Robust Topology Optimization: Minimization of Expected and Variance of Compliance

Robust topology optimization has long been considered computationally intractable as it combines two highly expensive computational strategies. This paper considers simultaneous minimization of expectancy and variance of compliance in the presence of uncertainties in loading magnitude, using exact formulations and analytically derived sensitivities. It shows that only a few additional load cases are required, which scales in polynomial time with the number of uncertain load cases.The approach is implementedusing the level set topologyoptimizationmethod. Shape sensitivities are derived using the adjoint method. Several examples are used to investigate the effect of including variance in robust compliance optimization. Nomenclature A = elasticity tensor C = compliance of the structure EC = expected compliance Em = Young’s modulus fi = load with uncertain magnitude Hϕ = Heaviside function K = stiffness matrix k = current iteration number m = order of the stiffness matrix n = number of uncertain loads p = solution to adjoint problem Pfi = probability density function for load i RbC = combined robust compliance objective function t = fictitious time domain u = displacement ui = displacement field for the single load of magnitude σi uμ = displacement field for the mean loading conditions VarC = variance of compliance Vol = volume constraint Vn = velocity function normal to the boundary v = virtual displacement w = nondimensional factor ΓS = boundary of the structure Δt = time step ε = strain tensor ςm = sensitivity of expected compliance ςv = sensitivity of compliance variance η = weighting factor for the multiple objectives κi;j = entry in the inverse stiffness matrix λ = Lagrange multiplier μi = mean magnitude of uncertain load i σ = covariance σi = standard deviation of uncertain load i ϕx = level set implicit function Ω = design domain ΩS = structural domain I