The Interval Uncertain Optimization Strategy Based on Chebyshev Meta-model

Abstract

This paper proposes a new design optimization method for structures subject to uncertainty. Interval model is used to account for uncertainties of uncertain-but-bounded parameters. It only requires the determination of lower and upper bounds of an uncertain parameter, without necessarily knowing its precise probability distribution. The interval uncertain optimization problem containing interval design variables and/or interval parameters will be formulated as a nested double-loop procedure, in which the outer loop optimization updates the midpoint of interval variables while the inner loop optimization calculates the bounds of objective and constraints. However, the nested double-loop optimization strategy will be computationally prohibitive, and it may be trapped into some local optimal solutions. To reduce the computational cost, the interval arithmetic is applied to the inner loop to directly evaluate the bounds of interval functions, so as to eliminate the optimization of the inner loop. The Taylor interval inclusion function is introduced to control the overestimation induced by the intrinsic wrapping effect of interval arithmetic. Since it is hard to evaluate the high-order coefficients in the Taylor inclusion function, a Chebyshev meta-model is proposed to approximate the Taylor inclusion function. Two numerical examples are used to demonstrate the effectiveness of the proposed method in the uncertain design optimization.