Application of Order-Preserving Functions to the Modeling of Computational Mechanics Problems with Uncertainty

In many engineering problems the shape of the structure is not exactly known. In that situation it is possible to consider a family of shapes which belong to the interval set Ω ∈ [Ω,Ω]. In order to find upper and lower bound of the solution, which depend on uncertain (interval) shape, it is possible to use properties of topological derivative or differentials which are positive definite (in many cases positive definite differentials are simply positive). Numerical examples are related to the heat transfer, imprecise probability, and integration on manifolds.