Nondeterministic "Possibilistic" Approaches for Structural Analysis and Optimal Design

The development of methods to take Into account uncertainties in structural analysis and in design optimization is attracting both the scientific and the industrial communities. In this domain possibilistic methods in which uncertainties are defined by fuzzy numbers appear as an alternative to the classical probabilistic methods such as the Monte Carlo simulations or the stochastic finite element method. The principal difficulty of possibilistic methods is that they lead to the resolution of systems of equations in which the coefficients are defined by intervals. Several approaches for the direct solution of such interval linear equations systems are presented and compared. The vertex method is taken as reference. It is shown that the problems that are solved are mathematically different according to the method used. For static linear analysis a cost-effective iterative solution of the vertex is proposed. It is based on Neumann series expansions and solves an optimization subproblem to compute extrema of the structural responses. The effectiveness of the method is illustrated by the solution of truss problems, classical in the optimization literature. Extension of the method to inverse problems of design is also considered.