Complex Envelope Vectoization fo the solution of mid-high fequency acoustic poblems A. Sestiei Depatment of Mechanical and Aeospace Engineeing Univesity of Rome la Sapienza
Pesentation layout - Low fequency mapping of HF vibation CEDA - CEV appoach and govening equations - Unde-sampling - Limits of CEV and emaks - Application of CEV to benchmaks - Test cases and esults - Application of CEV to Bounday Element Fomulation - Test cases and esults - A efeence model fo futue developments WKB - Conclusions
LOW FREQUENCY MAPPING OF HF VIBRATIONS CEDA* Assuming a focing tem of adiant fequency ω, the equation of motion fo a one-dimensional undamped stuctue is: 2 L[ u] m u f The complex envelope displacement is defined as follows: jk0x u E[ u] u ju e ~ u is the Hilbet Tansfom of u, i.e. ~ u and u + j u is the analytical displacement. E: envelope opeato x u x d *Cacatea and Sestiei: Complex envelope displacement analysis: a quasi-static appoach to vibations. JSV, vol. 2012, 1997.
LOW FREQUENCY MAPPING OF HF VIBRATIONS cont d By applying the envelope opeato to the motion equation and expessing the physical displacement in tems of the complex envelope one obtains: L[ 2 jk0x u ] m u f f j f e being L = E L E 1 u admits an invese which is given by: 0 u 2 Re{ u e j k x } CEDA equation So, once the CEDA equation is solved, one can econstuct the physical displacement
CEDA LAY-OUT IN THE WAVENUMBER DOMAIN Physical signal spectum Analytic signal spectum Shifted signal spectum complex envelope
LOW FREQUENCY MAPPING OF HF VIBRATIONS CEDA* When computing the complex envelope displacement we can ecove the physical displacement by the invese tansfomation
LOW FREQUENCY MAPPING OF HF VIBRATIONS CEDA Example Fo the exact solution 100 points used against 16 points in CEDA
CEV: VECTORIZATION PROCESS CEV: VECTORIZATION PROCESS Because of the difficulties to extend CEDA to multi-dimensional and viboacoustic This slide systems shows schematically anothe appoach the vectoization was poposed: pocedue: CEV each ow - Complex o column Envelope of the Vectoization. discete suface solution is tansfeed to a stip, so that, at the end, we have a new discete signal that we can manipulate conveniently as a one-dimensional system Spectum of the esponse fo a two-dimensional system k y k x
CEV VECTORIZATION PROCEDURE Note: in the above vectoization pocedue the spectum of the esponse signal is not anymoe concentated aound a single wavenumbe k 0 but athe speads into the whole wavenumbe domain Vectoized signal Fouie Tansfom
LOW FREQUENCY MAPPING OF HF VIBRATIONS CEV* FEM Stiffness matix K and mass matix M M u K u p If the excitation is hamonic with fequency ω, one has - ω 2 M + K u = p Au p * Giannini, Cacatea, Sestiei. High fequency vibation analysis by CEV. JASA, 1216, 2007
CEV cont d In CEV a set of complex envelope signals u and is poduced fom u p Fom Au p? A u p CEV equation
LOW FREQUENCY MAPPING OF HF VIBRATIONS CEV* 2 M K u p DMA Univesità di Roma La Sapienza Au p u and p can be viewed as sampled values of continuous signals us and ps. Because of the vectoized spead in the wavenumbe domain, in CEV two sets of complex envelope signals u s p s ae poduced fom us ps, each one chaacteized by a naow wavenumbe spectum. u s u s 1,2,..., This is obtained by filteing the spectum Uk of us by a set of naow spectal windows W k and shifted by the quantity k towads the wave numbe oigin to poduce U. P
The Envelop opeato fo high dimensional poblems Space High Feq. Signal Fouie Tansfom Definition of Bands Fo each band Window Set of low fequency signals obtained by IFT Wavenumbe shift Wavenumbe shift
LOW FREQUENCY MAPPING OF HF VIBRATIONS CEV cont d The th complex signal u is obtained by invese Fouie Tansfom and. } { } { 1 1 k k W k k U F k U F s u
Set of low wavenumbe signals High Feq. Space Signal The opeation is evesible povided that the whole set of windowed signals ae consideed } 2Re{ 1 s k j P e s u s u } { } { 1 1 k k W k k U F k U F s u 1,2,..., P s u s u DMA Univesità di Roma La Sapienza
NUMERICAL RESULTS 34x34 DMA Univesità di Roma La Sapienza
How to build A M, K fom the physical matix A M, K The explicit fom of the motion equation is: It can be tuned into continuous fom by using ps and us to obtain: p u A s p d u s a I, 1 s k j I p d u e s a F k k W k k P k k U k k W k k A Assume the kenel to have the fom: a s,σ = a s - σ. By i Fouie tansfoming the motion equation, ii applying the window W, and iii shifting the esult towad the oigin by k : Invese Fouie Tansfom h k k hk p u A DMA Univesità di Roma La Sapienza
Reconsideing its discete countepat N n1 a j k s mn mn e un pm with a a DMA Univesità di Roma La Sapienza How to build A M, K fom the physical matix A M, K cont d s e A u j k s k h hk h k sampling inteval of the dummy vaialble s p This expession pemits to ecove the envelope matix A M, K fom the physical matix A M, K
CEV EQUATION summay Fom the discete equation of a consevative dynamic poblem Au p by: load tansfomation: i Fouie tansfom F, ii application of bandpass windows W, iii wave numbe shift opeation towad the oigin S, iv invese Fouie tansfom F -1 : p F 1 S W F p p E p matix tansfomation DMA Univesità di Roma La Sapienza A u p CEV Equation
LIMITS OF CEV AND REMARKS The condition as,σ = as-σ is stictly valid fo infinite systems. Fo finite systems it is not tue. Howeve it can be shown that it holds in an aveage sense, i.e. if the esponse is spatially aveaged.
CEV EQUATIONS FOR NON CONSERVATIVE SYSTEMS DMA Univesità di Roma La Sapienza Fo non consevative systems A and u ae complex, the steps pesented above can be maintained povided that u, p and A ae witten diffeently, as follows: 0 p p A A A A A u u u R I I R I R p u A p u A CEV
UNDERSAMPLIG Fo example, fo a 6 x 6 matix A, and a eduction atio = 2, we may - The new poblem, poviding the complex envelope, has size NxN, identical to choose the oiginal one and has the same eigenvalues of the oiginal one. 1 0 0 - Each spectal window has a low wavenumbe spectum, thus a fine mesh is unnecessay and actually non 1 0convenient 0 fo the CEV application. 0 1 0 A suitable expansion matix R is intoduced to solve the poblem at a low computational cost. By omitting 0the 1 0supescipt fo the sake of simplicity: u 0 0 1 Ru ed 0 0 1 R is ectangula and admits a pseudo-invese R + such that so that, in this case it is R + = R T / R + R=I while RR + I The opeation A ed R AR p ed R p obtained A Rby AR aveaging ; its u elements R u ed ed DMA Univesità di Roma La Sapienza implies a patition of the oiginal matix into squae submatices, eplacing each submatix with a single value A ed u ed p ed
RECOVERING THE PHYSICAL SOLUTION The invese elationship povides the physical solution u Ru ed
Steps in CEV: summay DMA Univesità di Roma La Sapienza A M and K is detemined by a standad FEM The windowed envelop matix is detemined fom A is not affected by the filteing opeation The windowed envelope load is obtained by FT of the focing vecto, windowing the wavenumbe component by W, and shifting the spectum by The IFT povides p A is detemined by A ed R A R ed p is detemined by ed p R p ed k. Fom the CEV equation, u is detemined, and the physical esponse ecoveed
LIMITS OF CEV AND REMARKS DMA Univesità di Roma La Sapienza Looking at the spectum of a point load flat in the fequency domain, the Leakage filteed effect spectum of the is filteing not shaply pocess, ectangula, implying due that to CEV the use acts of coectly a finite numbe on the foced of points, pat i.e. of the a tuncated solution FT but of does the not signal tackle leakage efficiently the homogeneus pat A A If the matix is not educed, the eigenvalues of ae equal to those of the physical poblem A and the Filte envelope spectum solution shifted matches to the pefectly the physical solution. wavenumbe oigin This is a dawback especially fo low damping. Howeve the contibution of diffeent windows having diffeent phases, educes significantly When the envelope matix is educed, the eigenvalues of the educed poblem ae diffeent lowe fom those of the physical poblem: an eo is intoduced mistuning such spead. Response of the filteed load it is not Howeve: the eigenvalues of the educed stictly confined poblem ae in the a subset filte of the eigenvalues of the non educed poblem, bandwidth unde the assumption that the CEV solution is smooth. Using a set of windows in the wavenumbe domain, it is possible to appoximate sufficiently well the eigenvalues of the oiginal poblem thus obtaining a easonable esponse.
LIMITS OF CEV AND REMARKS CEV can be consideed successful when the modes of the system do not play a key ole. Paticulaly: when the damping is elatively high when a diect field is pepondeant with espect to the evebeant field fo high fequency poblems with an acceptable damping when an extenal acoustic poblem no modes is consideed*.
Application 1: 2D cavity with a point souce DMA Univesità di Roma La Sapienza FE discetization: gid 64x64 4096 DOF CEV discetization: gid 16x16 512 DOF Damping 0.01
Application 2: bending plate 1x1x0.002 m with a point souce FE gid: 64x64 12288 DOF. CEV DOFS: 384 Reduction ate 32 Damping 0.01
Application 3: foam plate 1x1x0.05 m with a point souce FE discetization: gid 64x64 4096 DOF CEV discetization: gid 16x16 512 DOF Damping 0.17
A benchmak fo the Maie Cuie ITN 4 6
30 3 3 5 BENCHMARK FOR A PRIN NATIONAL PROJECT DMA Univesità di Roma La Sapienza Reduction fom 32226 dofs to 786 dofs atio 41 Plate dimensions: 1-600x400x3 mm, 2-300x400x3, 3-400x400x3. 60 Damping 0.001 Damping 0.04
Fomulation of the extenal poblem DMA Univesità di Roma La Sapienza APPLICATION OF CEV TO BEM FORMULATION FEM Velocity field on the bounday Pessue on the Bounday BEM Radiation into the field
APPLICATION OF CEV TO BEM FORMULATION Integal fomulation of a vibo-acoustic poblem, unde steady conditions, by the fee-space Geen function Whateve the type of elements used, in matix fom one has This poblem can be solved by following the same steps shown befoe, i.e.: - a FEM is used to detemine the esponse v CEV can be used but not convenient - a BEM is used to detemine the matix T o - a BEM is used to detemine the matix T - a BEM is used to compute c c c ed T T ed T ed T ed moe convenient computationally
APPLICATION OF CEV TO BEM FORMULATION The CEV method applied to the bounday element fomulation povides The mistuning does not affect the BEM-CEV fomulation because the opeato T is not diectly elated to the stuctual opeato: thus thee is no eo in the natual fequencies location. NO MISTUNING
TEST CASES: pulsating sphee = 0.1m, v n = 0.01 m/s Extenal field. Reduction facto 8: fom 1016 to 127 DOFS Pessue field on the suface f = 800 Hz Extenal field Intenal field
TEST CASES: extenal field geneated by a loaded box Foce specta ae flat between 700 and 2500 Hz Pessue spectum at a distance d=15m fom the box Reduction facto 21: fom 1176 to 112 DOFS Point spectum at 15 m fom the box: Compaison between efeence solution and CEV fo diffeent eduction factos 21 168 Note that fo eduction facto 168, the CEV Dofs ae only 14
TEST CASES: extenal field adiated by a benchmak Flat specta foces Pessue spectum at a efeence point of the field Reduction facto 19: fom 855 to 45 DOFS
TEST CASES: extenal field adiated by a benchmak
TEST CASES: extenal field adiated by a benchmak
A efeence model fo futue developments WKB* Possible connections with CEV, VTCR, DEA, WBM, WIA? WKB Eikonal establishes a elationship between the exact equations of waves and the ay tacing appoximation. Consideing the wave equation with a hamonic excitation, it is: whee nx is the efaction index: c 0 /cx By factoizing ψ into amplitude and phase ψx = Ax e jsx, substituting into the wave equation and getting eal and imaginay pats: * Wenzel, Kames and Billouin
WKB cont. DMA Univesità di Roma La Sapienza Let ε = 1/k 2 : By expessing the solution in the fom and using a petubation appoach up the fist ode:
WKB cont. At ode zeo, i.e. λ 0, i.e. k DMA Univesità di Roma La Sapienza ay tacing Equal phase sufaces ae those ove which Sx ae constant: ays ae those lines intesecting othogonally equal phase sufaces. The unit vecto along the ay is: poducing: that can be solved iteatively.
WKB cont. Tuning this appoach to discete fom: with: i.e. L S a = p LS acts on the wave opeato by modulating exponentially its coefficients, just as CEV and othes methods do.
Compaison between WKB and CEV WKB CEV - It uses diffeent odes of petubation - It uses seveal windows centeed on to appoximate the solution k to shift the HF components and appoximate the solution - Sx is unknown - The phase modulating tem is assigned } 2Re{ 1 s k j P e s u s u } { } { 1 1 k k W k k U F k U F s u DMA Univesità di Roma La Sapienza
Compaison between WKB and VTCR/WBM* WKB VTCR/WBM In WKB the solution is epesented by Both VTCR and WBM use shape waves amplitudes modulated by phase functions that ae exact solution of the functions, and diffeent petubation govening diffeential equations - odes ae used to appoximate the modes epesented by a supeposition solution. of wave shape functions in WBM, local modes, supeposition of appopiate waves in VTCR. In VTCR a two-scale appoximation is used in the weak fomulation. Only the slow scale wave amplitude is discetized while the fast vaying scale spatial shape of the wave is analytically descibed. * WBM: Wave based method, VTCR: Vaiational theoy of complex ays
Compaison between WKB and DEA/WIA* WKB DEA/WIA In WKB the zeo-th ode appoximation povides the ay tacing technique. Using highe odes of petubation it is possible to efine the solution. In DEA the ay dynamics is descibed by a set of basis functions. When using the lowest basis function the ay tacing appoximation, and SEA, is obtained In WIA the displacement field is obtained by a supeposition of tavelling waves. By neglecting phase dependencies, only an enegy beam is associated to each wave. By expanding each beam by a Fouie seies, an enegy balance equations is obtained. It povides the SEA equations if the seies is aested to the fist tem. *DEA: Dynamic Enegy analysis, WIA: Wave intensity appoach
CONCLUDING REMARKS 1 CEV does not use an enegy fomulation, but pefoms a tansfomation leading to a new vaiable that has a low wavenumbe content and capable of accounting coectly fo the bounday conditions. As such, it is paticulaly appopiate to deal with high fequency poblems. The envelope mass and stiffness matices ae detemined diectly fom the FE matices, so that any commecial FE o BE code can be used. A eduction technique is applied to these matices to get a new poblem whose dimensions ae much smalle than the oiginal one, with a elevant saving of time computation. The appoach is paticulaly convenient when the modes do not play a elevant ole: e.g. extenal acoustic poblems
CONCLUDING REMARKS 2 CEV can be applied to both extenal and intenal vibo-acoustic poblems, with eos bounded within 3 db fom the efeence solution. Unlike the application of CEV with FEM, in the application to BEM the mistuning poblem is absent. Thus, CEV can be efficiently used to detemine the intenal acoustic field. The eduction facto can be inceased at will, without affecting significantly the quality of the solution: howeve, inceasing the eduction facto it is necessay to incease the numbe of spectal filtes. Shall we define a common goal and tasks to addess pedictive methods, with clea tasks and goals?
Time aveage? Fequency aveage? Ensemble aveage? Diect field? Revebeant field? Wave intefeence? Thank you fo you attention Any question?
Weak fom of the kenel Fo an infinite system as,σ = as-σ feee space Geen function Fo finite systems it does geneally hold. Howeve it can hold in a weak o aveage fom, i.e. By FT as,σ By FT as-σ a s, σ = as σ a s, σ a s σ e ik xx ds = Ak x,σ e ik xx ds = Ak x e ik xσ i.e., to satisfy the convolution condition it must be Ak x,σ = Ak x e ik xσ Ak x = Ak x,σ e ik xσ By IFT, it is possible to detemine as as: as = F 1 Ak x, σ e ik xσ
SPECTRAL WINDOW DECOMPOSITION DMA Univesità di Roma La Sapienza W 1 W 2 W 3 W 4 W 5 W 6 W 1 LOW FREQUENCY SOLUTION N. 1.... W 6.. LOW FREQUENCY SOLUTION N. 6 FORWARD SHIFTING MOVE THE SOLUTIONS AT THE RIGTH WN- +
HIGH FREQUENCY PROBLEMS & SPECTRAL CHARACTERISTICS δs FOURIER TRANSFORM space step a k MAX maximum wavenumbe
CEDV SOLUTION: DMA Univesità di Roma La Sapienza DESCRIBE THE RESPONSE BY A SET OF NARROW BAND SIGNALS LOW FREQUENCY SIGNAL HIGH FREQUENCY SIGNAL WAVENUMBER AXIS ORIGIN
NUMERICAL RESULTS 34x34 DMA Univesità di Roma La Sapienza
NUMERICAL RESULTS, PLATE 34x34 vs 16x16 COMPARISON BETWEEN VECTORIZED SOLUTIONS ENVELOPE PHYSICAL
CEV: GOVERNING EQUATIONS d s x s x 1 ~ ~ ˆ s jx s x s x x jk e s x s x 0 ˆ New equation of motion, 0 s f d x e s I a s jk Equation of motion fo the discete countepat 0 i j l jk ij ij e M M 0 i j l jk ij ij e K K f x K M 2
CEV: ENVELOPE VECTORIZATION 2 M K w f ~ jk0x w E[ w] w jw e 2 M K w f M K ij ij M K ij ij e e jk0l ji jk0l ji
Spectum of the vectoized displacement of high dimensional poblems