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Shape Derivative for Penalty-Constrained Nonsmooth–Nonconvex Optimization: Cohesive Crack Problem

Author

Listed:
  • Victor A. Kovtunenko

    (Karl-Franzens University of Graz
    Siberian Division of the Russian Academy of Sciences)

  • Karl Kunisch

    (Karl-Franzens University of Graz
    Radon Institute, Austrian Academy of Sciences, RICAM Linz)

Abstract

A class of non-smooth and non-convex optimization problems with penalty constraints linked to variational inequalities is studied with respect to its shape differentiability. The specific problem stemming from quasi-brittle fracture describes an elastic body with a Barenblatt cohesive crack under the inequality condition of non-penetration at the crack faces. Based on the Lagrange approach and using smooth penalization with the Lavrentiev regularization, a formula for the shape derivative is derived. The explicit formula contains both primal and adjoint states and is useful for finding descent directions for a gradient algorithm to identify an optimal crack shape from a boundary measurement. Numerical examples of destructive testing are presented in 2D.

Suggested Citation

  • Victor A. Kovtunenko & Karl Kunisch, 2022. "Shape Derivative for Penalty-Constrained Nonsmooth–Nonconvex Optimization: Cohesive Crack Problem," Journal of Optimization Theory and Applications, Springer, vol. 194(2), pages 597-635, August.
    • Handle: RePEc:spr:joptap:v:194:y:2022:i:2:d:10.1007_s10957-022-02041-y
    DOI: 10.1007/s10957-022-02041-y
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    References listed on IDEAS

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    1. M. Bergounioux, 1997. "Use of Augmented Lagrangian Methods for the Optimal Control of Obstacle Problems," Journal of Optimization Theory and Applications, Springer, vol. 95(1), pages 101-126, October.
    2. Nina Ovcharova & Joachim Gwinner, 2014. "A Study of Regularization Techniques of Nondifferentiable Optimization in View of Application to Hemivariational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 162(3), pages 754-778, September.
    3. Kristian Bredies & Dirk A. Lorenz & Stefan Reiterer, 2015. "Minimization of Non-smooth, Non-convex Functionals by Iterative Thresholding," Journal of Optimization Theory and Applications, Springer, vol. 165(1), pages 78-112, April.
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