Fuzzy stochastic finite element method
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The fuzzy stochastic finite element method (FSFEM) may be developed
on the basis of an uncertain variational formulation considering fuzzy random fields.
The evaluation of the steady state condition of the uncertain energy
functional leads to the fuzzy stochastic differental equation system
Beside the FE-discretization of the structure the FSFEM
needs a suitable representatin of fuzzy random fields. Heuristic or
spectral approaches may be used. Heuristic methods base on point
discretizations of the fuzzy random fields in a set of fuzzy random
vectors, such as the midpoint method. Furthermore, spectral methods
comprise expansions of the fuzzy random fields dependent of fuzzy
random vectors. In both cases the fuzzy random fields are represented
with the aid of their marginal fuzzy probability distribution functions
and their fuzzy correlation functions (Fig. 1).
Fig.1 Fuzzy correlation function
The
key of the numerical treatment of the FSFEM is the representation of
marginal fuzzy probability distribution functions and fuzzy correlation
functions with the aid of their trajectories (real-valued propability
distribution functions and real-valued correlation functions).
Furthermore, the trajectories are specified by the fuzzy bunch
parameters. The fuzzy stochastic response is determined by means
of α-level optimization in the bunch parameter space. The repeated computation of the stochastic fundamental solution required by this method is based on the Monte-Carlo simulation.
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.
