- Email: [email protected]
Shape optimization of sheet metal parts in a flexible production environment
Journal of Materials Processing Technology 115 (2001) 136±141
Shape optimization of sheet metal parts in a ¯exible production environment U. Dirksen*, N. Austerhoff, F. Maevus, M. Kleiner Chair of Forming Technology, Lehrstuhl fur Umformtechnik, Universitat Dortmund, Baroper Str. 301, 44227 Dortmund, Germany
Abstract The design of complex bending parts can be considered as a cyclic process of evaluating the manufacturing process and the making of design modi®cations. This procedure has to be applied when a complex workpiece cannot be manufactured with the available bending processes in its original design, because a bending sequence free of collisions could not be found or technical limits of the manufacturing process have been reached. Considering the application of many different combinational and ¯exible bending methods, a large number of design modi®cations are possible. Even a product designer with great experience is not able to imagine all those possibilities and therefore fails to use the existing manufacturing potential. In order to prevent the modi®cations from becoming a trial-and-error process, the product designer has to be assisted in the design-cycle. Therein an iterative re®nement of the bent component guides the product designer to a workpiece, which ful®ls all constraints and uses the whole manufacturing potential. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Shape optimization; STEP; Evolutionary algorithms
1. Introduction In recent years, the application of sheet metal parts in production has become more and more important. Parts which were manufactured by cutting or casting in the past are more and more replaced by sheet metal bending [1] parts. For sheet metal bending parts this development has led to increased complexity of their form and in the same way product design and planning of production has become a more complicated and dif®cult task than ever. The design of complex bending parts can be considered as a cyclic process of evaluating the manufacturing process and the making of design modi®cations. This procedure has to be applied when a complex workpiece cannot be manufactured with the available bending processes in its original design, because a bending sequence free of collisions could not be found or technical limits of the manufacturing process have been reached. Considering the application of many different combinational and ¯exible bending methods, a large number of design modi®cations are possible. Even a product designer with life-long experience is not able to imagine all those possibilities and therefore fails to use the existing manufacturing potential. In order to prevent the modi®cations from becoming a trial-and-error process or a random *
Corresponding author. Tel.: 49-231-755-2680; fax: 49-231-755-2489. E-mail address: [email protected] (U. Dirksen).
search, the product designer has to be assisted in the designcycle to optimize the shape and to achieve a ®nal product which can be manufactured with low costs and the required quality. In the following section, the design and the components of such a design-cycle are presented. After an overview of the design-cycle is given the parts belonging to it are described in further detail. 2. Design-cycle The structure of the design-cycle is shown in Fig. 1. As a workpiece example for the better instruction of the several manufacturing problems, a tray for A4 paper is chosen. Its preliminary design is depicted in more detail in Fig. 2. The design-cycle starts with a draft which already considers all functional requirements on the workpiece and a design speci®cation to ensure all these requirements. In the speci®cation, all constraints for dimensions and features have to be de®ned, examples are the allowed maximal tolerances for the width and height of the workpiece or the attribute to be stackable. When the speci®cation is completed, an evaluation of the form of the bent component is carried out to verify whether the workpiece can be manufactured or whether modi®cations are necessary. This evaluation includes the computation of a bending sequence, determination of eventually required handling operations, and the calculation of
0924-0136/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 1 ) 0 0 7 5 3 - 1
U. Dirksen et al. / Journal of Materials Processing Technology 115 (2001) 136±141
Fig. 1. The design-cycle of a sheet metal part.
Fig. 2. Graphical user and tools interface (GUTI) of the design-cycle.
137
138
U. Dirksen et al. / Journal of Materials Processing Technology 115 (2001) 136±141
coordinates for the gauges. In the main, this is done by the CAD-independent software module FBS (®nd bending sequence) [2]. The main purpose of FBS is to search for and ®nd a bending sequence. Apart from that it veri®es and ensures that the bending sequence will not provoke a collision throughout the entire procress, which includes positioning of the workpiece in the tool or press brake as well as the removal of the workpiece and the bending process itself. As bending methods, air bending, two-roll bending, and swivel bending are considered. The input for FBS consists of a description of workpiece, of machine, available tools, and a list of other relevant data. Among the tools, the ¯exible tool system [3,4] developed at the department, holds a key position. It allows the adjustment of the die width within a limited range in order to produce different ®nal geometries without any tool changes. For each bend of the sheet metal part at least one simulation data record exists. For instance, for air bending it indicates the die width, punch displacement, loaded and unloaded bend angle, and the used tools. This technological information is determined by means of the semianalytic process simulations DIBESI (die bending simulation) [3] and SWIBESI (swivel bending simulation) [5] which simulate the forming process taking into account real-plastic material behavior. With a growing number of bending methods and existing tools to be considered, the runtime for the determination of a practicable bending sequence increases a lot in time. In order to reduce the time, the machine learning system MOBAL [6] has been integrated in the evaluation phase. Its primary function is to propose for each bend a bending method and its secondary function is to propose partial bending sequences which may be positive or negative. The knowledge base of the machine learning system is built-up by workpiece examples and is extended while the system is working. For every new designed product, the manufacturing results will be returned to the system and added to the knowledge base. The more workpiece examples are included, the more valid are the proposals returned from MOBAL and the time for the evaluation of the sheet metal part is shortened. After the evaluation, it is known if the workpiece is bendable or not. In the latter case, a design optimizer is started to generate alternative shapes, which ful®l the requirements laid down in the design speci®cation. The variants are re-evaluated to determine if the workpiece is now producible and the manufacturing requirements are assured, or if the design-cycle has to be continued. In this case, further modi®cations are necessary and the design optimizer is provided with information from the evaluation process whether the evaluated variant has been an improvement or a change for the worse. The described cycle is continued until a product has been found which ful®lls all requirements. To assist the human product designer, an interactive graphical user and tools interface (GUTI, Fig. 2) is located in the center of the design-cycle. Therein, the sheet metal
part is de®ned or can be loaded from a STEP ®le, the design speci®cation is done and alternative workpiece shapes, interim solutions of the optimization process, and the virtual manufacturing environment are visualized. 3. Product data representation The above-described design-cycle consists of many different components which need all data belonging to the product. Fig. 3 shows a product-oriented survey of this data. Beside geometry and topological data it also contains planning and manufacturing data. It has to be emphasized that every design-cycle component has read access to this data as well as modify access. In order to meet these demands a central data base is needed in which the data is stored and is accessible both for commercial and individually developed software. This demand has been realized and in order to meet the above-mentioned requirement and others the international standard ISO 10303 STEP (standard for the exchange of product model data) has been started. This standard de®nes data structures called generic resources for all kind of product data and combines them to application protocols which belong to special application contexts such as automotive or shipping industries. Up to the present day, an application protocol (AP) has not been de®ned for sheet metal bending. Because the creation
Fig. 3. A product oriented data representation.
U. Dirksen et al. / Journal of Materials Processing Technology 115 (2001) 136±141
139
Fig. 4. The application reference model of application protocol 207.
of a new AP is very complex and time-consuming, we have chosen the approach to adjust an existing AP. An examination of the available APs showed that AP 207 sheet metal die planning and design is well suited as a base for an AP for sheet metal bending. In Fig. 4, a simpli®ed version of the application resource model of AP 207 is shown. With this model the workpiece can be de®ned in different geometry models in order to get different abstraction levels of the workpiece. Furthermore, the de®nition of relationships from the shape de®nition to item de®nitions allows access to product data through the geometry. In addition to that version management has been made available which is important for the storage of interesting workpiece variants for later use. Last but not least, the storage of forming stage data is possible. Modi®cations of the AP 207 have to be done for describing process data like workpiece handling or machine data and manufacturing limits. Because STEP is just capable of handling product data it is not possible to store all the tool, machine, and material data which was needed to manufacture the sheet metal part. This data is stored in a relational database and is accessed via references incorporated in the new AP.
have to be applied. For this purpose a design optimizer will be developed, the major features of which are given by:
4. Design optimizer
The ®rst item describes what are ®xed and variable defaults for dimensions of the sheet metal part while the second describes requirements such as parallelism, orthogonality and symmetry between geometrical, and topological
If the preliminary draft of the bent component is not bendable at all design modi®cations to the workpiece shape
a design specification language; criteria for a comparison of the original workpiece and the computed variants; a shape optimization algorithm. The design speci®cation language is used to identify and specify the characteristics of the sheet metal part. As primary characteristics of a sheet metal part, the geometrical, topological, feature-based and technological data, tolerances, and material are considered. These data are generally speci®ed inside a CAD system and can be loaded by the GUTI when needed. Independent from the actual form of the draft are restrictions based on the design requirements and the function of the workpiece. They de®ne the tolerable modi®cations to the workpiece shape and specify which defaults are obligatory and which limits have to be taken into account. In the main, the following restrictions have been de®ned: defaults for dimensions and bend radii; relationships between bends and outer edges, legs of the bent component, features, and planes in the three-dimensional space.
140
U. Dirksen et al. / Journal of Materials Processing Technology 115 (2001) 136±141
elements. The main characteristics of the tray for A4 papers (Fig. 2) are the following: the minimum dimensions for the A4 paper format; the demand that the trays can be piled up one onto another; the restriction that the piled-up trays are not allowed to move relative to each other once they are in the stack. These characteristics have to be expressed in terms of the restrictions described above. This is accomplished by selecting elements from the workpiece in the GUTI and assigning the corresponding properties from menu-oriented dialogues. The selected properties will then be converted to the design speci®cation language which is realized in the logic-oriented programming language Prolog. Therein, a set of predicates is de®ned for declaring dimension restrictions and de®ning relationships. These predicates can be combined if more complex and abstract primitives shall be realized and allows the user to adjust the speci®cation language to his/her own application context. First, only a few restrictions for the design speci®cation should be prescribed, so that in a ®rst iteration step many variants will be generated. Then the new variants are visualized in the GUTI, so that the product designer is able to verify if they are considered to be valid or if the design speci®cation has to be re®ned. Fig. 5 displays a variant of the tray which is invalid. As it is indicated in the picture its side walls are not orthogonal to the base which is a clear contradiction to the requirement that trays should be stackable. By adding the restriction that the side walls have to be orthogonal to the base, this variant will not appear as a suggestion in further cycles. Apart from the described design speci®cation, the comparison of the original design with the generated variants is essential for the successful use of the design optimizer. For the preliminary draft the design optimizer just takes into account the collision data and the manufacturing limits while for the variants also the information whether the modi®cation has been an improvement or not is included. The main characteristics of the collision data are the following: the time of the occurrence of the first collision; number of collisions; degree of collision. The main objective in searching a valid design is to increase the number of bendings free of collisions or to
Fig. 5. A variant which may be generated by the design optimizer because of an incomplete design specification.
minimize the number of collisions. If a collision occurs the degree of collision has to be known for the evaluation of the shape modi®cation. A decreasing degree means an improvement, an increasing degree a deterioration of the previous design. This comparison is important for the optimization algorithm which design is the most demanding part of the design optimizer. The compliance with the above-de®ned restrictions often results in the necessity to modify more than one part of the geometry or topology in the same step. The optimization algorithm should take into account the design speci®cation to avoid invalid designs. Because the number of bends of the bent component is changeable, the number of the design variables to be considered by the optimization algorithm are changing as well. Another problem is the large number of possible design modi®cations. The successfully choice of one out of them can become a very complicated task. The application of evolutionary algorithms (EAs) as optimization algorithms is subject of a research project ®nanced by the DFG (German Research Association). EAs realize a directed search and re¯ect the robustness, ef®ciency, and ¯exibility of the natural evolution in computer applications. In Refs. [7±10], the application of EAs in shape and layout optimization has been successfully applied. As advantages of EA, Nissen [11] and Hammel and BaÈck [10] summarize the applicability for complex search spaces, the robust behavior when multicriteria, non-continuous, and non-deterministic functions are used and the limited requirements on the objective function. 5. Conclusion The presented design-cycle assists product designers in the design of complex sheet metal bending parts in a ¯exible production environment which is characterized by bending methods, such as air bending, two-roll bending, and swivel bending. The iterative re®nement of the bent component in the proposed design-cycle guides the product designer to a workpiece, which is bendable, which meets all the speci®ed constraints, and which uses better the available manufacturing potential. References [1] M. Munk, Einstieg in die 3D-Welt, in: Wirtschaftliche Blechumformung durch simulation, EuropaÈische Forschungsgesellschaft fuÈr Blechverarbeitung e. V., Vol. T19, 1999. [2] U. Dirksen, Effiziente Bestimmung von Biegefolgen fuÈr Gesenkbiegeteile, Master's Thesis, University of Dortmund, 1997. [3] H. Sulaiman, Erweiterung der Einsetzbarkeit von Gesenkbiegepressen durch die Ent-wicklung von Sonderwerkzeugen, Dr.-Ing. Dissertation, University of Dortmund, 1995. [4] B. Heller, Realisierung und Erprobung einer online-RuÈckfederungskorrektur fuÈr das flexible Freibiegen von Feinblechen, Internal Report, Chair of Forming Technology, University of Dortmund, 1995.
U. Dirksen et al. / Journal of Materials Processing Technology 115 (2001) 136±141 [5] R. Warstat, Optimierung der ProduktqualitaÈt und Steigerung der FlexibilitaÈt beim CNC-Schwenkbiegen, Dr.-Ing. Dissertation, University of Dortmund, 1995. [6] Y. Rusli, Einsatz von Maschinellem Lernen in der Umformtechnik am Beispiel der Biegeumformung, Master's Thesis, Chair of Forming Technology, University of Dortmund, 1996. [7] J. Cai, G. Thierauf, G. de Wendt, Optimierung von Stahlgittermasten mit Evolutionsstrategien, Stahlbau 67 (3) (1998) 3, pp. S183-S190. [8] K. Trint, Strukturoptimierung mit geschachtelten evolutionsstrate-
141
gien, Dr.-Ing. Dissertation, Technische UniversitaÈt, Berlin, 1997. [9] T. Nguyen, T. Huang, Advances in Genetic Programming: Evolable 3D Modeling for Model-based Object Recognition Systems, A Bradford Book, MIT Press, Cambridge, MA, 1994, pp. S459±S475. [10] U. Hammel, T. BaÈck, Optimierung in der simulation: evolutionaÈre algorithmen. Technical Report 38/98, DFG-Sonderforschungsbereich 531, 1998. [11] V. Nissen, EinfuÈhrung in evolutionaÈre algorithmen, Computational Intelligence, Viesweg, 1997.
Shape optimization of sheet metal parts in a ¯exible production environment U. Dirksen*, N. Austerhoff, F. Maevus, M. Kleiner Chair of Forming Technology, Lehrstuhl fur Umformtechnik, Universitat Dortmund, Baroper Str. 301, 44227 Dortmund, Germany
Abstract The design of complex bending parts can be considered as a cyclic process of evaluating the manufacturing process and the making of design modi®cations. This procedure has to be applied when a complex workpiece cannot be manufactured with the available bending processes in its original design, because a bending sequence free of collisions could not be found or technical limits of the manufacturing process have been reached. Considering the application of many different combinational and ¯exible bending methods, a large number of design modi®cations are possible. Even a product designer with great experience is not able to imagine all those possibilities and therefore fails to use the existing manufacturing potential. In order to prevent the modi®cations from becoming a trial-and-error process, the product designer has to be assisted in the design-cycle. Therein an iterative re®nement of the bent component guides the product designer to a workpiece, which ful®ls all constraints and uses the whole manufacturing potential. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Shape optimization; STEP; Evolutionary algorithms
1. Introduction In recent years, the application of sheet metal parts in production has become more and more important. Parts which were manufactured by cutting or casting in the past are more and more replaced by sheet metal bending [1] parts. For sheet metal bending parts this development has led to increased complexity of their form and in the same way product design and planning of production has become a more complicated and dif®cult task than ever. The design of complex bending parts can be considered as a cyclic process of evaluating the manufacturing process and the making of design modi®cations. This procedure has to be applied when a complex workpiece cannot be manufactured with the available bending processes in its original design, because a bending sequence free of collisions could not be found or technical limits of the manufacturing process have been reached. Considering the application of many different combinational and ¯exible bending methods, a large number of design modi®cations are possible. Even a product designer with life-long experience is not able to imagine all those possibilities and therefore fails to use the existing manufacturing potential. In order to prevent the modi®cations from becoming a trial-and-error process or a random *
Corresponding author. Tel.: 49-231-755-2680; fax: 49-231-755-2489. E-mail address: [email protected] (U. Dirksen).
search, the product designer has to be assisted in the designcycle to optimize the shape and to achieve a ®nal product which can be manufactured with low costs and the required quality. In the following section, the design and the components of such a design-cycle are presented. After an overview of the design-cycle is given the parts belonging to it are described in further detail. 2. Design-cycle The structure of the design-cycle is shown in Fig. 1. As a workpiece example for the better instruction of the several manufacturing problems, a tray for A4 paper is chosen. Its preliminary design is depicted in more detail in Fig. 2. The design-cycle starts with a draft which already considers all functional requirements on the workpiece and a design speci®cation to ensure all these requirements. In the speci®cation, all constraints for dimensions and features have to be de®ned, examples are the allowed maximal tolerances for the width and height of the workpiece or the attribute to be stackable. When the speci®cation is completed, an evaluation of the form of the bent component is carried out to verify whether the workpiece can be manufactured or whether modi®cations are necessary. This evaluation includes the computation of a bending sequence, determination of eventually required handling operations, and the calculation of
0924-0136/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 1 ) 0 0 7 5 3 - 1
U. Dirksen et al. / Journal of Materials Processing Technology 115 (2001) 136±141
Fig. 1. The design-cycle of a sheet metal part.
Fig. 2. Graphical user and tools interface (GUTI) of the design-cycle.
137
138
U. Dirksen et al. / Journal of Materials Processing Technology 115 (2001) 136±141
coordinates for the gauges. In the main, this is done by the CAD-independent software module FBS (®nd bending sequence) [2]. The main purpose of FBS is to search for and ®nd a bending sequence. Apart from that it veri®es and ensures that the bending sequence will not provoke a collision throughout the entire procress, which includes positioning of the workpiece in the tool or press brake as well as the removal of the workpiece and the bending process itself. As bending methods, air bending, two-roll bending, and swivel bending are considered. The input for FBS consists of a description of workpiece, of machine, available tools, and a list of other relevant data. Among the tools, the ¯exible tool system [3,4] developed at the department, holds a key position. It allows the adjustment of the die width within a limited range in order to produce different ®nal geometries without any tool changes. For each bend of the sheet metal part at least one simulation data record exists. For instance, for air bending it indicates the die width, punch displacement, loaded and unloaded bend angle, and the used tools. This technological information is determined by means of the semianalytic process simulations DIBESI (die bending simulation) [3] and SWIBESI (swivel bending simulation) [5] which simulate the forming process taking into account real-plastic material behavior. With a growing number of bending methods and existing tools to be considered, the runtime for the determination of a practicable bending sequence increases a lot in time. In order to reduce the time, the machine learning system MOBAL [6] has been integrated in the evaluation phase. Its primary function is to propose for each bend a bending method and its secondary function is to propose partial bending sequences which may be positive or negative. The knowledge base of the machine learning system is built-up by workpiece examples and is extended while the system is working. For every new designed product, the manufacturing results will be returned to the system and added to the knowledge base. The more workpiece examples are included, the more valid are the proposals returned from MOBAL and the time for the evaluation of the sheet metal part is shortened. After the evaluation, it is known if the workpiece is bendable or not. In the latter case, a design optimizer is started to generate alternative shapes, which ful®l the requirements laid down in the design speci®cation. The variants are re-evaluated to determine if the workpiece is now producible and the manufacturing requirements are assured, or if the design-cycle has to be continued. In this case, further modi®cations are necessary and the design optimizer is provided with information from the evaluation process whether the evaluated variant has been an improvement or a change for the worse. The described cycle is continued until a product has been found which ful®lls all requirements. To assist the human product designer, an interactive graphical user and tools interface (GUTI, Fig. 2) is located in the center of the design-cycle. Therein, the sheet metal
part is de®ned or can be loaded from a STEP ®le, the design speci®cation is done and alternative workpiece shapes, interim solutions of the optimization process, and the virtual manufacturing environment are visualized. 3. Product data representation The above-described design-cycle consists of many different components which need all data belonging to the product. Fig. 3 shows a product-oriented survey of this data. Beside geometry and topological data it also contains planning and manufacturing data. It has to be emphasized that every design-cycle component has read access to this data as well as modify access. In order to meet these demands a central data base is needed in which the data is stored and is accessible both for commercial and individually developed software. This demand has been realized and in order to meet the above-mentioned requirement and others the international standard ISO 10303 STEP (standard for the exchange of product model data) has been started. This standard de®nes data structures called generic resources for all kind of product data and combines them to application protocols which belong to special application contexts such as automotive or shipping industries. Up to the present day, an application protocol (AP) has not been de®ned for sheet metal bending. Because the creation
Fig. 3. A product oriented data representation.
U. Dirksen et al. / Journal of Materials Processing Technology 115 (2001) 136±141
139
Fig. 4. The application reference model of application protocol 207.
of a new AP is very complex and time-consuming, we have chosen the approach to adjust an existing AP. An examination of the available APs showed that AP 207 sheet metal die planning and design is well suited as a base for an AP for sheet metal bending. In Fig. 4, a simpli®ed version of the application resource model of AP 207 is shown. With this model the workpiece can be de®ned in different geometry models in order to get different abstraction levels of the workpiece. Furthermore, the de®nition of relationships from the shape de®nition to item de®nitions allows access to product data through the geometry. In addition to that version management has been made available which is important for the storage of interesting workpiece variants for later use. Last but not least, the storage of forming stage data is possible. Modi®cations of the AP 207 have to be done for describing process data like workpiece handling or machine data and manufacturing limits. Because STEP is just capable of handling product data it is not possible to store all the tool, machine, and material data which was needed to manufacture the sheet metal part. This data is stored in a relational database and is accessed via references incorporated in the new AP.
have to be applied. For this purpose a design optimizer will be developed, the major features of which are given by:
4. Design optimizer
The ®rst item describes what are ®xed and variable defaults for dimensions of the sheet metal part while the second describes requirements such as parallelism, orthogonality and symmetry between geometrical, and topological
If the preliminary draft of the bent component is not bendable at all design modi®cations to the workpiece shape
a design specification language; criteria for a comparison of the original workpiece and the computed variants; a shape optimization algorithm. The design speci®cation language is used to identify and specify the characteristics of the sheet metal part. As primary characteristics of a sheet metal part, the geometrical, topological, feature-based and technological data, tolerances, and material are considered. These data are generally speci®ed inside a CAD system and can be loaded by the GUTI when needed. Independent from the actual form of the draft are restrictions based on the design requirements and the function of the workpiece. They de®ne the tolerable modi®cations to the workpiece shape and specify which defaults are obligatory and which limits have to be taken into account. In the main, the following restrictions have been de®ned: defaults for dimensions and bend radii; relationships between bends and outer edges, legs of the bent component, features, and planes in the three-dimensional space.
140
U. Dirksen et al. / Journal of Materials Processing Technology 115 (2001) 136±141
elements. The main characteristics of the tray for A4 papers (Fig. 2) are the following: the minimum dimensions for the A4 paper format; the demand that the trays can be piled up one onto another; the restriction that the piled-up trays are not allowed to move relative to each other once they are in the stack. These characteristics have to be expressed in terms of the restrictions described above. This is accomplished by selecting elements from the workpiece in the GUTI and assigning the corresponding properties from menu-oriented dialogues. The selected properties will then be converted to the design speci®cation language which is realized in the logic-oriented programming language Prolog. Therein, a set of predicates is de®ned for declaring dimension restrictions and de®ning relationships. These predicates can be combined if more complex and abstract primitives shall be realized and allows the user to adjust the speci®cation language to his/her own application context. First, only a few restrictions for the design speci®cation should be prescribed, so that in a ®rst iteration step many variants will be generated. Then the new variants are visualized in the GUTI, so that the product designer is able to verify if they are considered to be valid or if the design speci®cation has to be re®ned. Fig. 5 displays a variant of the tray which is invalid. As it is indicated in the picture its side walls are not orthogonal to the base which is a clear contradiction to the requirement that trays should be stackable. By adding the restriction that the side walls have to be orthogonal to the base, this variant will not appear as a suggestion in further cycles. Apart from the described design speci®cation, the comparison of the original design with the generated variants is essential for the successful use of the design optimizer. For the preliminary draft the design optimizer just takes into account the collision data and the manufacturing limits while for the variants also the information whether the modi®cation has been an improvement or not is included. The main characteristics of the collision data are the following: the time of the occurrence of the first collision; number of collisions; degree of collision. The main objective in searching a valid design is to increase the number of bendings free of collisions or to
Fig. 5. A variant which may be generated by the design optimizer because of an incomplete design specification.
minimize the number of collisions. If a collision occurs the degree of collision has to be known for the evaluation of the shape modi®cation. A decreasing degree means an improvement, an increasing degree a deterioration of the previous design. This comparison is important for the optimization algorithm which design is the most demanding part of the design optimizer. The compliance with the above-de®ned restrictions often results in the necessity to modify more than one part of the geometry or topology in the same step. The optimization algorithm should take into account the design speci®cation to avoid invalid designs. Because the number of bends of the bent component is changeable, the number of the design variables to be considered by the optimization algorithm are changing as well. Another problem is the large number of possible design modi®cations. The successfully choice of one out of them can become a very complicated task. The application of evolutionary algorithms (EAs) as optimization algorithms is subject of a research project ®nanced by the DFG (German Research Association). EAs realize a directed search and re¯ect the robustness, ef®ciency, and ¯exibility of the natural evolution in computer applications. In Refs. [7±10], the application of EAs in shape and layout optimization has been successfully applied. As advantages of EA, Nissen [11] and Hammel and BaÈck [10] summarize the applicability for complex search spaces, the robust behavior when multicriteria, non-continuous, and non-deterministic functions are used and the limited requirements on the objective function. 5. Conclusion The presented design-cycle assists product designers in the design of complex sheet metal bending parts in a ¯exible production environment which is characterized by bending methods, such as air bending, two-roll bending, and swivel bending. The iterative re®nement of the bent component in the proposed design-cycle guides the product designer to a workpiece, which is bendable, which meets all the speci®ed constraints, and which uses better the available manufacturing potential. References [1] M. Munk, Einstieg in die 3D-Welt, in: Wirtschaftliche Blechumformung durch simulation, EuropaÈische Forschungsgesellschaft fuÈr Blechverarbeitung e. V., Vol. T19, 1999. [2] U. Dirksen, Effiziente Bestimmung von Biegefolgen fuÈr Gesenkbiegeteile, Master's Thesis, University of Dortmund, 1997. [3] H. Sulaiman, Erweiterung der Einsetzbarkeit von Gesenkbiegepressen durch die Ent-wicklung von Sonderwerkzeugen, Dr.-Ing. Dissertation, University of Dortmund, 1995. [4] B. Heller, Realisierung und Erprobung einer online-RuÈckfederungskorrektur fuÈr das flexible Freibiegen von Feinblechen, Internal Report, Chair of Forming Technology, University of Dortmund, 1995.
U. Dirksen et al. / Journal of Materials Processing Technology 115 (2001) 136±141 [5] R. Warstat, Optimierung der ProduktqualitaÈt und Steigerung der FlexibilitaÈt beim CNC-Schwenkbiegen, Dr.-Ing. Dissertation, University of Dortmund, 1995. [6] Y. Rusli, Einsatz von Maschinellem Lernen in der Umformtechnik am Beispiel der Biegeumformung, Master's Thesis, Chair of Forming Technology, University of Dortmund, 1996. [7] J. Cai, G. Thierauf, G. de Wendt, Optimierung von Stahlgittermasten mit Evolutionsstrategien, Stahlbau 67 (3) (1998) 3, pp. S183-S190. [8] K. Trint, Strukturoptimierung mit geschachtelten evolutionsstrate-
141
gien, Dr.-Ing. Dissertation, Technische UniversitaÈt, Berlin, 1997. [9] T. Nguyen, T. Huang, Advances in Genetic Programming: Evolable 3D Modeling for Model-based Object Recognition Systems, A Bradford Book, MIT Press, Cambridge, MA, 1994, pp. S459±S475. [10] U. Hammel, T. BaÈck, Optimierung in der simulation: evolutionaÈre algorithmen. Technical Report 38/98, DFG-Sonderforschungsbereich 531, 1998. [11] V. Nissen, EinfuÈhrung in evolutionaÈre algorithmen, Computational Intelligence, Viesweg, 1997.