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Correspondence to S Kravanja Faculty of Civil Engineering University of Maribor Slovenia 17 SI 2000 Maribor Slovenia E mail stojan kravanja uni mb si CCC 0029 5981 98 020329 36 17 50Received 12 August 1996 1998 John Wiley optimization topology optimization discrete variable optimization Mixed Integer Non linear Programming MINLP the Modifi ed OA ER algorithm MINLP strategy hydraulic gate sliding gate roller gate Aswan 1 INTRODUCTION This paper describes the Mixed Integer Non linear Programming MINLP approach to the synthesis of roller and sliding gate structures i e the simplest types among vertical lift hydraulic steel gates see Figure 1 Roller and sliding gates are also regarded as the most frequently Figure 1 Vertical lift hydraulic steel gate structure manufactured types of hydraulic steel gates for headwater control They are used to regulate the water stream on hydro electric plants dams or spillways As hydraulic steel gates are very special structures only a few authors have discussed their optimization e g Kravanja et al 1 3 Jongeling and Kolkman4 as well as Almquist et al 5 Particular interest was shown in the optimization not of these roller and sliding gates but of similar structures In such investigations Vanderplaats and Weisshaar6 as well as Gurdal et al 7 optimizedstiff enedlaminated compositepanels Butler8 and Ringertz9 optimized stiff enedpanels Farkas and Jarmai10 optimizedwelded rectangular cellular plates Finckenoret al 11 treated skin stringer cylinders and Gendy et al 12 stiff ened plates Almost all authors used Non linear Programming NLP techniques Gurdal et al 7 proposed the genetic algorithm while Kravanja et al 1 3 introduced MINLP algorithms and strategies to the simultaneous topology and para meter optimization of the gate In Parts I of this three part series of papers a general view of the MINLP approach to the simultaneous topology and parameter optimization of structures is presented Part II describes the extension to the simultaneous standard dimension optimization Based on the superstructure approach defi ned in Parts I and II the main objective of this paper Part III is the MINLP synthesis of roller and sliding hydraulic steel gate structures obtained at minimal gate costs and subjected to defi ned design material stress defl ection and stability constraints As the MINLP approach enables simultaneous topology parameter and standard dimension optimization a number of gate structural elements girders and plates the gate global geometry intermediate distances between structural elements and all continuous and standard dimensions are obtained simultaneously This last part of the three part series of papers is divided into three main sections 1 Section 2 describes how diff erent topology and standard dimension alternatives are postu lated and how their interconnection relations are formulated by means of explicit logical constraints in order to perform topology and standard dimension alterations within the optimization procedure 330S KRAVANJA Z KRAVANJA AND B S BEDENIK Int J Numer Meth Engng 43 329 364 1998 1998 John Wiley b minimal structure with only fi xed structural elements girders As the modelling of vertical girders is simplifi ed and needs no special interconnection logical constraints the modelling of discrete decisions regarding horizontal girders proved to be more sophisticated 2 2 1 Modelling of topology alterations Let us consider the vertical cross section of the gate element superstructure with fi xed and alternative horizontal girders see Figure 3 a The number of fi xed and alternative girders and their locations in the superstructure can be described by the following logical constraints M m M ym M 9 9 ym ym 1 m 2 3 M 1 10 yM 9 1 11 Logical constraint 9 defi nes the minimal M and maximal M 9 number of structural elements girders While number M represents the number of fi xed structural elements the diff erence between the maximal and minimal number of elements M 9 M gives the number of alternative structural elements Constraint 10 defi nes the direction of the removal of alternative elements from the top down the superstructure From Figure 3 is evident that the mostupperelementis the fi xed one whichis setby the constraint 11 It then furtherfollows from 334S KRAVANJA Z KRAVANJA AND B S BEDENIK Int J Numer Meth Engng 43 329 364 1998 1998 John Wiley here they are proposed to be calculated by constraints 21 The next constraints 24 28 express normal and shear stresses acting in the section of horizontal girders The combinations between these stresses are not decisive because p 9acts in the middle of the girder and q 9acts at both girder ends Constraints 29 31 explain the local buckling constraints for the plate sections of girder webs Constraints 29 are activated when the water load acts on the outer side of the skin plate and constraints 30 are actuated when water acts on the inner side of the skin plate The plate buckling reduction coeffi cient s dependent on the plate slenderness ratio jP is defi ned by EUROCODE 3 17 s 0 6 Jj2 P 0 13 In this paper the buckling coeffi cient s is determined by the equation s 1 0 22703 1 75365j1 Lateral buckling of horizontal girders is verifi ed by constraints 32 in the case of compressed girder fl anges when water acts on the inner side of the skin plate All stability constraints 29 32 are defi ned according to EUROCODE 3 The horizontalgirder defl ectionis expressed by means of the inequality constraint 33 All constraints of the lower and upper horizontal girders are defi ned in the same manner as intermediate girders with the exception that diff erent coeffi cients of the eff ective width ln mare used here see constraints 34 and 35 THE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESIS PART III339 1998 John Wiley m 2 3 M 1 for the girder m 1 2 M 2 for the lower and m 2 3 M 1 for the upper skinplate element p2 1 3n m 1 p1n m k p1 3n m 1 2 p1 3n m 1 p1n m k p1 3n m 1 p23 m M for the girder and m M 1 for the skin plate element p2 1 3n M 1 p1n M k p1 3n M 1 2 p1 3n M 1 p1n M k p1 3n M 1 p23 m 1 2 M 1 70 dhsn hn M hsn n 1 2 N 71 hn M hsn L4 dg n 1 2 N 72 hn m hn m 1 L4 dg n 1 2 N m 2 3 M 1 73 hwn 1 H8 74 hwn hwn 1 h0n 1 n 2 3 N 75 H505 n N h0n 76 hgn h0n h gn h sn n 1 2 N 77 hn M h0n h gn n 1 2 N 78 tfn bwn tsn tfn 1 bwn 1 tsn 1 n 2 3 N 79 3 4 Pure integer logical constraints The set of constraints 80 82 represents integer logical linear constraints Constraints 80 determine the number of fi xed structural elements fi xed horizontal girders and skin plate elements for each gate element Constraints 81 make sure that the number of horizontal girders and skin plate elements of the upper gate elements is equal to or higher than the one of the lower gateelements Constraints 82 explainthat the upper alternative intermediate horizontalgirders and skin plate elements disappear in the direction from the top to the bottom of the gate before the lower ones do This in eff ect reduces the combinatorics of the MINLP problem since it THE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESIS PART III345 1998 John Wiley m 2 3 M 1 82 3 5 Interconnection logical constraints Interconnection logical mixed integer linear constraints start with constraint 83 which defi nesthe gate partitionin the horizontal direction Constraints 84 87 defi ne vertical distance hsnbetween the uppermost selected intermediate horizontal girder and the sill of the gate element Consequently the uppermost fi xed horizontal girder is always connected to the uppermost selected intermediate alternative horizontal girder and disconnected from the disappearing ones Constraints 88 91 determine the minimal openings between fl anges of horizontal girders dg 2 2 v V yn v n 1 83 hsn hn m h61 4 1 yn m 1 h61 4 1 yn m h61 4 yn m 1 n 1 2 N m 2 M 2 84 hsn hn m h61 4 1 yn m 1 h61 4 1 yn m h61 4 yn m 1 n 1 2 N m 2 M 2 85 hsn hn M 1 h61 4 1 yn M 1 n 1 2 N 86 hsn hn M 1 h61 4 1 yn M 1 n 1 2 N 87 hn 2 bfn 2 2 h 08 bf 08n hm 3 4 M 1 89 hn M bf upn hn m bfn m 2 hm 1 2 M 1 90 hn M bf upn hn M 1 bfn M 1 2 h if not the bounds are set to zero bfn m b61 fn m yn m n 1 2 N m 1 2 M 92 bfn m b 08 9 fn m yn m n 1 2 N m 1 2 M 93 twn m t61 wn m yn m n 1 2 N m 1 2 M 94 twn m t 08 9 wn m yn m n 1 2 N m 1 2 M 95 dhn m d61 hn m yn m n 1 2 N m 1 2 M 1 96 dhn m d 08 9 hn m yn m n 1 2 N m 1 2 M 1 97 bwin m b61 wn yn m n 1 2 N m 1 2 M 98 bwin m b 08 wn yn m n 1 2 N m 1 2 M 99 tfin m t61 fn yn m n 1 2 N m 1 2 M 100 tfin m t 08 fn yn m n 1 2 N m 1 2 M 101 3 6 2 ogical relations for common variables Mixed integer linear constraints 102 105 defi ne the logical relations of the common design variables These constraints enforce upon the common design variables of the horizontal girders bwin mand tfin mthe values of the corresponding common variables of the entire gate bwnand tfn bwin m bwn b61 wn 1 yn m n 1 2 N m 1 2 M 102 bwin m bwn b61 wn 1 yn m n 1 2 N m 1 2 M 103 tfin m tfn t61 fn 1 yn m n 1 2 N m 1 2 M 104 tfn m tfn t61 fn 1 yn m n 1 2 N m 1 2 M 105 3 7 ogical constraints for standard variables 3 7 1 Standard dimension logical constraints Standard dimension logical constraints defi ne four standard dimensions thickness of the skin plate tsn fl ange thickness of the horizontal girder tfn the web thickness of the outer horizontal girder t065 wn m and the web thickness of the inner horizontal girder t wn m Each standard dimension is determined by a scalar product between its corresponding vector of binary variables y and its assigned vector of discrete constants q constraints 106 108 110 111 113 and 114 where only one discrete value can be selected y 1 see constraints 107 109 112 and 115 Note thatd 03denotescorrosionadditionto the sheet ironplates In the synthesis these dimensionsare fi rst treated as continuous and later as standard dimensions The bound logical constraints 94 and 95 for the design variable twn mare therefore suitable for the standard dimensions t065 wn m and t wn m As standard dimensions tsn and tfncorrespond to the common standard design variables for THE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESIS PART III347 1998 John Wiley m 1 or m M 110 t065 wn m p P q 1 d 03 yn p t065 UP wn m 1 yn n 1 2 N m 1 or m M 111 p P yn p 1 n 1 2 N 112 t wn m r R q 3 d 03 yn r t UP wn m 1 yn n 1 2 N m 1 2 M 113 t wn m r R q 3 d 03 yn r t UP wn m 1 yn n 1 2 N m 1 2 M 114 r R yn r 1n 1 2 N 115 3 8 Constraints for the MILP phase only Additional MILP constraints 116 121 considered only in the MILP phase of the MINLP optimization are included in the model in order to perform the optimization faster and more effi ciently To enforce faster MINLP convergence equal vertical distances between the selected horizontal girders are proposed dhn m hn M h 08 M 9 1 d61 1 yn M 1 n 1 2 N m 1 2 M 1 116 dhn m hn M h 08 M 9 1 d61 1 yn M 1 n 1 2 N m 1 2 M 1 117 dhn m hn M h 08 M 9 2 d61 yn M 1 d61 1 yn M 2 n 1 2 N m 1 2 M 2 118 348S KRAVANJA Z KRAVANJA AND B S BEDENIK Int J Numer Meth Engng 43 329 364 1998 1998 John Wiley m 1 2 M 2 119 dhn m hn M h 08 M 9 3 d61 yn M 2n 1 2 N m 1 2 M 3 120 dhn m hn M h 08 M 9 3 d61 yn M 2 n 1 2 N m 1 2 M 3 121 3 9 Heuristic constraints Heuristic constraints additionally increase the effi ciency of the MINLP optimization and are added only for the MILP phase Constraints 122 125 show that the thicknesses of each upper gate element must not exceed the ones of the lower gate elements Each upper main gate element is higher than its adjoining lower main gate elements see constraints 126 tsn tsn 1 n 2 3 N 122 tfn tfn 1 n 2 3 N 123 twn m twn 1 m n 2 3 N m 1 or m M 124 twn m twn 1 m 2 n 2 3 N m 2 3 M 1 125 hon hon 1n 2 3 N 126 3 10 Initialization of variables The initialization of continuous and binary variables is also included see list of symbols in AppendixI Only design variableshave to be initialised while non design variables are calculated on the basis of the initialized design variables 4 SYNTHESIS OF THE INTAKE GATE IN ASWAN Acomprehensivecomparativedesign research work has been performedon some of the construc ted roller and sliding gates This paper shows the example of the synthesis of the already existing roller gate named Intake Gate see Figure 6 The Intake Gate was constructed and erected in Aswan II Egypt by the Slovenian company Metalna19 from Maribor Eight identical turbine intakes were erected each regulated by an Intake Gate consisting of three vertical main elements The IntakeGate was designed by means of a classicalmethod where a simplifi ed static system for the structural analysis of the gate is used The material actually used i e St 44 2 has been considered in the optimization The technical data of the Intake Gate and the economic data for the optimization are presented in Tables I IV The actual self manufacturing and transportation costs in the amount of 70518 have been calculated for the entire gate at the actual topology of 4 4 4 11 4 horizontal girders for each of the three main gate elements and 11 vertical girders for the entire gate with the corresponding number of skin plate elements The task of the synthesis of the Intake Gate is to fi nd the optimal topology geometry as well as continuous and standard sizes with respect to the minimum of self material and labour costs subjected to design stress defl ection and stability constraints The synthesis is performed THE MINLP OPTIMIZATION APPROACH TO STRUCTURAL SYNTHESIS PART III349 1998 John Wiley b horizontal cross section through three steps it begins with the generation of the gate s superstructure continues with the development of the gate s MINLP optimization model and ends by solving the defi ned MINLP problem 4 1 Simultaneous topology and parameter optimization of the Intake Gate Let us fi rst consider the synthesis of the Intake Gate followed by a simultaneous topology and continuous parameter optimization Such synthesis has been performed by means of the MINLP approach introduced in Part I This approach enables the obtaining of the optimal number of horizontal and vertical girders and skin plate elements as well as all continuous optimal dimensions simultaneously Thegenerated superstructureof the IntakeGate is the same as the one in Figure 2 It comprises n main gate elements n 1 2 3 m horizontal girders per each nth gate element m 1 2 6 an odd number 3 2v of vertical girders for the entire gate v 1 2 3 and the corresponding number m 1 2 2v of skin plate elements per each main gate element 350S KRAVANJA Z KRAVANJA AND B S BEDENIK Int J Numer Meth Engng 43 329 364 1998 1998 John Wiley b minimal gate topology 4 4 4 5 only fi xed structural elements are included alternative elements are removed determine the number of 3 2 l Vy1 v vertical girders for the entire gate The number of skin plate elements is given automatically when the number of horizontal and vertical girders is known Each gate topology is therefore expressed by the correspondingvector of binaryvariables y Myn m y1 lN i e y My1 m y2 m y3 m y1 lN i e y My1 1 y1 2 y1 3 y1 4 y1 5 y1 6 y2 1 y2 2 y2 3 y2 4 y2 5 y2 6 y3 1 y3 2 y3 3 y3 4 y3 5 y3 6 y1 1 y1 2 y1 3 N Note that the smallest fi rst index represents the lowest main gate element and the smallest second index the lowest horizontal girders In order to obtain the whole linear approximation for the MILP master problem of the modifi ed OA ER algorithm the fi rst NLP has to be performed for the entire superstructure and the initial vector of binary variables has to include the full set of structural elements i e y M1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1N All further vectors of binaryvariables obtainedby the MILP steps clearly show which structural elements are selected and which are rejected through the optimization The gate has been modelled by means of the proposed general optimization model for roller and sliding gate structures GATOP GATe OPtimization The self manufacturing costs such as material welding sheet iron cutting and anti corrosion resistant painting as well as transporta tion costs have been accounted for in the economic type of objective function subjected to given design material stress defl ection and stability constraints For real comparison between the obtainedoptimal gate and the actually erected IntakeGate identical economic parameters static system material stress and defl ection allowances as well as all functions from the structural analysis of the Intake Gate have been considered as constraints in the model As standard dimensions are in this chapter not explicitly considered in the optimization standard dimension logical constraints are removed from the model The MINLP model of the Intake Gate for 352S KRAVANJA Z KRAVANJA AND B S BEDENIK Int J Numer Meth Engng 43 329 364 1998 1998 John Wiley y2 m y3 m y1 vNTopology 1y1 M1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1N6 6 6 9Initial topology 2y2 M1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 1 0 1 1 0 0N4 4 5 5 3y3 M1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 0 0N5 5 5 5 4y4 M1 1 1 0 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 0 0N4 5 5 5Optimal topology 5y5 M1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 1 0 0N4 4 4 5 Table VII Convergence to the optimal result of the Intake Gate MINLPMINLPResultho1ho2ho3tsn bwn tfin m iterationsubphase topologymmmm 1Initialization6 6 6 9 1 NLP559566 6 6 92 6672 6672 6670 966 21 MILP476044 4 5 52 3852 6153 0000 922 2 NLP493944 4 5 52 4672 5333 0000 968 32 MILP469845 5 5 52 6672 6672 6670 966 3 NLP495205 5 5 52 6062 6062 7880 942 43 MILP495004 5 5 52 3852 7982 8170 940 4 NLP493874 5 5 52 3852 7782 8370 947 54 MILP501204 4 4 52 5092 6172 8741 011 5 NLP496304 4 4 52 5382 6992 7630 997 continuous sizes contains 1045 in equality constraints 335 continuous and 27 binary 0 1 variables The synthesis of the Intake Gate was carried out by a user friendly version of the MINLP computer package TOP Topology Optimization Program 1 The computer package TOP is a more general version of the computer package PROSYN an MINLP process synthesizer see Kravanja and Grossmann 20 21 TOP is the implementation of many advanced optimization tec