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Optimal reliability for components under thermomechanical cyclic loading

Bittner L., Gottschalk H.

Control and Cybernetics

2016

Vol. 45, no. 4

421--455

We consider the existence of optimal shapes in a context of the thermo-mechanical system of partial diﬀerential equations (PDE) using the recent approach based on elliptic regularity theory (Gottschalk and Schmitz, 2015; Agmon, Douglis and Nirenberg, 1959,1964; Gilbarg and Trudinger, 1977). We give an extended and improved deﬁnition of the set of admissible shapes based on a class of suﬃciently diﬀerentiable deformation maps applied to a baseline shape. The obtained set of admissible shapes again allows to prove a uniform Schauder estimate for the elasticity PDE. In order to deal with thermal stress, a related uniform Schauder estimate will be derived for the heat equation. Special emphasis is put on Robin boundary conditions, which are motivated by the convective heat transfer processes. It is shown that these thermal Schauder estimates can serve as an input to the Schauder estimates for the elasticity equation (Gottschalk and Schmitz, 2015). This is needed to prove the compactness of the (suitably extended) solutions of the entire PDE system in some state space that carries a C2-Hölder topology for the temperature ﬁeld and a C3-Hölder topology for the displacement. From this, one obtains the property of graph compactness, which is the essential tool to prove the existence of optimal shapes. Due to the topologies employed, the method works for objective functionals that depend on the displacement and its derivatives up to third order, as well as on the temperature ﬁeld and its derivatives up to second order. This general result in shape optimization is then applied to the problem of optimal reliability, i.e. the problem of ﬁnding shapes that have minimal failure probability under cyclic thermomechanical loading.