 Hz rang agr within tight tolrancs in all tstd cass. Thus, FEM appars mor suitabl for simulations in th audibl frquncy rang, assuming th typical ar siz of a human bing, du to its rlativly limitd computational burdn and th availability of commrcial cods. Howvr, ths prliminary rsults show also that th gnralizd finit lmnt formulation can b convnint in short-wav acoustic problms with th aim of simulating th auditory apparatus including th whol had or invstigating th ultra-sound fild, i.. whn th wav lngth is shortr than th charactristic dimnsion of th problm. Th analysis rsults concrning th natural frquncis of th auditory canal ar consistnt with som publishd studis [2] and [24] which idntify thr modal frquncis in th audibl rang with th first modal on includd in th 3 to 4 Hz rang. As futur dvlopmnts, th implmntation and intgration of this mthod in a commrcial cod can b plannd. Th application of GFEM to a complt ar modl including bon conduction in th had can b considrd. Morovr GFEM can b proposd for convnint application to othr filds (.g. ultrasounds in mdicin). 5 ACKNOWLEDGMENTS Th support and contribution of Prof. Stfano Brrttini and Dr. Luca Bruschini of th U.O. Finit Elmnt Formulations Applid to Outr Ear Modling 371

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