Abstract
A numerical coupling of two recent methods in shape and topology optimization of structures is proposed. On the one hand, the level set method, based on the shape derivative, is known to easily handle boundary propagation with topological changes. However, in practice it does not allow for the nucleation of new holes. On the other hand, the bubble or topological gradient method is precisely designed for introducing new holes in the optimization process. Therefore, the coupling of these two methods yields an efficient algorithm which can escape from local minima. It have a low CPU cost since it captures a shape on a fixed Eulerian mesh. The main advantage of our coupled algorithm is to make the resulting optimal design more independent of the initial guess.
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References
Allaire, G., Shape Optimization by the Homogenization Method, Springer Verlag, New York (2001).
Allaire, G., De Gournay, F., Jouve, F. and Toader, A.-M., Structural optimization using topological and shape sensitivity via a level set method, Control and Cybernetics, 34, 59–80 (2005).
Allaire, G., Jouve, F. and Toader, A.-M., A level set method for shape optimization, C. R. Acad. Sci. Paris, Série I, 334, 1125–1130 (2002).
Allaire, G., Jouve, F. and Toader, A.-M., Structural optimization using sensitivity analysis and a level set method, J. Comp. Phys., 194(1), 363–393 (2004).
Bendsøe, M., Methods for Optimization of Structural Topology, Shape and Material, Springer Verlag, New York (1995).
Bendsøe, M. and Sigmund, O., Topology Optimization. Theory, Methods, and Applications, Springer Verlag, New York (2003).
Burger, M., A framework for the construction of level set methods for shape optimization and reconstruction, Interfaces and Free Boundaries, 5, 301–329 (2003).
Burger, M., Hackl, B. and Ring, W., Incorporating topological derivatives into level set methods, J. Comp. Phys., 194(1), 344–362 (2004).
Céa, J., Garreau, S., Guillaume, P. and Masmoudi, M., The shape and topological optimizations connection, IV WCCM, Part II (Buenos Aires, 1998), Comput. Methods Appl. Mech. Engrg., 188, 713–726 (2000).
Eschenauer, H. and Schumacher, A., Bubble method for topology and shape optimization of structures, Structural Optimization, 8, 42–51 (1994).
Garreau, S., Guillaume, P. and Masmoudi, M., The topological asymptotic for PDE systems: The elasticity case, SIAM J. Control Optim., 39(6), 1756–1778 (2001).
De Gournay, F., PhD Thesis, Ecole Polytechnique (2005).
Liu, Z., Korvink, J.G. and Huang, R., Structure topology optimization: Fully coupled level set method via FEMLAB, Struct. Multidisc. Optim., 29(6), 407–417 (2005).
Mohammadi, B. and Pironneau, O., Applied Shape Optimization for Fluids, Clarendon Press, Oxford (2001).
Murat, F. and Simon, S., Etudes de problèmes d’optimal design, in Lecture Notes in Computer Science, Vol. 41, pp. 54–62, Springer Verlag, Berlin (1976).
Osher, S. and Sethian, J.A., Front propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys., 78, 12–49 (1988).
Pironneau, O., Optimal Shape Design for Elliptic Systems, Springer-Verlag, New York (1984).
Sethian, J.A., Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Materials Science, Cambridge University Press (1999).
Simon, J., Differentiation with respect to the domain in boundary value problems, Num. Funct. Anal. Optimz., 2, 649–687 (1980).
Sokołowski, J. and Żochowski, A., On the topological derivative in shape optimization, SIAM J. Control Optim., 37, 1251–1272 (1999).
Sokołowski, J. and Żochowski, A., Topological derivatives of shape functionals for elasticity systems, Mech. Structures Mach., 29(3), 331–349 (2001).
Sokołowski, J. and Zolesio, J.P., Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer Ser. in Comp. Math., Vol. 10, Springer, Berlin (1992).
Wang, M.Y., Wang, X. and Guo, D., A level set method for structural topology optimization, Comput. Methods Appl. Mech. Engrg., 192, 227–246 (2003).
Wang, X., Yulin, M. and Wang, M.Y., Incorporating topological derivatives into level set methods for structural topology optimization, in Optimal Shape Design and Modeling, T. Lewinski et al. (eds), Polish Acad. of Sc., Warsaw, pp. 145–157 (2004).
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Allaire, G., Jouve, F. (2006). Coupling the Level Set Method and the Topological Gradient in Structural Optimization. In: Bendsøe, M.P., Olhoff, N., Sigmund, O. (eds) IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials. Solid Mechanics and Its Applications, vol 137. Springer, Dordrecht . https://doi.org/10.1007/1-4020-4752-5_1
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DOI: https://doi.org/10.1007/1-4020-4752-5_1
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