Collocation methods for fuzzy uncertainty propagation in heat conduction problem

Abstract

Based on the combination of collocation technology and fuzzy theory, this paper proposes a full grid fuzzy collocation method (FGFCM) and a sparse grid fuzzy collocation method (SGFCM) for fuzzy uncertainty propagation in heat conduction problem. Converting fuzzy parameters into interval variables by level-cut strategy, the Legendre polynomial series provides a surrogate function for temperature response. To calculate the expansion coefficients, FGFCM evaluates the deterministic solutions directly on the full tensor product grids, whereas Smolyak algorithm is introduced in SGFCM to reduce the number of collocation points. According to the smoothness property of surrogate function and fuzzy decomposition theorem, the interval bounds and membership functions of uncertain temperature response are derived, respectively. Comparing result with traditional Monte Carlo simulation and parameter perturbation method, two numerical examples evidence the remarkable accuracy and effectiveness of proposed methods for fuzzy temperature field prediction in engineering.