Calibration of the 2-Phase Bubble Tracking Model for Liquid Mercury Target Simulation with Machine Learning Surrogate Models

The Spallation Neutron Source (SNS) at Oak Ridge National Laboratory is one of the most powerful accelerator-driven neutron sources in the world. The intense protons strike on SNS’s mercury target to provide bright neutron beams, which also leads to severe fluid-structure interactions inside the target. Prediction of resultant loading on the target is difficult particularly when helium gas is injected into mercury to reduce the loading and mitigate the pitting damage on vessel walls. A 2-phase material model that incorporates the Rayleigh-Plesset (R-P) model is expected to address this multi-physics problem. However, several uncertain parameters in the R-P model require intensive simulations to determine their optimal values. With the help of machine learning and the measured target strain, we have studied the major uncertain parameters in this R-P model and developed a framework to identify optimal parameters that significantly reduce the discrepancy between simulations and experimental strains. The preliminary results show the possibility of using this mercury/helium mixture and surrogate models to predict a better match of target strain response when the helium gas is injected.


Introduction
A two-phase constitutive material model that incorporates the Rayleigh-Plesset (R-P) model [1] for mercury/helium mixture in extreme environments to predict the stress/strain response of target vessel has been develop [2], which is one of the important tools to facilitate the target's fatigue analysis [3] and to improve the target's design due to the absence of long-term diagnostics.Without helium gas injection, the equation of state (EOS) material model was used to simulate the mercury in the target vessel and had good predictions of the vessel strain response [4].A further machine learning study of these model parameters [5] illustrated how to utilize the modern computational method to finely tune the EOS model parameters for a better strain response prediction.Work presented in this paper focused on identifying the major parameters in the two-phase mercury model and their reasonable ranges for machine learning application.Preliminary validation results will be presented at the end of this paper to show the capability of this two-phase mercury/helium model.

R-P model and its integration in Sierra
Before introducing bubbles in mercury, the traditional finite element model validated at FTS includes the solid stainless-steel vessel, and the flowing mercury which relies on the EOS material model and its tensile threshold to describe the dynamic behaviour [4].Experimental measurements shows that small helium bubbles injection into mercury can significantly reduce the stress/strain response on the stainless-steel vessel, more importantly to reduce the pitting damages [6].However due to the strain reduction effect from injected bubbles, the simplified EOS mercury model is no more applicable to predict target vessel's strains.Regarding that a two-phase bubble tracking model is developed to include the gas phase bubbles, along with their dynamic behaviours, to address this complex multiphysics problem [7].This two-phase bubble tracking material model is based on R-P model, which is Equation ( 1) describes the relation of radius of bubble R, the evolution of bubble size ( ̇ and  ̈) and its environmental pressure.L0 is the initial density of the fluid surrounding the bubble,  is the kinematic viscosity of the surrounding fluid,  is the adiabatic index of the bubble gas,  is the surface tension of the fluid, 0 is the initial pressure of the bubble, and  is normally the real-time pressure far from the bubble but is used in the R-P model as the pressure in the surrounding fluid that contains many bubbles.More details of this basic bubble dynamics and Equation (1) can be found in references [1,2,7].The dynamics of the gas-fluid mixture was integrated into a material subroutine in the Sierra finite element code [7] to enable the large-scale parallel computing of many simulations for building the surrogate models in next sections.

Identification of major model parameters
As for the two-phase bubble tracking model itself, several physical parameters, such as bubble sizes and their group distribution, are still difficult to measure directly.By leveraging the measured strain data for the target with helium bubbles injected, several simulations have been run and compared with the measured strain to evaluate the model parameters' impact on the strain response.The parameters in Equation ( 1), the surface tension (), adiabatic index (), and viscosity () and the bulk modulus () of the bubble-mercury mixture were selected in sensitivity study with variations in ranges [0.1 2.0] N/m, [1.46 1.94], [0.001 0.02] N•s/m 2 and [21.03 37.85] GPa respectively.The nominal values of these four parameters in previous study [7] fell in corresponding range.We vary one parameter at a time and fix the others at their nominal values, resulting in 24 combinations of these 4 parameters (see Table 1).Previous bubbles distribution study [7] showed that 8 to 10 families are sufficient to capture the true bubble distribution, so 8 families were used in the four parameters sensitivity study.Figure1 illustrates such a distribution, which combined with the 4 parameters in Table 1 as two-phase mercury model input.The corresponding cumulative gas volume ratio is shown in Figure 1, all the 24 cases listed in Table 1 used the same 8-family bubbles distribution.Strains on sensors A, C, G, P and N (Figure 2) were used to verify parameters' sensitivity.3 indicated that within the allowable variation ranges, the bulk modulus had most significant impact than the other three parameters on the strain response.The green curve in Figure 3, along with the green region surrounding it, stood for the mean strain history and its 95% confidence interval from experimental measurement for sensor A at beam power level of 1.4 MW with helium gas injected.The other three model parameters, , , and  generated more concentrated strain curves when they varied in corresponding ranges, indicating a much less impact on the strain response.On the other sensors such as sensor C, G, N and P (locations shown in Figure 2), bulk modulus also showed as the most dominant parameter on affecting the strain response.Thus, the bulk modulus was the primary one for parameter calibration.
Next, we fix all the four material parameters, i.e. bulk modulus K=37.85 GPa (this value returned a better match strain curve in previous sensitivity study), surface tension =0.47N/m, adiabatic index =1.66and viscosity =0.0015N•s/m 2 , and study the bubbles distribution's impact on the strain response.Variations of bubbles distribution focused on the shape of bubbles cumulative volume curve and the value of cumulative volume itself.Curves in Figure 4 illustrate the bubbles distribution in 4 different patterns, labelled as Curve 1, 2, 3 and 4 respectively.Major difference between Curve 1 and 2 is the curve convexity.Besides the 8-family bubbles, 3, 5, 6, 10, 12 and 15-family bubbles were tested to further under-stand their impact on the strains.Simulation results in Figure 5 showed that the bubbles family number has minor or moderate impact on the strain response.The black curves in Figure 5 depicted the averaged strain measured at sensor A from experiments.For the same accumulate gas volume, bubbles distributed in less or equal to 8 families have larger variation on the strain response than those distributed in ≥10 families.Discrete bubbles distribution in 8 or less families should provide more flexibility to match the experimental strains, and more computational efficient in this regarding.We fix the number of families at 8, and represent the bubble distribution as a beta distribution with two parameters a and b.We will calibrate the shape of the bubble distribution by calibrating these parameters.
Increasing the total gas volume fraction, such as from 0.002 to 0.01, can significantly reduce the strain amplitude as shown in Figure 5 (c).This trend shown in simulation agreed with the experimental observation, in which higher gas flow rate alleviated the vessel's strain and damage as well.The gas volume fraction is a key parameter in this gas-mercury mixture model and will be calibrated in this study.

Preliminary machine learning results
Regarding to the impact on the simulation strain data, the bulk modulus, gas cumulative volume, and two beta distribution parameters a and b for controlling the bubble distribution curve were adopted as major parameters in this machine learning study.Variation of bulk modulus was extended in range [10 50] GPa as an initial search.The gas cumulative volume was initialized in range [0.0003 0.003] which was correlated to the real measured gas flow rate varying from 0.8 to 3.2 slpm in target tests.The two curve shape control parameters, a and b were in intervals [0.1 2.0] and [0.1 8.0] respectively to render a wide range of 8-family bubble distributions including right-skewed (Curve 1) and left-skewed distributions (Curve 4, Figure 4).In the first batch of finite element simulation, 150 sets of randomly selected model parameters from these 4 items were launched in HPC at NERSC and the simulations completed a 1.0e-3 second dynamic analysis.Among the 150 simulation strain datasets, 130 were used for training and the remaining 20 sets were used for model validation.A framework was built to efficiently collaborate the high-fidelity simulations, machine learning and inverse optimization, for current and future applications.
In the preliminary machine learning study, sparse polynomial approximations [8] were used to train these 150 simulation strains on different sensors.Results in Table 2 showed that the training errors vary from 7-11%, validation errors vary from 9-13%.Results in Table 2 also showed that using polynomial degree of 7 returned the minimum in both training and validation errors, which indicated that an overfitting might occur for polynomial degree higher than 7 for the limited 150 training datasets, while lower degree of polynomial resulted in underfitting for this surrogate model.

Conclusion
The complex 2-phase gas-mercury mixture material model included more unknown parameters than traditional EOS-based mercury material model.With several simulations, we identified the major parameters in this 2-phase bubble tracking model that had larger impact on the sensor strains.The bulk modulus in the material model, the gas volume, and the two parameters to adjust the bubbles distributions were the major model parameter objects for further machine learning study.Varying these four model parameters in appropriate ranges generated the necessary training datasets for building surrogate models.First batch of 150 simulation results were collected by using the framework and workflow was created in this project.Initial machine learning results showed promising surrogate models from sparse polynomial approximations that had training errors varied from 7-11% and validation errors varied from 9-13%.More simulations and machine learning methods will be applied through this framework to build a better surrogate model for this complex mercury target with gas injection included, which then will be incorporated in inverse optimization to support our model calibration effort.

Figure 2 .
Figure 2. Selected sensors and their location.

Figure 3 .
Figure 3. Impact of 4 parameters on sensor strains, a) bulk modulus, b) surface tension, c) adiabatic index, d) viscosity.Strain results shown in Figure3indicated that within the allowable variation ranges, the bulk modulus had most significant impact than the other three parameters on the strain response.The green curve in Figure3, along with the green region surrounding it, stood for the mean strain history and its 95% confidence interval from experimental measurement for sensor A at beam power level of 1.4 MW with helium gas injected.The other three model parameters, , , and  generated more concentrated strain curves when they varied in corresponding ranges, indicating a much less impact on the strain response.On the other sensors such as sensor C, G, N and P (locations shown in Figure2), bulk modulus also showed as the most dominant parameter on affecting the strain response.Thus, the bulk modulus was the primary one for parameter calibration.

Table 1 .
Tuning parameters in mercury material model.

Table 2 .
Preliminary model training result.