Uncertainty and Optimization in Structural Mechanics by El Hami Abdelkhalak; Bouchaib Radi; & Bouchaïb Radi

Author:El Hami, Abdelkhalak; Bouchaib, Radi; & Bouchaïb Radi
Language: eng
Format: epub
Publisher: John Wiley & Sons, Incorporated
Published: 2013-04-02T16:00:00+00:00

The constraint Φβ + t n = F(x) ensures that the point F(x) is found on the normal and the other constraints ensure the attainability of x in relation to the initial MOP problem. The sub-problem (4.5) is termed the NBI’s sub-problem and can be expressed as (NBIβ) since β is the parameter characterizing the sub-problem. The solutions of these sub-problems are called “NBI points”. It should be noted that if the start is not taken in F*, the first constraint must be written in the form Φβ + t n = F(x) − F*(x). The idea is to solve (NBIβ) for the various values of β and to find the different points on the boundary ∂h. In fact, it is the manner in which the points P1, P2 and P3 in Figure 4.3 have been determined using β1, β2 and β3, respectively. To best illustrate these concepts, we will focus on a bidimensional example; l = 2.

The four essential stages of the NBI method are:

– Determination of the individual minima of each objective function (individual minima). This entails solving the problem separately, that is finding the minimum of f1(x), which verifies the different constraints, which we have termed , and then searching for , which minimizes f2(x).

– Drawing of the CHIM. Calculate and then draw the CHIM linking with , as shown in Figure 4.4(a)

– The drawing of the normal vector in relation to the CHIM pointing toward the start. Take , by carrying out a change of reference , we obtain the graph 4.4(b)

– The formulation and resolution of the NBI optimization problem (NBIβ).

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