**4.4. Optimization of sheet shape**

Optimization is the action of obtaining the preferable results during the part design. In the CAE-based application of optimization, several situations can cause the numerical noise (wrinkling). When the numerical noise exists in the design analysis loop, it will create many artificial local minimums. In this case, the minimization of local thinning condition in the blank sheet metal was tested with a cost function of the optimization system was chosen to minimize the thinning ratio of 20% thinnest element.

$$\text{i.e.}\text{Cost function: }\mathbf{f} = \sum\_{i=1}^{n} \left\| \frac{\mathbf{t} - \mathbf{t}\_{0}}{\mathbf{t}\_{0}} \right\|^{2}$$

**Table 3.** Levels of pressure for different dies

**Figure 14.** Optimisation of complex shape part

minimize the thinning ratio of 20% thinnest element.

**4.4. Optimization of sheet shape** 

Die cavity Beginning instability (MPa) Critical (MPa) Experimental (MPa) D1 4,90 6,74 5,2 D2 2,85 2,85 3,0 D3 5,10 6,86 5,3

Optimization is the action of obtaining the preferable results during the part design. In the CAE-based application of optimization, several situations can cause the numerical noise (wrinkling). When the numerical noise exists in the design analysis loop, it will create many artificial local minimums. In this case, the minimization of local thinning condition in the blank sheet metal was tested with a cost function of the optimization system was chosen to where 0t is the initial thickness and t the final thickness.

In this case a significant design variable for formability of blank during hydroforming process and the design (D and d) constraints were defined:

50 D 250mm 20 d 100mm . The experimental final shape is shown in Figure 10a. ≤ ≤ ≤≤ The comparison of the force versus the maximum displacement with the initial and optimized blank shape is present in Figure 10b. Good agreement between the optimum shape and the experimental values. Figures 10c and 10d compare the initial and the optimum blank shape. Successfully decreased the cost function (thinning ratio) from 50% to 20% is obtained without wrinkling (Figure 10e and 10f) (see Ayadi et al.,2011).

## **5. Reliability analysis**

Recently, RBDO has become a popular philosophy to solve different kind of problem. In this part, we try to prove the ability of this strategy to optimize loading path in the case of THP where different kind of nonlinearities exist (material, geometries and boundary conditions). The aim of this study is to obtain a free defects part with a good thickness distribution, decrease the risk of potential failures and to let the process insensitive to the input parameters variations. For more detailed description of the RBDO methodology and variety of frameworks the reader can be refer to the following references (Youn et al., 2003; Enevoldsen et al., 1994; El Hami et al., 2011). The RBDO problem can be generally formulated as:

$$\begin{cases} \text{Min } \mathbf{f}\left(\mathbf{d}, \mathbf{X}\right) \\ \text{subject to } \mathbf{P}\left[\mathbf{G}\_{\mathbf{i}}\left(\mathbf{d}, \mathbf{X}\right) \leq 0\right] - \Phi\left(-\mathbb{B}\_{\mathbf{t}\_{i}}\right) \leq 0 \text{ i} = 1, \dots, \text{np} \\ \mathbf{d}^{\mathsf{L}} \leq \mathbf{d} \leq \mathbf{d}^{\mathsf{U}}, \ \mathbf{d} \in \ \mathbf{R}^{\mathsf{ndv}} \text{ and } \mathsf{X} \in \ \mathbf{R}^{\mathsf{nvr}} \end{cases} \tag{12}$$

where f(d,X) is the objective function, d is the design vector, X is the random vector, and the probabilistic constraints is described by the performance function Gi(X), np, ndv and nrv are the number of probabilistic constraints, design variables and random variables, respectively, <sup>β</sup>ti is the prescribed confidence level which can be defined as ( ) <sup>−</sup> β = −Φ <sup>i</sup> 1 t f P where f P is the probability of failure and Φ is the cumulative distribution function for standard normal distribution.

The process failure state is characterized by a limit state function or performance function G(X), and G(X)=0 denotes the limit state surface. The m-dimensional uncertainty space in thus divided into a safe region ( ) Ω= > <sup>s</sup> {X:G X 0 ( ) } and a failure region ( ) Ω= ≤ <sup>f</sup> {X:G X 0 ( ) } (see Radi et al.,2007).

### **5.1. Definition of the limit state functions**

The risk of failure is estimated based on the identification of the most critical element for necking and severe thinning. For this reason fine mesh was used in this study to localize the plastic instability or the failure modes in one element. Some deterministic finite element simulations show that always severe thinning is localised in element 939 in the centre of the expansion region and necking in element 1288 as shown in Figure 22.

Hydroforming Process: Identification of the Material's Characteristics and Reliability Analysis 19

The first limit state function was taken to be the difference between the maximum stress and

( ) { } } {

σ =η σ

( ) { } } {

( ) = =

w 2

F d

= = ε −ε

σ =η σ

f 2

( )

f 2

shown in Figure 25, it can be given by the following expression:

 ε =φ ε ( )

forming stress limit. The role of this constraint is to maintain the maximum stress on the critical element below σ<sup>f</sup> . The second limit state function is used to control the severe thinning in the tube, to define this function we use the FLD plotted in the strain diagram as

= =σ −σ

where ε<sup>1</sup> is the major strain in the critical element and εth is the thinning limit determined

The objective function consists to reduce the wrinkling tendency, this function is inspired

() ( )

<sup>2</sup> N N <sup>2</sup> i ii w w 1w i1 i1

<sup>1</sup> is the maximum stress in the most critical element and σf the corresponding

=σ −σ

c f 1

c

G x,y F (14)

(15)

th f 1

G x,y (13)

**Figure 16.** Forming limit stress diagram

where σ<sup>c</sup>

the corresponding FLSD as shown in Figure 25:

from the FLD curve as shown in Figure 17.

from the FLD and given by the following expression:

Since the strain\stress of element 939\1288 is the critical strain\stress of the hydroformed tube, then the reliability of these two elements represented in reality the reliability of the hydroforming process.

In this work, the limit state functions take advantage from the FLSD and the FLD of the material to assess the risk or the probability of failure of necking and severe thinning. From these curves we distinguish mainly two zones: feasible region: when a tube hydroforming process can be done in secure conditions and unfeasible region when plastic instability can appear as shown in Figures 24-25. In reality, the FLSD and FLD was used in several papers (Kleiber et al, 2004; Bing et al., 2007) as failure criteria in the aim to assess the probability of failure.

The limit state function depends on the variable of the process. Mathematically, this function can be described as Z G x,y = ( ) { } } { , where {x} presents a vector of deterministic variables and {y} is a vector of random variables.

**Figure 15.** Location of the critical elements for severe thinning and necking

**Figure 16.** Forming limit stress diagram

hydroforming process.

failure.

**5.1. Definition of the limit state functions** 

variables and {y} is a vector of random variables.

**Figure 15.** Location of the critical elements for severe thinning and necking

The risk of failure is estimated based on the identification of the most critical element for necking and severe thinning. For this reason fine mesh was used in this study to localize the plastic instability or the failure modes in one element. Some deterministic finite element simulations show that always severe thinning is localised in element 939 in the centre of the

Since the strain\stress of element 939\1288 is the critical strain\stress of the hydroformed tube, then the reliability of these two elements represented in reality the reliability of the

In this work, the limit state functions take advantage from the FLSD and the FLD of the material to assess the risk or the probability of failure of necking and severe thinning. From these curves we distinguish mainly two zones: feasible region: when a tube hydroforming process can be done in secure conditions and unfeasible region when plastic instability can appear as shown in Figures 24-25. In reality, the FLSD and FLD was used in several papers (Kleiber et al, 2004; Bing et al., 2007) as failure criteria in the aim to assess the probability of

The limit state function depends on the variable of the process. Mathematically, this function can be described as Z G x,y = ( ) { } } { , where {x} presents a vector of deterministic

expansion region and necking in element 1288 as shown in Figure 22.

The first limit state function was taken to be the difference between the maximum stress and the corresponding FLSD as shown in Figure 25:

$$\begin{cases} \mathbf{G}\left(\left\{\mathbf{x}\right\}, \left\{\mathbf{y}\right\}\right) = \sigma\_{\mathbf{f}} - \sigma\_{\mathbf{1}}^{c} \\ \sigma\_{\mathbf{f}} = \mathfrak{n}\left(\sigma\_{\mathbf{2}}\right) \end{cases} \tag{13}$$

where σ<sup>c</sup> <sup>1</sup> is the maximum stress in the most critical element and σf the corresponding forming stress limit. The role of this constraint is to maintain the maximum stress on the critical element below σ<sup>f</sup> . The second limit state function is used to control the severe thinning in the tube, to define this function we use the FLD plotted in the strain diagram as shown in Figure 25, it can be given by the following expression:

$$\begin{cases} \mathbf{G}\left(\left\{\mathbf{x}\right\}, \left\{\mathbf{y}\right\}\right) = \mathbf{F}\_{\text{th}} = \sigma\_{\text{f}} - \sigma\_{\text{1}}^{\text{c}}\\ \sigma\_{\text{f}} = \mathfrak{n}\left(\sigma\_{\text{2}}\right) \end{cases} \tag{14}$$

where ε<sup>1</sup> is the major strain in the critical element and εth is the thinning limit determined from the FLD curve as shown in Figure 17.

The objective function consists to reduce the wrinkling tendency, this function is inspired from the FLD and given by the following expression:

$$\begin{cases} \mathbf{F}\_{\mathbf{w}} = \sum\_{i=1}^{N} \left( \mathbf{d}\_{\mathbf{w}}^{i} \right)^{2} = \sum\_{i=1}^{N} \left( \mathbf{e}\_{1}^{i} - \mathbf{e}\_{\mathbf{w}}^{i} \right)^{2} \\ \mathbf{e}\_{\mathbf{w}} = \boldsymbol{\Phi} \left( \mathbf{e}\_{2} \right) \end{cases} \tag{15}$$

where ε<sup>1</sup> is the major strain in element i , and ε<sup>w</sup> is the wrinkling limit value determined from the FLD, N is the number of elements.

Hydroforming Process: Identification of the Material's Characteristics and Reliability Analysis 21

In real metal forming processes the material properties of the blank may vary within a specific range and thus probably also impact the forming results. In this work, the material

where K is the strength coefficient value, n the work hardening exponent, ε<sup>0</sup> the strain parameter, and ε the true strain. Hardening variables ( ) K,n are assumed to be normal distributed with mean values μ and standard deviations σ . Friction problem plays also a key role in hydroforming process and present some scatter, to take account for this variation a normal distribution of the static friction coefficient is assumed. Finally, the initial thickness of the tube is considered as a random variable. Table 5 illustrates the statistical properties of

Variable Mean value Cov(%) Distribution type K(MPa) 530 5 Normal n 0.22 5 Normal h(mm) 1 5 Normal μ 0.1 5 Normal

We make the assumption that all the input parameters are considered to be statistically

σ= ε+ε ( )<sup>n</sup> <sup>K</sup> <sup>0</sup> (16)

of the tube is assumed to be isotropic elastic-plastic steel obeying the power-law:

**Figure 18.** Definition of the design variables

all random parameters.

independent.

**5.2. Definition of the random variables** 

**Table 5.** Statistical properties of random parameters

The success of a THP is dependent on a number of variables such as the loading paths (internal pressure versus time and axial displacement versus time), lubrication condition, and material formability. A suitable combination between all these parameters is important to avoid part failure due to wrinkling, severe thinning or necking. Koç et al. (Koç et al.,2002) found that loading path and variation in material properties has a significant effect on the robustness of the THP and final part specifications. In this work, we define the load path as design variables to be optimized.

**Figure 17.** Forming Limit Diagram

The load path given the variation of the inner pressure vs. time is modelled by two points ( ) 1 2 P ,P displacement is imposed as a linear function of time, for axial displacement we interest only on the amplitude D . Table 4 illustrates the statistical properties of the design variables.


**Table 4.** Statistical properties of the control points described the load path

**Figure 18.** Definition of the design variables

design variables to be optimized.

**Figure 17.** Forming Limit Diagram

variables.

from the FLD, N is the number of elements.

where ε<sup>1</sup> is the major strain in element i , and ε<sup>w</sup> is the wrinkling limit value determined

The success of a THP is dependent on a number of variables such as the loading paths (internal pressure versus time and axial displacement versus time), lubrication condition, and material formability. A suitable combination between all these parameters is important to avoid part failure due to wrinkling, severe thinning or necking. Koç et al. (Koç et al.,2002) found that loading path and variation in material properties has a significant effect on the robustness of the THP and final part specifications. In this work, we define the load path as

The load path given the variation of the inner pressure vs. time is modelled by two points ( ) 1 2 P ,P displacement is imposed as a linear function of time, for axial displacement we interest only on the amplitude D . Table 4 illustrates the statistical properties of the design

Variable Mean value Cov(%) Distribution type

( ) 1P MPa 15 5 Normal

( ) <sup>2</sup> P MPa 35 5 Normal

D(mm) 8 5 Normal

**Table 4.** Statistical properties of the control points described the load path

### **5.2. Definition of the random variables**

In real metal forming processes the material properties of the blank may vary within a specific range and thus probably also impact the forming results. In this work, the material of the tube is assumed to be isotropic elastic-plastic steel obeying the power-law:

$$\mathbf{G} = \mathbf{K} \left( \mathbf{e} + \mathbf{e}\_0 \right)^n \tag{16}$$

where K is the strength coefficient value, n the work hardening exponent, ε<sup>0</sup> the strain parameter, and ε the true strain. Hardening variables ( ) K,n are assumed to be normal distributed with mean values μ and standard deviations σ . Friction problem plays also a key role in hydroforming process and present some scatter, to take account for this variation a normal distribution of the static friction coefficient is assumed. Finally, the initial thickness of the tube is considered as a random variable. Table 5 illustrates the statistical properties of all random parameters.


**Table 5.** Statistical properties of random parameters

We make the assumption that all the input parameters are considered to be statistically independent.

#### **5.3. Evaluation of the probability of failure**

Consider a total number of m stochastic variables denoted by a vector <sup>=</sup> { } <sup>T</sup> X x ,x , ,x 12 m , in probabilistic reliability theory, the failure probability of the process is expressed as the multi-variant integral:

$$\mathbf{P}\_{\mathbf{f}} = \mathbf{P}\left(\mathbf{G}\_{\mathbf{j}}\left(\mathbf{x}\right) < 0\right) = \int\_{\varOmega\_{\mathbf{j}}} \mathbf{f}\_{\mathbf{x}}\left(\mathbf{x}\right) d\mathbf{x}\_{1} \dots d\mathbf{x}\_{n} \quad \text{where } \varOmega\_{\mathbf{j}} : \left[\mathbf{x} \in \mathfrak{R}^{n} : \mathbf{G}\_{\mathbf{j}}\left(\mathbf{x}\right) < 0\right] \tag{17}$$

Hydroforming Process: Identification of the Material's Characteristics and Reliability Analysis 23

uncertainty associated to the parameters defined previously. The method presented here seems more suitable since optimization of the metal forming processes is time consuming and require many evaluations of the probabilistic constraints, additional it can be used in

In order to verify the quality of the response surface, a classical measure of the correlation between the approximate models and the exact value given by finite element simulations of the limit state function is used and shows that the approximation models can predict with a high precision the real response. Before proceeding to the reliability analysis and optimization process, the main effect plot is drawn to show how each of the variables affect severe thinning and necking. It is observed that the strength coefficient, work hardening exponent and initial thickness of the tube has the most significant impact on the severe

Figure 28 shows a finite element model (FEM) that was defined to simulate the THP. It is formed of the die that represents the final desired part, a punch modelled as rigid body and meshed with 4-node, bilinear quadrilateral, rigid element called R3D4. The tube is composed of 1340 elements (4-node, reduced integration, doubly curved shell element with five integration points through the shell section, called S4R). Since the part is symmetrical, the only quarter-model was used. The numerical simulations of the process are carried out using the explicit dynamic finite element code ABAQUS\Explicit©. The dynamic explicit

In this work, we aim to optimize the loading path under the variation of some parameters, here the objective function consist to minimize the wrinkling tendency and the probabilistic constraints was defined to avoid severe thinning and necking. We can formulate the RBDO

> ( ) ( )

thinning i necking i

(19)

 ≤ ≤ ≤ ≤

s.c to P G p,X 0 Pa P G p,X 0 Pa

where ( ) <sup>≤</sup> <sup>i</sup> P G p,X 0 and i Pa are the probability of constraint violation and the allowable

A probabilistic methodology was developed and applied to optimize THP with respect to probabilistic constraint. The methodology combined an optimization strategy and probabilistic analysis. A routine is prepared with MATLAB with the use of the toolbox optimization strategy based a successive quadratic programming. The probabilistic

( )

Min f p

conjunction with an optimization procedure.

thinning and necking plastic instabilities.

algorithm seems more suitable for this simulation.

**5.5. Formulation of the optimization problem** 

 

probability of the ith constraint violation, respectively.

**5.4. Finite element model** 

problem as follows:

where f P is the process failure probability, ( ) <sup>x</sup> f X is the joint probability density function of the random variables X . A reliability analysis method was generally employed since been very difficult to directly evaluate the integration in Equation (20). In the case when the problem presents a high non linearity, the use of the classical method to assess the probability of failure becomes impracticable.

Evaluation of the probability of failure is metal forming processes remain still a complicated and computational cost due to the lot of parameters that can be certain and the absence of an explicit limit state function. The appliance of the direct Monte Carlo seems impractical.

Therefore various numerical techniques have been proposed for reducing the computational cost in the evaluation of the probability of failure (Donglai et al., 2008; Jansson et al., 2007). Monte Carlo simulations coupled with response surface methodology (RSM) is used to assess the probability of failure. To build the objective function and the limit state functions given by Equations (16), (17) and (18), RSM is used based on the use Latin Hypercube design (LHD). The LHD was introduced in the present work for its efficiency, with this technique; the design space for each factor is uniformly divided. These levels are the randomly combined to specify n points defining the design matrix. Totally 50 deterministic finite element simulations were run, from these results we find a suitable approximation for the true functional relationship between response of interest y and a set of controllable variables that represent the design and random variables. Usually when the response function is not known or non Linear, a second order is utilized in the form:

$$\mathbf{y} = \boldsymbol{\beta}\_0 + \sum\_{i=1}^n \boldsymbol{\beta}\_i \mathbf{x}\_i + \sum\_{i=1}^n \boldsymbol{\beta}\_{ii}^2 \mathbf{x}\_i^2 + \sum\_{i$$

where ε represents the noise or error observed in the response, y such that the expected response is y − ε and β 's the regression coefficients to be estimated. The least square technique is being used to fit a model equation containing the input variables by minimizing the residual error measured by the sum of square deviations. To assess the probability of failure, the limit state functions are then estimated for a new more consequent sample (1 million) starting from the model given by the response surface methodology and the probability of failure is given then by =f fail total P N N . N is the number of failing points fail and Ntotal is the total number of simulations. This methodology will be implemented in the optimization process to optimize the loading paths with taking into account of the uncertainty associated to the parameters defined previously. The method presented here seems more suitable since optimization of the metal forming processes is time consuming and require many evaluations of the probabilistic constraints, additional it can be used in conjunction with an optimization procedure.

In order to verify the quality of the response surface, a classical measure of the correlation between the approximate models and the exact value given by finite element simulations of the limit state function is used and shows that the approximation models can predict with a high precision the real response. Before proceeding to the reliability analysis and optimization process, the main effect plot is drawn to show how each of the variables affect severe thinning and necking. It is observed that the strength coefficient, work hardening exponent and initial thickness of the tube has the most significant impact on the severe thinning and necking plastic instabilities.

### **5.4. Finite element model**

22 Metal Forming – Process, Tools, Design

multi-variant integral:

**5.3. Evaluation of the probability of failure** 

where f P is the process failure probability, ( ) <sup>x</sup>

probability of failure becomes impracticable.

( ) ( ) ( ) Ω = <= j

Consider a total number of m stochastic variables denoted by a vector <sup>=</sup> { } <sup>T</sup> X x ,x , ,x 12 m , in probabilistic reliability theory, the failure probability of the process is expressed as the

the random variables X . A reliability analysis method was generally employed since been very difficult to directly evaluate the integration in Equation (20). In the case when the problem presents a high non linearity, the use of the classical method to assess the

Evaluation of the probability of failure is metal forming processes remain still a complicated and computational cost due to the lot of parameters that can be certain and the absence of an explicit limit state function. The appliance of the direct Monte Carlo seems impractical.

Therefore various numerical techniques have been proposed for reducing the computational cost in the evaluation of the probability of failure (Donglai et al., 2008; Jansson et al., 2007). Monte Carlo simulations coupled with response surface methodology (RSM) is used to assess the probability of failure. To build the objective function and the limit state functions given by Equations (16), (17) and (18), RSM is used based on the use Latin Hypercube design (LHD). The LHD was introduced in the present work for its efficiency, with this technique; the design space for each factor is uniformly divided. These levels are the randomly combined to specify n points defining the design matrix. Totally 50 deterministic finite element simulations were run, from these results we find a suitable approximation for the true functional relationship between response of interest y and a set of controllable variables that represent the design and random variables. Usually when the response

> = = <= <sup>=</sup> <sup>β</sup> <sup>+</sup> <sup>β</sup> <sup>+</sup> <sup>β</sup> <sup>+</sup> <sup>β</sup> + ε nn n 2 2 0 i i ii i ij i j i 1 i 1 i jj 1

where ε represents the noise or error observed in the response, y such that the expected response is y − ε and β 's the regression coefficients to be estimated. The least square technique is being used to fit a model equation containing the input variables by minimizing the residual error measured by the sum of square deviations. To assess the probability of failure, the limit state functions are then estimated for a new more consequent sample (1 million) starting from the model given by the response surface methodology and the probability of failure is given then by =f fail total P N N . N is the number of failing points fail and Ntotal is the total number of simulations. This methodology will be implemented in the optimization process to optimize the loading paths with taking into account of the

y x x xx (18)

function is not known or non Linear, a second order is utilized in the form:

f j X 1n P P G x 0 f x dx dx Ω ∈ℜ < { ( ) } <sup>n</sup> where : x : G x 0 j j (17)

f X is the joint probability density function of

Figure 28 shows a finite element model (FEM) that was defined to simulate the THP. It is formed of the die that represents the final desired part, a punch modelled as rigid body and meshed with 4-node, bilinear quadrilateral, rigid element called R3D4. The tube is composed of 1340 elements (4-node, reduced integration, doubly curved shell element with five integration points through the shell section, called S4R). Since the part is symmetrical, the only quarter-model was used. The numerical simulations of the process are carried out using the explicit dynamic finite element code ABAQUS\Explicit©. The dynamic explicit algorithm seems more suitable for this simulation.

#### **5.5. Formulation of the optimization problem**

In this work, we aim to optimize the loading path under the variation of some parameters, here the objective function consist to minimize the wrinkling tendency and the probabilistic constraints was defined to avoid severe thinning and necking. We can formulate the RBDO problem as follows:

$$\begin{cases} \text{Min } \mathbf{f} \text{(p)}\\ \text{s.c. to } \mathbf{P} \Big[ \mathbf{G}\_{\text{thimning}} \Big( \mathbf{p}\_{\prime} \mathbf{X} \big) \leq 0 \Big] \leq \mathbf{P} \mathbf{a}\_{\text{i}}\\ \qquad \mathbf{P} \Big[ \mathbf{G}\_{\text{neching}} \Big( \mathbf{p}\_{\prime} \mathbf{X} \big) \leq 0 \Big] \leq \mathbf{P} \mathbf{a}\_{\text{i}} \end{cases} \tag{19}$$

where ( ) <sup>≤</sup> <sup>i</sup> P G p,X 0 and i Pa are the probability of constraint violation and the allowable probability of the ith constraint violation, respectively.

A probabilistic methodology was developed and applied to optimize THP with respect to probabilistic constraint. The methodology combined an optimization strategy and probabilistic analysis. A routine is prepared with MATLAB with the use of the toolbox optimization strategy based a successive quadratic programming. The probabilistic

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 process and the quality of the final part.

Hydroforming Process: Identification of the Material's Characteristics and Reliability Analysis 25

the reliable design but this, it's not true for necking, in fact optimization based on reliability

Design D mm ( ) ( ) 1P MPa ( ) <sup>2</sup> P MPa <sup>β</sup><sup>1</sup> <sup>β</sup><sup>2</sup>

DDO 7 18 35.4643 1.6418 4.6112

RBDO β = 2 7 17.1111 35 1.9995 4.0376

RBDO β = 2.5 7 16.1165 35 2.5004 3.5679 RBDO β = 3 7.1132 14.9451 35.0035 3.0005 3.1669

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

Design Deterministic β = 2 β = 2.5 β = 3 % of decrease 35.7754 33.7909 30.0629 23.4896 SQP iteration 10 19 21 19

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

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

> ( ) ( ) ( ) <sup>−</sup> − − − μ − μ μσ = − + + σ σσ

( ) ( ) ( ) −μ −μ μσ = − − − σ σσ 1 x x

x x k k <sup>1</sup> f x k, , exp 1 k 1 k (20)

f x 0, , exp exp (21)

1 1 <sup>1</sup>

analysis try to find a tradeoffs between the desired reliability confidence.

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

function and the iterations number for the different cases.

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

better thickness distribution in the tube (El Hami et al.,2012).

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

shape parameter k 0 ≠ is:

**Figure 19.** Finite Element Model

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