Stochastic stress-based topology optimization of structural frames based upon the second deviatoric stress invariant

摘要

This work pertains to the stochastic stress-based topology optimization of frame structures considering uncertainty. Specifically, this paper presents an investigation of the second deviatoric stress invariant, J(2), as the measure of stress in the domain where the objective is to minimize the maximum of the expected values of the stress invariant. Analytical expressions for the expected value of the J(2) invariant are developed employing the perturbation approach. The premise being that the J(2) invariant eliminates the square root operator that would otherwise be present if using the von Mises stress, and by eliminating this operator, the nonlinearity in the functional mapping between the random input (magnitude and direction of external forces) and uncertain output (expected values of stress) is reduced. Hence, an improved accuracy can be achieved with an analytical approximation of a given order. The analytical expressions of the expected value of the J(2) invariant to the second order are obtained based on a Taylor series expansion along with the associated sensitivities for gradient-based optimization. The analytical expressions are implemented in an optimization scheme and applied for the design of three ground structures considering different variability in the input random variables. For each example, the relative error between the maximum of the expected values of the J(2) invariant obtained using the analytical expressions and corresponding values evaluated by performing stochastic finite element analysis, whereby the input distribution is sampled using Monte Carlo methods, finite element simulation performed for each sample realization, then computing the expected value of the stress measure using straightforward statistical expressions, and finally estimation of the maximum via the p-norm, were observed to be less than 2%, even for a coefficient of variation of 0.4. The optimized designs were again analyzed using the von Mises stress and the relative error again computed with respect to those obtained from Monte Carlo sampling and finite element simulation. In general, the relative error using the analytical expressions of the von Mises stress was greater than, except in a couple instances, those obtained using the J(2) invariant.