Interval uncertain optimization of structures using Chebyshev meta-models

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 will be formulated as a nested double loop procedure, in which both the design variables and structural parameters are regarded as interval numbers. In practice, the nested double loop optimisation will be computationally prohibitive, so the linear optimization model is widely used. However, the linear optimization model induces large error for strong nonlinear model. To improve the accuracy and without increasing computation cost much, the interval arithmetic is applied to the inner loop to directly evaluate the bounds of interval functions, replacing the linear model. The interval arithmetic is easily subject to overestimation due to its intrinsic wrapping effect, so the Taylor inclusion function will be introduced to compress the overestimation in interval computations. 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. A typical 25-bar space truss structure optimization problem with interval uncertainties are used to demonstrate the effectiveness of the proposed method in the uncertain design optimization of structures.