【正文】 如果基于其中一個(gè)。) ，零件A周圍和接著零件B周圍參考點(diǎn)所連接而成的軌跡。例如兩個(gè)多邊形，A 和 B， 明可夫斯基和詳細(xì)說(shuō)明了A和B上每一個(gè)頂點(diǎn)的總和。 明可夫斯基和 零件的外形被近似嵌套在每個(gè)多邊形的n 個(gè)頂點(diǎn)上，在CCW方向上有限連續(xù)。對(duì)于排樣設(shè)計(jì)中零件間布局的特殊問題則根據(jù)問題報(bào)告采用增加旋轉(zhuǎn)計(jì)算方法 [7, 16]和模擬退火 [11]， 但是迄今為止并沒有能夠被實(shí)際應(yīng)用的精確的計(jì)算方法。從根本上來(lái)講， 它僅是一種解決位置關(guān)系的方法，這樣的外形有缺陷，但不會(huì)重疊。這些計(jì)算方法基于建筑幾何學(xué)中一個(gè)外形從另外一個(gè)上‘發(fā)展’出來(lái)。當(dāng)解決較復(fù)雜設(shè)計(jì)問題時(shí) (也就是在2D平面上將較多不同零件嵌套在一起)，它不能保證最佳排樣方法，但是可以根據(jù)獲得的計(jì)算結(jié)果進(jìn)而總結(jié)為一個(gè)較好的解決方法。盡管差別很小，但在大批量生產(chǎn)中每個(gè)零件所浪費(fèi)的材料會(huì)累計(jì)進(jìn)而導(dǎo)致較多材料損失。 (由于對(duì)稱)，然后從中選出最佳排樣方法。[7]，在設(shè)計(jì)中決定零件傾斜程度和帶料寬度以及合適的材料利用率。增量旋轉(zhuǎn)法是一種流行的排樣設(shè)計(jì)方法[610, 16]。這種方法適合不相互重疊的矩形[3]、拉深多邊形[4, 5]、已知相互關(guān)聯(lián)的外形[6]。通過計(jì)算機(jī)介紹的設(shè)計(jì)過程所得出的步驟。前期工作曾經(jīng), 帶料排樣設(shè)計(jì)問題需要通過手工來(lái)解決。后與上一工位零件相嵌套，以及兩個(gè)不同零件間的相互嵌套。嵌套法對(duì)于零件之間的布局是一個(gè)重要問題從以往的經(jīng)驗(yàn)來(lái)看零件間相互嵌套常常能夠改善材料的利用率，就像在單一帶料中將每個(gè)零件層疊在一起。此項(xiàng)任務(wù)較為復(fù)雜，盡管如此，在設(shè)計(jì)中改變搭邊值以后能夠改變步距 (帶料中鄰近零件之間的距離) 以及帶料寬度。加工設(shè)計(jì)階段決定材料的利用率，搭邊大小決定(通常很長(zhǎng))工具的壽命。即使微小的數(shù)值的改進(jìn)也能使材料利用率提高。當(dāng)一個(gè)或多個(gè)零件布置在帶料上時(shí)，設(shè)計(jì)者選擇零件的布置方法，以及帶料的寬度，以及多個(gè)零件在一起的情況下，他們的位置關(guān)系。很明顯，使用最理想的排樣設(shè)計(jì)對(duì)于提高公司的競(jìng)爭(zhēng)力是至關(guān)重要的。但材料不能被完全利用到零件上，因?yàn)榱慵灰?guī)則的外形必須被包含在帶料內(nèi)。關(guān)鍵字：沖壓，模具設(shè)計(jì)，最佳化緒論在沖壓生產(chǎn)中，能夠快速生產(chǎn)不同復(fù)雜程度的薄片金屬零件，特別是在大產(chǎn)量的情況下，能夠高強(qiáng)度生產(chǎn)。,或是將兩個(gè)不同的工件嵌套在一起。這種計(jì)算方法可以預(yù)示在帶料中結(jié)構(gòu)廢料的位置及形狀，以及工藝廢料的位置和最佳寬度。Applying this method to the example leads to the optimal translation vector of (, ), giving the strip layout shown in , with a material utilization of %.Interestingly, while it appears that the pairs of parts could be pushed closer together for a better layout, doing so decreases utilization.Optimal Strip Layout for Part A Paired with It Layout Optimization of Different Parts Paired TogetherVery often parts made from the same material are needed in equal quantities, for example, when leftand righthand parts are needed for an assembly. Blanking such parts together can speed production, and can often reduce total material use. This strip layout algorithm can be applied to such a case with equal ease. Consider a second sample part, B, shown in . The relevant Minkowski sum for determining relative position translations, , is shown in . In this case, contains 15 edges, whose utilization values are shown in . Again, multiple local maxima occur while traversing particular edges of . The optimal layout occurs with a translation vector of (, ), shown in , giving a utilization value of %. Strip width is and pitch is in this example. 中文翻譯沖壓中多工件的最佳排樣, 機(jī)械工程系麥克馬斯特大學(xué)漢密爾頓, 安大略湖20001030摘要在沖壓生產(chǎn)中，生產(chǎn)成本受材料利用率影響最大，材料支出占整個(gè)生產(chǎn)成本的75%。)1. Select the relative position of B with respect to A. The Minkowski sum defines the set of feasible relative positions ().2. ‘Join’ A and B at this relative position. Call the bined blank C.3. Nest the bined blank C on a strip using the Minkowski sum with the algorithm given in [14] or [15].4. Repeat steps 13 to span a full range of potential relative positions of A and B. At each potential position, evaluate if a local optima may be present. If so, numerically optimize the relative positions to maximize material uti Layout Optimization of One Part Paired with ItselfThe first step in the above procedure is to select a feasible position of blank B relative to A. This position is defined by translation vector t from the origin to a point on , as shown in . During the optimization process, this translation vector traverses the perimeter of.Relative Part Translation Nodes on , showing Translation Vector t.Initially, a discrete number of nodes are placed on each edge of . The two parts are temporarily ‘joined’ at a relative position described by each of the translation nodes, then the bined blank is evaluated for optimal orientation and strip width using a singlepart layout procedure (., as in [14] or [15]). In this example, consists of 12 edges, each containing 10 nodes, for a total of 120 translation nodes. The position of each node is found via linear interpolation along each edge , where is vertex I on the Minkowski sum with a coordinate of ( , ). Defining a position parameter s such that s = 0 at and s = 1 at , coordinates of each translation node can be found as:(2)(3)If m nodes are placed on each edge, ,the position parameter values for the node, , are found as: (., –A) is slid around A, the reference vertex on –A will trace out the path shown as the heavy line in . This path is the Minkowski sum . Methods for calculating the Minkowski sum can be found in putational geometry texts such as [17, 18]. Sample Part A to be Nested.Minkowski Sum (heavy line) of sample Part (light line).The significance of this is that if the reference vertex on –A is on the perimeter of , A and –A will touch but not overlap. The two blanks are as close as they can be. Thus, for a layout of a pair of blanks with one rotated 180186。 (due to symmetry), the orientation giving the best utilization is selected.The disadvantage of this method is that, in general, the optimal blank orientation will fall between the rotation increments, and will not be found. Although small, this inefficiency per part can accumulate into significant material losses in volume production.Metaheuristic optimization methods have also been applied to the strip layout problem, both simulated annealing [11, 12] and genetic programming [13]. While capable of solving layout problems of great plexity (. many different parts nested together, general 2D nesting of sheets), they are not guaranteed to reach optimal solutions, and may take significant putational effort to converge to a good solution.Exact optimization algorithms have been developed for fitting a single part on a strip where the strip width is predetermined [14] and where it is determined during the layout process [15]. These algorithms are based on a geometric construction in which one shape is ‘grown’ by another shape. Similar versions of this construction are found under the names ‘nofit polygon’, ‘obstacle space’ and ‘Minkowski sum’. Fundamentally, they simplify the process of determining relative positions of shapes such that the shapes touch but do not overlap. Through the use of this construction (in this paper, the particular version used