In order to develop a suitable method for processing convex fuzzy input parameters the concept of α-level discretization is adopted. All fuzzy input parameters are discretized using the same sufficient high number of number of α-levels αk, k=1,...,r. The core procedure is called α-level optimization and operates according to a modified evolution strategy and is particularly suitable for non-linear problems. With the aid of the mapping model z=f(x1,..,xn) it is possible to compute elements in the result space. The mapping of all elements of Xαk yields the crisp subspace Zαk.
Once the largest zj,αk_r and the smallest element zj,αk_l of the crisp subspace Zαk have been found, two points of the membership function are known (Fig. 1).
Fig. 1: Mapping of the fuzzy input parameters 1 and 2 onto the result parameter
must be satisfied. The objective functions are satisfied by the optimum points. By this procedure the fuzzy results are yield α-level by α-level. In case of fuzzy structural analysis the mapping model f(x) is represented by the computational model M, e.g.
- finite element model
- non linear dynamic analysis
multi body systems
- Möller, B, and Beer, M (2004) Fuzzy Randomness - Uncertainty in Civil Engineering and Computational Mechanics, Springer, Berlin Heidelberg New York.
- Möller, B, Graf, W, and Beer, M (2000) Fuzzy structural analysis using alpha-level optimization, Computational Mechanics 26:547–565.
- Möller, B, and Beer, M (1997) Application of Fuzzy Modeling in Civil Engineering, In: Proceedings of the Second International ICSC Symposium on Fuzzy Logic and Applications ISFL 97, edited by N. Steele. ETH Zürich, pages 345–351.