A Weight Balanced Multi-Objective Topology Optimization for Automotive Development

Transcription

1 A Weght Balanced Mult-Objectve Topology Optmzaton for Automotve Development Nkola Aulg 1, Emly Nutwell 2, Stefan Menzel 1, Duane Detwler 3 1 Honda Research Insttute Europe GmbH, Offenbach/Man, Germany 2 Oho State Unversty SIMCenter, Columbus, OH, USA 3 Honda R&D Amercas, Raymond, OH, USA Abstract Topology optmzaton n the feld of automotve development strves for conceptual car components whch are effcently desgned for multple, partly conflctng loadngs. Structures desgned to manage nonlnear loadng condtons whch are often seen n crash events typcally requre a maxmzaton of energy absorpton whle statc loadngs typcally requre a mnmzaton of complance of the structure. The hybrd cellular automata algorthm, whch ams for a unform dstrbuton of the nternal energy densty, offers an effcent soluton for the sngle objectve problems but may fal where large energy dfferences occur n dfferent load cases. Recently, t was demonstrated that a mult-objectve optmzaton can be performed by lnearly weghtng and careful scalng of the load case energy levels. In ths paper the approach s appled to the practcal example of a vehcle control arm structure whch s subject to two complance load cases and an energy maxmzaton load case and compared to a sequental optmzaton approach of the dscplnes. Results demonstrate the practcal feasblty of the proposed methods. Partcularly, the scaled weghtng approach yelds a set of non-domnated trade-off solutons, facltatng the selecton of a sutable balance between energy absorpton and stffness requrements. 1 Introducton In the vehcle desgn process, a dffcult challenge s to develop a lghtweght structure whch can meet both statc stffness requrements for vehcle performance targets such as NVH and rde handlng as well as generatng a structure that s effcent wth regard to passve crash safety requrements. These requrements are often n conflct, and consderng all requrements smultaneously has been a hstorcal challenge. Such a mult-dscplnary desgn process [1] that nvolves dfferent dscplnes s often tackled not only usng dfferent software tools, but also requres several members of the desgn team workng together. Ths can result n conflctng desgns that must be resolved late n the desgn process potentally addng cost and weght to the overall structure. The feld of topology optmzaton refers to algorthms that am at fndng the optmum layout of a lghtweght structure. In contrast to shape optmzaton, whch targets fne-tunng of a rather advanced structural desgn, topology optmzaton provdes the engneer or desgner wth a structural concept early n the desgn process. Concretely, topology optmzaton approaches address the problem of fndng the optmal dstrbuton of materals and vod wthn a pre-defned desgn space [2]. For the defned load cases, topology optmzaton provdes the geometrc layout, ncludng features such as holes of the structure defned by the desgn space. Topology optmzaton has been establshed as an effcent method to develop structures that are effcent for stffness requrements, but the feld of crashworthness (.e. energy absorpton) remans a challengng applcaton. The complexty due to the nonlnear nature of these crash type loads lends tself to heurstc approaches such as the hybrd cellular automata approach [3,4]. Recently, a methodology focusng on the concurrent optmzaton of load cases subject to stffness and energy absorpton objectves was proposed and tested on a smply supported beam example [5]. In ths paper, the method s appled and evaluated on a practcal model, specfcally the model of a vehcle control arm. The structure developed by ths concurrent optmzaton methodology s compared to a baselne approach of performng the statc and dynamc optmzaton sequentally. Secton 2 descrbes the topology optmzaton method appled to the control arm. In Sec. 3, the model of the control arm, the desgn space, and ts load cases are descrbed. In Sec. 4 the parameters of the optmzaton s descrbed and the results are presented and dscussed. The paper s concluded n a summary n Sec. 5.

2 2 Weghted Crashworthness Topology Optmzaton Method Ths secton presents the topology optmzaton approach to address both mnmum complance and maxmum energy absorpton n a concurrent mult-load case optmzaton. A bref ntroducton of topology optmzaton methods and targeted objectve functons s presented as well. Frequently appled for topology optmzaton are densty-based methods [6]. These methods operate on a fnte element mesh of the desgn space and assgn a varable to each element. The orgnally bnary varable that descrbes presence or absence of materal wthn an element s replaced by a contnuous densty varable. Materal propertes of the element can be obtaned based on the densty values by usng, for example, the popular Sold Isotropc Materal wth Penalzaton (SIMP) scheme [7]. In ths case, a materal property of the element, commonly the Young s modulus, s nterpolated by a power law approach, accordng to E p E0, (1) wth densty ρ wth =1, N, the number of elements N and the Young s modulus of the orgnal materal E 0. To avod ntermedate denstes these are penalzed by a penalzaton exponent p. A typcal objectve of lnear statc topology optmzaton s to maxmze the stffness of a structure,.e. to mnmze ts complance. The mnmum complance problem can be stated as: mn c u T f subject to: Ku f V V f 0 mn 1, 1,, N, wth the complance c, the dsplacement vector u and the load vector f. In the equlbrum condton, K s the stffness matrx. A constrant V f s mposed on the porton of the desgn space flled wth materal,.e. the volume fracton V of the structure. In order to avod numercal nstabltes a mnmum densty ρ mn s defned. In crashworthness topology optmzaton, several other objectves are of mportance such as avodng ntruson nto the cabn space, keepng acceleraton curves smooth, or achevng hgh energy absorpton. Ths work s focused on the latter,.e. t s amed for a maxmzaton of the energy absorbed by the structure for the appled load. Ths can be formulated as: max EA subject to: r 0 V V 0 mn f 1, 1,, N, wth the total energy absorbed by the structure EA and the resdual (2) (3) r of the fnte element analyss. For complance problems, gradent-based methods based on analytcal desgn varable senstvtes are commonly appled. However, due to the complexty of crashworthness problems, analytcal senstvty nformaton s dffcult to obtan. Addtonally, the nonlnearty of the problem mght render a gradent descent problematc. Ths s why for crashworthness topology optmzaton, usually heurstc methods are appled. A popular heurstc approach s referred to as Hybrd Cellular Automata (HCA). It was frst devsed as a method for the remodelng of bone growth, but t also provdes effcent solutons for lnear statc problems [8, 9]. Ths heurstc approach was subsequently extended to nonlnear loadng problems [3, 4]. The workng prncple of the HCA s the target of achevng a unform dstrbuton of a feld varable throughout the structure by mnmzng the devaton from a set-pont. A new desgn s obtaned teratvely by the update rule:

3 new * K P S S, (4) where K P s a scalng parameter, S s the feld varable and S * s the feld varable set-pont. Accordngly, when the change of the structure s approxmatng zero, the feld varable s more unformly dstrbuted. The HCA approach s able to consder several load cases n a mult-load case approach. In the multload case approach, a lnear combnaton of the elemental feld varables s appled: S L l1 w S l l, (5) wth weght w l and feld varable S l referrng to load case l and the number of load cases L. The HCA approach has been demonstrated to yeld effcent structures; however, the dscplnes of crashworthness and lnear statcs are typcally consdered n separate work. In [5] t s proposed to perform a mult-load case optmzaton n the mult-dscplnary case of concurrently dealng wth a statc and a nonlnear load case. Ths approach lnearly weghts the stran energy densty as a feld varable obtaned from the statc load case and the nternal energy densty as a feld varable occurrng n the nonlnear load case. Due to the large dfferences n energy levels for lnear and nonlnear load cases, a scalng factor needs to be ntroduced such that the preference of the load case can be separated from the energy level: S L l1 w S l l L l1 p l S s l l. (6) In ths notaton, the user s preference p l for the load case s separated from the weght used n the optmzaton by refactorng the energy scale factor s l. Accordng to [5], the scalng factor s chosen such that the energy levels of all load cases are scaled down to the level of the load case wth the lowest energy. For ths purpose the energy of the full desgn space,.e. ρ =1 for all elements s used: s l mn N full Sl 1 N full Sl' l ' 1 (7) The procedure of applyng the method can be descrbed as follows: 1. Defne user preferences p for all load cases. 2. Compute scalng factor s l from energy levels for all load cases, accordng to (7). 3. Compute weghts w l =p l /s l used for the lnear combnaton of the feld varables. 4. Run mult-load case HCA optmzaton. A mult-objectve study can be performed by applyng several dfferent preferences/weghtngs. Each of the load cases s treated as one separate optmzaton objectve, such that a set of trade-off solutons n objectve space s obtaned. 3 Model of Vehcle Control Arm In ths work we consder the optmzaton of a vehcle control arm structure. In ths secton we descrbe the LS-Dyna model and the assocated load cases. In general, the stffness of the control arm s mportant for NVH, durablty, and rde comfort and steerng response; however, durng a load retenton scenaro, t can be subject to substantal deformaton whch the structure must effcently manage. Thus, durng normal operaton, mnmum complance s requred, whle durng a loadng event whch results n yeldng of the part, energy absorpton should be maxmzed. We represent these requrements by two lnear load cases (mnmum complance), and one nonlnear load case (maxmze energy absorpton).

4 Fg.1: The LS-Dyna control arm model (left) and the desgn space for the optmzaton (rght). Fg.2: The LS-Dyna model of the statc load cases wth dsplacement contours shown. Fg. 1 shows the geometry of the control arm represented as an LS-DYNA fnte element model. The volume connectng the bushng mounts of the control arm s defned as the desgn space for the topology optmzaton. The volume of the entre control arm structure ncludng the desgn space s meshed usng tetrahedral elements wth a mesh sze of 3 mm nomnal. The materal of the control arm desgn space as well as the suspenson and chasss at the ends are modelled wth a pecewse lnear elastc-plastc alumnum materal model represented wth *MAT24. Ths materal s smplfed to a *MAT1 (equvalent Young s modulus and densty) when the statc load cases are solely consdered. The control arm has a mass of 4.025kg of whch the full densty desgn space s 3.506kg. There are two statc load cases defned. The loads are appled as shown n Fg.2 drectly to the control arm structure. The load cases are defned n z- and y-drecton respectvely, and are referred to n the remander of ths paper as load case FZ and load case FY. A statc load of 10N s dstrbuted to the ball jont mount pont and bushng mount structure for the FZ and FY load cases respectvely, and the constrants are defned as shown n Fg 2. These load cases represent normal operatng condtons for whch mnmum complance s the desred optmzaton objectve. Care s taken that the lnear assumpton holds and the deformaton stays wthn the lnear regon of the materal model. There s one nonlnear load case defned (Fg. 3) where the control arm s subjected to a prescrbed dsplacement n the x-drecton whch s ncreased from 0 to 110mm wthn 200ms. Ths model represents a laboratory test desgned to ensure that the suspenson components meet a mnmum load requrement. Ths laboratory test s nspred by extreme loadng condtons to the suspenson system of the vehcle. A test fxture s mounted to the wheel hub, and a chan attaches the test fxture to a loadng actuator whch then apples the load. The relevant suspenson components (damper, te rod, knuckles) are modelled usng a smplfed beam representaton n order to model realstc boundary condtons. The rubber bushngs of the model are represented wth sold elements usng *MAT_181_SIMPLIFIED_RUBBER/FOAM. Although ths controlled laboratory test s not nspred by any crash requrement, t provdes a smple but deal load case for the objectve of maxmzng energy absorpton. The lnear statc load cases are analysed wth LS-Dyna Implct; the nonlnear load case s analysed wth LS-Dyna Explct.

5 Fg.3: The LS-Dyna model of the nonlnear load case of the control arm. A prescrbed dsplacement s appled to the chan. The rghtmost fgure shows the deformed structure when the load s appled to the full desgn space. 4 Experments The target of the work s to fnd a conceptual desgn that balances the dfferent load cases,.e., the objectves of mnmum complance and maxmum energy absorpton assocated wth the respectve load requrements. Ths secton descrbes the conducted experments and dscusses achevement of the targeted objectves. In order to obtan baselnes for comparson, each objectve was consdered separately. Secondly, a sequental approach s appled based on the dea of frst optmzng the complance objectves, followed by a second optmzaton for the energy absorbng targets. Fnally, we apply the multobjectve approach based on scaled weghtng. All experments mplement the software package LS- TaSC [10,11], the topology optmzaton software desgned to work wth LS-DYNA. LS-TaSC provdes a straghtforward nterface to mplement the HCA algorthm descrbed n Sec. 2, ncludng the capablty to weght multple load cases. Unless stated otherwse, all optmzaton runs converged wth respect to mass redstrbuton. The mass target 1,718 kg of the desgn space, correspondng to a volume fracton of V f =0.49. The neghbourhood radus was defned at 9mm based on the mesh sze of 3mm n the desgn space. The move lmt was reduced from a default value of 0.1 to 0.02 n order to allow the optmzaton algorthm to evolve the structure wthout generatng nstabltes. The LS-TaSC parameters are summarzed n Tab. 1. All results are evaluated as obtaned from the optmzaton, post-processng s not consdered wthn the scope of ths paper. The secton s concluded wth a dscusson on the acheved objectve values n Sec HCA Parameter (LS-TaSC Settng) Parameter Value Mass Fracton 0.49 Mnmum Length Scale (Neghbor Radus) 9mm Move Lmt 0.02 Convergence tolerance Table 1: LS-TaSC parameter settngs. 4.1 Separate Optmzaton of Dscplnes Intally, the statc and the nonlnear load cases were run separately n order to understand the resultng structures and obtan solutons that represent the practcal optmum that can be acheved wth the appled optmzaton tool. The scalng factor for the statc load cases s determned accordng to (6) by measurng the overall nternal energy of the desgn space as n the LS-Dyna MATSUM database. Although the scalng method s proposed for combnng nonlnear and statc loads, t can of course be used as well to consstently obtan approprate scalng factors and weghts for pure statc mult-load case topology

6 optmzaton. Snce both load cases are consdered of equal mportance, the same preference s chosen for both, so that nether load case domnates the structure. The nternal energy of the FZ load case s approxmately 2.2 tmes that of the FY load case; therefore, a weght of 0.69 was appled for the FY load case and a weght of 0.31 was appled for the FZ load case The optmzaton objectve for the statc load cases s complance, whch s measured by the stroke at the load pont. Fg.4: Loadng and soluton resultng from mult-load case optmzaton of the statc load cases (left) and loadng and soluton resultng from sngle-load case optmzaton of the nonlnear load case (rght). The nonlnear load case s appled to the same desgn space that s defned for the statc load cases. For ths loadng, rather than mnmum complance, the desred outcome s maxmzng energy absorpton for the whole structure under ths loadng. Energy absorpton was calculated by measurng the force n the chan and ntegratng t over the stroke. LS-TaSC was run for both dscplnes separately. The resultng structures are shown n Fg. 4. A leadng motvaton for ths study s that often the structures that are optmzed for stffness load cases perform very poorly for energy absorpton and vce versa. Ths s a sgnfcant challenge for applcatons n whch both requrements are mportant desgn crtera. In order to demonstrate ths ssue, the optmzed structure that was generated for the complance load case s also evaluated wth respect to the nonlnear load case and vce versa. The results are dscussed n Sec Sequental Optmzaton of Dscplnes As an ntal attempt to combne lnear and nonlnear load requrements (complance and energy absorpton), a sequental approach s used. The entre desgn space s optmzed subject to the statc load cases for mnmum complance, as descrbed n Sec. 4.1 but wth a reduced mass fracton of 0.3. The resultng desgn was then removed from the desgn space. A mass fracton of 0.30 s then assgned to the remanng structure for the optmzaton of the dynamc load case, such that the overall mass target of kg s acheved. The optmzaton process and the resultng structure are shown n Fg. 5. The deformaton of the structure subject to the dynamc load s shown n Fg. 8 (left). Fg.5: Sequental approach to combne lnear (mnmze complance) and nonlnear (maxmze energy absorpton) load cases for a sngle structure. 4.3 Mult-objectve Optmzaton of Dscplnes Usng Scaled Weghtng Although the sequental process addresses both stffness and energy absorpton as objectves, there s lttle control as to the balance of the load cases. Wth the target to balance the dscplnes of dynamc and statc loads, a mult-objectve optmzaton approach s appled followng the approach

7 descrbed n Sec. 2. All load cases are optmzed concurrently, based on user defned preferences. From the energy levels of the load cases and the defned preferences, sutable weghts are computed. Fg.6: Dependence of the objectve value on the preference for the statc FY (left), the statc FZ load (md) and the nonlnear load case (rght). In order to obtan a set of trade-off solutons that wll balance the desgn to dfferent degrees, dfferent sets of preferences are defned. The preference for the dynamc load s chosen as: p 0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9. dyn The statc load cases preferences are computed based on the dynamc load case preference: p FY p 1 p ) / 2. FZ ( dyn These are chosen equally snce no statc load case s preferred over the other. From the preferences, the weghts for the optmzaton can be determned usng the scalng factors computed from the nternal energy. Ths data s obtaned from analyzng the structure wth the complete desgn space flled wth materal. In practce, topology optmzaton analyzed wth LS-TaSC requres a consstency of the desgn space. For ths reason, geometry, elements, and element numberng of the desgn space need to be exactly consstent for the dfferent load cases. An optmzaton study s conducted such that a topology optmzaton result s generated for each preference value. Accordngly, a study consstng of nne dfferent optmzatons s conducted. Fg.7: The objectve values of the structures obtaned from runnng the concurrent optmzaton of nonlnear and lnear loads for dfferent user preferences. Also results from Sec. 4.1 and Sec 4.2 are shown. The two lnear load cases objectves are combned by averagng.

8 Fg.8: Resultng deformed structures of trade-off from runnng the sequental topology optmzaton of stffness and energy absorpton (left), and concurrent scaled weghtng topology optmzaton, resultng from an equal preference of complance and energy absorpton (rght). The expectaton s that for ncreasng the preference of a partcular load case, the respectve objectve value shows a clear trend of mprovement. In fact, the stroke of the statc load cases s decreasng for the statc load cases and the energy absorpton s ncreased for the dynamc load case when the respectve preference s ncreased. Ths can be observed n the plots of the resultng preference dependences n Fg. 6. Possbly due to the heurstc nature of the HCA approach as well as the hgh number of varables, the observed trend s not deal. Yet, the proposed approach shows a clearly defned regularzaton of the objectve values based on the preferences. Ideally, the optmzaton study would result n nne non-domnated trade-off solutons unformly dstrbuted on the Pareto front. A non-domnated soluton s a soluton that s, when compared to any of the other solutons, better n at least one of the objectves. Snce three load cases are consdered, the user s confronted wth solutons dstrbuted n a three dmensonal objectve space. Choosng the best soluton from such a set s, n tself, a non-trval task, snce wth respect to the objectve values, none of the non-domnated solutons s better than another. In ths work, we are manly nterested n balancng the dscplnes of energy absorpton and complance. Therefore, n order to facltate the task of choosng one trade-off soluton, the strokes, as measure of complance, are combned to one objectve by averagng. The solutons n the resultng two dmensonal objectve space are shown n Fg. 7. Note that one domnated soluton s omtted n the plot, so that a set of eght non-domnated trade-off solutons remans. The approach provdes the user wth the freedom to choose the soluton wth the most approprate balance for the desgn applcaton. For further analyss n ths work, we smply choose the soluton for whch the preferences are equally dstrbuted between the dynamc and the statc load cases,.e. the result for p dyn =0.5 s chosen as representatve trade-off. The resultng structure s shown n Fg. 8 (rght). 4.4 Dscusson Tab. 2 and Fg. 9 present the results of the dfferent experments. All values are normalzed wth respect to mass to allevate small dfferences n the achevement of the mass target. Intutvely, optmzaton of the separate dscplnes s expected to yeld the best result n ther respectve objectve value. Thus these results are consdered as a practcal Baselne Result and assgned 100% performance. As descrbed n Sec. 4.1, the baselne structures were also evaluated for the opposng load cases,.e. the optmzed structure that was generated for the complance load case s also evaluated wth respect to the nonlnear load case and vce versa. The performance s shown desgnated as Opposng Result. It demonstrates that applyng a structure optmzed for the complance load cases performs poorly for energy absorpton load case as well as a structure optmzed for the energy absorpton load case performs poorly for the complance load cases. Amng for a balance of the load cases, an approach mplementng a sequental desgn optmzaton process was appled as descrbed n Sec Ths s an ntutve, practcal approach, also motvated by the fact that t can account for the use of dfferent tools for dfferent optmzaton objectves. However, there are nherent dsadvantages to ths approach ncludng gvng a preference for the frst

9 load objectve, and lack of control other than the ntal optmzaton parameters as to the balance between the load cases. The result of the sequental approach s shown n Fg. 9 and Tab. 2 as Sequental Result. In ths case, the statc FY and the dynamc load case perform nearly as well as the structures developed exclusvely for ether the statc or nonlnear load cases. The reducton of the desgn space for the nonlnear optmzaton dd not have a too severe mpact on the performance of the resultng structure. However, degradaton s noted for the statc FZ load case. Ths approach does show an overall mprovement n the balance of all load cases over the desgns that were developed wthout consderng the conflctng loads. Yet, there s lttle control on balancng the load cases. If a dfferent soluton s desred wth regards to the balance between the conflctng objectves, the entre process must be repeated wth adjusted mass fractons that are at best an educated guess for each load case type. Ths motvates the approach to concurrently optmze all load cases. The trade-off soluton can be nfluenced by a user preference whch s mostly decoupled from the energy levels of the load cases by usng the proposed scalng. Systematcally, several runs can be performed to obtan a set of trade-off structures from whch the user can choose accordng to the applcaton. The chosen trade-off for ths case s that of the optmzaton that assgns equal preferences between statc and nonlnear objectves. It s shown n Fg. 9 and Tab. 2 as Concurrent Result. Although the structure s not very dfferent from the result of the sequental optmzaton (Fg. 8), t shows better objectve values. Of partcular note s that the FY loadng case for the concurrent optmzaton outperforms the baselne result whch consder the statc loads separately. Ths s possbly due to smlar optmal solutons for the dynamc load case and the FY load case whch effectvely ncreases the weght preference of the FY case over that of the baselne result. Overall, the concurrent soluton s the best balanced structure found n terms of achevng all the objectves defned for ths structure. Optmzaton Run Statc FZ Statc FY Dynamc [mm -1 / kg ] [%] [mm -1 / kg] [ %] [knm / kg] [%] Baselne Result Opposng Result Sequental Result Concurrent Result % Table 2: Optmzaton results. 100% 80% 60% 40% Baselne Result Opposng Result Sequental Results Concurrent Results 20% 0% Statc FZ Statc FY Dynamc Fg.9: Summary of topology optmzed control arm structure objectves normalzed to baselne.

10 For further more practcal evaluaton of the results, there s a need for post-processng, whch s not consdered wthn the scope of ths paper. Here, structures that stll contan a small amount of ntermedate denstes are compared. The next step s a post-processng of the results to a structure consstng only of materal and vod. Ths s another non-trval problem especally for the dynamc load case. The mass fracton as well as the deformaton mode must be mantaned requrng the manual nvolvement of an engneer, and s a possble drecton for future research. 5 Summary In ths paper a topology optmzaton wth regard to statc and nonlnear requrements was conducted. The hybrd cellular automata algorthm was appled for topology optmzaton of both objectves. Experments were performed on a practcal LS-Dyna model of a vehcle control arm. The results confrm the sutablty of the appled algorthm for the consdered optmzaton objectves. Two approaches for balancng the mult-dscplnary requrements were evaluated. The frst trade-off structure s obtaned by sequentally optmzng statc and dynamc load cases. Another structure s obtaned by a mult-objectve study, performng a topology optmzaton of statc and dynamc load cases concurrently, usng a scaled weghtng approach for the energy aggregaton. Results show that both methods are sutable to obtan feasble trade-off solutons, yet the scaled weghtng approach provdes a better soluton and the possblty for the engneer to specfy preferences that provde more control on balancng the objectves. 6 Acknowledgements We thank Wllem Roux from LSTC, for support, especally wth regard to the mplementaton of the varable move lmt functonalty for LS-TaSC. 7 Lterature [1] Duddeck, F.: Multdscplnary optmzaton of car bodes. Structural and Multdscplnary Optmzaton 35(4), 2008, [2] Sgmund, O. and Maute, K.: Topology Optmzaton Approaches, Structural and Multdscplnary Optmzaton, 48, Sprnger Berln Hedelberg, 2013, [3] Patel, N. M.: Crashworthness desgn usng topology optmzaton, Ph. D. thess, Unversty of Notre Dame, [4] Patel, N. M., Kang, B.-S., Renaud, J. E. and Tovar, A.: Crashworthness desgn usng topology optmzaton, Journal of Mechancal Desgn 131, 2009, [5] Aulg, N., Nutwell, E., Menzel, S., Detwler, D.: Towards mult-objectve topology optmzaton of structures subject to crash and statc load cases, Engneerng Optmzaton 2014, CRC Press, 2014, [6] Bendsøe, M. and Sgmund, O.: Topology Optmzaton Theory, Methods and Applcatons, 2nd ed., Sprnger Verlag Berln, [7] Bendsøe, M.: Optmal shape desgn as a materal dstrbuton problem, Structural optmzaton 1, 1989, [8] Tovar, A.: Bone remodelng as a hybrd cellular automaton optmzaton process, Ph. D. thess, Unversty of Notre Dame, [9] Tovar, A., Patel, N.,M. Nebur, G. L., Sen, M. and Renaud J. E.: Topology optmzaton usng a hybrd cellular automaton method wth local control rules, Journal of Mechancal Desgn 128(6), 2006, [10] Roux, W.: Topology desgn usng LS-TaSC TM verson 2 and LS-DYNA, 8th European LS-DYNA Users Conference, [11] Lvermore Software Technology Corporaton: The LS-TaSC TM Software, Topology and Shape Computatons Usng the LS-Dyna Software, User s Manual, Verson 3.0, from: