Research On Non-Robabilistic Reliability-Based Topology Optimization Design Of Interval Parameter Structures

A variety of uncertainties, such as material properties, geometric dimensions, external loads and boundary conditions, are inherent in practical engineering problems. The uncertainties usually result in the failure of structural designs. The design optimization based on the reliability theory is an effective approach to avoid the structural failure in presence of uncertainties. Based on the interval model, the research mainly focuses on the non-probabilistic reliability-based topology optimization of continuum structures under static or dynamic external loads.Firstly, the definition and measurement of non-probabilistic reliability index based on interval model are studied. Based on the global convergent version of the method of moving asymptotes, an algorithm is proposed to solve the interval non-probabilistic reliability index. Examples show that the proposed algorithm is efficient and stable, especially for complex problems.Secondly, for the case that uncertainties only come from external loads, the external loads are expressed into an equivalent form by the sum of uncorrelated loads. Based on the definition of interval non-probabilistic reliability index, the reliability-based topology optimization model of continuum structures is established using equivalent loads. And explicit descriptions of non-probabilistic reliability constraints are presented. Different from the previous two-stage optimization process, the inner optimization process that finds the worst condition case of the reliability problem is not required in this model, which greatly improves the efficiency of structural optimization.For the case that uncertainties come from geometric dimensions, material properties, external loads, etc, an approach of constructing the equivalent loads for uncertainties is proposed. In the method, all the structural uncertainties are transferred into equivalent loads for uncertainties, and the original uncertain structure is converted into a deterministic structure. We also present the feasibility condition to guarantee the accuracy of structural optimization. From the results of topology optimization, we can find that uncertainties have a large effect on the structural optimization, and that reliability-based topology optimization usually suggests more reasonable and more reliable structural topologies different from deterministic optimization.In order to consider the structural geometrically nonlinearity, a novel equivalent static loads method for uncertainties is proposed using the traditional equivalent static loads method for reference. The equivalent static loads are redefined to consider uncertainties. In the new definition, equivalent static loads can generate the same interval displacement response fields in linear static analysis as actual loads of nonlinear static analysis. Based on the interval non-probabilistic reliability index, two linear non-probabilistic reliability-based optimization models are formulated using the new equivalent static loads respectively for size optimization and topology optimization. In order to avoid directly solving the nested double-loop reliability optimization problem, the performance measure approach is utilized to convert constraints on the reliability index into constraints on the concerned performance. To deal with the mesh distortion of geometrically nonlinear analysis in topology optimization, low density elements are temporarily removed before geometrically nonlinear response analysis.Finally, for the reliability-based topology optimization of interval parameter structures under dynamic loads, another definition of equivalent static loads for uncertainties is proposed to consider structural dynamic characteristics. In the new definition, equivalent static loads for uncertainties can generate the same interval displacement fields in static response analysis as original dynamic loads of dynamic response analysis. Based on the dynamic response interval analysis method, the new equivalent static loads for uncertainties for dynamic response problems is constructed. Using the equivalent static loads for uncertainties, the original dynamic response reliability-based topology optimization problems are converted into the static response problems, which greatly reduces the solving difficulty. Based on the definition of interval non-probabilistic reliability index, we respectively formulated two reliability-based topology optimization models. One is to minimize the structural material volume with displacement reliability constraints. The other is to minimize the structural compliance with constraints on the material volume and bounds of uncertainties. Numerical examples demonstrate the applicability and validity of the proposed model and numerical techniques.