Scece Joural of Appled Mathematcs ad Statstcs 5; 3(6): 63-74 Publshed ole December, 5 (http://www.scecepublshggroup.com//sams) do:.648/.sams.536.6 ISSN: 376-949 (Prt); ISSN: 376-953 (Ole) Applcato of Lat Hypercube Samplg Based Krgg Surrogate Models Relablty Assessmet Lu Chu *, Eduardo Souza De Curs, Abdelkhalak El Ham, Mohamed Ed Laboratory of Optmzato ad Relablty Mechacal Structure, Departmet of Mechacs, Natoal Isttute of Appled Scece of Roue, Roue, Frace Emal address: Lu.chu@sa-roue.fr (Lu Chu) To cte ths artcle: Lu Chu, Eduardo Souza De Curs, Abdelkhalak El Ham, Mohamed Ed. Applcato of Lat Hypercube Samplg Based Krgg Surrogate Models Relablty Assessmet. Scece Joural of Appled Mathematcs ad Statstcs. Vol. 3, No. 6, 5, pp. 63-74. do:.648/.sams.536.6 Abstract: Relablty assessmet s oe of the ecessary ad crtcal parts structural desg uder ucertates. The methods for structural relablty assessmet am at evaluatg the probablty of lmt state by cosderg the fluctuato of actg loads, varato of structural compoet or system, ad complexty of operatg evromet. Lat Hypercube samplg () method as advaced Mote Carlo smulato () has hgher effcecy samplg. It wll be chose ad appled ths paper order to obta a effectve database for buldg Krgg surrogate models. I ths paper, we propose a effectve method to have relablty assessmet by Lat Hypercube samplg based Krgg surrogate models. Ths method keeps the certa level of accuracy predcto of the respose of a structural fte elemet model or other explct mathematcal fuctos. Keywords: Lat Hypercube Samplg, Krgg Models, Relablty Assessmet. Itroducto Ucertaty s a evtable ssue the process of maufacture, frastructure, ad egeerg desg. Quatfyg ad propagatg the ucertaty the smulato or desg process as a key compoet of rsk aalyss, robustess evaluato or relablty based optmzato attracts atteto of researchers ad desger []. The tradtoal determstc model s ot effectve for structural aalyss because of avodg the effects of ucertates the parameters. To cosder parameter fluctuato the real operato evromet, the Mote Carlo smulato () s chose to perform the stochastc smulato. It s oe of most popular dscrete algorthmc for ucertaty aalyses ad used wth creasg frequecy. (samplg based approach) s useful for several reasos. Frst, the samplg based approach covers the full rage of each ucerta varable a complcated system. Secod, modfcato of the model s ot requred, ad drect estmates of dstrbuto fuctos are provded. I addto, the process of samplg, a varety of sestvty aalyss procedures are avalable. Last but ot the least, aalyss procedures ca be developed ad allow the propagato of results through systems of lked models []. However, ca reach a certa level of accuracy oly f a very large umber of teratos are performed. It s obvous that methods become computatoal prohbtve whe smulato model s complcated. To be more effcet tha the radom samplg method, several mproved methods wth dfferet samplg techques have bee developed. Importace samplg (weghted samplg), s expected to reduce error to zero f the probablty desty fucto s properly selected [3]. L. M. Berler [4] appled mportace samplg Mote Carlo method sequetal problems of Bayesa updatg. P. Beaurepare [5] attempted performg mportace samplg techque relablty based optmzato of structure. I the lteratures, the frst-order sestvty method, as a varace reducto techque, s also utlzed to accelerate estmato covergece [6]. The varace reducto techques are especally mportat whe s appled to estmate small falure probablty [7]. Lat Hypercube Samplg () method operates by subdvdg the sample space to smaller regos ad samplg wth these regos. The produced samples more effectvely fll the sample space ad therefore reduce the
Scece Joural of Appled Mathematcs ad Statstcs 5; 3(6): 63-74 64 varace of computed statstcal estmators [8]. Ste M [9] had research the large sample propertes of smulatos usg Lat hypercube samplg. Owe [] ad Hutgto [] tested the lmtato of Lat Hypercube samplg. Improved have bee developed, Stock R [] proected samples oto a kow subspace to mmze tegrated mea square error ad maxmze etropy. Ima [3] had efforts to reduce spurous correlatos, Flora [4] rearraged the matrx of samples based o a trasformato of the rak umber matrx. Further, methods for costructg orthogoal are proposed to possess ehaced space fllg propertes [5], utlzed teratve optmzato methods are appled order to reduce spurous correlatos [6]. A surrogate model ca be thought of as a regresso to a set of data, where the data s a set of put-output parg obtaed by evaluatg a black-box model of the complex system [7]. Surrogate models may be classfed to two geeral categores based o ther proposed: local ad global models. A example of optmzato usg local s Respose Surface Method expressed (RSM) as polyomal surrogate models. The krgg models are terpolato models based o the assumpto that there s a spatal correcto betwee the values of the fucto to be approxmated [8]. Wth these models a kow or sampled value of the lmt state fucto to be approxmated s exactly predcted. The Krgg models do ot assume a uderlyg global fuctoal form as assumed the polyomal regresso models ad ca approxmate arbtrary fuctos wth hgh accuracy global as well as local approxmatos [9]. Krgg model was orgally developed spatal statstcs by Krge. Mathero [] had cotrbuto mathematcal formulato of Krgg model. I the poeerg work of Sacks et al [], Krgg model was chose to predct determstc fuctos physcal processes. A uvarate krgg model combg a regresso term wth a zero-mea statoary Gaussa dsturbace process was developed by Hadcock []. Va Beer [3] started wth the applcato of Krgg model radom smulato. The track record of applcato of Krgg models radom smulato holds great promse. Ths paper presets a effectve method to have relablty assessmet by Lat Hypercube samplg based Krgg surrogate models. For that purpose, Lat Hypercube Samplg s utlzed to buld a relable database for approxmatg the respose of a structural fte elemet model or other explct mathematcal fuctos. Oce the database s defed, we compare the relatve performace of approxmato methods to ft the probablstc respose of orgal models: amely, respose surface method (frst order ad secod order polyomal regressos) ad Krgg models. The, relablty assessmet s predcted by the surrogate models, whch heavly reduced computatoal cost ad also kept the certa level of accuracy.. Lat Hypercube Samplg Method A compromse method of advaced s Lat hypercube samplg () approach. Ths approach dvdes the rage of each varable to dsot tervals of equal probablty, ad oe value s radomly selected from each terval [4]. It mproves stablty ad also matas the tractablty of radom samplg. Cosder a statstc system descrbed by the relato: = F( X) () { } X = X, X,, X () Where the radom vector X possesses depedet compoets defed over the sample space S descrbg put radom varables wth margal cumulatve dstrbuto fuctos P. F s the operator commoly represets computer smulato such as fte elemet model wth propagato of ucertates the system. Tradtoal Mote Carlo methods rely o the so-called Smple Radom Samplg (SRS) whch realzato of X, deoted x k, k =,, N (samples), are geerated as depedet dstrbuto realzato o sample space by x = P ( U ) ; =,, (3) k X Where U are uformly dstrbuted samples o [, ]. The realzatos of x are the appled to the system y = F( x) ad y s statstcally evaluated. Correlated radom varables are ot cosdered because of Prcpal Compoet Aalyss ad Nataf or Roseblatt trasformatos ca be appled to produce a set of ucorrelated radom varables from correlated relatoshps betwee varables. dvdes the rage of each vector compoets X, X,, X to dsot subsets of equal probablty. Samples of each vector compoets are draw from the respectve subset accordg to, x = P ( U ) (4) k X Where =,, ; =,, m; refers to the total umber of vector compoets or dmesos of vector. m s the umber of subset a desg. Subscrpt k deotes a specfc sample, Where Uare uformly dstrbuted samples o ξ, ξ, ξ =, ξ = (5) m m The samples x [ x, x, x ] =, k =,, N are k k k k assembled by uformly radomly groupg the terms of the geerated vector compoets. I Lat Hypercube Samplg approach, the rage of all radom put varables s dvded to tervals wth equal probablty, whch are restrcted wth the respectve terval avod the dsadvatage of clusterg together, as demostrated Fg..
65 Lu Chu et al.: Applcato of Lat Hypercube Samplg Based Krgg Surrogate Models Relablty Assessmet.8.8.6.6.4.4....4.6.8..4.6.8 Fg.. Comparso betwee ad samplgexample : Cosder the cubc polyomal fucto gve by as referece [5]. Mea Value 7 65 6 55 Stadard Devato 8 6 4 5 5 5 5 Number of samplg 5 5 5 Number of samplg Probablty.8.6.4. 5 5 5 Probablty Desty.5..5 5 5 5-3 - 3.5 Probablty.8.6.4. 5 5 5 Probablty Desty..5 5 5 5-3 - 3 Fg.. Results of Example (Fg.a ad Fg.b are the mea value ad stadard devato records of ad ; Fg.c ad Fg.d are the probablty ad probablty desty of wth dfferet umber of samplg; Fg.e ad Fg.f are the probablty ad probablty desty of wth dfferet umber of samplg.)
Scece Joural of Appled Mathematcs ad Statstcs 5; 3(6): 63-74 66. Probablty.8.6.4. 5 5 5 Probablty Desty.5..5 5 5 5-5 5 5-5 5 5. Probablty.8.6.4. 5 5 5 Probablty Desty.5..5 5 5 5-5 5 5-5 5 5 Fg. 3. Results of Example (Fg 3.a ad Fg 3.b are the probablty ad probablty desty of wth dfferet umber of samplg; Fg 3.c ad Fg 3.d are the probablty ad probablty desty of wth dfferet umber of samplg.) = F(X) = X X α X X + X X (6) The fucto possesses three radom varables: X, Xad α, the probablty desty dstrbuto of X, Xad α are Gaussa dstrbuto (5, ), Log-ormal dstrbuto (,.5), ad uform dstrbuto (, ), respectvely. Example : Suppose the lmt state fucto expressed as, = ML E (7) Where M, L, E are three radom varables. Here we defed the probablty desty dstrbuto of M, L, E are Gaussa dstrbuto (, ), Log-ormal dstrbuto (,.5), ad uform dstrbuto (., ), respectvely. The results of Example ad Example of ad dfferet umber of samplg are preseted Fg. ad Fg. 3. To beg geeratg the, a terval of each feature s chose at radom. The tersecto of these tervals the mult-dmesoal feature space s a small hypercube, from whch a sample s take at radom. Next, type of terval s selected at radom for each feature. A sample s produced at radom from that small hypercube. Ths cotues utl N samples have bee geerated. Each terval of each parameter s sampled exactly oce the process. I cotrast to radom samplg, the etre rage of each feature s always represeted a. Ubased estmates of the sample meas of the outcomes are obtaable. By comparg Fg.c wth Fg.e, t s obvous that method s easer to have coverget ad accurate result statstc wth smaller umber of samplg Example tha method. Fg.d ad Fg.e support the same cocluso. However, Example, whe the probablty desty of output result s very cocetrated a small rage, the advatage of s ot evdet the as Fg. 3. To compare wth more precsely, here we defed relatve error ca be calculated as Eq. (8) e = ( p p ) =, e d = ( f f) = (8) Therefore, the relatve errors probablty ad probablty desty ca be calculated as e ad e d. We dvded the rage of to tervals, accordgly pots the probablty ad probablty desty ca be obtaed ad expressed as p ad f, where p ad fare the probablty ad probablty desty of samplg or. Fg. 4 presets relatve errors of ad Example ad Example. I Fg 4.a ad Fg 4.b, the relatve errors probablty desty ad probablty of are smaller tha that of whe they have same umber of samplg. However, Fg 4.c ad Fg 4.d, the advatage of s ot absolutely evdet.
67 Lu Chu et al.: Applcato of Lat Hypercube Samplg Based Krgg Surrogate Models Relablty Assessmet Relatve Error of Probablty Desty 5 x -6 4 3 5 5 5 Number of Samplg Relatve Error of Probablty 4 x -3 3 5 5 5 Number of Samplg Relatve Error of Probablty Desty. x -6.8.6.4. 5 5 5 Number of Samplg Relatve Error of Probablty.8.6.4. x -3 5 5 5 Number of Samplg Fg. 4. Relatve errors of ad Example ad Example (Fg 4.a ad Fg 4.b are the relatve errors probablty desty ad probablty of ad Example ; Fg 4.c ad Fg 4.d are the relatve errors probablty desty ad probablty of ad Example.) 3. Surrogate Models A surrogate model ca be thought of as a regresso to a set of data, where the data s a set of put-output parg obtaed by evaluatg a black-box model of the complex system [6]. Surrogate models may be classfed to two geeral categores based o ther proposed: local ad global models. A example of optmzato usg local s Respose Surface Method expressed (RSM) as polyomal surrogate models. RSM sequetally fts local frst ad secod order regresso models to a small rego of the overall search space, as Eq. (9) ad Eq. (). F( β : x) = β + β x (9) = F( β : x) = β + β x + β x x = = = () I the other had, a global surrogate model s a fucto that approxmates the system across the desg space. Krgg models ft a spatal correlato fucto to a data set cosstg of put-output pars obtaed by evaluatg the uderlyg fucto. G( x) = F( β : x) + z( x) () Where F( β : x) s a determstc compoet defed by a regresso model that gves a approxmato to G( x ) mea value ad z( x ) s a statoary Gaussa process wth zero mea ad covarace, Cov[ z( x), z( x )] = σ R( θ : x, x ) () That terpolates the errors betwee the regresso model predctos F( β : x) ad the true lmt state fucto values G( x ) at the m realzatos of the vector of basc radom varables x, wth σ the costat process varace ad R s a prescrbed correlato fucto. Several correlato fuctos are avalable, such as the expoetal, lear ad Gaussa correlato fuctos, the most wdely used correlato fucto for structural relablty problems s the asotropc Gaussa correlato fucto Wth ( θ :, ) = exp( θ ) = R x x d (3) d = x x the dstace betwee the evaluato pot x ad the referece pot x the th drecto of the basc radom varables space ad θ = [ θ,..., θ ] T a vector of parameters that defe the verse of the correlato legth each drecto. A krgg terpolato model s completely defed by a vector of regresso coeffcets β, a vector of correlato parameter θ ad the varace σ of the statoary Gaussa process. These parameters are estmated by fttg the Krgg model to a sample of support pots. Where F s the regresso matrx ad y s the vector of true lmt state fucto values. The matrx R defes the correlato betwee each par of support pots accordg to the prescrbed correlato fucto. θ has to be frst estmated usg the method of maxmum lkelhood: { L θ } ˆ θ = argm ( ) (4) θ
Scece Joural of Appled Mathematcs ad Statstcs 5; 3(6): 63-74 68 m L( θ) = R( θ) σ ( θ) (5) Its predcto at a gve pot of the space of basc radom varables ca be obtaed, ˆ T ( ) ( ) ˆ T G x = f x β + r( x) ˆ γ (6) ˆ γ = R ( y Fβ) (7) ˆ T () ( m) r( x) = [ R( θ : x, x ),..., R( θ : x, x )] (8) A vector wth the correlatos betwee the predcto pot ( k ad the m realzatos x ) ( k =,..., m) of the vector of basc radom varables used the Krgg model fttg correspods to the expected or mea value of the Krgg model predcto, a estmate for the varace or ucertaty assocated wth the model predctos ca be gve by: T T T σ G = σ + u( x) ( F R F) u( x) r( x) R r( x) (9) T u( x) = F R r( x) f( x) () : = ( β ) β F x = x β () To ga some sght to the behavor of the Krgg surrogate models, we created Krgg models by database of ad, whch has dfferet amout of samplg. Zero-order, frst-order ad secod-order Krgg models are appled to predct the results whe the put varables are chaged Example ad Example. I Example, the rages of put varables ( X, Xad α ) are (4, 7), (., ), ad (.,.8) respectvely. They are totally cluded the rage of ad Secto. I Example, we set up the rages of put varables ( M, L, E) as (.5, 3.5), (.7,.3), ad (.,.8) respectvely. It sures that the rages of put varables surrogate Krgg models do ot exceed the scope of ad. Here, the relatve errors are calculated by comparg the dfferece betwee the results predctg by Krgg models wth the exact results from the aalytcal fucto Example ad Example. = y y r = = y e () Where y s the result predcted by Krgg models, ys the exact result from the aalytcal fucto, they have same put ( X, X ad α ) Example, ad ( M, L, E ) Example. Table ad Table 3 preset tme cost of Krgg models fttg ad predctg Example ad Example. Table ad Table 4 provde the values of relatve errors, whch demostrate the dfferece betwee the results predcted by Krgg models ad the exact results whe the put varables are certa. We fd that the umber of samplg or whch s the orgal database for Krgg models s the most mportat factor to tme cost. The samplg or wll be trasferred to Krgg models as the orgal database to defe parameters Krgg models. Accordg to the crease of umbers of samplg, the computatoal cost Krgg models wll sharply grow. Ths cocluso s proved both Table ad Table 3. I Table, the relatve errors from zero-order, frst-order ad secod-order Krgg model are very small. I Example, Krgg models have satsfed accuracy as surrogate models to predct. I addto, whe the umber of samplg ad s same, orgal database provded by s always better tha that of Example. Alog to the crease sze of orgal database from ad, results predcted by Krgg models are more precse. However, the computatoal cost also sharply creases as Table. Therefore, for creatg Krgg surrogate model, there s a trade-off betwee accuracy ad computatoal cost. Table. Tme cost of Krgg models fttg ad predctg Example. KM KM KM ().65.438.33 (5).396.3353.3499 ().5545.5945.64 () 6.656 6.768 9.99 (5) 6.89 58.4 65. ().756.747.85 (5).3356.3488.338 ().5767.5835.69 () 6.6484 6.836 6.855 (5) 5.465 58.48 59. Uts Table for tme cost are secod. The results are tested the same computer. Table. Relatve error of Krgg models predcto Example. KM KM KM () 4.436 e-4 5.566 e-4 9.7 e-4 (5).669 e-7.47 e-7.959 e-5 ().467 e-7.43 e-6 4.5837 e-6 () 3.96 e-8.744 e-8 3.446 e-8 (5) 7.8 e-8 3.97 e-8 3.4849 e-8 () 9.57 e-5. e-4.59 (5) 3.57 e-6 4.994 e-4.637 e-4 () 4.783 e-6.6464 e-5 7.97 e-6 ().45 e-7 8.899 e-8 7.5655 e-6 (5) 5.85 e-7 7.44 e-7.7 e-5 The results predcted by Krgg model Example do ot have the same level of accuracy as Example. I Table 4, we ca fd the relatve errors are uable to be gored. To make Krgg surrogate model to be more approprate complcated stuato, we stll have a lot of work to do. I order to track the property of Krgg models, tests of zeroorder, frst order ad secod-order Krgg models whch are based o the database of ad are performed ad recorded Fg. 5. Frstly, the results predcted by Krgg models whch are based o database of are
69 Lu Chu et al.: Applcato of Lat Hypercube Samplg Based Krgg Surrogate Models Relablty Assessmet more approxmated to the exact results of aalytcal fucto Example tha that based o database of. Therefore, for creatg Krgg surrogate models, method s a more effectve ad compettve method tha method. Secodly, at the begg of the predcto, Krgg models have large fluctuato ad the predcted results are ot precse whe compared wth the exact results, ths part should be take to cosderato ad chose to be removed or fltered. Lastly, compared the results predcted by zeroorder, frst order ad secod-order Krgg models whch are based o the database of wth dfferet umber of samplg, choce of best Krgg model depeds o the certa stuato. It s dffcult to defe whch Krgg model s the best. 8 6 4 KM - 5 5 Exact 4 3 KM - 5 5 Exact - Iput Varable - Iput Varable KM - 5 KM - - - 5-5 -3 Iput Varable - Iput Varable KM - 5 KM - - -5 - -5 - Iput Varable - Iput Varable Fg. 5. Krgg models predcto Example (Fg 5.a, Fg 5.c ad Fg 5.e are the results predcted by zero-order, frst-order ad secod-order Krgg models respectvely whch are based o database of ; Fg 5.b, Fg 5.d ad Fg 5.f are the results predcted by zero-order, frst-order ad secod-order Krgg models respectvely whch are based o database of ). Table 3. Tme cost of Krgg models fttg ad predctg Example. KM KM KM ().74.88.858 (5).3934.366.369 ().7373.466.89 () 4.667 4.6 6.435 (5) 7.7 7.8 7.45 ().699.86.674 (5).338.366.379 ().3988.5859.689 () 5.97 3.8678 4.643 (5) 6.85 3.89 43.57 (Uts Table 3 for tme cost are secod. The results are tested the same computer.) Table 4. Relatve error of Krgg models predcto Example. KM KM KM ().766 e6.4 e4.638 e5 (5).8833 e3 4.59 e5.888 e5 () 7.386 e3 5.375 e4.9754 e4 () 9.784 3.9539 e5 5.6788 e3 (5).7566 e4 473.6334.6469 e3 ().839 e5.55 e4 4.943 e5 (5).9474 e5.94 e4.9898 e3 ().578 e5 9.486 e4.367 e5 () 9.46 e3.64 e5 3.857 e3 (5).4799 e5 6.487 e3.736 e4
Scece Joural of Appled Mathematcs ad Statstcs 5; 3(6): 63-74 7 4. Example Fte Elemet Model Whe the lmt state fucto s mplct ad each determstc samplg s computatoal expesve, we propose creatg Krgg surrogate models to have relablty assessmet, s chose as a effectve method to provde orgal database for Krgg models. Modal frequecy s a mportat area of structural dyamcs whch has deservedly receved much atteto. Usually, researchers ad desgers detfy the basc modal frequeces of a specfc structural system ad avod the perodc loadg cocde wth them order to prevet the damage or falure of resoace. The dyamc equato ca be wrtte as, [ M ]{ x} + [ C]{ x} + [ K ]{ x} = { F} ɺɺ ɺ (3) Where [ M ] s the mass matrx descrbg the dstrbuto of mass, t s about the structural degree of freedom, { xɺ } ad { xɺɺ } are the frst ad secod dervatves of the dsplacemet wth respect to tme. Note that the force appled to the system s ow a fucto of tme. Whle mass ad stffess of a structure are measured ad relatvely easly derved, the mechasm whereby eergy s lost through dampg s less easly modeled. The vscous dampg model represeted by matrx [ C ] s commoly but by o meas exclusvely used, beg proportoal to velocty. If there s o dampg, the equato of moto s [ M ]{ x} + [ K ]{ x} = { F} ɺɺ (4) For free (uforced) vbratos the followg relatoshp s obeyed [ M ]{ x} + [ K ]{ x} = ɺɺ (5) The soluto to whch ca be wrtte the form { x} { ψ} e w t = (6) Where ω s the resoat frequecy. Substtutg back to the vbrato equato leads to the well-kow egevalue problem [ K ]{ ψ } = λ [ M ]{ ψ } (7) Where λ = ω, ad { ψ } ca be thought of the mode shapes correspodg to the system atural frequeces{ ω }. Whle the egevalues have a exact relatoshp wth the resoat frequeces, the egevectors are arbtrarly scaled; t s commo practce to defe a uquely scaled set of egevectors such that The result s T [ ] [ M ][ ] [ I ] φ φ = (8) T [ φ] [ K ][ φ] dag( λ) = (9) Where [ φ ] s the matrx of mass ormalzed ege-vector. I ths paper, our fte elemet model of wg structure, as preseted Fg., s costructed by ANSS Parameter Desg Laguage. The parameters the orgal determstc model are correspodg wth geometrcal propertes ad materal propertes. Where S s the parameter represetg the rato of area betwee the two arfol sectos, t s.5 as tal. L ad D as preseted the Fg. 6, are 6.5 m ad.4 m respectvely. For materal property, oug s module s 7e Pa, Posso rato s.33, ad physcal desty s 7 kg/m 3. Natural frequecy / Hz Mmum stress / N/M Maxmum stress / N/M Fg. 6. Fte elemet model of wg structure. Table 5. Results of determstc fte elemet model. 3 4 5 6468 97798 9869 44798 5788.43e.75e.88e.94e.48e.49e.89e3.43e3.63e3.39e3 The results of modal frequeces of wg structure the determstc fte elemet model are as preseted Table 5. Lat Hypercube samplg method s performed the determstc fte elemet model to calculate the modal frequeces. The parameters correspodg wth geometry (S, D, L) ad materal property (E, P, R) are as put varables the process of Lat Hypercube samplg, whle the modal frequeces of specfc wg structure are output varable each samplg terato. We chose uform dstrbuto as probablty dstrbuto for put varables. The rages of S, D, L are (,.5), (, 3) ad (4, ) respectvely. The rages of E, P, R are (le, e), (,.5) ad (, 8) respectvely. The uts for put varables keep uchagg as the determstc fte elemet model. We had samplgs for each put varables by method. Fg.7 provdes the records of fve modal frequeces the process of stochastc smulato. To be more obvous, the accumulatve probabltes of fve modal frequeces of wg structure are preseted Fg.8 as umercal statstcs. The evaluato of the stochastc smulato fte elemet model of wg by Lat Hypercube samplg method s preseted Table. The mea value, stadard devato, skewess, ad also the mmum ad maxmum of fve modal frequeces for wg structure by are obtaed ad cocluded Table.
7 Lu Chu et al.: Applcato of Lat Hypercube Samplg Based Krgg Surrogate Models Relablty Assessmet Modal frequecy x 6.5 F F F3 F4 F5.5 3 4 5 6 7 8 9 Iterato umber Fg. 7. Records of modal frequeces the process of Lat Hypercube samplg. Cumulatve probablty.8.6.4. Modal frequecy Modal frequecy Modal frequecy 3 Modal frequecy 4 Modal frequecy 5 Mea value / *e5 Hz Stadard devato / *e5 Hz Table 6. Results of Lat Hypercube Samplg method. F F F3 F4 F5.48595.563.46 3.48 4.783.6679.88.99.6633.66 Skewess / *e5 Hz.57.9.59.9678.958.5.5 Value of modal frequecy x 6 Fg. 8. Cumulatve probablty of fve modal frequeces. Mmum / *e5 Hz.889.33.474.744.69 Maxmum / *e5 Hz.9 5.79 9.498.67 8.737 Probablty Desty 3.5 x -5 3.5.5 () () KM KM KM Probablty Desty 3 x -5.5.5 () () KM KM KM.5.5-5 5 5 Frst Modal Frequecy x 4-5 5 5 Frst Modal Frequecy x 4 Probablty Desty.5 x -5.5.5 () KM(-) KM(-) KM(-5) Probablty Desty x -5.5.5 () KM(5) KM() KM() 5 5 Frst Modal Frequecy x 4 5 5 Frst Modal Frequecy x 4 Fg. 9. Tests of Krgg models as surrogate models for fte elemet model (Fg 9.a ad Fg 9.b are the results of probablty desty of fst modal frequecy predcted by Krgg models whch are based o ad, respectvely; Fg 9.c presets the results predcted by secod-order Krgg model based o dfferet umber of ; Fg 9.d presets the results predcted by secod-order Krgg model wth dfferet amout of pots).
Scece Joural of Appled Mathematcs ad Statstcs 5; 3(6): 63-74 7 I order to detfy oe Krgg model as a surrogate model to replace tme-cost computato of orgal model, we should fd aswers for three questos: frstly, whch Krgg model s the most approprate to ths specfc case; secodly, how may should be provded as orgal database for Krgg model; lastly, we set up how may pots Krgg model for predcto. These three questos ca be solved Fg. 9. Fg 9.a presets the results of probablty desty of fst modal frequecy predcted by zeroorder, frst-order ad secod-order Krgg models based o ad compared wth the results of ad. It s obvous that secod-order Krgg model s more approxmated to the orgal fte elemet model. Fg 9.b whch presets the results predcted by Krgg models based o supports the same pot. The, secodorder Krgg model s chose as surrogate model to have predcto. I Fg 9.c, we fd that the results predcted by secodorder Krgg model based o dfferet amout of, amely, ad 5, are close to each other ad coverget very well to the precse result. I the other had, havg predcto to pots ad pots by defed secod-order Krgg model has the same level of accuracy ad stablty as demostrated Fg 9.d. Krgg model as a surrogate model, t, the secod order regresso provdes coverget ad accurate predcto results. I addto, the advatages of Krgg model are ot oly at ther accuracy, but also reflect at tme-savg process. The 5 ad performg calculato of modal frequeces of wg structure the fte elemet model costs 955.49 s, ad f samplg, t also cost 37.37 s; whle the surrogate model, fttg secodorder Krgg model oly costs 9.63 s, ad predct the correspodg result of put radom samplg, t oly costs.73 s. The advatage of tme-savg s very compettve as a surrogate model. The coveece of Krgg surrogate model s sgfcat relablty assessmet. Mechacal resoace may cause volet swayg motos ad eve catastrophc falure mproperly costructed structures cludg brdges, buldgs ad arplaes, a pheomeo kow as resoace dsaster. It s the tedecy of a mechacal system to respod at greater ampltude whe the frequecy of ts oscllatos matches the system's modal frequecy of vbrato. Desgers struggle to avod ths physcal pheomeo happeg the operato stuato. However, the tradtoal determstc model gores the effects of ucertates the real complcated operato evromet. To propagate the ucertaty the determstc model of complcated system, the calculato expese s a heavy burde. It s ecessary to create a approprate surrogate model to have predcto ad provde relablty assessmet. Mea value of modal frequecy 6 x 5 5 4 3 M M M3 M4 M5 Varace of modal frequecy x.5.5 V V V3 V4 V5 µ µ B of modal frequecy 4 8 6 B B B3 B4 B5 Dfferece of Mea value.5 x 5.5.5 M M3 M43 M54 4 µ -.5 µ Fg.. Results predcted by Krgg model based o for relablty assessmet (Fg.a ad Fg.b preset meda value ad varace for fve modal frequecy respectvely; Fg. c provdes results of B for fve modal frequecy; Fg.d presets the dfferece betwee two eghbor modal frequeces).
73 Lu Chu et al.: Applcato of Lat Hypercube Samplg Based Krgg Surrogate Models Relablty Assessmet I the example of fte elemet model of wg structure, the parameters correspodg wth geometry (S, D, L) ad materal property (E, P, R) are supposed to be ucerta ad fluctuate a specfc rage order to smulate the ucertates the real stuato. To be geeral, the type of probablty dstrbuto s chose to be Gaussa dstrbuto, as( µ, σ ), ( µ, σ ), ( µ, σ ), ( µ, σ ), ( µ, σ ) ad S S D D L ( µ R, σ R) for the parameters respectvely. The stadard devato s settled by % of the mea value for each parameter to smulate the fluctuato. Secod order Krgg model based o s appled as surrogate model. The, the relato betwee the probablty property (mea value, varace, etc.) of fve modal frequeces ad mea value of put varables (the mea value of the put varables ths paper Sychroous crease, they ca be set as ay other tedecy for relablty assessmet). I Fg.a ad Fg.b, we ca fd that both mea value ad varace for all fve modal frequecy wll reduce whe the mea value of put varables crease. B s the result of mea value dvded by the stadard devato for every modal frequecy. I cotrary wth mea value ad varace of modal frequecy, B has postve gradet as Fg.c. The dfferece betwee two eghbor modal frequeces s a mportat parameter structural desg. Fg.d presets dfferece of mea value betwee the secod ad frst modal frequecy, betwee the thrd ad the secod modal frequecy, ad betwee the ffth ad the forth modal frequecy has the same tedecy, they all have egatve gradet. I cotrary, the dfferece of mea value betwee the fourth ad the thrd modal frequecy has postve gradet. Therefore, Fg. provdes essetal formato for structural desgers. 5. Cocluso L. method s more effectve tha method for provdg comprehesve orgal database to Krgg models.. Whe the relato betwee the put varables ad output results ca be expressed as polyomal, Krgg models have satsfed accuracy as surrogate models to predct. Alog wth the crease the umber of samplg, the computatoal cost of creatg Krgg models sharply grows, the same tme, the results predcted by Krgg models ca be more precse. Therefore, there s a trade-off betwee computatoal expese ad accuracy. 3. Comparso betwee zero-order, frst-order ad secodorder Krgg model s ot evdet, whch oe s the most approprate model depeds o the specfc relato betwee the put varables ad output results. 4. Krgg models as surrogate models are very promsg relablty assessmet. Its advatages of tme-savg ad hgh level of accuracy are compettve ucertaty aalyss. E E P P Refereces [] Doald L. Phllps, Day G. Mark. Spatal ucertaty aalyss: propagato of terpolato errors spatally dstrbuted models. Ecologcal Modellg, Volume 9, Issues 3, 5 November 996, Pages 3-9. [] L. Goel, R. Bllto. Mote Carlo smulato appled to dstrbuto feeder relablty evaluato. Electrc Power Systems Research, Volume 9, Issue 3, May 994, Pages 93-. [3] R.E. Melchers. Importace samplg structural systems. Structural Safety, Volume 6, Issue, July 989, Pages 3-. [4] L. Mark Berler, Chrstopher K. Wkle. Approxmate mportace samplg Mote Carlo for data assmlato. Physca D: Nolear Pheomea, Volume 3, Issues, Jue 7, Pp 37-49. [5] P. Beaurepare, H. A. Jese, G. I. Schuëller, M. A. Valdebeto. Relablty-based optmzato usg brdge mportace samplg. Probablstc Egeerg Mechacs, Volume 34, October 3, Pp 48-57. [6] Muoz Zuga Mguel, Garer Jossel, Remy Emmauel. A orgal sestvty statstc wth a ew adaptve accelerated Mote-Carlo method. Proceda - Socal ad Behavoral Sceces, Volume, Issue 6,, Pages 77-773. [7] ag Lu, M. ousuff Hussa, Gray Ökte. Optmzato of a Mote Carlo varace reducto method based o sestvty dervatves. Appled Numercal Mathematcs, Volume 7, October 3, Pages 6-7. [8] Mchael D. Shelds, Krubel Teferra, Adam Hap, Raymod P. Daddazo. Refed Stratfed Samplg for effcet Mote Carlo based ucertaty quatfcato. Relablty Egeerg & System Safety, Volume 4, October 5, Pp 3-35. [9] Ste M. Large sample propertes of smulatos usg Lat hypercube samplg. Techometrcs 987. [] Owe A. A cetral lmt theorem for Lat hypercube samplg. J. R. Stat Soc B 99, 54(): 54-5. [] Hutgto D. Improvemets to ad lmtatos of Lat hypercube samplg. Probablstc Egeerg Mechacs, 998, 3(4): 45-53. [] Stock R. A study o algorthms for optmzato of Lat hybercubes. J Stat Pla Iferece 6, 36(9): 33-347. [3] Ima R. A dstrbuto-free approach to ducg rak correlato amog put varables. Commo Stat: Smul Comput 98: (3):3-334. [4] Flora A. A effcet samplg scheme: updated Lat hypercube samplg. Probablstc Egeerg Mechacs, 99, 7: 3-3. [5] Coppa T, Lucas T. Effcet early orthogoal ad spacefllg Lat hypercubes. Techometrcs 7, 49(), 45-55. [6] Novak D. Correlato cotrol small-sample Mote Carlo type smulatos : a smulated aealg approach. Probab Eg Mech 9, 4, 45-46.
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