Methods For Generating Perturbation Vectors For Topography Optimization of Structures

Transcription

1 Methods For Geeratg Perturbato Vectors For opograph Optmzato of Structures Jua Pablo Leva Vaderplaats Research ad Developmet, Ic, 4700 Gardebrook 5, NOVI, MI 48375, USA, Abstract Methods to automatcall geerate perturbato vectors for topograph optmzato of structures are preseted he perturbato vectors are created so that grds of desgable regos, tpcall modeled wth shell or composte fte elemets, ca move ether ormal to ther orgal locatos or a specfed drecto Maufacturg requremets such a mmum szes of bead patters, maxmum heghts ad trastoal dstaces betwee desgable grds ad o-desgable grds are cosdered Kewords: opograph Optmzato, Shape Optmzato, Perturbato Vectors 3 Itroducto opograph optmzato s a optmzato techque that allows mprovg the curvature of structures that are tpcall, but ot ecessarl, assembled wth shell or composte elemets opograph optmzato ca be treated as a specal tpe of shape optmzato (grd locato optmzato) A popular wa to mplemet shape optmzato s b usg perturbato vectors A perturbato vector s a vector that pots where the grds assocated to t would move f ts correspodg desg varable s 0 ad all rest of the shape desg varables are 00 I ths tpe of shape optmzato mplemetato the optmzer searches for the best soluto b searchg for the best lear combato of perturbato vectors scaled b ther correspodg desg varables I topograph optmzato, grds of the desgable rego are allowed to move ether ormal to the shell or composte elemets or a specfed drecto A mportat applcato of topograph optmzato s bead patter optmzato to crease the stffess of shell structures opograph optmzato mplemeted as a specal case of shape optmzato requres the creato of specalzed perturbato vectors ad ther assocated desg varables hese specalzed perturbato vectors are amed here topograph perturbato vectors here are several was to create topograph perturbato vectors, for stace CJ Che from Vsteo Corporato developed a method [] whch ever other elemet the desgable rego s allowed to move perpedcular to ts orgal posto Bra Voth, form Altar Egeerg, has also developed a method to geerate topograph perturbato vectors Hs method ad perhaps ehacemets to t are used to geerate shape optmzato data for the Optstruct software; ufortuatel Voth has ot publshed the methods he developed, but results of them ca be see Optstruct brochures he methods preseted here were developed for the structural optmzato program GENESIS [] ad the were mplemeted so that the ca be used wth other exstg shape or szg optmzato capablt 4 Procedure o Geerate opograph Perturbato Vectors he proposed procedure to geerate topograph petrubato vectors requres three basc steps hese steps are explaed ext 4 Surface preparato hs step cossts reorderg the odes of all the elemets the topograph rego so that ther assocated orms are cosstet wth the eghbor orms hs step s ol for teral calculatos; the elemets themselves are ot chaged 4 Normal drecto calculato hs step cossts of calculatg the orms assocated to each grd o the topographcall desgable rego he grd orms are calculated as a weghted average of the orm of all elemets that are coected to the grd hs step s optoal because occasoall the perturbato vectors ca be costructed usg a predefed drecto stead of the ormal drecto 43 Perturbato calculato hs step cossts of calculatg the perturbato vectors hese vectors are calculated usg the orms calculated o step ad parameters that detf a desrable basc shape he created perturbato vectors ca optoall referece smultaeousl multple grds for mprovg maufacturablt ad effcec hs paper wll focus ths step 5 Grd Locato Update Equatos he basc equatos used shape optmzato to terall calculate the updated grd locatos [3] are: X X 0 + XP () Y Y 0 + YP Z Z 0 + ZP where X, Y ad Z are the updated coordates of the grd X0, Y0 ad Z0 are the tal coordates of the grd XP, YP ad ZP are the compoets of the th perturbato vector correspodg to grd Fall, cotas the value of the desg varable

2 6 Specalzato of Grd Locato Update Equatos I ths work, the grd locato update equatos are specalzed for topograph optmzato he specalzato start b allowg oe desg varable per desgable grd ad make all perturbatos that affect each grd to pot o the ormal drecto at the grd, as show ext: Z 0 Z Y 0 Y x X 0 X () where x, ad z are the compoets of the ormal vector of the surface at grd s the magtude of the perturbato at grd assocated to desg varable Defg {G}, {G0}, [N], [] ad {} as follows: ad usg Eq () elds to: {G} {G0} + [N][]{} (3) For smplct, the [N] ad [] matrces wll be referred as the ormal matrx ad the topograph matrx, respectvel he multplcato of the ormal matrx ad the topograph matrx produces the [P ] matrx I ths paper, ths matrx wll be amed the basc topograph perturbato matrx he colums of [P ] are the basc topograph perturbato vectors [P ] [N][] (4) Combg Eqs (3) ad (4) elds the followg equato: {G} {G0} + [P ]{} (5) Because the ormal matrx s kow, from Step, the calculato of the basc topograph perturbato matrx s reduced to calculate the topograph matrx wo methods are preseted ext to geerate the topograph matrx he frst method focuses o buldg the colums of the matrx ad wll be amed here the basc method he secod method wll focus o buldg the rows of the topograph matrx ad wll be amed here the lk method 7 Basc Method (Perturbato Based Method) o create a perturbato vector, sa {P}, t s ol eeded to create a perturbato patter hat perturbato patter ca be stored colum of the topograph matrx he same perturbato patter ca be repeated to other grds to create all perturbatos Fgure a Perturbato Patter Fgure b opograph Perturbato Vector { } { } [ ] { } z x z x z x z x o o o o z x z x z x z x N Go G,,,,

3 he smplest patter that ca be defed s δ, where δ s Kroecker delta hs patter wll make the topograph matrx to be detcal to be the dett matrx hs matrx works well o some problems, but others t could produce dstorted meshes o avod ths, t s possble to create perturbato patters that spa multple grds A smple patter that acheves that s the followg: {( D / D ) H f D D 00 f D > D (6) I the above equato D s the dstace betwee grds ad ; D s a predefed fluece dstace ad H s a scale factor that represets the magtude of the perturbato for Aother useful patter s the followg: H ( (D D) /(D D) ) H 00 f D D f D < D D f D > D (7) I the above equato D s the dstace betwee grds ad ; D s a predefed fluece dstace were the perturbato s keep costat, D s a fluece dstace ad H s a scale factor that represets the magtude of the perturbato for 8 Lk Method (Grd Based Method) B defg a depedet desg varable D as: D Dv + + (8) ad b defg {D} as the vector that cotas all depedet desg varables he followg equato ca be wrtte: {D} []{} (9) I ths case equatos (3) ca be re-wrtte as: {G} {G0} + [N]{D} (0) B workg wth the depedat desg varables D, the terms ca be seeg as weghtg factors of the depedet desg varables hs weghtg factors should be bult to act as flters that help avodg mesh dstortos Oe possble set s preseted ext: {( D / D ) H f D D 00 f D > D () where D s the dstace betwee grds ad, D s a predefed fluece dstace ad H s scale factor that affect all terms of the row assocated to grd Aother useful set of weghtg factors s the followg: H ( (D D) /(D D) ) H 00 f f D < f D D D > D D D () I the above equato D s the dstace betwee grds ad ; D s a predefed fluece dstace where all varables that are close eough are gve the same weghtg factors ad D s a predefed fluece dstace to reduce to zero the fluece of the desg varable o grds that are far awa H s a scale factor that affects all desg varables assocated to grd It should be metoed here that although Eqs (6) ad (7) look smlar to Eqs () ad () the are ot he scale factor the frst two equatos affects the colums of the topograph matrx whle the scale factors Eqs () ad () affect the rows hat dfferece turs out to be mportat for fdg a procedure for properl scalg the perturbato vectors

4 9 opograph Matrx Codesato B mapulatg the rows ad/or colums of the basc topograph perturbato matrx ad/or the topograph matrx several useful results ca be obtaed For example, to eforce that desg varables ad alwas get the same value oe ca add colums ad of the topograph matrx ad locate the resultg colum ad elmatg colum (groupg) B makg two rows of the basc topograph perturbato matrx be the same, oe ca eforce that the two grd move the same magtude Groupg ca be used to reduce the umber of desg varable ad/or to produce mmum sze cotrol Groupg alog wth row mapulatos ca be used to eforce dfferet tpes of smmetres Smmetres however, ca be better eforced as a secod separate procedure, a procedure that repeats the rows of the master grd the slave grds (smmetrc grds) 0 Maufacturg Cosderatos 0 Maxmum Grd Movemets For maufacturg reasos ver ofte grds caot be allowed to move as much as the optmzer would attempt to move them So s mportat to fd was to lmt how much the grds ca move For the Lk method, a smple procedure s used; t volves scalg the rows of [] usg the followg expresso: H H max (3) where Hmax s the maxmum dstace that the grds are allowed to move ether the ormal drectos or a alteratve user predefed drecto For the basc method the procedure s ot as eas ad ot alwas s possble to fd a set of scalars that produce the desred effect For the cases where that s possble, the procedure volves solvg a sstem of equatos that could be costl Results usg Eq (3) for two dfferet maxmum heghts are show ext: Fgure a Small Maxmum Heght Fgure b Large Maxmum Heght 0 Mmum Bead Patter Dmesos opograph optmzato ca be utlzed to desg bead patter o metal sheets Ofte, to maufacture ths tpe of structures t s requred that bead patters mata a certa mmum sze, so t s mportat to fd was to cotrol the mmum sze of the optmzed results Mmum sze ca be acheved b usg dfferet D values equatos (7) or () ad b groupg varables that are close to each other (for example D<D) he followg fgures show some examples usg dfferet D values Eq (): Fgure 3a Large Mmum bead sze Fgure 3b Small Mmum bead sze Fgure 3a Smaller Mmum bead sze he results show Fgs 3a, 3b ad 3c were obtaed usg, 55 ad 69 desg varables respectvel 03 rasto dstaces o get smooth results betwee the desgable grds ad the o-desgable oes s mportat for maufacturg ad for mesh qualt Smoothg ca be optoall acheved b elmatg the perturbato vectors that drectl desg these grds ad keepg the row that desgs these grds so the ca move he trasto es ca be further smoothed b usg the terms D (7) or () wth D>D beg (D-D) the trasto e As a result of ths the trasto grds wll be able to move but ot full Fal Step o Buldg the opograph Matrx he elmatos of row ad/or the repetto of colums dscussed above ca be smbolcall wrtte usg two matrces: [R] ad [C] If we call [R] a 3m matrx wth m< ad [C] a q matrx wth q<, the fal topograph perturbato matrx ca be wrtte as: [P ] [R][N][][C] (4) Eq (4) represets the codesed basc topograph perturbato matrx hs matrx cotas the fal perturbato vectors hs matrx s the topograph perturbato matrx hs matrx affect m sets of grds usg q desg varables hs matrx practce s ot costructed usg the expresso above because that would volve ma uecessar operatos, lke multplg b zero Istead, the fal perturbatos are costructed ad kept sde the program a sparse matrx format

5 Examples wo optmzato examples are preseted to llustrate the use of topograph optmzato wth the proposed approach Both problems use a 8x40 mm plate he materal propertes of the plate are E 07,000 N/mm ad ν 03 he plate s modeled usg a 466 degrees of freedom fte elemet mesh that cotas 779 grds ad 70 quadrlateral elemets he two examples show extreme desg optos he frst example shows two cases where there are as ma desg varables as desgable grds he secod example shows a case were all grd each of sx topograph regos are desged b oe depedet desg varable Example he frst example ams at maxmzg the torsoal rgdt of the plate he thckess of the plate s 0 mm he corer grds of the tp are loaded wth vertcal loads of opposte drectos of 00 N each that produce a overall torso load of 800 Nmm For maufacturg reasos, oe of the edges of the plates are allowed to chage ad the maxmum grd chage the vertcal drecto s 00 mm wo cases are studed I case A, the grds are ol allowed to move the postve drecto of the orm I case B, the grds are allowed to move both drectos he optmzato problem s to mmze the stra eerg wth a volume costrat of 735mm 3 (% above the tal volume of 70 mm 3 ) For both cases, oe topograph rego that cotas all grds s used Fgure 4 Ital Desg Results for Example I both cases, 663 grd perturbato vectors ad 663 depedet desg varables were automatcall geerated I case A, the tal stra eerg was reduced from 33E- Nmm to 9963E-3 Nmm (5% mprovemet) O case B, the stra eerg was reduced to 849E-3 Nmm (36% mprovemet) I both cases, the volume costrat was actve (735mm 3 ) Fgs 5a ad 5b show the optmzed cofguratos for the two cases I both cases, beads patters followg +/-45 degrees drectos were obtaed hese patters seem reasoable for creasg torsoal stffess Fgure 5a opograph Optmzato Results for Case A Fgure 5b opograph Optmzato Results for Case B 3 Example he secod example ams at maxmzg the bedg rgdt of the plate he thckess of the plate s 06 mm he tp edge s loaded wth evel dstrbuted vertcal load of 0 N/mm (360 N) he locatos of the grds o the larger edges of the plates are ot allowed to chage, whereas the locatos of the grd of the short edges are he Optmzato problem s to mmze the stra eerg wth a volume costrat of 5mm 3 (8% above tal volume of 43 mm 3 ) I ths case, sx topograph regos were used Fgure 6 Ital Desg 4 Results for Example Sx topograph regos (strp of elemets alog the loger drecto) produced 40 grd perturbato vectors wth correspodg 6 depedet desg varables I the fal desg, 4 desg varables took a postve value ad the rest took values close to zero Fg 7 shows the fal optmzed cofgurato he tal stra eerg was reduced from 495 N-mm to 994 N-mm (395% mprovemet) I the fal desg, the volume costrat was actve (volume creased from 43 mm 3 to 5 mm 3, a 8% crease) Fgure 7 opograph Optmzato Results for Example

6 3 Practcal Cosderatos Oce the topograph perturbato vectors are bult ad used to solve a optmzato problem two problems could arse he frst oe s that the desg varable ma ot move because tal sestvtes could be zero he frst problem occurs aturall flat plates where movg the varables to a postve or egatve drecto s the same o solve that problem s eas: stead of settg the tal value of all desg varables to zero the are set to small radom value hs has the mor weakess that the tal desg s ot the same as the orgal desg, whch s usuall preferred stadard shape optmzato problems he secod problem s mesh dstorto Although usg perturbato patters such the oe Eq (8) or flters such as the oes Eq () help reducg mesh dstorto the are ot capable of completel elmate the problem ad whe ths problem occurs t s ot eas to fx t wthout a smoothg algorthm he ol eas fx s to put a lmt o how much the grds move, that could be acheved usg a smaller maxmum heght costrat (Hmax) or reducg the bouds of the desg varables 4 Numercal Cosderatos Of the two methods preseted, the basc ad the lk, the lk method tured out to be better because t ca deal a geeral wa wth the commo requremet of maxmum heght costrats For problems where maxmum heght s ot a requremet the basc methods work as well It s terestg to meto here that topograph optmzato tpcall takes about 7 to 5 desg ccles to coverge hs effcec however comes mostl from the approxmate problem alread bult the GENESIS program 5 Coclusos A geeral procedure to automatcall geerate topograph perturbato vectors for shape optmzato of structures has bee preseted Maufacturg requremets such a mmum szes of bead patters, maxmum heghts ad trastoal dstaces betwee desgable grds ad o-desgable grds were cosdered Usg automatcall created perturbato vectors smplfes the shape optmzato process ad better ad more ovatve desgs ca be foud 6 Ackowledgemets he author wshes to thak hs colleagues Bra C Watso ad Iku Kosaka for ther cotrbutos to ths work 7 Refereces Che CJ, Mare S ad Usma M Improved Fuel ak Desg Usg Optmzato AMD-Vol 7 Desg Optmzato wth Applcatos Idustr ASME, 997, pp GENESIS Structural Optmzato Software User s Maual, Verso 7 Vaderplaats Research ad Developmet, Ic Colorado Sprgs, CO, USA, December 00 3 Leva, JP ad Watso, BC Shape Optmzato the Geess Program Optmzato Idustr II, Baff, Caada, Ju 6-0, 999