A BAYESIAN OPTIMIZATION METHODOLOGY FOR THE INVERSE DERIVATION OF VISCOPLASTICITY MODEL CONSTANTS IN HIGH STRAIN-RATE SIMULATIONS
Authors: RYAN SHANNON , BERK JULIAN , RANA SANTU , MCDONALD BRODIE , VENKATESH SVETHA 
Published: September 2022

Inverse strategies, which incorporate numerical simulation of the experiment and iterative adjustment of model constants until measured signals can be numerically reproduced, are widely employed in finite element modelling, e.g., Finite Element Model Updating (FEMU) [1], Constitutive Equation Gap Method [2], etc. These strategies permit the utilisation of more extensive data than classically used, e.g., digital image correlation (DIC) for surface strain. Such inverse strategies are typically employed on thermomechanical tests, however a limited number of studies have investigated these methods applied to functional experiments, that is experiments which have utility beyond the derivation of constitutive model constants. For example, Walls et al. [3] performed inverse modelling to optimise constants of energetic materials cylinder test and wedge test. Portone et al. [4] performed model identification and tuning of existing model constants based on inverse modelling of a penetration vs. time experiment. These approaches, however, may result in model constants which are not generalisable – that is, for experimental conditions different to those used in the inverse approach, the identified constants will be inaccurate. Here we present a methodology for deriving constitutive model constants for use in explicit finite element simulations of dynamic processes from multiple functional experiments. The resulting model should be better able to generalise to previously unseen conditions, provided that those conditions are within the range bounded by the initial experiments used for the optimisation. We demonstrate the optimisation methodology using high hardness armour steels across three types of experiments that induce a wide range of loading condition: ballistic depth of penetration [5], rod-on-anvil [6], and near-field blast deformation [7]. We employ an inverse methodology in which (1) a random initial guess of material constitutive model parameter is made (for a predefined constitutive model), x0, (2) simulations of the functional experiments defined above are conducted in LS-DYNA with the selected constitutive model and model constants, and (3) the simulation results are compared with the experimental measurements to determine the total simulation error. An optimisation function is tasked with modifying the model constants and repeating this process until a minimum error is obtained, from which the optimal constitutive model constants can be identified. As hydrocode simulations are computationally expensive, we utilise Bayesian optimization (BO), one of the most sample efficient optimisation techniques available, for this task. We perform an initial optimisation to select a plasticity model from: Zerilli- Armstrong, Johnson-Cook, and two modifications of the Johnson-Cook model. The best performing option, a modified Johnson-Cook model (MJC), is then subject to a more comprehensive optimisation process utilising a leave-one-out (LOO) methodology to maximise the generalisability of the identified solution. In multiple runs of the optimisation the best identified MJC constants do not exhibit any obvious clustering, suggesting that the solution is non-unique, i.e., there are a range of hyperparameter combinations that can generate comparable results in LS-DYNA. This is not unexpected as the constants of the MJC model are not independent - for instance the strain hardening response can be well defined by varying subsets of its empirical constants. A comparison of the dynamic stress-strain response described by the different constantsidentified in each optimisation run are plotted in Figure 1, together with examples of thesimulated experimental results. For comparison, baseline results are also plotted which utilise constitutive models and constants derived from conventional mechanical characterisation experiments or alternative optimisation approaches. To evaluate the generalisability of the optimised MJC constants we simulate a new experimental condition not included in the optimisation process - dynamic compression with a split Hopkinson pressure bar (SHPB) apparatus from [6][8]. The best simulation results utilising the BO-identified constitutive model constants were found to provide comparable results to the baseline model derived specifically from these experiments, and superior results compared to other baseline models, confirming the generalisability of the derived model constants.