Collocation methods for uncertain heat convection-diffusion problem with interval input parameters

• Collocation methods are developed to the interval uncertainty analysis.
• High-order Legendre polynomial series is adopted for temperature approximation.
• Continuously-differentiable property of polynomial function is used for extrema.
• Smolyak-type sparse grids are constructed to avoid the curse of dimensionality.

This paper proposes a full grid interval collocation method (FGICM) and a sparse grid interval collocation method (SGICM) to solve the uncertain heat convection-diffusion problem with interval input parameters in material properties, applied loads and boundary conditions. The Legendre polynomial series is adopted to approximate the functional dependency of temperature response with respect to the interval parameters. In the process of calculating the expansion coefficients, FGICM evaluates the deterministic solutions directly on the full tensor product grids, while the Smolyak sparse grids are reconstructed in SGICM to avoid the curse of dimensionality. The eventual lower and upper bounds of temperature responses are easily predicted based on the continuously-differentiable property of the approximate function. Comparing results with traditional Monte Carlo simulations and perturbation method, the numerical example evidences the remarkable accuracy and effectiveness of the proposed methods for interval temperature field prediction in engineering.