Visualization of Pareto Optimal Fronts for Multiple Objectives with D-SPEX

Transcription

1 10 th Internatonal LS-DYNA Users Conference Optmzaton (1) Vsualzaton of Pareto Optmal Fronts for Multple Objectves wth D-SPEX Katharna Wtowsk and Marko Thele DYNAmore GmbH Stuttgart, Germany Tushar Goel Lvermore Software Technology Corporaton Lvermore, CA, USA Abstract Most engneerng optmzaton problems requre mult-objectve optmzatons that have no unque optmum because the objectves often conflct. LS-OPT 3.3 offers the capablty to smultaneously compute many Pareto optmal solutons usng a mult-objectve genetc algorthm. The optmzaton post processng tool D-SPEX provdes advanced features to vsualze these Pareto optmal solutons and to approxmate the Pareto optmal front usng the movng least squares method. In addton, the movng least squares method may now be used n D-SPEX as an addtonal opton for buldng response surfaces as well as for the computaton of vrtual hstores. 4-39

2 Optmzaton (1) 10 th Internatonal LS-DYNA Users Conference Introducton Most engneerng optmzaton problems requre mult-objectve optmzatons that have no unque optmum because the objectves often conflct, e.g. the power and the consumpton of a vehcle. So no sngle desgn can be consdered optmal wth respect to all objectves, nstead the optmum s a set of solutons, the so-called Pareto optmal solutons. For a Pareto optmal soluton, there s no desgn that provdes better solutons for any objectve wthout worsenng another objectve. LS-OPT 3.3 offers the possblty to compute multple Pareto optmal solutons for mult crtera optmzaton problems usng a genetc algorthm ([1]). Usually they are calculated meta model based. In order to ad the engneer n understandng dfferent trade-offs, the optmzaton post processng tool D-SPEX provdes some features to vsualze the Pareto optmal solutons. Introducton to D-SPEX D-SPEX s a Matlab based post processng tool that specalzes n the vsualzaton of optmzaton data provded by LS-OPT. It has been developed n cooperaton wth AUDI AG for the occupant safety department but s currently used n varous dscplnes by AUDI, Damler and other clents. Installaton packages for D-SPEX are avalable for Wndows and Lnux. Although D-SPEX s a Matlab based tool, Matlab s not needed n order to run D-SPEX. D-SPEX s ntended to provde features that are not currently mplemented n the LS-OPT vewer. Therefore, t s a complement to the vsualzaton capabltes of LS-OPT. Its prmary focus s on the vsualzaton of meta-models although t also provdes features to vsualze stochastc results. D-SPEX s also thought of as a testng platform for new features that mght evolve n LS-OPT. For more nformaton on the features of D-SPEX, see [4]. In the followng, the latest development of D-SPEX, the vsualzaton of Pareto optmal solutons, s presented. Vsualzaton of Pareto Optmal Solutons The frst approach to vsualze the Pareto optmal front n D-SPEX s to vsualze the Pareto optmal solutons computed by LS-OPT. If the optmzaton problem has more than three objectves, all dmensons may not be vsualzed at once. Hence, 2D or 3D secton planes have to be selected for vsualzaton (Fgure 1), accordng to the vsualzaton of response surfaces. The objectves that are not vsualzed are set to fxed values. In contrast to response surfaces, the Pareto optmal solutons data set provded by LS-OPT may not have contnuty among objectves. If only the ponts that lay exactly on the secton plane are selected, only a few or no ponts wll be avalable for plottng. Therefore, a set of ponts wthn a small dstance ε to the selected secton s plotted. The bandwdth ε may be selected by the user as a percentage of the desgn space (Fgure 1). To provde an mpresson of the proxmty of the selected ponts to the secton plane, the ponts are color-coded from black to whte dependng on the dstance to the plane. The closer the pont s to the plane, the darker the color s, see Fgure 4 and Fgure

3 10 th Internatonal LS-DYNA Users Conference Optmzaton (1) selected objectves value for fxed objectve bandwdth of plotted ponts Fgure 1: D-SPEX graphcal user nterface for vsualzaton of Pareto optmal fronts The D-SPEX vsualzaton results are presented usng a smple example wth two varables x1 and x2, and three objectves f1, f2 and f3. The objectves are defned as f ( x1 a ) + ( x2 b ) ) ( x1 a ) + ( x2 b ) ), = = sn, whereas a and b are real numbers that shft the functon n x1 and x2 drecton, respectvely. Hence we get conflctng objectves. The objectve f1 s dsplayed n Fgure 2. The Pareto optmal solutons set for ths example s obtaned usng LS-OPT and a three-dmensonal vsualzaton of ths set s shown n Fgure 3. Fgure 2: Objectve f1 4-41

4 Optmzaton (1) 10 th Internatonal LS-DYNA Users Conference Fgure 4 and Fgure 5 show some 2D secton planes. To gve an example, f3 and f2 are set to a fxed value, respectvely. Comparng these fgures to Fgure 3, where all dmensons of the Pareto optmal solutons are dsplayed, the vsualzaton method of the sectons becomes obvous. To see the nfluence of the bandwdth of the dsplayed ponts, both sectons are dsplayed for a 10% and 1% bandwdth. So far, only the relaton between dfferent objectves are vsualzed, but f the engneer selects a sutable desgn, t s mportant to know whch varable constellaton leads to that desgn, hence D-SPEX offers two possbltes explaned below to vsualze the varable values n combnaton wth the objectves. 1. As a further dmenson, a varable, response or composte may be selected to color the ponts. Examples are dsplayed n Fgure 6, Fgure 7 and Fgure 8. To be comparable to Fgure 4 and Fgure 5, the same sectons are selected for Fgure 7 and Fgure 8. As explaned above, a set of Pareto optmal ponts wthn a small dstance ε to the selected secton s plotted. Instead of the dfferent gray values, the transparency of the ponts s used to vsualze the dstance from the selected secton. 2. To vsualze the relaton between objectves and varables, responses or compostes, t s also possble to plot objectves aganst varables, responses or compostes n any combnaton. In cases where enough Pareto optmal ponts are avalable, lke n the shown example, the vsualzaton of these ponts gves a suffcent mpresson of the Pareto optmal front. For other cases as are frequently encountered wth the drect smulaton based GA for engneerng problem, a meta-model may be used to approxmate the surface. Fgure 3: Pareto optmal ponts, 3 objectves 4-42

5 10 th Internatonal LS-DYNA Users Conference Optmzaton (1) Fgure 4: secton planes f3 = , bandwdth 10% (left) and 1% (rght) Fgure 5: secton planes f2 = , bandwdth 10% (left) and 1% (rght) Fgure 6: Pareto optmal ponts, 3 objectves, color varable x1 4-43

6 Optmzaton (1) 10 th Internatonal LS-DYNA Users Conference Fgure 7: secton planes f3 = , bandwdth 10% (left) and 1% (rght) Fgure 8: secton planes f2 = , bandwdth 10% (left) and 1% (rght) Approxmaton of Pareto Optmal Ponts usng Movng Least Squares The meta-models used e.g. for the approxmaton of the response surfaces by LS-OPT may not be able to capture the Pareto optmal front nformaton, because Pareto optmal fronts may be dsconnected, as Fgure 6 shows. Therefore, D-SPEX uses the movng least squares method to approxmate the Pareto optmal fronts ([2]). Fgure 9: Pareto optmal ponts wth approxmated surface, f3 =

7 10 th Internatonal LS-DYNA Users Conference Optmzaton (1) The dea of the movng least squares method s to compute a local approxmaton for each evaluaton pont usng a weghted polynomal least squares approxmaton, e.g., a lnear approxmaton. The ponts wth small dstance to the evaluaton pont have a large weght, the ponts wth large dstance have a small weght. If the weght s zero, the pont s not consdered. If enough ponts wth non zero weghts are not avalable, approxmaton values are not computed and dsconnected surfaces may be found. An example of a Pareto optmal front approxmated usng the movng least squares method s dsplayed n Fgure 9. The Movng Least Squares Method for Approxmaton To assess the qualty of the movng least squares (MLS) approxmaton, the method was mplemented for response surfaces and compared to the feedforward neural network (FF) and the radal bass functon network (RBF) used by LS-OPT. The SPRESS (Square root of the Predcted Resdual Sum of Squares), whch measures the predcton capablty of any meta model, and the RMS error (square root of resdual sum of squares) ([3]), were compared (Table 1) for FF, RBF and MLS approxmatons by means of a smple crash worthness desgn wth two load cases. Subject of ths nvestgaton s the ht of a car on a sngle pole. In the frst load case the car hts the pole n the mddle of ts front part. The thcknesses of the bumper, the hood, and the bottom are to be determned. For the second load case an antmetrc scenaro s nvestgated. In contrast to the frst load case the httng pont s now moved from the mddle to the rght as shown n the fgures below. For both load cases the consdered responses are the mnmal x dsplacement of two selected nodes (Intru1, Intru2), the HIC value at a selected node and the absolute ntruson that s calculated from the dfference of Intru1 and Intru2. The objectve of both load cases s the respectve HIC value whch has to be mnmzed. For both load cases constrants are ntroduced for the absolute ntruson. SPRESS RMS error Response FF RBF MLS FF RBF MLS Intru2_md e Intru1_md e HIC_md Intruson_absolute_md Intru2_rght HIC_rght Intru1_rght Intruson_absolute_rght Table 1: Predcton capablty of Feed forward neural networks (FF), Radal Bass Functons (RBF) and Movng Least Squares (MLS), 5 ponts used for approxmaton. Judgng from ths example, t can be sad that the movng least squares approxmaton has a comparable qualty to the radal bass approxmaton and feed forward neural networks. Fgure 4-45

8 Optmzaton (1) 10 th Internatonal LS-DYNA Users Conference 10 and Fgure 11 show examples of the surfaces. The uprght black lnes vsualze the dstance from the ponts to the surface. Fgure 10: Feedforward Neural network (left) compared to Movng Least Squares (rght) Fgure 11: Radal Bass Functons (left) compared to Movng Least Squares (rght) The Movng Least Squares Method for Vrtual Hstores A vrtual hstory s an estmate of a hstory for a specfc parameter combnaton by extendng the meta-model concept to hstores. It means that meta-models are created at a number of dscrete ponts n tme of the hstores. So the ndvdual meta-models can be used to assemble vrtual hstores. So far, D-SPEX provded only lnear meta models to approxmate vrtual hstores, ([4]), but snce the movng least squares approxmatons results n a good approxmaton, they were also appled to compute the vrtual hstores n D-SPEX to get better approxmatons. 4-46

9 10 th Internatonal LS-DYNA Users Conference Optmzaton (1) - vrtual hstory - computed hstory wth same varable constellaton better ft Fgure 12: vrtual hstory, lnear approxmaton (left) and movng least squares approxmaton (rght) better ft Fgure 13: vrtual hstory, lnear approxmaton (left) and movng least squares approxmaton (rght) In Fgure 12 and Fgure 13, both approxmatons are compared usng the example explaned n the secton above. Fgure 12 shows the dsplacement of a node over tme, Fgure 13 shows the force at the pole. The black lne s the computed hstory wth the same varable constellaton as the vrtual hstory (blue curve). So the movng least squares approxmaton provdes better results, at least for ths example. Conclusons In cases wth two or three objectves, t s suffcent to vsualze the Pareto optmal soluton computed by LS-OPT. In cases wth more objectves, there are often not enough ponts per dmenson avalable to get an mpresson of the Pareto optmal front only by vsualzng the ponts, so t s necessary to compute an approxmatng surface. The movng least squares method turned out to be sutable to approxmate surfaces that may be dsconnected. However further work wll be necessary n order to provde a soluton that s sutable for the engneerng needs. 4-47

10 Optmzaton (1) 10 th Internatonal LS-DYNA Users Conference References [1] Goel, T., Stander, N., Mult-Objectve Optmzaton Usng LS-OPT, 2007 [2] Kunle, M., Entwcklung und Untersuchung von Movng Least Square Verfahren zur numerschen Smulaton hydrodynamscher Glechungen, 2001 [3] Stander, N., Roux, W., Eggleston, T., Crag, K., LS-OPT User s Manual, Verson 3.3, Lvermore Software Technology Corporaton, Lvermore, 2008 [4] Wtowsk, K., Thele, M., Seer G., Mühlhuber, W., van den Hove, M., New Features n D-SPEX wth Applcaton,