Abstract
We consider the probability of failure for components made of brittle materials under one time application of a load, as introduced by Weibull and Batdorf-Crosse. These models have been applied to the design of ceramic heat shields of space shuttles and to ceramic components of the combustion chamber in gas turbines, for example. In this paper, we introduce the probability of failure as an objective functional in shape optimization. We study the convexity and the lower semi-continuity properties of such objective functionals and prove the existence of optimal shapes in the class of shapes with a uniform cone property. We also shortly comment on shape derivatives and optimality conditions.
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Notes
We note that in [6] a different set of conditions is referred to as Neumann boundary conditions, hence the term natural boundary conditions in order to avoid confusion.
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Acknowledgments
We would like to thank Patricia Hülsmeier and Christoph Ziegeler for making their Ph.D. Theses available to us. We are grateful to Rolf Krause from ICS Lugano for interesting discussions. We also thank the referees for reading the submitted article very carefully and providing many suggestions for improvement.
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Communicated by Zenon Mroz.
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Bolten, M., Gottschalk, H. & Schmitz, S. Minimal Failure Probability for Ceramic Design Via Shape Control. J Optim Theory Appl 166, 983–1001 (2015). https://doi.org/10.1007/s10957-014-0621-8
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DOI: https://doi.org/10.1007/s10957-014-0621-8