IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v194y2022i2d10.1007_s10957-022-02041-y.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-022-02041-y
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-022-02041-y?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Kai Tu & Haibin Zhang & Huan Gao & Junkai Feng, 2020. "A hybrid Bregman alternating direction method of multipliers for the linearly constrained difference-of-convex problems," Journal of Global Optimization, Springer, vol. 76(4), pages 665-693, April.
    2. Peng, Bo & Xu, Hong-Kun, 2020. "Proximal methods for reweighted lQ-regularization of sparse signal recovery," Applied Mathematics and Computation, Elsevier, vol. 386(C).
    3. Ouayl Chadli & Joachim Gwinner & M. Zuhair Nashed, 2022. "Noncoercive Variational–Hemivariational Inequalities: Existence, Approximation by Double Regularization, and Application to Nonmonotone Contact Problems," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 42-65, June.
    4. M. Hintermüller & I. Kopacka, 2011. "A smooth penalty approach and a nonlinear multigrid algorithm for elliptic MPECs," Computational Optimization and Applications, Springer, vol. 50(1), pages 111-145, September.
    5. Daria Ghilli & Karl Kunisch, 2019. "On a Monotone Scheme for Nonconvex Nonsmooth Optimization with Applications to Fracture Mechanics," Journal of Optimization Theory and Applications, Springer, vol. 183(2), pages 609-641, November.
    6. Hao Jiang & Daniel P. Robinson & René Vidal & Chong You, 2018. "A nonconvex formulation for low rank subspace clustering: algorithms and convergence analysis," Computational Optimization and Applications, Springer, vol. 70(2), pages 395-418, June.
    7. Daria Ghilli & Karl Kunisch, 2019. "On monotone and primal-dual active set schemes for $$\ell ^p$$ ℓ p -type problems, $$p \in (0,1]$$ p ∈ ( 0 , 1 ]," Computational Optimization and Applications, Springer, vol. 72(1), pages 45-85, January.
    8. Yaohua Hu & Chong Li & Kaiwen Meng & Xiaoqi Yang, 2021. "Linear convergence of inexact descent method and inexact proximal gradient algorithms for lower-order regularization problems," Journal of Global Optimization, Springer, vol. 79(4), pages 853-883, April.
    9. Mahdi Boukrouche & Domingo Tarzia, 2012. "Convergence of distributed optimal control problems governed by elliptic variational inequalities," Computational Optimization and Applications, Springer, vol. 53(2), pages 375-393, October.
    10. Minh N. Dao & Joachim Gwinner & Dominikus Noll & Nina Ovcharova, 2016. "Nonconvex bundle method with application to a delamination problem," Computational Optimization and Applications, Springer, vol. 65(1), pages 173-203, September.
    11. Carolin Natemeyer & Daniel Wachsmuth, 2021. "A proximal gradient method for control problems with non-smooth and non-convex control cost," Computational Optimization and Applications, Springer, vol. 80(2), pages 639-677, November.
    12. Yarui Duan & Song Wang & Yuying Zhou, 2021. "A power penalty approach to a mixed quasilinear elliptic complementarity problem," Journal of Global Optimization, Springer, vol. 81(4), pages 901-918, December.
    13. Nina Ovcharova & Joachim Gwinner, 2016. "Semicoercive Variational Inequalities: From Existence to Numerical Solution of Nonmonotone Contact Problems," Journal of Optimization Theory and Applications, Springer, vol. 171(2), pages 422-439, November.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:194:y:2022:i:2:d:10.1007_s10957-022-02041-y. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.