陳申岳 S-Y. Chen, 2007, Gradent-Based Structural and CFD Global Shape Optmzaton wth SmartDO and the Response Smoothng Technology, Proceedng of the 7 th World Congress of Structural and Multdscplnary Optmzaton (WCSMO7), COEX Seoul, 21 May 25 May 2007, Korea.
Gradent-Based Structural and CFD Global Shape Optmzaton wth SmartDO and the Response Smoothng Technology Shen-Yeh Chen 1 1 Presdent, FEA-Opt Technology, Chunan, Maol, Tawan, S-Y.Chen@FEA-optmzaton.com 1. Abstract A new general purpose desgn optmzaton commercal pacage SmartDO s appled to the structural and CFD global shape optmzaton problems. Usng the new technques called the Response Smoothng Technology mplemented n SmartDO, t s possble to use the gradent-based approach, and stll overcome the numercal nose caused by meshng and other dscretzaton n FEA or FVM related applcaton. Ths maes SmartDO able to perform global shape optmzaton wth FEA and CFD applcatons n an effcent and systematc way. One structural and one CFD ndustry examples are llustrated. The results are compared wth the ones optmzed by conventonal gradent-based algorthms. The new approach shows sgnfcant mprovement over the tradtonal one. 2. Keywords Global Optmzaton, Shape Optmzaton, Structural Optmzaton, CFD Optmzaton 3. Introducton Structural shape optmzaton methodologes are farly mature. However, one of the most mportant ssues n structural shape optmzaton s stll how to overcome the numercal nose cause by the meshng uncertanty of descretzaton approaches, le fnte element analyss or fnte volume methods. Many approaches have been proposed to deal wth the problems mentoned above, ether drectly or ndrectly. The frst approach s to control the meshng, so that the mesh pattern remans the same before and after the desgn changes. Some researchers uses the nodal coordnates as the desgn varables drectly [1-2], or addng connectons and constrants between nodes [3-4]. Others use the controllng parameters n CAD or geometrc model as desgn varables, and relate the nodal coordnates wth the blendng nterpolaton or parameter of the geometry [5-9]. Two methods of mesh controllng seem to be popular n the commercal structural optmzaton pacages today. The frst method s called the fcttous load method or the Natural Shape Functon approach [10]. An mproved verson of ths approach, called the Hybrd Natural Approach [11], replaces fcttous load wth dsplacements, and couples wth geometry parameters to smplfy the process of decdng desgn varables. The second method uses doman elements or super elements to defne the perturbed mesh doman wth a few controllng ey nodes [12], whch does not need the parametrc nformaton from the CAD system. Ths approach requres the orgnal mesh to be dvded nto several doman elements. A new approaches called the Contour Natural Shape Functon recently proposed by Chen et. al. use the so-called contour lne to rebuld CAD-le controllng net outsde the exstng mesh, wthout any parametrc nformaton from the CAD, and don t need to splt the mesh nto dfferent doman ether [13]. A thorough lterature survey regardng the mathematcal model shape changes s also conducted n the same paper [13]. These methods are also referred as mesh-morphng. The second approach tres to focus on the optmzaton solver and algorthms. Whle the statstcs-based approaches le genetc algorthms, has been appled on global optmzaton problems wth certan relablty and stablty [14,15], there have been very few researches devoted n the gradent-based approach to deal wth the nose phenomena rased from the CAE-based problem. Chen [16] uses the Lagrange nterpolaton together wth the Desgn of Experment to construct the global response surfaces, and use genetc algorthms to search the global mnmum. The approach was successfully appled to solve the optmzaton of ballstc sheld wth explct dynamc fnte element analyss. Snyman [17] uses central fnte dfference to calculate the gradent of the objectve functon. Ths approach s able to overcome the nose n the objectve functon. However, there s no CAE-based example n the paper. Ths paper deal wth the numercal nose observed n the CAE-based or FEA-based optmzaton problems usng the Response Smoothng Technology mplemented n the commercal pacage SmartDO [18]. Two examples, one for the structural shape optmzaton, and one for the CFD-based shape optmzaton, wll be llustrated later. 4. The Response Smoothng Technology and Implementaton 4.1. Numercal Nose
In the CAE-based or FEA-based applcaton of desgn optmzaton, the desgn-analyss loop usually loos le Fgure 1. Several stuatons can cause the numercal nose. One of the sources s the meshng. When the mesh s created n free form, the mesh pattern s lely to change before and after the desgn changes (for example, from a mesh wth n nodes nto a mesh wth n+m nodes). Fgure 1 The Desgn-Analyss Loop When the numercal nose exsts n the desgn-analyss loop, the relatonshp between the model response and the desgn varables changes appears smlar to Fgure 2. Ths wll create many artfcal local mnma, even the problem tself s actually convex and has only one local mnmum. For methods that use the mesh-morphng, the desgn-analyss loop becomes Fgure 3. Ths can elmnate the numercal nose caused by mesh, but there are stll some other sources of nose. Fgure 2 The Numercal Nose Appears n the CAE-Based Optmzaton
Fgure 3 The Desgn-Analyss Loop wth Mesh Morphng One example s the nose caused by tang maxmum stress from the structural analyss. Another example s caused by ntegraton of secton pressure or flow rate from a CFD model. If these sources are combned wth the meshng nose, the problem wll be even more severe. 4.2. The Response Smoothng Technology Generally, the problems of nonlnear programmng can be formulated as equaton (1) fnd x = x, x... x Desgn Varables to mnmze subjected to f( x) { } 1 2 g ( x) G ( x) = 1 0 0 g x L x x U NDV = 1,..., NINEQC = 1,..., NDV Objectve Functon (Inequalty) Constrants Lower/Upper Bounds (1) Most of the gradent-based soluton algorthms requres equaton (1) to be smooth and well-behaved, and can only solve the local optmum. Snce the numercal nose s consdered n ths paper, the solver wll have to search for global mnmum. In order to acheve ths goal, equaton (1) s modfed as fnd to mnmze subjected to x = { x, x... x } 1 2 w f( x) + Φ * g ( x) G ( x) = 1 φ 0 0 g NDV = 1,..., NINEQC Desgn Varables Objectve Functon (Inequalty) Constrants (2) x L x x U = 1,..., NDV Lower/Upper Bounds Here w, Φ and φ are propretary formulatons whch tae nto account the values of objectve functon and constrants from the desgn hstory. The new formulaton s then solved by usual nonlnear programmng technques. The purpose of these three addtonal functons s to push the desgn pont out of the local mnmum, and avod the search path n the desgn hstory. Fgure 4 shows the schematcs of how the search path wll loo le.
Fgure 4 Schematcs of the Response Smoothng Technology 4.3. The Computer Program SmartDO The Response Smoothng Technology n the prevous secton s mplemented n the commercal pacage SmartDO. SmartDO uses the standard Tcl/T shell as the user nterface. The users communcate SmartDO and external pacages through Tcl/T scrpt. Because the optmzer s embedded nto Tcl/T scrpt, SmartDO can utlze full functonalty of Tcl/T. Therefore t s powerful for couplng wth any external CAE pacages. The basc archtecture of SmartDO s shown n Fgure 5. 5. Numercal Examples Fgure 5 The Basc Archtecture of SmartDO 5.1. Example for Structural Shape Optmzaton Fgure 6 shows the front vew, sde vew and sotropc vew of the orgnal desgn a gant hoo. The hoo was use on some crtcal msson for lftng materals. The totally heght of the hoo s about 1.8 m, and the total weght about 700 g. The hoo s made of structural steel.
Fgure 6 The Front Vew (Left), Font Vew (Center) and Isotropc Vew (Rght) of the Hoo It s very crtcal to reduce the weght of the hoo, otherwse t may be dffcult to fnd the lftng facltes able to provde the capablty. Wth the rsng expense of steel globally, the cost of the materal s another ssue. It wll be also dangerous to operate a hoo wth such a huge weght.. The goal of the desgn s to mnmze the weght of the hoo as much as possble, whle the maxmum equvalent stress should reman the same or be lower. The analyss wll have to be performed by the customer s exstng commercal pacage. The parametrc model s bult for the commercal pacage, but only free meshng s possble. Totally twenty-three desgn varables are used as shown n Fgure 7, whch ln to the coordnates of certan ponts, and the dmenson at certan locaton. The model s bult n three-dmensonal geometry. Fgure 7 The Desgn Varables of the Hoo There are at least two sources of numercal nose n ths problem, namely, tang maxmum equvalent stress of the model and free meshng. Also, from experence, local mnmum s expected.
SmartDO was used to solve ths problem, wth the Response Smoothng Technology opton turned on. Fgure 8 shows the front vew and sde vew of the optmal desgn. For comparson purpose, Fgure 9 shows the orgnal desgn (left), the desgn optmzed by the tradtonal Methods of Feasble Drectons [19] (MFD, center), and the desgn optmzed by SmartDO wth the Response Smoothng Technology (RST, rght). The desgn n the center only has very mnor changes n each desgn varable and dmenson (about less than 1%), and the reducton n weght s only 0.09% (wth maxmum stress satsfyng the stress constrant). However, the desgn by SmartDO reduces the weght by 20%, and the maxmum stress s a lttle less than then orgnal desgn. From the result t s obvous that the Method of Feasble Drectons was trapped n the local mnmum due to numercal nose, and the Response Smoothng Technology was able to escape from the local mnmum. Fgure 8 Dfferent Vew of the Optmum Desgn by SmartDO Fgure 9 Orgnal Desgn (Left) and Optmal Desgn by Method of Feasble Drectons (Center) and Response Smoothng Technology (Rght) 5.2. Example for CFD Shape Optmzaton Ths example s for the desgn of an ndustral ejector. An ndustral ejector s a faclty used to mx two flows and compress them nto hgher pressure. Fgure 10 shows the two-dmensonal desgn drawng of an ejector. The prmary flow enters the nozzle wth specfc pressure, and s ejected nto the flow path. The prmary flow wll usually reach supersonc speed when passes through the throat of the nozzle. After exstng the nozzle and enterng the mxng zone, the prmary flow s expanded, whch wll create negatve (sucton) pressure, and cause the secondary flow to flow and mx n. Fnally
the mxed flow wll exst the ejector wth a (hopefully) hgher pressure. Fgure 10 The Two-Dmensonal Drawng of an Ejector The ejector s smplfed and modeled n a commercal pacage as shown n Fgure 11. The model s axal symmetrc, wth the symmetrc axs noted as Fgure 11. The boundary P-n1 presents the nlet of prmary flow, and the P-n2 as the nlet of secondary flow. P-out s the outlet of the mxed flow, and all other boundares are modeled as statc walls. The pressure and temperature of P-n1 and P-out s gven, and the flow rate and temperature of P-n2 s also gven. The desgn varables and the geometry are defned n Fgure 12. There are totally 13 desgn varables, selected as D2, D4, D5, D6, D10, D11, D12 L4, L9, ( L10-L9), ( L11-L10), (L5-L11) and (L6-L5). Each desgn varable has ts lower and upper bound. Fgure 11 Smplfed CFD Model of the Ejector
Fgure 12 Desgn Varables and Geometry Parameters of the Ejector For the current example, the goal s to maxmze the compresson rato, whch s defned as Pressure of P out C r = (3) Pressure of P n2 The constrants are mostly geometrc constrants, whch lmt the range and relatonshp between the desgn varables. From the ntal senstvty study by manual adjustng the desgn varables, t was found that the problem had very serous problems of numercal nose. There are two reasons for the numercal nose. Frst, the mesh s created n free form. Second, the pressure at P-n2 has to be ntegrated along the lne, and then the average s taen. Table 1 shows the CFD results of the orgnal desgn, and those optmzed by Response Smoothng Technology (RST) and by the Method of Feasble Drectons (MFD). The result shows that, whle the desgn optmzed by RST can ncrease about 390% of the compresson rato compared to the orgnal desgn, the one optmzed by MFD has only 5% mprovement. Table 2 shows the fnal desgn varables of all three confguratons. It can be seen that for certan desgn varables, RST and MFD s gong n the opposte drecton of change. Also, most changes of desgn varables from MFD are very nsgnfcant. Ths example has proved that RST can at least escape from local mnmum n many stuatons. Table 1 CFD Result for the Ejector Model, Orgnal, Response Smoothng Technology (RST), and Method of Feasble Drectons (MFD) model P-n1 P-n2 P-out Cr Orgnal 957.68 1.96 4.00 2.04 RST 957.55 0.40 4.00 9.94 MFD 957.40 1.86 4.00 2.15
Table 2 Comparson of the Fnally Desgn Varables for the Model, Orgnal, Response Smoothng Technology (RST), and Method of Feasble Drectons (MFD) Orgnal RST MFD D2 5.30 5.06 5.41 D4 50.30 52.35 49.94 D5 80.30 94.81 80.81 D6 201.47 202.02 201.47 D10 160.30 158.98 161.11 D11 135.30 131.35 135.02 D12 80.30 101.07 80.36 L4 280.96 312.80 281.05 L5 1385.92 1365.65 1386.13 L6 2195.92 2173.20 2196.25 L9 520.96 520.37 520.92 L10 640.96 640.77 641.28 L11 1225.92 1209.19 1226.23 6. Conclusons A new numercal optmzaton approach, the Response Smoothng Technology, for dealng wth numercal nose n CAE-based optmzaton s developed. It was appled on structural and CFD shape optmzaton. The examples have shown that, the new approach can at least escape from the local mnmum of CAE-based applcaton. And the reducton of the objectve functon values s sgnfcantly larger than the one optmzed by the tradtonal methods. Snce the algorthms were embedded nto Tcl/T shell, the ntegraton wth other pacages s flexble and easy. 7. Acnowledgement Ths research was partally sponsored by Industral Technology Research Insttute of Tawan and Insttute of Nuclear Energy Research of Tawan. Ther assstance s greatly apprecated. 8. References 1. O.C. Zenewcz, O.C. and J.C. Campbell, Shapng Optmzaton And Sequental Lnear Programmng, n Optmum Structural Desgn, edted by Gallagher, 1973, 109-126. 2. Ramarshnan, C.V., and Francavlla, A., Structural Shape Optmzaton Usng Penalty Functons, Journal of Structural Mechancs, 1975 (3), 403-422. 3. E.J. Huang, K.K. Cho, J.W. Hou, and Y.M. Yoo, Shape Optmal Desgn Of Elastc Structures, n Proceedng of Internatonal Symposum for Optmal Structural Desgn, 1981. 4. L. Holzletner, and K.G. Mahmoud, Structural Shape Optmzaton Usng MSC/NASTRAN And Sequental Quadratc Programmng, Computers & Structures, 1999(70), 487-514. 5. K.H. Chang, and K.K. Cho, A Geometry-Based Parameterzaton Method For Shape Desgn Of Elastc Sold, Mechancs of Structure and Machnes, 1992(20), 257-264. 6. D.A. Tortorell, A Geometrc Representaton Schemes Sutable For Shape Optmzaton, Mechancs of Structure and Machnes, 1993(21), 95-121. 7. S. Chen, and D.A. Tortorell, Three-Dmensonal Shape Optmzaton Wth Varatonal Geometry, Structural Optmzaton, 1997(13), 81-94. 8. E. Hardee, K-H. Chang, J. Tu, K.K. Cho, I. Grndeanu, and X. Yu, A Cad-Based Desgn Parameterzaton For Shape Optmzaton Of Elastc Solds, Advances n Engneerng Software, 1999(301), 185-199. 9. W. Annccharco, and M. Cerrolaza, Fnte Elements, Genetc Algorthms And β -Splnes : A Combned Technque For Shape Optmzaton, Fnte Element n Analyss and Desgn, 1999(33), 125-141. 10. A.D. Belegundu, and S.D Rajan, A Shape Optmzaton Approach Based On Natural Desgn Varables And Shape Functons, Computer Method n Appled Mechancs and Engneerng, 1988(66), 87-106. 11. S.D. Rajan, S-W. Chn, and L. Gan, Toward a Practcal Desgn Optmzaton Tool, Mcrocomputer n Cvl
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