Robust topology optimization with hybrid uncertainties using level set methods

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Topology optimization has been experiencing great popularity in a diversity of engineering areas. Parameters involved in most topology optimization problems are under deterministic assumptions. However, in practical applications, uncertainties are inevitably existing due to various reasons, such as manufacturing tolerances, loads, material properties, geometric dimensions and boundary conditions, as well as aging within the whole life circle of structural service. In particular, in the conceptual design by the topology optimization, more reliable results can be expected if uncertainties are taken into account, as the performance of a topological design varies with the uncertainties. In this setting, the deterministic assumption may result in a design that is unfeasible. Hence, it is of great importance to incorporate uncertainties into the topology optimization to account for unavoidable variations. Probability models have been widely used to describe the uncertainties of parameters in structures, which in general require a sufficient number of samples to completely construct the distributions. However, in engineering, it is very difficult to gain complete information to accurately describe the probability distributions, while it is relatively easy to get their interval bounds for limited information. In practice, it is recognized that a structure often involves uncertainties of multiple sources, in which some uncertain parameters can be regarded as random variables and the others can be modelled as interval variables. Hence, a design problem under random-interval hybrid uncertainties consists of both the aleatory and epistemic uncertainties at the same time. In this thesis, the hybrid uncertainties will be considered in topology optimization problems to achieve robust designs. The detailed contents are outlined as follows: Chapter 1 provides a brief introduction for this research. Chapter 2 gives the background and a literature review. Chapter 3 describes the details of a parametric level set method (PLSM) based on compactly supported radial basis functions (CSRBFs). Some efficient random-interval hybrid uncertain analysis methods are developed in Chapter 4. In the following Chapters, the uncertainty analysis methods are then employed to formulate robust topology optimizations for structures with hybrid uncertainties, as follows: In Chapter 5, robust topology optimization methods based on orthogonal polynomials are developed for both static and dynamic continuum structures with hybrid uncertainties. In Chapter 6, robust topology optimization methods based on dimension reduction methods are developed for the multi-scale design of static and dynamic structures with hybrid uncertainties. Finally, conclusions and prospects are given in Chapter 7.
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