Course by Hanno Gottschalk - Optimal reliability: Stochastic failure time processes and geometry optimization  02 - 03 June 2015

10:00 - 12:00


Hanno Gottschalk, University of Wuppertal

1.    The first lecture gives an outline on the engineering practice of fatigue life calculation for mechanical components under cyclic loading. The elasticity PDE and the weak solution theory are developed, physical failure mechanisms as Low Cycle Fatigue are reviewed. The combination of Neuber- and Glinker shake-down, Ramberg Osgood equantion and Coffin-Manson Basquin equation are explained with an emphasis to mathematical properties. Stochastic failure time processes are introduced on the basis of the first occurrence time of spatio-temporal point processes. A specific failure time model is introduced based on the Weibull distribution. It is explained, how the demand for optimal reliability, i.e. the choice of a form that minimizes failure probabilities, in a natural way leads to a problem in the field of shape optimization.  
2.    The second lecture reviews the fundamentals of shape optimization including  definitions of admissible shapes, shape compactness, the relation to the state equation and the requirement of lower semicontinuity. These features are combined into an abstract proof of the existence of optimal shapes. The role of regularity theory in the existence proof of shapes with optimal reliability properties is explained. We shortly introduce the theory of strong solutions and boundary regularity (Schauder esimates) that allow to show the crucial graph compactness properties needed in the abstract existence proofs. Uniform Schauder estimates are provided and the proof of existence of forms with minimal failure probabilities is completed.
3.    In the last lecture, an exposition of the notion of shape derivatives and shape gradients is given. We sketch a proof, how the existence of shape derivatives for very singular objective functionals, motivated by failure probabilities, can actually be constructed. Implications to optimization on the infinite dimensional manifold of shapes are given.

Name University Dates of visit
Hanno Gottschalk University of Wuppertal 02/06/2015 - 03/06/2015
Total Guests : 1
Name University Dates of visit
Total Guests : 0