Abstract
In this paper we present a new framework to approach the problem of structural shape and topology optimization. We use a level-set method as a region representation with a moving boundary model. As a boundary optimization problem, the structural boundary description is implicitly embedded in a scalar function as its “iso-surfaces.” Such level-set models are flexible in handling complex topological changes and are concise in describing the material regions of the structure. Furthermore, by using a simple Hamilton���Jacobi convection equation, the movement of the implicit moving boundaries of the structure is driven by a transformation of the objective and the constraints into a speed function that defines the level-set propagation. The result is a 3D structural optimization technique that demonstrates outstanding flexibility in handling topological changes, the fidelity of boundary representation, and the degree of automation, comparing favorably with other methods in the literature based on explicit boundary variation or homogenization. We present two numerical techniques of conjugate mapping and variational regularization for further enhancement of the level-set computation, in addition to the use of efficient up-wind schemes. The method is tested with several examples of a linear elastic structure that are widely reported in the topology optimization literature.
Similar content being viewed by others
References
Allaire, G. 1997: The homogenization method for topology and shape optimization. In: Rozvany, G. (ed.) Topology Optimization in Structural Mechanics, pp. 101–133
Allaire, G.; Jouve, F.; Taoder, A.-M. 2002: A level-set method for shape optimization. C. R. Acad. Sci. Paris, Ser. I 334, 1–6
Bendsoe, M.P. 1989: Optimal shape design as a material distribution problem. Struct. Optim. 1, 193–202
Bendsoe, M.P. 1997: Optimization of Structural Topology, Shape and Material. Berlin: Springer
Bendsoe, M.P. 1999: Variable-topology optimization: Status and challenges. In: Wunderlich, W. (ed.) Proceedings of the European Conference on Computational Mechanics (held in Munich, Germany)
Bendsoe, M.P.; Haber, R. 1993: The Michell layout problem as a low volume fraction limit of the homogenization method for topology design: an asymptotic study. Struct. Optim. 6, 63–267
Bendsoe, M.P.; Kikuchi, N. 1988: Generating optimal topologies in structural design using a homogenisation method. Comput. Methods Appl. Mech. Eng. 71, 197–224
Bendsoe, M.P.; Sigmund, O. 1999: Material interpolations in topology optimization. Arch. Appl. Mech. 69, 635–654
Breen, D.; Whitaker, R. 2001: A level set approach for the metamorphosis of solid models. IEEE Trans. Visual. Comput. Graphics 7(2), 173–192
Bulman, S.; Sienz, J.; Hinton, E. 2001: Comparisons between algorithms for structural topology optimization using a series of benchmark studies. Comput. Struct. 79, 1203–1218
Diaz, A.R.; Bendsoe, M.P. 1992: Shape optimization of structures for multiple loading conditions using a homogenization method. Struct. Optim. 4, 17–22
Eschenauer, H.A.; Kobelev, H.A.; Schumacher, A. 1994: Bubble method for topology and shape optimization of structures. Struct. Optim. 8, 142–151
Eschenauer, H.A.; Schumacher, A. 1997: Topology and shape optimization procedures using hole positioning criteria. In: Rozvany, G. (ed.) Topology Optimization in Structural Mechanics, pp. 135–196 . Wien: Springer
Lewinski, T.; Sokolowski, J.; Zochowski, A. 1999: Justification of the bubble method for the compliance minimization problems of plates and spherical shells. Proc. 3rd World Congress of Structural and Multidisciplinary Optimization (WCSMO-3) (held in Buffalo, NY)
Lin, C.Y.; Chao, L.-S. 2000: Automated image interpretation for integrated topology and shape optimization. Struct. Multidisc. Optim. 20, 125–137
Mlejnek, H.P. 1992: Some aspects of the genesis of structures. Struct. Optim. 5, 64–69
Osher, S.; Sethian, J.A. 1988: Front propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79, 12–49
Osher, S.; Fedkiw, R. Level set methods: an overview and some recent results. J. Comput. Phys. 169, 475–502
Peng, D.; Merriman, B.; Osher, S.; Zhao, H.-K.; Kang, M. 1999: A PED-based fast local level set method. J. Comput. Phys. 155, 410–438
Reynolds, D.; McConnachie, J.P.; Bettess, W.; Christie, C.; J. Bull, W. 1999: Reverse Adaptivity – a new evolutionary tool for structural optimization. Int. J. Numer. Methods Eng. 45, 529–552
Rozvany, G. 1989: Structural Design via Optimality Criteria. Dordrecht: Kluwer
Rozvany, G.; Zhou, M. 1991: The COC algorithm, Part II: topological, geometrical and generalized shape optimization. Comput. Methods Appl. Mech. Eng. 89, 309–336
Rozvany, G.; Zhou, M.; Birker, T. 1992: Generalized shape optimization without homogenization. Struct. Optim. 4, 250–252
Rozvany, G. 2001: Aims, scope, methods, history and unified terminology of computer aided topology optimization in structural mechanics. Struct. Multidisc. Optim. 21, 90–108
Sapiro, G. 2001: Geometric Partial Differential Equations and Image Analysis. Cambridge: Cambridge University Press
Sethian, J.A.; Wiegmann, A. 2000: Structural boundary design via level set and immersed interface methods. J. Comput. Phys. 163(2), 489–528
Sethian, J.A. 1999: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge: Cambridge University Press
Shu, C.-W. 1988: Total-variation-diminishing time discretization. SIAM J. Sci. Stat. Comput. 9, 1073–1084
Shu, C.-W.; Osher, S. 1988: Efficient implementation of essentially non-oscillatory shock capture schemes. J. Comput. Phys. 77, 439–471
Sigmund, O. 2000: Topology optimization: A tool for the tailoring of structures and materials. Philos. Trans.: Math. Phys. Eng. Sci. 358, 211–228
Sigmund, O. 2001: A 99 topology optimization code written in Matlab. Struct. Multidisc. Optim. 21, 120–718
Sokolowski, J.; Zochowski, A. 1999: On the topological derivative in shape optimization. SIAM J. Control Optim. 37(4), 1251–1272
Sokolowski, J.; Zolesio, J.P. 1992: Introduction to Shape Optimization: Shape Sensitivity Analysis. New York: Springer-Verlag
Suzuki, K.; Kikuchi, N. 1991: A homogenization method for shape and topology optimization. Comput. Methods Appl. Mech. Eng. 93, 291–381
Wang, M.Y.; Wang, X.M.; Guo, D.M. 2003: A level set method for structural topology optimization. Comput. Methods Appl. Mech. Eng. 192(1-2), 227–246
Wang, M.Y.; Wang, X.M. 2003: A multi-phase level set model for multi-material structural optimization. Proc. 5th World Congress of Structural and Multidisciplinary Optimization (WCSMO5) (held in Lido di Jesolo, Italy)
Xie, Y.M.; Steven, G.P. 1997: Evolutionary Structural Optimization. New York: Springer
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang , X., Wang , M. & Guo , D. Structural shape and topology optimization in a level-set-based framework of region representation. Struct Multidisc Optim 27, 1–19 (2004). https://doi.org/10.1007/s00158-003-0363-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-003-0363-y