For tantalum metal, the definitions of these seven calibration parameters and the minimum/maximum bounds for each are provided in Table 2. We begin by inferring the parameter values for the seven dimensionless parameters (ș, ț, Ȗ, y0, y, s0, s) and estimating the discrepancy bias of the PTW model. The procedure involves 10,000 Markov Chain Monte-Carlo (MCMC) runs exploiting the parameter ranges as defined in Table 2. The Bayesian inference is initiated with 500 runs at seven different levels. The GPM surrogates used to replace the PTW simulations are trained with a 100 run Latin Hypercube maxi-min design-of-computer-experiment. Table 2: The upper and lower bounds of the seven Parameters of the PTW model. Symbol Description Minimum Maximum ș Initial strain hardening rate 2.78 × 10-5 0.0336 ț Material constant in thermal activation energy term (relates to the temperature dependence) 0.438 1.110 Ȗ Material constant in thermal activation energy term(relates to the strain rate dependence) 6.96 × 10-8 6.76 × 10-4 y0 Minimum yield stress (at T = 0 K) 0.00686 0.0126 y Maximum yield stress (at T ≈ melting) 7.17 × 10 -4 0.00192 s0 Minimum saturation stress (at T = 0 K) 0.0126 0.0564 s Maximum saturation stress (at T ≈ melting) 0.00192 0.00616 5.2 Investigation of Forecasting Error for PTW Model The investigation of forecasting errors can be facilitated by excluding some of the available measurements, known as hold-out experiments, during calibration. Consider a PTW model which is calibrated and bias corrected with less than the available number of experimental measurements, thereby furnishing us with hold-out experiments. This calibrated and bias corrected PTW model constitutes our best possible estimate for the given amount of experiments. We can now attempt to predict the hold-out experiments by exploiting the availability of this best possible PTW model. The difference between the PTW forecasts and the hold-out experiments yields the forecasting error of this particular PTW model for a given number of calibration experiments. One of the reasons that PTW models are especially suitable for the illustration of forecasting errors is the availability of a large number of measurements conducted at varying control parameter settings. The measurement points (ni in Table 1) from all six datasets are combined to obtain a pool of 142 data points, constituting stress measurements and the associated value for strain, strain rate and temperatures. We 441
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