Topology optimization of structures under constraints on first passage probability

Abstract

A new method is proposed to incorporate the first passage probability into stochastic topology optimization using sequential compounding method (Kang and Song 2010). Parameter sensitivities of the first passage probability in the probabilistic constraint are derived to facilitate the use of gradient-based optimizer for efficient topology optimization. The proposed method is applied to building structures subjected to stochastic ground motion to find optimal bracing systems which can resist future realization of stochastic excitations while achieving a desired level of reliability. Optimal design of a lateral load-resisting system of a structure is one of the essential tasks in structural engineering as it is directly linked to building safety and operation. In particular, reliable operation and safety under stochastic excitations by natural hazards such as earthquake, wind loads are major design objectives. However, deterministic description of future realization of a random process is frequently limited because only a set of few time histories are available. Therefore, a probabilistic prediction of structural responses based on random vibration analysis is much needed in the process for optimal design. To address this issue, the authors performed a study of topology optimization of structures under stochastic excitations (Chun et al. in review). In the study, random vibration analysis by a discrete representation method (Der Kiureghian 2000) and structural reliability theory were integrated into topology optimization framework. In addition, the authors developed the system reliability-based topology optimization framework under stochastic excitations (Chun et al. 2013) to consider system failure events with statistical dependency using the matrix-based system reliability method (Song and Kang 2009). The developed method helps satisfy probabilistic constraints on a system failure event, which consists of multiple limit-states defined in terms of different locations, failure modes and time points as it optimizes a structural system. Chun et al. (in review) has evaluated an instantaneous failure probability of the structure subjected to random excitations at a discrete time point. However, a more practical application in engineering can be achieved if the failure probability is evaluated for exceedance event over a time interval. This helps promote the use of the proposed stochastic topology optimization framework for the design of lateral load-resisting system under stochastic excitations. Thus, in this paper, a stochastic topology optimization framework is proposed to handle probabilistic constraints on the first passage probability. 1. RANDOM VIBRATION ANALYSIS USING DISCRETE REPRESENTATION METHOD 1.1. Discrete representation of stochastic process The discrete representation method (Der Kiureghian 2000) discretizes a continuous 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 2 stochastic process with a finite number of standard normal random variables. For example, a zero-mean Gaussian process f(t) can be discretized as: