A Performance Measure Approach to composites reliability: a transmission loss application

Transcription

1 10 th World Congress on Structural and Multdscplnary Optmzaton May 19-24, 2013, Orlando, Florda, USA A Performance Measure Approach to compostes relablty: a transmsson loss applcaton Roberto d Ippolto 1, Taylor Newll 1, Bram van der Heggen 1, Nck Tzannetaks 2 1 Noess Solutons N.V., Leuven, Belgum, roberto.dppolto@noesssolutons.com, taylor.newll@noesssolutons.com, bram.vanderheggen@noesssolutons.com 2 LMS Internatonal, Leuven, Belgum, nck.tzannetaks@lmsntl.com 1. Abstract Although the aerospace producton process s much better controlled than the process n other ndustres, t remans true that very small manufacturng varablty exsts n the geometrcal parameters (flange thcknesses, hole dameters ) as well as n materal propertes. In the current desgn process, the effect of ths manufacturng varablty s usually compensated for by applyng safety factors. Ths s not an deal stuaton, as t may lead to slghtly over-desgned structures. A much more promsng approach s to nclude probablstc models of desgn varables nto the mechancal smulaton process. Then, wth a new methodology based on relablty analyss, engneers can obtan a better understandng of the actual effect of the manufacturng tolerances and of varablty n materal propertes. Based on the analyss results, the robustness and relablty of the desgn can be assessed and mproved f needed. In ths paper, the above-mentoned probablstc approach s demonstrated on a stffened composte panel of an arcraft and ts acoustc performances n terms of transmsson loss are assessed. The frequency dependent transmsson loss of ths composte structure s optmzed wth respect to two ply thcknesses and four materal propertes n order to obtan a hgh transmsson loss throughout dfferent frequences for good acoustc nsulaton. The obectve of ths study s to maxmze the mnmum transmsson loss throughout the entre acoustc frequency spectrum. The structural behavor of the composte panel s calculated wth the Fnte Element Method, whle the acoustc performance s predcted wth the Boundary Element method. The combnaton of both modelng technques allows accurate predcton of the panel s transmsson loss. Durng the optmzaton process, the parameters are ntally optmzed usng determnstc approaches: frst a global search and then refned by a local search, wthout any consderaton for parameter varablty. The soluton found by the determnstc approaches s then used as the startng pont for the probablstc approach to mprove the relablty and robustness. Therefore, the Performance Measure Approach has been used. The two optmzaton approaches use surrogate models constructed on the results of a Desgn of Experments. In ths way, optmal solutons were obtaned wth a fracton of the otherwse requred number of mechancal smulaton runs. The determnstc approaches mproved the transmsson loss from 8.8 db of the reference confguraton to 13.6 db. The probablstc approach mproved the relablty from 3.27σ of the benchmark confguraton to 4.0σ wth a transmsson loss of stll 13.6 db. The probablstc approach adopted for the current problem has enabled the dentfcaton of an optmal materal and ply confguraton that mproves the relablty and robustness of the optmal soluton found by a determnstc optmzaton. Therefore, the soluton s less susceptble to varablty n materal propertes and manufacturng tolerances and less lkely to volate constrants found by the determnstc optmzaton, whle practcally mantanng the transmsson loss level found by the determnstc counterpart 2. Keywords: compostes, relablty, optmzaton, performance based approach 3. Introducton The use of large, long-dstance cruse arcraft s ncreasng the mportance of passenger comfort, not only n terms of acceptable nose and vbraton levels but also for ar qualty aspects, addressng the subectve human percepton of comfort. As a consequence, arcraft manufacturers are consderng the desgn and producton of fuselage structures offerng hgher standards of comfort, as well as actve and passve safety characterstcs. Consequently, a bg effort s beng put n research addressng new materals and manufacturng solutons that could result very attractve not only from the structural pont of vew (weght and producton cost reductons), but also for cabn comfort and health montorng. In fact, these last aspects are drectly related to crewmember complance ssues (due to longer trps) and to passenger expectatons, vewng comfort as one of the man arcraft qualty ndcators. The new materal propertes and ther applcaton n the desgn of aeronautc structures may lead to a greater 1

2 exteror/nteror nose transmsson. Thus, t s mportant to have a valdated desgn approach to select and test possble solutons that overcome these nose transmsson problems. Structural analyss technques are tradtonally based on determnstc methods (such as fnte element method) and are not able to take nto account random characterstcs of nput parameters lke materal propertes, geometry and appled loads. Thus the uncertantes and the varabltes that characterze composte propertes are not properly taken nto account. Ths can lead to overszed structures and dampng/soundproofng treatments n order to be able to satsfy the requrement for structural and acoustc performance. Ths penalzes weght savng advantages that are one of the maor reasons n the use of composte materals n aeronautc and aerospace applcatons. In last years, many dfferent methods have been developed n order to nclude uncertan model propertes n the FE analyss and to gve the opportunty to evaluate the effect of these uncertantes n the analyss results. In the present work, a probablstc approach has been used to descrbe the dynamc behavor of a composte rbbed panel subect to ncdent sound pressure n order to assess the transmtted nose nsde the arcraft cabn. In a probablstc fnte element approach, desgn parameters are ntroduced as stochastc varables to model randomness and spatal varablty of the geometrcal and mechancal propertes. The probablstc characterzaton of the acoustc response due to the ntroduced varablty s assessed and s used to compute the probablty that a value representatve of the performance of the system doesn t exceed a selected threshold. In the next sectons an overvew of the applcaton case wll be gven, together wth a bref overvew of the probablstc technques used to perform the analyss. Also, detals on the analyss mplementaton wll be explaned, together wth the enablng tools used to ntegrate a probablstc approach n the standard composte desgn process. 4. Test case descrpton In ths paper, the above-mentoned probablstc approach s demonstrated on a stffened composte panel of an arcraft and ts acoustc performances n terms of transmsson loss are assessed (Fgure 1). Transmsson loss through a panel s a measure of the reducton n sound power (or sound pressure) that occurs from one sde to the other sde of the panel, and s defned as follows (Fgure 2). Fgure 1: Vew of the composte panel from the Recever Room n the physcal experment set up. Fgure 2: Schematcs of physcal set up to measure Transmsson Loss of the composte panel. The frequency dependent transmsson loss of ths composte structure s optmzed wth respect to two ply thcknesses and four materal propertes n order to obtan a hgh transmsson loss throughout dfferent frequences for good acoustc nsulaton. The obectve of ths study s to maxmze the mnmum transmsson loss throughout the entre acoustc frequency spectrum. (1) The structural behavor of the composte panel s calculated wth the Fnte Element Method, whle the acoustc performance s predcted wth the Boundary Element method (Fgure 3). The combnaton of both modelng technques allows accurate predcton of the panel s transmsson loss. 2

3 Fgure 3: Schematcs of numercal set up showng BEM baffled prncple As mentoned before, physcal varables have been modeled n the analyss proect wth probablty dstrbuton functons n order to characterze the varabltes concerned wth the materal propertes of the varous layers [2]. A total of 8 nput parameters have been consdered, as reported n Table 1. Property Dstrbuton Mean Standard Devaton Ply_thckness_1 Normal 0.21 mm mm Ply_thckness_2 Normal 0.21 mm mm Mat1_YoungModulus_E1 Normal 6.2 E 10 Pa Pa Mat1_YoungModulus_E2 Normal 6.2 E 10 Pa Pa Mat2_YoungModulus_E1 Normal Pa Pa Mat2_YoungModulus_E2 Normal Pa Pa Table 1: Statstcally characterzed materal propertes for the composte panel 5. The Relablty Problem Statement To better understand the methodology used n the analyss, a bref overvew of the relablty problem statement s presented here. In the relablty theory, varabltes of an engneerng desgn can be characterzed by the varatons of a random system parameter set. These random varables are modeled wth probablty dstrbuton functons and are representatve of the uncertan model parameters. Thus, gven a set of random varables T, wth 1n, the probablty dstrbuton of each random varable s descrbed ether by ts cumulatve dstrbuton functon (CDF, x F ) or by ts probablty densty functon (PDF, f x ) and s often bounded by tolerance lmts on the system parameter values. For each realzaton of the set of nput random varables, the performance of the system has to be evaluated. Ths performance s generally descrbed by Performance Functons (PF, G x 1... m structural desgn, are usually the selected falure crtera. Thus, f consdered to fal f x 0 wth ) that, for a G x s one of the m system PFs, the system s G for at least one ndex. The probablty of falure of the system for every th performance functon s then the mult dmensonal ntegral of the ont PDF functon of the set of random varables over the falure doman. Ths s expressed by the ntegral [3] : F G The event space n 0 P G x 0 f x dx dx where x : G x 1 n : 0 (2) s the regon of the stochastc space where only falure events occur. Thus, the ntegral of the ont probablty densty functon th falure crteron. P f x over yelds the probablty of falure 0 x f, FG P G In general, gven a fxed performance ndex (or measure) 0 wth 1... m f P, of the structure for the g of the system, Eq (2) can be generalzed by 3

4 computng the probablty of the performance functon assumng values smaller than the selected threshold. F G g PG x g f x In ths case, the event space n dx1 dx where x : G x g G n G : (3) G represents the regon of the stochastc space where each performance parameter of the structure s below the prescrbed quantty g. The man am of the relablty analyss s to estmate the probablty of falure of a structure, gven a set of nput random varables and a set of falure crtera defnng the safe and fal regons or. Ths estmaton s usually carred on by approxmatng the mult dmensonal ntegral wth approprate technques. These nclude samplng methods and lmt state approxmatons, usually computed by transformng the problem defnton from the parameter space nto the standard normal space Y. The total probablty of falure of the structure s gven by the ntegral of the ont PDF G f x over the unon of all the falure domans. These falure domans are defned by the correspondng Performance Functons G x and ther threshold values g (performance ndexes or measures). In the next sectons an overvew of the technques used to approxmately estmate the multdmensonal ntegral and used to explot ths estmaton for optmzaton procedures s presented The Relablty-based Desgn Optmzaton Model In engneerng desgn, the tradtonal determnstc desgn optmzaton model has been successfully appled to systematcally mprove the system desgn process, yeldng a reducton of the costs and an mprovement of the fnal qualty of the products. However, uncertantes are present n ether engneerng smulatons and n manufacturng processes. Ths calls for dfferent optmzaton models that can yeld not only an mprovement n the desgn, but also a hgher level of confdence. Thus, a relablty-based desgn optmzaton (RBDO) model for robust and cost-effectve desgns can be defned usng mean values of the random system parameters as desgn varables and optmzng the cost subect to prescrbed probablstc constrants (e.g. a maxmum on the allowed probablty of falure) by solvng a mathematcally nonlnear programmng problem. As a result, the RBDO soluton provdes not only an mproved desgn but also a hgher level of confdence n the desgn. The general RBDO model can be defned as where mn Cost d d subect to Pf, PG x 0 P f, wth 1m the cost functon can be any functon of the mean values of the nput parameters and P, s the probablstc constrant that can be defned for each falure mode and needs to be satsfed. f For ths optmzaton problem, the constrant defnton s now expressed n terms of probablty dstrbutons and thus needs to be evaluated, for each optmzaton step, wthn the probablty framework. Thus, for each teraton of the optmzaton loop, an estmaton of the probablstc constrant n terms of ts multdmensonal ntegral (see Eq. (2) and (3)) has to be computed. For ths purpose, dfferent methods exst. Most of them apply a transformaton of the nput parameter space to the standard normal space Y. In ths space, each probablty of falure P, can be represented n terms of the relablty ndex [[3] ] as Where as P x 0 P P G y f, P G f, 0 1 P s the standard normal CDF (zero mean and standard devaton 1). Ths equaton can be generalzed F G g P G x g G f, f (4) (5) (6) 4

5 Where G F s the CDF of the th system response. The same approach can be used also to express the probablstc constrant of Eq. (4) n a dfferent notaton. In ths case, the probablty of falure P f, wll be the target probablty of falure P f, and the relablty ndex the target relablty ndex t,. 1 P P f, t, t, f, Usng Eq. (6), the second condton of Eq. (4) can be rewrtten as t, Pf, t Pf, FG 0, The relaton expressed n Eq. (8) n terms of the relablty ndex, can also be expressed n terms of the performance measure through nverse transformaton. In fact, usng Eq. (7) and Eq (8), the target probablty of falure can be expressed n terms of the target performance measure as 1 P F 1 g FG f, G t, where g s named target probablstc performance measure. It represents the value of the performance functon equvalent to the target relablty ndex t,. The expresson of the probablstc constrant of Eq. (8) and (9) can be used n the optmzaton problem of Eq. (4) to equvalently replace the orgnal defnton. (7) (8) (9) Fgure 4: Two step optmzaton procedure: frst optmze wthout consderng varaton n desgn parameters, then optmze for relablty and robustness takng nto account the varaton n desgn parameters Performance Measure Approach The formulaton of the general RBDO model of Eq. (4) that uses Eq. (8) to descrbe the probablstc constrant s called Relablty Index Approach (RIA) [4] [3]. An alternatve and more effectve way of formulatng the general RBDO model s to use the nverse formulaton of the probablstc constrant of Eq. (9), known as Performance Measure Approach (PMA) [5] [6]. The RBDO model can be re-wrtten as 5

6 mn Cost d d subect to g g ( k) ( k) T ( k) In ths case, at a gven desgn T measure d g s performed usng g d wth 1m (10), the evaluaton of the target probablstc performance ( k ) 1 d F G f x dx1 dxn G To facltate the estmaton of the ntegral n Eq. (11), a transformaton from the parameter space to the standard normal uncorrelated space Y s generally used [7]. Ths permts to easly evaluate the ntegral by transformng the ont PDF functon of the nput random varables n a multdmensonal ont normal PDF, whch can be easer evaluated. Thus, for each teraton of the optmzaton process the followng problem has to be solved mn G u s.t. y y β t, wth 1m Where y s defned by the transformaton Y=T() from the nput parameters space to the standard normal space Y. The approxmate soluton of ths problem yelds an estmaton of the performance measure, gven by g G y, * Wth the formulaton used n Eq. (13), the performance measure g, whch depends on the mean values of the nput random parameters (also called nomnal pont), has to be estmated for each teraton. The target of the optmzaton s to fnd the nomnal pont that can satsfy all the probablstc constrants n terms of the performance measure. Ths approach yelds an mproved convergence rate of the PMA methodology wth respect to the RIA formulaton. In fact, PMA mnmzes a complex cost functon subect to a smple constrant, whch rases less numercal problems and therefore yelds a better algorthm performance, whle the RIA algorthm mnmzes a smple cost functon subect to a complex constrant. Both these methods use a Lmt State Approxmaton to make an approxmaton of the exact Lmt State Functon (LSF); refer to [3] [8] for a more elaborate descrpton. In partcular, the method consdered n ths paper s the Frst Order Relablty Method (FORM) and t has been used for both RIA and PMA. Ths method has the advantage of beng fast and relatvely accurate n the analyss for less than 15 nput varables [9]. However, t s recognzed that such an approach can lead to large approxmaton errors for a hgher number of varables, and also to hgh computaton tmes when usng a gradent-based senstvty computaton [10]. Consderng such hgh-dmensonal problems s however outsde the scope of ths paper Desgn Of Experments and Response Surfaces Desgn Of Experments (DOE) [12] s a general approach to nvestgate complex correlatons between nput parameters and output response quanttes. The experments are set up n such a way that a maxmum amount of nformaton s obtaned n a mnmum amount of computaton tme. It s often combned wth Response Surface Methodology (RSM) [13] : a meta-model of certan order s estmated from the expermental data, whch yelds nsght n the functonal relatonshp between the nput parameters and the true response and output quanttes. DOE methodology was ntroduced n England n the 19 th century and frst used n agrcultural studes. The methodology then dssemnated to the rest of the world. Taguch appled DOE n the automotve ndustry n the 1980 s to mprove the product qualty and producton process. Nowadays, DOE methodology s wdely used n all felds of Computer Aded Engneerng. The selecton of a DOE method depends on the avalable computatonal power and the expected order (lnear, quadratc wth/wthout cross terms...) of the RSM model that s requred to accurately represent the actual functonal performances. For the present paper, two DOE methods have been used: a Latn Hypercube desgn of experment (DOE) method and a full Random one. Latn Hypercube DOE s a method that belongs to the category t (11) (12) (13) 6

7 of sem-random methods desgn n whch the ponts are chosen followng a sem-random process. However, a method that uses random ponts throughout the desgn, could contan many cluster ponts. Ths s certanly not an nterestng stuaton for purposes of exploraton. It s of greatest nterest, however, to get ponts to fll n as much as possble the area of nvestgaton. In statstcal samplng, randomzaton structure s made through a Latn-Hypercube, whch s defned as sample contanng a grd postons, f and only f there s only one sample n each row and each column. A Latn-Hypercube DOE generalzes ths dea by forcng only a sample sze of each axs-algned plan n more dmensons. Dvdng the sze of each factor n n equdstant levels, one obtans the nxn unt wth a number of experments equal to the number of levels. The ponts of the plan are accomplshed by selectng a permutaton of levels for each dmenson and combnng wthn the plan. In ths way the maxmum number of levels used because each level s wthn the plan. Therefore, f one or more factors are less mportant than others, each pont wll be able to provde nformaton about the nfluence of the other factors on the response. In ths way none of the tme-consumng computer experments become useless. In normal random methods, the clusterng of experments n specfc parts of the doman provdes good nformaton of the specfc regon but not on the overall desgn doman. The LHD DOE method s extremely effectve f one wants to analyze the doman wth a fxed number of experments, dependng on the complexty of the case study, thus havng a drect control on the computatonal cost of the DOE. Fgure 5: Vsualzaton of a Latn Hypercube typcal desgn for 2 factors and 5 experments DOEs are often combned wth Response Surface Methodology (RSM) [14] : a meta-model of a certan order s estmated from the DOE data, yeldng nsght n the functonal relatonshp between the nput parameters and the output quanttes. For the present case, a response surface model has been created on the bass of the DOE results for the output performances. Ths response model has then been used to perform a fast optmzaton procedure and to locate, wth suffcent accuracy, a new possble optmal composte panel confguraton to mprove the transmsson loss performance. The man advantage of ths approach s that the calculaton of the DOE generally has a fxed computatonal effort that depends on the number of parameters, the type of non lnearty expected and few other consderatons. On the other sde, a global or local optmzaton process has no fxed computatonal effort that can be estmated, snce ths depends on the satsfacton of a convergence crteron for the specfed algorthm (e.g. on the dmensonalty of the problem for gradent based algorthms lke Sequental Quadratc Programmng - SQP - or Non Lnear Programmng by Quadratc Lagrangan - NLPQL). For ths paper, gven the computatonal tme requred for one sngle executon of the complete analyss, the LH DOE approach lmts the total effort that can be spent and reles on the creaton of a response model to consderably speed up the optmzaton process. On the other hand, the results of the optmzaton process obtaned usng the response model have to be verfed wth a sngle fnal drect smulaton, n order to assess the error between the analytcal response model and the smulaton analyss only for the optmal pont. 6. Analyss Case Implementaton From the CAD model of the composte panel, two knds of meshes are created. One s the fnte element model (FEM) for the structural analyss and the other s the boundary element model (BEM) for the acoustc analyss. The structural analyss was conducted usng NASTRAN R to obtan the vbraton modes, whereas the sound radaton was computed usng the BEM n LMS Vrtual.Lab R Acoustcs (Fgure 6). The FEM and BEM are both determnstc smulatons where f the nputs are the same so s the output. The uncertanty s ncorporated n the desgn varables.e. the two ply thckness and the 4 materal propertes (Fgure 7). The results from the BEM s used to compute the transmsson loss n the post-process (Fgure 6). The process ntegraton workflow tools has enabled the automatc executon of the dfferent analyss phases n order to automatcally terate durng the optmzaton process and fnd the optmal composte panel desgn. In ths 7

8 way, the generaton of new combnatons of nput parameters can be fully automated. Ths ntegrated process captures the varous tasks that are usually performed manually and automates them to reduce user nterventon to the mnmum. In the current case, Optmus R drves the executon of LMS Vrtual.Lab as showed n Fgure 8. In ths workflow, Optmus generates new panel confguratons (n terms of materal propertes and ply geometry), changes the geometry (see Fgure 6) and updates the acoustc and structural meshes. In ths way the analyss fles are submtted, frst to Nastran to compute a Crag-Bampton modal soluton and have an orthogonalzed combned mode set between normal modes (related to the natural vbraton of the body) and statc modes (related to the deformaton due to localzed loadng). The Crag-Bampton Method combnes the constrant modes wth a lmted number of the lowest Egen modes of the structure, the fxed-nterface normal modes, n order to allow the reduced model to correctly represent both statc and dynamc behavor. The results of the Crag Bampton analyss are then used as structural modes n the acoustc analyss together wth the BEM approach to compute the acoustc soluton n terms of Transmsson Loss n the range between 100 Hz and 1kHz. The mnmum n ths frequency nterval s used as performance measure of the system and delvered as result of the sngle analyss run. These outputs are read and then ncorporated n the optmzaton routnes to fnd the most sutable desgn among all admssble desgn. Ths ntegrated process ncludes the batch executon of LMS Vrtual.Lab to change the geometry, update the mesh and perform the acoustc analyss. Optmus then extracts the results that are needed for the optmzaton process. Fgure 6: Smulaton usng acoustc boundary element method and structural fnte element method models. Desgn Param eters 2 ply thcknesses 4 m ateral propertes System /Com ponent Perform ances Sm allest Transmsson loss value n the range 100Hz-1kHz Total of 6 varables Fgure 7: Uncertantes n the analyss process 8

9 Fgure 8: Process ntegraton workflow wth Noess Optmus 5. Results A two step approach was employed n the determnaton of an optmal desgn of the composte rb panel. The parameters were ntally optmzed usng determnstc approaches: frst a global search and then refned by a local search, wthout any consderaton for parameter varablty. The soluton found by the determnstc approaches was then used as the startng pont for the probablstc approach to mprove the relablty and robustness (Fgure 4). Optmus R developed at Noess Solutons was used to perform both the determnstc and probablstc approach of optmzaton. The optmzaton was performed on the Radal Bass Functon Network constructed from 31 random samples and 50 Latn Hypercube samples. The global search was performed usng Self-Adaptve Evoluton whch s an mplementaton of genetc algorthm, and the local search refnement was done usng the Nonlnear Programmng Quadratc Search (NLPQL) whch s an mplementaton of Sequental Quadratc Programmng [14]. Then, n the probablstc approach, Frst Order Relablty Method (FORM) was used n Performance Measure Approach (PMA). The maxmzaton of transmsson loss was performed usng NLPQL wth relablty ndex constrant to be greater than 4.0. The determnstc approaches mproved the transmsson loss from 8.8 db of the reference confguraton to 13.6 db (Table 2). The probablstc approach mproved the relablty from 3.27σ of the benchmark confguraton to 4.01σ wth a transmsson loss of stll 13.6 db (Table 2). 9

10 Table 2: Results of two step optmzaton 5. Concluson The probablstc approach has enabled the dentfcaton of an optmal materal and ply confguraton that mproves the relablty and robustness of the optmal soluton found by a determnstc optmzaton. Therefore, the soluton s less susceptble to varablty n materal propertes and manufacturng tolerances and less lkely to volate constrants found by the determnstc optmzaton, whle practcally mantanng the transmsson loss level found by the determnstc counterpart. 6. References References should be lsted at the end of the paper and consecutvely numbered. Refer to references n the text wth reference number n brackets as [1]. Style the reference lst accordng to the followng examples. [1] Mathur, G.P., Chn, C.L., Smpson, M.A and Lee, J. T., Structural Acoustc Predcton and Interor Nose Control Technology, NASA-CR , The Boeng Company, Long Beach, CA [2] Thacker, B.H., Rha, D.S., Hall, D.A., Auel, T.R., Prtchard, S.D., Capabltes and Applcatons of Probablstc Methods n Fnte Element Analyss, Ffth ISSAT Internatonal Conference on Relablty and Qualty n Desgn, Las Vegas, NV, 1999 [3] Melchers, R.B., Structural Relablty Analyss and Predcton, 2nd Edton, [4] Sudret, B., Der Kureghan, A., Stochastc Fnte Element Methods and Relablty, A state of the art report, Report N. UCB/SEMM-2000/08 Department of Cvl and Envronmental Engneerng, Unversty of Calforna, Berkley, November [5] Youn, B.D., Cho, K.K., Park, Y.H., Hybrd Analyss Method for Relablty-Based Desgn Optmzaton, Journal of Mechancal Desgn, vol 125, pp , June 2003, ASME. [6] B.D. Youn, K.K. Cho, Lu Du, Adaptve Probablty Analyss Usng An Enhanced Hybrd Mean Value Method, Journal of Structural and Multdscplnary Optmzaton, vol. 29, no. 2, 2004, pp , Sprnger Berln Hedelberg. [7] Lu P-L, Der Kureghan A, Multvarate Dstrbuton Models wth Prescrbed Margnals and Covarances, Probablstc Engneerng Mechancs, 1 (2), 1986, pp [8] d Ippolto, R., Donders, S., Tzannetaks, N., Van der Auweraer, H., Vandeptte, D., Integraton of Probablstc Methodology n the Aerospace Desgn Optmzaton Process, 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamcs and Materals Conference, Austn, Texas, USA, Apr , 2005, AIAA [9] d Ippolto, R., Donders, S., Tzannetaks, N., Van de Peer, J., Van der Auweraer, H., An overvew of lmt state algorthms and ther applcablty to fnte element relablty analyss, 9th Internatonal Conference 10

11 On Structural Safety And Relablty ICOSSAR, Rome, Italy, June 19-23, [10] Schuëller, G.I., Pradlwarter, H.J., Koutsourelaks, P.S., A Crtcal apprasal of relablty estmaton procedures for hgh dmensons, Probablstc Engneerng Mechancs, Vol. 19, pp , [11] Myers, R.H., Montgomery, D.C., Response Surface Methodology: Process and Product Optmzaton Usng Desgned Experments, Wley & Sons, [12] Lorenzen, T.J.; Anderson, V.L.: Desgn of Experments, a no-name approach, Marcel Dekker, Inc., New York, USA, [13] Khur, A.I.; Cornell, J.A.: Response Surfaces, Desgn and Analyss, Marcel Dekker, Inc., New York, USA, second edton, [14] Noess Solutons, OPTIMUS, Rev. 10.6, Aprl [15] Van der Auweraer, H.; Donders, S.; Hadt, R.; Brughmans, M.; Mas, P.; Jans, J.: New approaches enablng NVH analyss to lead desgn n body development, Proc. EIS NVH Symposum New Technologes and Approaches n NVH, Coventry, UK, November 3,