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Optimal Design of Brittle Composite Materials: a Nonsmooth Approach

Author

Listed:
  • Marina Prechtel

    (University of Erlangen-Nuremberg)

  • Günter Leugering

    (University of Erlangen-Nuremberg)

  • Paul Steinmann

    (University of Erlangen-Nuremberg)

  • Michael Stingl

    (University of Erlangen-Nuremberg)

Abstract

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.

Suggested Citation

  • Marina Prechtel & Günter Leugering & Paul Steinmann & Michael Stingl, 2012. "Optimal Design of Brittle Composite Materials: a Nonsmooth Approach," Journal of Optimization Theory and Applications, Springer, vol. 155(3), pages 962-985, December.
    • Handle: RePEc:spr:joptap:v:155:y:2012:i:3:d:10.1007_s10957-012-0094-6
    DOI: 10.1007/s10957-012-0094-6
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    References listed on IDEAS

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    1. Sundaram,Rangarajan K., 1996. "A First Course in Optimization Theory," Cambridge Books, Cambridge University Press, number 9780521497190, October.
    2. G. S. Liu & J. Y. Han & J. Z. Zhang, 2001. "Exact Penalty Functions for Convex Bilevel Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 110(3), pages 621-643, September.
    3. Sundaram,Rangarajan K., 1996. "A First Course in Optimization Theory," Cambridge Books, Cambridge University Press, number 9780521497701, October.
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