外文翻译--混合整数非线性规划（MINLP）方式的滚子和滑门结构

外文翻译原文 ABSTRACT Part III of this three-part series of papers describes the synthesis of roller and sliding hydraulic steel gate structures performed by the Mixed-Integer Non-linear Programming (MINLP) approach. The MINLP approach enables the determination of the optimal number of gate structural elements (girders, plates), optimal gate geometry, optimal intermediate distances between structural elements and all continuous and standard crossectional sizes. For this purpose, special logical constraints for topology alterations and interconnection relations between the alternative and fixed structural elements are formulated. They have been embedded into a mathematical optimization model for roller and sliding steel gate structures GATOP. GATOP has been developed according to a special MINLP model formulation for mechanical superstructures (MINLP-MS), introduced in Parts I and II. The model contains an economic objective function of self-manufacturing and transportation costs of the gate. As the GATOP model is non-convex and highly non-linear, it is solved by means of the Modified OA/ER algorithm accompanied by the Linked Two-Phase MINLP Strategy, both implemented in the TOP computer code. An example of the synthesis is presented as a comparative design research work of the already erected roller gate, the so-called Intake Gate in Aswan II in Egypt. The optimal result yields 29)4 per cent of net savings when compared to the actual costs of the erected gate. . 1998 John Wiley & Sons, Ltd. KEY WORDS: structural synthesis； optimization； topology optimization； discrete variable optimization； Mixed-Integer Non-linear Programming； MINLP； the Modified 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 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.,1D. Jongeling and Kolkman. as well as Almquist et al. Particular interest was shown in the optimization not of these (roller and sliding gates) but of similar structures. In such investigations, Vanderplaats and Weisshaar. as well as Gurdal et al. optimized stiffened laminated composite panels, Butler. and Ringertz. optimized stiffened panels, Farkas and Jarmai1. optimized welded rectangular cellular plates, Finckenor et al.1. treated skin stringer cylinders and Gendy et al.1. stiffened plates. Almost all authors used Non-linear Programming (NLP) techniques. Gurdal et al. proposed the genetic algorithm, while Kravanja et al.1D. introduced MINLP algorithms and strategies to the simultaneous topology and parameter 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 defined 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 defined design, material, stress, deflection 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 different topology and standard dimension alternatives are postulated 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. 2. Section 3 represents a general MINLP optimization model for roller and sliding gate structures GATOP. 3. Finally, in Section 4, the proposed superstructure MINLP approach is applied to the synthesis of an already erected roller gate, the so-called Intake Gate in Aswan II in Egypt. 2. SUPERSTRUCTURE ALTERNATIVES AND THE MODELLING OF THEIR DISCRETE DECISIONS 2.1. Postulation of topology and standard dimension alternatives The first step in the synthesis of the gate is the generation of an MINLP superstructure in which different topology/structure and standard dimension alternatives are embedded to be selected as the optimal result. The gate superstructure also contains the representation of structural elements which may construct each possible structure alternative as well as different sets of discrete values, defined for each standard dimension alternative. As both the roller and sliding gates have almost the same structure, it was reasonable to propose a superstructure, which could well be useful for both of them. 2.1.1. Topology alternatives The gate superstructure typically includes a representation of main gate elements, where each gate element is composed of various structural elements, such as horizontal girders, vertical girders, stiffeners and plate elements of the skin plate, see Figure 2. The superstructure comprises n main gate elements, n3N, each containing m horizontal girders, m3M, the (3#2v) number of vertical girders through the entire gate, v3., and the corresponding (m!1)(2#2v) number of skin-plate elements. To each mth horizontal girder of the nth main gate element an extra binary variable yn,m is assigned. The number of horizontal girders and corresponding plate elements of the skin plate, distributed over the nth main gate element, can therefore be determined simply by,m n my. Note that the proposed minimal number of identical vertical girders is 3 and that they can take only odd numbers. If a binary variable yv is assigned to each v, v V., the number of vertical girders can be obtained by (3+2 vyv). In the same way an even number (2+2 . vyv) of equal horizontal partitions of the entire gate is proposed. In the case of vertical girders, we can see that the structural elements can also be determined by suitable linear combinations of binary variables. Among the maximal number Mmax of horizontal girders per each main gate element, the upper and lower girders together with the minimal number Mminof intermediate horizontal girders and the adjoining skin-plate elements are treated as fixed structural elements, which are always present in the optimization. All other (Mmax Mmin) intermediate horizontal girders with the corresponding number of adjoined skin-plate elements are then regarded as alternative structural elements, which may either disappear or be selected. Since only alternative structural elements participate in the discrete optimization, the size of the discrete decisions is significantly reduced. Each possible combination between selected alternative structural elements and fixed structural elements forms an extra gate topology/structure alternative. 2.1.2. Standard dimension alternatives Four standard dimensions are additionally defined to represent the standard thicknesses of sheet-iron plates: the thickness of the skin-plate tsn for each nth main gate element, the .flange thickness of the horizontal girder tfn, the web thickness of the outer horizontal girder,outwnmt m=1or m=M ,and the web thickness of the inner horizontal girder int,wnmt,1(m(M. Since the thickness tsn has a common value for the entire skin-plate of the nth main gate element and the tare the same for all horizontal girders of the nth main gate element, i.e. they correspond to the common standard design variables for the superstructure or its part ,st comtd from the special MINLP-MS model formulation in Part II. Similarly, the web thicknesses ,outwnmt, which take a common value for both outer horizontal girders of the nth main gate element, and,innwnmt , which are the same for all the inner horizontal girders, correspond to the common standard design variables ,st comacd of the alternative structural elements. An extra set of discrete values of standard dimension alternatives and a special set of the same size of binary variables are introduced for each mentioned standard dimension. Each standard dimension tsn shall be expressed within the given i standard dimension alternatives, i I, standard dimension tfnby k alternatives, k K, standard dimension ,outwnmtby p alternatives, p P, and standard dimension ,innwnmtby r alternatives, r R. The vector of i binary variables yn,i and the vector of i discrete values qn,I are assigned to the variable tsn, the vectors of k elements yn,k and qn,k to the variable tfn, the vectors of p elements yn,pand qn,kto the variable,innwnmtand the vectors of r elements yn,vand qn,v to the variable ,innwnmtConsequently, the overall vector of binary variables assigned to the gate superstructure is y=yn,m ,y,v,yn,I,yn,p,yn,v 2.2. Modeling of discrete decisions The postulated gate superstructure of topology and standard dimension alternatives can be formulated as an MINLP problem using a special MINLP model formulation (MINLP-MS) for simultaneous topology, parameter and standard dimension optimization of mechanical superstructures, described in Part II. As can be seen from the (MINLP-MS), the objective function is typically subjected to structural analysis and logical constraints. While structural analysis constraints represent a mathematical model of a synthesized structure, logical constraints are used for the explicit modeling of logical decisions. Modeling of discrete decisions to determine topology alternatives is an objective of the highest importance for the synthesis. In order to perform these decisions within the MINLP optimization, interconnection logical constraints Dy+R(d, p) r are proposed. While variables, their bounds and most of the constraints of the MINLP-MS model formulation are represented in Part II, interconnection logical constraints and the constraints defining topology alterations are described in this section. The latter ones are derived from the following basic integer or mixed-integer logical constraints: (a) Multiple choice constraints for selecting among a set of units I: Select exactly M units: . iiIyM (1) Select M units at the most: . iiIyM (2) Select at least M units: . iiIyM (3) (b) If then conditions: if unit k is selected then unit i must be selected: yk -yi 0 (4) (c) Activation or deactivation of continuous variables, inequalities or equations: 1. example to relate continuous variable x to the scalar value U: x=Uy (5) if 1 , 0 0y x u i f y x 2. an opposite relation: X=U（ 1-y） (6) 1 0 , 0y x i f y x u 3. example for the bounds of continuous variable x: x1,0y x xupy (7) if 1 , 01 , 0 0 0upy x x x i f y x (d) Integer cuts constraint eliminates unnecessary integer combinations yk= , 1, .,kiy i m 0,1me.g. those found at previous MINLP iterations: k| B I | - 1k kiii B I i N Iyy (8) where kkiB I |y 1i , kkiN I |y 0i In order to describe the modeling of discrete decisions, a general gate superstructure from Figure 2 is addressed in which the defined structural elements are typically horizontal and vertical girders. As the modeling of vertical girders is simplified and needs no special interconnection logical constraints, the modeling of discrete decisions regarding horizontal girders proved to be more sophisticated. 2.2.1. Modeling of topology alterations Let us consider the vertical cross-section of the gate element superstructure with fixed and alternative horizontal girders, see Figure 3(a). The number of fixed and alternative girders and their locations in the superstructure can be described by the following logical constraints: m i n m a xmmMM y M (9) 1 , 2 , 3 , . . . , 1mmy y m M (10) max 1My (11) Logical constraint (9) defines the minimal (Mmin) and maximal (Mmax) number of structural elements (girders). While number Mmin represents the number of fixed structural elements, the difference between the maximal and minimal number of elements (Mmax-Mmin) gives thenumber of alternative structural elements. Constraint (10) defines the direction of the removal of alternative elements: from the top down the superstructure. From Figure 3 is evident that the most upper element is the fixed one, which is set by the constraint (11). It then further follows from constraint (10) that all the rest fixed elements are located at the bottom of the superstructure. Hence, constraints (9)-(11) represent the explicit model for topology alterations of horizontal girders. 2.2.2. Modeling of interconnection relations Interconnection relations between alternative and fixed structural elements within the superstructure require special attention paid to the structural synthesis performed by the MINLP approach. Interconnection relations either restore the connections between currently selected (existing) structural elements or cancel the relations between currently rejected (disappearing) structural elements. Since MINLP methods optimize the topology and parameters simultaneously, it is necessary to define these interconnection relations in an explicite equational form, so that they can enable interconnections and disconnections between the elements during the optimization process. Special interconnection logical constraints for interconnection relations between the alternative and fixed structural elements have been proposed. They will be embedded into the MINLP optimization model of the gate structure, enabling the latter to thus become self-sufficient for automatic topology and parameter optimization. The modeling of interconnection logical constraints, however, requires additional effort, since most element constraints include functions not only of their own variables but also of the variables belonging to their adjoining structural elements. Such an example is, e.g. the constraint of the moment of inertia In,mof the mth horizontal girder of the nth main gate element (see equation (23) in the following section), which includes the substituted expression (S6) of the skin-plate effective width ,nmsb with the heights (between girders and the sill) hm-1and hm+1of both adjoining girders. The constraints of the mth intermediate horizontal girder are typically functions of three heights: hm-1, hm and hm+1, and two vertical distances between horizontal girders: 1mhd,mhdand1mhd The distance mhdis simply defined by the constraint 1mh m md h h m=2,3 ,M-1 (12) The problem arises if hm+1is not defined when the adjoining upper alternative girder to the mth horizontal girder does not exist. For example, let us consider the third girder in Figure 3(a) which is the uppermost existing intermediate element. In order to define h4so as to fulfil the constraints of girder 3, h4should temporarily become equal to the height of the uppermost fixed girder h6 =hM (Figure 3(b). The main idea is to set all heights of non-existing intermediate girders (girders 4 and 5 in Figure 3(a) to the value of h6by means of the logical constraints .mmuph h md h y , m=2,3, ,M-1 (13) , .mmlow exh h md h y , m=2,3, ,M-1 (14) Note, that constraints (13) and (14) restore the upper muphdand lower ,mlowexhd bounds of the distancemhdwhen the corresponding girder exists (ym=1) and set it to zero, otherwise. When the distance is zero, it follows from constraint (12) that hm becomes equal to h. In this way all distances and heights are defined for any girder that becomes the uppermost selected intermediate one and re-establishes its connection to the uppermost fixed girder. As the uppermost selected intermediate girder is connected to the uppermost fixed girder (e.g. girder 3 to girder 6 in Figure 3(a), the latter should also, in the similar way, be connected to the former one (girder 6 to girder 3 in Figure 3(a). Constraints for the uppermost fixed girder are then just functions of two heights: hM and hM-1and a distance1Mhd . The problem arises if some intermediate girders do not exist, e.g. girders 4 and 5 in Figure 3(a). In such cases, hM-1should not be considered. Instead, the height hS(h3in Figure 3(a) of the upper selected intermediate girder should be defined and substituted for hM-1. The vertical distance dhs of the uppermost selected intermediate girder is then defined by the constraint: hs M sd h h (15) The selection of the height hs among all hm can be performed by the following logical constraints: 11(1 ) .u p u ps m s m s mh h h y h y , m=2,3, ,M-2 (16) 11(1 ) .u p u ps m s m s mh h h y h y , m=2,3, ,M-2 (17) 11(1 )ups M s Mh h h y (18) 11(1 )ups M s Mh h h y (19) Constraints (16) and (17) set hS to the height hm of that mth existing horizontal girder (ym=1), which has the existing adjoining lower girder (ym-1=1) and the disappearing adjoining upper girder (ym+1=0). However, for mM!1 the next upper girder always exists, since it is fixed, i.e. (yM =1). In this case we need additional constraints, i.e. (18) and (19), which set hS to the height hM-1. 3. THE MINLP OPTIMIZATION MODEL FOR ROLLER AND SLIDING HYDRAULIC STEEL GATE STRUCTURESDGATOP An MINLP mathematical optimization model for roller and sliding hydraulic steel gate structures GATOP (GATe OPtimization) has been developed. The model has proven efficient for the synthesis of roller and sliding gates. As an interface for mathematical modeling and data inputs/outputs GAMS (General Algebraic Modeling System) by Brooke et al.14 ,a high-level language has been used. The first version of GATOP was developed to perform NLP optimization problems of fixed gate structures, while the dead weight of the gate structure was considered in the objective function, see Reference 15. The new GATOP version is a much more general one: many alternative horizontal girders, vertical girders and plate elements of the skin-plate are now simultaneously represented in a composite form of the gate superstructure. Thus, the new GATOP model is formulated as an MINLP problem performing gate synthesis. In the process of the development of the GATOP model, the following assumptions and simplifications have been considered: 1. A simplified static system for roller and sliding gates is to be used. It includes independent simply supported horizontal girders that are combined with independent clamped skin-plate elements. Such a static system is convenient for gates in which the horizontal girders are much longer than their intermediate vertical distances. Vertical girders have the same height as horizontal ones. 2. In the above case, the horizontal girders transmit almost all the water load, so that the participation of vertical girders can be neglected. Although the vertical girders are not analysed, they are nevertheless considered as a geometrical and economic fact in the objective function. 3. Only the water load, i.e. the hydrostatic pressure on the skin plate, is taken into consideration, while the dead weight, friction and buoyancy are neglected. 外文翻译译文 摘要 执行的水力 钢门结构综合 非线性规划 方法使能门结构元素 (大梁，板材 )，优选的门几何、结构元素和所有连续和标准 剖面图 大小之间的结构元素和所有连续和标准尺寸的最佳 优选 中间距离。 为此，拓扑结构改变和互联联系的特别逻辑限制在选择和被修理的结构元素之间被公式化。 他们被埋置了入路辗和滑的钢门结构 GATOP 一个数理优化模型。 GATOP 根据已经制定了一个特殊的 MINLP 机械上层 上层 建筑（ MINLP - MS）的第一和第二部分介绍了模型的制定。该模型包含一个自我制造和运输费用的经济大门的目标函数。由于 GATOP 模型非凸，高度非线性的，它是 由链接解决两相的 MINLP 战略，无论是在顶级计算机代码中实现办公自动化陪同的 改性 推理算法的手段。一个综合的例子是作为一个比较设计已经竖立滚子门，所谓的二进水口闸门埃及阿斯旺的研究工作。最优结果产生净储蓄占百分之 二十九点 四的时候相比，在门口竖立的实际成本。 关键词：结构合成，优化，拓扑优化，离散变量优化，混合整数非线性规划 ；的 MINLP；修正 办公 /推理算法 ；MINLP（非线性规划）的 战略液压启闭闸门，推拉门 ；辊闸门 ；阿斯旺 1。简介 本文介绍了混合整数非线性规划（ MINLP）的方式来滚子和滑门 结构，即在垂直升降水工钢闸门合成最简单的类型，见图 1。滚子和闸也被视为最频繁对水源的控制水工钢闸门制造类型。它们用来规管水力发电厂，大坝或溢洪道的水流。 由于水工钢闸门是非常特殊的结构，只有一个已经讨论了一些作者的优化，例如 卡尔文尼伽、蒋格林 和 柯克曼 以及阿尔姆基斯特等人。特别有趣的是表现在没有这些（辊闸），但类似的结构优化。在这种调查， 万德普拉斯 和 维斯哈尔 以及 戈达尔 等 。 加筋复合材料层合板优化 ， 巴特勒和 瑞格提斯 加筋板的优化，法卡斯和 加麻依 优化焊接矩形蜂窝板， 菲克肯 等。斯特林格气瓶和治疗皮肤 茛蒂 等。加劲 板。几乎所有使用的非林程式学（ NLP）技术作家。 茛蒂 等。提出了遗传算法，而 科若文 等 .-推出的 MINLP 算法和拓扑结构的同时和 参数 的闸门优化策略。 图 1。垂直升降水工钢闸门结构 在我对这个文件三个部分组成的系列部分，一个是的 MINLP 方法和参数的同步拓扑结构优化的一般看法是主办。第二部分描述了标准尺寸的同时优化扩展。根据在第一和第二部分所定义的上层建筑的方法，本文件（第三部分）是合成和滑动滚轮的 MINLP 水工钢闸门结构，取得了以最小的成本，受到 明确的 设计，材料，应力的主要目标， 偏转 和稳定性约束。 随着的 MINLP 方法能够同时拓扑结构，参数和标准尺寸的优化，一门结构构件（梁，板），门全球性的几何形状，结构之间的所有连续元素和中间的距离和数量的标准尺寸可同时得到。 这种对论文三部分组成的系列的最后一部分，分为三个主要部分： 1。第 2 节介绍了如何 不同 拓扑结构和标准尺寸的替代品 是假设 以及如何联网关系是由明确的逻辑约束手段，制定以执行在优化过程的拓扑结构和标准尺寸的改变。 2。第 3 代表滚子，滑动闸门结构 GATOP 一般的 MINLP 优化模型。 3。最后，在第 4 节，建议上层建筑的 MINLP 方法应用到一个已经竖立滚筒门 ，所谓的二进水口闸门埃及阿斯旺合成 。 2。上部结构及其替代品和离散决策建模 2.1。推导的拓扑结构和标准尺寸的替代品 在合成大门 的第 一步是一个混合整数非线性规划上层建筑中 不同 拓扑 /结构和标准尺寸的嵌入式替代品被选定为最佳结果的产生。选定为最佳效果。门上层建筑也包含了结构元素可建造每一个可能的结构替代以及 不同 套离散值，每个标准尺寸替代 不同 代表性。但由于该轮和滑动闸门几乎相同的结构，它是合理的建议上层建筑，这很可能是为他们两个非常有用。 2.1.1。拓扑结构的替代品 门上层建筑通常包括一门主要元素，其中每个元素 是门各种结构元素，如梁横向，纵向梁， 加劲 和 面 板块内容，代表组成，见图 2。上层建筑包括 N 迈门元素， n N，每个包含米的水平梁， m M，纵向大梁（ 3+2V）通过整个大门， v V, v3 的数量，以及相应的（ m-1） （ 2+2V）的数目皮板元素。 图 2。门的上层建筑，由三个主要元素门，建造水平和每个包含六纵九梁 对每一个 m 个第 n 个元素的水平梁大门额外二进制变量 yn,m是劲分配。横向梁和相应的 面 板单元板，在第 n 个元素的分布大门，因此被确定数量可以简单地用 mYn,m 表示。 附注 k 劲，该垂直梁相同数量最 少为 3，他们可以只需要奇数。如果一个二进制变量 yv是分配给每个 v， v V 的，纵向大梁可以 用（ 3+2 v yv ）表示。 以同样的方式一个偶数（ 2+2 v yv）的同等水平将得到整个分区的建议。在纵向大梁的情况下，我们可以看到，结构构件，也可确定合适的二元变量的线性组合 。 其中最大数量 Mmax 的水平梁，每门各主要元素，上下梁连同中间水平梁和毗邻的 面 板元素的最小数 Mmin 是当作固定结构的部分，它们总是在优化本。所有其他（ Mmax -Mmin）的中间与邻接 面 横板单元对应的数字，然后把梁结构元素作为替代，这 可能不是消失或者被选中。由于唯一的选择结构元素的离散优化参与，离散决定大小 显著 减少。选定的替代结构之间的每个元素和 固定 结构元素可能组合形式一门额外的拓扑 /结构的选择。 2.1.2。标准尺寸的替代品 四个标准尺寸此外 定义 代表铁皮板的标准厚度： 面 板吨 tsn 每 n 个元素的正门厚度， 水平梁 的 法兰 厚度 tfn，外层的水平梁腹板厚度吨,outwnmt， m=1 或 m=M，和内部横向梁腹板厚度吨,innwnmt， 1mM 自厚度 tsn 有一个对整个 面 的第 n 个元素的大门板共同的价值和分别为第 n 个元素的所有大门梁的水平相同，即它们对应到上层建筑或部分从特殊吨的 MINLP - MS 在第二部分制定模型的通用标准设计变量。同样， 门 页厚度吨,outwnmt， 它 为第 n 个元素外正门水平梁的共同价值，这是所有内部横向梁相同，对应的共同的 ,st comaed可替代的标准设计变量结构元件。一种额外的标准离散值设置一个替代品和一维二元变量的大小相同，介绍了一套特别的标准尺寸为每个提及。 每个标准 尺寸 tsn 应表示在给定的标准尺寸 i 中选择替代尺寸 ， i I，被 K的替代， k K，标准尺寸吨标准尺寸由 P 选择 ， p P 的，由 R 所替代 ， r R。 i 的二进制变量载体 yn,r 和离散值 qn,r 分配给变量载体 tfn时， p 元素 yn,p 和 qn,p的变量载体,outwnmt和 R 元素 的载体 yn,r 和 qn,k 分配给 变量,innwnmt。因此，分配到门上层建筑整体向量变量是 y= yn,m ,y1,v ,yn,I yn,k ,yn,p ,yn,r 2.2。离散决策建模 该拓扑结构和标准尺寸的替代假设门上层建筑可归结为一个混合整数非线性规划 MINLP 模型采用一种特殊配方（的 MINLP - MS）的拓扑结构的同时，参数和机械超标准尺寸的优化结构第二部分所述的问题。由于可以从（的 MINLP - MS）的出现，其目标函数通常受到结构分析和逻辑约束。虽然结构分析制约代表了一个综合结构的数学模型，逻辑约束是用于显式建模的逻辑判断 决定确定的离散拓扑建模是一种替代品为合成最为重要的目标。为了执行内部的 MINLP 优化 这些决定，互连逻辑约束 Dy+R（ d,p） r 是建议。而变量，其范围和对的 MINLP - MS 模型制定的限制大多数是在第二部分，互连逻辑约束和约束丹宁拓扑结构的改变代表介绍本节。后者的是来源于以下基本的整数或混合整数逻辑约束： （一）在我设置的单位选择多个选择的限制： 精确地选择测绘单位： 选择 M 单位至多： 选择至少 M 单位： （二）如果当时条件： 如果单位 k 为单位选定，