Modeling uncertainty in three-dimensional heat transfer problems

We present a generalized polynomial chaos method to solve the steady and unsteady heat transfer problems with uncertainty in boundary conditions, diffusivity coefficient and forcing terms. The stochastic inputs and outputs are represented spectrally by employing the orthogonal polynomial functionals from the Askey scheme, as a generalization of the original polynomial chaos idea of Wiener [1]. A Galerkin projection in random space is applied to derive the equations in weak form, and a parallel spectral/hp element method is employed to solve the resulting set of deterministic equations. Simulations in three-dimensional domains with stochastic dimension 38 and about 150 million unknowns are presented here for the first time.