Topology Design using LS-TaSC Version 2 and LS-DYNA

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1 Topology Desgn usng LS-TaSC Verson 2 and LS-DYNA Wllem Roux Lvermore Software Technology Corporaton, Lvermore, CA, USA Abstract Ths paper gves an overvew of LS-TaSC verson 2, 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 global constrants, geometrc defntons such as symmetry and castng drectons, and shells are gven. 1 Overvew The goal of topology optmzaton s to fnd the shape of a structure wth the maxmum utlty of the materal. For dynamc problems lke crashworthness smulatons, ths s acheved by desgnng for a unform nternal energy densty n the structure whle keepng the mass constraned. 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 nformaton s grouped usng the followng three headngs: Cases These store the load case data such as, the LS-DYNA nput deck and executable to use. The Cases data therefore contan the nformaton on how to smulate the use of part. Part The rest of the problem defnton data such as the part ID, geometrc defntons such as beng an extruson, and the desred mass s gven here. Constrants These constrants on the global behavor of the structure, such as the stffness and the complance. Completon These are methodology data such as the convergence crterons. 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 and there s no node or element numberng conventon as n other approaches. 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. 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].

2 2 Methodology The typcal goal of topology optmzaton s to obtan maxmum utlty of the materal. Obtanng unform nternal energy densty n the structure s used as the obectve for optmzaton. 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 optmzaton problem s formulated as, mn x N subect to : L = = 1 N = 1 C x mn * ( U ( x ) U ), w (7) C x ρ( x ) V M l C u, 1.0. * = 1,2,..., J 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 ( x ) s computed as, where K s a scalng factor and s updated as, x t = K U t * * ( U )/ U. * U denotes the nternal energy densty set pont. The desgn varable x = x + x. t+ 1 t t 3 Control Parameters The algorthm, and therefore the fnal desgn, can be controlled usng some parameters: Mass Fracton The optmal desgn s created by deletng a fracton of the orgnal part. The Mass Fracton parameter controls the amount of mass preserved. The effect of dfferent values of ths parameter s shown n Fgure 1 usng the default value of the other parameters. Convergence Tolerance Parameter The analyss s termnated when the change n the Mass Redstrbuton (vew ths hstory n the vewer panel) for the teraton s less than the Convergence Tolerance. The Mass Redstrbuton can be nterpreted as the porton of materal moved n an teraton; f the Mass Redstrbuton s 1.0 (not actually possble), then all possble mass have been redstrbuted. Proxmty Tolerance Parameter Ths parameter descrbes a dstance controllng the neghborhood sze of the elements. The desgn varable at an element s updated usng the result at the element averaged together wth that of ts neghbors. Smaller values of ths parameter yeld fner-graned structures as show n Fgure 2.

$Fgure 1 : The effect of the mass fracton bound. Dfferent desgns created usng dfferent mass fractons are shown.$ 3, the software wll remove 70% of the orgnal structure. Fgure 2 Effect of the proxmty tolerance parameter. The mesh s 100 x 250.

3, the software wll remove 70% of the orgnal structure. Fgure 2 Effect of the proxmty tolerance parameter. The mesh s 100 x 250.

3 Fgure 1 : The effect of the mass fracton bound. Dfferent desgns created usng dfferent mass fractons are shown. The mass fracton s the fracton of orgnal mass that must be kept. So for a mass fracton of 0.3, the software wll remove 70% of the orgnal structure. Fgure 2 Effect of the proxmty tolerance parameter. The mesh s 100 x 250. The small crcles show the neghborhood sze. A parameter value of zero results n a checkerboard pattern. 4 New capabltes n verson 2 The followng new capabltes were added for verson 2.

$Ths s actually a method of computng the mass fracton, because the mass of the structure s ncreased or decreased untl the constrants are satsfed.$

4 Global constrants The global constrants are used to constrant responses caused by the whole part, such as a dsplacement or reacton force. Ths s actually a method of computng the mass fracton, because the mass of the structure s ncreased or decreased untl the constrants are satsfed. Shells The thckness of shells can be redesgned to have a unform dstrbuton of the nternal energy densty. Unlke for the solds, a 0/1 behavor s not enforced, and the goal s therefore not to create a new geometry, but to fnd a thckness dstrbuton. Geometrc constrants and multple parts These are constrants on the fnal geometry allowng practcal parts, whch can be manufactured, to be desgned. The GUI nterface showng the defnton of multple parts and geometrc constrants s shown n Fgure 3. The dfferent types of geometry constrants are: o extrusons o castng, one sded o castng, two sded and o symmetry. Fgure 3 Dfferent parts, each wth multple geometry constrants, can be desgned smultaneously. Ths allows parts wth complex geometres to be desgned. 5 Collaboraton wth other CAE Companes Beta CAE Systems S.A., the author of the ANSA preprocessor, s lookng nto CAE work flow procedures ncorporatng LS-TaSC. In addton, Detrot Engneered Products, author of MeshWorks Morpher, s nvestgatng methods of usng ther mesh refnement technologes. Contact these partes for more nformaton. 6 Examples 6.1 Example wth global constrants together wth multple geometry defntons The problem as shown Fgure 4 conssts of a beam beng subected to two mpact load cases as shown.

The problem also has two global constrants: for each load case t s requred that the dsplacement s less than 100 unts.

5 The problem has two symmetry constrants and an extruson constrant. It s requred to be an extruson n the z-drecton and to be symmetrc about both the YZ and ZX plane as shown n the fgure. The problem also has two global constrants: for each load case t s requred that the dsplacement s less than 100 unts. The resultng optmal desgn s shown n Fgure 5 wth the hstores of the constrant values for the teratons as shown n Fgure 6. Fgure 4 Desgn problem wth two load cases and three geometrc constrants Fgure 5 Optmal desgn. The part s an extruson symmetrc about two planes.

Fgure 6 Constrant hstores. The fnal desgn requred a trade-off between the two constrants. Ths s because the goal of topology s to get a good concept desgn, not to satsfy the constrants exactly. 6.2 Castng example Ths example s a sold part to be manufactured as a castng, whch was accordngly mposed as a castng geometry defnton.

6 Fgure 6 Constrant hstores. The fnal desgn requred a trade-off between the two constrants. Ths s because the goal of topology s to get a good concept desgn, not to satsfy the constrants exactly. 6.2 Castng example Ths example s a sold part to be manufactured as a castng, whch was accordngly mposed as a castng geometry defnton. Weght (materal) was to be removed from the structure to obtan the best use of the materal. The procedure accordngly computed an optmal desgn by strengthenng stressed regons of the structure and removng redundant materal. Alternatve, ths desgn procedure can also be vewed as computng a structure wth the best load path for the gven structural use. The geometry and loadng condtons for ths component are shown n Fgure 7. The FE model has about elements and a sngle lnear mplct load case as shown was consdered. Fgure 7: The ntal geometry and loadng condtons.

$The mass fracton specfed n the hstory s the amount of materal that the user specfed should be kept n ths case half (a mass fracton of 0.5) of the structure was scheduled to be retaned.$

7 The convergence hstory for the multple-load example s shown n Fgure 8. The ntal and fnal structures are shown n Fgure 9, whle fnal desgn s shown n Fgure 9. The mass fracton specfed n the hstory s the amount of materal that the user specfed should be kept n ths case half (a mass fracton of 0.5) of the structure was scheduled to be retaned. The element fracton used s the fracton of the orgnal number of elements n the part used at any pont n the desgn cycle ths value wll therefore converge to close to the requested mass fracton. The mass redstrbuton s the fracton of the overall mass moved around n the desgn cycle a small number ndcates convergence of the procedure. Fgure 8: Convergence hstory. The problem s converged when the mass dstrbuton s nearly zero. At that tme the fracton of elements used s also close the requested mass fracton.

For a shell problem such as ths, the thckness of each shell element s a desgn varable. Fgure 10 The geometry and loadng condtons of the shell example.

8 Fgure 9: Fnal desgn for castng problem. The outlne of the orgnal desgn s also shown. 6.3 Shell example The geometry and loadng condtons for the shell example are shown n Fgure 10. For a shell problem such as ths, the thckness of each shell element s a desgn varable. Fgure 10 The geometry and loadng condtons of the shell example. The left sde s bult-n, whle a downward load s appled to the rght, back corner. The fnal geometry wth the fnal thckness s shown n Fgure 11. Shell elements wth a thckness of less than 5% of the startng desgn were deleted n the fnal desgn. The fnal part s stronger on the sde where the load s appled and at the left rear where the bendng moment s the largest.

Fgure 11: The fnal geometry and thcknesses for the shell problem 7 Current Development Current mprovements and extensons are: Forgng geometry defnton.

The structure wll be forced to meetng basc frequency constrants. Support of more LS-DYNA materal models. The program used to only consder only *MAT_PIECEWISE_LINEAR_PLASTICITY.

9 Fgure 11: The fnal geometry and thcknesses for the shell problem 7 Current Development Current mprovements and extensons are: Forgng geometry defnton. Ths s smlar to a castng defnton, except holes are not allowed to extend through the structure; nstead a web of a certan thckness must reman. See Fgure 12 for an example. Frequency constrant. The structure wll be forced to meetng basc frequency constrants. Support of more LS-DYNA materal models. The program used to only consder only *MAT_PIECEWISE_LINEAR_PLASTICITY. The user wll be able to use *MAT_ELASTIC as well as others materal models as requested by ndustry. Dfferent felds for the obectve. Currently the desgn s done to obtan a unform nternal energy densty. Ths extenson allows plastc strans and Von Mses stresses also to be consdered. Fgure 12 Forgng defnton desgn. Notce the web that s requred to use a forgng manufacturng process. 8 Summary LS-TaSC computes the shape of a structure wth the maxmum utlty of the materal. It has been developed for non-lnear structures analyzed n an ndustral envronment. The tool s also sutable for

10 large lnear problems. Ths tool has been extended to shells, global constrants, multple parts, symmetry, and extruson constrants. References [1] Lvermore Software Technology Corporaton, LS-TaSC : A Topology and Shape Computatons for LS-DYNA, User s Manual, Verson 1.0, Lvermore Software Technology Corporaton, Lvermore, CA, [2] Hallqust JO. LS-DYNA theoretcal manual, Lvermore Software Technology Corporaton, Lvermore, CA, [3] Goel T, Roux WJ, and Stander N. A Topology Optmzaton Tool for LS-DYNA users: LS- OPT/Topology. 7th European LS-DYNA Users Conference, Salzburg Austra, May [4] NM Patel, Crashworthness Desgn Usng Topology Optmzaton, PhD thess, Unversty of Notre Dame, [5] RT Haftka, Z Gurdal, MP Kamat, Elements of Structural Optmzaton, Kluwer Academc Publshers, Dordrecht, The Netherlands, 2 nd ed., [6] SN Patnak, DA Hopkns, Optmalty of Fully-Stressed Desgn, Computer Methods n Appled Mechancs and Engneerng, 165, , 1998.

LS-TaSC Version 2.1. Willem Roux Livermore Software Technology Corporation, Livermore, CA, USA. Abstract

LS-TaSC Version 2.1. Willem Roux Livermore Software Technology Corporation, Livermore, CA, USA. Abstract 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,