A new probabilistic tool for the determination and optimization of multiphase equation of state parameters: Application to tin

¹ CEA, DAM, Valduc, 21120 Is-sur-Tille, France
² LPMA, UFR de Mathématiques Paris 7, 75013 Paris, France
³ CEA, DAM, Bruyères-le-Châtel, 91297 Arpajon, France

^a e-mail: emmanuel.fraizier@cea.fr

Abstract

A thermodynamically consistent Equation Of State (EOS) was developed to predict and analyse the behaviour of multiphase metals under shock wave loading. Assuming the Mie-Gruneisen hypothesis together with the Birch (for example) formulation, the EOS gives the relation between pressure P, temperature T and atomic volume V. Experimental data (P,V,T) for each phase are provided mainly by X-ray diffraction measurements with diamond anvil cells. In this work, mathematical tools are designed to optimize the determination of the EOS parameters and evaluate uncertainty. The general EOS form is y = f_ϑ(x) where y = P, x = (V,T) and ϑ is the parameter vector to calibrate. Using experimental data (x_i,y_i), the least square (non-linear) regression provides an optimal value ϑ^∗ for the fit parameters. The measurement errors on y and x give biased estimation of ϑ^∗ with the standard method. Assuming centered and known variance laws for the errors, a statistical procedure is proposed to estimate ϑ^∗ and determine confidence intervals. Thanks to a Bayesian approach it is possible to introduce physical interval knowledge of the parameters in this procedure. Moreover, various EOS f_ϑ^∗ formulations are evaluated with a chi-squared type statistical test. The present method is applied on experimental data for multi phase tin (β and γ phases and liquid state) in order to provide an optimized multi-phase model. Furthermore, the method is used to design further experimental campaign and to evaluate the gain of new experimental data with the corresponding estimated errors.