Topology optimization of continuum structures with stress constraints and uncertainties in loading

Abstract

SummaryTopology optimization using stress constraints and considering uncertainties is a serious challenge, since a reliability problem has to be solved for each stress constraint, for each element in the mesh. In this paper, an alternative way of solving this problem is used, where uncertainty quantification is performed through the first-order perturbation approach, with proper validation by Monte Carlo simulation. Uncertainties are considered in the loading magnitude and direction. The minimum volume problem subjected to local stress constraints is formulated as a robust problem, where the stress constraints are written as a weighted average between their expected value and standard deviation. The augmented Lagrangian method is used for handling the large set of local stress constraints, whereas a gradient-based algorithm is used for handling the bounding constraints. It is shown that even in the presence of small uncertainties in loading direction, different topologies are obtained when compared to a deterministic approach. The effect of correlation between uncertainties in loading magnitude and direction on optimal topologies is also studied, where the main observed result is loss of symmetry in optimal topologies.

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.…”

“…However, with the load position uncertainty considered, the iteration number increases remarkably due to involving the design sensitivity diversities in the solution process and the compliance minimization value becomes somewhat larger than the conventional deterministic counterpart. That is, excessive computational cost is surely necessary in the RTO problem under considering the loading uncertainties . Within the plot, some of the intermediate topology designs are provided to illustrate the evolutionary histories of the structural material layouts converging to the final optimums.…”

“…For the past decades, structural topology optimization problems with consideration of load uncertainties have received increasing interests in the design optimization community. Thus far, a variety of approaches and techniques have been proposed to characterize the uncertain loadings in their various aspects, and a comprehensive study of the associated problems under the load magnitude and/or direction uncertainties has been made in the literature . Usually, these loading uncertainties are assumed as the uncertain‐but‐bounded design parameters, and then the optimal design problem is expressed to minimize the worst objective function induced by the uncertain loading conditions (worst‐case method).…”

Robust topology optimization of compliant mechanisms with uncertainties in output stiffness

“…One key aspect of designs obtained by means of optimization is their extreme sensitivity to any change in the parameters considered during optimization. Thus, slight changes in the magnitude and/or direction of the loads or in the distribution of material properties can lead to large changes in the design or in the effective response of the optimized component.…”