A moment-based stochastic edge-based smoothed finite element method for electromagnetic forming process
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moment-based stochastic edge-based smoothed finite element method for electromagnetic forming process YANG Qin, WANG Bing, LI She & CUI XiangYang
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State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, China Received August 27, 2019; accepted November 22, 2019; published online May 27, 2020
In this paper, a novel stochastic method named as the moment-based stochastic edge-based finite element method (MSES-FEM) is proposed to deal with the uncertain electromagnetic problems. First, electromagnetic and mechanical field are formulated by smoothed Galerkin Weak Form under edge-based smoothed finite element method (ES-FEM) scheme. The moment analysis is then applied to obtain the first four moments of the responses and to observe the effects of each random variable on electromagnetic field responses. The maximum entropy theory is employed to calculate the probability density functions (PDFs) of the responses. A quasi-static electromagnetic problem and a practical electromagnetic forming problem (EMF) are performed. The proposed method successfully solves stochastic electromagnetic forming analysis under the uncertain parameters. Numerical results obtained by the proposed MSES-FEM are quite satisfactory with the ones by the Monte Carlo simulation (MCS). MSES-FEM, electromagnetic forming problem, the moment analysis, the maximum entropy theory, probability density function Citation:
Yang Q, Wang B, Li S, et al. A moment-based stochastic edge-based smoothed finite element method for electromagnetic forming process. Sci China Tech Sci, 2020, 63, https://doi.org/10.1007/s11431-019-1489-2
1 Introduction Recently, the applications of electromagnetic theory in practical engineering problems have attached much attention among the researchers for its special properties. Electromagnetic force has been widely used to solve the material processing, such as electromagnetic forming [1–4], electromagnetic riveting [5,6] and impulse welding [7]. Thus a number of numerical methods so far have been developed to solve electromagnetic problems. The finite difference method (FDM) [8] was the first method applied to electromagnetic field analysis, but it was gradually replaced by FEM [9–15] due to the regular grid of FDM cannot simulate the problem with complex geometry satisfactorily, and it is difficult for FDM to improve the calculation accuracy by introducing the higher-order terms of Taylor series. *Corresponding author (email: [email protected])
The FEM is the most widely used method due to its versatility and effectiveness. However, it has an essential drawback that the model is overly-stiff and difficult to treat mesh distortion under large deformation. To overcome the shortcomings of FEM, a class of smoothed finite element methods stemming from FEM have emerged [16]. These methods include cell-based (CS-FEM) [17], node-based (NS-FEM) [18–20] and edge-based (ES-FEM) [21,22] smoothed finite element method which used the strain smoothing technique [23] in each smoothing domains
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