Introducing Loading Uncertainty in Topology Optimization

Uncertainty is an important consideration in structural design and optimization to produce robust and reliable solutions. This paper introduces an efficient and accurate approach to robust structural topology optimization. The objective is to minimize expected compliance with uncertainty in loading magnitude and applied direction, where uncertainties are assumed normally distributed and statistically independent. This new approach is analogous to a multiple load case problem where load cases and weights are derived analytically to accurately and efficiently compute expected compliance and sensitivities. Illustrative examples using a level-set-based topology optimization method are then used to demonstrate the proposed approach. Nomenclature A = material property tensor a = element volume Ex = expected value of x f = surface tractions ffg = loading vector fi = entry in loading vector H = Heaviside function J = structural compliance [K] = stiffness matrix m = stiffness matrix order n = number of loads with uncertainty Px = probability density function of x p = body forces u = displacement field fug = displacement vector V = velocity function Vol = material volume limit v = virtual displacement field w = weighting factor = shape parameter = structural boundary t = time step " = strain tensor = positive value = angle of load application x, y = horizontal and vertical directions i;j = entry in the inverse stiffness matrix = constant to enforce volume constraint = mean loading magnitude = mean loading direction = standard deviation of loading magnitude = standard deviation of loading direction = implicit function , s = structural domain I