Fuzzy stochastic analysis
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With the aid of the fuzzy stochastic analysis it is possible to map fuzzy random input parameter onto fuzzy random response parameters.
If the uncertainty of input parameters of a structural analysis is described with the aid of fuzzy random functions, the following crisp mapping problem has to solve
This means, the fuzzy random functions 
(structural input parameters) are mapped onto the fuzzy random functions 
(structural response parameters). As fuzzy functions and random functions (random processes) are special cases of fuzzy random functions, these uncertainty models are also accounted for within the mapping. The structural model M represents numerically the functional connection f =M between the input parameters and the responses. The crisp mapping f is refered to as deterministic fundamental solution.
In the case of fuzzy stochastic structural analysis a nonlineaer static or dynamic structural analysis is applied as deterministic fundamental solutionis. On the other hand the fuzzy stochastic finite element method is obtained, if the deterministic fundamental solution is based on the finite element method.
For the numerical solution, all fuzzy random functions are specified with the aid of their multi-dimensional fuzzy probability distribution functions. Every multi-dimensional fuzzy probability distribution function represents an assessed bunch of real-valued multi-dimensional probability distribution functions
called trajectories. The trajectories are obtained by the aid of the α-discretization in the space of the fuzzy bunch parameters 
. Thus, the fuzzy stochastic analysis is transfered in a set of real-valued stochastic analyses. For the stochastic analyses any arbitrary stochastic algorithm can be used. If e.g. the Monte Carlo simulation is applied a sample for every discrete response parameter is obtained. On the basis of these samples trajectories 
of the fuzzy probability distribution functions of discrete fuzzy random responses can be estimated. In order to determine the fuzzy probability distribution functions a set of trajectories has to be computed with the aid of the α-level optimization.
(structural input parameters) are mapped onto the fuzzy random functions (structural response parameters). As fuzzy functions and random functions (random processes) are special cases of fuzzy random functions, these uncertainty models are also accounted for within the mapping. The structural model M represents numerically the functional connection f =M between the input parameters and the responses. The crisp mapping f is refered to as deterministic fundamental solution. In the case of fuzzy stochastic structural analysis a nonlineaer static or dynamic structural analysis is applied as deterministic fundamental solutionis. On the other hand the fuzzy stochastic finite element method is obtained, if the deterministic fundamental solution is based on the finite element method.
For the numerical solution, all fuzzy random functions are specified with the aid of their multi-dimensional fuzzy probability distribution functions. Every multi-dimensional fuzzy probability distribution function represents an assessed bunch of real-valued multi-dimensional probability distribution functions
called trajectories. The trajectories are obtained by the aid of the α-discretization in the space of the fuzzy bunch parameters . Thus, the fuzzy stochastic analysis is transfered in a set of real-valued stochastic analyses. For the stochastic analyses any arbitrary stochastic algorithm can be used. If e.g. the Monte Carlo simulation is applied a sample for every discrete response parameter is obtained. On the basis of these samples trajectories of the fuzzy probability distribution functions of discrete fuzzy random responses can be estimated. In order to determine the fuzzy probability distribution functions a set of trajectories has to be computed with the aid of the α-level optimization.
Fig. 1: Algorithm of the fuzzy stochastic analysis
References
- Möller, B, and Beer, M (2004) Fuzzy Randomness - Uncertainty in Civil Engineering and Computational Mechanics, Springer, Berlin Heidelberg New York.
- Möller, B (2004) Fuzzy randomness - a contribution to imprecise probability, Special Issue of ZAMM 84(10–11):754–764.
- Sickert, J, Beer, M, Graf, W, and Möller, B (2003) Fuzzy probabilistic structural analysis considering fuzzy random functions, In: 9th Int. Conference on Applications of Statistics and Probability in Civil Engineering, edited by A. Der Kiureghian and S. Madanat and J. M. Pestana. Millpress, Rotterdam, pages 379–386.
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