Robust topology optimization for structures under interval uncertainty

Abstract

This paper proposes a new non-probabilistic robust topology optimization approach for structures under interval uncertainty, as a complementarity of the probabilistic robust topology optimization methods.Firstly, to avoid the nested double-loop optimization procedure that is time consuming in computations, the interval arithmetic is introduced to estimate the bounds of the interval objective function and formulate the design problem under the worst scenario. Secondly, a type of non-intrusive method using the Chebyshev interval inclusion function is established to implement the interval arithmetic. Finally, a new sensitivity analysis method is developed to evaluate the design sensitivities for objective functions like structural mean compliance with respect to interval uncertainty. It can overcome the difficulty due to non-differentiability of intervals and enable the direct application of gradient-based optimization algorithms, e.g. the Method of Moving Asymptotes (MMA), to the interval uncertain topology optimization problems. Several numerical examples are used to demonstrate the effectiveness of the proposed RTO method.

Robust topology optimization under load position uncertainty

“…Whenever the external load is, for some reason, imposed away from the nominal loading point, the horizontal bar will properly convey a part of the external load between the two main paths to balance their load‐transferring capacities. In addition, this optimal configuration result is also appreciably different from the RTO design with considering the load direction uncertainty, wherein an arch‐type structure was often obtained for providing the necessary lateral stiffness …”

“…The first example is a simple but heuristic topology optimization problem resulting in a single‐bar configuration in the loading point fixation condition . Figure shows a rectangular panel subjected to a single concentrated force of 10 kN applied nominally at the midpoint of the bottom edge, which is, however, subject to an uncertainty quantification ±10 mm laterally along the boundary.…”

“…As we know, there are generally two main approaches often used to deal with the topology optimization problems under uncertainties: RBTO and RTO, respectively. 7,13,[23][24][25][26] From the aforementioned illustration, it is recognized that the loading position of F 1 is an uncertain-but-bounded parameter to the structural layout design. Usually, the precise statistical…”

“…For the present topology optimization problem of the compliance minimization, the position uncertainty of an applied load should be properly incorporated into the optimal design process such that a robust structural configuration is constituted in the design domain to provide the necessary diversities of the load transfer paths, which are required to be consistent with the uncertainty quantification. To this end, it was usually needed to determine the worst loading condition among all the possible uncertain loading positions with a lower-level optimization formulation, and then the corresponding compliance sensitivities were utilized in Equation (23). However, it is important to note that this solution process for identifying the critical load case is extremely time consuming because the design objective of the lower-level optimization problem is generally a nonconvex function.…”

“…The first example is a simple but heuristic topology optimization problem resulting in a single-bar configuration in the loading point fixation condition. 13,14,21,23,24 Figure 2 shows a rectangular panel subjected to a single concentrated force of 10 kN applied nominally at the midpoint of the bottom edge, which is, however, subject to an uncertainty quantification ±10 mm laterally along the boundary. That is, the applied vertical force may not be applied exactly at the specific point The DTO in the classical sense of the loading point fixation is first implemented and the optimal material layout is shown in Figure 4A, which is a single-bar configuration to transfer the external load to the restraint boundary in the most straight and shortest way.…”