A probabilistic approach for optimising hydroformed structures using local surrogate models to control failures

https://doi.org/10.1016/j.ijmecsci.2015.04.002Get rights and content

Highlights

  • An efficient probabilistic approach is proposed for optimising hydroforming process.

  • Plastic instabilities are controlled locally by defining local element patches where the failure may potentially initiate.

  • Local approximations reduce the error accumulation and increase the efficiency of the probabilistic approach.

  • Global sensitivity analysis is performed to discard the non-influential factors and reduce the problem dimensionality.

Abstract

A probabilistic approach is proposed to optimise hydroformed structures by taking into account the potential variabilities. An efficient implementation requires an appropriate strategy for uncertainty representation and propagation. Moreover, the probability of failure associated to each failure mode must be accurately estimated. To this end, the failure modes are controlled locally only at the highly strained regions which reduces the problem complexity and increases the precision of the generated surrogate models. In this study, finite element simulations with material formability diagrams are used to predict the critical zones in which failure modes may initiate. The predicted zones agree well with the experimental and numerical simulations. By this simplification, the optimisation problem is formulated differently while retaining the relevant physical features of the process. To illustrate this strategy, tee-shaped tube hydroforming process is proposed due to its complexity to demonstrate the benefits of the probabilistic approach. The optimisation problem is formulated within deterministic and probabilistic frameworks to determine the optimal loading paths. It will be shown that probabilistic optimum allows better process mechanics and improved thickness distribution in the hydroformed tube. This approach can be extended to other metal forming processes and easily implemented for industrial products within reasonable computational time.

Introduction

During the last decade, hydroforming process has become one of the most important advanced manufacturing technologies used for producing a large variety of components and structures. The process finds various applications in the aerospace industry, such as panels, fuselage parts and in the appliance industry, such as fitting, joints, knobs and handles. The process offers significantly improved component stiffnesses in addition to lower cost compared to traditional techniques such as stamping, forging or welding. Since the application of the hydroforming process is relatively new (compared to other metal forming processes such as stamping and forging), a know-how with many trial efforts and numerical simulations such as optimisation strategies coupled with finite element analyses (FEA) is requested to achieve a good performance of the process. More precisely, it appears that monitoring and controlling the process is necessary in order to obtain parts that satisfied the desired specifications. Indeed, final shapes obtained by hydroforming process are highly dependent on the applied loads, material properties, lubrication conditions and tube geometry leading this process generally more sensitive to the input parameters than other metal forming processes.

In the literature, deterministic design optimisation (DDO) approaches which assume precise knowledge of the parameters involved in hydroforming process have been extensively used to determine specific loading conditions to be defined for the process. Several strategies have been proposed in which the main objective was to determine the optimum loading curves necessary to hydroform a “good” part, free of defects (i.e., wrinkling, bursting and severe thinning). Fann and Hsiao [1] used the conjugate gradient optimisation method to optimise a tee-shaped tube. Aydemir et al. [2] proposed an adaptive simulation approach to optimise loading path in which process parameters are adjusted during the simulation via a fuzzy knowledge based controller. Imaninejad et al. [3] sought the internal pressure and axial feed of a tee-joint design by chosen to minimise the thickness variation with restrictions about the effective stress. Jansson et al. [4] proposed an adaptive optimisation method using the response surface method (RSM) as metamodelling technique. This strategy allows to produce an optimal solution from a single simulation within reasonable computational time. Abedrabbo et al. [5] used genetic algorithm search methods to identify the best loading paths in combination with the nonlinear structural finite element code LS-DYNA. Di Lorenzo et al. [6] developed a cascade optimisation procedure based on a steepest descent method and the RSM utilising a moving least squares approximation to find the optimal internal pressure curve. Ahmadi Brooghani et al. [7] proposed an efficient technique to optimise loading path based on the multilevel RSM in which the optimisation process can be continued as a multilevel process. Manabe et al. [8] developed an in-process fuzzy control system for T-shape hydroforming to determine adaptively suitable loading pah. Xu et al. [9] showed that the formability is significantly improved by pulsating load and that bursting can be effectively inhibited even under the severe nonuniform deformation. Other approaches and strategies have been applied with the intent of optimising the loading paths, see, e.g., [10], [11], [12], [13] and the references therein. The aforementioned works highlight that the proper control of the process parameters in terms of press setting (i.e., internal pressure and axial displacement) is of fundamental importance for the development of optimised tube hydroforming (THF) process in order to achieve the desired characteristics of the final component. As earlier mentioned, the cited works formulated the optimisation problem under deterministic framework. However, performance of manufacturing processes is greatly affected by the presence of uncertainties which have various sources such as loads, material properties, friction coefficients and geometric characteristics. In some cases, the uncertainty ranges can be quite large for several parameters which may affect the process stability. For example, Karthik et al. [14] have shown that for different coils of the same material, the strain hardening parameter can have coil-to-coil variation up to 14%. In the same research paper, it was shown that the variability which may affect the anisotropy coefficients from one coil to another is important. For deep drawing process, Gantar and Kuzman [15] have shown that variability associated to material properties and initial sheet thicknesses are dominant variables, influencing wrinkling and necking behaviours. This can explain the fact that in practice, one may frequently observe that the specifications of the manufactured products deviate from the desired ones. Thus, the major sources of uncertainty must be incorporated to raise the manufacturing accuracy and guarantee the process reproducibility.

In practice, manufacturer engineer tries to deal with the uncertainties by choosing reasonably conservative parameters. Mostly, the notion of safety factor was applied to take into account implicitly the uncertainties, however the selection of such value is not a trivial task and requires expert opinion. Moreover, several authors have shown that even with definition of safety factor, this does not necessarily correspond to a low probability of failure. This is due to the complicated relationship between the factor of safety and the probability of failure which depends on the uncertainties in parameters. For these reasons, the definition of safety factor fails to address the problem of properly and consistently dealing with uncertainties as shown in previous work [16]. Therefore, it is more appropriate to explicitly integrate the uncertainty within the framework of reliability-based design optimisation (RBDO). RBDO has been successfully tested in many practical cases, for example for design optimisation in the automotive and aircraft industries among others [17], [18], [19], [20], [21], [22], [23]. However, the use of such approach for complex problems still remains limited by the computing time and is up to now far from being a practical tool for all engineering applications and in particular for 3D metal forming processes. Very few works have been reported in the literature to integrate variability at the early design stage [24], [25], [26]. However, a limited number of random parameters are mostly considered which seems insufficient to better treat problems arising from realistic industrial forming design and production processes. Thus, the purpose of this paper is to implement a probabilistic approach with efficient strategy to manage uncertainties and solve rigourously the complex hydroforming process within reasonable computational time. In this contribution, the potential failure modes will be controlled locally only in the regions of interest in which the potential plastic instabilities may initiate. This alternative allows to estimate the probabilities of failure associated to each plastic instability with a high precision.

As already mentioned, practical implementation of the RBDO framework is far from trivial for problems which involve complex physical phenomena. Thus, let us illustrate in Fig. 1 the different steps required to solve the problem efficiently. The proposed approach consists primarily of three steps: in the first step, we need to prepare the 3D finite element (FE) model used to simulate the process in question. Then, the design variables to be optimised as the random parameters must be defined. It should be noted that when several factors are involved in such process, screening or global sensitivity analysis can be performed in order to reduce the problem dimensionality and avoid numerical difficulties in optimisation iterative procedure. Then, the performance function as the constraints required to control the process should be defined and formulated. In the second step, a design of experiment (DOE) is defined to construct the surrogate models by running a limited number of FE simulations and post-processing the outputs to get the requested quantities. Then, the generated surrogate models will be used to propagate the uncertainties from the random parameters to the specifications defined previously. Therefore, when such uncertainty is propagated through the surrogate model, a model output becomes a distribution rather than a single value. By determining the output probability density function (PDF) of the desired specification, one may evaluate the probability of failure with a high precision by fixing unacceptable limit defined by using a suitable criterion. In the third step, the formulation of the optimisation problem can be stated as the choice of the suitable optimisation algorithm. Optimisation problem consists in defining the objective function as the probabilistic constraints which control the potential failure modes by means of suitable criteria in order to maximise the process stability. Finally, FE simulation with optimal solution is performed to check surrogate models׳ predictions and validate the results.

To implement the probabilistic approach described above, RBDO offers an appropriate framework for design under uncertainty since it is intended to solve the optimisation problem subjected to probabilistic constraints [27]. During the last decade, RBDO has been intensively studied for large scale problems due to its significance, its conceptual and mathematical complexity. However, despite these considerable advances some difficulties still persist mainly for complex engineering problems as manufacturing processes. Such difficulties are related to an extremely high computational cost, nonlinearities of the problem in question originating from material behaviour, geometry and contact problems. This leads to studied additionally specialised modelling strategies such as metamodelling, sensitivity analysis or approximation strategies, as well as reliability and optimisation coupling or decoupling-strategies for solving RBDO problems efficiently [28], [29], [30]. In order to reduce the computational costs, the most popular alternative consists in replacing the original model by a surrogate one which is much faster to evaluate. Various surrogates have been used amongst which are quadratic response surface, support vector machines, neural networks and kriging. Within the framework of the proposed study, we provide a comparison between two metamodelling techniques in order to evaluate their performances in estimating the probability of failure: the popular RSM and the least square support vector regression (LSSVR). Despite its weakness to deal with high nonlinear problems, the former was widely used to solve various practical engineering problems including metal forming processes, crashworthiness and reliability of structures due to its simplicity [31], [32], [33]. The second one is based on the support vector machine theory proposed by [34]. The latter has proved its efficiency for solving high nonlinear problems in many engineering applications [35], [36], [37], [38], [39]. It should be noted that the complexity of the proposed problem arises from the nonlinearity of the limit state function (LSF) which separates the failure domain from the safe one which makes the evaluation of the probability of failure associated to each plastic instability a challenging task. Both the metamodelling techniques are compared and analysed from precision and time computation using test function from the literature. Then, an illustrative example about THF process is presented to demonstrate the capability and the efficiency of the probabilistic approach to deal efficiently with metal forming processes in general. The novel contribution of this paper is to control the potential plastic instabilities locally which enhances the precision related to the assessment of the probabilities of failure in the regions of interest and provides useful information concerning the failure initiation.

The following paper is structured as follows. In Section 2 the theoretical background related to the LSSVR is introduced. Then, we compare the capability of the proposed surrogate models to evaluate the probability of failure via test function from the literature and to solve probabilistic optimisation problem. Section 3 presents in detail the problem of tee-shape THF process as the global sensitivity analysis (SA) to determine the most important factors which drive the output uncertainty of the requested specifications. Section 4 illustrates the DDO as the RBDO frameworks and discusses in depth the obtained results. Finally, in Section 5 we give concluding remarks and make some comments on a possible extension of the proposed approach.

Section snippets

Numerical example

This section starts with a brief review of the basic concepts related to the LSSVR used to construct the surrogate models as the DOE used to generate the sampled points. For the RSM, the reader can refer to the following reference for basic theory [40]. Then, a numerical example is given in order to check the capability of the proposed metamodelling techniques to approximate the real model and in assessing the probabilistic constraints. Finally, an optimisation problem from the literature is

Finite element model

Fig. 4a shows the half FE model with rigid and deformable bodies, it is composed of the die which represents the desired geometry, left and right punches to feed material at the expanded region and the initial tube. The die and the punches are assumed to be rigid while the tube is assumed to be deformable. Due to the symmetry of the problem, it is possible to model only a quarter of the FE model. To enhance the reliability of the FE model, numerical convergence test is performed in order to

Optimisation problem formulations

The optimisation problem consists in seeking the optimum design variables vector reduced at X=(P2,P3,D2,T2). In other side, parameters vector is reduced at p=(K,n,h0,r45,r90). Using those simplifications, the DDO problem is formulated as follows:FindX=(X1,,Xk)TMinimiseXC^(X)subjectto:g^wi(X)=ε1wiξ(ε2wi)0;i=1,,9g^ni(X)=ε1njη(ε2ni)0;j=1,,7g^d(X)0XklXkXku,k=1,,4and a typical RBDO problem is formulated as follows:FindX=(X1,,Xk)TMinimiseXC^(X)subjectto:p^f=Pr[gwi(X;p)]pftarget,i=1,,9p^f

Conclusion

A probabilistic approach for solving the challenging hydroforming manufacturing process when accounting for the uncertainty that can affect some parameters has been presented. We have shown that RBDO provides a consistent framework for dealing with uncertainty for optimising hydroforming process. The probabilistic approach allows to overcome the drawback of the DDO as was shown and ensures a minimal level of reliability that can be defined by the designer. To reduce computational costs related

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