Abstract
Filtering has been a major technique used in homogenization-based methods for topology optimization of structures. It plays a key role in regularizing the basic problem into a well-behaved setting, but it has the drawback of a smoothing effect around the boundary of the material domain. In this paper, a diffusion technique is presented as a variational approach to the regularization of the topology optimization problem. A nonlinear or anisotropic diffusion process not only leads to a suitable problem regularization but also exhibits strong “edge”-preserving characteristics. Thus, we show that the use of nonlinear diffusions brings the desirable effects of boundary preservation and even enhancement of lower-dimensional features such as flow-like structures. The proposed diffusion techniques have a close relationship with the diffusion methods and the phase-field methods from the fields of materials and digital image processing. The proposed method is described and illustrated with 2D examples of minimum compliance that have been extensively studied in recent literature of topology optimization.
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References
Allaire, G. 2001: Shape Optimization by the Homogenization Method. Berlin Heidelberg New York: Springer
Allaire, G.; Jouve, F.; Taoder, A.-M. 2004: Structural optimization using sensitivity analysis and a level-set method. J Comput Phys194, 363–393
Aubert, G.; Kornprobst, P. 2000: Mathematical Problems in Image Processing. Berlin Heidelberg New York: Springer
Bendsoe, M.P.; Sigmund, O. 2003: Topology Optimization: Theory, Methods and Applications. Berlin Heidelberg New York: Springer
Bendsoe, M.P. 1999: Variable-topology optimization: Status and challenges. Proceedings of the European Conference on Computational Mechanics (W. Wunderlich editor) (held in Munich, Germany)
Bendsoe, M.P.; Kikuchi, N. 1988: Generating optimal topologies in structural design using a homogenisation method. Comput Methods Appl Mech Eng71, 197–224
Bendsoe, M.P.; Sigmund, O. 1999: Material interpolations in topology optimization. Arch Appl Mech69, 635–654
Bourdin, B. 2001: Filters in topology optimization. Int J Numer Methods Eng50, 2143–2158
Bourdin, B.; Chambolle, A. 2003: Design-dependent loads in topology optimization. ESAIM Control Optim Calc Var9, 19–48
Cahn, J.; Hilliard, J.E. 1958: Free energy of a nonuniform system. I. Interfacial free energy. J Chem Phys28(1), 258–267
Cea, J.; Malanowski, K. 1970: An example of a max-min problem in partial differential equations. SIAM J Control8, 305–316
Charbonnier, P.; Blanc-Feraud, L.; Aubert, G.; Barlaud, M. 1997: Deterministic edge-reserving regularization in computed imaging. IEEE Trans Image Process6(2), 298–311
Cheng, K.-T.; Olhoff, N. 1981: An investigation concerning optimal design of solid elastic plates. Int J Solids Struct17, 305–323
Diaz, R.; Sigmund, O. 1995: Checkerboards patterns in layout optimization. Struct Optim10, 10–45
Eyre, D. 1993: Systems of Cahn-Hilliard equations. SIAM J Appl Math53(6), 1686–1712
Guo, X.; Gu, Y. 2004: A new density-stiffness interpolation scheme for topology optimization of continuum structures. Eng Comput21(1), 9–22
Haber, R.B.; Jog, C.S.; Bendsoe, M.P. 1996: A new approach to variable-topology shape design using a constraint on perimeter. Struct Optim11, 1–12
Leo, P.H.; Lowengrub, J.S.; Jou, H.J. 1998: A diffuse interface model for microstructural evolution in elastically stressed solids. Acta Mater46(6), 2113–2130
Murat, F. 1972: Théorèmes de non-existence pour des problèmes de contrôle dans les coeffcients. C R Acad Sci ParisA-274, 395–398
Murat, F. 1977: Contre-exemples pour divers problèmes où le contrôle intervient dans les coeffcients. Ann Mat Pura Appl112, 49–68
Osher, S.; Santosa, F. 2001: level-set methods for optimization problems involving geometry and constraints: frequencies of a two-density inhomogeneous drum. J Comput Phys171, 272–288
Petersson, J. 1999: Some convergence results in perimeter-controlled topology optimization. Comput Methods Appl Mech Eng171, 123–140
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 Eng89, 309–336
Rozvany, G.; Zhou, M.; Birker, T. 1992: Generalized shape optimization without homogenization. Struct Optim4, 250–252
Ruiter, De M.J.; Keulen, van F. 2000: Topology optimization: Approaching the material distribution problem using a topological function description. In Topping, B.H.V. (ed.), Computational Techniques for Materials, Composites and Composite Structures, 111–119. Edinburgh, UK: Civil-Comp
Samson, C.; Blanc-Feraud, L.; Aubert, G.; Zerubia, J. 2000: A variational model for image classification and restoration. IEEE Trans Pattern Anal Mach Intell22(5), 460–472
Sapiro, G. 2001: Geometric Partial Differential Equations and Image Analysis. Cambridge: Cambridge University
Sethian, J.; Wiegmann, A. 2000: Structural boundary design via level-set and immersed interface methods. J Comput Phys163, 489–528
Sigmund, O.; Petersson, J. 1998: Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim16(1), 68–75
Tikhonov, A.N.; Arsenin, V.Y. 1997: Solutions of Ill-Posed Problems. Washington: Winston and Sons
Wang, M.Y.; Wang, X.M.; Guo, D.M. 2003: A level set method for structural topology optimization. Comput Methods Appl Mech Eng192(1–2), 227–246
Wang, X.M.; Wang, M.Y.; Guo, D.M. 2004: Structural shape and topology optimization in a level-set based framework of region representation. Struct Multidisc Optim27(1–2), 1–19
Wang, M.Y.; Zhou, S.W. 2004: A variational method for structural topology optimization. Proc. of Second International Conference on Structural Engineering, Mechanics and Computation (held in Cape Town, South Africa)
Wang, M.Y.; Zhou, S.W. 2003: A phase-field method for structural topology optimization. submitted for publication. preprint
Warren, J.A. 1995: How does a metal freeze? A phase-field model of alloy solidification. IEEE Comput Sci Eng2(2), 38–48
Weickert, J. 1997: A review of nonlinear diffusion filtering, Scale-Space Theory in Computer Vision. ter Haar Romeny, B. et al. (ed.), Lect Notes Comput Sci1252, 3–28, Berlin: Springer
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Wang, M., Zhou, S. & Ding, H. Nonlinear diffusions in topology optimization. Struct Multidisc Optim 28, 262–276 (2004). https://doi.org/10.1007/s00158-004-0436-6
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DOI: https://doi.org/10.1007/s00158-004-0436-6