International Journal of Machine Tools and Manufacture
A hybrid multi-fidelity approach to the optimal design of warm forming processes using a knowledge-based artificial neural network
Introduction
Reducing the weight of vehicles is one of the primary means by which their fuel consumption and hazardous air pollutants can be reduced. The portability of various products such as appliances, computers, machinery, etc. can be also improved by the realization of lightweight structures. Thus, there has been an increasing trend toward the use of lightweight materials (i.e., aluminum, magnesium, advanced high strength steel (AHSS), and composites) in automotive, aerospace, appliance, and energy industries. Due to the inferior formability of some lightweight materials such as aluminum and magnesium at room temperature, however, the justification of using lightweight structural parts to replace steel is very difficult and questionable in the context of conventional fabrication processes like forging and stamping.
As one of the technologies that enables improvement in the formability of lightweight materials, the warm forming process is attracting more and more interest from industries. It was reported that at elevated temperature levels (200–350 °C) aluminum–magnesium alloys (0–6.6% Mg) achieved an elongation up to 300% [1]. Many preliminary studies have proven a significant increase in formability with 5XXX and 6XXX series of aluminum and AZ31 and AZ61 magnesium alloys [2], [3], [4]. A significant effect of both strain rate hardening and strain hardening on formability was confirmed by some early experimental studies. The improvement of formability at elevated temperatures was attributed mainly to the enhanced strain rate sensitivity that comes with increasing temperature [5], [6].
Attempts on warm forming of lightweight materials have not, however, gone beyond lab-scale experiments on simple cups and a few prototype trials in the industry, because of various underlying unknowns. These unknowns include the effect of temperature distribution and control on the forming limits and lack of understanding of complex interface conditions. To broaden the fundamental understanding of the warm forming mechanism and to shorten product and process design lead time, finite element analysis (FEA) techniques have been tried in recent decades. The effectiveness of using FEA for warm forming analysis has been verified through case studies where FEA results were compared with experimental findings for simple part cases [7], [8], [9], [10]. In addition, optimization techniques, such as sensitivity analysis [11], [12], genetic algorithm [13], and artificial neural network (ANN) [14] have been increasingly applied to metal forming process design and control in combination with FEA. When such optimization methods were used for the industrial-size warm forming FEA models, however, it was recognized that the time required to complete such simulations could be impractically long due to the large number of design variables and the complex features of the full scale models. Thus, in order to exploit the benefits of FEA as a virtual experimentation and/or prototype process effectively, and to eliminate costly trial and error procedures of current practices, alternative process and tooling optimization techniques become necessary.
To attempt to achieve computational efficiency, while preserving reliability in design optimization problems, there has been some research on the multi-fidelity technique. This combines results from a small number of expensive and accurate high-fidelity analyses with those from a large number of cheaper and less accurate low-fidelity analyses. Hutchison et al. [15] coupled a detailed aerodynamic model with a simple algebraic model in the optimization of a high-speed civil transport (HSCT) wing. They first utilized the simple model to estimate the drag components. Then, the deviation of the responses from the detailed model was corrected by a constant scaling function obtained at the nominal design point. This approach allowed the optimization to proceed effectively without excessive computational expense. Heftka [16] demonstrated a global-local approximation method with a simple beam example. In his study, a crude FEA model captured the general trend of a buckling load with respect to the cross sectional area. A linearly varying scaling factor calculated using the derivative of a more refined FEA model enabled the inexpensive and reliable predictions of the bucking load in a wider range of design space. Neural network techniques were also used to correlate the low- and the high-fidelity models. Watson and Gupta [17] used a neural network to predict the differences between the two models in microwave circuit design. Leary et al. [18] developed a knowledge-based artificial neural network (KBNN) incorporating the low-fidelity analysis result into the network structure. Using the knowledge acquired from the cheap function when training the network, an improved level of accuracy could be achieved over the entire design domain using only a small number of expensive results.
This paper attempts to develop a hybrid multi-fidelity system to determine the optimal temperature distribution of tooling elements in a rectangular cup forming process. Less accurate and less costly isothermal FEA was used as the low-fidelity model, and was combined with expensive and accurate non-isothermal FEA (i.e., high-fidelity model) to reach the optimal design point in a short computational time. The non-linear relationship between the design variables (i.e., regional temperatures on the tooling) and the response (i.e., part depth value before failure) was established using (a) a polynomial function and (b) a KBNN [18]. The performance of each algorithm was evaluated by comparing the approximations from the developed multi-fidelity systems to the high-fidelity analysis results. The effect of regional temperatures on forming performance was investigated by constructing the response surfaces near the optimal design point. In addition, detailed deformation characteristics of the material at different temperature conditions were further discussed from the temperature and thickness strain distributions on the formed parts.
Section snippets
Development of a multi-fidelity system
The design problem of this paper is the optimization of warm forming temperature distribution. We wish to maximize the formability of an aluminum alloy by varying the temperature distribution on the tooling system in a range of 25–350 °C. Since the classical optimization procedures that are based on an iteration algorithm require hundreds of high-fidelity analyses to obtain reliable responses, we cannot afford the high computational cost for complex design problems such as a warm forming
Determination of the optimal temperature distribution
After obtaining the reliable multi-fidelity prediction system based on the KBNN technique, the next goal is to find the optimal input pattern (i.e., optimal temperature distribution of the tooling elements) to match a given output specification (i.e., maximum part depth value before failure). As illustrated in Fig. 7, the exact same algorithm by which we learned the weights and biases of the original network in Fig. 4 was applied. The original input neuron was considered as the new hidden
Conclusions
In order to reduce the computational expense for the optimal temperature design of warm forming processes, a hybrid multi-fidelity system combining the accurate high-fidelity analyses with the efficient low-fidelity approximations was developed along with an ANN technique.
The basic knowledge about the temperature effects on warm forming performance was obtained from a large number of isothermal FEA, which is efficient but not very accurate or complete. This prior knowledge was then utilized in
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