**5.6. Results and discussion**

Optimization problem is solved with different reliability level target or allowable probability of failure: = ⇔ β == ⇔ β ==⇔ β = ff f P 2.28% 2;P 0.62% 2.5;P 0.13% 3 .

Table 6 resume the results obtained in the case of deterministic and reliability design for different values of the reliability index. The resolution of the problem shows that the deterministic design presents a high probability of failure for necking but an acceptable probability of failure in severe thinning, the benefits of RBDO is to ensure a level of reliability for both necking and severe thinning. The results of the optimization are reported in Table 6.

where β1 is the reliability level for necking and β2 for severe thinning. As shown is table 3, for the deterministic design we have a high reliability level for severe thinning compared to


the reliable design but this, it's not true for necking, in fact optimization based on reliability analysis try to find a tradeoffs between the desired reliability confidence.

**Table 6.** Optimal parameters for different design

24 Metal Forming – Process, Tools, Design

**Figure 19.** Finite Element Model

**5.6. Results and discussion** 

in Table 6.

process and the quality of the final part.

methodology allow to take account to the variability in metal forming process particularly is known that theses uncertainty have a significant impact on the success or the failure of the

In general manner the RBDO is solved in two spaces physical space for the design variables and normal space when we assess the reliability index. In order to avoid calculation of the reliability and the separation of the solution in two spaces which leads to very large computational time especially for large scale structures and for high nonlinear problem like hydroforming process, the transformation approach that consist in finding in one step the probability of failure based on the predicted models and optimal design is used. In this methodology, a deterministic optimization and a reliability analysis are performed

Optimization problem is solved with different reliability level target or allowable

Table 6 resume the results obtained in the case of deterministic and reliability design for different values of the reliability index. The resolution of the problem shows that the deterministic design presents a high probability of failure for necking but an acceptable probability of failure in severe thinning, the benefits of RBDO is to ensure a level of reliability for both necking and severe thinning. The results of the optimization are reported

where β1 is the reliability level for necking and β2 for severe thinning. As shown is table 3, for the deterministic design we have a high reliability level for severe thinning compared to

sequentially, and the procedure is repeated until desired convergence is achieved.

probability of failure: = ⇔ β == ⇔ β ==⇔ β = ff f P 2.28% 2;P 0.62% 2.5;P 0.13% 3 .

The main drawback of RBDO is that it requires high number of iterations compared to deterministic approach to converge. Table 7 shows the percentage decrease of the objective function and the iterations number for the different cases.


**Table 7.** Decrease of the objective function and number of iterations

Figure 20 presents the thickness distribution in an axial position obtained with deterministic approach and for the optimization strategy with the consideration to the probabilistic constraints. With a probabilistic approach satisfactory results are obtained to achieve a better thickness distribution in the tube (El Hami et al.,2012).

To show the effects of the introduced variability on the probabilistic constraints, a probabilistic characterization of severe thinning and necking when β=0 has been carried out. The generalized extreme value distributions type I (k=0) for severe thinning and type III (k<0)for necking seem fit very well the data. The probability density function for the generalized extreme value distribution with location parameter , scale parameter σ, and shape parameter k 0 ≠ is:

$$\mathbf{f}\left(\mathbf{x}|\mathbf{k},\mu,\sigma\right) = \left(\frac{1}{\sigma}\right) \exp\left(-\left(1+\mathbf{k}\frac{\left(\mathbf{x}-\mu\right)}{\sigma}\right)^{-\frac{1}{\mathbf{k}}}\right) \left(1+\mathbf{k}\frac{\left(\mathbf{x}-\mu\right)}{\sigma}\right)^{-1-\frac{1}{\mathbf{k}}}\tag{20}$$

For k 0 = , corresponding to the Type I case, the density is:

$$\mathbf{f}\left(\mathbf{x}\middle|0,\mu,\sigma\right) = \left(\frac{1}{\sigma}\right) \exp\left(-\exp\left(-\frac{\left(\mathbf{x}-\mu\right)}{\sigma}\right) - \frac{\left(\mathbf{x}-\mu\right)}{\sigma}\right) \tag{21}$$

The parameters that characterize these distributions are summarized in Table 8. Then we can simply assess the probability of failure of the potential failure modes to show how uncertainties can affect the probability of failure.

Hydroforming Process: Identification of the Material's Characteristics and Reliability Analysis 27

Using the Nelder-Mead (NM) simplex search method, a flow stress curve (Swift's model) that best fits the stress-strain of the used anisotropic material could be determined with consideration global response (force/displacement). The local behaviour (stress/strain) of the welded joints and the HAZ is identified numerically using ABAQUS solver from global results (force/depth) of nanoindentation tests. The identified hardening coefficients are introduced by Swift model. From the simulations carried out, it is clear the influence of the plastic flow behaviour of the WT in the final results (thickness distribution, stress instability,

It is also clear that to predict with more accuracy the results, the model used for simulation has to be as realistic as possible. Therefore, future work in this area will include the experimental identification approach of the hardening model coupled with damage. Indeed, we think that measurements of displacements and strains without contact can improve results quality. The suggested model coupled with ductile damage can contribute to the

The plastic deformation of a circular sheet hydraulically expanded into a complex female die was explored using experimental procedure and numerical method using ABAQUS/EXPLICIT code©. As future work, one can study others optimization techniques without using derivatives to make a numerical comparison between these different techniques and integration of adaptive remeshing procedure of sheet forming processes.

In the second part of this work, an efficient method was proposed to optimize the THP with taking into account the uncertainties that can affect the process. The optimization process consists to minimize an objective function based on the wrinkling tendency of the tube under probabilistic constraints that ensure to decrease the risk of potential failure as necking and severe thinning. This method can ensure a stable process by determining a load path that can be insensitive to the variations that can affect input parameters. Construction of the objective function and reliability analysis was done based on the response surface method (RSM). The study shows that the RSM is an effective way to reduce the number of

Probabilistic approach revealed several advantages and promoter way than conventional deterministic methodologies, however, probabilistic approach need precise information on the probability distributions of the uncertainty and is sometimes scarce or even absent. Moreover, some uncertainties are not random in nature and cannot be defined in a

tube circularity and critical thinning and rupture).

simulations and keep a good accuracy for the optimization.

*LMR, INSA de Rouen, St Etienne de Rouvray, France* 

deduction of forming limit diagrams.

probabilistic framework.

*LMMI, FST Settat, Settat, Morocco* 

*GAMMA3, UTT, Troyes, France* 

**Author details** 

A. El Hami

A. Cherouat

B. Radi

**Figure 20.** Thickness variation in an axial position


**Table 8.** Statistical parameters of the extreme value distribution
