Simulation and design of an IPHI-based neutron source,first steps toward SONATE

Abstract

Facilities providing bright thermal neutron beams are of primary importance for various research topics. At CEA-Saclay, a compact accelerator driven neutron source, SONATE, is investigated in taking advantage of the IPHI accelerator able to deliver a 3 MeV proton beam with an intensity up to 100 mA. To optimize the performances of such a neutron source, it is necessary to maximize the thermal neutron flux while minimizing the contribution of other particles. In this work, optimization has been performed using the Monte Carlo code TOUCANS, a neutron transport code based on Geant4 developed at CEA-Saclay. This latter has been coupled to PROMETHEE, a software allowing multi-objective optimization for many simulation software. In this work the Kriging metamodel based approach is used to optimize a neutron beamdump. To take into account the various constraints, noise on the detection system and radiation protection issues, several beamdump configurations are evaluated. The variation of beamdump parameters makes it possible to identify the set of optimal solutions, the Pareto front. It allows to focus on the set of best choices and to choose wisely the best configurations. After describing the validation of TOUCANS on experimental tests performed from 2016 to 2022, the capability of such an approach will be presented.

Keywords

Monte Carlo Geant4 Simulation black-box optimization Kriging active learning

1. Introduction

Neutron beams play an essential role in materials science and mainly come from research reactors or spallation sources. Spallation sources have a high cost making them difficult to build by a single country. Most of the aging research reactors are scheduled to be shut down and new ones are not considered either for political reasons or their high cost [7]. To overcome this future shortage of neutrons, an alternative technology at a lower cost, Compact Accelerator driven Neutron Sources (CANS) are being developed with performances that could approach reactor ones.

A CANS based on the IPHI (High Intensity Proton Injector) accelerator is under development at CEA-Saclay to replace the Orphée reactor, shut down in 2019. IPHI delivers a 3 MeV proton beam with an intensity up to 100 mA. Until now, the tests were performed with beryllium targets which are part of a target-moderator-reflector (TMR) assembly. This latter provides thermal neutrons thanks to a polyethylene moderator from which they are extracted and collimated before arriving at an experimental setup. In this work, the neutron beam from the TMR assembly is sent to an experimental room having an imaging system and a neutron beamdump. The purpose of the beamdump is to minimize fast and thermal back-scattered neutrons along with gamma radiation which can induce noise on the image. In addition, space constraints and radiation protection issues have to be taken into account making the beamdump optimization difficult. The performances of the experimental setup are evaluated with Monte Carlo simulations driven by parameters of the different elements (geometries, materials, etc). To optimize CANS shielding and performances, different approaches have already been investigated using for example genetic algorithms [9,10]. In this work, to tackle this multi-objective optimization problem, the PROMETHEE software is used [16]. This graphical software environment for parametric computer simulations relies on the Funz environment [17] which contains a set of specific R-based [14] algorithms for sparse active learning.

This paper describes the use of such multi-objective optimization method in the design of the experimental setup. Section 2 starts with the presentation and validation of the Geant4-based simulation code named TOUCANS on the latest experiment performed at IPHI. Then, Section 3 describes the multi-objective optimization procedure performed with PROMETHEE. Finally, Section 4 provides an example of the coupling between TOUCANS and PROMETHEE applied to the design of a neutron beamdump.

2. Geant4 simulation verification and validation

The simulation code TOUCANS (TOolkit for a Unified Compact Accelerator Neutron Sources design) based on Geant4 [1] (version 10.07.p01) is used in this work. TOUCANS is developed at CEA-Saclay (CEA/DRF/IRFU/DPhN) since 2019. Recently, it has been benchmarked against Tripoli-4^®, a reference code used by the French nuclear industry [5]. Improvement brought to the Geant4 neutron High-Precision package, regarding the thermal scattering law (TSL) data treatment, reduces the differences between Tripoli-4^® and TOUCANS/Geant4 to less than a few % [19]. TOUCANS has also been benchmarked against the experimental configurations of 2016 [21] and 2019 [20]. In 2022, a new experimental campaign was performed with a 50 kW beryllium target. Thermal neutron flux measurements were done using the gold foil activation technique. As the Fig. 1(b) shows, the simulated thermal neutron flux (E_n < 512 meV – cadmium cut) from TOUCANS agree well with the experimental data at the measurement point within the uncertainties.

Fig. 1.

Left: Sketch of the 2022 TMR experimental configurations. Right: Thermal neutron flux (E_n < 512 meV) as a function of the distance, for the TMR configuration of 2022. Geant4 simulations are compared with experimental data.

3. Multi-objective optimization: Kriging metamodel

TOUCANS simulations are based on input parameters defined in an input file corresponding to the characteristics of the installation to optimize. These characteristics correspond to the geometries and materials making up the TMR assembly such as the collimator, the imaging system and the beamdump presented in Fig. 2. The optimization of such systems requires to deal with a large number of input parameters d that need to be tuned to satisfy m outputs, called objectives.

Fig. 2.

Illustration of the beamdump configuration from Geant4 simulation. Left: simplified modelisation of the installation, from the production of neutrons by the TMR assembly on the left to the stopping of the beam by the beamdump in the experimental room. Right: geometric characteristics of the beamdump resulting from a preliminary study. The parameters set by the technical constraints are indicated in blue. The continuous variable parameters for which an optimal value is sought are indicated in red.

Correlations between the input parameters could be present and nonlinear effect could impact the objective values making the optimization difficult. A Monte Carlo simulation being by definition non-deterministic, no information about properties of the considered function are available, it’s a black-box function. In addition, Monte Carlo simulations are very time consuming and computationally expensive therefore optimization studies based on a complete exploration of the input space is impossible when the dimensionality of the input space is high. To reduce this computational cost, an approach based on metamodels (surrogate models) of the black-box function to discover is used. They provide a mathematical approximation of the input/output relationship with few evaluations and use it to predict the objectives on the research domain. This allows to understand and quantify the impact of input parameters on multiple objectives.

The general principles of a metamodel-based optimization are described using the Kriging metamodel [11] (also known as gaussian process regression [15]). This procedure is based on the Efficient Global Optimization algorithm (EGO) described in [8]. This algorithm could be split in four steps represented in Fig. 3 and described in the following:

Fig. 3.

Flowchart of the optimization procedure performed with PROMETHEE. Methods, algorithms and criterions used are shown on the right.

(1)

Sensitivity analysis: The goal is to evaluate the impact of each input parameter on each objective. This is performed here with the Morris method [12] which is of the type “one-factor-at-a-time” i.e. only one parameter is varied between each simulation giving an elementary effect Δ. A parameter R is chosen to create a d-dimensional R-level grid corresponding to a discretization of the input space. Therefore, $R (d + 1)$ simulations are performed to estimate the elementary effect attributable to each input parameter. For each of them, the mean of the absolute value of the elementary effects $μ^{*} = \frac{1}{R} \sum_{r = 1}^{R} | Δ^{r} |$ and the standard deviation of the elementary effects $σ = \frac{1}{R - 1} \sum_{r = 1}^{R} {(Δ^{r} - μ)}^{2}$ are calculated. They respectively represent the influence on the objectives and the nonlinear effects/correlation of the inputs between them on the objectives.

(2)

Initial design: Generate a set of n initial points specified by a space-filling experimental design. Latin Hypercube Sampling (LHS) is used in d dimensions giving $n = R \times d$ points. It ensures the full coverage of the range of the input parameters and allows to accurately reproduce the one-dimensional projection of the input sampling design.

(3)

Metamodel training: After evaluating the function on the initial design with TOUCANS, the parameters of the metamodel are fitted to the data using a Maximum Likelihood Estimation (MLE). The metamodel used here is the Kriging metamodel which is a spatial interpolation method, characterized by its mean $μ_{K} (x)$ and covariance function (kernel) $c (x, x^{'})$ . The goal of Kriging method is to predict the output $y^{'} (x^{'})$ of the TOUCANS code at a point not yet observed or simulated. Note that the choice of such metamodel, which can hold a gaussian noise to interpolate is consistent with Monte-Carlo solvers like Geant4.

(4)

Infill criterion optimization: A new point $x_{n + 1}$ is then suggested based on a so-called infill criterion which is based on the metamodel. This latter determines the balance between exploration (search for new possible optimum) and exploitation (local improvement around existing observation). In the mono-objective case, the EGO-algorithm is based on the Expected Improvement (EI) criterion [8]. EI is defined as the difference between the current minimum of the observations and the new function value provided by Kriging metamodel. It is then the expectation of this quantity since the new function value is a random variable.

A new observation $y (x_{n + 1})$ is obtained by running this new simulation. Covariance parameters are then re-estimates and the Kriging metamodel updated. This is repeated until the stopping criterion/convergence is met.

Several methods are available to perform each step. For step 1, several sensitivity methods are described in [13]. For steps 2 to 4, an overview of available packages can be found in R and Python languages, in references [3] and [2], respectively.

To illustrate the optimization procedure, a 1-dimensional example is shown in Fig. 4. An objective function for which the global minimum is to find is given by the blue line. At the first iteration, an initial design is generated and the simulation is performed giving five observations of the function represented by the red dots. A Kriging metamodel can then be set up, the black line represents this metamodel and the dotted black lines represent its uncertainty given by the covariance function. From there, the EI can be estimated using the formula from [8]. At iteration 1, the algorithm initially focuses on the current local minimum at $x \approx 0.55$ where the calculated EI is the highest. Therefore, a new observation point is performed at this maximum EI at iteration 2. Then, the algorithm explores region where the uncertainty is the highest here at $x \approx 0.20$ . At iteration 3, three additional points are added around the global minimun. At this stage, the global minimum is precisely determined as the EI tends to 0.

Fig. 4.

Optimization procedure on a 1-dimensional example. Top: the objective function to find the global minimum is in blue. The design points observed are in red. Black lines represent the mean $μ_{K} (x)$ of the Kriging metamodel. Black dotted lines represent the uncertainty given by the covariance function $c (x, x^{'})$ . The green curve corresponds to the calculated expected improvement. Kriging metamodel and EI calculation were performed with the DiceKriging R-package [18].

4. Metamodel-based sequential procedure applied to a beamdump design

The procedure described in Section 3 is applied to the design of a neutron beamdump in its environment represented in Fig. 2.

A neutron beam arrives from the left, goes through a collimator, interacts with the imaging system before ending up in the beamdump. The objectives of the latter is to minimize the background noise on the imaging system and minimize the equivalent dose rate outside the room.

The beamdump is a hollow box made up of 3 envelope layers with a square opening as represented in Fig. 2(b). A layer of borated polyethylene (PE) to slow down and stop fast neutrons flux, is wrapped between two lead layers to attenuate gamma radiation produced by neutron capture. The experimental room and the beamdump maximum length being fixed by technical constraints, the thickness of the layers must be optimized to minimize neutron diffusions and the equivalent dose rate outside the room. The input parameters of the beamdump are the transverse dimension to the neutron beam and for each of the 3 layers, the thickness of the bottom of the box and its lateral thickness.

Two methods have been used to optimize this beamdump. The first method is based on grid search algorithm. The second method corresponds to the sequential procedure described in Section 3. Both cases have been performed to assess the beamdump efficiency on the objectives. Using the first method, a beamdump configuration is found whose input parameter values are given in Table 1.This design makes it possible to respect the equivalent dose rates outside the room without generating significant noise on the image, it will be referred to as the reference design. The second method is then applied and its results are presented in the following.

Table 1
Values of input parameters and objectives for the beamdump found by applying grid search algorithm and optimization algorithm. All input parameter values are in centimeters. The uncertainties are not shown because they are negligible

Input Objectives

xyDimension Side1 (lead) Side2 (borated PE) Side3 (lead) Bottom1 (lead) Bottom2 (borated PE) Bottom3 (lead) Dose (a.u.) Noise (a.u.)

box-constraints $[140, 240]$ $[0, 5]$ $[0, 30]$ $[0, 5]$ $[0, 5]$ $[0, 30]$ $[0, 30]$

Standard method 175 3 25 1 3 25 1 $13.07 \cdot 10^{- 2}$ $9.48 \cdot 10^{- 3}$

PROMETHEE (iteration 5) 178 1 27 2 1 24 1 $11 \cdot 10^{- 2}$ $8.59 \cdot 10^{- 3}$

	Input	Objectives
box-constraints	$[140, 240]$	$[0, 5]$	$[0, 30]$	$[0, 5]$	$[0, 5]$	$[0, 30]$	$[0, 30]$
Standard method	175	3	25	1	3	25	1	$13.07 \cdot 10^{- 2}$	$9.48 \cdot 10^{- 3}$
PROMETHEE (iteration 5)	178	1	27	2	1	24	1	$11 \cdot 10^{- 2}$	$8.59 \cdot 10^{- 3}$

The results of sensitivity analysis (step 1) is performed with the Morris method for equivalent dose rate as objective are shown in Fig. 5. According to [12], the limit of non-linearity of the inputs is represented by the line $μ^{*} = σ$ . Above this limit, interactions between inputs are significant, indistinguishable from nonlinear effects. Figure 5 shows that all input parameters defined in our problem are relevant to this study. Parameters which have the highest impact are the side and the bottom of borated PE (Side2 and Bottom2) and the least impact for the internal lead layer. This is not surprising for an incident neutron beam.

Fig. 5.

Sensitivity analysis: estimated means $μ^{*}$ and standard deviations σ of the distributions of elementary effects in the example of the beamdump design. The blue dotted line corresponds to $μ^{*} = σ$ .

Then steps 2 to 4 are performed. There are seven continuous input parameters as represented in Fig. 2(b) and two objectives to optimize. An initial design is generated with parameter $R = 20$ which gives $n = 140$ and as many observations after simulation with the TOUCANS code. When considering multiple objectives, steps 3 and 4 of the optimization algorithm needs to be modified. Rather than gathering objectives into a scalar problem, an approach where metamodels could be fitted independently to each objective [3] is used. Minimizing all the objectives at once is often not possible because of conflicting interests. It is then necessary to determine the set of optimal solutions over the box-constrained domain, called a Pareto set. The image of the Pareto set in the objective space is called the Pareto front, which is used to select the best solution (see Fig. 6).

Fig. 6.

Pareto front example illustrating initial design points observed (in red) and improvements for two possible new observations (green and blue). The dotted red line represents the Pareto front i.e. the set of non-dominated points. The red hatched area represents the zone where points are dominated. The white area represents the zone where new solutions are sought.

Fig. 7.

Multi-objective optimization results for the beamdump design with five iterations. Left: initial design of experiments represented with blue dots, new observations are in orange diamond and reference point in red triangle. Right: design of experiments given by GPareto package relative to the reference design represented with black line.

With objective dimension higher than one, the EI changes to the Expected Hypervolume Improvement (EHI) [6]. In this work, the GPareto package [4] is used to estimate the EHI. The results of this optimization are given in Fig. 7 where the reference point corresponds to the beamdump of reference. Applying this new optimization algorithm in this case, the Pareto front obtained is represented in Fig. 7(a). Blue dots represent the observations of the initial design given by LHS, five additional observations suggested by the algorithm are indicated in orange diamonds and the reference point is indicated in red triangle. This figure shows that the initial design is sufficiently representative to have a majority of the objective points dominated but allows a single point to be non-dominant, even with respect to the reference point. After few iterations, several observations make it possible to slightly improve the Pareto front and so to find a better configuration than the reference point. The final result is given in Table 1 for the last iteration converged.

The image of this Pareto front in the space of variables is illustrated in Fig. 7(b). For each input parameter, the design points are shown relatively to the reference design points represented as a black horizontal line. The reference point is not the one towards which the beamdump configuration must necessarily converge. Different configurations can give very similar objectives, in particular when correlations between input parameters are important. This result illustrates that the convergence towards a common optimum value mainly concerns the input parameters that have the highest impact (see Fig. 5). Among these parameters, those concerning the borated PE layer (Side2 and Bottom2) quickly converge towards the values of the reference beamdump. This confirms the optimum thickness of polyethylene found by the grid search algorithm with respect to the fixed space constraints. The transverse dimension of the beamdump (xyDimension) also converges towards a common value. The box constraints for this input parameter are limited below by the minimum beamdump layer thickness and above by the room dimensions. This makes the range of variation quite small. The suggested values for the lead layer parameters are quite different, probably due to a lower impact and correlation effects. Nevertheless, this shows that it is possible to optimize theses input parameters compared to the beamdump of reference.

To summarize, the grid search algorithm gives a satisfactory beamdump configuration. Using the optimization procedure with PROMETHEE, a few steps of the algorithm allows to find better configurations with objectives that are simultaneously optimized compared to the previous one. The number of simulations carried out by the optimization algorithm is much lower compared to the grid search algorithm, and thus reducing the computation time.

5. Conclusion

In this paper the TOUCANS code used to design a CANS based on IPHI accelerator has been validated against a new experimental configuration with a 50 kW beryllium target tested in 2022. Since it has also been validated against experiments with the 2016 and 2019 TMR configurations, this validates neutron transport using TOUCANS and gives confidence in the reliability of the TOUCANS code to model SONATE. A multi-objective optimization procedure is used here to design an experimental installation and so to find the best set of parameters allowing to successfully reach multiple objectives at a time. Starting from a set of input parameters it aims to find the set of the best solution with the help of the Pareto front. This optimization procedure was applied to the design of a beamdump with constraints. Multiple configurations improving the Pareto front could be found with only few algorithm steps, reducing computational costs. Several infill criterions are proposed in the GPareto package according to the optimization problem. In the future this procedure will be applied to other parts of the installation.

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	Input							Objectives

	xyDimension	Side1 (lead)	Side2 (borated PE)	Side3 (lead)	Bottom1 (lead)	Bottom2 (borated PE)	Bottom3 (lead)	Dose (a.u.)	Noise (a.u.)
box-constraints	$[140, 240]$	$[0, 5]$	$[0, 30]$	$[0, 5]$	$[0, 5]$	$[0, 30]$	$[0, 30]$
Standard method	175	3	25	1	3	25	1	$13.07 \cdot 10^{- 2}$	$9.48 \cdot 10^{- 3}$
PROMETHEE (iteration 5)	178	1	27	2	1	24	1	$11 \cdot 10^{- 2}$	$8.59 \cdot 10^{- 3}$