Abstract
This paper presents two approaches for the automated layout of threedimensional objects in space. The goal is to achieve high packing densities and fitting of objects in predefined design spaces while satisfying technological side constraints. The focus is on small-sized problem instances (up to 20 objects) with complex, possibly non-convex shapes. Linear programming methods form the common ground of our approaches.
The first approach is a global optimization algorithm based on the branch-and- bound paradigm. We introduce a discretization of the configuration space of all possible arrangements which facilitates a complete enumeration of solutions. The bounding procedure then allows for a drastic reduction of the search space. We use a limited number of discrete object orientations within this method.
The second approach is a local optimization scheme which starts out from a given initial arrangement and is capable to perform continuous object rotations. It is based on a linearization of orthonormal rotation matrices. We also present a perspective for combining global optimization and continuous object rotations. Examples in this paper are taken from the automobile industry but applications are not limited to this area. Various objective functions may be optimized, including the volume and the location of the center of gravity. We also show how to integrate a wiring area estimation into the global optimization procedure.
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References
Aronov, B., Sharir, M.: On translational motion planning in 3-space. Proc. 10th Annu. ACM Symposium on Computational Geometry, pages 21–30, 1994.
Basch, J., Guibas, L. J., Ramkumar, G. D., Ramshaw, L.: Polyhedral Tracings and their Convolution. Algorithms for Robotic Motion and Manipulation: 1996 Workshop on the Algorithmic Foundations of Robotics, pages 171–184. A K Peters, Ltd., 1997.
Bischoff, E. E., Wäscher, G., editors: EJOR; Special Issue on Cutting and Packing, volume 84. Elsevier, 1995.
Cagan, J., Degentesh, D., Yin, S.: A simulated annealing-based algorithm using hierarchical models for general three-dimensional component layout. Computer-Aided Design, 30(10):781–790, 1998.
Fekete, S., Schepers J.: A new exact algorithm for general orthogonal d-dimensional knapsack problems. Proceedings 5th European Symposium on Algorithms ESA′97, LNCS 1284, pages 144–156. Springer-Verlag, 1997.
Flemming, U., Woodbury, R.F.: Software environment to support early phases in building design (SEED): Overview. ASCE Journal of Architectural Engineering, 1(4): 147–152, 1995.
Ghosh, P.K.: A solution of polygon containment, spatial planning, and other related problems using minkowski operations. Computer Vision, Graphics, and Image Processing, 49:1–35, 1990.
Heckmann, R. and Lengauer, T.: Computing closely matching Upper and Lower Bounds on Textile Nesting Problems. European Journal of Operational Research, 108:473–489, 1998
Kolli, A., Cagan, J., Rutenbar, R.: Packing of generic, three-dimensional components based on multi-resolution modelling. Proceedings of the 1996 ASME Design Engineering Technical Conferences and Computers in Engineering Conference: Design Automation Conference [96-DETC/DAC-1479], Irvine, 18-22 August 1996.
Landon, M. D., Balling, R.J.: Optimal packaging of complex parametric solids according to mass property criteria. ASME Journal of Mechanical Design, 116:375–381, 1994.
Lengauer, T.: Combinatorial Algorithms for Integrated Circuit Layout. Applicable Theory in Computer Science. Wiley-Teubner, Chichester-Stuttgart, 1990.
Milenkovic, V.J.: Rotational Polygon Overlap Minimization. Proceedings of the 13th Annual Symposium on Computational Geometry, ACM, Nice, France, June 4-6, 1997.
Schäfer, M., Lengauer, T.: Automated layout generation and wiring area estimation for 3D electronic modules. ASME Journal of Mechanical Design (to appear)
Sandgren, E., Dworak, T.: Part layout optimization using a quadtree representation. Advances in Design Automation 1988: Proceedings of the 14th ASME Design Automation Conference, pages 211–219, Kissimmee, 25-28 September 1988.
Scheithauer, G., Terno, J.: Modelling of packing problems. Optimization, 28:63–84, 1993.
Szykman, S., Cagan, J.: Constrained three-dimensional component layout using simulated annealing. ASME Journal of Mechanical Design, 119(1):28–35, 1997.
Szykman, S., Cagan, J., Weisser, P.: An integrated approach to optimal three dimensional layout and routing. ASME Journal of Mechanical Design, 120(3):510–512, 1998.
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Lengauer, T., Schäfer, M. (2003). Combinatorial Optimization Techniques for Three-Dimensional Arrangement Problems. In: Jäger, W., Krebs, HJ. (eds) Mathematics — Key Technology for the Future. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55753-8_6
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DOI: https://doi.org/10.1007/978-3-642-55753-8_6
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