Non-probabilistic reliability-based topology optimization of geometrically nonlinear structures using convex models

This paper describes a non-probabilistic reliability-based topology optimization method for the design of continuum structures undergoing large deformations. The variation of the structural system is treated with the multi-ellipsoid convex model, which is a realistic description of the parameters being inherently uncertain-but-bounded or lacking sufficient probabilistic data. The formulation of the optimal design is established as a volume minimization problem with non-probabilistic reliability constraints on the geometrically nonlinear structural behaviour. In order to circumvent numerical difficulties in solving the nested double-loop optimization problem, a performance measure-based approach is employed to transform the constraint on the reliability index into one on the concerned performance. In conjunction with an efficient adjoint variable scheme for the sensitivity analysis of reliability constraints, the optimization problem is solved by gradient-based mathematical programming methods. Three numerical examples for the optimization design of planar structures are presented to illustrate the validity and applicability of the proposed method. The obtained optimal solutions show the importance of incorporating various uncertainties in the design problem. Moreover, it is also revealed that the geometrical nonlinearity needs to be accounted for to ensure satisfaction of the reliability constraints in the optimal design of structures with large deformation.