Modified Integrated Multi-Point Approximation And GA Used In Truss Topology Optimization

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1 Joural of Muldscplary Egeerg Scece ad echology (JMES) Vol. 4 Issue 6, Jue Modfed Iegraed Mul-Po Appromao Ad GA sed I russ opology Opmzao Adurahma M. Hasse 1, Mohammed A. Ha 2 Mechacal ad Idusral Egeerg versy of rpol, o rpol, ya a.hasse@uo.edu.ly 1, m.ha@uo.edu.ly 2 Asrac Geec algorhm (GA) s a kd of opmzao mehod whch does o requre he dfferealy ad couy of he fuco or varales, ad hus ca e used for prolems f oly he adapao values of speces ca e oaed. I s approprae o use GA solvg complcaed prolem as srucural opology opmzao. However, GA requres he calculao all he adapao values of he gve speces, whch meas a eremely large umer of srucural aalyses are eeded o coduc f GA s drecly appled o opology opmzao, especally for large-scale prolems. A modfed egraed mul-po appromae coceps ad GA for russ opology opmzao cludg couously cross-secoal sze varales ad dscree opology varales s proposed. he prmal russ opmzao prolem s a frs rasformed o a seres of appromaely medvarale prolems usg he proposed appromae formula; afer ha he opology varales ad cross-secoal areas of he russ are deermed y GA ad dual mehod respecvely hrough a layered sraegy. he fal opmzao soluo ca e reached afer few eraos. he umercal eamples show he russ opmzao wh opology varales ca e oaed afer less srucural aalyss compared wh ohers, whch preses ha he proposed modfed egraed mulpo appromao has hgher qualy ha he former egraed mulpo appromao. Keywords opology Srucural Opmzao; Mul-po appromao; GA. I. INRODCION opology opmzao s recogzed as oe of he mos challegg prolem srucural syhess. I s usually a mahemacal programmg prolem wh dscree varales, ad may es sgular opma. Alhough appromao coceps are very mpora meag ad wll remarkaly effec o he effcecy of he logarhms of srucural opmzao, s hardly o esalsh a hgh qualy appromae prolem for opology srucural opmzao ecause of s dscree properes. Geerally, o dscree srucures such as russes he opology opmzao s cocered wh fdg a opmal dsruo of maeral wh a specfed doma. Oe of he approaches of he Ha Huag Mechacs of Maerals BAA Beg, Cha hhuag@uaa.edu.c opology opmzao prolems s so-called groud srucure mehod [1,2]. he groud srucure approach was frs proposed y Dor e al. [3]. I he groud srucure prolem wh all possle par-wse ercoeco wh fed odal locaos, he opmal opology soluo s o e seleced. he mos dffcules such approach are ha here may es may local opmal soluos. he orgal mul-po appromao (MA) [4] s oe of he mos effecve appromao fucos for srucural szg opmzao, u s o loger effecve whe desg varale k approaches o zero, whch s always happeed russ opology opmzao. Wh wolevel appromao coceps ad GA [5], ad he egraed mul po appromao (IMA) [6] formula was pu forward o solve russ opology opmzao prolems, he orgal prolem s rasformed o a seres of frs-level eplc appromae prolems usg IMA, ad he a layered opmzao sraegy s roduced. he opology varales of he russes are opmzed hrough GA he eeral layer, ad he cross-secoal areas of ars are opmzed he eral layer hrough a seres of secod level appromae prolems ha ca e solved y he duel mehod. I hs work, he egraed mul-po appromae fuco s furher suded, hus he modfed egraed mulpo appromao (MIMA) s proposed, whch MA ad IMA are comed as oe fuco wh wo raches for codos whe a ar ess or removed respecvely. he ew appromao fuco o oly ca avod he sgulary of he orgal MA for opology opmzao, u also keeps he sasfed accuracy feaures for szg prolems. he umercal eamples show he russ opmzao wh opology varales ca e oaed afer oale few srucural aalyss, whch preses ha he proposed MIMA has hgher qualy ha he former IMA. II. MODIFIED INEGRAED MIPOIN APPROXIMAION FNCION he Iegraed mulpo appromao model, [6] whch s accuracy has ee esed hrough a seres of eplc ad mplc fucos, [7]. he model ca e represeed here as follows: - JMESN

2 Joural of Muldscplary Egeerg Scece ad echology (JMES) Vol. 4 Issue 6, Jue Fd M S X 1, 2,, 1, 2,, W f 1 g 0 (1 ) (1 ) 0 or 1 1,, m 1,, where X s he vecor of cross-secoal sze varales, s he vecor of opology varales, s he group umer of lked ar, m s he umer of respose cosras, ad (1) are he upper ad lower ouds of he sze varales, ad s a very small value usually akes (0.01 )used o susue he cross-secoal sze of a removed ar, f (X ) s wegh of ars group, ad g (X ) represes a respose cosra fuco. A. he layered sraegy o solve he appromae prolems o solve he prolem (1), a seres of appromae prolem called as he frs-level appromae prolems are creaed hrough roducg a Modfed Iegraed Mul-Po Appromae (MIMA) fuco. he MIMA fuco s esalshed y mprovg he former Iegraed Mulpo Appromae (IMA) fuco. A p- h sage he appromae prolem ca e saed as: - Fd M S X 1, 2,, 1, 2,, W f 1 ( g 0 (1 ) (1 ) 0 or m 1,, J1 1,, 1,, ma, (2) Where J 1 s he umer of acve respose cosra of he orgal prolem (1); ad are ( he upper ad lower ouds of he sze varales prolem (2) afer cosderg he move lms ad ( p ) a he p-h sage; ad g p ( p ) represes he appromae cosra fuco cosruced wh he proposed Modfed Iegraed Mul-po Appromae fuco, whch s ased o he formao of prmal fuco g( X) a mulple kow pos, ad effecve eve f some ars are removed (e.g. k k ha, meas he ars k-h group are removed). Bu he g p eplc appromae fuco was formed ased o MA srucural opmzao prolem whch s hgher qualy as s used for cross-secoal sze opmzao he ecomes sgular ad o loger effecve f he desg varale k approaches o zero or s susued y very small value k, whch ca e happeed russ opology opmzao for hs reaso IMA was proposed [6], whch s a weghed represeao of he aylor epasos wh respec o ermedae varales. IMA s sll effecve eve f some ars are removed so ca e used for opology opmzao. I order o cosruced a appromae fuco ha has sasfed accuracy oh sze ad opology opmzao a Modfed Iegraed Mulpo Appromao (MIMA) s proposed hs paper for g p (2), whch egraes MA ad IMA as oe fuco wh wo raches for codos whe he correspodg ar ess or s removed respecvely, whch ca e wre as: - H g( g( X ) g com,, h 1 1 Where, (3) 1 g( X ) 1r r r. O O O ro g com,, 1 g( X ) r e M ( ) 1 rm Where, X (=1,,H; =1,,) are he kow pos; H s he umer of pos o e coued; s he umer of desg varales a po; g(x ) s he fuco values ad g s he appromaed values. Ad, h (X) s he weghg fuco, whch ca e deermed as: - h h l h H l1 H S1 Sl h l ( X X S 1,..., H, ) ( X X ) S l 1,..., H he epoe r O ad r M are he adapve parameers o corol he o-leary of he appromao, hey o e foud from he followg equaos respecvely: - 1 g( X ) 1rO ro ro g H g X. ( X ) H X r O 1 1 g ( X H ) g( X ) r M g( X ) r.(1 M ( H ) e ) 1 However, he proposed MIMA s epeced o e of he advaages of oh MA ad IMA. he layered sraegy o solve he frs-level appromae prolems (2) passes hrough a seres of a layered opmzao, where he opology varales of he russes are opmzed usg GA echque he eeral layer, JMESN

3 he he prolem rasferred o he secod-level appromao where he cross-secoal areas of he ars are opmzed he eral layer ad solved y he duel mehod. he requred srucural aalyss for russ opology opmzao ca e dramacally decreased as GA s oly used o solve he appromae prolems he eeral layer where o srucural aalyss s eeded. O he oher had, a relavely small umer of speces s ake GA as he desg varales of cross-secoal areas are deermed eral layer. III. NMERICA EXAMPES Dffere eamples from leraure are chose o demosrae he valdy ad compare he effcecy of he modfed egraed mulpo appromao fuco wh he mos well-kow eamples 10-ar ad he 72-ar russes, ad oher eample s a groud russ srucure: A. Eample 1. he e-bar russ For he srucure respose, he 10-ar russ, Fg.1, s loaded y 1*10 5 ls. a ode 2 ad 4 Y dreco, where he dsplaceme cosras 2, ad he sress lms 25*10 3 ps, he Youg s modulus s E=1*10 7 l./ 2 ad he specfc wegh 0.1 l./ 3. Ial cross-secoal area of each ar s 30 2 ad he area lower lm s equal o he Opmzed e-ar, russ s show Fg. 2. Comparso s made for oh szg ad opology opmzao. he erao hsory daa s show ale (1). ale (2) shows he fal opmum desg varale resuls ad umer of aalyss. Resuls of he proposed MIMA mehod reaches he opmum soluos afer a relavely less umer of eraos compared wh ohers. he proposed mehod shows very effce resuls. Fg. 1. he e Bar russ Joural of Muldscplary Egeerg Scece ad echology (JMES) Vol. 4 Issue 6, Jue Fg. 2. he Opmzed e-bar russ ABE I. Aalyss ABE II. RSS. IERAION HISORY DAA FOR 10-BAR RSS Weghs (ls) opology Szg opology Szg opology Varales FINA OPIMM DESIGN VARIABE RESS, FOR10-BAR Weghs (ls) opology Szg opology Szg B. Eample 2. he 72-Bar russ opolog Aalyss Weghs [ls.] For he srucure respose, he 72-ar russ, Fg. 3. Where he dsplaceme cosras 0.25, for oher deals see ref. [9]. he Opmzed 72-ar, russ s show Fg. 4. he erao hsory resuls are show ale. (3). he fal opmum desg varale resuls are ale (4) Clearly, he prese sudy has remarkale umer of aalyss for he opology y JMESN

4 comparg wh he former oe ad he fal opmal desg varale resuls have cosderale precse agreeme. Joural of Muldscplary Egeerg Scece ad echology (JMES) Vol. 4 Issue 6, Jue ABE IV. RSS FINA OPIMM DESIGN VARIABE RESS FOR 72-BAR Fg. 3. he 72-Bar russ Fg. 4. he opmzed 72-Bar russ ABE III. IERAION HISORY DAA FOR 72-BAR RSS Weghs (ls) Aalyss opology Szg opology Szg opology Varales Weghs (ls) opology Szg opology Szg opology Aalyss Weghs [ls.] C. Eample 3. he e-node, 2D russ groud srucure he proposed mehod s appled o he e-ode russ Fg. 5, wh groud srucure wh all possle ercoeco a oal of 34 memers; hs eample s ake from referece [10]. he parameers used for hs eample are as follows: he srucure s loaded y 10*10 3 ls a ode 3, 5, & 7 Y dreco, where he dsplaceme cosras 2, ad he sress lms 25*10 3 ps, he Youg s modulus s E=1*10 7 l/ 2 ad he specfc wegh 0.1l/ 3. Fg. 5. e-node russ groud srucure JMESN

5 Opmzed e-ode, russ groud srucure s show Fg. 6, as well as he opmzed soluo from ref. [6] & [10] for comparso. he opmzed soluo ale 5 shows prese sudy has same opology as ref. [10], ad overlappg memers are show he opmum soluo. he cross-secoal areas are dffere for hose overlappg memers, u are almos decal for ohers; he oecve fuco s comparale. Fg. 6. he Opmzed e-node russ groud srucure ABE V. CROSS-SECIONA AREA OF HE OPIMIZED EN-NODE RSS GROND SRCRE Cross-secoal area ( 2 ) Memer No. MIMA IMA, Ref. [6] Ref. [10] No. of Aalyss Weghs [l] IV. CONCSION Oe ca coclude from he resuls ased o he coduced compuaoal russes eamples, ha he effcecy of he proposed MIMA mehod for russ opology ad sze opmzao furher mproved. I also preses ha he MIMA has a hgher qualy of appromao compared o IMA, as GA s us used o solve he appromae prolems. he proposed mehod ca reach he opmum soluo of a prolem wh oh opology ad sze desg varales afer few srucural aalyses, whch s eve comparale o solve he opmzao prolem wh oly he cross-secoal sze as varale. Moreover, he groud russes srucure resuls also assure ha he proposed model resuls are sasfyg ad comparale for opology opmzao. Joural of Muldscplary Egeerg Scece ad echology (JMES) Vol. 4 Issue 6, Jue V. REFERENCES [1]. Krsch Opmal opologes of russ srucures Appl. Mech. Rev. 42, (1989). [2] B. H. V. oppg Mahemacal programmg echques for shape ad layou opmzao of skeleal srucures shape ad layou opmzao of srucural sysems ad opmaly crera mehods; Edo y G. I. N. Rozavay. Sprger, Vea (1992). [3] Dor, W. C. Gomory R. E. ad Greeerg H. J Auomac desg of opmal srucures J. Mech. 3, [4] H. Huag, & R. W. Xa wo-level mulpo cosra appromao cocep for srucural opmzao ; Srucural Opmzao 9, (1995). [5] H. Huag, A. Hasse, H. a ad Y. Dog russ opology opmzao y usg wo level mulpo appromao ad GA CJK-OSM-3 Jo Symposum o Opmzao of Srucural ad Mechacal Sysems, Ocoer 30 Novemer 2, 2004, Kaazawa, Japa, [6] Adurahma Hasse, Mohammed A. Ha, Ha Huag "Srucural opology Opmzao usg Iegraed Mul-Po Appromao"; Ieraoal Joural of Egeerg Research & echology (IJER); Vol. 5 Issue 11, Novemer- 2016, [ISSN: ], hp:// IJERV5IS [7] Adurahma Hasse ad H. Huag Sudy o accuracy of he comed mul-po appromao mehods used srucural opology opmzao CJK-OSM-3 Jo Symposum o Opmzao of Srucural ad Mechacal Sysems, Ocoer 30 Novemer 2, 2004, Kaazawa, Japa. [8] Dog Yogfag ad Huag Ha russ opology opmzao y usg mul-po appromao ad GA Chese Joural of Compuaoal Mechacs; Vol.21-No.6; [9]. Schm ad H. Mura Appromao coceps for effce srucural syhess ; NASA CR-2552; [10] KaGal Desg of russ-srucure for Mmum Wegh usg Geec Algorhms Deparme of Mechacal Egeerg, Ida Isue of echology, Kapur, PIN , Ida, KaGal Repor No.99001, de@k.ac.. JMESN