CURVE FITTING ON EMPIRICAL DATA WHEN BOTH VARIABLES ARE LOADED BY ERRORS

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1 TOME VI (ye 8) FASCICULE (ISSN ) CURVE FITTING ON EMPIRICAL ATA WHEN BOTH VARIABLES ARE LOAE BY ERRORS ANRÁS NYĺRI LÁSZLÓ ÖNÖZSY Pofesso emets etmet of Fd d Het Egeeg Uvesty of Msoc H-55 Msoc-Egyetemváos Hgy Resech feow Chst oe Lbotoy fo Mthse Modeg of Metgc Pocesses Uvesty of Leobe A-87 Leobe Ast ABSTRACT: A method hs bee deveoed to ft cve dffeetbe to the th ode devtves oto set of emc dt whe both deedet d deedet vbes e oded by eos of om obbty dstbto. The osed method c sccessfy be sed dffeet feds of mechc/mte scece d egeeg s we s the fed of mesh geeto techqes to ovde gds fo comtto fd dymcs (CF) smtos. EYWORS: smoothg ocede oyoms cve fttg cotos fctos mesh geeto. INTROUCTION The motce of the smoothg d cve fttg obem wee yed mott oe Whtte s wo [] the mdde of the th cety. Nyí ws deveoed smoothg ocede d e eqto system sove method [45] whe the emc dt sets hve bee oded by dom eos. These methods wee sccesfy bt secod-ode cotos mesh geeto techqe by öözsy [6] to deteme othogo cve coodte etwo fo comtto fd dymcs (CF) smtos. A method ws fthe deveoed by Nyí [7] to costct Hemte oyoms fttg o to 4 8 ots of fctos esectvey to the th ode devtves. Usg smoothg ocede wth ote cve fttg method s oe of the most eevt sse cety the fed of mechc/mte scece d egeeg ctos. Fo eme the dscotos hse dgm fomto hs to be coed wth the coesodg tsot eqtos fo modeg sodfcto ocesses sg dst stees [89]. The osed method c hve beefc effect o these ds of obems s we esecy whe dscotos fcto hs to be sbsttted by cotos oe. 7

2 ANNALS OF THE FACULTY OF ENGINEERING HUNEOARA JOURNAL OF ENGINEERING. TOME VI (ye 8). Fscce (ISSN ). THE EQUATION OF SMOOTHING The m of ths e s to ft th ode dffeetbe cve of oto set of emc dt oded by eos of om dstbtos o both vbes. The bshed es [45] oy the deedet vbes cses of d dmesos wee sosed hvg eos. The hyothess w be mted tht the obbty dstbto of the eo foows the om w d ot y moe ssmto w be t coceg the css of the ect fcto fom whch the ves dffe f thee s y. The oot of the smoothg s the sm of sqes of the dffeeces betwee the th emc dt d the obted ves d the dvded dffeeces of those w be the mmm. Let the ( ξ η ) emc dt be gve the ( y) ξ < ξ whee the. The mmm of the sm w be soght fo S ( ) ( η ) q ( ξ ) th dvded dffeece s coodte system (.) Itodcg the Lgge fcto. (.) the fst devtve of ths s d d d ( ) ( ) λ (.) λ ( ) ( ) ( λ ) λ λ ( ) ( ) λ λ (.4) λ (.5) wth ths the dvded dffeece c be wtte s foows [ ( )] f the (.6) f the f the m[ ]. The mbes > q > e vesey ooto to the sqe of the stdd devtos. The t sm of the Eq. (.) beogs to ot s S ( ) ( η ) q ( ξ ) The ecessy codtos fo the mmm of ths sm e S. (.7) S. (.8) 8

3 ANNALS OF THE FACULTY OF ENGINEERING HUNEOARA JOURNAL OF ENGINEERING. TOME VI (ye 8). Fscce (ISSN ) 9. THE FIRST CONITION OF SMOOTHING Let s dffeette S esect to gettg ( ) η S (.) whee ( ) ( ) [ ]. The e eqto system (L.E.S.) hs to be soved s s foows ( ) ( ) [ ] η. (.) The comct fom of the system s η (.). the f the f the whee f The coeffcets e ( ) [ ] µ (.). the f the f the f µ µ µ ( ) ( ) [ ] µ (.b) whee { } m the f the f the f µ > µ < the f the f the f µ > µ < { }. the f m the f the f µ > µ < The gothm fo the soto of the bded L.E.S. hs bee fod e [4].

4 ANNALS OF THE FACULTY OF ENGINEERING HUNEOARA JOURNAL OF ENGINEERING. TOME VI (ye 8). Fscce (ISSN ). THE SECON CONITION OF SMOOTHING Fo stsfyg the secod codto et s dffeette the S ccodg to ( ) ( ) ξ q (.) d deotg ths by f ( ) ξ q f (.) ( ) µ f f [ ] T µ [ ] T f f f... f whee. the f the f the f µ µ µ The devtve teso of f vecto ccodg to s f (.) d the etes of t e q f whee the tevs fo e s sme s the cse of Eq. (.). We hve obted bded stcte o-e system of eqtos s we hve hd evosy. Fthemoe et s dffeette the Eq. (.) ( ) [ ] ( ) { ( ) [ ] ( ) ( ) ( ) [ ] ( )} ( ) [ ] d the o-e system of eqto ( ) ( ) b f t hs to be soved by teto fo t d m m m t. The eemets of the ( ) teso c be omted s ( ) ( ). f f t

5 ANNALS OF THE FACULTY OF ENGINEERING HUNEOARA JOURNAL OF ENGINEERING. TOME VI (ye 8). Fscce (ISSN ) 4. THE CHOICE OF THE SMOOTHING PARAMETERS Fo the se of smcty et s choose the metes deedet of.e. d q q. Let s defe the foowg vectos todcg the foowg [...] T [ ] T ξ [ ] T... ξ...ξ ( ξ η; q) ( ) η; η ( ) ( ) ( ) ( ) fctos whch gve the eesso fo Eq. (.) s ξ; q ξ (4.) S q (4.) d the codtos fo gettg the mmm ve e S S. Let s omte e d e t whch the mmm s eched the S S ( e e; q). Choosg bty mbes e κ ρ > d eme the foowg S e sm-tot whch gees wth ( ; κ ρ) ( ; q) κ ( ; ) ρ ( q) e e e e e; S f κ d ρ q. I ode to sech the mmm of S s the fcto of d q we hve to ccte the devtves If κ κ d ρ ρ S κ S ρ q q q the the devtves w be zeo. Choosg κ d q ρ the κ d S q ρ. q ( q ) S( κ ρ) d S hs ts mmm. At fst et s cosde the cse whe the odtes e smoothed oy S ( ) ( η ) If the the soto w coesod to the codtos.e. t s. Ths s Gss est sqe oyom d ths method s eteso of t. I ths cse d deotes the coesodg ves. beogs to q. The secod cse whe thee w be o smoothg the d ( ). degee bo whch mmzes the sm ( η ).

6 ANNALS OF THE FACULTY OF ENGINEERING HUNEOARA JOURNAL OF ENGINEERING. TOME VI (ye 8). Fscce (ISSN ) Net emto w be the cse whe d q e vbes d et s cosde to the foowg fcto whee ( ) ( ) ( q) q U( q) s beogg to q. Fo S we obt coseqety d S U d S q κ ρ ( q ) U( q ) (4.) ( q ) U( q ) U. q Choosg these metes ccodgy the sm S w be mmzed so tht the sms of ( η ) d ( ξ ) w be the smest. If the eos of the emc dt e ow.e. ( ) ( ) g η fctos ( ) ( q) g ξ wth oe d q c be detemed. e gve the comtg the 5. THE CURVE FITTING Afte beg cometed to the smoothg ocede the obted set of ves c be sed by yg the fowd d bcwd Newto I. d II. m th ode oyoms. Oe c ft oto m ots of oe of the oyoms o m m P ( ) (5.) Q m ( ) ( ). (5.) m Usg the devtves of those oyoms t evey eghbog ot oe c ft Hemte oyoms [7] og smoothy to the eqed ode of dffeet qotets [7]. Fy the ( µ ) th devtve of the fcto s whee ( µ ) ( µ ) [ ]! µ... (5.) [] [ ]... (5.4) mes the sm of the ossbe eemets odcts of. λ () λ λ... λ λ λ λ λ λ λ. (5.5) λ

7 ANNALS OF THE FACULTY OF ENGINEERING HUNEOARA JOURNAL OF ENGINEERING. TOME VI (ye 8). Fscce (ISSN ) 6. THE ALGORITHM OF SMOOTHING PROCEURE Fo the foowg eme we sose tht thee e two e d owe bds est comed to the m dgo of the e eqto system ( ) theefoe d. Fst of the dvded dffeeces w be comted d ech ste c be fod Tbe -. ( )( ) [ ( )] [ ( )] [ ( )] ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) TABLE. THE FIRST ERIVATIVE OF THE LANGRANGE FUNCTION USING EQUATION (.5) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

8 ANNALS OF THE FACULTY OF ENGINEERING HUNEOARA JOURNAL OF ENGINEERING. TOME VI (ye 8). Fscce (ISSN ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) TABLE. FIRST ERIVATIVES FOR SATISFYING THE CONITIONS OF SMOOTHING PROCEURE ( ) TABLE. SECON ERIVATIVES FOR SATISFYING THE CONITIONS OF SMOOTHING PROCEURE Afte owg the dvded dffeeces the fst devtve of the Lgge oyom d the othe devtves fo stsfyg the codtos of smoothg ocede we hve to sove the e eqto system ccodg to Eqs. (.)-(.b). The coeffcets of the e eqto system fo stsfyg the fst codto of smoothg ocede e s foows 4

9 ANNALS OF THE FACULTY OF ENGINEERING HUNEOARA JOURNAL OF ENGINEERING. TOME VI (ye 8). Fscce (ISSN ) f [ ( )] µ [ ( ) ( )] f the µ the µ. Fo stsfyg the secod codto of smoothg we hve to sove o-e system of eqto theefoe the eemets of the Jcob mt Eq. (.) hs to be costcted ccodg to f q whee d the ecessy devtves e ( ξ ) f q [ ( )] ( ) [ ( )] ( ). 7. NUMERICAL EXAMPLE FOR SMOOTHING PROCEURE A comete mec eme hd bee fod Tbe ξ η TABLE 4. THE ( ξ η ) SET OF EMPIRICAL ATA ( ) ( ) TABLE 5. ELEMENTS OF THE LINEAR EQUATION SYSTEM FOR THE FIRST CONITION 5

10 ANNALS OF THE FACULTY OF ENGINEERING HUNEOARA JOURNAL OF ENGINEERING. TOME VI (ye 8). Fscce (ISSN ) The dvded dffeeeces [ ( )]. [ ( )]. 6 [ ( )]. 8 d the dgo eemets of the e eqto system [ ( )] [ ( ) ( )] [ ( ) ( )] [ ( ) ( )] 6. 4 [ ( ) ( )] ( ) TABLE 6. ERIVATIVES FOR CONSTRUCTING THE JACOBIAN MATRIX 6

11 ANNALS OF THE FACULTY OF ENGINEERING HUNEOARA JOURNAL OF ENGINEERING. TOME VI (ye 8). Fscce (ISSN ) TABLE 7. ERIVATIVES FOR CONSTRUCTING THE JACOBIAN MATRIX [ ( )] ( ) TABLE 8. ERIVATIVES FOR CONSTRUCTING THE JACOBIAN MATRIX 7

12 ANNALS OF THE FACULTY OF ENGINEERING HUNEOARA JOURNAL OF ENGINEERING. TOME VI (ye 8). Fscce (ISSN ) [ ( )] ( ) TABLE 9. ERIVATIVES FOR CONSTRUCTING THE JACOBIAN MATRIX TABLE. ERIVATIVES FOR CONSTRUCTING THE JACOBIAN MATRIX 8

13 ANNALS OF THE FACULTY OF ENGINEERING HUNEOARA JOURNAL OF ENGINEERING. TOME VI (ye 8). Fscce (ISSN ) TABLE. ERIVATIVES FOR CONSTRUCTING THE JACOBIAN MATRIX f q TABLE. IAGONAL ELEMENTS OF THE JACOBIAN MATRIX 8. NUMERICAL EXAMPLE FOR CHOOSING THE SMOOTHING PARAMETERS ξ η q TABLE. THE SET OF EMPIRICAL AN SMOOTHE ATA SYSTEM 9

14 ANNALS OF THE FACULTY OF ENGINEERING HUNEOARA JOURNAL OF ENGINEERING. TOME VI (ye 8). Fscce (ISSN ) TABLE 4. QUANTITIES FOR CHOOSING THE SMOOTHING PARAMETER q q TABLE 5. QUANTITIES FOR CHOOSING THE SMOOTHING PARAMETER FIGURE. CHOOSING THE SMOOTHING PAREMETERS

15 ANNALS OF THE FACULTY OF ENGINEERING HUNEOARA JOURNAL OF ENGINEERING. TOME VI (ye 8). Fscce (ISSN ) 9. NUMERICAL EXAMPLE FOR MESH GENERATION FIGURE. A GRI FOR COMPUTATIONAL FLUI YNAMICS (CF) SIMULATIONS. CONCLUSIONS The de of the eseted method hs bee bsed o eteso of Gss est sqe oyom. It s cbe to smooth emc dt system whch e oded by eos of om obbty dstbto d to ft cve dffeetbe to the th ode devtves oto set of smoothed emc dt system. The osed method c be sed dffeet feds of mthemtc d egeeg sceces s we s the fed of mesh geeto techqes esecy whe dscotos fcto hs to be sbsttted by cotos oe.

16 ANNALS OF THE FACULTY OF ENGINEERING HUNEOARA JOURNAL OF ENGINEERING. TOME VI (ye 8). Fscce (ISSN ) REFERENCES/BIBLIOGRAPHY [] E. Whtte: The Ccs of Obsevto Bce & So Ltd. Lodo 954. [] A. Nyí: Soto of the Thee-meso Fow the Bded Sce of Hydc Mche bsed o Potet Theoy ( Hg) octo Thess of the Hg Acdemy of Sceces Uvesty of Msoc etmet of Fdd Het Egeeg Msoc Hgy 99. [] A. Nyí: A Method of Smoothg Emc Fcto t No-Eqdstt Pots Poceedgs of the 9 th Cofeece o Fd Mchey Bdest Hgy 99. [4] A. Nyí: Sfce Fttg d New ect Method fo Sovg Boc Bd Le System A Iteto Jo Comtes & Mthemtcs wth Acto Vo [5] A. Nyí: Smoothg Thee-meso Set of Emc t Msct Uvesty of Msoc etmet of Fd- d Het Egeeg Msoc Hgy. [6] L. öözsy: Comtto of Two-meso She Fows wth the Soto of Tbet Votcty Tsot Eqto ( Hg) Ph.. thess Uvesty of Msoc etmet of Fd d Het Egeeg Msoc 4. [7] A. Nyí: Hemte Iteotg Poyoms Msoc Mthemtc Notes Vo. 6 No. Msoc Hgy [8] A. Ldwg M. Gbe-Petze F. Mye A. Ishmz M. W: A Wy of Cog Tey Phse gm Ifomto wth Mthse Sodfcto Smtos Mt. Sc. Eg. A [9] A. Ldwg A. Ishmz M. Gbe-Petze F. Mye M. W R. Tze W. Schützehöfe: How To Combe Tey Phse gm Ifomto wth Mthse Sodfcto Smtos Poceedgs of the 5th ece Iteto Cofeece o Sodfcto Pocessg Sheffed U [] A. Ishmz M. Gbe-Petze F. Mye M. W A. Ldwg: Mthse Mtcomoet Modeg of Sodfcto Pocesses: Cog Sodfcto etcs wth Themodymcs Iteto Jo of Mtes Resech 8. ( tg)