0 th World Congress on Structural and Multdscplnary Optmzaton May 9-24, 203, Orlando, Florda, USA On the optmal topologes consderng uncertan load postons János Lógó, Erka Pntér 2, Anna Vásárhely 3 Department of Structural Mechancs, Budapest Unversty of Technology and Economcs H- Budapest, Műegyetem rkp. 3, Hungary logo@ep-mech.me.bme.hu, 2 epnter@mal.bme.hu, 3 anna.vasarhely@freemal.hu. Abstract A new numercal method s presented for the contnuum type topology optmzaton problems n the case of uncertan loadng postons. The optmzaton problem s a volume mnmzaton one subjected to probablstc complance constrants. In addton to the optmzaton procedure a parametrc study s presented to nvestgate the layout theory. It s proved that not only statcally determnate but statcally undetermnate structure can be the optmal layout. 2. Keywords: Optmal layout, topology optmzaton, probablstc loadng, uncertan pont of applcaton, multply load case. 3. Introducton The more than century old topology optmzaton has a relatvely young new research drecton, namely consderng probablstc data n the topology desgn. Uncertanty s typcally lmted to the loadng, although recent works have consdered extensons to support condtons, and materal propertes, etc.. [6-8]. In ths paper the loadng postons are taken as stochastc varables and all the other data are determnstc. The lnearly elastc dscretzed structure s modelled mechancally by plane stress dsk elements. To make the optmzaton method robust an equvalent determnstc problem s derved [-3,5]. The elaborated technque can be descrbed as follows: t s assumed that the load postons are gven by ether ther dstrbuton functon, mean value and covarance matrx or by the smple values of the probablty of the occurrence of a force at a certan locaton. The frst case, where the statstcal nformaton (dstrbuton functons, mean values and varatons) are gven, s always fnshed by a smple calculaton whch results n the probablty values of the occurrence of a force at a certan locaton, that s practcally n the second case. Hence each load s consdered n an extended loadng doman. Snce the loadng postons are not known precsely, an equvalent loadng system should also be created around the expected locaton of each force to perform a "smulaton". Accordng to the orgnal dstrbuton assumpton, mean value and varaton of the pont applcatons, an extended force system s set up for each possble loadng doman wth the orgnal magntude of the force and gven (or calculated) probablty values. Each load s ndependent and acts as an ndependent load case n the orgnal loadng doman. Applyng these forces at these "base" ponts as loads the stochastc desgn problem becomes a determnstc one. By the use of the elements of ths force system one by one, the dsplacement vectors can be calculated from the usual lnear equlbrum equatons wth several load cases. Snce the materal s lnearly elastc, sotropc and homogenous, the addtve propertes of the dsplacements and the recprocty theorem can be appled. Usng these vectors and the assgned probablty values the expected dsplacement and ts varaton can be calculated. By the use of these data the orgnal complance value, whch s probablstc due to the poston uncertantes, can be substtuted wth a determnstc one applyng the Kataoka theorem [4]. Usng ths complance formulaton ether a mn-max objectve functon s formed, whch s composed by the expected complance and a certan type of varance of the complance due to the ndependent load cases or a volume mnmzaton problem s created what s subjected to several complance constrants. Here the later one s used as base problem. At frst case the constrants are the volume lmtaton and the sde values of the desgn varables. In the case of Gaussan dstrbuton of the dsplacement feld the unconstrant problem objectve functon s smplfed to a functon mnmzaton also due to the Kataoka theorem [4]. An extended SIMP type algorthm s elaborated for the soluton method. To valdate the model a determnstc mnmum complance truss desgn s performed analytcally and numercally.
Several numercal examples are presented. 4. Notes on the layout theory The mnmum weght desgn as an objectve was a rather popular topc durng golden ages of the optmzaton (e.g. durng the 50-s to 70-s of the last century). The authors nvestgated whether the statcally determnate or undetermnate structure gves the optmal layout [8-0] wth mnmum weght. It s known n engneerng desgn that a statcally determnate structure s not senstve for knematc loadng, but any change n statc loadng may produce an unexpected collapse. In ths way the statcally undetermnate structures can be more safe for unexpected load cases (see: the structure of the bones). Ths queston s rather dffcult whenever the loadng uncertanty s nvestgated. In ths case the load can be consdered as a quantty gven n an nterval. In ths paper at frst t s proved trough two smple examples that one can construct several (nfnte number) alternatve statcally undetermnate structures havng the same volume and complance value f a statcally determnate structure exsts. 4.. Smple examples for equvalent determnate versus undetermnate structures The frst example s a 3-bar truss -as a base structure- wth a vertcal force at the top (Fg.). The materal s lnearly elastc (for sake of smplcty the Young s modulus E=) and the vertcal load s 00. The members are supported by hnges at the bottom. The total complance s calculated as follow: C F L 2 = n = EA where F = the elastcally calculated force n member, L = length of member, A = cross-sectonal area of member. One can see n Table. that mechancally same (same volume and same complance) structures can be composed by smple modfcaton of the number of the members (doubled the bars and mechancally equvalently decrease the cross-sectonal areas of the members). Here a 6 and 2-bar structures were calculated as examples. () Fgure : Alternatve truss laouts n the case of a vertcal pont load at the top Table : Comparatve values of the optmal 3, 6 and 2-bar structures n the case of a vertcal pont load at the top Complance (external Pot. energ.) Complance of the bars (stran energy) length pc secton (cm2) volume Normal force stress top dsplacement 2,82842 3,57 332,89 47,4-3,0025 0,0572 5,72 5,79083764 2,82842 6 0,785 332,89 23,57-3,0025 0,0572 5,72 5,79083764 2,82842 2 0,3925 332,89,785-3,0025 0,0572 5,72 5,79083764 σ=200 Young's 20000N/mm2 2
A very smlar example can be calculated f the top vertcal force (00) s modfed and a horzontal force (57,74) s added ( see Fg 2.) All the other data and the way of the calculatons are the same. The results of the calculatons can follow from the detals of Table 2. Fgure 2: Alternatve truss problems n the case of two pont loads at the top Table 2: Comparatve values of the optmal 3, 6 and 2-bar structures n the case of two pont loads at the top Complance (external Pot. energ.) Complance of the bars (stran energy) length pc secton (cm2) volume Normal force stress top dsplacement 2,828427,57 444,063 0,58 00 0,0572 5,72 8,852022 2,828427 2,57 888,26 9,92 57,74 0,0572 3,82957 0,680824 total 332,89 0,033 9,532957 9,532845 6 bar truss 2,828427 0,785 222,035 3,65 00 0,0572 5,72 0,022858 2,828427 2 0,785 444,063 9,96 57,74 0,0572 3,82957 0,34042 2,828427 2 0,785 444,063 37,8 0,033 4,743564 2,828427 0,785 222,035 50,79 4,4260 total 332,89 9,532957 9,532845 2 bar truss 2,828427 0,3925,058,82 00 0,0572 5,72 0,0367 2,828427 2 0,3925 222,035 0 57,74 0,0572 3,82957 0 2,828427 2 0,3925 222,035 4,98 0,033 0,70206 2,828427 2 0,3925 222,035,79 0,95399 2,828427 2 0,3925 222,035 8,59 2,37782 2,828427 2 0,3925 222,035 23,57 3,82723 2,828427 0,3925,058 25,39 2,2234 total 332,89 9,532957 9,53220 3
As a concluson of the calculaton above one can state f a statcally determnate structure exsts as a soluton of a determnstc problem wth a sngle load case, several (nfnte number) statcally equvalent undetermnate structures can be derved wth the same weght and the complance. 4.2. Mnmum volume desgn of structures accordng to the optmal layout theory In the case of probablstc loadng the magntude, the lne of acton, the drecton and the pont of applcaton of the load can be uncertan. Here trough a smple example t s proved that not only one type of layout can be optmal. There wll be sngular layout solutons for certan case or the optmal layout can be changed f the magntude of the horzontal load s uncertan. Fgure 3: Three-bar truss problem n the case of two pont loads at the top Table 3.a: Comparatve values of the optmal 3, 6 and 2-bar structures n the case of two pont loads at the top 3 bar truss V H A A2 A3 Vol 3-bar truss 8,660254 0 0 2,598 0 4,499867998 8,660254 0,5 0,258 2,585 0,258 5,50935338 8,660254 0,58 2,434 0,58 6,5398666 8,660254,5 0,967 2,52 0,967 7,595373338 8,660254 2,407,753,407 8,664285066 8,660254 2,5,90,237,9 9,742546849 8,660254 3 2,45 0,59 2,45 0,82764203 8,660254 3,5 2,964 0,038 2,964,928793 8,660254 4 3,278 0 3,278 3,2 8,660254 4,5 3,69 0 3,69 4,476 8,660254 5 3,999 0 3,999 5,996 8,660254 5,5 4,49 0 4,49 7,676 8,660254 6 4,879 0 4,879 9,56 8,660254 6,5 5,379 0 5,379 2,56 8,660254 7 5,99 0 5,99 23,676 8,660254 7,5,98 0,98 47,924 4
Table 3.b: Comparatve values of the optmal 3-6 and 2-bar structures n the case of two pont loads at the top 2 bar truss V H A Volu of 2-bar truss A2 A3 8,660254 0 2 8 0 2 8,660254 0,5 2,09 8,076 0 2,09 8,660254 2,079 8,36 0 2,079 8,660254,5 2,79 8,76 0 2,79 8,660254 2 2,39 9,276 0 2,39 8,660254 2,5 2,50 9,996 0 2,499 8,660254 3 2,72 0,88 0 2,72 8,660254 3,5 2,98,92 0 2,98 8,660254 4 3,28 3,2 0 3,28 8,660254 4,5 3,69 4,476 0 3,69 8,660254 5 3,999 5,996 0 3,999 8,660254 5,5 4,49 7,676 0 4,49 8,660254 6 4,879 9,56 0 4,879 8,660254 6,5 5,379 2,56 0 5,379 8,660254 7 5,99 23,676 0 5,99 8,660254 7,5,98 47,924 0,98 Accordng to the papers of Rozvany and Maute [2] or Slva et al [] the optmal layout s a two leg structure wth a well-defned nclnaton angle f the horzontal force s uncertan. Here a very specal case s studed where the ntal layout s based on the optmal layout comng from the above cted papers. In addton to the two legs structure an addtonal vertcal leg s consdered formng a statcally undetermnate structural layout. The problem s a mnmum volume desgn of a three legs structure wth constraned complance the formulaton (eq..) s the same and smaller than a gven bound)-. The member forces are calculated from the equlbrum equatons takng nto account the compatblty equatons as well. The top load s determnstc wth gven value, whle the top horzontal force s probablstc. It s modeled on the way that ths force can be any value n a gven nterval (Fg. 3) as t s ndcated n the above cted papers [, 2]. The optmalty condton to determnate the layout s that the horzontal force can not exceed the expected lower and upper bounds (±7,5). Fgure 4: Optmal cross-sectonal areas and mnmum volumes of three and two bar truss problems 5
The constraned mathematcal programmng problem s form wth the dea that the unknowns are the cross-sectonal areas of the members and the two sde legs are n 30 degree nclnaton angle. There are two load cases (the horzontal force n each case can change ts drecton). The problem numercally s solved by a sequental quadratc programmng algorthm of MATCAD 5. One can follow the numercal values of the optmal cross-sectonal areas n the case of three-bar truss (Table 3.a) and the numercal values of the optmal cross-sectonal areas n the case of two-bar structure (Table 3.b), respectvely. Graphcally these results are presented n Fgure 4. One can see that n case of H=0 a vertcal bar s the optmal layout whle -3.5<H<0<H<3.5 the optmal layout s a three-bar structure. Otherwse the optmal layout s a two legs structure. A very smlar suspcon was presented n the almost forgotten paper of Nagtegaal and Prager [3]. Here the authors nvestgated the queston of the optmal layout n the case of two alternatve loads wth same pont of applcaton. A necessary and suffcent condton for global optmalty was derved for the plane truss where the loadngs were created on the way that the load factors for plastc collapse under one or the other load were not to exceed a gven value. The results were one, two or three bar trusses dependng on the loadng domans. The optmal layout problem of a mnmum weght truss desgn problem wth a sngle vertcal force load presented by Save [0] n the case of stress constrants. The conclusons of hs results and optmal layouts comng from the results obtaned from our examples are n good agreement. 5. Probablstc Complance Desgn n the Case of Uncertan Loadng Postons The determnstc complance desgn procedure of a lnearly elastc 2D structure (dsk) n plane stress s known from lterature. Ths topology optmzaton problem s gven for sngle load as follows: subject to G g = p g g W = γ ga t = mn! (2.a) T uf C 0; tg + tmn 0; ( forg =,..., G), tg tmax 0; ( forg =,..., G). (2.b-d) Here the ground element thcknesses t g are the desgn varables wth lower bound t mn and upper bound t max, respectvely. By the use of the FE (fnte elements) dscretzaton, each ground element (g=,, G) contans several sub-elements (e=,, E s ), whose stffness coeffcents are lnear homogeneous functons of the ground T element thckness t g. Furthermore γ g s the specfc weght and Ag the area of the ground element g. u s the nodal dsplacement vector assocated wth the loadng F. The dsplacements u can be calculated from Ku = F, where K s the system stffness matrx. p s the penalty parameter ( p ) and the gven complance value s denoted by C. The above constraned mathematcal programmng problem can be solved by the use of an approprate SIMP algorthm (e.g. Lógó[]). 5.. Multply load cases and uncertan loadng magntudes The above problem n case of several load cases should be extended by addtonal complance constrants () representng the ndependent loadngs ( F ). By the use of a generalzed complance desgn concept (Lógó []) the new constrants () T () ( C ) P u F 0 q (3) can be ntroduced nstead of eq.(2.b). Here 0 q s a gven expected probablty value what gves nformaton about the possblty of a falure. Followng the upperbound theorem of Kataoka [4] eq.(3) can be substtuted by the followng determnstc expresson whch s convex and determned for each ndependent load case: n j = ( ) f u C+ Φ q b K b 0. (4) () () - () T () () j j ov Here u = E( u ), j =,..., n s the expected value of the dsplacement under the ndependent force () () j j 6
( =,..., m) ( ) () () T () () () () F n the drecton of ths load, b = f, f2,..., fk,..., f n, K ov s the covarance matrx of these dsplacements. The number of the ndependent load cases depends on the Then the penalzed mnmum weght problem subjected to probablstc complance constrant due to the uncertan loadng magntude has the form: G g = p g g W = γ ga t = mn! (5.a) subject to n () () - () T () () f u C+ Φ ( q) b Kovb 0; = n ( m) ( m) - ( m) T ( m) ( m) f u C+ Φ ( q) b Kov b 0; = tg + tmn 0; ( forg =,..., G), tg tmax 0; ( forg =,..., G). Ths type of constraned mathematcal programmng problem can be solved by usng an approprate optmalty crtera algorthm (see e.g. Lógó[5]). 5.2. Uncertan loadng postons Here a smplfed mechancal model s created on bass of the orgnal loadng doman. Let us consder the desgn problem gven n Fgure 5. Snce the loadng postons are not known precsely an equvalent loadng system should be also created around the expected locaton x of each force f to perform the smulaton. Accordng to the orgnal dstrbuton assumpton, the mean value and the standard devaton of the pont applcaton are determned by the force system fj ( j =,.., k) wth the orgnal magntude f - for sake of smplcty and to descrbe the loadng domans- seven ponts as base ponts are used wth symmetrcal adjustment ( f, f2, f3, f 4). (The mnmum number of the ponts s three.) Each load s ndependent and a well-defned probablty value wj ( j =,.., k = 7) s assgned to them (n practce t can take as desgn nformaton). The determnaton of ths probablty value wj ( j =,.., k) s based on the orgnal dstrbuton and t can be calculated wth a smple computaton. In ths way the loadng s gven by these doubled parameters - w ( j =,.., k = 7), ( f, f2, f3, f4) - and appled as ndependent load cases. The modfed topology desgn problem s gven n Fgure 5 f the orgnal load and the supports are located on the same lne. j (5.b) Fgure 5: The desgn doman wth the modfed loadngs and the correspondng probabltes 7
If the applcaton ponts and the supports can not be connected wth a sngle lne the surrogate model of the loadng s based on a force and uncertan moment system at the expected locaton of the orgnal load.applyng these forces at these base ponts as loads the stochastc desgn problem becomes a determnstc one after ths transformaton. By the use of the element fj ( j =,.., k) of these force system one by one, the dsplacement vectors u j ( j =,.., k) can be calculated from the Ku j = f j lnear equatons. Snce the materal s lnearly elastc the addtve propertes of the dsplacements and the recprocty theorem can be appled. Usng these vectors and 2 the assgned probablty values wj ( j =,.., k) the expected dsplacement u and ts varaton D ( u ) can be calculated n the followng form: k = jwj j= u u ; (6.a) k 2( ) ( ) 2 2 = j j j= D u u w u. (6.b) These computed values are used to compose the element of the mathematcal programmng problem eq.(5). Due to the nature of ths type of loadng the covarance matrx s dagonal. ( ) ( ) ( ) K = D u, D u,..., Dn un (7) 2 2 2 ov 2 2 Interchangng the expected complance calculaton by the generalzed expected stran energy formulaton the penalzed mnmum weght problem subjected to probablstc complance constrant has the form: G g = p g g W = γ ga t = mn! (8.a) subject to n () T () - () T () () u Ku C+ Φ ( q) b 0; Kovb = n ( mt ) ( m) - ( m) T ( m) ( m) u Ku C+ Φ ( q) b 0; Kov b = tg + tmn 0; ( forg =,..., G), tg tmax 0; ( forg =,..., G). (8.b-d) 6. Numercal example To demonstrate the method ntroduced above the example problem of Rozvany and Maute [2] s used to create the base problem (Fgure 6.a). The pont of applcaton of the vertcal load s uncertan. The geometry s gven by Fgure 6.a: Base problem for the SIMP-type soluton Fgure 6.b: Surrogate loadng 8
L=40 whle the determnstc magntude of load s V=50. The values (-e to +e) demonstrate the devaton of the pont of applcaton of the force V. The surrogate loadng s represented by an addtonal horzontal force system H correspondng to the eccentrcty e. (Here for demonstratve reason H s 50.) Due to the nature of ths problem three ndependent load cases have to consdered. These load cases are: (V=50, H=-50), (V=50, H=0) and (V=50, H=50) Fgure 6.b-. The complance lmt s 60000. The expected probablty s q=0,9. The obtaned optmal topology can be seen n Fgure 7. One can see that a statcally undetermnate structure s the optmal layout. The whte lnes demonstrate the center lne of the truss members. The nclnaton angle s 36. Fgure 7: Optmal layout 7. Conclusons If the load s probablstc, surrogate determnstc load cases are suggested to model the uncertan pont of applcatons. Mnmum three ndependent load cases need to model the uncertanty connected to an uncertan pont of applcaton of the orgnal load. The surrogate loadng system s problem dependent. In case of probablstc loadng the optmal layout can be statcally undetermnate structure. To make more approprate models need some addtonal nvestgatons on the topc. 8. Acknowledgements The present study was supported by the Hungaran Natonal Scentfc and Research Foundaton (OTKA) (grant K 885). 9. References [] J. Lógó, New Type of Optmalty Crtera Method n Case of Probablstc Loadng Condtons, Mechancs Based Desgn of Structures and Machnes, 35, 2, 47-62, 2007. [2] J. Lógó, J., Ghaem, M. and Vásárhely, A. Stochastc complance constraned topology optmzaton based on optmalty crtera method, Perodca Polytechnca-Cvl Engneerng, 5, 2, 5-0, 2007. [3] J. Lógó, M. Ghaem and M. Movahed Rad, Optmal topologes n case of probablstc loadng: The nfluence of load correlaton, Mechancs Based Desgn of Structures and Machnes, 37, 3, 327-348, 2009. [4] S. Kataoka, A Stochastc Programmng Model, Econometra, 3, 8-96, 963. [5] J. Lógó, SIMP type topology optmzaton procedure consderng uncertan load poston, Perodca Polytechnca-Cvl Engneerng, 56, 2, 23-220, 202. [6] P.D. Dunnng, H.A. Km and G. Mullneux, Introducng Loadng Uncertanty n Topology Optmzaton, AIAA Journal, 49(4), 760-768, 20. [7] J.K. Guest and T. Igusa, Structural optmzaton under uncertan loads and nodal locatons, Comp. Meth. Appl. Mech. Eng., 98, 6-24, 2008. [8] M. Jalalpour, T. Igusa and J.K. Guest, Optmal desgn of trusses wth geometrc mperfectons: Accountng for global nstablty, Internatonal Journal of Solds and Structures, 48(2), 30-309, 20. [9] J. Barta, On the Mnmum Weght of Certan Redundant Structures, Acta Tech. Acad. Sc. Hung.; 8, 67-76, 957. [9] G. Sved, The Mnmum Weght of Certan Redundant Structures, Australan Journal of Appled Scences, 5, -9, 954. [0] M.A. Save, Remarks on Mnmum-Volume Desgns of a Three-bar Truss, Journal of Structural Mechancs,,,, 0-0, 983. 9
[] M. Slva, D, Tortorell, J.A. Norato, C. Ha and H.R. Bae, Component and system relablty-based topology optmzaton usng a sngle loop method, Internatonal Journal for Structural and Multdscplnary Optmzaton, 4,, 87-06, 200 [2] G.I.N. Rozvany and K. Maute. Analytcal and numercal solutons for a relablty-based benchmark example.structural and Multdscplnary Optmzaton, 43, 6, 745-753, 20. [3] J.C. Nagtegaal and W. Prager Optmal layout of a truss for alternatve loads, Internatonal Journal of Mechancal Scences,5, 7, 583-592, 973. [4] G.I.N. Rozvany, On symmetry and non-unqueness n exact topology optmzaton, Structural and Multdscplnary Optmzaton, 43, 3, 297-37, 20. 0