Modelling And Topology Optimization Of Structures With Uncertainties

The inevitable uncertainties in structure need to be treated according to their features and sources. When they are bounded without distribution information and cannot be described with probability model with sufficient samples, the non-probabilistic convex model is suitable due to the fact that it provide a smooth and convex bound reference for such uncertainties. However, improper modeling of the uncertainties may give rise to misleading non-probabilistic reliability analysis, thus result in either unsafe or over-conservative designs. This paper presents a construction method for minimum-volume ellipsoidal convex model based on semi-definite programming (SDP). In this method, the uncertain parameters are first divided into groups according to their sources. For each individual group of uncertainties, the minimum-volume ellipsoid problem is reformulated into a semi-definite programming (SDP) problem and thus can be efficiently solved to its global optimum. It is also applied in non-probabilistic reliability analysis and design optimization of structures with bounded uncertainties. The effectiveness and efficiency of the present techniques for convex model construction and the corresponding reliability analysis are demonstrated with numerical examples of structural topology optimization problems with bounded variations arising from different sources.Under the classification of uncertainty sources, the reason why geometric uncertainties are different from load uncertainties and material uncertainties, especially the possible topological changes arising from photolithography or additive manufacturing, is that they present a new challenge:How to account for such uncertainties in modelling, analysis and optimization. This thesis address this issue by developing an implicit characterization method with stochastic level set perturbation based on Karhunen-Loeve expansion. The analysis is treated by a pseudo spectral version of polynomial chaos expansion (PCE). In the robust topology optimization stage, an adjoint-variable shape sensitivity scheme is developed to account for geometric variations and topological changes. The numerical example concerning with compliance shows an exponential convergence of PCE. The effectiveness of the optimization method is demonstrated in compliance minimization problems. This robust topology optimization method also shows length-scale control effects.