A phase-field model for compliance shape optimization in nonlinear elasticity.

*(English)*Zbl 1251.49054Summary: Shape optimization of mechanical devices is investigated in the context of large, geometrically strongly nonlinear deformations and nonlinear hyperelastic constitutive laws. A weighted sum of the structure compliance, its weight, and its surface area are minimized. The resulting nonlinear elastic optimization problem differs significantly from classical shape optimization in linearized elasticity. Indeed, there exist different definitions for the compliance: the change in potential energy of the surface load, the stored elastic deformation energy, and the dissipation associated with the deformation. Furthermore, elastically optimal deformations are no longer unique so that one has to choose the minimizing elastic deformation for which the cost functional should be minimized, and this complicates the mathematical analysis. Additionally, along with the non-uniqueness, buckling instabilities can appear, and the compliance functional may jump as the global equilibrium deformation switches between different bluckling modes. This is associated with a possible non-existence of optimal shapes in a worst-case scenario. In this paper, the sharp-interface description of shapes is relaxed via an Allen-Cahn or Modica-Mortola type phase-field model, and soft material instead of void is considered outside the actual elastic object. An existence result for optimal shapes in the phase field as well as in the sharp-interface model is established, and the model behavior for decreasing phase-field interface width is investigated in terms of \(\Gamma \)-convergence. Computational results are based on a nested optimization with a trust-region method as the inner minimization for the equilibrium deformation and a quasi-Newton method as the outer minimization of the actual objective functional. Furthermore, a multi-scale relaxation approach with respect to the spatial resolution and the phase-field parameter is applied. Various computational studies underline the theoretical observations.

##### MSC:

49Q10 | Optimization of shapes other than minimal surfaces |

74P05 | Compliance or weight optimization in solid mechanics |

49J20 | Existence theories for optimal control problems involving partial differential equations |

##### Keywords:

shape optimization; nonlinear elasticity; phase-field model; buckling deformations; \(\Gamma \)-convergence
PDF
BibTeX
XML
Cite

\textit{P. Penzler} et al., ESAIM, Control Optim. Calc. Var. 18, No. 1, 229--258 (2012; Zbl 1251.49054)

**OpenURL**

##### References:

[1] | G. Allaire, Shape optimization by the homogenization method, Applied Mathematical Sciences146. Springer-Verlag, New York (2002). · Zbl 0990.35001 |

[2] | G. Allaire, Topology Optimization with the Homogenization and the Level-Set Method, in Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, NATO Science Series II : Mathematics, Physics and Chemistry170, Springer (2004) 1-13. · Zbl 1320.74089 |

[3] | G. Allaire, E. Bonnetier, G. Francfort and F. Jouve, Shape optimization by the homogenization method. Numer. Math.76 (1997) 27-68. Zbl0889.73051 · Zbl 0889.73051 |

[4] | G. Allaire, F. Jouve and A.-M. Toader, A level-set method for shape optimization. C. R. Acad. Sci. Paris, Sér. I334 (2002) 1125-1130. · Zbl 1115.49306 |

[5] | G. Allaire, F. Jouve and H. Maillot, Topology optimization for minimum stress design with the homogenization method. Struct. Multidisc. Optim.28 (2004) 87-98. · Zbl 1243.74148 |

[6] | G. Allaire, F. Jouve and A.-M. Toader, Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys.194 (2004) 363-393. · Zbl 1136.74368 |

[7] | G. Allaire, F. de Gournay, F. Jouve and A.-M. Toader, Structural optimization using topological and shape sensitivity via a level set method. Control Cybern.34 (2005) 59-80. Zbl1167.49324 · Zbl 1167.49324 |

[8] | S.M. Allen and J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall.27 (1979) 1085-1095. |

[9] | L. Ambrosio and G. Buttazzo, An optimal design problem with perimeter penalization. Calc. Var.1 (1993) 55-69. · Zbl 0794.49040 |

[10] | R. Ansola, E. Veguería, J. Canales and J.A. Tárrago, A simple evolutionary topology optimization procedure for compliant mechanism design. Finite Elements Anal. Des.44 (2007) 53-62. |

[11] | J.M. Ball, Convexity conditions and existence theorems in nonlinear elasiticity. Arch. Ration. Mech. Anal.63 (1977) 337-403. · Zbl 0368.73040 |

[12] | J.M. Ball, Global invertibility of Sobolev functions and the interpenetration of matter. Proc. R. Soc. Edinb. A88 (1981) 315-328. Zbl0478.46032 · Zbl 0478.46032 |

[13] | B. Bourdin and A. Chambolle, Design-dependent loads in topology optimization. ESAIM : COCV9 (2003) 19-48. · Zbl 1066.49029 |

[14] | A. Braides, \Gamma -convergence for beginners, Oxford Lecture Series in Mathematics and its Applications2. Oxford University Press, Oxford (2002). |

[15] | M. Burger and R. Stainko, Phase-field relaxation of topology optimization with local stress constraints. SIAM J. Control Optim.45 (2006) 1447-1466. · Zbl 1116.74053 |

[16] | A. Chambolle, A density result in two-dimensional linearized elasticity, and applications. Arch. Ration. Mech. Anal.167 (2003) 211-233. · Zbl 1030.74007 |

[17] | Y. Chen, T.A. Davis, W.W. Hager and S. Rajamanickam, Algorithm 887 : CHOLMOD, supernodal sparse Cholesky factorization and update/downdate. ACM Trans. Math. Softw.35 (2009) 22 :1-22 :14. |

[18] | P.G. Ciarlet, Three-dimensional elasticity. Elsevier Science Publishers B. V. (1988). · Zbl 0648.73014 |

[19] | A.R. Conn, N.I.M Gould and P.L. Toint, Trust-Region Methods. SIAM (2000). · Zbl 0958.65071 |

[20] | S. Conti, H. Held, M. Pach, M. Rumpf and R. Schultz, Risk averse shape optimization. SIAM J. Control Optim. (to appear). · Zbl 1230.49039 |

[21] | B. Dacorogna, Direct methods in the calculus of variations. Springer-Verlag, New York (1989). · Zbl 0703.49001 |

[22] | T.A. Davis and W.W. Hager, Dynamic supernodes in sparse Cholesky update/downdate and triangular solves. ACM Trans. Math. Softw.35 (2009) 27:1-27 :23. |

[23] | G.P. Dias, J. Herskovits and F.A. Rochinha, Simultaneous shape optimization and nonlinear analysis of elastic solids, in Computational Mechanics - New Trends and Applications, E. Onate, I. Idelsohn and E. Dvorkin Eds., CIMNE, Barcelona (1998) 1-13. |

[24] | X. Guo, K. Zhao and M.Y. Wang, Simultaneous shape and topology optimization with implicit topology description functions. Control Cybern.34 (2005) 255-282. · Zbl 1167.65433 |

[25] | Z. Liu, J.G. Korvink and R. Huang, Structure topology optimization : Fully coupled level set method via femlab. Struct. Multidisc. Optim.29 (2005) 407-417. · Zbl 1243.74155 |

[26] | J.E. Marsden and T.J.R. Hughes, Mathematical Foundations of Elasticity. Prentice-Hall, Englewood Cliffs (1983). · Zbl 0545.73031 |

[27] | L. Modica and S. Mortola, Un esempio di \Gamma - -convergenza. Boll. Un. Mat. Ital. B (5)14 (1977) 285-299. |

[28] | P. Pedregal, Variational Methods in Nonlinear Elasticity. SIAM (2000). Zbl0941.74002 · Zbl 0941.74002 |

[29] | J.A. Sethian and A. Wiegmann, Structural boundary design via level set and immersed interface methods. J. Comput. Phys.163 (2000) 489-528. Zbl0994.74082 · Zbl 0994.74082 |

[30] | O. Sigmund and P.M. Clausen, Topology optimization using a mixed formulation : An alternative way to solve pressure load problems. Comput. Methods Appl. Mech. Eng.196 (2007) 1874-1889. · Zbl 1173.74375 |

[31] | J. Sikolowski and J.-P. Zolésio, Introduction to shape optimization, in Shape sensitivity analysis, Springer (1992). Zbl0761.73003 · Zbl 0761.73003 |

[32] | M.Y. Wang and S. Zhou, Synthesis of shape and topology of multi-material structures with a phase-field method. J. Computer-Aided Mater. Des.11 (2004) 117-138. |

[33] | M.Y. Wang, X. Wang and D. Guo, A level set method for structural topology optimization. Comput. Methods Appl. Mech. Eng.192 (2003) 227-246. · Zbl 1083.74573 |

[34] | M.Y. Wang, S. Zhou and H. Ding, Nonlinear diffusions in topology optimization. Struct. Multidisc. Optim.28 (2004) 262-276. · Zbl 1243.74160 |

[35] | P. Wei and M.Y. Wang, Piecewise constant level set method for structural topology optimization. Int. J. Numer. Methods Eng.78 (2009) 379-402. Zbl1183.74222 · Zbl 1183.74222 |

[36] | Q. Xia and M.Y. Wang, Simultaneous optimization of the material properties and the topology of functionally graded structures. Comput. Aided Des.40 (2008) 660-675. |

[37] | S. Zhou and M.Y. Wang, Multimaterial structural topology optimization with a generalized Cahn-Hilliard model of multiphase transition. Struct. Multidisc. Optim.33 (2007) 89-111. · Zbl 1245.74077 |

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.