Hirvijoki, E.

Sanchis, L.

Zonta, F.

English

MPG.PuRe

Backward Monte Carlo method for fast-ion-loss simulationsFilippo Zonta1, Lucia Sanchis1, Eero Hirvijoki1 and the ASDEX Upgrade Team*1 Department of Applied Physics, Aalto University,P.O. Box 11100, 00076 AALTO, FinlandIn fusion devices, the fast-ion population plays a crucial role in the plasma performance as itprovides one of the primary sources of plasma heating. It is therefore of vital interest to predictthe behaviour and the possible confinement losses of such particles with tested and validatedmodels [1–3]. The standard approach, based on the Forward Monte Carlo method (FMC) [4, 5],often suffers of poor statistics and low accuracy when analyzing small target domains of inter-est, wasting a large amount of resources for markers that do not contribute to the fast-ion losssignal. As a remedy, with the Backward Monte Carlo (BMC) algorithm [6–8], the probabilityof a marker reaching a specific area is tracked backwards in time starting at the region of inter-est. The calculation of the probability involves an iterative process in which, during every step,points in phase-space are advanced according to their forward equations of motion and the sta-tistical spread from the Monte Carlo collision operator taken into account deterministically. Inthe present work, we propose a novel 5D guiding-center BMC-based scheme for simulation offast-ion losses, integrated into the Monte Carlo orbit-following code ASCOT [4, 5]. The escapeprobability obtained with the BMC-scheme is used as information for importance sampling ofthe initial distribution of markers mitigating the artificial diffusion error that was shown previ-ously in [8] and at the same time solving the statistics problem of the standard FMC scheme.Backward Monte Carlo methodDiscretizing time as the regularly spaced ordered set {0,∆t, ...,m∆t, ...,M∆t}, the functionΦ is used to describe the probability of a particle at phase-space position z and time index mto reach the target domain Ω during a time interval τ ∈ [m∆t,M∆t]. The probability Φ can beevaluated formally with the following recursive formula [8]Φ(z,m−1) =∫RdΦ(ϕ∆t,(m−1)∆t(z,r),m) exp[−r2/2](2π)d/2dr, (1)where (z, t) 7→ (ϕ∆t,t(z,R), t+∆t) is the particle integrator map combining both the Hamiltonianmotion ϕ∆t,tU (z) and the collisional contribution, evaluated with the Euler-Maryama schemeϕ∆t,tC (z,R) = z+K(z, t)∆t +Σ(z, t) ·R√∆t, (2)*See the author list of H. Meyer et al 2019 Nucl. Fusion 59 11201447th EPS Conference on Plasma Physics P5.1012where K(z, t) and Σ(z, t) are the guiding center friction and diffusion coefficients [9, 5], and R∼N (0,I) is a random variable following a standard multivariate distribution. In the numericalimplementation of the BMC scheme, a 5D phase-space mesh is constructed to represent theprobability Φ. Since the particle push-forward map is continuous, the function Φ is representedas a linear interpoland Φ(z,m) = ∑i Φmi Λi(z). Eq. (1) is then discretized by integrating thestochastic part numerically with a Gauss-Hermite quadratureΦm−1i = π−1/2∑kwk ∑jΦmj Λj(ϕ∆ti (√2rk)), (3)where rk and wk are the standard nodes and weights for the Gauss-Hermite integration of thetype∫∞−∞ exp(−r2) f (r)dr ≈ ∑k wk f (rk). Fast-ion loss signals are then computed as an integralof the source distribution weighted with the event probability of hitting the target domain Ω inthe given time interval [0, t]I(Ω, t) =∫S(z′)E[P(zτ ;Ω, t) | z0 = z′]dz′, (4)In principle, the BMC scheme could be used directly in (4). In the present work, it is adoptedfor performing importance sampling of the source distribution to improve the statistics of anexisting FMC simulation.Test cases for the new algorithmthe BMC scheme is applied to an AUG H-mode discharge where the fast-ion source (fig. 1)is introduced by means of the neutral beam injector (NBI) system. In particular, the magneticand kinetic profiles correspond to the shot AUG33143 with the beams Q7 and Q8 producedby the BBNBI code [10], assuming an energy of 93 keV and power of 2.5 MW for each beam.1 1.5 2 2.5R [m]-1.5-1-0.500.511.5z [m]00.511.522.531021BBNBIAUGNBI #8NBI #7#ions/m2a)-2 -1 0 1 2R [m]-2-1012z [m]00.511.522.51021BBNBIAUGNBI #8NBI #7#ions/m2b)Figure 1: Initial distribution of beams Q7 andQ8 projected onto the toroidal (a) and toroidal(b) cross sections.For this example, the target domain is set to asmall subset of the wall near the divertor. Thetarget hit probability is computed with boththe BMC and FMC schemes, with differenttotal integration time. The result is illustratedin Fig. 2. Where BMC manages to produce ameaningful-appearing evolution for the prob-ability backwards in time, FMC seems to pro-duce only Monte Carlo noise, i.e., insuffi-cient statistics. The benefits of the new BMC-IS method are made apparent by the hit ratestatistics in Table 1.47th EPS Conference on Plasma Physics P5.10121.0 1.5 2.0 2.5R [m]−1.5−1.0−0.50.00.51.01.5z [m]10−410−310−210−1a) BMC, T = 5×10−6 s1.0 1.5 2.0 2.5R [m]−1.5−1.0−0.50.00.51.01.5z [m]10−410−310−210−1b) BMC, T = 5×10−4 s1.0 1.5 2.0 2.5R [m]−1.5−1.0−0.50.00.51.01.5z [m]10−410−310−2c) FMC, T = 5×10−4 sFigure 2: Comparison of the probability from BMC and FMC for different total times T . Timestep is ∆t = 1×10−6 s with 100 sub-cycles. Grid size is 64 nodes in all dimensions. The illus-trated quantity is the mean value of Φ over the φ , p‖ and p⊥ dimensions.Mesh elements T=2×10−5s T=1×10−4s T=5×10−4sFMC — 10 10 11BMC-IS 325 485 188 86BMC-IS 645 920 853 829BMC-IS 1285 1909 1101 1097Table 1: Number of recorded hits as a function of the total simulation times T for FMC andBMC-IS, using 105 input markers. Major time step is ∆t = 1×10−6 s with 100 sub-cycles.While the FMC manages only a few hits to the target, the BMC-IS provides a significantly largernumber of recorded hits. These results are likely to improve if irregular meshes mimicking thefield lines and the shape of the target domain are adopted.In order to check quantitatively the consistency of the BMC scheme, the probability matrixhas been computed for different mesh sizes and then compared to a FMC simulation usingmarkers on a few common nodes. Figure 3 shows the resulting mean error of each mesh re-finement after 5 major time steps with respect to the benchmark FMC solution. The error isconsistently reduced by increasing the mesh density, indicating that the probability from theBMC calculation converges consistently towards the reference solution.Summary and conclusionsThis paper introduced a novel Backward Monte Carlo scheme to improve statistics in simu-lations of fast-ion losses, especially to small targets. The method was implemented within the47th EPS Conference on Plasma Physics P5.1012 0.001 0.01 0.1 8  16  32  64  128mean errmesh points per dimensionFigure 3: Convergence test for the BMC scheme.ASCOT suite of codes and tested in realistic AUG configurations using NBI particle distribu-tions. In the test cases considered, the approximate target hit probability from the fast BackwardMonte Carlo calculation was used to perform importance sampling of the initial marker pop-ulation. Using the re-sampled markers in Forward Monte Carlo simulations then significantlyimproved the statistics of target hits in comparison to not using the information from the Back-ward Monte Carlo calculation. These encouraging results are anticipated to improve further ifirregular meshes, that adapt to target and plasma geometry, are explored.Ackwoledgments Academy of Finland grant Nos. 320058, 324759, 315278. Euratom pro-gramme 2014-2018 and 2019-2020 grant No. 633053References[1] Garcia-Munoz et al. . Plasma Physics and Controlled Fusion, 55(12):124014, December2013.[2] Kurki-Suonio et al. Plasma Physics and Controlled Fusion, 59(1):014013, January 2017.[3] Van Zeeland et al. Nuclear Fusion, 55(7):073028, July 2015.[4] Hirvijoki et al. Computer Physics Communications, 185(4):1310–1321, April 2014.[5] Särkimäki et al. Computer Physics Communications, 222:374–383, January 2018.[6] Bormetti et al. European Journal of Applied Mathematics, 29(1):146–187, 2018.[7] Zhang et al. Physics of Plasmas, 24(9):092511, September 2017.[8] E. Hirvijoki, 2019, arXiv:1905.04952.[9] Hirvijoki et al. Physics of Plasmas, 20(9):092505, September 2013.[10] Asunta et al. Computer Physics Communications, 188:33–46, March 2015.47th EPS Conference on Plasma Physics P5.1012

Backward Monte Carlo method for fast-ion-loss simulations

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Backward Monte Carlo method for fast-ion-loss simulations

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