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 StinglEmail author


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.


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Marina Prechtel
    • 1
  • Günter Leugering
    • 1
  • Paul Steinmann
    • 2
  • Michael Stingl
    • 3
    Email author
  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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