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# A gradient-based decomposition approach to optimize pressure path and counterpunch action in Y-shaped tube hydroforming operations

Rosa Di Lorenzo, Giuseppe Ingarao, Francisco Chinesta

► To cite this version:

Rosa Di Lorenzo, Giuseppe Ingarao, Francisco Chinesta. A gradient-based decomposition approach to optimize pressure path and counterpunch action in Y-shaped tube hydroforming operations. International Journal of Advanced Manufacturing Technology, 2009, 44 (1-2), pp.49 - 60. 10.1007/s00170-008-1813-x. hal-01004836

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Submitted on 2 Apr 2017

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# A gradient-based decomposition approach to optimize pressure path and counterpunch action in Y-shaped tube hydroforming operations

Rosa Di Lorenzo · G. Ingarao ·
Francisco Chinesta

**Abstract** In tube hydroforming, the concurrent actions of pressurized fluid and mechanical feeding allows obtaining tube shapes characterized by complex geometries such as different diameters sections and/or bulged zones. Main process parameters are material feeding history (i.e., the punches velocity history), internal pressure path during the process, and (in T- or Y-shaped tube hydroforming) counterpunch action. What is crucial, in such processes, is the proper design of operative parameters aimed to avoid defects (for instance underfilling or ductile fractures). Actually, the design of tube hydroforming operations is mainly aimed to prevent bursting or buckling occurrence and such issues can be pursued only if a proper control of process parameters is performed. In this paper, a design procedure for Y-shaped tube hydroforming operations was developed. The aim of the presented approach is to calibrate both internal pressure history during the process and counterpunch action in order to reach a sound final component. The approach utilized to optimize the aforementioned parameters is founded on gradient-based techniques and the optimization problem here addressed depends on a considerable number of design variables. In order to reduce the total number of numerical simulations/experiments necessary to reach the optimal values of the design variables, the basic idea of this paper is to develop a sort of

decomposition approach aimed to take into account subsets of design variables in the most effective way. The proposed decomposition approach allows avoiding about 50% of the numerical simulations necessary to solve the same problem by traditional gradient technique.

**Keywords** Y-shaped tube hydroforming · Gradient-based techniques · Decomposition approach · Optimization

## 1 Introduction

The main innovations which have characterized the research developments in the field of forming technologies regard both the introduction of new processes, aimed to replace traditional ones, and the application of innovative design procedures able to optimize the most important process parameters.

In fact, some alternative forming processes are nowadays widely utilized in mechanical and automotive industries, since they proved their effectiveness in reducing production costs. One of the most significant example of such innovations is the application of hydroforming of sheet and tubes. Actually, in the field of sheet stamping, the possibility to avoid the use of expensive dies by replacing them with the action of a pressurized fluid is very attractive since costs reduction is highly required.

In tube hydroforming, the concurrent actions of pressurized fluid and mechanical feeding allows to obtain tube shapes characterized by complex geometries such as different diameters sections and/or bulged zones. The main process parameters are material feeding history (i.e., the punches velocity history), internal pressure path during the process and (in T- or Y-shaped tube hydroforming) counterpunch action. What is crucial in such processes is

R. Di Lorenzo · G. Ingarao (✉)
Dipartimento di Tecnologia Meccanica, Produzione e Ingegneria Gestionale, Università di Palermo, Palermo, Italy
e-mail: p.ingarao@dtpm.unipa.it

F. Chinesta
Laboratoire de Mecanique des Systemes et des Procedes UMR 8106 CNRS-ENSAM-ESEM,
Paris, France
the proper design of operative parameters aimed to avoid defects (for instance underfilling or ductile fractures). Actually, the design of tube hydroforming operations is mainly aimed to prevent bursting or buckling occurrence, and such issues can be pursued only if a proper control of process parameters is performed. The studies on tube hydroforming are mainly focused on process optimization of loading paths [1, 2] also utilizing intelligent techniques [3, 4] proving that a proper calibration of the internal pressure history allows the prevention of typical defects such as buckling, underfilling, or ductile fractures.

Many researchers also studied three dimensional tube hydroforming operations developing numerical investigations and experimental validation. It was assessed that the production of T- or Y-shaped tubes generally requires also the control of a counterpunch action with the aim to reduce thinning in tube-bulged zones. In operations aimed to obtain Y-shaped tubes, material feeding action is provided by two punches whose strokes have to be generally regulated in different ways [5, 6]. In particular, the authors experienced the effectiveness of gradient-based optimization procedures in designing the typical process parameters in T-shaped tube hydroforming and the experimental validations supported such conclusion [7].

More recently, some authors investigated the estimation of process parameters using a conical die geometry [8], while others studied the effect of the loading paths on the bulged shape and the wall thickness distribution in tube bulge hydroforming [9]. Moreover, some results on analytical models to determine the variation in the stresses and strains along the tube wall as the internal pressure increases to expand the circular tube into a square cross-section were presented [10].

Finally, in the last years, some results on the application of fuzzy control strategies were proposed [11] and, also, on the possibility to improve formability by pulsating hydroforming [12].

It has to be underlined that in the optimization of loading paths in Y-shaped tube hydroforming design field, some authors presented the results of the application of Augmented Lagrangian and Response Surface Methods for loading paths optimization [5]. Such results prove that a significant number of numerical simulations are necessary to reach good results in terms of final product shape; namely, the application of ALM led to a total number of numerical simulations which was higher than 150. Other applications of optimization strategies [such as the utilization of Sequential Approximate Optimization (SAO) algorithm] to tube hydroforming optimization cases were presented, but even in this cases, more than 100 simulations were developed [13].

On the other hand, the optimization of complex tube hydroforming operations such as the ones aimed to produce Y-shaped tubes may depend on different variables. In this

paper, both the internal pressure path and the counterpunch action were taken into account. Thus, an optimization problem depending on a large set of variables was formulated and a gradient-based approach was followed to reach optimal solutions. As a consequence, the implementation of the optimization procedure required a great number of experimental simulations. In this way, a decomposition procedure was developed in order to reduce the computational effort to optimize the investigated process. More in detail, the paper presents the strategy followed to divide the investigated optimization problem into smaller subproblems by splitting the chosen design variable domains in different subdomains over which the optimization has to be carried out. The aim of the proposed approach is the determination of good solutions both for pressure path and counterpunch force path along the process. On the other hand, the aim is also the reduction of the computational effort necessary to reach the solutions in order to acquire a design tool which could have an industrial applicability.

In the following sections, the problem formulation will be presented, and the decomposition strategy will be analysed before presenting the results of the application of the proposed procedure to different cases.

## 2 Design assumptions and optimization problem formulation

In this paper, the analyzed optimization problem (design of a tube hydroforming operation to manufacture a Y-shaped tube) is characterized by an objective function which is not analytically linked to the design variables. Two main phases have to be developed in order to reach an optimal solution: the former is the modeling phase, and the latter concerns the computational aspect. Figure 1 shows the typical steps of these two phases.

In the present application, the modeling phase was developed with reference to the tube hydroforming operation whose dies geometry is shown in Fig. 2.

The tube initial thickness was equal to 2 mm, while the external diameter of the bulge zone was equal to 50 mm. The investigated material is a steel whose flow rule is expressed by the following equation that was experimentally determined:

$$ \sigma = 653\epsilon^{0.253} \qquad (1) $$

### 2.1 Modeling phase

The objective function was designed in order to represent a double aim to be pursued: bursting danger reduction and tube shape accuracy.
<figure><img src="image_2.png" /><figcaption>Fig. 1 The phases of an optimization procedure</figcaption></figure>

In the Y-tube hydroforming operation design, many factors influence the aforementioned objectives: internal pressure path, punches velocity history, counterpunch action etc. Some authors presented and demonstrated an effective way to calibrate the two punch actions [10]. In the presented application, such hypothesis was followed, namely, the punches velocity was kept constant, but one of the punches has a total stroke which is half of the other one. The optimization developed in this study aims to optimize the internal fluid pressure path and the counter-punch force. The pressure vs. process time curve shape was designed taking into account the available knowledge on the tube hydroforming process. Thus, such shape consists of two linear paths with different slopes which are fully defined by three points coordinates (see Fig. 3).

In particular, the time instants corresponding to points P1, P2, and P3 in Fig. 3 were preliminary fixed according to a numerical analysis on the investigated process. As a consequence, the internal pressure path curve is fully

<figure><img src="image_3.png" /><figcaption>Fig. 2 The die geometry in the investigated process</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3 The hypothesized pressure vs. process time curve shape</figcaption></figure>

defined if the three pressure values, corresponding to the three chosen points, are fixed.

The counterpunch action was modeled with three design variables as well, in particular, a linear path was chosen and two force values ($F_1$ and $F_2$) and starting the time($T_C$) of counterpunch action were optimized (see Fig. 4).

According to these hypotheses, the design problem here addressed can be translated into an optimization problem characterized by six design variables and an objective function to be optimized which includes both bursting prevention and underfilling minimization.

In this paper, the chosen objective function was formulated as consisting of two terms: the former is the maximum thinning on the tube walls ($t\%$, which is related to the aim to reduce bursting danger) and the latter measuring the difference between the desired Y region volume and the obtained one ($u$, measuring a sort of underfilling with respect to an ideal Y-shape).

$$f(x_k) = t\% + Ku \quad (2)$$

In this function, $K$ is the weight necessary to calibrate the contribution of the underfilling term. Such objective function has to be minimized to accomplish the design aims. It has to be underlined that the chosen objective function considers both fracture risk and underfilling. As the latter aspect is concerned, the definition of the term $u$ in Eq. 2 allows taking into account and to evidence both wrinkling and buckling phenomena since such term is

<figure><img src="image_5.png" /><figcaption>Fig. 4 The hypothesized force vs. process time curve shape</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 5 “Technological splitting criterion”</figcaption></figure>

measured as a difference of two volumes (desired bulge volume and obtained one).

It is necessary to underline that the reference ideal final shape was fixed on the basis on some simple considerations on the invariability of the volume in the plastic deformation process. Thus, once the stroke values of the axial punches were fixed, the final bulge height was immediately determined.

## 2.2 Computational phase

Once the analyzed process was modeled in terms of design variables and objective function, an integration between numerical simulation and an optimization algorithm was necessary in order to update the design variable values toward the objective function minimization. As already mentioned, gradient-based algorithms proved their effectiveness in metal-forming optimization [14–17]; in particular, the authors have experienced the utilization of steepest descent method in tube hydroforming optimization [18]. The analytical formulation of a gradient-based optimization procedure can be summarized in the following steps (k denotes the method iteration number):

- identify the design variables by a vector $\mathbf{x}_k$ at iteration k;

- choose the initial values: $\mathbf{x}_k \in R^n$ with $k=0$;

- calculate the gradient of the objective function $f(\mathbf{x}_0)$: if the convergence is reached the algorithm can be stopped;

- or else calculate an updated value of the design variables $\mathbf{x}_{k+1} = \mathbf{x}_k + \alpha_k \mathbf{d}_k$ (where the scalar $\alpha_k \ge 0$ is called “step size” or “step length” at iteration k and indicates the entity of design variables adjustment at iteration k; $\mathbf{d}_k$ is the direction of movement, i.e., the direction along which the objective function goes toward a minimum;

- verify that $f(\mathbf{x}_{k+1}) < f(\mathbf{x}_k)$;

- repeat steps 4 and 5 until convergence is reached.

Such general approach can be refined according to different techniques with respect both to the gradient calculation and to the definition of step size and step direction. The procedure proposed in this application was the steepest descent method. Such method is based on the hypothesis that if a minimum of the objective function is required, then the search direction is given by the opposite of the function gradient; thus, the following expression holds:

$$ \mathbf{x}_{k+1} = \mathbf{x}_k - \alpha_k \nabla f(\mathbf{x}_k) \quad (3) $$

A finite difference method was utilized in order to calculate the gradient; fixing a perturbation of the design variables, it was possible to calculate the gradient as follows:

$$ \frac{\delta f(\mathbf{x}_k)}{\delta \varepsilon_j} = \frac{f(\mathbf{x}_k + \varepsilon_j) - f(\mathbf{x}_k)}{\varepsilon_j} \quad (4) $$

where $\varepsilon_j$ is defined as $[00:\varepsilon:0]$ i.e., a column vector whose components are null except the one of the jth row which is equal to $\varepsilon$.

Thus, the calculation of the gradient required the evaluation of the objective function values for each value and for each perturbation of the design variables. Such evaluation of the objective function was obtained through the solution of direct problems, i.e., numerical simulations aimed to evaluate the values assumed by the objective function at the variation of the design variables. The numerical simulations were all solved by utilizing the LS-DYNA explicit commercial code. As the step size evaluation is concerned, a line search procedure was utilized in order to determine the most performing value.

## 3 The decomposition approach

If an industrial application of the optimization procedure is foreseen, the computational effort has to be limited, even in

<figure><img src="image_7.png" /><figcaption>Fig. 6 “Gradient splitting criterion”</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 7 “Static procedure”</figcaption></figure>

the cases taking into account a large number of design variables. In fact, problems characterized by a large number of design variables generally lead to a large number of direct problems to be solved to reach the solution. This is true both for gradient-based approaches and for response surface methodologies and even for methods based on artificial intelligence [4, 5, 13, 19, 20]. Some authors have proposed optimization approaches based on subproblem identification for structural problems characterized by a large number of variables [21].

The basic idea of this paper is focused on the possibility to decompose the design variables space in subdomains, each consisting of a fewer number of variables. The idea to utilize a decomposition approach for a metal forming process optimization is quite new, and the foreseen advantages are related to the possibility to apply an optimization technique on different small subdomains instead of taking into account a single problem with many variables.

Working on optimization problems with reduced dimensions (i.e., variables), allows the computational effort to decrease, but should lead to an increased risk of entrapment into local minimum points. For these reasons, a proper calibration of the optimization procedure is

<figure><img src="image_9.png" /><figcaption>Fig. 8 “Dynamic procedure”</figcaption></figure>

<table><caption>Table 1 The investigated cases</caption><thead><tr><th>Case Id</th><th>Splitting criterion</th><th>Subdomain sequencing policies</th><th>Starting variables subset</th></tr></thead><tbody><tr><td>1</td><td>Gradient</td><td>Static</td><td></td></tr><tr><td>2</td><td>Gradient</td><td>Dynamic</td><td></td></tr><tr><td>3</td><td>Technological</td><td>Static</td><td>Pressure</td></tr><tr><td>4</td><td>Technological</td><td>Static</td><td>Counterpunch</td></tr><tr><td>5</td><td>Technological</td><td>Dynamic</td><td>Pressure</td></tr><tr><td>6</td><td>Technological</td><td>Dynamic</td><td>Counterpunch</td></tr></tbody></table>

necessary. In particular, the following issues have to be accomplished:

- choice of the most performing variables splitting criterion;

- identification of the proper sequence of subdomains investigations;

- determination of the linkage between the solutions from different subdomains.

In this paper, different scenarios were taken into account analyzing two splitting criteria and two subdomains sequencing policy.

### 3.1 Splitting criteria

As mentioned the design variables vector for the analysed problem can be defined as follows:

$$ \begin{bmatrix} P_1 \\ P_2 \\ P_3 \\ F_1 \\ F_2 \\ T_C \end{bmatrix} \qquad (5) $$

The former three components of the vector refer to internal pressure curve definition, while the latter ones regard the counterpunch force curve. The main assumption of the splitting procedure is that the design space was divided into two subspaces each consisting of three variables. Moreover, the splitting was applied before the development of the

<figure><img src="image_10.png" /><figcaption>Fig. 9 Numerical result of the starting solution</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Fig. 10** Internal pressure and counterpunch forces curves evolution (cases 1 and 2)</figcaption></figure>

optimization procedure, and the possibility to develop a permutation of the variables belonging to different subspaces was not investigated.

Two splitting criteria were considered: the former based on technological considerations and the latter based on gradient calculation. The first criterion is based on the assumption that the six variables can be divided into two subsets according to a technological perspective: one subset consists of three pressure-related variables, while the other contains three counterpunch force-related variables (see Fig. 5). Such scenario will be addressed as “technological splitting criterion” in the following.

On the other hand, a division based on gradient calculation was considered: the partial derivatives of the objective function (Eq. 4) were calculated with respect to each of the six variables. Thus, it was possible to estimate which variables mainly contributed to function reduction, i.e., along which directions the function decreases more. As a consequence, the subsets identification was performed as follows:

- the three highest partial derivatives were identified;

- the first variables subset contains the variables related to these derivatives (such subset will be indicated as “gradient 1 subset” in the following);

- the second subset contains the three variables which are related to the partial derivatives which less contribute to function decreasing (such subset will be indicated as “gradient 2 subset” in the following).

Such scenario will be addressed as “gradient splitting criterion” (see Fig. 6).

## 3.2 Subdomains sequencing

Once the subsets to be analyzed were defined, the optimization procedure was developed. Actually, the

optimization over the two subsets could be implemented basing on two different policies. If the optimization is performed concurrently on the two subsets, then a parallel approach is developed. In the present application, a parallel approach was not useful since the two separated optimal solutions could not be merged into a unique solution due to the technological relation between the two variables sets.

Thus, in the application presented in this paper, a serial sequencing approach was utilized. In particular, the gradient-based optimization was applied over the domain identified by the first subset of variables and subsequently on the other one.

It has to be underlined that the interaction between the subproblems has to be properly taken into account; otherwise, some inadequate local minimum points could be achieved.

In this way, two different sequencing possibilities were considered on the basis of the number of steepest descent iterations carried out, on the selected subset, before applying the procedure to the other variable subset. In particular, the former sequencing procedure (called “static procedure”) was characterized by the achievement of an optimum on the current subset which was a “partial” optimum since it depended exclusively on the subdomain taken into account. In other words, some steepest descent iterations were developed before changing the variables subset. Once the variable subset was changed, the steepest procedure was reiterated until a new “partial” optimum was reached. This procedure was then iterated until convergence (see Fig. 7 which reports the flow diagram of the static sequencing procedure).

As the latter sequencing procedure (called “dynamic procedure”) is concerned, the following procedure was implemented:

- only one steepest descent iteration was developed on the first variable subset obtaining an improvement of the objective function;

- the second steepest descent iteration was applied on the second variable subset;

<figure><img src="image_12.png" /><figcaption>**Fig. 11** Numerical result (cases 1 and 2)</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Fig. 12 Pressure curve evolutions in case 3</figcaption></figure>

— the previous steps were iterated shifting from one variable subset to the other until the convergence was reached.

In order to clarify such sequencing procedure, the following Fig. 8 shows its flow diagrams.

It is worth pointing out that both the static and the dynamic procedure are characterized by some peculiarities. In particular, the static procedure is characterized by a relevant design variable variation between the shifting from one subset to the other. Also, a low level of interaction between the subproblems is present. On the contrary, the dynamic procedure does not present a relevant design variable variation before shifting, while a very high level of interaction between the subproblems arises.

## 4 Analysis of the investigated case studies

The decomposition approach was applied to some different case studies at the variation of the splitting criteria and of the subdomains sequencing policies. Table 1 summarizes the investigated cases.

As it can be observed within the technological splitting criterion, both pressure and counterpunch force subsets were considered as starting variable subsets (i.e., the first set which undergoes to the optimization procedure).

In all the investigated cases, the starting values of the variables were chosen in such a way to obtain a defective

<figure><img src="image_14.png" /><figcaption>Fig. 13 Counterpunch force curve evolutions in case 3</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 14 Thinning distribution in case 3 (percent values)</figcaption></figure>

component in order to test the effectiveness of the developed procedures in reaching optimal solutions evolving from bad conditions.

Figure 9 shows the final component obtained utilizing such starting values; as it can be observed, a heavy wrinkling defect is present at the tube radius, and the obtained bulge is very far from the ideal profile.

In the following subsections, the six investigated cases are presented together with the obtained results

### 4.1 Case1 and 2

Both cases 1 and 2 are based on a gradient-splitting criterion; in particular, the partial derivatives of the six variables were calculated, and the three highest ones were identified. In particular, the highest derivatives corresponded to the variables which are the components of the following vector.

$$ \begin{bmatrix} P_2 \\ F_2 \\ T_C \end{bmatrix} \qquad (6) $$

Such vector represents the first subset to be analyzed (“gradient 1 subset”)

<figure><img src="image_16.png" /><figcaption>Fig. 15 Counterpunch force curve evolutions in case 4</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>**Fig. 16** Pressure curve evolutions in case 3</figcaption></figure>

The gradient splitting criterion application led to unsatisfactory results; in fact, a local minimum was suddenly reached, and it was impossible to get out from this stationary point and iterate the optimization procedure. Once the first iteration was developed, both the dynamic and the static procedures were applied, but no further improvements of the objective function were obtained.

The following Fig. 10 shows the internal pressure and counterpunch forces curves evolution after the unique iteration which provided improvements in cases 1 and 2.

As it can be observed, an improvement with respect to the starting solution was obtained; actually, a significant objective function reduction was determined in the aforementioned iteration.

As it can be observed from Fig. 11, an acceptable maximum thinning (equal to 11%) was reached at the upper corner radius. The maximum thinning is reached at the corner radius, which implies the most relevant expansion of the material in the bulge zone; as a consequence, the localized deformation is higher leading to the most relevant thinning. Moreover, the final shape of the tube is not accurate enough. In fact, even if the desired bulge height was reached, the fillet radii are not accurate. As a consequence, the underfilling term of the objective function was quite high.

<figure><img src="image_18.png" /><figcaption>**Fig. 17** Finale result of case 4 (thinning distribution)</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>**Fig. 18** Pressure curve evolutions in case 5</figcaption></figure>

The results obtained in cases 1 and 2 can be summarized as follows: the gradient splitting criterion can lead to a relevant objective function reduction by a few numbers of numerical simulations (just one iteration was run), but at the same time, the risk of entrapment into unsatisfactory local minimum points is very elevated.

## 4.2 Cases 3 and 4

Both in cases 3 and 4, a technological splitting criterion was applied, and a static sequencing procedure was implemented. The two cases differed only for the starting variable subset; in case 3, pressure variable subset was taken into account as starting set, while in case 4, counterpunch force variables one was the starting subset.

Thus, in case 3, the steepest descent procedure was iterated until a partial optimum was reached on pressure variable subset. Figure 12 shows the pressure curves evolution along the iterations of case 3.

Once the partial optimum was obtained (i.e., no further improvements are gained through the pressure variables subset), the optimization shifted to the other variables subset (counterpunch variables). In case 3, such step led to the evolution reported in Fig. 13. Also, in such step, an optimal curve was obtained which, coupled with the optimal pressure curve obtained in the previous step, was the final result of case 3.

The combination of pressure and counterpunch force curves allowed to obtain a final component characterized by an excessive maximum thinning (related to fracture occurrence, see Fig. 14). Such result can be justified, since

<figure><img src="image_20.png" /><figcaption>**Fig. 19** Counterpunch force curve evolutions in case 5</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>**Fig. 20** Iterations of the procedure in case 5</figcaption></figure>

in the first step of the optimization, high pressure levels are reached due to the optimization focused only on pressure curves which implies that the algorithm tries to minimize the objective function with no calibration of counterpunch action. Since the counterpunch action was not taken into account, the algorithm increased the pressure levels to compensate for the high counterpunch forces and, as a consequence, excessive thinning was unavoidable.

On the other hand, as case 4 is concerned, the steepest descent procedure was initially applied on counterpunch force variable subset. Such optimization developed along three iterations leading to the curves evolution shown in Fig. 15.

As it can be noticed, a progressive reduction of counterpunch forces regime was obtained with respect to the initial solution. Once the partial optimum was obtained on counterpunch force variable subset, the optimization was applied on pressure variable subset leading to the evolution shown in Fig. 16 which was characterized by a slight variation in comparison to the initial pressure curve.

Also, in case 4, the coupling of the optimal curves was the final result of the optimization. The associated final part, which was characterized by a very low final thinning value, is reported in Fig. 17.

Nevertheless, such result was not satisfactory since a significant wrinkling phenomenon was reached and also an incorrect final bulge shape was obtained. This outcome was probably related to the limited action of the counterpunch which depended on the optimization step which was performed only on the counterpunch force variables without taking into account the internal pressure effect.

On the basis of the above considerations, it can be assessed that neither the static procedure starting with pressure variables nor the one starting with counterpunch force variables can lead to satisfactory results since the two sets of variables cannot be optimized one by one.

In other words, the static sequencing procedure leads to two different local minimum points in the two analyzed

cases. In fact, the process mechanics of the investigated operation implies that acting only on pressure curves provides excessive thinning since the algorithm searches for a solution compensating the high counterpunch forces values, while acting only on counterpunch forces, leads to a lowering of such forces since the algorithm looks for conditions which avoid excessive thinning (due to high pressure levels).

Thus, the static sequencing procedure can be considered ineffective in optimizing the chosen problem. In this way, the following section describes the results of the application of the aforementioned dynamic procedure.

### 4.3 Cases 5 and 6

The two cases here addressed are characterized by a dynamic sequencing policy and by a technological splitting criterion. The difference between cases 5 and 6 is the starting subset: in case 5, the first analyzed subset contains pressure variables, while in case 6, the starting subset contains counterpunch force variables.

Both the analyzed cases led to performing results as it will be discussed in the following.

In particular, as case 5 is regarded, the lowest value of the objective function among all the investigated cases was obtained. The first step of this case consisted of the application of just one iteration of the steepest descent method on the pressure subset variables. The evolution from the initial values is illustrated in Fig. 18 in which such step corresponds to “iteration1.”

As it can be observed, the algorithm provided higher pressure levels with respect to the initial ones which were coherent with the current counterpunch forces (at this stage, the counterpunch forces were set at their initial values which, as mentioned, were quite high).

The following step of the dynamic sequencing procedure is focused on the steepest descent application to counterpunch forces by fixing the pressure curve to the one obtained in the initial step (iteration1 in Fig. 18).

The algorithm application provided the counterpunch forces evolution reported in the following Fig. 19 (iteration1 also in this case).

<figure><img src="image_22.png" /><figcaption>**Fig. 21** Final result of case 5</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>Fig. 22 Final result of case 6</figcaption></figure>

Iteration 1 of this step led to a lowering of counterpunch forces which was an expected evolution with respect to the initial conditions and also related to the current pressure values.

Since a dynamic procedure was developed, again, pressure values were taken into account and then again the procedure shifted to counterpunch forces optimization. The results of this iterative approach are indicated by “optimal” both in Fig. 18 (for pressure values) and in Fig. 19 (for counterpunch forces values).

The objective function evolution obtained in case 5 is illustrated in Fig. 20 in order to evidence the performances of the proposed algorithm.

The “optimal” curves allowed to obtain a final tube with a maximum thinning of about 18.5% and characterized by a bulge shape which can be considered accurate enough with respect to the ideal desired profile (see Fig. 21). What is more, the Y-bulge zone did not present an upper boundary surface; nevertheless, the ideal bulge height was achieved. In fact, the Y-bulge zone is “open,” but the final deformation reached the desired level.

Such results proved that the dynamic procedure is very performing as case 6 development confirmed.

In fact, starting from counterpunch force variables and applying the same procedure of case 5, the final result is very satisfactory too both in terms of maximum thinning and of desired final shape (see Fig. 22a and b, respectively). Actually, the final pressure and counterpunch force curves are very similar to the ones obtained in case 5.

## 5 Discussion of the results

The analysis of the cases described in the previous section led to some assessments which can be summarized as follows:

- a splitting criterion based on gradient calculation is not performing: both in cases 1 and 2, no satisfactory results were obtained since local minimum points were reached and neither a static nor a dynamic sequencing policy allows to get out from such local solutions;

- a technological splitting criterion seems to be more performing but not if coupled with a static subdomain sequencing policy: cases 3 and 4, in fact, did not reach good solutions. A static procedure focused on pressure variables, as starting variable subset led to excessive thinning, since a static procedure focused on counterpunch force variables provided an inaccurate final bulge shape;

- a technological splitting criterion coupled with a dynamic subdomain sequencing policy provided very satisfactory results: actually both cases 5 and 6 gave sound final components and similar final pressure and counterpunch force curves proving that such approach is not significantly influenced by which variable subset is analyzed as starting subset.

What is more, the most performing cases permitted to reach sound solutions by developing a total number of numerical simulations which is lower than 25.

It has to be underlined that the main aim of the proposed investigation was the assessment of a reliable procedure allowing reaching a final hydroformed tube whose quality can be considered high from a technological point of view. Such solution should be reached trying to reduce the number of direct problems to be solved in order to enhance the possible industrial applicability of the method.

It is worth pointing out that the implementation of splitting and sequencing strategies was actually aimed to

<figure><img src="image_24.png" /><figcaption>Fig. 23 Comparison of the optimal counterpunch force curves in cases 5, 6, and “gradient-6”</figcaption></figure>
<figure><img src="image_25.png" /><figcaption>**Fig. 24** Comparison of the optimal pressure curves in cases 5, 6, and “gradient-6”</figcaption></figure>

reduce the total number of direct problems. In fact, in the analyzed problem, the investigated six design variables could have been optimized by developing a complete steepest descent procedure over the global set of six design variables (i.e., no subset identification and sequencing should have been necessary).

Of course, such scenario requires an increased number of numerical simulations, and as it will be demonstrated in the following, no significant improvements of the solution can be reached.

In fact, in order to prove the real effectiveness of the proposed approach, a steepest descent application was developed on the whole set of six variables (such procedure will be addressed by “gradient-6” in the following). Of course, such optimization starts from the same starting solutions utilized in the other cases discussed above.

The obtained optimal solution in this “gradient-6” approach is quite similar to the ones reached in cases 5 and 6. Actually, the objective function minimum value reached with “gradient-6” procedure is the same obtained in cases 5 and 6.

Moreover, as it can be noticed in Figs. 23 and 24, the optimal pressure and counterpunch force values are almost the same in the three compared cases except for the values of $P_3$ and $F_2$ variables which slightly differ from each other.

Such difference is related to the final component characteristics:

- “gradient-6” final component reaches a maximum thinning of 15.6% (lower than the ones obtained in cases 5 and 6) at the upper corner radius as previously obtained;

- “gradient-6” final component is related to an increased underfilling occurrence with respect to cases 5 and 6 (see Fig. 25, which compares the final profiles of the three cases).

Such results are of course related to the lower pressure ($P_3$) and counterpunch force ($F_2$) levels which characterize “gradient-6” solution in comparison with cases 5 and 6.

According to the above considerations, it can be assessed that the “gradient-6” procedure leads to a solution which is not significantly different from the ones of cases 5 and 6, and what is more, it requires a number of direct problem solutions which is about 50% higher than the ones necessary in cases 5 and 6. Thus, such approach is not effective, while the splitting and sequencing procedure proves its usefulness.

## 6 Conclusions

In the paper, the optimization of a Y-shaped tube hydro-forming operation was developed. In particular, the internal pressure and the counterpunch action were optimized by applying a decomposition approach integrated with a steepest descent optimization algorithm.

Different approaches were tested in order to investigate the possibility to solve the whole problem through a decomposition into subproblems analyzed over design variable subdomains.

The aim of the presented study was double:

<figure><img src="image_26.png" /><figcaption>**Fig. 25** Comparison of the final profiles in cases 5, 6, and “gradient-6”</figcaption></figure>
- determine the best strategy to split the variable domain into subdomains;

- identify the most performing sequencing policy to investigate the different subdomains in order to reach the final solution.

The analyzed cases allowed reaching the conclusion that a technological splitting criteria to identify the variable subdomains is very performing, and the best sequencing policy is a dynamic approach.

Moreover, the most relevant advantages presented by the chosen procedure consist in the obtainment of quite good performance from a technological point of view by a reduced number of numerical simulations in comparison with a steepest descent approach. Such reduction makes the proposed approach attractive for a potential industrial applicability.

The possible developments of the proposed approach concern the possibility to utilize a different optimization procedure based, for instance, on response surface methodology. Moreover, the authors are experiencing the possibility to integrate gradient-based approaches and response surface methods, and a possible development of the present work should be focused on such integration in order to improve the final result.

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