×

zbMATH — the first resource for mathematics

Minimum stress optimal design with the level set method. (English) Zbl 1244.74104
Summary: This paper is devoted to minimum stress design in structural optimization. We propose a simple and efficient numerical algorithm for shape and topology optimization based on the level set method coupled with the topological derivative. We compute a shape derivative, as well as a topological derivative, for a stress-based objective function. Using an adjoint equation we implement a gradient algorithm for the minimization of the objective function. Several numerical examples in 2-d and 3-d are discussed.

MSC:
74P15 Topological methods for optimization problems in solid mechanics
74P10 Optimization of other properties in solid mechanics
PDF BibTeX Cite
Full Text: DOI
References:
[1] Allaire, G.; Jouve, F.; Toader, A.-M., A level set method for shape optimization, C R acad sci Paris Sér I, 334, 1125-1130, (2002) · Zbl 1115.49306
[2] Allaire, G.; Jouve, F.; Toader, A.-M., Structural optimization using sensitivity analysis and a level set method, J comp phys, 194/1, 363-393, (2004) · Zbl 1136.74368
[3] Osher, S.; Santosa, F., Level set methods for optimization problems involving geometry and constraints: frequencies of a two-density inhomogeneous drum, J comp phys, 171, 272-288, (2001) · Zbl 1056.74061
[4] Sethian, J.; Wiegmann, A., Structural boundary design via level set and immersed interface methods, J comp phys, 163, 489-528, (2000) · Zbl 0994.74082
[5] Wang, M.Y.; Wang, X.; Guo, D., A level set method for structural topology optimization, Comput methods appl mech eng, 192, 227-246, (2003) · Zbl 1083.74573
[6] Allaire, G.; Jouve, F.; de Gournay, F.; Toader, A.-M., Structural optimization using topological and shape sensitivity via a level set method, Control and cybern, 34, 59-80, (2005) · Zbl 1167.49324
[7] Burger, M.; Hackl, B.; Ring, W., Incorporating topological derivatives into level set methods, J comp phys, 194/1, 344-362, (2004) · Zbl 1044.65053
[8] Wang, X.; Yulin, M.; Wang, M.Y., Incorporating topological derivatives into level set methods for structural topology optimization, (), 145-157
[9] de Gournay, F., Velocity extension for the level-set method and multiple eigenvalues in shape optimization, SIAM J control optim, 45, 1, 343-367, (2006) · Zbl 1108.74046
[10] Allaire, G.; Jouve, F., A level-set method for vibration and multiple loads structural optimization, Comput methods appl mech eng, 194, 3269-3290, (2005) · Zbl 1091.74038
[11] de Gournay F, Allaire G, Jouve F. Shape and topology optimization of the robust compliance via the level set method, vol. 14. ESA in/COCV;2008. pp. 43-70. · Zbl 1245.49054
[12] Murat, F.; Simon, S., Etudes de problèmes d’optimal design. lecture notes in computer science, vol. 41, (1976), Springer Berlin, p. 54-62
[13] Pironneau, O., Optimal shape design for elliptic systems, (1984), Springer New York · Zbl 0496.93029
[14] Sokołowski, J.; Zolesio, J.P., Introduction to shape optimization: shape sensitivity analysis. Springer series in computational mathematics, vol. 10, (1992), Springer Berlin · Zbl 0761.73003
[15] Allaire, G.; Jouve, F.; Maillot, H., Topology optimization for minimum stress design with the homogenization method, Struct multidiscip optim, 28, 87-98, (2004) · Zbl 1243.74148
[16] Duysinx, P.; Bendsoe, M., Topology optimization of continuum structures with local stress constraints, Int J numer meth eng, 43, 1453-1478, (1998) · Zbl 0924.73158
[17] Cheng, G.D.; Guo, X., Study on topology optimization with stress constraints, Eng optim, 20, 129-148, (1992)
[18] Lipton, R., Design of functionally graded composite structures in the presence of stress constraints, Int J solids struct, 39, 2575-2586, (2002) · Zbl 1032.74020
[19] Stainko, R.; Burger, M., A one-shot approach to topology optimization with stress constraints, ()
[20] Achtziger, W., Topology optimization subject to design-dependent validity of constraints, (), 177-191
[21] Pereira, J.T.; Fancello, E.A.; Barcellos, C.S., Topology optimization of continuum structures with material failure constraints, Struct multidiscip optim, 26, 50-66, (2004) · Zbl 1243.74157
[22] Osher, S.; Sethian, J.A., Front propagating with curvature dependent speed: algorithms based on hamilton – jacobi formulations, J comp phys, 78, 12-49, (1988) · Zbl 0659.65132
[23] Osher, S.; Fedkiw, R., Level set methods and dynamic implicit surfaces. applied mathematical sciences, vol. 153, (2003), Springer New York · Zbl 1026.76001
[24] Sethian, J.A., Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision and materials science, (1999), Cambridge University Press Cambridge · Zbl 0973.76003
[25] Allaire G, Conception optimale de structures. Mathématiques and applications, vol. 58, Heidelberg: Springer; 2006.
[26] Céa, J., Conception optimale ou identification de formes, calcul rapide de la dérivée directionnelle de la fonction coût, Math model numer anal, 20, 3, 371-402, (1986) · Zbl 0604.49003
[27] Eschenauer, H.; Kobelev, V.; Schumacher, A., Bubble method for topology and shape optimization of structures, Struct optim, 8, 42-51, (1994)
[28] Céa J, Garreau S, Guillaume P, Masmoudi M. The shape and topological optimizations connection. In: IV WCCM, part II, Buenos Aires, 1998. Comput Methods Appl Mech Eng 2000; 188; 713-726. · Zbl 0972.74057
[29] Garreau, S.; Guillaume, P.; Masmoudi, M., The topological asymptotic for PDE systems: the elasticity case, SIAM J control optim, 39, 6, 1756-1778, (2001) · Zbl 0990.49028
[30] Sokołowski, J.; Żochowski, A., On the topological derivative in shape optimization, SIAM J control optim, 37, 1251-1272, (1999) · Zbl 0940.49026
[31] Sokołowski, J.; Żochowski, A., Topological derivatives of shape functionals for elasticity systems, Mech struct Mach, 29, 3, 331-349, (2001)
[32] Amstutz, S., Topological sensitivity analysis for some nonlinear PDE system, J math pures appl, 85, 4, 540-557, (2006) · Zbl 1090.35053
[33] Novotny, A.; Feijóo, R.; Taroco, E.; Padra, C., Topological sensitivity analysis, Comput methods appl mech eng, 192, 7-8, 803-829, (2003) · Zbl 1025.74025
[34] Pommier, J.; Samet, B., The topological asymptotic for the Helmholtz equation with Dirichlet condition on the boundary of an arbitrarily shaped hole, SIAM J control optim, 43, 3, 899-921, (2004) · Zbl 1080.49028
[35] Ammari, H.; Kang, H., Reconstruction of small inhomogeneities from boundary measurements. lecture notes in mathematics, vol. 1846, (2004), Springer Berlin
[36] Ammari, H.; Kang, H.; Nakamura, G.; Tanuma, K., Complete asymptotic expansions of solutions of the system of elastostatics in the presence of an inclusion of small diameter and detection of an inclusion, J elasticity, 67, 2, 97-129, (2002) · Zbl 1089.74576
[37] Ammari, H.; Vogelius, M.; Volkov, D., Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter. II. the full Maxwell equations, J math pures appl, 80, 8, 769-814, (2001) · Zbl 1042.78002
[38] Burger, M., A framework for the construction of level set methods for shape optimization and reconstruction, Interfaces and free boundaries, 5, 301-329, (2003) · Zbl 1081.35134
[39] Sigmund, O., On the design of compliant mechanisms using topology optimization, Mech struct Mach, 25, 493-524, (1997)
[40] Jouve F, Mechkour H. Level set based method for design of compliant mechanisms. Eur J Comput Mech (2008), to appear. · Zbl 1237.70007
[41] Pedregal, P., Fully explicit quasiconvexification of the Mean-square deviation of the gradient of the state in optimal design, Electr res announce AMS, 7, 72-78, (2001) · Zbl 0980.49021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.