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

Published online by Cambridge University Press:  23 December 2010

Patrick Penzler ,
Martin Rumpf  and
Benedikt Wirth
Patrick Penzler
Affiliation:
Institute for Numerical Simulation, Bonn University, Endenicher Allee 60, 53115 Bonn, Germany. benedikt.wirth@ins.uni-bonn.de
Martin Rumpf
Affiliation:
Institute for Numerical Simulation, Bonn University, Endenicher Allee 60, 53115 Bonn, Germany. benedikt.wirth@ins.uni-bonn.de
Benedikt Wirth
Affiliation:
Institute for Numerical Simulation, Bonn University, Endenicher Allee 60, 53115 Bonn, Germany. benedikt.wirth@ins.uni-bonn.de
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Abstract

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 Γ-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.

Keywords

Shape optimizationnonlinear elasticityphase-field modelbuckling deformationsΓ-convergence
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 1 , January 2012 , pp. 229 - 258
Copyright
© EDP Sciences, SMAI, 2010

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References

G. Allaire, Shape optimization by the homogenization method, Applied Mathematical Sciences 146. Springer-Verlag, New York (2002).
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 Chemistry 170, Springer (2004) 1–13.
Allaire, G., Bonnetier, E., Francfort, G. and Jouve, F., Shape optimization by the homogenization method. Numer. Math. 76 (1997) 27–68. Google Scholar
Allaire, G., Jouve, F. and Toader, A.-M., A level-set method for shape optimization. C. R. Acad. Sci. Paris, Sér. I 334 (2002) 1125–1130. Google Scholar
Allaire, G., Jouve, F. and Maillot, H., Topology optimization for minimum stress design with the homogenization method. Struct. Multidisc. Optim. 28 (2004) 87–98. Google Scholar
Allaire, G., Jouve, F. and Toader, A.-M., Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194 (2004) 363–393. Google Scholar
Allaire, G., de Gournay, F., Jouve, F. and Toader, A.-M., Structural optimization using topological and shape sensitivity via a level set method. Control Cybern. 34 (2005) 59–80. Google Scholar
Allen, S.M. and Cahn, J.W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27 (1979) 1085–1095. Google Scholar
Ambrosio, L. and Buttazzo, G., An optimal design problem with perimeter penalization. Calc. Var. 1 (1993) 55–69. Google Scholar
Ansola, R., Veguería, E., Canales, J. and Tárrago, J.A., A simple evolutionary topology optimization procedure for compliant mechanism design. Finite Elements Anal. Des. 44 (2007) 53–62. Google Scholar
Ball, J.M., Convexity conditions and existence theorems in nonlinear elasiticity. Arch. Ration. Mech. Anal. 63 (1977) 337–403. Google Scholar
Ball, J.M., Global invertibility of Sobolev functions and the interpenetration of matter. Proc. R. Soc. Edinb. A 88 (1981) 315–328. Google Scholar
Bourdin, B. and Chambolle, A., Design-dependent loads in topology optimization. ESAIM : COCV 9 (2003) 19–48. Google Scholar
A. Braides, Γ -convergence for beginners, Oxford Lecture Series in Mathematics and its Applications 2. Oxford University Press, Oxford (2002).
Burger, M. and Stainko, R., Phase-field relaxation of topology optimization with local stress constraints. SIAM J. Control Optim. 45 (2006) 1447–1466. Google Scholar
Chambolle, A., A density result in two-dimensional linearized elasticity, and applications. Arch. Ration. Mech. Anal. 167 (2003) 211–233. Google Scholar
Chen, Y., Davis, T.A., Hager, W.W. and Rajamanickam, S., Algorithm 887 : CHOLMOD, supernodal sparse Cholesky factorization and update/downdate. ACM Trans. Math. Softw. 35 (2009) 22 :1–22 :14. Google Scholar
P.G. Ciarlet, Three-dimensional elasticity. Elsevier Science Publishers B. V. (1988).
A.R. Conn, N.I.M Gould and P.L. Toint, Trust-Region Methods. SIAM (2000).
S. Conti, H. Held, M. Pach, M. Rumpf and R. Schultz, Risk averse shape optimization. SIAM J. Control Optim. (to appear).
B. Dacorogna, Direct methods in the calculus of variations. Springer-Verlag, New York (1989).
Davis, T.A. and Hager, W.W., Dynamic supernodes in sparse Cholesky update/downdate and triangular solves. ACM Trans. Math. Softw. 35 (2009) 27 :1–27 :23. Google Scholar
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.
Guo, X., Zhao, K. and Wang, M.Y., Simultaneous shape and topology optimization with implicit topology description functions. Control Cybern. 34 (2005) 255–282. Google Scholar
Liu, Z., Korvink, J.G. and Huang, R., Structure topology optimization : Fully coupled level set method via femlab. Struct. Multidisc. Optim. 29 (2005) 407–417. Google Scholar
J.E. Marsden and T.J.R. Hughes, Mathematical Foundations of Elasticity. Prentice-Hall, Englewood Cliffs (1983).
Modica, L. and Mortola, S., Un esempio di Γ − -convergenza. Boll. Un. Mat. Ital. B (5) 14 (1977) 285–299. Google Scholar
P. Pedregal, Variational Methods in Nonlinear Elasticity. SIAM (2000).
Sethian, J.A. and Wiegmann, A., Structural boundary design via level set and immersed interface methods. J. Comput. Phys. 163 (2000) 489–528. Google Scholar
Sigmund, O. and Clausen, P.M., Topology optimization using a mixed formulation : An alternative way to solve pressure load problems. Comput. Methods Appl. Mech. Eng. 196 (2007) 1874–1889. Google Scholar
J. Sikolowski and J.-P. Zolésio, Introduction to shape optimization, in Shape sensitivity analysis, Springer (1992).
Wang, M.Y. and Zhou, S., Synthesis of shape and topology of multi-material structures with a phase-field method. J. Computer-Aided Mater. Des. 11 (2004) 117–138. Google Scholar
Wang, M.Y., Wang, X. and Guo, D., A level set method for structural topology optimization. Comput. Methods Appl. Mech. Eng. 192 (2003) 227–246. Google Scholar
Wang, M.Y., Zhou, S. and Ding, H., Nonlinear diffusions in topology optimization. Struct. Multidisc. Optim. 28 (2004) 262–276. Google Scholar
Wei, P. and Wang, M.Y., Piecewise constant level set method for structural topology optimization. Int. J. Numer. Methods Eng. 78 (2009) 379–402. Google Scholar
Xia, Q. and Wang, M.Y., Simultaneous optimization of the material properties and the topology of functionally graded structures. Comput. Aided Des. 40 (2008) 660–675. Google Scholar
Zhou, S. and Wang, M.Y., Multimaterial structural topology optimization with a generalized Cahn–Hilliard model of multiphase transition. Struct. Multidisc. Optim. 33 (2007) 89–111. Google Scholar