  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.
