Robust topology optimisation with generalised probability distribution of loading

This paper presents an objective function formulation of robust topology optimisation. Noting that the most common objective for topology optimisation is minimisation of compliance, the robust objective in this paper considers minimisation of a weighted sum of expected and variance of compliance. The uncertainties considered are in the magnitude and direction of loading. What is unique about the approach is that the directional uncertainties are represented by angle rather than the resolution of the Cartesian components. The four components of the objective, i.e. expected and variance of compliance for uncertainties in magnitude and directions are each considered for all uncertainty distributions described by first and second moments. The formulations for expected compliance is reasonably straight forward and can be computed fast by considering one or three additional load cases. The formulations for variance is somewhat more complex however, they are analytically exact and can be computed very fast. Noting that a numerical robust optimisation approach is typically associated with high computational cost, the analytical formulation for high dimensional topology optimisation is significant.