Free-Form Grid Shell Design Based On Genetic Algorithms

Transcription

1 Free-Form Grd Shell Desgn Based On Genetc Algorthms ABSTRACT Mlos Dmcc Stuttgart Unversty Jan Knppers Stuttgart Unversty In the 1st century, as free-form desgn grows n popularty, grd shells are becomng a unversal structural soluton, enablng the conflaton of structure and skn (façade) nto one sngle element (Kolarevc 003). Ths paper presents some of the results of a comprehensve research project focused on the automated desgn and optmzaton of grd structures over some predefned free form shape, wth the goal of generatng a stable and statcally effcent structure. It shows that by combnng desgn and FEM software n an teratve, Genetc Algorthmsbased optmzaton process, stress and deformaton n grd shell structures can be sgnfcantly reduced, materal can be saved and stablty enhanced. 7 acada 011 _proceedngs ntegraton through computaton

2 Fg. 1 1 Introducton At the end of the 0th century we wtnessed the appearance of the frst steel free-form grd shell structures entrely composed of unque structural members, snce there was no longer any substantal dfference n cost between producng 1000 unque objects and 1000 dentcal ones (Kolarevc 003). In the 1st century the feld of free-form grd shell structural desgn s beng developed further, but structural desgn and optmzaton technques are stll mostly based on the tral-and-error approach. In smpler terms, we developed a varety of technques that enable us to generate optcally acceptable trangular, quadrangular or hexagonal grds over a gven free-form surface, but when ther statcal effcency s brought to attenton there are no ready answers about how to optmze the grd. Ths paper shows how by changng the member dsposton,.e., by performng geometrcal and topologcal optmzaton of the grd shell, substantal dfferences n statcal performance can be acheved. In order to not lmt the creatvty of archtects, the dea was to generate the best structural soluton over some already defned shape. Instead of form-fndng we are tryng to fnd the best geometry and topology of a grd shell, whle keepng t on the specfc surface durng the process. The proposed method of structural optmzaton s constructed as a C++ based plug-n for Rhnoceros 3D, one of the man NURBS (Non Unform Ratonal B-Splnes) geometry based modelng tools used by archtects for free-form desgn today. The algorthm communcates teratvely wth FEM software for statc analyss. In ths case Oasys GSA commercal FEM software s used. Grd Formaton Before the optmzaton algorthm explanaton, the method of automatc grd generaton over a gven free-form NURBS surface has to be addressed. Ths s mportant n order to understand how dfferent grd shell solutons are generated n the process of fndng the most effcent one. For ths purpose, and wthn the presented research, the decson was made to use Vorono Dagrams (De Berg et al. 1997), for two man reasons. Frst, NURBS surfaces are mathematcally represented over two parameters (uv) and algorthms for Vorono dagram generaton n D (n plane) can be therefore mapped onto the surface, usng a drect xy-uv transformaton. Second, dependng on the dsposton of Vorono ponts, a large number of dfferent, natural lookng structures can be generated, but also structures wth a regular grd pattern (lke trangular, quadrangular and hexagonal). Therefore, Vorono ponts generated over a gven NURBS surface are basc varables. As depcted n Fgure 1, we take the surface, generate a Vorono dagram over t and what we do next s relax the Vorono structure. For the process of relaxaton the Force Densty Method (Gründg et al. 000) s expanded to work for any knd of grd, and addtonally to always keep the grd on the surface, whle relaxng t. By relaxng a Vorono structure we got foam-lke grd that we called Voronax (Vorono + Relax). The Voronax grd has polygons (cells) wth much more smlar corner angles and edge lengths, whch are, from a structural pont of vew, more acceptable for the grd shell desgn. The advantage of ths complexty s that Voronax grds can easly change ther densty, whle beng optcally smooth and structurally acceptable. They keep the topology of the Vorono dagram whch means that on average ther polygons have ~ 6 edges (Sack 1999; Urruta 1999). We can use that to see what dstrbuton of densty (dstrbuton of structural members) s statcally favorable. 3 Basc Plug-n Structure The goal of ths research s to make a unversal method for grd shell optmzaton; one that s adaptable, easly expandable and wth a large number of varables, (.e., wth an easy defnton of boundares and settngs wthn whch we want our soluton to be generated). Therefore a plug-n was developed so that the user can: Fgure 1. Vorono dagram and voronax structure 73 form, geometry and complexty

3 Fg. Fg. 3 Fg. 4 1) Choose the surface over whch the grd wll be generated ) Choose the basc pattern of the grd (e.g. Delaunay trangulaton (De Berg et al. 1997), quadrangular, Vorono, Voronax) 3) Set a support combnaton (e.g. all four edges, two edges, fully restraned, movable) 4) Set a load combnaton (any load combnaton defnable n FEM software) 5) Set materal propertes 6) Set cross-secton of the structural members 7) Defne the ftness functon (e.g. mnmze Von Mses stress, mnmze deformaton, maxmze load bucklng factor) 8) Defne one or more penalty functons (e.g. lmt the length of a member, lmt the sze of a polygon, lmt the stress generated n one member) 9) Set GAs parameters (e.g. crossover and mutaton probablty, number of ndvduals, number of generatons) Each one of these settngs (Fgure ) can be easly expanded and redefned. When they are chosen, the optmzaton process begns and the algorthm converges toward the best soluton for that [combnaton of nput settngs, whatever they are]. 4 Genetc Algorthms Fgure. Input parameters, expandable and changeable Fgure 3. Basc GAs loop Fgure 4. Basc loop for one grd shell soluton Genetc Algorthms (GAs) are chosen as a sutable method for mult-objectve and hghly nonlnear optmzaton. It s a stochastc method, based on the prncple of evoluton, wthn whch a random populaton of ndvduals s generated (grd shells n our case) at the begnnng. The best ndvduals, accordng to ther ftness, are then chosen for reproducton and wth specfc crossng technques, solutons are combned to brng new offsprng and n that way form a new generaton. The crossng methods ensure the hertage of good genes, thus enablng the whole process to converge toward the best ftness soluton. Specfc mutaton algorthms enable random alteraton of ndvduals n order to ntroduce dversty and ensure a better exploraton 74 acada 011 _proceedngs ntegraton through computaton

4 Fg. 5 of the search space, thus avodng convergence to local optma. Ths loop (Fgure 3) then contnues untl the satsfactory soluton s found. In our case, we are searchng for a grd shell structure wth mnmum materal usage (mnmum weght) and mnmum potental energy of the system. Grd shells can be evaluated optcally or statcally, accordng to the defned ftness functon, and n ths paper the focus s on the statcal optmzaton. More on the bascs of the Genetc Algorthms applcaton can be found n Genetc Algorthms n Search, Optmzaton and Machne Learnng (Goldberg 1989). 4.1 BASIC LOOP Genetc Algorthms work wth a chromosome representaton. In ths research the chromosome s formed as a strng of real-valued numbers whch are later on transformed nto the uv coordnates on the surface. Ths s done wth a specfc set of decodng functons. The uv coordnates are used to generate ponts from whch a Vorono dagram (over a gven surface) s calculated and eventually relaxed, resultng n a Voronax grd structure. Each grd shell n the algorthm goes through an eleven step process depcted n Fgure 4. Frst, the basc GAs operatons (selecton, crossng, mutaton) are performed, followed by the decodng part (or generaton) where the chromosome s transformed nto a grd shell and prepared for FEM statc analyss. Step 8 refers to an automatc call of the FEM software where the statc analyss of the generated grd shell s performed. When the needed results are obtaned (e.g. forces, moments, deformatons, etc.) the evaluaton accordng to the chosen ftness functon s carred out, and the soluton s penalzed f t volates any of the specfed constrants. The ftness value and the volaton of constrants are then combned and scaled nto one fnal ftness value of the generated ndvdual soluton. In a usual optmzaton there are 50 grd shells n a generaton, and the process lasts for generatons, thus sometmes generatng more than 30,000 solutons. All the solutons are kept n specfc text fles that enable ther recreaton,.e., extracton and drawng of any of the generated grd shells n the process. 5 Optmzaton In order to llustrate the optmzaton process, and what ts contrbuton s, a surface shown n Fgure 5 s chosen. It s a free-form vertcal wall, the edges of whch are restraned,.e., the structural jonts of the generated solutons on the edges are restraned from movement or rotaton n all drectons. In Fgure 5 we also see a basc cross-secton used for the optmzaton, the crcular hollow secton: CHS 193x5.0. The dea s to perform a geometrcal and topologcal optmzaton of the grd, and therefore all generated members have the same secton. In that way we can look for the mnmal stress or mnmal dsplacement soluton by changng the geometry and keepng the mass of the structure relatvely the same. The load appled s the selfweght of the structural members and a horzontal surface load. The horzontal load s appled by calculatng the surface of each cell (structural polygon), and dstrbutng t to the structural jonts (Fgure 5). Wthn the research, experments were done wth properly orented rectangular cross-sectons and wth proper wnd load (normal to the surface at all ponts). An optmzaton wth these settngs however ntroduces a dfferent set of problems whch are not the focus of ths paper, and that s why, for the presented optmzaton, the settngs were smplfed usng a crcular secton and horzontal load. Ths however has no effect on the effcency of the optmzaton process, snce t works for any knd of nput parameter combnaton. The most mportant part of the GAs optmzaton s the ftness functon. In ths case the goal s to mnmze Von Mses stress (σv) n the structure. For each structural member n the grd shell the smplfed verson of Von Mses stress (Equatons 1-4) s calculated at both of ts ends (denoted Fgure 5. Surface, cross-secton and load 75 form, geometry and complexty

5 as 0 and 1). Those values are summed up for all (n) structural members resultng n a ftness value (F(x)) for the entre structure, whch we are tryng to mnmze (Equaton 5). Eq. 1 σ = σ + + v x 3τ xy 3τ xz F M σ = ± M ± F τ = x y z y Eq. x Eq. 3 xy Eq. 4 A Wy Wz Ay τ xz Fz = A z Mnmze: Eq. 5 n F( x) = [ σ,,0 + σ,,1] = 1 v v Here we also ntroduce another ftness functon developed wthn the research, whch wll be used only for comparson purposes. Namely, for each jont n the structure ts dsplacement (movement) s calculated (d) as a vector n space, derved from the movements n all three (x,y,z) drectons (Equaton 6). The magntude of all jont movements s then summed up, resultng n a total dsplacement of the structure (Equaton 7). Fg. 6 Eq. 6 Eq. 7 d = x + y + z F( x) = n d = VORONAX OPTIMIZATION The Voronax pattern optmzaton s performed wth a 150 pont chromosome. That means that for each ndvdual soluton, 150 ponts are generated over a surface, turned nto a Vorono dagram, whch s then relaxed resultng n a Voronax grd structure. In Fgure 6 there are two graphs showng the convergence of the optmzaton process after 550 generatons (7,500 generated ndvdual grd shell solutons). The graph on the top shows the progress of the average ftness value n each generaton (calculated from 50 ndvduals). The graph bellow shows ftness values of the best ndvdual soluton (grd shell) n each generaton. It can be seen how both graphs show a constant descent of the total Von Mses stress generated n the structure and a steady convergence. In the mddle column, depcted from the front vew, there s: 1. The worst generated soluton, created randomly n one of the frst generatons, havng 113 GPa as the total amount of Von Mses stress and 13.4m of total jont dsplacement.. For comparson, a hexagonal structure s used, representng bascally a unform verson of the Voronax grd. The reason for ths s that Voronax keeps the topology of the Vorono structure after relaxaton, whch means that on average ts polygons have ~ 6 edges and jonts have a 3-member connecton (as n a hexagonal grd). Ths unformly dstrbuted grd only shows a slghtly better performance (101 GPa and 7.58m) than the worst generated soluton. 3. The best generated soluton from one of the latest generatons has the smallest amount of Von Mses stress generated n ts members (38 GPa),.e., three tmes smaller than the worst generated soluton and a 6 tmes smaller amount of dsplacement (.m). In Fgure 6, on the rght-hand sde, there s a colour analyss of ths Voronax grd soluton, showng the dstrbuton of the grd densty (from blue=sparse to red=dense). There s a number of dfferent ways of how ths nformaton can be used n grd shell desgn. Followng the advce of the GAs algorthm we can use dfferent technques, from controlled relaxaton to a combnaton of dfferent patterns, to acheve a statcally effcent desgn. The followng s an examnaton of such a desgn. 5. INTERPRETATION Fgure 6. Results of the optmzaton process We can generate a unform quadrangular structure over our free-form wall as shown on the left-hand sde n Fgure 7. Then we can try to nterpret the ntenton of the GAs optmzaton process. It can be seen that the best structural soluton offered has an enlarged grd densty around the convex parts 76 acada 011 _proceedngs ntegraton through computaton

6 Fg. 8. Fg. 7 (red area n two representatons n the mddle of Fgure 7 thus stffenng them up, and stretchng the cells over the dagonal between the two convex parts (yellow area). Usng ths nformaton we can try to generate a quadrangular structure wth a smlar number of jonts and members, as depcted on the rght-hand sde of the fgure. By dong so, we get a quadrangular structure wth 13% less generated stress and a 5% smaller amount of dsplacement. By combnng dfferent patterns (trangular, quadrangular, hexagonal) we can develop dfferent solutons, knowng the dstrbuton of grd densty (hence stffness) that produces optmal results accordng to the desred crtera. 6 Con c lu son Ths paper presents an automated method of grd shell optmzaton that offers optmal structural solutons over some gven free-form surface. The focus s on the fact that no approxmaton or pure tral and error method has to be nvolved n the structural desgn process f we use the proposed optmzaton method. The advantage of the Voronax structure s that t can be easly nterpreted most of the tme. For example, n Fgure 8, there are results of the optmzaton done over two flat vertcal surfaces, wth the same load combnaton appled as n the examples above (self-weght of the str-uctural members + horzontal load). In the example on the left, the jonts are restraned on four corners of the structure, and n the mddle of the surface edges on the structure depcted on the rght (restraned areas are marked red). For each opton the best soluton obtaned n an optmzaton process can be seen, and next to t a look through the last generaton s depcted. Namely, f we take all 50 solutons of one generaton and lne them up one behnd the other, we can get a comprehensve pcture of the ntenton of the optmzaton process. It can be seen how the center part n both cases has larger cells, stablzed wth the O-shaped formaton of denser cells n the case on the left and the X-shape formaton n the case on the rght. These experments are a part of the comprehensve research done wth dfferent shapes, ftness functons, penalty functons, support and load combnatons and dfferent patterns. Optmzatons are done not only as sngle-objectve but also as mult-objectve ones, showng that, dependng on the free-form shape and grd pattern, we can generate grd shells that have up to 6 tmes less Von Mses stress and up to 10 tmes less dsplacement when compared to a regular (unform) structure, generated wth the same number of structural members and over the same gven surface. R e f e re n c e s De Berg, M. et al Computatonal Geometry, Berln: Sprnger Verlag, Fg. 8 Goldberg, D Genetc Algorthms n Search, Optmzaton & Machne Learnng. Readng, Massachusetts: Addson Wesley. Gründg, L. et al A Hstory of the Prncpal Developments and Applcatons of the Force Densty Method n Germany Proceedngs of the IASS. Chana-Crete. Kolarevc, B Archtecture n the Dgtal Age Desgn and Manufacturng. NewYork: Spon Press. Sack J. R., and J. Urruta Handbook of Computatonal Geometry, North Holland. Fgure 7. Interpretaton of the GAs optmzaton Fgure 8. Dfferent support combnatons 77 form, geometry and complexty