12 th Internatonal LS-DYNA Users Conference Optmzaton(1) LS-TaSC Verson 2.1 Wllem Roux Lvermore Software Technology Corporaton, Lvermore, CA, USA Abstract Ths paper gves an overvew of LS-TaSC verson 2.1, a topology optmzaton tool usng LS-DYNA for the analyss of nonlnear structural behavor. The focus s on ts capabltes, current development drectons, and ntegraton nto an ndustral desgn envronment. Examples of usng the new developments such as dynamc load scalng are gven. Overvew The goal of topology optmzaton s to fnd the shape of a structure wth the best use of the materal. An alternate vew of the process s that of selectng the best load path for the specfed use of the structure. It must of course be possble to manufacture the fnal structure, and the tool can mpose varous requrements ensurng ths. The overall LS-TaSC [1] process conssts of () the desgn problem defnton, () performng the desgn optmzaton teratvely usng LS-DYNA [2], and () post-processng the results. The topology desgn problem s defned by () the allowable geometrc doman, () how the part wll be used, and () propertes of the part such as manufacturng constrants. Addtonally, you have to specfy methodology requrements such as termnaton crtera and management of the LS-DYNA evaluatons. The ntal parts specfy the desgn doman the optmum parts computed wll be nsde the boundares delmted by the ntal parts. The parts must be modeled usng sold or shell elements. The part may contan holes; a structured mesh s accordngly not requred. Geometry constrants such as beng an extruson or a castng drecton may be specfed. The use of the part s descrbed by LS-DYNA nput deck. The desgn process ams for a unform nternal energy densty n the structure as computed by LS-DYNA usng ths nput deck. The fnal shape of the part s descrbed by the subset of the ntal elements used. Ths s outputted n the form of an LS-DYNA nput deck. Verson 2.1 was created from verson 2 by addng features such as forgng geometry defntons and dynamcs load case weghng as descrbed n later sectons together wth an enumeratons of the current capabltes. Methodology The typcal goal of topology optmzaton s to obtan maxmum utlty of the materal. Obtanng unform nternal energy densty n the structure together used as the obectve for optmzaton. 1
Optmzaton(1) 12 th Internatonal LS-DYNA Users Conference The obectve s typcally modfed by the use of the SIMP (sold sotropc materal wth penalzaton) algorthm [3] to ensure that the structure s a sold. Ths follows the formulaton proposed by Patel [4], wth the resultng mplementaton beng smlar to the fully-stressed desgn and unform stran energy densty approaches (Haftka and Gurdal [5], Patnak and Hopkns [6]). The use of an element s controlled by changng the amount of materal n the element. Ths s acheved by assgnng a desgn varable to the densty of each element. The materal s parameterzed usng a so-called densty approach. In ths approach, a desgn varable s drectly lnked to the ndvdual materal element such that each cell has ts own materal model. The desgn varable x, also known as relatve densty, vares from 0 to 1 where 0 ndcates vod and 1 represents the full materal. The materal propertes correspondng to the values of desgn varables are obtaned usng an approprate nterpolaton model as descrbed n the manual [1]. The optmzaton problem s formulated as, NL * mn UwxU (), x 1 N * subect to: () xvm 1 l u C C C, 2,1,..., J x x mn where U represents the nternal energy densty of the th element, V s the volume of th element, U * represents nternal energy densty set pont, and C s the th constrant. There are L load cases wth a total of J constrants. The superscrpts l and u represent lower and upper bounds on the constrants, respectvely. The change n the desgn varable of th varable (Δx ) s computed as, where K s a scalng factor and varable s updated as, x t x x. t1 t.0.1 / *. t x KUU * U t * U denotes the nternal energy densty set pont. The desgn Overvew of current capabltes LS-TaSC s developed for the topology optmzaton of non-lnear problems nvolvng dynamc loads and contact condtons. It s used to fnd a concept desgn for structures analyzed usng LS- DYNA. 2
12 th Internatonal LS-DYNA Users Conference Optmzaton(1) General capabltes Sold desgn usng frst-order hexahedrons, tetrahedral, and pentahedral elements Shell thckness desgn usng frst-order quadrlateral and trangular elements Global constrants Multple load cases, ncludng dynamc load case weghng Tght ntegraton wth LS-DYNA Large models wth mllons of elements Geometry defntons Multple parts Extrusons Symmetry Castng, one sded Castng, two sded Forgng Postprocessng Desgn hstores LS-PREPOST plots of the geometry evoluton and the fnal desgn New capabltes n verson 2.1 The followng capabltes were added to create verson 2.1: Dynamc load case weghng Dynamc load case weghng s used to ensure that a part performs equally well for all desgn scenaros. Forgng geometry defntons Ths geometry defnton s set to obtan a part that can be manufactured usng a forgng process. Mnor features: Castngs can have nteror holes. Pentahedral elements are supported. The memory footprnt s reduced more than a factor of 2 and an opton s provded whch can be set to reduce memory use by a further factor of 2. *MAT_ELASTIC s supported for the desgn part. Lghtly used elements can be kept nstead of deleted. The SIMP algorthm can be swtched on and off. Coordnate systems are no longer lmted to DIR=X. Restartng was mproved to be faster by usng more archved results. A frnge plot of the materal utlzaton as consdered n the desgn process can be vewed. The fracton of the orgnal number of elements used n the desgn can be vewed as a hstory. 3
Optmzaton(1) 12 th Internatonal LS-DYNA Users Conference Forgng Ths geometry defnton s to create a structure that can be manufacture usng a forgng process. Materal s removed only from the sdes of the structure. Ths s smlar to a two-sded castng defnton, except that a mnmum thckness of materal wll be preserved. The geometry defnton wll therefore not create holes through the structure. Ths capablty s avalable only for solds. Dynamc load case weghng It can happen that a sngle load case domnates the topology of the fnal desgn makng the structure perform badly for the other load cases. Ths can be resolved by assgnng dfferent weghts to the load cases, but t s dffcult to know good weghng values n advance. Dynamc weghng of the load cases s used to select the load case weghts based on the responses of the structure as the desgn evolves, thereby resultng n a desgn that performs well for all load cases. The dynamc weghng s done by defnng a desred relatonshp between the responses of all the load cases. The algorthm wll scale the load case weghts to acheve ths relatonshp. Say we have constrant C 1 from the frst load case and constrant C 2 from the second load case, then we wrte our desred behavor as k1 C1 offset 1 k2c2 offset wth C the constrant value, k a scale factor, and an offset added. The weght w of load case s adusted to change constrant C. The target value s computed as n kc offset Ct arg et n from whch we compute w ( Ct arg et kc offset )/ C / w wth the dervatve approxmated as ±1 and a maxmum bound s place on w to ensure convergence n a reasonable number of teratons. The fnal weghts found are not sutable for restartng. They can be examned though for an ndcaton of good values of the weghts, but usually the fnal weghts found usng dynamc weghng are too large. Examples Forgng example Ths example s a sold part to be manufactured as a forgng, whch was accordngly mposed as a forgng geometry defnton ncludng a mnmum web thckness. The geometry and loadng condtons for ths component are shown n Fgure 1. The FE model has about 60 000 elements and a sngle lnear mplct load case as shown was consdered. 4
12 th Internatonal LS-DYNA Users Conference Optmzaton(1) Fgure 1: The ntal geometry and loadng condtons. The fnal desgn s shown n Fgure 2. Note the web that s requred for forgng manufacturng. Flanges and a rb were created to carry the bendng load effcently. Fgure 2 Desgn wth forgng geometry defnton. Note the web that s requred for forgng. Dynamc load scalng Ths s a nonlnear structure compressed by an mpactor n two load cases as shown n Fgure 3. A symmetry geometry defnton requrng symmetry around the center was mposed to remove 5
Optmzaton(1) 12 th Internatonal LS-DYNA Users Conference the need for a rght load addtonal to the shown left load. The center load case domnates the geometry of the fnal desgn f the load cases are not scaled wthout load case scalng the desgn s much stffer for the center load than for the left load. It s dffcult to know up front whch values of the load case weghts to use n order to have a balanced desgn. In ths example t s shown how dynamc load scalng creates a balanced desgn. Fgure 3 The geometry and loadng condtons of the dynamc loadng example. A left load, a center load, and symmetry around the center are appled. In the followng pctures we show the results from a standard study wth both load case havng equal weghts, and a study wth dynamc load balancng. In Fgure 4 to the left we show the reacton forces for the standard desgn study n whch the load cases were weghed equally. The dynamc load balancng was then set to have the fnal results for these two reacton force to be the same for both load cases, whch allowed us to acheve the reacton force curves as shown to the rght of Fgure 4 for the balanced desgn. Fgure 4: The reacton forces for the two load cases. To the left are the results from the orgnal problem statement, from whch t can be seen that the reacton forces dffers greatly for the two load cases. To the rght are the results wth dynamc load scalng, from whch t can be be seen that the reacton forces are now smlar. The desgns obtaned have dfferent topologes as shown n Fgure 5. 6
12 th Internatonal LS-DYNA Users Conference Optmzaton(1) Fgure 5 Fnal desgns. The desgn usng equal weghng s shown on the left, whle the desgn usng dynamc weghng s shown on the rght. Note the bottom of the desgns dffer: the dynamc scalng has the pllars connected usng a truss structure n order to provde more support for the offset load. The convergence of the maxmum reacton force value for both load cases s shown n Fgure 6. The fnal weghts are 0.12 for the center load case and 7.2 for the rght load case. Fgure 6 Convergence of the reacton forces. The plots are that of the maxmum reacton force for each load case versus the teraton number. The equally weghted desgn study s shown to the left, whle the results for dynamc weghng are shown to the rght. 7
Optmzaton(1) 12 th Internatonal LS-DYNA Users Conference Current Developments Improvements currently under nvestgaton are: Shape optmzaton. Both sold and shells structures wll be consdered. For sold structures the outer surface wll be modfed to releve stress concentratons, whle shell structures wll receve geometrc features to be stffer and for bucklng specfc desgns. Integraton wth geometry. The user nterface wth be unfed wth the LS-PrePost capabltes, whch wll gve a smoother ablty for tasks such as defnng coordnate systems and vsualzng results. Summary LS-TaSC computes the shape of a structure wth the best use of the materal. It has been developed for non-lnear structures analyzed n an ndustral envronment and s accordngly sutable for large lnear problems. Ths tool consders solds and shells, global constrants, multple parts, and manufacturng constrants. Ths tool has been extended to load case weghng and forgng geometry defntons. References [1] Lvermore Software Technology Corporaton, LS-TaSC : A Topology and Shape Computatons for LS-DYNA, User s Manual, Verson 2.1, Lvermore Software Technology Corporaton, Lvermore, CA, 2011. [2] Hallqust JO. LS-DYNA theoretcal manual, Lvermore Software Technology Corporaton, Lvermore, CA, 1998. [3] MP Bendsøe, O Sgmund, Materal Interpolaton Schemes n Topology Optmzaton, Archves of Appled Mechancs, 69, 635-654, 1999. [4] NM Patel, Crashworthness Desgn Usng Topology Optmzaton, PhD thess, Unversty of Notre Dame, 2004. [5] RT Haftka, Z Gurdal, MP Kamat, Elements of Structural Optmzaton, Kluwer Academc Publshers, Dordrecht, The Netherlands, 2 nd ed., 1990. [6] SN Patnak, DA Hopkns, Optmalty of Fully-Stressed Desgn, Computer Methods n Appled Mechancs and Engneerng, 165, 215-221, 1998. 8