Inverse parameter estimation to predict material parameters of the\xa0Cowper\u2013Symonds constitutive equation in electrohydraulic forming process

Abstract

The development of a reliable numerical simulation is essential for understanding high-speed forming processes such as electrohydraulic forming (EHF). This numerical model should be created based on the accurate material properties. However, dynamic material properties at strain rates exceeding 1000 s^{-1} cannot be easily obtained through an experimental approach. Thus, this study predicted two material parameters in the Cowper–Symonds constitutive equation based on inverse parameter estimation, such that the parameters predicted using the numerical simulation corresponded well with those obtained from the experimental results. The target material was a 1-mm-thick Al 5052-H32 sheet. The comparison target included the final deformation shape of the sheet in the EHF-free bulging test at three input voltages of 6, 7, and 8 kV. For the inverse parameter estimation, the posterior distribution for the two parameters included a likelihood and a prior distribution. For the likelihood construction, a reduced-order surrogate model was developed in advance to substitute the numerical simulation based on ordinary Kriging and principal component analysis. Moreover, the error distribution of the bulge height between the experiment and reduced-order surrogate model was obtained. The prior distribution at 7 kV was defined as a uniform distribution, and the posterior distribution at 7 kV was employed as a prior distribution at 6, 7, and 8 kV. Furthermore, Markov chain Monte Carlo sampling was employed and the Metropolis–Hastings algorithm was adopted to obtain the samples following the posterior distribution. After the autocorrelation calculation for sample independence, the lag with an autocorrelation of pm ,0.02 interval was selected and every lag^{mathrm {th}} sample was obtained. The total number of acquired samples was 10^{5}, and the mean values were calculated from the obtained samples. Consequently, the numerical simulation with mean values displayed good agreement with the experimental results.