Structural Optimization Using OPTIMIZER Program

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1 SprngerLnk - Book Chapter ق.ظ 1 of /03/12 11:30 Book Chapter large verson Structural Optmzaton Usng OPTIMIZER Program Book III European Conference on Computatonal Mechancs Publsher Sprnger Netherlands DOI / Copyrght 2006 ISBN (Prnt) (Onlne) DOI / _709 Page 709 Subject Collecton Engneerng SprngerLnk Date Thursday, June 05, 2008 PDF (46.9 KB) Free Prevew III European Conference on Computatonal Mechancs Solds, Structures and Coupled Problems n Engneerng: Book of Abstracts / _709 C. A. Motasoares, J. A. C. Martns, H. C. Rodrgues, Jorge A. C. Ambróso, C. A. B. Pna, C. M. Motasoares, E. B. R. Perera and J. Folgado M. H. Abolbashar 2, M. Majd and M. R. Mahpeykar (2) Manufacturng & Automotve Engneerng Research Center, Ferdows Unversty of Mashhad, PO Box , Mashhad, Iran Abstract OPTIMIZER s a user-frendly desgn optmzaton study tool that helps users to optmze almost any optmzaton problems. There are several optmzaton algorthms n OPTIMIZER program such as Genetc Algorthm (GA), Constrant Steepest Descent (CSD) and Constrant Quas- Newton (SQP). The OPTIMIZER can only solve problems that have an explct mathematcal expresson both for cost functon and constrants. To extend the OPTIMIZER capablty for other applcatons, t s lnked wth an analyss software lke ANSYS. In ths paper, several structural optmzaton problems are solved usng OPTIMIZER and the results are compared wth other reported solutons. Furthermore, the effectveness of the above-mentoned methods for the selected problems s presented. M. H. Abolbashar Emal: abolbash@ferdows.um.ac.r Fulltext Prevew (Small, Large)

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3 III European Conference on Computatonal Mechancs Solds, Structures and Coupled Problems n Engneerng C.A. Mota Soares et.al. (eds.) Lsbon, Portugal, 5 8 June 2006 STRUCTURAL OPTIMIZATION USING OPTIMIZER PROGRAM M.H. Abolbashar 1, M. Majd, and M.R. Mahpeykar 1 Manufacturng & Automotve Engneerng Research Center, Ferdows Unversty of Mashhad, PO Box , Mashhad, Iran abolbash@ferdows.um.ac.r Keywords: Optmzaton program, Optmzer, Structural optmzaton. Abstract. OPTIMIZER s a user-frendly desgn optmzaton study tool that helps users to optmze almost any optmzaton problems. There are several optmzaton algorthms n OPTIMIZER program such as Genetc Algorthm (GA), Constrant Steepest Descent (CSD) and Constrant Quas-Newton (SQP). The OPTIMIZER can only solve problems that have an explct mathematcal expresson both for cost functon and constrants. To extend the OPTIMIZER capablty for other applcatons, t s lnked wth an analyss software lke ANSYS. In ths paper, several structural optmzaton problems are solved usng OPTIMIZER and the results are compared wth other reported solutons. Furthermore, the effectveness of the above-mentoned methods for the selected problems s presented.

4 1 INTRODUCTION There s several optmzaton software developed for specal tasks. For example, VsualDOC [1, 2] s a flexble desgn optmzaton study tool that helps couplng optmzaton capabltes wth analyss program. GENESIS [3, 4] s a fully ntegrated fnte element analyss and numercal optmzaton program for analyss and desgn of a wde range of structures. For analyss wth GENESIS, statc, normal modes, drect and modal frequency response and heat transfer calculatons are avalable. The choce of analyss s based on the nput data provded by the user. Begnnng wth OPTIMIZER 1.0 [5], t contans 13 optmzaton algorthms of varous orders. For example, for mult dmensonal optmzaton t provdes fve algorthms, namely Steepest Descent, Conjugate Gradent, DFP, BFGS and Newton. Currently OPTIMIZER2.0 [6] s ndependent of OPTIMIZER1.0. In ths paper, the features and capabltes of OPTIMIZER2.0 wll be dscussed. Moreover, OPTIMIZER2.0 s lnked to the ANSYS analyss program to extend ts ablty for solvng the optmzaton problem wth mplct functons. 2 GRAPHICAL USER INTERFACE (GUI) The basc advantages of the GUI are to provde a fully automated, graphcal desgn envronment that s easy to use for the users. Interacton wth the program can be acheved usng a spreadsheet or wndow based forms. User should specfy the desgn varables, objectve functon, and constrants, f any. The user also specfes whch method to use and any necessary control parameters. At the post-processng stage, the user may retreve the hstory of the optmzaton process and the plot of the desgn varables, constrants volaton and cost functon varaton durng the optmzaton teratons. Fgure 1 shows a vew of the OPTIMIZER2.0 GUI for defnng a project catalog ncludng desgn varables, cost functon and constrants. Fgure 1: OPTIMIZER2.0 GUI for defnng a project 2

5 3 OPTIMIZATION ALGORITHMS OPTIMIZER2.0 contans 10 algorthms. For lnear constraned optmzaton, OPTIMIZER2.0 provdes Smplex7 method. For problems wth quadratc cost functon and lnear constrants, t provdes quadratc programmng (QP) [7]. For nonlnear constraned t provdes sx algorthms. They are Constrant Steepest Descent (CSD) [7], Constrant Quas- Newton (SQP) [7], Sequental Lnear Programmng (SLP) [7], Methods of Feasble Drectons (MFD) [8], Gradent projecton (GP) [8] and Optmalty Crtera [8]. For nongradent base optmzaton t provdes Genetc Algorthm (GA) [9] and Evolutonary Programmng (EP) [10] methods. These algorthms (see Fgure 2) have been tested and demonstrated to be both effcent and relable for many optmzaton problems. Fgure 2: Optmzaton algorthms 4 LINKING WITH ANSYS In many cases of desgn optmzaton, there s no explct mathematcal expresson for the objectve and/or constrants functon. Therefore, the problem may be solved by an analyss software to get responses values. For ths type of problems, OPTIMIZER2.0 s lnked wth ANSYS6.1 [11]. 5 VERIFICATION EXAMPLES Three problems have been solved to demonstrate the OPTIMIZER2.0 capablty. They are weght mnmzaton of a ten-rod truss, a cantlever beam and a three-rod truss that are presented n the followng. 3

6 5.1 Example 1: Weght mnmzaton of a ten rod truss [3, 12] The frst example wll be gven here where ANSYS solves the analyss problem and OPTIMIZER2.0 s used to perform optmzaton process usng CSD method. The same problem has been solved by GENESIS [3] software. The problem s defned as: Mnmze the mass of the ten rod truss structure shown n Fgure 3, subject to stress constrants n each element. The ten desgn varables are the tenrod cross sectonal areas wth the ntal value of 5.0 area unts n2. A lb force s appled at the nodes 4 and 2 as shown n Fgure 3. The ten desgn varables have lower and upper bounds of and 1000 area unts n2, respectvely. The stress responses are constraned to be between and Ps. Modulus of elastcty and Posson's rato set to 1e07 Ps and 0.1, respectvely. The optmum desgn varables obtaned from OPTIMIZER2.0 and reported from GENESIS along wth the error are presented n Table 1. The error s calculated as (GE- OP)/GE 100 where GE and OP stand for GENESIS and OPTIMIZER2.0 result, respectvely. As t s seen n Table 1, the maxmum error of desgn values between GENESIS and OPTIMIZER2.0 s only 0.46%. The hstory of the objectve functon and the maxmum constrant volaton are shown n Table 2. As t s seen n ths Table, the optmum objectve functon obtaned by OPTIMIZER2.0 s smaller than that of GENESIS and the optmum maxmum constrant volaton at optmum for GENESIS and OPTIMIZER2.0 s the same. It may be concluded that the OPTIMIZER2.0 s more effectve for solvng ths problem. Fgure 3: Ten-rod truss Varable Desgn varable value ERROR Label (GENESIS) (OPTIMIZER2.0) (%) A E E A E E A E E A E E A E E A E E A E E A E E A E E A E E Table 1: Optmum desgn varable values for example 1 4

7 Maxmum Constrant Volaton (%) Objectve functon NO. (GENESIS) (OPTIMIZER2.0) (GENESIS) (OPTIMIZER2.0) ERROR (%) Table 2: Iteraton hstory for example Example 2: Desgn of a cantlevered beam [13] The cantlevered beam shown n Fgure 4 s to be desgned for mnmum materal volume, the desgn varables are the wdth b and heght h at each of N segments, where here N = 5. The beam s subject to lmts on stress (calculated at the left end of each segment), deflecton under the load, and the geometrc requrement that the heght of any segment does not exceed twenty tmes the wdth. The desgn problem s defned as: N Mnmze : V = bhl (1) Subject to : σ 1 0 = 1,N σ h 20b 0 = 1,N y b h y N = 1,N = 1,N σ and σ are the element and the lmt stresses, Where and y are the element deflecton and the lmt deflecton, respectvely. Other parameters are shown n the Fgure. Ths problem was solved by the modfed feasble drectons method usng DOT [14] optmzaton software and method of feasble drectons usng OPTIMIZER2.0 software. Optmum desgn varables and objectve functon values obtaned from DOT and OPTIMIZER2.0 program are presented n Table 3. It s shown that the error between the optmum objectve functon obtaned by OPTIMIZER2.0 wth respect to DOT s %. Therefore, OPTIMIZER2.0 has provded a reasonably good result for ths problem. = 1 yn (2) (3) (4) (5) (6) 5

8 P h l 1 l 2 l 3 l 4 l 5 L Fgure 4: Cantlevered beam b P = N E = 200Gpa L = 500cm 2 σ = N/cm y = 2.5 cm Desgn varables values (DOT) (OPTIMIZER2.0) Intal Optmum Intal Optmum b b b b b h h h h h Varable Label Objectve Functon ERROR (%) Table 3: Optmum desgn varables and objectve functon values for example Example 3: Desgn of a three-rod truss [3, 15] The thrd example s to desgn a three-rod truss for the mnmum weght. Alternatvely, the problem can be stated as: Mnmze the mass of the three rod truss structure shown n Fgure 5, subject to stress constrants n each element and dsplacement constrants on grd 4. Grd 4 dsplacement s constraned to be between -0.2 and 0.2 n and stresses are constraned to be between and Ps n both load cases. The three desgn varables are the three-rod cross sectonal areas. There are two load cases of 20,000 and 19,000 (lb) appled at node 4. In ths example, ANSYS provdes analyss and CSD method n OPTIMIZER2.0 s used to perform optmzaton. Ths problem has been also solved by GENESIS software [3]. 6

9 Y E =1e07 Ps υ = 0.33 A 1 A 2 A X P 2 =20,000 lb P 1 =19,000 lb Fgure 5: Three-rod truss Optmum desgn varables and objectve functon values obtaned from GENESIS and OPTIMIZER2.0 program are presented n Table 4. As shown n ths Table, the optmum objectve functon obtaned by OPTIMIZER2.0 s smaller than that of GENESIS. The optmum maxmum constrant volaton at optmum for GENESIS and OPTIMIZER2.0 s 0.1% and 0.02%, respectvely. It s also seen that the number of teratons for GENESIS and OPTIMIZER2.0 s 10 and 8, respectvely. It may be concluded that the OPTIMIZER2.0 has provded a more accurate soluton for ths problem. Varable Intal Optmum values ERROR Label Value (GENESIS) (OPTIMIZER2.0) (%) A E E A E E A E E Objectve functon Maxmum Constrant Volaton (%) Number of Iteratons Table 4: Optmum desgn varables and objectve functon values for example 3. 6 CONCLUSIONS There s a clear need for a general-purpose optmzaton program that wll provde a graphcal user nterface. OPTIMIZER2.0 s such a desgn optmzaton study tool. Solvng three examples, t s shown that the OPTIMIZER2.0 s a good optmzaton tool. In an attempt to extend the capablty of the OPTIMIZER2.0 for the problem wth mplct functons, t s seen that the optmum desgn may be obtaned wth a reasonable accuracy wth respect to the analytcal soluton and/or results obtaned by the smlar programs. 7

10 REFERENCES [1] Vsual DOC 4.0, VR&D's, Next Generaton Desgn Optmzaton Software System, 1767 S. 8th Street, Sute 200, Colorado Sprngs, CO 80906, [2] Vanderplaats, G.N., VsualDOC - A Flexble Desgn Optmzaton Software System, Vanderplaats research & Development, Inc., 1767 S. 8th Street, Sute 100, Colorado Sprngs, Co 80906, Sept., [3] GENESIS User s Manual (Verson 6.0), Vanderplaats Research & Development, Inc., 1767 S. 8th Street, Sute 100, Colorado Sprngs, Co 80906, January [4] New Feature n GENESIS 7.5, VR&D's, Inc., [5] Abolbashar, M.H., OPTIMIZER 1.0, Mechancal Engneerng Department, Ferdows Unversty of Mashhad, PO Box , Iran, [6] Abolbashar, M.H., OPTIMIZER 2.0, Mechancal Engneerng Department, Ferdows Unversty of Mashhad, PO Box , Iran, [7] Arora, J.S., Introducton to Optmum Desgn, McGraw-Hll, Inc., Sngapore, [8] Haftka, R.T., Elements of Structural Optmzaton, Thrd Revsed and expanded edton, Kluwer Academc Publshers Group, USA, [9] Goldberg, D.E., Genetc algorthm n search, optmzaton and machne learnng, Addson Wesley, Readng, MA, [10] Kt Po Wong Cameron Alge, Basc Evolutonary Programmng for Statc Dspatch of Cogeneraton, Artfcal Intellgence and Power Systems Research Group, Department of Electrcal and Electronc Engneerng, the Unversty of Western Australa, [11] Lawrence, K.L., ANSYS Tutoral Release 7.0 (and Release 6.1), Mechancal and Aerospace Engneerng Unversty of Texas at Arlngton, [12] Zhang, W.H. and Fleury, C., A modfcaton of convex approxmaton methods for structural optmzaton, Computers & Structures Vol. 64, No. 1-4, 1997, pp [13] Vanderplaats, G.N., Numercal Optmzaton Technques for Engneerng Desgn, Vanderplaats Research & Development, Inc., 1767 S. 8th Street, Sute 100, Colorado Sprngs, Co 80906, [14] DOT Users Manual (Verson 4.2), Vanderplaats Research & Development, Inc., Colarado Sprngs, CO, [15] Rao, S.S., Optmzaton Usng Fuzzy Set Theory, n: Structural Optmzaton: Status and Promse, (ed. Kamat, P. K.), Amercan Insttute of Aeronautcs and Astronautcs, Inc.,