Analyses for lower and upper bounds of sobol' total sensitivity index based on derivative

A main sensitivity index S_i was set as the lower bound of S_i^tot and the new upper bound of S_i^tot was built based on the derivative. On the basis of functional analysis and Euler-Lagrange equation, the new upper bound of S_i^tot based derivative was analyzed for different variable distribution types, such as uniform, normal, exponential, triangular, Beta distribution etc. The derived process and formulas were presented in detail. Several numerical and engineering examples were used to verify the precision and efficiency of the presented bounds, which may provide the accurate bounds of S_i^tot.