Comparison of robust, reliability-based and non-probabilistic topology optimization under uncertain loads and stress constraints

TL;DR

This paper compares three different approaches to topology optimization under uncertain loads and stress constraints: robust, reliability-based, and non-probabilistic, highlighting their differences in handling uncertainties and computational methods.

Contribution

It provides a comprehensive comparison of three distinct uncertainty handling methods in topology optimization, detailing their data requirements and computational strategies.

Findings

Robust approach uses mean and standard deviation of loads with first-order perturbation.

Reliability-based approach requires full probability distributions and reliability constraints.

Non-probabilistic approach considers worst-case loads using bounds and nested optimization.

Abstract

It is nowadays widely acknowledged that optimal structural design should be robust with respect to the uncertainties in loads and material parameters. However, there are several alternatives to consider such uncertainties in structural optimization problems. This paper presents a comprehensive comparison between the results of three different approaches to topology optimization under uncertain loading, considering stress constraints: 1) the robust formulation, which requires only the mean and standard deviation of stresses at each element; 2) the reliability-based formulation, which imposes a reliability constraint on computed stresses; 3) the non-probabilistic formulation, which considers a worst-case scenario for the stresses caused by uncertain loads. The information required by each method, regarding the uncertain loads, and the uncertainty propagation approach used in each case is quite different. The robust formulation requires only mean and standard deviation of uncertain loads; stresses are computed via a first-order perturbation approach. The reliability-based formulation requires full probability distributions of random loads, reliability constraints are computed via a first-order performance measure approach. The non-probabilistic formulation is applicable for bounded uncertain loads; only lower and upper bounds are used, and worst-case stresses are computed via a nested optimization with anti-optimization. The three approaches are quite different in the handling of uncertainties; however, the basic topology optimization framework is the same: the traditional density approach is employed for material parameterization, while the augmented Lagrangian method is employed to solve the resulting problem, in order to handle the large number of stress constraints.