  In this paper, the problem of load uncertainty in compliance problems is
addressed where the uncertainty is described in the form of a set of finitely
many loading scenarios. Computationally more efficient methods are proposed to
exactly evaluate and differentiate: 1) the mean compliance, or 2) any
scalar-valued function of the individual load compliances such as the weighted
sum of the mean and standard deviation. The computational time complexities of
all the proposed algorithms are analyzed, compared with the naive approaches
and then experimentally verified. Finally, a mean compliance minimization
problem, a risk-averse compliance minimization problem and a maximum compliance
constrained problem are solved to showcase the efficacy of the proposed
algorithms. The maximum compliance constrained problem is solved using the
augmented Lagrangian method and the method proposed for handling scalar-valued
functions of the load compliances, where the scalar-valued function is the
augmented Lagrangian function.
