6 th Chia-Japa-Korea Joit Symposium o Optimizatio of Structural ad Mechaical Systems Jue 22-25, 200, Kyoto, Japa Freeform Optimizatio: A New Capability to Perform Grid by Grid Shape Optimizatio of Structures Jua Pablo Leiva Vaderplaats Research ad Developmet, Ic. 4700 Gardebrook, Ste. 5 Novi, MI 48375, USA jp@vrad.com Abstract This paper presets a ew capability for performig a special type of shape optimizatio. Traditioal shape optimizatio is ofte performed by usig perturbatio vectors that are liked with grid coordiates ad desig variables which a optimizer ca chage. I this work, these shape perturbatio vectors are split so that each grid, ad/or a set of earby grids, ad/or a set of grids liked by a maufacturig costrait, has its ow desig variable. This split produces great variability i the aswers. I this work, possible distortios that could happe durig a optimizatio ru are preveted with automatically geerated distortio costraits ad/or by mesh smoothig. To distiguish this capability from stadard shape optimizatio, we amed this capability freeform optimizatio. This capability could be see as a geeralizatio of topography optimizatio sice it ca reproduce most of topography results, but it is differet tha topography optimizatio sice it ca be used to desig ay type of structure icludig solids ad trusses whereas topography is mostly used for shell structures. Several examples which show the applicatio of freeform are preseted. Oe example will show the use of freeform optimizatio to fid optimal rib locatios i a solid structure. Other examples will demostrate the use of freeform optimizatio to fid optimal bead locatios i shell structures. Keywords: Freeform, Shape, Topography, Optimizatio. Itroductio Traditioal shape optimizatio has bee successfully utilized to optimize a variety of structures for may years. However, i today s competitive eviromet, where ew desigs are expected to perform better tha previous oes ad at the same time to be more ecoomical, traditioal shape optimizatio has started to be isufficiet. This situatio was similar for sizig optimizatio some years ago. Our respose the, as software developers, was to improve o sizig optimizatio by developig topometry optimizatio []. Topometry optimizatio is essetially a automatically geerated elemet-by-elemet sizig optimizatio. So whe faced with the challege to improve o shape optimizatio, we thought why ot create somethig like a grid-coordiate by grid-coordiate optimizatio? Researchig the literature we ca fid some papers o which each grid-coordiate is desiged. The problem with this approach is that it ca easily distort fiite elemet meshes. I additio, this approach does ot give eough cotrols to be useful. Because of that, we have bee usig topography optimizatio for shell structures ad occasioally for solid structures. Topography optimizatio works by automatically creatig perturbatio vectors which i geeral are ormal to the desigable surface areas [2]. I Geesis, for shells, topography works directly [3]. For solids it eeds to use a exteral shell ski that is used to desig the surface grids of the structure ad mesh smoothig to move the iterior odes. The use of topography optimizatio has give good results but has several limitatios. The good results ca be attributed to the fact that it uses perturbatio vectors which are well defied. The most oticeable limitatios are that: a) topography optimizatio ca easily distort the mesh, b) topography oly works with certai types of elemets ad c) topography ca ot produce beads with variable heights (user cotrolled maximum heights). Recetly, we have developed a method to overcome the first limitatio. The method uses iterally created distortio costraits. Distortio costraits will be discussed later i this paper as they are ew ad very importat to freeform optimizatio. Traditioal shape optimizatio that uses perturbatio vectors does ot have the last two limitatios described. I other words, it is ot limited to ay type of elemets ad perturbatio vectors ca take almost ay shape we wat. It should be metioed here that there are several ways to create perturbatio vectors ad they ca be created either iterally i the software (for example usig the Domai elemets [4]) or usig a desig pre-processor like Desig Studio [5]. A questio arises: Ca we take advatage of the best of topography (each grid has its ow variable) ad the best of shape optimizatio (perturbatio vectors are kow, have cotrolled shape ad work with all elemets)? The implatatio of freeform is a attempt to aswer that questio. The idea behid freeform is the followig: The program takes a existig perturbatio vector ad the breaks it ito idividual perturbatios so that each grid has its ow perturbatio vector ad its correspodig desig variable. Because of this, we ca defie freeform optimizatio capability as a grid-by-grid shape optimizatio capability.
2. Approximatio Cocepts Before goig ito the details o how freeform has bee implemeted, a explaatio of how the Geesis program solves a optimizatio problem is preseted ext. I Geesis, a structural optimizatio problem is solved usig the approximatio cocepts approach. I this approach, a approximate aalysis model is created ad optimized at each desig cycle. The desig solutio of the approximate optimizatio is the used to update the fiite elemet model, ad a full system aalysis is performed to create the ext approximate aalysis model. The sequece of desig cycles cotiues util the approximate optimum desig coverges to the actual optimum desig. I the mid-seveties Schmit et al. itroduced approximatio cocepts for traditioal structural optimizatio [5-6]. These cocepts, i the eighties ad early ieties, were refied to improve the quality of approximatios [7-9]. The approximate problem is solved usig either the BIGDOT [] or DOT [0] optimizers. The purpose of usig the approximatio cocepts approach is to reduce the umber of desig cycles to reduce time. With these approximatios a good egieerig aswer ca be typically foud i 0 desig cycles. 3 Shape Optimizatio 3. The Perturbatio Vector Shape optimizatio is used to fid the optimal locatio of grids i a structure. I the Geesis software, shape optimizatio uses perturbatio vectors. The perturbatio vectors dictate how much ad i which directio the coordiates of the grids ca chage whe its correspodig desig variable chages i oe uit. The effect of oe perturbatio vector is show i Fig. below. Figure. Perturbatio Vector The followig figure shows how a perturbatio vector affects the locatio of the grids. A value of 0.0 o the desig variable causes o movemet. A value of.0 causes the structure to move the same as the perturbatio vector. If the desig variables take a value larger tha.0, the the grid movemets are larger tha the perturbatio. I Fig. 2, below, it is show how a shape would look if the desig variable associated to the perturbatio vectors took a value of 5.0. Figure 2. Shape Chages 3.2 Grid Locatio Update Whe there are multiple perturbatio vectors, the grid locatios are updated by addig to the iitial coordiate a liear combiatio of all perturbatio vectors ad its correspodig desig variables. The equatios used i shape optimizatio to iterally calculate the ew locatios of the grids are: X i = X io + DV j * XP ij j () Y i = Y io + DV j * YP ij j Z i = Z io + DV j * ZP ij j I equatios (), X i, Y i ad Z i are the updated coordiates of the grid i. X io, Y io ad Z io are the iitial coordiates of the grid i. XP ij, YP ij ad ZP ij are the compoets of the j th perturbatio vector correspodig to grid i. Fially, DV j is the value of desig variable j. These equatios are also used by topography ad freeform optimizatio.
4. Freeform Optimizatio We ca defie freeform optimizatio as a grid-by-grid shape optimizatio capability i which a give perturbatio vector is split to icrease the variability of the aswers. Besides the stadard costraits, three special costraits ca be used to improve the quality of aswers: grid fractio costraits, distortio costraits ad maufacturig costraits. 4. Split Perturbatios The process to split perturbatios is quite simple. For each grid it desigs, a ew perturbatio is created usig the compoets associated with that grid while the other compoets are igored. Figure 3. Perturbatio Split 4.. Split Perturbatio Example The followig example shows a perturbatio vector that is split ito multiple oes. A flat plate is used. Fig. 4(a) shows the iitial perturbatio vector that is used to simultaeously desig 25 grids. Fig. 4(b-d) shows 3 of the 25 perturbatio vectors geerated. I these figures, the color bars represet the magitude of the perturbatios. Red idicates that the grid has moved to its maximum locatio ad blue idicates that the grids have ot moved. (a) Iitial Pert. Vector (b) st Pert. Vector (c) 2d Pert. Vector (d) 25th Pert. Vector Figure 4. Split Perturbatio 4.2 Grid fractio Costraits Grid fractio costraits are used to limit the umber of grids that ca move i the desigable space. These costraits typically create more ecoomical ad clea desigs. There are several optioal expressios to implemet this type of costrait. Here we will oly show oe of them: i= UBgfr gfr( x, K, x) = 0.0 (2) UBgfr where, gfr is the grid fractio costrait, x i is the ith desig variable, is the umber of desig variables i the freeform regio ad UB gfr is the upper boud value of the costrait. Typical values for the grid fractio upper bouds are 0.2, 0.35 ad 0.5. These values are for limitig the umber of grids that move to 20%, 35% ad 50%, respectively. The third example, i sectio 8 of this paper, shows the effect of usig differet grid fractio costraits. 4.3 Distortio Costraits The mai drawback of shape optimizatio ad ay other special case of it such as topography or freeform optimizatio is that the elemets of the mesh ca get distorted. Distorted elemets reduce the accuracy of the fiite elemet results ad produce aswers that are typically stiffer tha the correct aswers. Moreover, distorted elemets might yield to egative or zero jacobias of the stiffess matrices producig either usolvable system of equatios or icorrect aswers. For these x i
reasos, it is importat to prevet this problem. The traditioal way to solve this problem is by either usig mesh smoothig where the grids of the mesh are iterally moved to reduce distortio or by simply reducig the scope of the desig variables. The first method is very useful, while the secod method is icoveiet ad limitig because it reduces the chace to improve the desig. I the Geesis program, we added the ability to automatically create a distortio costrait so the problem is attacked directly. Each type of measurable distortio parameters, such as the jacobia of the stiffess matrix, aspect ratio, iterior agle ad warpig, are cosidered. The type of equatio used is the followig: Distortiol ( x, K, x) UBgdl gdl ( x, K, x) = 0.0 (3) UB ad/or LBgdl Distortiol ( x, K, x) gdl ( x,, x) = 0.0 LB gdl K (4) where, gd l is a grid distortio costrait associated to Distortio l, x i is the ith desig variable, is the umber of desig variables i the freeform regio, UB gdl is the upper boud value of the costrait ad LB gdl is the lower boud value of the costrait. The values for the distortio bouds are elemet depedet ad distortio type depedet. These costraits are geerated automatically. These costraits are also used with shape ad topography optimizatio. 4.4 Maufacturig Costraits If there are maufacturig costraits, such as symmetry costraits, the split process is chaged to esure that valid perturbatios are created. Examples of maufacturig costraits are: plaar symmetries, cycle symmetries ad extrusio. Examples i sectios 6 to 9 show the use of maufacturig costraits. 4.5 The Freeform Optimizatio Problem The freeform optimizatio problem with distortio ad grid fractio costraits ca be stated as: Mi F( x, x, K, x ) x i such that : g ( x, x, K, x ) 0; j gfr ( x, x, K, x ) 0; k gd ( x, x, K, x ) 0; l 2 2 2 2 j =, m l =, d xli xi xui; i =, where F is the objective fuctio, g j are the stadard costraits, g frk are the grid fractio costraits, g dl are the distortio costrais, x i is the ith desig variable ad xl i ad xu i are the lower ad upper side costraits. This optimizatio problem is solved usig approximatio cocepts described i sectio 2. 5. Advatages ad Disadvatages of Freeform Optimizatio with respect to Other Optimizatio Types 5. Freeform vs. Shape Optimizatio The mai advatage of freeform over shape optimizatio is that it ca produce more variability tha shape optimizatio. The mai disadvatage of freeform over shape optimizatio is that it has more chaces tha shape optimizatio to distort the fiite elemet meshes. gdl k =, gf 5.2 Freeform vs. Topography Optimizatio The mai advatage of freeform over topography optimizatio is that it works with ay type of elemets. Topography is mostly used with shells ad occasioally with solids. Aother advatage of freeform over topography is that freeform ca geerate beads with variable heights. See sectio 9 of this paper for a example which illustrates this latter metioed advatage. 5.3 Freeform vs. Topology Optimizatio The mai advatage of freeform over topology optimizatio is that it ca add extra material outside the borders. Topology optimizatio ca oly take material away. The mai advatage of topology over freeform is that topology ca produce iterior cavities ad that it ca ot distort meshes. (5)
6. Example. Rib Patter Optimizatio of a Solid Structure 6. Descriptio of the Problem This example demostrates the use of freeform optimizatio to fid the optimal locatio of rib patters i a solid structure. The overall dimesios of the structure are 40 mm x 8 mm x.5 mm. The structure is assembled with 8640 hexahedral elemets ad 988 grids. The Youg s modulus is 207,000 MPa ad the Poisso s ratio is 0.3. There are three torsio load cases. I the first load case, oe ed is fixed o its two corers while the other ed is subject to a pair of loads that produce a twist. I the secod load case, the loads ad boudary coditios are reversed from the first load case. I the third load case, the four corers of the structure are costraied while the ceter of the structure is loaded i torsio. The three load cases ca be see i Fig. 5. Figure 5. Load cases A perturbatio vector that ca move the top ad bottom surface grids is applied as show i Fig. 6. The colors i this figure show the magitude of the perturbatio. The red color represets the maximum movemet which is.2 mm, the blue color represets o movemet. I Fig. 6 grids i red are desigable while the grids i blue are ot desigable. Fig. 6 ca show the compoets of the perturbatio vector i the top. The compoets i the bottom are similar to the oes i the top but istead of poitig upward, they poit dowward makig the perturbatio vector symmetric with respect to the midsize x-y plae that passes through the ceter of the structure. Figure 6. Perturbatio Vector The objective fuctio of the problem is to miimize the sum of the ormalized strai eergy of the three torsio load cases. A grid fractio costrait of 0.45 is used to reduce the umber of grids that move. I additio, maufacturig costraits are used to obtai triple symmetry about the ceter of the structure. 6.2 Results Fig. 7 shows four views of the freeform optimizatio results. The fial desig is triple symmetric, as imposed, ad the 45% grid fractio seems to be satisfied. The output file of the program, ot show here, cofirms that. Figure 7. Freeform Results
Usig the provided perturbatio vector, show i Fig. 6, freeform optimizatio automatically geerated 608 perturbatio vectors ad 608 correspodig desig variables. These 608 perturbatio vectors cotrolled the desig of 4650 surface grids. A plot of the objective fuctio versus the desig cycle umber ca be see i Fig. 8(a). The objective fuctio was reduced from 3.0 to 0.64. I other words, the fial structure is early 5 times stiffer tha the origial oe. Fig. 8(b) shows that the ormalized mass chaged approximately 60%. I this case, the optimizatio process took 9 desig cycles to coverge. At the ed of the optimizatio there were o violated costraits. (a) Objective fuctio (b) Normalized Mass Figure 8. Optimizatio History Results 6.3 Validatio of the Problem To validate the freeform optimizatio results of this problem, a topology optimizatio ru is performed o a equivalet problem. Sice topology optimizatio ca ot add material, two layers of elemets are added to the top ad bottom of the structure. The topology model is assembled with 8,368 hexahedral elemets ad 22,228 grids. The topology model uses the same objective fuctio ad maufacturig costraits. I additio, a mass fractio costrait of 0.45 is used. The topology desigable area was limited to the added material area. Figure 9. Freeform Optimizatio vs. Topology Optimizatio Desigs 6.4 Compariso of Freeform ad Topology Optimizatio Results Freeform ad topology optimizatio iitial ad fial desigs are show i Fig. 9. From this figure, it ca be see that both freeform ad topology optimizatio results suggest similar rib patters. This topology optimizatio result validates the freeform results.
7. Example. Bead Patter Optimizatio Usig Differet Types of Maufacturig Costraits 7. Descriptio of the Problem This example demostrates the use of freeform optimizatio for fidig the optimal locatio of bead patters usig differet types of maufacturig costraits. The example uses a plate structure. The overall dimesios of the plate are 40 mm by 8 mm. The thickess of the plate is mm. The structure is assembled with 720 quadrilateral elemets ad 779 grids. The Youg s modulus is 207,000 MPa ad the Poisso s ratio is 0.3. The plate has two corers costraied i oe ed ad it is subjected to torsio loads i the other ed, as show i Fig. 0(a), below. A simple perturbatio vector is created usig a uiform patter as show i Fig. 0(b). The objective fuctio of the problem is to miimize the strai eergy to maximize the stiffess of the structure. There are three alterative desigs ad they are described below. (a) Loadig Coditio (b) Perturbatio Vector Figure 0. Iputs to the problem 7.2 Alterative Desig. Symmetrical Desig I this case a maufacturig costrait is used to eforce symmetry with respect to the x-z plae that passes through the ceter of the plate. A bead fractio costrait of 0.33 is used to limit the amout that the grids ca move. I additio, the grids are restricted to move oly upward. (a) Sigle Symmetry Desig (b) Objective ad Max Costrait Violatio History Plots Figure. Freeform Results for Sigle Symmetry Desig For this alterative, freeform optimizatio automatically geerated 35 perturbatio vectors ad 35 correspodig desig variables. Fig. (a) above, shows the fial desig. Fig. (b) shows that it took 2 desig cycles to coverge ad that there are o costrait violatios i the last desig cycle. The strai eergy decreased approximately 35 percet. 7.3 Alterative Desig 2. Double Symmetry Desig I this case, two maufacturig costraits are used to eforce double symmetry with respect to the ceter of the plate. A bead fractio costrait of 0.33 is used to limit the amout that the grids ca move. I additio, the grids are restricted to move oly upward. (a) Double Symmetry Desig (b) Objective ad Max Costrait Violatio History Plots Figure 2. Freeform Results for Double Symmetry Desig
For this alterative, freeform optimizatio automatically geerated 80 perturbatio vectors ad 80 correspodig desig variables. Fig. 2(a), above, shows the fial desig. Fig. 2(b) shows that it took 2 desig cycles to coverge ad that there are o costrait violatios i ay desig cycle. The strai eergy decreased approximately 35 percet. 7.4 Alterative Desig 3. Uiform Desig I this case, two maufacturig costraits are used. Oe maufacturig costrait is used to eforce symmetry with respect to the x-z plae ad aother to geerate a uiform profile throughout the x directio. A bead fractio costrait, equal to 0.50, is used to limit the amout that the grids ca move. Here the equality costrait is used to iduce the optimizer to make the aswers more sharp. (a) Simple Symmetry ad Uiform Profile Desig (b) Objective ad Max Costrait Violatio History Plots Figure 3. Freeform Results for Simple Symmetry ad Uiform Profile Desig For this alterative, freeform optimizatio automatically geerated 8 perturbatio vectors ad 8 correspodig desig variables. Fig. 3(a), above, shows the fial desig. Fig. 3(b) shows that it took 2 desig cycles to coverge ad that there are o costrait violatios i the last desig cycle. The strai eergy decreased approximately percet. 7.5 Comparig the Alterative Desigs Alterative desigs ad 2 are similar i a way that both produced beads which are orieted i 45 degrees with respect to the mai directio of the plate. These two results are reasoable as whe there are torsio loads experiece has show, that they ca be carried to the supports more effective with members distributed i 45 degrees. I desig 3, where the beads are forced to be alog the mai directio (x), the results are cosistet with the maufacturig requiremet. The first two alterative desigs are more efficiet to carry torsio loads tha the last desig, they ca improve the objective fuctio more (35% versus % improvemet). 8. Example. Bead Patter Optimizatio of Curve Structure Usig Differet Types of Grid Fractio Costraits 8. Descriptio of the Problem This example demostrates the use of freeform with differet bead fractio costraits. The example uses a curve shell structure assembled with 3200 quadrilateral elemets ad 338 grids. The overall projected-dimesios of the structure are 60 mm by 50 mm by 20 mm. The curvature of the structure is obtaied by makig the height of the structure a quadratic fuctio of x, where x is measured alog its legth. The thickess of the plate is mm. The Youg s modulus is 207,000 MPa ad the Poisso s ratio is 0.3 The shell has two corers costraied i oe of the short sides ad it is subjected to torsio loads applied to the opposite short side. The objective fuctio of the problem is to miimize the strai eergy. Three alterative grid fractio costraits are imposed: 0.5, 0.20 ad 0.25. These grid fractio costraits are to allow the grids to move up to 5%, 20% ad 25% respectively. (a) Grid fractio=5% (b) Grid fractio=20% (c) Grid fractio=25% Figure 4. Freeform Results Usig Differet Grid Fractio Costrait Values 8.2 Comparig the Alterative Desigs Figures 4 (a-c) show that whe the grid fractio costraits are icreased, the umber of beads icreases. These results are expected as the role of bead fractio is precisely to do that.
9. Example. Bead Patter Optimizatio Usig No-Uiform Maximum Height 9. Descriptio of the Problem This example demostrates the use of freeform to obtai bead patters that ca have o-uiform heights. The objective fuctio of the problem is miimizig the strai eergy. There is a grid fractio costrait of 0.3. There are maufacturig costraits: the first oe is to eforce symmetry with respect to the plae x-z that passes through the ceter of the plate ad which divides it i two throughout the legth ad the secod is to obtai logitudial beads. To obtai bead patters that are ot uiform i height, the oly thig to do is geerate a perturbatio vector that represets a upper boud for the desig. The iitial desig is show i Fig. 5(a). The provided o-uiform perturbatio vector is show i Fig. 5(b). (a) Iitial Desig (b) No-Uiform Perturbatio Vector (c) Fial Desig Figure 5. Optimal Locatio of Bead Patters Usig Freeform Optimizatio 9.2 Results The fial desig show i Fig. 5(c) has mirror symmetry, logitudial symmetry ad bead patters with variable height, as required. It is iterestig to metio here that this result ca ot be obtaied with topography optimizatio. 0. Summary ad Coclusios This paper has described freeform optimizatio which is a ew capability to optimize structures. This ew capability ca be defied as a grid-by-grid shape optimizatio capability. Three special costraits that improve the results were discussed: distortio costraits, grid fractio costraits, ad maufacturig costraits. This capability ca be used with ay type of structure but is particularly useful to fid ribs i solid structures ad uiform or o-uiform bead patters i shell structures. Refereces. Leiva, J. P., Topometry Optimizatio: A New Capability to Perform Elemet by Elemet Sizig Optimizatio of Structures, preseted at the 0th AIAA/ISSMO Symposium o Multidiscipliary Aalysis ad Optimizatio, Albay, NY, August 30 - September, 2004. 2. Leiva, J.P., Methods for Geeratio Perturbatio Vectors for Topography Optimizatio of Structures 5th World Cogress of Structural ad Multidiscipliary Aalysis ad Optimizatio, Lido di Jesolo, Italy, May 9-23, 2003. 3. GENESIS User's Maual, Versio.0, VR&D, Colorado Sprigs, CO, November, 2009. 4. Leiva, J.P., ad Watso, B.C., Automatic Geeratio of Basis Vectors for Shape Optimizatio i the Geesis Program, 7th AIAA/USAF/NASA/ ISSMO Symposium o Multidiscipliary Aalysis ad Optimizatio, St. Louis, MO, Sep 2-4, 998, pp. 5-22. 5. Desig Studio for Geesis, Step-by-Step Example Maual, Versio.0. Vaderplaats Research ad Developmet, Ic., Colorado Sprigs, CO, October 2009. 6. Schmit, L. A., ad Farshi, B., Some Approximatio Cocepts for Structural Sythesis, AIAA J., Vol. 2(5), 974, pp 692-699. 7. Schmit, L. A., ad Miura, H., Approximatio Cocepts for Efficiet Structural Sythesis, NASA CR-2552, March 976. 8. Vaderplaats, G.N., ad Salajegheh, E., New Approximatio Method for Stress Costraits i Structural Sythesis, AIAA J., Vol. 27, No. 3, 989, pp. 352-358. 9. Cafield, R. A., High Quality Approximatios of Eigevalues i Structural Optimizatio of Trusses, AIAA J.,Vol 28, No. 6, 990, pp. 6-22. 0. Vaderplaats, G. N., Numerical Optimizatio Techiques for Egieerig Desig; with Applicatios, 3rd Ed., Vaderplaats Research & Developmet, Ic., 999.. DOT, Desig Optimizatio Tools User s Maual, Versio 5.0. Vaderplaats Research ad Developmet, Ic., Colorado Sprigs, CO, Jauary 999. 2. Vaderplaats, G., Very Large Scale Optimizatio, preseted at the 8th AIAA/USAF/NASA/ISSMO Symposium at Multidiscipliary Aalysis ad Optimizatio, Log Beach, CA September 6-8, 2000.