Studies On Several Critical Issues In Confidence Structural Robust Design And Optimization

Non-probability robust design optimization is usually formulated as a Bi-level optimization model. In the upper level program, the aim is to find the best design within the feasible domain and the lower-level program is to examine the feasibility of a given design by finding the worst case structural response. It is the global optimality of the lower level problem that is the key point for solving the Bi-level optimization model. It is strongly pointed out that global optimality must be ensured for the lower-level program (at least theoretically) otherwise the feasibility of a given design may not be determined exactly. Moreover, the reliability of the so called "optimal robust design" cannot be assured. This issue has not been well addressed in the literatures.Non-convex lower level programs are often encountered in practical engineering applications. The classical gradient-based optimization method is not applicable for the lower level problem, because it often fails to solve the non-convex programs to global optimalily. A so-called "Confidence Robust Design and Optimization" method is proposed for truss structures in order to make sure the reliability of the optimal structural design, surrounding which several methods have been deeply discussed. The main results are listed below:Firstly, the extreme structural response analysis of truss structures is considered, taking the possible uncertainties of Young's moduli of the bars into consideration. The uncertainty of Young's modulus is described by a non-probability but bounded interval model. With the use of this uncertain model, it is shown that the structural responses such as nodal displacement and bar stress will attain their extreme values when Young's moduli of the bars take either their upper or lower bound value. Based on this conclusion and Simultaneous Analysis and Design (SAND) Formulation, a linear mixed 0-1 programming problem is constructed to find the extreme values of the structural response by the well-established branch and bound approach with global optimality.Secondly, a Bi-level program formulation is proposed for the confidence robust optimal design of trusses under load uncertainties, which can be described by ellipsoid models. Based on duality theorems, a confidence upper bound of the extremal structural response is obtained by solving a standard convex Linear Semi-definite Programming (LSDP) problem, which can be solved with global optimality to assure the feasibility of a given design. Based on the sensitivity analysis of the lower level SDP problem, the upper level program is performed by the classical gradient-based nonlinear optimization algorithm. Several numerical examples demonstrate the effectiveness of the proposed approach.In the last, two new formulations are presented for the confidence robust structural optimization under non-probabilistic stiffness uncertainties. With the use of quadratic embedding technique of uncertainty and the S-procedure lemma, a single-level Nonlinear Semi-definite Programming (NSDP) formulation is also proposed and its mathematical properties are analyzed. Furthermore, a Bi-level program formulation for confidence robust design is proposed. In order to ensure the strict feasibility of the optimal solution, in the lower-level of program, the constraints are imposed on the confidence upper bounds of the structural responses by constructing the "bounding ellipsoid", which can be obtained efficiently by solving some convex LSDP problems. The upper level programs are then solved by employing the classical gradient-based nonlinear optimization algorithms with the use of the SDP sensitivity analysis methods.