Uncertainty quantification for a 1D thermo-hyperelastic coupled problem using polynomial chaos projection and p-FEMs Dedicated to our friend, Prof. Ernst Rank, on the occasion of his 60th birthday.

Numerical solutions of non-linear stochastic thermo-hyperelastic problems at finite strains are addressed. These belong to a category of non-linear coupled problems that impose challenges on their numerical treatment both in the physical and stochastic spaces. Combining the high order finite element methods (FEMs) for discretizing the physical space and the polynomial chaos projection (PCP) method for discretizing the stochastic space, a non-intrusive scheme is obtained manifesting an exponential convergence rate. The method is applied to a 1-D coupled, stationary, thermo-hyperelastic system with stochastic material properties. We derive exact stochastic solutions that serve for comparison to numerical results, allowing their verification. These demonstrate that stochastic coupled-problems intractable by standard Monte-Carlo (MC) methods may be easily computed by combining high-order FEMs with the PCP method controlling discretization errors.