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1、Stamping Die Strip Optimization for Paired PartsV.Vamanu, T.j.NyeMechanical Engineering DepartmentMcMaster UniversityHamilton, OntarioOctober 30, 2000AbstractIn stamping, operating cost are dominated by raw material costs, which can typically reach 75% of total costs in a stamping facility. In this

2、paper, a new algorithm is described that determines stamping strip layouts for pairs of parts such that the layout optimizes material utilization efficiency. This algorithm predicts the jointly-optimal blank orientation on the strip, relative positions of the paired blanks and the optimum width for

3、the strip. Examples are given for pairing the same parts together with one rotated 180, and for pairs of different parts nested together. This algorithm is ideally suited for incorporation into die design CAE systems.Keywords: Stamping, Die Design, Optimization, Material Utilization, Minkowski Sum,

4、Design ToolsIntroductionIn stamping, sheet metal parts of various levels of complexity are produced rapidly, often in very high volumes, using hard tooling. The production process operates efficiently, and material costs can typically represent 75% of total operating costs in a stamping facility 1.

5、Not all of this material is used in the parts, however, due to the need to trim scrap material from around irregularly-shaped parts. The amount of scrap produced is directly related to the efficiency of the stamping strip layout. Clearly, using optimal strip layouts is crucial to a stamping firms co

6、mpetitiveness.The degree of this trim loss is determined at the tooling design stage when the strip layout is created. As a part or parts are laid out on the strip, the designer chooses the orientation of the part(s), width of the strip, and, in the case of multiple parts blanked together, their rel

7、ative positions. Ideally, the material utilization is maximized. The value of even tiny improvements in material utilization can be great; for example, in a stamping operation running at 200 strokes per minute, a savings of just 10 grams of material per part will accumulate into a savings of more th

8、an a tonne of raw material per eight-hour shift. The material utilization is set during the tooling design stage, and remains fixed for the (usually long) life of the tool. Thus, there is significant value in determining the optimal strip layout before tooling is built.This task is complicated, howe

9、ver, since changing each variable in the layout can change both the pitch (distance along the strip between adjacent parts) and strip width simultaneously. Evaluating layout efficiency manually is extremely challenging, and while exact optimal algorithms have been described for the layout of a singl

10、e part on a strip, so far only approximate algorithms have been available for the layout of pairs of parts together. Nesting solutions for pairs of parts is an important problem since it is empirically known that nesting pairs of parts can often improve material utilization compared to nesting each

11、part on a separate strip. This paper addresses the common cases in which a given part is nested with a second copy of itself rotated at 180, and when two different parts are nested together. In this paper we describe a new algorithm that provides the optimal strip layout for these two cases.Previous

12、 WorkOriginally, strip layout problems were solved manually, for example, by cutting blanks from cardboard and manipulating them to obtain a good layout. The introduction of computers into the design process led to algorithmic approaches. Perhaps the first was to fit blanks into rectangles, then fit

13、 the rectangles along the strip2. Variations of this approach have involved fitting blanks into non-overlapping composites of rectangles 3, convex polygons 4,5 and known interlocking shapes6. A fundamental limitation exists with this approach, however, in that the enclosing shape adds material to th

14、e blank that cannot be removed later during the layout process. This added material may prevent optimal layouts from being found.A popular approach to performing strip layout is the incremental rotation algorithm 6-10, 16. In it, the blank, or blanks, are rotated by a fixed amount, such as 27, the p

15、itch and width of the layout determined and the material utilization calculated. After repeating these steps through a total rotation of 180 (due to symmetry), the orientation giving the best utilization is selected. The disadvantage of this method is that, in general, the optimal blank orientation

16、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.Meta-heuristic optimization methods have also been applied to the strip layout problem, both simulated annealing 11, 12 an

17、d genetic programming 13. While capable of solving layout problems of great complexity (i.e. many different parts nested together, general 2-D nesting of sheets), they are not guaranteed to reach optimal solutions, and may take significant computational effort to converge to a good solution.Exact op

18、timization 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 t

19、his construction are found under the names no-fit 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

20、 used is the Minkowski sum), efficient algorithms can be created that find the globally optimal strip layout.For the particular problem of strip layout for pairs of parts, results have been reported using the incremental rotation algorithm 7, 16 and simulated annealing 11, but so far no exact algori

21、thm has been available. In what follows, the Minkowski sum and its application to strip layout is briefly introduced, and its extension to nesting pairs of parts is described.The Minkowski SumThe shape of blanks to be nested is approximated as a polygon with n vertices, numbered consecutively in the

22、 CCW direction. As the number of vertices increases, curved edges on the blank can be approximated to any desired accuracy. Given two polygons, A and B, the Minkowski sum is defined as the summation of each point in A with each point in B,(1)Intuitively, one can think of this process as growing shap

23、e A by shape B, or by sliding shape B (i.e., B rotated 180) around A and following the trace of some reference point on B. For example, Fig.1 shows an example blank A. If a reference vertex is chosen at (0, 0), and a copy of the blank rotated 180 (i.e., A) is slid around A, the reference vertex on A

24、 will trace out the path shown as the heavy line in Fig.2. This path is the Minkowski sum . Methods for calculating the Minkowski sum can be found in computational geometry texts such as 17, 18. Sample Part A to be Nested.Minkowski Sum (heavy line) of sample Part (light line).The significance of thi

25、s 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 180 relative to the other, defines all feasible relative positions between the pair of blanks. A cor

26、ollary of this property is that if the Minkowski sum of a single part is calculated. With its negative, i.e.,. (A complete explanation of these properties of the Minkowski sum is given in 15.) These observations were the basis for the algorithm for optimally nesting a single part on a strip.The situ

27、ation when nesting pairs of parts is more complex, since not only do the optimal orientations of the blanks and the strip width need to be determined, but the optimal relative position of the two blanks needs to be determined as well. To solve this problem, an iterative algorithm is suggested:Given:

28、 Blanks A and B (where B=A when a blank is paired with itself at 180)1. Select the relative position of B with respect to A. The Minkowski sum defines the set of feasible relative positions (Fig.2).2. Join A and B at this relative position. Call the combined blank C.3. Nest the combined blank C on a

29、 strip using the Minkowski sum with the algorithm given in 14 or 15.4. Repeat steps 1-3 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 util

30、ization.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 Fig.3. During the optimization process, this translat

31、ion 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 combined blank is

32、evaluated for optimal orientation and strip width using a single-part layout procedure (e.g., 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

33、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:(4)Calculating the utiliza

34、tion at each of the 120 nodes on Fig.3 gives the results shown in Fig.4. In this figure, the curve is broken as the translation vector passes the end of each edge of to show how utilization can change during the traversal of each edge. While some edge traversals show monotonic changes in utilization

35、, others show two or even three local maxima. Discovering these local optima is the reason why a number of translation nodes are needed.Optimal Material Utilization for Various Translations Between Polygons A and A.As a progression is made around, when local maxima are indicated, a numerical optimiz

36、ation technique is invoked. Since derivatives of the utilization function are not available(without additional computational effort),an interval-halvingApproach was taken 19. The initial interval consists of the nodes bordering the indicated local maximal point. Three equally-spaced points are place

37、d across this interval (i.e. at 1/4, 1/2 and 3/4 positions), and the utilization at each is calculated. By comparing the utilization values at each point, a decision can be made as to which half of the interval is dropped from consideration and the process is repeated. This continues until the desir

38、ed accuracy is obtained.Applying this method to the example leads to the optimal translation vector of (747.894, 250.884), giving the strip layout shown in Fig.5, with a material utilization of 92.02%.Interestingly, while it appears that the pairs of parts could be pushed closer together for a bette

39、r layout, doing so decreases utilization.Optimal Strip Layout for Part A Paired with Itself.Layout Optimization of Different Parts Paired TogetherVery often parts made from the same material are needed in equal quantities, for example, when left-and right-hand parts are needed for an assembly. Blank

40、ing 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 Fig.6. The relevant Minkowski sum for determining relative position translations, is shown in F

41、ig.7. In this case, contains 15 edges, whose utilization values are shown in Fig.8. Again, multiple local maxima occur while traversing particular edges of. The optimal layout occurs with a translation vector of (901.214, 130.314), shown in Fig.9, giving a utilization value of 85.32%. Strip width is

42、 1229.74 and pitch is 1390.00 in this example.Sample Part B to be Nested.Minkowski Sum (heavy line) of Sample Parts (light and dashed lines).Optimal Material Utilization for Various Translations Between Polygons A and B.ConclusionsIn the stamping operation, production costs are dominated by material

43、 costs, so even tiny per-part gains in material utilization are worth pursing. This paper has presented a new algorithm for creating optimal strip layouts for pairs of parts nested together. This algorithm takes advantage of the Minkowski sum calculation to both find feasible relative positions betw

44、een the pairs of parts, and to determine the optimal orientation and strip width for the strip layout.When evaluating combinations of layouts, it should be kept in mind that all permutations should be considered. For example, the strip layout process for the sample parts in this paper would consider

45、 strip layouts for A alone, A paired with itself, B alone, B paired with itself, and A and B paired together. The designer would then consider total raw material costs, tooling construction costs and press operating costs since blanking parts together requires larger tools and presses and changes pr

46、oduction rates.There are opportunities to extend this algorithm, as well. One obvious extension is to include optimization over relative rotations between the pairs of parts, i.e., changing the orientation of part B relative to A on the strip. A second opportunity is to study the utilization functio

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