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We consider the existence of optimal shapes in the context of thethermomechanical system of partial differential equations (PDE) using therecent approach based on elliptic regularity theory. We give an extended andimproved definition of the set of admissible shapes based on a class ofsufficiently differentiable deformation maps applied to a baseline shape. Theobtained set of admissible shapes again allows one to prove a uniform Schauderestimate for the elasticity PDE. In order to deal with thermal stress, arelated uniform Schauder estimate is also given for the heat equation. Specialemphasis is put on Robin boundary conditions, which are motivated fromconvective heat transfer. It is shown that these thermal Schauder estimates canserve as an input to the Schauder estimates for the elasticity equation. Thisis needed to prove the compactness of the (suitably extended) solutions of theentire PDE system in some state space that carries a c2-H\"older topology forthe temperature field and a C3-H\"older topology for the displacement. Fromthis one obtains he property of graph compactness, which is the essential toolin an proof of the existence of optimal shapes. Due to the topologies employed,the method works for objective functionals that depend on the displacement andits derivatives up to third order and on the temperature field and itsderivatives up to second order. This general result in shape optimization isthen applied to the problem of optimal reliability, i.e. the problem of findingshapes that have minimal failure probability under cyclic thermomechanicalloading.

Optimal Reliability for Components under Thermomechanical Cyclic Loading