Journal of Optimization Theory and Applications

, Volume 155, Issue 3, pp 962–985 | Cite as

Optimal Design of Brittle Composite Materials: a Nonsmooth Approach

  • Marina Prechtel
  • Günter Leugering
  • Paul Steinmann
  • Michael Stingl


Our goal is to design brittle composite materials yielding maximal energy dissipation for a given static load case. We focus on the effect of variation of fiber shapes on resulting crack paths and thus on the fracture energy. To this end, we formulate a shape optimization problem, in which the cost function is the fracture energy and the state problem consists in the determination of the potentially discontinuous displacement field in the two-dimensional domain. Thereby, the behavior at the crack surfaces is modeled by cohesive laws. We impose a nonpenetration condition to avoid interpenetration of opposite crack sides. Accordingly, the state problem is formulated as variational inequality. This leads to potential nondifferentiability of the shape-state mapping. For the numerical solution, we derive first-order information in the form of subgradients. We conclude the article by numerical results.


Shape optimization Variational inequality Nonsmooth optimization 



The authors gratefully acknowledge the funding of the German Research Council (DFG), which, within the framework of its ‘Excellence Initiative’ supports the Cluster of Excellence ‘Engineering of Advanced Materials’ at the University of Erlangen-Nuremberg.


  1. 1.
    Kohn, R., Strang, G.: Optimal design and relaxation of variational problems. Commun. Pure Appl. Math. 39(1), 113–137 (1986) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bendsøe, M., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71, 197–224 (1988) CrossRefGoogle Scholar
  3. 3.
    Jog, S., Haber, R., Bendsøe, M.: A displacement based topology design method with self-adaptive layered materials. In: Bendsøe, M., Mota Scares, C. (eds.) Topology Design of Structures, pp. 219–238. Kluwer, Dordrecht (1992) Google Scholar
  4. 4.
    Allaire, G., Kohn, R.: Optimal design for minimum weight and compliance in plane stress using extremal microstructures. Eur. J. Mech. A, Solids 12(6), 839–878 (1993) MathSciNetMATHGoogle Scholar
  5. 5.
    Tartar, L.: Estimation fines des coefficients homogeneises. In: Kree, P. (ed.) Ennio de Giorgi’s Colloquium, pp. 168–187. Pitman, London (1985) Google Scholar
  6. 6.
    Francfort, G., Murat, F.: Homogenization and optimal bounds in linear elasticity. Arch. Ration. Mech. Anal. 94, 307–334 (1986) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Gibiansky, L., Cherkaev, A.: Microstructures of composites of extremal rigidity and exact bounds on associated energy density. In: Cherkaev, A., Kohn, R. (eds.) Topics in the Mathematical Modelling of Composite Materials. Progress in Nonlinear Differential Equations an Their Applications, vol. 31B, pp. 273–317. Birkhauser Boston, Cambridge (1997) CrossRefGoogle Scholar
  8. 8.
    Milton, G.: On characterizing the set of possible effective tensors of composites: the variational method and the translation method. Commun. Pure Appl. Math. 43, 63–125 (1990) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Allaire, G., Kohn, R.: Optimal bounds on the effective behavior of a mixture of two well-ordered elastic materials. Q. Appl. Math. 51, 643–674 (1993) MathSciNetMATHGoogle Scholar
  10. 10.
    Pedersen, P.: On optimal shapes in materials and structures. Struct. Multidiscip. Optim. 19, 169–182 (2000) CrossRefGoogle Scholar
  11. 11.
    Hild, P., Münch, A., Ousset, Y.: On the active control of crack growth in elastic media. Comput. Methods Appl. Mech. Eng. 198(3–4), 407–419 (2008) MATHCrossRefGoogle Scholar
  12. 12.
    Khludnev, A., Leugering, G.: On elastic bodies with thin rigid inclusions and cracks. Math. Methods Appl. Sci. 33(16), 1955–1967 (2010) MathSciNetMATHGoogle Scholar
  13. 13.
    Münch, A., Pedregal, P.: Relaxation of an optimal design problem in fracture mechanic: the anti-plane case. ESAIM Control Optim. Calc. Var. 16, 719–743 (2010) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Yin, X., Travitzky, N., Greil, P.: Three-dimensional printing of nanolaminated Ti3AlC2 toughened TiAl3–Al2O3 composites. J. Am. Ceram. Soc. 90(7), 2128–2134 (2007) CrossRefGoogle Scholar
  15. 15.
    Kato, J.: Material optimization for fiber reinforced composites applying a damage formulation. Ph.D. thesis, Institut für Baustatik und Baudynamik der Universität Stuttgart (2010) Google Scholar
  16. 16.
    Kato, J., Lipka, A., Ramm, E.: Multiphase material optimization for fiber reinforced composites with strain softening. Struct. Multidiscip. Optim. 39, 63–81 (2009) CrossRefGoogle Scholar
  17. 17.
    Kato, J., Ramm, E.: Optimization of fiber geometry for fiber reinforced composites considering damage. Finite Elem. Anal. Des. 46, 401–415 (2010) CrossRefGoogle Scholar
  18. 18.
    Kovtunenko, V.: Nonconvex problem for crack with nonpenetration. Z. Angew. Math. Mech. 85(4), 242–251 (2005) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Khludnev, A., Kovtunenko, V.: Analysis of Cracks in Solids. WIT Press, Ashurst (1999) Google Scholar
  20. 20.
    Benedict, B., Sokolowski, J., Zolesio, J.: Shape optimization for contact problems. In: Thoft-Christensen, P. (ed.) System Modelling and Optimization. Lecture Notes in Control and Information Sciences, vol. 59, pp. 790–799. Springer, Berlin (1984) CrossRefGoogle Scholar
  21. 21.
    Sokolowski, J., Zolesio, J.: Shape sensitivity analysis of contact problem with prescribed friction. Non-Linear Anal. 12, 1399–1411 (1988) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Sokolowski, J.: Shape sensitivity analysis of nonsmooth variational problems. In: Zolésio, J. (ed.) Boundary Control and Boundary Variations. Lecture Notes in Control and Information Sciences, vol. 100, pp. 265–285. Springer, Berlin (1988) CrossRefGoogle Scholar
  23. 23.
    Inglis, H., Geubelle, P., Matous, K., Tan, H., Huang, Y.: Cohesive modeling of dewetting in particulate composites: micromechanics vs. multiscale finite element analysis. Mech. Mater. 39(6), 580–595 (2007) CrossRefGoogle Scholar
  24. 24.
    Prechtel, M., Leugering, G., Steinmann, P., Stingl, M.: Towards optimization of crack resistance of composite materials by adjustment of fiber shapes. Eng. Fract. Mech. 78, 944–960 (2011) CrossRefGoogle Scholar
  25. 25.
    Haslinger, J., Maekinen, R.: Introduction to Shape Optimization. Cambridge University Press, Cambridge (2003) MATHGoogle Scholar
  26. 26.
    Hintermüller, M., Laurain, A.: Optimal shape design subject to elliptic variational inequalities. SIAM J. Control Optim. 49(3), 1015–1047 (2011) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Liu, G., Han, J., Zhang, J.: Exact penalty functions for convex bilevel programming problems. J. Optim. Theory Appl. 110(3), 621–643 (2001) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Outrata, J., Kocvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer, Dordrecht (1998) MATHGoogle Scholar
  29. 29.
    Prechtel, M., Leiva Ronda, P., Janisch, R., Hartmaier, A., Leugering, G., Steinmann, P., Stingl, M.: Simulation of fracture in heterogeneous elastic materials with cohesive zone models. Int. J. Fract. 168, 15–29 (2011) CrossRefGoogle Scholar
  30. 30.
    Leugering, G., Prechtel, M., Steinmann, P., Stingl, M.: A cohesive crack propagation model: mathematical theory and numerical solution. Commun. Pure Appl. Anal. (2011, accepted for publication) Google Scholar
  31. 31.
    Beremlijski, P., Haslinger, J., Kocvara, M., Outrata, J.: Shape optimization in contact problems with coulomb friction. SIAM J. Optim. 13, 561–587 (2002) MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Haslinger, J., Vlach, O.: Signorini problem with a solution dependent coefficient of friction (model with given friction): approximation and numerical realization. Appl. Math. 50, 153–171 (2005) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Geissler, G., Kaliske, M.: Time-dependent cohesive zone modelling for discrete fracture simulation. Eng. Fract. Mech. 77, 153–169 (2010) CrossRefGoogle Scholar
  34. 34.
    Langtangen, H.: Computational Partial Differential Equations: Numerical Methods and Diffpack Programming. Springer, Berlin (2003) Google Scholar
  35. 35.
    Wächter, A., Biegler, L.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006) MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Sundaram, R.: A First Course in Optimization Theory. Cambridge University Press, Cambridge (1996) MATHCrossRefGoogle Scholar
  37. 37.
    Schramm, H., Zowe, J.: A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results. SIAM J. Optim. 2(1), 121–151 (1992) MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Gill, P.E., Murray, W., Saunders, M.S.: SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Rev. 47(1), 99–131 (2005) MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Charlotte, M., Francfort, G., Marigo, J.J., Truskinovsky, L.: Revisiting brittle fracture as an energy minimization problem: comparison of Griffith and Barenblatt surface energy models. In: Benallal, A. (ed.) Proceedings of the Symposium on Continuous Damage and Fracture, pp. 1–12. Elsevier, Paris (2000) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Marina Prechtel
    • 1
  • Günter Leugering
    • 1
  • Paul Steinmann
    • 2
  • Michael Stingl
    • 3
  1. 1.Department of MathematicsUniversity of Erlangen-NurembergErlangenGermany
  2. 2.Department of Mechanical EngineeringUniversity of Erlangen-NurembergErlangenGermany
  3. 3.Department of MathematicsUniversity of Erlangen-NurembergErlangenGermany

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