Study of the Efficiency and Accuracy of Optimisation Algorithms within Inverse Identification of the Parameter Values of a Nonlinear Concrete Material Model

SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 19 | NUMBER: 2 | 2019 | DECEMBER © 2019 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 9 STUDY OF THE EFFICIENCY AND ACCURACY OF OPTIMISATION ALGORITHMS WITHIN INVERSE IDENTIFICATION OF THE PARAMETER VALUES OF A NONLINEAR CONCRETE MATERIAL MODEL Pe KRÁL1, Jiří KALA1, Pe HRADIL1 1Ins i u e o S uc u al Mechanics, Facul y o Ci il Enginee ing, B no Uni e si y o Technology, Ve eří 331/95, 602 00 B no, Czech Republic [email p o ec ed], [email p o ec ed], [email p o ec ed] DOI: 10.35181/ ces-2019-0013 Abs ac . The in e se iden i ica ion o he pa ame e alues o nonlinea ma e ial models, which ha e been de eloped o , in e alia, conc e e modelling, is cu en ly a p ocess ha is widely used and in es iga ed in he ield o esea ch and de elopmen . Today he e a e se e al app oaches ha can be employed o he in e se iden i ica ion p ocess. One o he mos signi ican o hese app oaches in ol es he use o op imisa ion algo i hms which, howe e , o en demons a e a ying le els o p ecision and e iciency wi hin speci ic asks. These aspec s a e he subjec o he esea ch p esen ed in his con ibu ion. Keywo ds Compu a ional model, conc e e, expe imen al da a, ou - poin bend, global op imisa ion, ma e ial model, objec i e unc ion, op imisa ion algo i hms. 1. In oduc ion The modelling o con inuum mechanics asks using nonlinea mechanics ools is cu en ly he main a ea o ocus a many scien i ic ins i u ions ([1], [2], [3] and [4]). The e m “use o nonlinea mechanics ools” e e s in pa icula o he use o geome ic and ma e ial (physical) nonlinea i ies wi hin con inuum mechanics asks. The necessi y o using geome ic o ma e ial nonlinea i ies wi hin nume ical calcula ions is dependen on he ype o s uc u e in ques ion and pa icula ly on he selec ed ma e ial om which i will be made. Mode n compu a ional sys ems based on he ini e elemen me hod ([5], [6], [7] and [8]), which a e cu en ly seeing widesp ead use in he in es iga ion o con inuum mechanics asks, con ain a se ies o me hods o he conside a ion o he nonlinea beha iou o s uc u es, along wi h many nonlinea ma e ial models ha can be employed o desc ibe he beha iou o p ac ically any ma e ial in he con ex o nume ical simula ions. Howe e , he applica ion o nonlinea ma e ial models wi hin calcula ions o a nume ical na u e gi es ise o a undamen al di icul y in e ms o he necessi y o de ine he pa ame e s o hese models co ec ly in o de o hem o unc ion p ope ly. This ask o en is no e y easy as nonlinea ma e ial models (and nonlinea ma e ial models o conc e e in pa icula ) e y o en include pa ame e s o a pu ely ma hema ical na u e o pa ame e s which can only be de i ed using special expe imen al da a. I ele an da a which would enable he alues o he ma e ial model’s pa ame e s o be de i ed di ec ly a e no a ailable, he p ocess o he in e se iden i ica ion o ma e ial pa ame e alues can cu en ly be used o deal wi h his p oblem ([9], [10], and [11]). Du ing in e se iden i ica ion, ou pu da a usually consis ing o expe imen al da a a e gene ally used o ob ain alues o ma e ial and o he pa ame e s. The in e se iden i ica ion p ocess is he e o e based on he combina ion o expe imen al da a wi h nume ical calcula ions and iden i ica ion app oaches. The goal is o ob ain he bes possible app oxima ion o expe imen al da a om nume ically simula ed da a. The mos widely used iden i ica ion app oaches oday a e me hods based on he exe cise o a i icial neu al ne wo ks [12] and op imisa ion algo i hms [13]. The capabili y o using op imisa ion algo i hms o pe o m he in e se iden i ica ion o pa ame e alues is o e ed by, e.g. op iSLang so wa e [14], which con ains a o al o i e op imisa ion algo i hms whose e iciency and accu acy can a y o di e en asks. The aim o his pape is o in es iga e he e iciency and accu acy o op iSLang op imisa ion algo i hms du ing SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 19 | NUMBER: 2 | 2019 | DECEMBER © 2019 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 10 he in e se iden i ica ion o alues o a small quan i y o pa ame e s o he modi ied e sion o a nonlinea ma e ial model o conc e e known as he Con inuous Su ace Cap Model, which is implemen ed in he explici ini e elemen compu a ional sys em LS-Dyna [15]. Fo his pu pose, a ask in he o m o a ou -poin bending es is ca ied ou on a high, s eel- ein o ced conc e e beam. To achie e he se goal, he applica ion o his conc e e model and o he ma e ial models wi hin he compu a ional model o a es c ea ed in LS-Dyna is equi ed as well as he use o expe imen al da a ob ained om a eal ou -poin bending es . These aspec s a e desc ibed in he ollowing chap e s. 2. Ma e ial Models A o al o h ee ma e ial models implemen ed in he LS- Dyna p og amme we e used in he compu a ional model o he ou -poin bending es . A modi ied e sion o he Con inuous Su ace Cap Model was used o model he beha iou o he high conc e e beam. The beha iou o he conc e e ein o cemen was modelled using he Plas ic Kinema ic Model and he beha iou o he washe s was modelled using he Linea Elas ic Model. 2.1. The Con inuous Su ace Cap Model The Con inuous Su ace Cap Model ([16] and [17]) is a nonlinea ma e ial model o conc e e based on elas o- plas ic cons i u i e heo y. The occu ence o plas ic de o ma ions is, wi hin he model, con olled by he achie emen o a yield su ace [18] whose unc ional ela ionship can be exp essed as: 22 123 2 3 1 1 (, , ) ( ) () (,), c YI J J J J F I FI κ =−ℜ (1) whe e he second membe on he igh side o he equa ion is a combina ion o he shea ailu e unc ion F (I1) and he ha dening model Fc (I1, κ ) ia a mul iplica i e o mula ion. The shea ailu e unc ion and he ha dening model can be exp essed ma hema ically as: 1 11 () exp , I FI I β αλ θ − =− + (2) 2 1 12 (()) (,) 1(() ()) c IL FI XL κ κκκ − =− − o 1(),IL κ > (3) 1 (,)1 c FI κ = o 1(),IL κ ≤ (4) whe e: ()L κκ = o 0, κκ > (5) 0 ()L κκ = o 0, κκ ≤ (6) 1 () () (). XLRFI κκ =+ (7) The desc ip ion o he pa ame e s con ained in Eqs. (1)-(7) is as ollows: I1 is he i s in a ian o he s ess enso , J2 and J3 a e he second and hi d in a ian s o he de ia o ic pa o he s ess enso , ℜ(J3) is he educ ion ac o acco ding o Rubin, κ is he ha dening pa ame e , α , β , λ, and θ a e ma e ial cons an s which a e de i ed om he expe imen al es ing o conc e e in iaxial comp ession, and R is he a io pa ame e o he ha dening model. As he Con inuous Su ace Cap Model is pa o an explici ini e elemen sol e , i enables he e ec o he s ain a e on he s ess s a e o be conside ed wi hin i s o mula ion. Howe e , his abili y o he model can be neglec ed du ing he calcula ions ia he pe inen se ing o he model pa ame e IRATE (IRATE = 0: calcula ion wi h he e ec o he s ain a e on he s ess s a e → he iscous componen o he model is swi ched o ; IRATE = 1: calcula ion wi h he e ec o he s ain a e on he s ess s a e → he iscous componen o he model is swi ched on). I can be concluded om he abo e ac s ha i he pa ame e IRATE equals ze o, he nume ically simula ed esponse o he model co esponds o s a ic (slow) loading, which means ha i is independen o he eloci y o loading used. Fo his eason, he Con inuous Su ace Cap Model can be used no only o he nume ical modelling o he dynamic loading o conc e e s uc u es bu also o model he quasi-s a ic o s a ic loading o conc e e s uc u es. Wi hin he s udy desc ibed in his pape , he pa ame e IRATE equals ze o was used because he s a ic esponse o he s uc u e was modelled. In o de o p e en he dependence o nume ical simula ion esul s on he ini e elemen mesh, an algo i hm is implemen ed in he ma e ial model which is based on he p inciple o he c ack band model and hus ul ils he unc ion o a localisa ion limi e . The modi ied e sion o he Con inuous Su ace Cap Model includes, abo e and beyond he basic e sion (25 ma e ial pa ame e s), a o al o 3 ma e ial pa ame e s whose nume ical alues mus be de ined. The alues o he o he pa ame e s a e gene a ed au oma ically based on he alues o hese h ee pa ame e s. On he basis o he in e se iden i ica ion o he alues o hese h ee pa ame e s, he e iciency and accu acy o selec ed op imisa ion algo i hms we e es ed in his pape . Desc ip ions and used uni s a e lis ed o he pa ame e s o he modi ied e sion o he Con inuous Su ace Cap Model in Tab. 1 [15]. Tab.1: Ma e ial pa ame e s o he modi ied e sion o he Con inuous Su ace Cap Model. Pa ame e Desc ip ion o he pa ame e Uni RO Mass densi y. Mg/mm3 FPC Uncon ined uniaxial comp essi e s eng h. MPa DAGG Maximum agg ega e size. mm 2.2. The Plas ic Kinema ic Model The Plas ic Kinema ic Model is a bilinea model based on elas o-plas ic cons i u i e heo y. I is sui able o modelling he beha iou o cons uc ion s eel o s eel SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 19 | NUMBER: 2 | 2019 | DECEMBER © 2019 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 11 eba , o o model he beha iou o plas ic ma e ial ha beha es in a simila manne o s eel. The model enables he modelling o a ma e ial wi h ha dening, which can be iso opic o kinema ic, o wi hou ha dening. As he model is pa o an explici ini e elemen sol e , i includes he iscous beha iou al componen . I hus enables, wi hin i s o mula ion, he conside a ion o he e ec o he s ain a e on he s ess s a e. Howe e , his abili y o he model can be neglec ed by using ze o alues o he ele an pa ame e s, as is he case wi h he Con inuous Su ace Cap Model. In calcula ions ca ied ou o his pape , he iscous componen o he model was neglec ed because he s a ic esponse o he s uc u e was modelled, and he o mula ion o he model wi hou ha dening was used. The desc ip ions and uni s used o he ma e ial pa ame e s o he Plas ic Kinema ic Model, whose alues needed o be de ined, a e lis ed in Tab. 2 [15]. Tab.2: Ma e ial pa ame e s o he Plas ic Kinema ic Model. Pa ame e Desc ip ion o he pa ame e Uni RO Mass densi y. Mg/mm3 E Young’s modulus o elas ici y. MPa PR Poisson’s a io. - SIGY Yield s eng h. MPa ETAN Tangen modulus (ETAN = 0: ma e ial wi hou ha dening). MPa 2.3. The Linea Elas ic Model As a mo e de ailed ma e ial model was no needed o he modelling o he beha iou o he washe s, he Linea Elas ic Model, o in o he wo ds a cons i u i e model espec ing gene alised Hooke’s Law, was used o his pu pose. The desc ip ions and uni s used o he ma e ial pa ame e s o he Linea Elas ic Model, whose alues needed o be de ined, a e lis ed in Tab. 3 [15]. Tab.3: Ma e ial pa ame e s o he Linea Elas ic Model. Pa ame e Desc ip ion o he pa ame e Uni RO Mass densi y. Mg/mm3 E Young’s modulus o elas ici y. MPa PR Poisson’s a io. - 3. Expe imen al Da a and he Compu a ional Model 3.1. Expe imen al Da a The expe imen al da a used in his pape a e he esul s o a ou -poin bending es ca ied ou on a high, s eel- ein o ced conc e e beam. This es was ca ied ou and desc ibed wi hin [19]. A schema ic ep esen a ion o he es can be seen in Fig. 1. I shows he geome y o he conc e e beam oge he wi h he geome y and loca ion o he ein o cing ba and he washe s a he poin s whe e he loading and suppo o he beam ook place. The ma e ial pa ame e s o he ha dened (28 days old) conc e e and s eel eba we e he ollowing, as s a ed in [19]: Pa ame e s o he conc e e: • Modulus o elas ici y: 20.68 GPa, • Poisson’s a io: 0.15, • Uniaxial comp essi e s eng h: 24.13 MPa, • Uniaxial ensile s eng h: 3.10 MPa. Pa ame e s o he s eel eba : • Young’s modulus o elas ici y: 210.00 GPa, • Yield s eng h: 344.75 MPa, • C oss-sec ional a ea o he ein o cing ba : 0.71x10-4 m2. Fig. 1: Schema ic ep esen a ion o he ou -poin bending es . Fig. 2: Measu ed expe imen al da a. As shown in Fig. 1, he beam was loaded wi h o ces P du ing he es . The in ensi y o he o ces inc eased linea ly o e ime un il 80 kN was achie ed. The eloci y o loading was e y slow, hus he loading was s a ic. Du ing he es , he e ical displacemen (de lec ion) o SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 19 | NUMBER: 2 | 2019 | DECEMBER © 2019 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 12 beam U was measu ed a midspan. The exac loca ion whe e he de lec ion o he beam was measu ed is ma ked wi h poin B in Fig. 1. The measu ed expe imen al da a a e shown in Fig. 2. 3.2. Compu a ional Model The compu a ional model o he ou -poin bending es was c ea ed in LS-Dyna p og amme, du ing which he es scheme om Fig. 1 was espec ed. The compu a ional model equi ed he c ea ion o ini e elemen models o he high beam, ein o cing ba and washe s, and he bounda y condi ions (suppo s and loading) had o be de ined. The ini e elemen model o he beam was c ea ed using 3D eigh -node explici s uc u al ini e elemen s. The ini e elemen mesh o ming he model o he beam was egula (see Fig. 3). As he in e se iden i ica ion p ocess equi ed he epea ed execu ion o nume ical simula ions o he ask, he symme y o he ask was exploi ed (only hal o he beam was modelled) and he consis ence o he ini e elemen mesh o he beam model was chosen in such a way ha he ime equi ed o he calcula ion was no oo high, which i migh o he wise ha e been due o he use o an explici ini e elemen algo i hm. The ein o cing ba was modelled using 3D wo-node explici beam ini e elemen s. The consis ence o he beam ini e elemen mesh was adap ed o he ini e elemen model o he beam in such a way ha con inui y be ween he beam and he ein o cing ba model was ensu ed. The dimensions o he ec angula c oss sec ion o he beam ini e elemen s we e en e ed in such a way ha he esul an c oss-sec ional a ea co esponds o he alue om he expe imen (0.71x10-4 m2) which is a pa o his pape . The washe s we e modelled using 3D eigh -node explici s uc u al ini e elemen s. The size o he ini e elemen s o ming he models o he washe s co esponded o he size o he ini e elemen s o ming he model o he beam (see Fig. 3). Fig. 3: Fini e elemen model o he in es iga ed ask. The bounda y condi ions we e, wi hin he compu a ional model o he in es iga ed ask, de ined in he places whe e washe s we e loca ed and on he axis o symme y o he high beam. The model o he bo om washe had i s bounda y condi ions de ined in such a way ha i s displacemen was p e en ed in he ho izon al and e ical di ec ion. The model o he op washe had i s bounda y condi ions de ined in such a way ha i s displacemen was p e en ed only in he ho izon al di ec ion because o he applica ion o loading. A he loca ion o he axis o symme y, he symme ic bounda y condi ions we e de ined o he model o he beam. Loading was applied o he model o he op washe in he o m o p essu e linea ly inc easing o e ime wi h a inal o ce o 80 kN. Loading was also applied by conside ing he s uc u e’s sel -weigh . 4. The In e se Iden i ica ion P ocess 4.1. Global Op imisa ion and he Op imisa ion Algo i hms Used Fo his pape , “global op imisa ion” was used in o de o execu e he in e se iden i ica ion o he alues o he pa ame e s o he conc e e ma e ial model. The aim o global op imisa ion was o ind alues o hose pa ame e s whose iden i ica ion we e equi ed and whose applica ion would esul in nume ically simula ed da a displaying he smalles possible de ia ion om he expe imen al da a used. This would ep esen he “ e e ence esponse” wi hin he in e se iden i ica ion p ocess. In o he wo ds, he global minimum o he selec ed objec i e unc ion was sough , along wi h op imal alues o he iden i ied pa ame e s [13]. Global op imisa ion was ca ied ou using op imisa ion algo i hms in he op iSLang p og amme, du ing which hei accu acy and e iciency we e e alua ed. Du ing he op imisa ion, he selec ed objec i e unc ion which cha ac e ised he calcula ion o he Roo -Mean- Squa e De ia ion (RMSD) was hus minimised. I s ma hema ical exp ession was de ined by he equa ion: () 2 , 1 min, n calc i e ,i i UU RMSD n = − =→  (8) whe e U calc,i was subs i u ed by nume ically simula ed e ical displacemen (de lec ion) alues co esponding o he pe inen loading o ce alues and U e ,i was subs i u ed by e e ence (expe imen ally measu ed) e ical displacemen alues co esponding o he same loading o ce alues. Pa ame e n equalled he numbe o da a de ining he esul an shape o he expe imen al loading cu e in Fig. 2 (n = 32). As sugges ed abo e, he iden i ied pa ame e s we e only pa ame e s o he Con inuous Su ace Cap Model. The pa ame e s o o he ma e ial models used we e no iden i ied because o hei negligible in luence on he esul an shape o he nume ically simula ed loading cu e, which was e i ied by es calcula ions. The iden i ied SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 19 | NUMBER: 2 | 2019 | DECEMBER © 2019 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 13 pa ame e s o med, wi hin global op imisa ion, a design ec o which can be exp essed as: {} T ,, .RO FPC DAGG= d X (9) Fig. 4: Nume ically simula ed da a o ma e ial pa ame e s om he expe imen . The iden i ied pa ame e s en e ed he design ec o as con inuous andom a iables wi h a dis ibu ion o p obabili y a in e als gi en by de ined bounda y alues. The limi alues o he iden i ied pa ame e s gi en by he speci ica ion o he ma e ial model o conc e e we e used as bounda y alues [15] (see Tab. 4). Ma e ial pa ame e alues om he expe imen we e used as ini ial alues o he iden i ied pa ame e s, which we e necessa y o he i s gene a ion o i e a ion o he ele an op imisa ion algo i hm. The nume ically simula ed loading cu e o hese pa ame e alues is shown in Fig. 4. Ma e ial pa ame e s which we e no subjec o iden i ica ion we e en e ed de e minis ically using alues om he expe imen (see Tab. 5). Tab.4: Ini ial and bounda y alues o iden i ied pa ame e s. Pa ame e Uni Ini ial alue Minimum bounda y alue Maximum bounda y alue RO Mg/mm3 2.400x10-9 2.100x10-9 2.450x10-9 FPC MPa 24.13000 20.00000 58.00000 DAGG mm 16.00000 8.00000 32.00000 Tab.5: Pa ame e alues o he ma e ial models which we e no subjec o iden i ica ion. Pa ame e Uni Plas ic Kinema ic Model Linea Elas ic Model RO Mg/mm3 7.850x10-9 7.850x10-9 E MPa 210000 210000 PR - 0.3 0.3 SIGY MPa 344.750 - ETAN MPa 0 - The in e se iden i ica ion p ocess using op imisa ion algo i hms equi ed he epea ed execu ion o nume ical simula ions o he in es iga ed ask, du ing which he objec i e unc ion was minimised. The necessa y numbe o epe i ions (o in o he wo ds he necessa y numbe o gene a ions o i e a ions) pe o med o ind he global minimum o he objec i e unc ion o which he op imal alues o he iden i ied ma e ial pa ame e s would co espond di e ed g ea ly o he di e en op imisa ion algo i hms (see Tab. 6). Fo he pu poses o his pape a o al o i e op iSLang op imisa ion algo i hms (b ie ly desc ibed below) we e used in o de o s udy hei e iciency and accu acy. Non-Linea P og amming by Quad a ic Lag angian (NLPQL) NLPQL ([14] and [20]) is a sequen ial algo i hm based on nonlinea quad a ic p og amming. This op imisa ion algo i hm is sui able o he solu ion o asks which u ilise smoo h, con inuous as well as di e en iable objec i e unc ions and cons ain s. The algo i hm u ilises quad a ic app oxima ion o he Lag angian unc ion and he linea iza ion o cons ain s. Simplex Me hod (Simplex) The Simplex Me hod ([14] and [21]) is an i e a i e op imisa ion algo i hm based on linea p og amming which is execu ed sys ema ically wi h he pu pose o de e mining an op imal solu ion om a se o easible solu ions. The me hod is sui able o he op imisa ion o a low numbe o pa ame e s. Adap i e Response Su ace Me hod (ARSM) ARSM ([14] and [22]) is a di ec op imisa ion algo i hm sui able o he op imisa ion o bo h low and high numbe s o pa ame e s. The main ad an age o he algo i hm is he ac ha i p o ides an o e iew o he beha iou o an objec i e unc ion wi hin he whole design space, i allows he simple inclusion o addi ional equi emen s in o he objec i e unc ion, and i also equi es a ela i ely small numbe o design poin s. E olu iona y Algo i hm (EA) The E olu iona y Algo i hm ([14] and [23]) is one o a g oup o op imisa ion algo i hms ha u ilise p ocesses inspi ed by biological e olu ion (i.e. occu ing in he na u al wo ld), such as mu a ion, ep oduc ion and ecombina ion. The E olu iona y Algo i hm included in op iSLang p og amme is speci ically based on he combina ion o a gene ic algo i hm wi h an e olu ion s a egy. Pa icle Swa m Op imisa ion (PSO) PSO ([14] and [24]) is one o a g oup o op imisa ion algo i hms inspi ed by na u al phenomena. To be speci ic, PSO is a me hod whose algo i hm is inspi ed by and ies o imi a e he beha iou o locks o bi ds looking o ood. 4.2. Resul s and Thei E alua ion A able compa ison o he esul s ob ained om global op imisa ion om i e di e en op imisa ion algo i hms is SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 19 | NUMBER: 2 | 2019 | DECEMBER © 2019 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 14 shown in Tab. 6. To be speci ic, Table 6 shows he esul an op imal alues o iden i ied pa ame e s o he modi ied e sion o he Con inuous Su ace Cap Model ob ained om he bes gene a ions o i e a ions o he op imisa ion algo i hms, including he ele an minimum alues o he objec i e unc ion which de e mine he accu acy o he indi idual op imisa ion algo i hms used. Fu he mo e, Table 6 also shows he numbe o i e a ions o gene a ions o he op imisa ion algo i hms necessa y o ind he global minimum o he objec i e unc ion and hus also he op imal alues o iden i ied ma e ial pa ame e s. These da a de e mine he e iciency o he op imisa ion algo i hms used. Figu e 5 shows a g aphic compa ison o he esul s, oge he wi h he expe imen al da a. Tab.6: Table compa ison o he esul s. Pa ame e Uni Op imal alue NLPQL (numbe o necessa y i e a ions o he algo i hm = 21) RO Mg/mm3 2.400x10-9 FPC MPa 20.00000 DAGG mm 8.00000 RMSD mm 0.0074627 Simplex (numbe o necessa y i e a ions o he algo i hm = 75) RO Mg/mm3 2.409x10-9 FPC MPa 20.00000 DAGG mm 8.00829 RMSD mm 0.0074597 ARSM (numbe o necessa y i e a ions o he algo i hm = 180) RO Mg/mm3 2.450x10-9 FPC MPa 20.00000 DAGG mm 8.00000 RMSD mm 0.0074581 EA (numbe o necessa y gene a ions o he algo i hm = 400) RO Mg/mm3 2.450x10-9 FPC MPa 20.00000 DAGG mm 28.10027 RMSD mm 0.0073995 PSO (numbe o necessa y gene a ions o he algo i hm = 400) RO Mg/mm3 2.438x10-9 FPC MPa 20.00007 DAGG mm 8.00000 RMSD mm 0.0074575 The esul s shown in Tab. 6 and in Fig. 5 show ha he accu acy o all o he i e op imisa ion algo i hms used was e y sa is ac o y as a e y good app oxima ion o he expe imen al da a was achie ed by he nume ical simula ions, du ing which op imal alues we e gained o he iden i ied pa ame e s o he ma e ial model o conc e e used in he indi idual op imisa ion algo i hms. Mo eo e , i can be seen ha he di e ences in accu acy be ween he indi idual algo i hms a e p ac ically negligible ( he cu es p ac ically o e lap). Howe e , when looking a he RMSD alues o he indi idual algo i hms in Tab. 6, i can be concluded ha he mos accu a e op imisa ion algo i hm o he gi en ask is unambiguously he E olu iona y Algo i hm (EA). Fig. 5: G aphic compa ison o he esul s. When looking a Tab. 6 i can addi ionally be s a ed ha he mos e icien algo i hm o he gi en ask was clea ly he Non-Linea P og amming by Quad a ic Lag angian (NLPQL) as his algo i hm needed only 21 i e a ions o ind he global minimum o he objec i e unc ion. I can also be s a ed ha , in con as , he leas e icien op imisa ion algo i hms we e he E olu iona y Algo i hm and Pa icle Swa m Op imisa ion (PSO), as hese algo i hms needed up o a o al o 400 gene a ions o ind he global minimum o he objec i e unc ion. I can be concluded om he abo e-men ioned ac s ha wi h ega d o he accu acy and he inding o he global minimum o he objec i e unc ion du ing global op imisa ion, he E olu iona y Algo i hm is he mos ad an ageous, e en hough his algo i hm is no he mos e icien wi h ega d o calcula ion ime consump ion. Mo eo e , he op imal alue o DAGG pa ame e o he E olu iona y Algo i hm di e ed signi ican ly om he DAGG alues o o he algo i hms (see Tab. 6). This di e ence was due o he exis ence o wo e y close minimum peaks o he objec i e unc ion, o which jus one can be classi ied as he global minimum. Gi en he smalles RMSD alue o he E olu iona y Algo i hm, i can be concluded ha he global minimum o he objec i e unc ion was jus ound using his algo i hm. In he case o o he algo i hms, a local minimum which is, howe e , e y close o he global minimum was ound. Fo his eason, he esul s o all algo i hms can be conside ed as compa able. Ano he conclusion is ha om he poin o iew o e iciency, he use o he NLPQL algo i hm and possibly also he Simplex Me hod can be e y ad an ageous o global op imisa ion as hese op imisa ion algo i hms also exhibi e y good accu acy, which is compa able wi h ha o he E olu iona y Algo i hm. SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 19 | NUMBER: 2 | 2019 | DECEMBER © 2019 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 15 5. Conclusion This pape ocused on he pe o ming o a s udy o he e iciency and accu acy o i e op imisa ion algo i hms included in op iSLang so wa e du ing he in e se iden i ica ion o alues o a small numbe o pa ame e s o he modi ied e sion o he Con inuous Su ace Cap Model implemen ed in he LS-Dyna compu a ional sys em. A ou -poin bending es ask was used o his pu pose. I was execu ed on a high, s eel- ein o ced conc e e beam o which a compu a ional model was c ea ed, and expe imen al da a we e ob ained. The esul s o he s udy showed ha he accu acy o all i e o he op imisa ion algo i hms used was e y sa is ac o y as a e y good app oxima ion o expe imen al da a was achie ed by he nume ical simula ions in which he op imal alues o iden i ied pa ame e s ob ained ia he indi idual op imisa ion algo i hms we e used. Mo eo e , he di e ences in accu acy be ween he indi idual algo i hms we e p ac ically negligible. Howe e , du ing global op imisa ion he mos accu a e op imisa ion algo i hm was he E olu iona y Algo i hm, hough i was no he mos e icien as a as calcula ion ime consump ion is conce ned. Also, he esul s o he s udy showed ha he use o he NLPQL algo i hm, and possibly he Simplex Me hod, can be e y ad an ageous o global op imisa ion wi h ega d o e iciency. The accu acy o hese algo i hms was also compa able o ha o he E olu iona y Algo i hm. 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