Uncertainty analysis and optimization by using the orthogonal polynomials
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Engineering problems are generally described by mathematic models, and the parameters in mathematic models are usually assumed to be deterministic when solving these models. However, many parameters are hard to obtain accurately in practical application, which leads to the uncertainty of parameters. The uncertain parameters may induce the response of theoretical analysis that is quite different from the actual instance. In order to characterize the response of system more accurately, the uncertainty analysis methods need to be introduced. For the design optimization, considering the uncertainty may help to improve the reliability and robustness of design solution. This thesis investigates both the aleatory (random) uncertainty and epistemic uncertainty (expressed by interval variables in the thesis), by using the Polynomial Chaos (PC) expansion theory and Chebyshev polynomials approximation theory, respectively. Since there are many cases that both types of uncertainty are existed simultaneously, the hybrid uncertainty is also investigated in this thesis. A new hybrid uncertainty analysis method based on the orthogonal series expansion is proposed in this study, which solves the two types of uncertainty in one integral framework. The design optimization under uncertainty is also investigated based on the proposed uncertainty analysis method. The detailed content of this thesis is shown as follows. The interval uncertainty analysis theory is firstly studied in this thesis. By using the Chebyshev polynomials that have high accuracy in the approximation theory of polynomials, a new Chebyshev inclusion function based on the Chebyshev series expansion is proposed. The Chebyshev inclusion function can compress the wrapping effect of interval arithmetic more efficiently than the traditional Taylor inclusion function, especially for the interval computation of non-monotonic functions. On the other hand, the Chebyshev inclusion function does not require the derivatives information which has to be given in the computation of Taylor inclusion function. Therefore, the proposed Chebyshev inclusion function is quite easier to implement than the Taylor inclusion function. The Chebyshev inclusion function is applied to solve the ordinary differential equations (ODEs) and differential algebraic equations (DAEs) with interval parameters, which are used to solve the mechanical dynamic systems with interval parameters. Secondly, the random uncertainty analysis based on the PC expansion is investigated, where the polynomials series are used to approximate the response of a system with respect to the random variables. The hybrid uncertainty analysis method using the orthogonal series expansion is proposed, termed as Polynomial-Chaos-Chebyshev-Interval (PCCI) method, which is the combination of PC expansion method and Chebyshev interval method. Since both the polynomials used in PC expansion and Chebyshev polynomials belong to the orthogonal polynomials, the PCCI method investigates the random uncertainty and interval uncertainty under one integral framework. Two types of evaluation index of hybrid uncertainty are also proposed in the PCCI method, which is then used in the analysis of vehicle dynamics containing hybrid uncertainty. Thirdly, considering the interval uncertain parameters or variables existed in the optimization problems, the interval optimization design is investigated. To improve the computational efficiency of traditional nested optimization procedure in uncertainty optimization, the interval arithmetic is employed to delete its inner loop optimization. A new Chebyshev polynomials-based surrogate model is proposed to improve the computational efficiency in further. The numerical examples for the vehicle suspension design and truss structure design indicate that the interval optimization method has a good balance between the accuracy and efficiency. The interval optimization method is also employed to solve the continuous structural topology optimization problem with uncertain load conditions, which gives a more robust solution than the traditional deterministic topology optimization method. Lastly, the hybrid uncertainty optimization model is proposed by combining the PCCI method and the classical optimization algorithms. To use the traditional mathematical programming method, the sensitivity of objectives and constraints with uncertain parameters are derived. For the application, the proposed hybrid uncertainty optimization method is used in the optimization of a planar truss and a space truss structures. Compared with the deterministic optimization and pure random uncertainty optimization, the hybrid uncertainty optimization provides a more feasible solution.
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