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Construction and application of an ellipsoidal convex model using a semi-definite programming formulation from measured data

机译:基于实测数据的半定规划公式构建椭球凸模型

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As a set theory-based convex model, the ellipsoidal model provides an attractive framework for treating uncertain-but-bounded variations in the structural reliability analysis and design optimization. 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 systematic study on the mathematical formulation for constructing the minimum-volume ellipsoidal convex model using a given set of sample data, and shows its application in existing methods of non-probabilistic reliability analysis and design optimization of structures with bounded uncertainties. 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. Further, a linear transformation based on the eigenvalue analysis is employed to map the ellipsoidal model into a standard uncertainty space. This uncertainty modeling technique enables a compact and differentiable bound description of the parameter variations. Moreover, it has another useful property, the affine invariance, which is shown to be necessary for meaningful definition of a non-probabilistic reliability index. 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. (C) 2015 Elsevier B.V. All rights reserved.
机译:作为基于理论的凸模型,椭球模型为处理结构可靠度分析和设计优化中的不确定但有界的变化提供了一个有吸引力的框架。但是,不确定性的不正确建模可能会引起误导性的非概率可靠性分析,从而导致设计不安全或过于保守。本文对使用给定的一组样本数据构建最小体积椭球凸模型的数学公式进行了系统的研究,并展示了其在现有的非概率可靠性分析方法和有限不确定性结构设计优化中的应用。在这种方法中,首先根据不确定性的来源将不确定性参数分为几类。对于不确定性的每个单独组,最小体积椭球问题被重新表述为半定规划(SDP)问题,因此可以有效地求解其全局最优值。此外,基于特征值分析的线性变换被用于将椭圆模型映射到标准不确定空间中。这种不确定性建模技术可以对参数变化进行紧凑且可区分的边界描述。而且,它还有另一个有用的属性,仿射不变性,对于非概率可靠性指标的有意义的定义,它是必需的。通过结构拓扑优化问题的数值示例,证明了本技术对凸模型构建和相应可靠性分析的有效性和效率,该问题具有源于不同来源的有限变化。 (C)2015 Elsevier B.V.保留所有权利。

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