Distribution of stochastic excitation model parameters
Chapter 7 Chapter 7 Conclusion
7.2 Future work
Regarding the development of simulation methods for solving first excursion problems, several directions may be pursued. The idea of importance sampling using elementary events is applicable for any linear system with additive Gaussian uncertainties. This includes excitation uncertainties modeled by Gaussian processes, as well as structural or material uncertainties modeled by random fields. In these cases, important sampling densities similar to the one proposed in this work can be formulated, where the main effort is to derive analytically the design points in terms of Green's functions. For example, an importance sampling density can be formulated for the case when the stochastic excitation is represented in the frequency domain, which will be useful for problems where the excitation is modeled with a target spectrum. Another important task is to investigate the choice of proposal distributions for subset simulation, since they govern the efficiency of the
method. The failure analysis results on the distribution of stochastic excitation model parameters, such as the moment magnitude and epicentral distance of a point-source model, may be used to calibrate stochastic ground motion models as well as the distributions of the model parameters.
From a global perspective, the 'science' of simulation methods, especially for problems with a large number of uncertain parameters, is not well-explored. There are many issues yet to be addressed that have an important bearing on what simulation methods have to offer. For example, is it possible to construct a simulation method that has the same robustness as standard Monte Carlo simulation but that is substantially more efficient? A formal treatment on the limits in robustness and efficiency of simulation methods should give important insights on how we should proceed in developing efficient simulation methods for complex systems with a large number of uncertain parameters. To this. end, an information-theoretic approach seems to be a promising direction.
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Appendix A Some Additional Observations on
the Failure Region of SDOF Time-invariant Linear Systems
In this appendix, we present some additional observations on the failure region of SDOF time- invariant linear systems subjected to filtered white noise excitation. The conclusions are expected to hold, however, in more general situations, such as for MDOF systems or filtered white noise excitation, since the essential property underlying these characteristics is the continuity of the unit impulse response function of the system. The discussion is based on continuous-time systems, for the sake of mathematical convenience in analysis, but analogous results hold for discrete-time systems.
We will denote the excitation on the interval [0, T] by w(t), which is assumed to be square- integrable on [0, T], that is, w E L2[0, T]. The system response corresponding to the excitation w will be denoted by y(t; w):
y(t;w)
=lot
h(t- r)w(r)dr (A. I)where h is the unit impulse response function. The failure region will then be F = { w E L2 [0, T] : jy{t;