Computational Mechanics

, Volume 64, Issue 4, pp 895–915 | Cite as

Shape gradients for the failure probability of a mechanic component under cyclic loading: a discrete adjoint approach

  • Hanno GottschalkEmail author
  • Mohamed Saadi
Original Paper


This work provides a numerical calculation of shape gradients of failure probabilities for mechanical components using a first discretize, then adjoin approach. While deterministic life prediction models for failure mechanisms are not (shape) differentiable, this changes in the case of probabilistic life prediction. The probabilistic, or reliability based, approach thus opens the way for efficient adjoin methods in the design for mechanical integrity. In this work we propose, implement and verify a method for the numerical calculation of the shape gradients of failure probabilities for the failure mechanism low cycle fatigue, which applies to polycrystalline metal. Numerical examples range from a bended rod to a complex geometry from a turbo charger in 3D.


Shape gradients Failure probabilities Adjoint equation for structural mechanics 

Mathematics Subject Classification

49Q12 74P10 65C50 



We thank the Siemens Gas Turbine Engineering Department for Probabilistic Design for valuable discussions and support, in Dr. Georg Rollmann and Dr. Sebastian Schmitz in particular. We also thank Dr. T. Seibel (FZ Jülich), Prof. Dr. T. Beck (Technical University of Kaiserslautern) and R. Krause (ICS Lugano) for prior joint work on probabilistic LCF, on which this paper is based. This work has been made possible by financial support under the AG Turbo Grant 4.1.13 funded by Siemens Power and Gas, the German Federal Ministry of Economic Affairs (BMWi) under the Grant No. 03ET7041J and by the BMBF project GIVEN under the Grant No. 05M2018. Let us also thank both anonymous referees for their help in improving this manuscript.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of WuppertalWuppertalGermany

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