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Plasma edge simulations with codes like SOLPS-ITER are widely employed to interpret fusion experiments. However, numerical errors appearing in such simulations are rarely investigated, despite their potential large impact on simulation results. These errors consist of the statistical error and the bias, both resulting from the finite number of employed EIRENE Monte Carlo particles and incomplete convergence, and the discretization error due to the finite resolution of the computational grids. In this contribution, the resulting numerical errors on simulations of pure deuterium and neon seeded H-mode EAST discharges are examined. The statistical error can be kept small compared to other numerical error contributions by averaging the plasma profiles. This allows investigating the bias and discretization errors using Richardson extrapolation. It is shown that grid refinement and the number of employed Monte Carlo particles have the largest influence on the result, in agreement with similar studies of an ITER deuterium case. For the first time, numerical error bars on the entire simulated target profiles are determined showing that the largest numerical error is 17.9%, mainly due to the plasma grid discretization. On top, also numerical errors on simulated neutral pressures are investigated in detail, for which the statistical error is dominant. The analysis demonstrates which setup is needed to keep numerical errors limited: the SOLPS-ITER averaging procedure should be employed including enough EIRENE particles, and the involved grids should be sufficiently refined to reduce discretization errors.

The neutral atoms in the plasma edge of nuclear fusion devices are typically modeled using either a fluid or kinetic approach. The kinetic approach is most accurate, but it has two main disadvantages. First, the usual solution of the high-dimensional kinetic equation using Monte Carlo techniques introduces statistical noise, which hampers the convergence of the coupled plasma-neutral model. Second, the computational time strongly increases for highly collisional regimes. For these reasons, deterministic fluid neutral models remain an attractive alternative, in particular for the highly collisional conditions where their accuracy is expected to be high. In recent years, efforts have been undertaken to improve the agreement between the fluid and kinetic approach by introducing consistent transport coefficients and consistent boundary conditions in the fluid models. In this work, these so-called advanced fluid neutral models are further enhanced by introducing different strategies to cope with the high heterogeneity of the ion-neutral collisionality encountered in realistic plasma-edge geometries, namely isotropic neutral flux limiters and an automated selection criterion for the optimal neutral boundary conditions. The validity of the resulting fluid neutral models is thoroughly assessed for various representative simulation cases with different geometries, divertor collisionalities, and wall materials, including, for the first time, simulations in a realistic ITER plasma edge geometry. Strong quantitative agreement between the fluid and kinetic models is achieved for cases with highest divertor collisionality.

ispartof: CONTRIBUTIONS TO PLASMA PHYSICS vol:62 issue:5-6 status: published

ispartof: Contributions To Plasma Physics vol:62 issue:5-6 status: published

Type I Edge Localized Modes (ELMs) occur naturally in H-mode plasmas, the operational regime envisaged for ITER, and lead to high temporal heat load peaks on plasma-facing components that might induce surface melting and deteriorated material properties due to recrystallization. ELM mitigation techniques are being developed that either attempt to fully suppress the ELM, e.g. using resonant magnetic perturbations (RMP), or to trigger the ELM at higher frequencies, intending to reduce its energy content and hence the heat load peak. In this paper, the effect of the increased ELM frequency achieved by ELM mitigation techniques on the ITER tungsten divertor monoblock thermal and recrystallization behavior is numerically analyzed by solving the unsteady heat conduction equation when repeatedly exposing the monoblock to ELM-like heat loads under ITER baseline burning plasma operating conditions. The impinging heat load is based on a set of empirical relations readily found in literature and the recrystallization fraction is computed using an empirical data fit which is based on the non-isothermal JMAK equation as function of temperature and time. Uncertainties such as the statistical spread on empirical relations and the unknown inter-ELM heat load, as well as the effect of the ELM frequency, are quantified by a parameter scan. Adopting the recently developed concept of a ‘recrystallization budget’ to determine the allowed monoblock heat loads [G. De Temmerman et al., PPCF 60 (2018) 0044018], it was found that ELM frequencies between 25 – 50 Hz are allowed for inter-ELM heat loads between 9 – 6 MW m − 2 , respectively, when assuming that the scaling for the peak ELM target energy density in [T. Eich et al., NME 12 (2017) 84-90] holds. This provides a new limit for ELM mitigation strategies in ITER.

Energy dissipation in the plasma edge is key for future tokamaks. The potential of neon as radiating seeding species in disconnected double null (DDN) configuration is assessed in EAST discharges in high confinement mode (H-mode). As the separation between the two separatrices in the studied DDN discharges is minimum 1.5 cm, the configuration is effectively a single null configuration, and the benefits of the double null topology are minimal. Neon seeding, on the other hand, has a favourable effect: both the target heat flux and the divertor temperature decrease more than five-fold with increased seeding rate in high-recycling conditions. Interpretive edge plasma simulations with SOLPS-ITER in support of ongoing transport analysis are presented. For the unseeded case the numerical results agree with the experimental data within a factor two for the target temperature conditions and measured neutral pressures in the active divertor. The key for achieving good agreement is a suitable selection of coefficients for anomalous transport and neutral conductances between the upper cryopump and the main chamber.

ispartof: CONTRIBUTIONS TO PLASMA PHYSICS vol:60 issue:5-6 status: published

ispartof: CONTRIBUTIONS TO PLASMA PHYSICS vol:60 issue:5-6 status: published

openaire: EC/H2020/633053/EU//EUROfusion We present a spatially hybrid fluid–kinetic neutral model that consists of a fluid model for the hydrogen atoms in the plasma grid region coupled to a kinetic model for atoms sampled at the plasma–void interfaces and a fully kinetic model for the hydrogen molecules. The atoms resulting from molecular dissociation are either treated kinetically (approach 1) or are incorporated in the fluid model (approach 2). For a low-density JET L-mode case, the hybrid method reduces the maximum fluid–kinetic discrepancies for the divertor strike-point electron densities and electron temperatures from approximately 150% to approximately 20% for approach 1 and to approximately 40% for approach 2. Although the simulations with purely fluid neutral model become more accurate for increasing upstream plasma density, we still observe a significant improvement by using the hybrid approach. When consuming the same CPU time in averaging the electron strike-point densities and temperatures over multiple iterations as for the simulations with fully kinetic neutrals, hybrid approach 1 reduces the statistical error with on average a factor 2.5. Hybrid approach 2 further increases this factor to approximately 3.3, at the expense of accuracy.

© 2016 EURATOM We compare the plasma sources (particle, parallel momentum and ion energy) due to plasma-neutral interactions computed with different fluid neutral models with the sources from a Monte Carlo simulation of the kinetic neutral equation. This is done for a fixed background plasma, which is representative for an ITER detached case. We illustrate that the reaction data from the AMJUEL-HYDHEL databases can be incorporated in the fluid models. A pure pressure-diffusion equation gives already accurate results for the particle source, but it is inaccurate for the momentum and energy source. A parallel momentum equation has to be added to achieve predictions of the momentum and energy source within 30% of accuracy. Newly developed boundary conditions, based on the diffusion approximation for incident neutrals, show to be crucial for accurate results of the fluid models close to the divertor target. The slight overestimation of the momentum and energy sources can be further reduced by adding a separate neutral energy equation. ispartof: Nuclear Materials and Energy vol:12 pages:869-875 status: published

As part of an effort to validate the edge plasma model in the SOLPS-ITER code suite under ITER-relevant divertor plasma and neutral conditions, we report on progress in the modeling of the Alcator C-Mod divertor plasma with the new code. We perform simulations with a complete drifts model and kinetic neutrals, including effects of neutral viscosity, ion-molecule collisions and Lyα-opaque conditions, but assuming a pure deuterium plasma. Through a series of simulations with varying divertor geometries, we show the importance of including neutal leakage paths through the divertor substructure on the divertor plasma solution. Moreover, the impact of drifts on inner-outer target asymmetries is assessed. Including both effects, we achieve excellent agreement between simulations and upstream and outer target Langmuir Probe data. In absence of strong volumetric losses due to e.g. impurity radiation in our simulations, the strong inner target detachment observed experimentally remains elusive in our modeling at present. Keywords: SOLPS-ITER, Alcator C-Mod, Divertor modeling

© EURATOM 2018. This paper examines the effect of grid resolution on the solution of a B2-EIRENE simulation of a partially detached ITER divertor plasma. Due to the large amount of required computational time in coupled finite-volume/Monte-Carlo codes, simulations of the plasma edge are typically run on coarse grids. However, new averaging techniques make simulations with finer grids feasible. Starting from the original numerical grid, simulations are performed on two successively finer grids. Results of the numerical error analysis reveal that the discretization errors are large. Peak values are particularly sensitive to grid resolution and can increase more than for the same model input parameters. However, this effect is partially offset by a modification of the operational space of the ITER divertor in this case. By choosing the numerical parameters more adequately, a total numerical error of only has been achieved within a feasible computational time. ispartof: NUCLEAR FUSION vol:59 issue:2 status: published

© 2019 Plasma edge transport simulations typically rely on Monte-Carlo simulations for the kinetic neutral transport. However, for detached operating conditions the dominance of charge-exchange reactions between plasma and neutrals leads to excessive computational costs of these Monte-Carlo simulations. Since the fluid assumption becomes valid in these highly-collisional conditions, the use of a fluid neutral model is interesting to reduce computational costs. In this paper, we show within the SOLPS-ITER code suite that while our recently developed fluid neutral model is able to accurately represent the neutral transport in the divertor, it is unable to capture the kinetic effects near the outer wall. Due to this model flaw, discrepancies in plasma state arise between the fluid plasma-neutral model and a simulation with kinetic neutrals after converging the model equations. As a solution, we present a spatially hybrid neutral model that captures these kinetic effects near the wall, while benefiting from the fluid approach in the divertor region. Moreover, by using the kinetic code in the region outside the plasma grid, the method is able to accurately account for neutral transport through the voids surrounding the plasma and into pumps. We show on a case with a simplified 'slab’ geometry that the results of the proposed hybrid model compare excellently to those of the kinetic solution. ispartof: NUCLEAR MATERIALS AND ENERGY vol:19 pages:28-33 status: published

While turbulent transport is known to dominate the radial particle and energy transport in the plasma edge, a self-consistent description of turbulent transport in mean-field transport codes remains lacking. Mean-field closure models based on the turbulent kinetic energy (k ⊥) and turbulent enstrophy (ζ⊥) have recently been proposed to self-consistently model this transport. This contribution analyses the diffusive particle transport relations of these models by means of the Bayesian framework for parameter estimation and model comparison. The reference data includes a set of isothermal simulations that not only include the scrape-off layer (SOL) but also the outer core region (in which a drift wave-like model is activated) and a set of anisothermal SOL simulations, both obtained with the TOKAM2D turbulence code. This analysis shows that the k ⊥(–ζ⊥) model does not appropriately capture the diffusion coefficients for these new data sets, presumably due to the strong flows in the diamagnetic direction that appear in these new cases. While flow shear is expected to quench the turbulence and the turbulent transport, its effect was not explicitly taken into account in the earlier k ⊥(–ζ⊥) transport models. As flow shear provides a new mechanism for the decorrelation of the turbulence, we propose to introduce an additional time scale in the diffusive transport relation as D ∼ k ⊥ / ( ζ ⊥ + Ω S ) . Inspiration is drawn from shear decorrelation times reported in literature to propose several new candidate models, which are then analysed in a Bayesian setting. This allowed identifying irrelevant terms for certain models and to rank all models according to the Bayesian evidence. While the new models accounting for shear do improve the match to the data, significant errors still remain. Also, no single model could be identified that performs best for all data sets.

© 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim In the quest for computationally affordable plasma edge transport simulations of reactor-relevant plasmas, we consider fluid-neutral models in this paper. To obtain good agreement with kinetic-neutral models in charge-exchange dominated regimes, model equations, transport coefficients, and boundary conditions have to be derived consistently from the kinetic description. In this paper, we describe a fully fluid plasma edge transport model that is aimed at reproducing the results of coupled finite-volume/Monte Carlo runs with the kinetic Monte Carlo code EIRENE as closely as possible. To this end, we incorporate the rate coefficients from the AMJUEL reaction database, and boundary conditions are based on the reflection data from TRIM. The fluid-neutral model is implemented into the SOLPS-ITER code suite to compare the plasma sources with those of EIRENE on a simplified slab geometry. For a detached case, the results show a nearly perfect match between the fluid and kinetic model, with errors locally ranging up to a maximum discrepancy of 10% for particle and electron heat sources and 30% for parallel momentum sources. These very localized errors can be largely attributed to the discretization error, mainly near the ionization peak. ispartof: CONTRIBUTIONS TO PLASMA PHYSICS vol:58 issue:6-8 pages:718-724 ispartof: location:FRANCE, Aix Marseille Univ, Marseille status: published

The ongoing pursuit for higher power densities in electronic devices poses severe demands on cooling solutions. To explore the full potential of liquid-cooled microchannel heat sinks, topology optimization has been proposed as a method to optimize the microchannel layout. This has led to novel tree-like microchannel layouts. However, deficiencies in the porous medium models used thus far often lead to designs with disconnected channels, that cannot be used in practice. In this contribution, a new modeling approach is proposed for microchannel based heat sink topology optimization. A virtual pressure is introduced in the solid phase of the topology. This pressure field together with a zero velocity field in the solid then can be used for the regularization of the binary design variable. As a result, continuous optimization techniques can be used. This leads to improved results compared to optimization based on porous medium models for the solid phase. Indeed, in the latter case small leakage flows through the solid need to be tolerated in the numerical implementation, leading to inaccurate thermal modeling or ill-conditioned matrix systems. The approach presented in this paper circumvents these problems. The absence of penetration in our new model is demonstrated by two benchmarks. Performance of this model for optimization is validated by a hydraulic inverse design problem. ispartof: Proceedings of the 16th International Heat Transfer Conference, IHTC-16 ispartof: 16th International Heat Transfer Conference, IHTC-16 location:Beijing date:10 Aug - 15 Aug 2018 status: published

Divertor design for next step fusion reactors heavily relies on 2D edge plasma modeling with codes as e.g. B2-EIRENE. While these codes are typically used in a design-by-analysis approach, in previous work we have shown that divertor design can alternatively be posed as a mathematical optimization problem, and solved very efficiently using adjoint methods adapted from computational aerodynamics. This approach has been applied successfully to divertor target shape design for more uniform power load. In this paper, the concept is further extended to include all contributions to the target power load, with particular focus on radiation. In a simplified test problem, we show the potential benefits of fully including the radiation load in the design cycle as compared to only assessing this load in a post-processing step.

The SOLPS-ITER package represents a renewed coupling of the up-to-date parallelized EIRENE Monte-Carlo code and the B2.5 edge plasma solver. Each part of the fluid-kinetic code system has been dismantled and analysed. For SOLPS-ITER, full backwards compatibility with previous SOLPS4.3 ITER reference simulations is critical. We present the results of the ongoing comprehensive benchmark activity and discuss how SOLPS-ITER can be tuned to reproduce ITER simulations performed with SOLPS4.3. In particular in SOLPS-ITER a quantitative “metric” to measure the level of code convergence has been implemented, with corresponding code diagnostics carried over from SOLPS4.3, allowing code run times to be accurately compared. Good agreement is found between the two codes in the tests so far. Remaining discrepancies in the solution in front of the divertor targets are likely related to different formulations of the equations and the different discretization schemes used by the codes.

Shape optimization methods, as commonly applied in aerodynamic design applications, have recently been adapted for use in nuclear fusion divertor target design. The resulting algorithms are very efficient compared to the standard use of numerical edge plasma simulations as analysis tools only. In this paper, we highlight some numerical aspects of the underlying algorithm, focusing on a correct, nine-point discretization of the fluxes and the need for an adjoint pressure correction equation. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

Particle and power exhaust is a key performance issue for next step fusion reactors. Divertors need to be designed to handle power exhaust and to provide sufficient Helium pumping capacity. Plasma edge transport codes such as B2-Eirene [1,2] are indispensable for the design of future divertor concepts, as they allow for the extrapolation of the current understanding of the edge plasma physics to fusion reactor relevant operating regimes. Due to the long simulation times, it is impossible to consider many different geometries and operating points during the design process. Thus, different design requirements and constraints, combined with the large number of control variables turn divertor design into a challenging problem. Over the past decades similar design challenges are faced both in aerodynamics and in structural mechanics. In aerodynamics, shape optimization was introduced to design airfoils for drag reduction or lift maximization [3]. In structural mechanics topology optimization was used to design sufficiently strong constructions at minimal weight and material costs [4]. The method was further developed in fluid mechanics applications to achieve optimal flow configurations [5]. All problems have been treated very effectively by using optimization approaches including adjoint PDE formalisms for computing design sensitivities. Recently, adjoint based automated design methods were explored for fusion reactor and heat sink design. These methods start from a cost functional, which is a measure for the performance of a certain design, and identify optimal solutions by minimizing this cost functional. The minimum is hereby achieved through gradient-based optimization algorithms, where adjoint sensitivity analysis is used to keep gradient evaluation costs sufficiently low. With a detailed parameterization of the divertor shape, and the use of several divertor coils, the number of design variables can easily exceed hundreds or even thousands. For topology optimization of heat sinks every grid cell contains the porosity as a control parameter to assess whether a cooling channel or a fin structure is desired. In previous work [6,7], it is demonstrated that shape optimization methods applied to divertor shape design can efficiently propose divertors which optimize target power load spreading. Typically, V-shaped targets are found. Using adjoint methods for sensitivity analysis, simulation times needed for the optimization cycle are of the order of only a few simulations of the corresponding analysis problem. Also radiation heat loads, computed by a Monte Carlo method, could be incorporated within this methodology [8]. A similar methodology is developed for magnetic configuration design. However, adjoint sensitivities of the plasma edge grid generator are difficult to obtain. Therefore in parts adjoint techniques were developed [9,10]. Again, the optimization methodology successfully alters the magnetic configuration to reduced heat peaks. Further, the perspectives of topology optimization for cooling designs was investigated and led to branch-like cooling channel patterns for micro-channel heat sinks [11]. In the present paper, the status, perspectives and challenges for optimization tools both in divertor configurations and cooling designs will be reviewed. [1] Reiter, D. et al. (2005). Fus.Sci.Techn. 47, 172–186. [2] Kukushkin, A.S., Pacher, H.D., Kotov, V., Pacher, G.W., Reiter, D. (2011). Finalizing the ITER divertor design: The key role of SOLPS modelling. Fusion Engineering and Design, 86, 2865-2873. [3] Jameson, A., Martinelli, L., Pierce, N. A. (1998). Optimum aerodynamic design using the Navier-Stokes equations. Theoretical and Computational Fluid Dynamics, 10, 213-237. [4] Bendsøe, M.P. and Kikuchi, N, (1988). Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering, 71:197–224. [5] Borrvall, T. and Petersson, J. (2003). Topology optimization of fluids in Stokes flow. International journal for numerical methods in fluids, 41:77–107, 2003. [6] Dekeyser, W., Reiter, D., Baelmans, M. (2014). Automated divertor target design by adjoint shape sensitivity analysis and a one-shot method. Journal of Computational Physics, 278, 117-132. [7] Dekeyser, W., Reiter, D., Baelmans, M. (2014). Divertor target shape optimization in realistic edge plasma geometry. Nuclear Fusion, 54, art.nr. 073022. [8] Dekeyser, W., Reiter, D., Baelmans, T. (2015). Designing divertor targets for uniform power load. Journal of Nuclear Materials, 463, 1243-1247. [9] M. Blommaert, M. Baelmans, W. Dekeyser, N. Gauger, and D. Reiter, “A novel approach to magnetic divertor configuration design”, Journal of Nuclear Materials, vol. 463, pp. 1220-1224, Aug. 2015. [10] Blommaert, M., Dekeyser, W., Baelmans, M., Gauger, N., Reiter, D. (2015). An automated approach to magnetic divertor configuration design. Nuclear Fusion, 55, art.nr. 013001. [11] Van Oevelen, T. and Baelmans, M. (2014). Numerical topology optimization of heat sinks”, Proc. of the 15th International Heat Transfer Conference, August 2014, Kyoto (Japan). journal_title: Nuclear Fusion article_type: paper article_title: Achievements and challenges in automated parameter, shape and topology optimization for divertor design copyright_information: © 2017 IAEA, Vienna date_received: 2016-05-01 date_accepted: 2016-10-25 date_epub: 2017-01-20 ispartof: Nuclear Fusion vol:57 issue:3 ispartof: location:Vienna, Austria status: published

We derive fluid neutral approximations for a simplified 1D edge plasma model, suitable to study the neutral behavior close to the target of a nuclear fusion divertor, and compare its solutions to the solution of the corresponding kinetic Boltzmann equation. The plasma is considered as a fixed background extracted from a detached 2D simulation. We show that the Maxwellian equilibrium distribution is already obtained very close to the target, justifying the use of a fluid approximation. We compare three fluid neutral models: (i) a diffusion model; (ii) a pressure-diffusion model (i.e., a combination of a continuity and momentum equation) assuming equal neutral and ion temperatures; and (iii) the pressure-diffusion model coupled to a neutral energy equation taking into account temperature differences between neutrals and ions. Partial reflection of neutrals reaching the boundaries is included in both the kinetic and fluid models. We propose two methods to obtain an incident neutral flux boundary condition fo...

Due to the conflicting requirements, complex physical processes and large number of design variables, divertor design for next step fusion reactors is a challenging problem, often relying on large numbers of computationally expensive numerical simulations. In this paper, we attempt to partially automate the design process by solving an appropriate shape optimization problem. Design requirements are incorporated in a cost functional which measures the performance of a certain design. By means of changes in the divertor shape, which in turn lead to changes in the plasma state, this cost functional can be minimized. Using advanced adjoint methods, optimal solutions are computed very efficiently. The approach is illustrated by designing divertor targets for optimal power load spreading, using a simplified edge plasma model (© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Far SOL plasma flow, and hence main chamber recycling and plasma surface interaction, are today still only very poorly described by current 2D fluid edge codes, such as B2, UEDGE or EDGE2D, due to a common technical limitation. We have extended the B2 plasma fluid solver in the current ITER version of B2-EIRENE (SOLPS4.3) to allow plasma solutions to be obtained up to the “real vessel wall”, at least on the basis of ad hoc far SOL transport models. We apply here the kinetic Monte Carlo Code EIRENE on such plasma solutions to study effects of this model refinement on main chamber fluxes and sputtering, for an ITER configuration. We show that main chamber sputtering may be significantly modified both due to thermalization of CX neutrals in the far SOL and poloidally highly asymmetric plasma wall contact, as compared to hitherto applied teleportation of particle fluxes across this domain. ispartof: Journal of Nuclear Materials vol:415 issue:1 pages:S584-S588 ispartof: location:San Diego status: published

Divertor exhaust system design and analysis tools are crucial to evolve from experimental fusion reactors towards commercial power plants. In addition to material research and dedicated vessel geometry design, improved magnetic configurations can contribute to sustaining the diverted heat loads. Yet, computational design of the magnetic divertor is a challenging process involving a magnetic equilibrium solver, a plasma edge grid generator and a computationally demanding plasma edge simulation. In this paper, an integrated approach to efficient sensitivity calculations is discussed and applied to a set of slightly reduced divertor models. Sensitivities of target heat load performance to the shaping coil currents are directly evaluated. Using adjoint methods, the cost for a sensitivity evaluation is reduced to about two times the simulation cost of one specific configuration. Further, the use of these sensitivities in an optimal design framework is illustrated by a case with realistic Joint European Torus (JET) configurational parameters.

Even on the most powerful supercomputers available today, computational nuclear fusion reactor divertor design is extremely CPU demanding, not least due to the large number of design variables and the hybrid micro-macro character of the flows. Therefore, automated design methods based on optimization can greatly assist current reactor design studies. Over the past decades, “adjoint methods” for shape optimization have proven their virtue in the field of aerodynamics. Applications include drag reduction for wing and wing-body configurations. Here we demonstrate that also for divertor design, these optimization methods have a large potential. Specifically, we apply the continuous adjoint method to the optimization of the divertor geometry in a 2D poloidal cross section of an axisymmetric tokamak device (as, e.g., JET and ITER), using a simplified model for the plasma edge. The design objective is to spread the target material heat load as much as possible by controlling the shape of the divertor, while maintaining the full helium ash removal capabilities of the vacuum pumping system. ispartof: pages:1-8 ispartof: EPJ Web of Conferences vol:33 pages:1-8 ispartof: European Energy Conference location:Maastricht date:17 Apr - 20 Apr 2012 status: published

Tokamak divertor design for next-step fusion reactors heavily relies on numerical simulations of the plasma edge. Currently, the design process is mainly done in a forward approach, where the designer is strongly guided by his experience and physical intuition in proposing divertor shapes, which are then thoroughly assessed by numerical computations. On the other hand, automated design methods based on optimization have proven very successful in the related field of aerodynamic design. By recasting design objectives and constraints into the framework of a mathematical optimization problem, efficient forward-adjoint based algorithms can be used to automatically compute the divertor shape which performs the best with respect to the selected edge plasma model and design criteria. In the past years, we have extended these methods to automated divertor target shape design, using somewhat simplified edge plasma models and geometries. In this paper, we build on and extend previous work to apply these shape optimization methods for the first time in more realistic, single null edge plasma and divertor geometry, as commonly used in current divertor design studies. In a case study with JET-like parameters, we show that the so-called one-shot method is very effective is solving divertor target design problems. Furthermore, by detailed shape sensitivity analysis we demonstrate that the development of the method already at the present state provides physically plausible trends, allowing to achieve a divertor design with an almost perfectly uniform power load for our particular choice of edge plasma model and design criteria.

Owing to the complex physics of the plasma edge, numerical simulation tools are indispensable for the evaluation of novel divertor concepts. Simulation-based divertor design is computationally extremely expensive, however, not in the least due to the large number of design variables. In this work, we show how shape optimisation methods can prove very valuable in partially automating the design process. We apply the continuous adjoint method to design divertor targets for maximum power load spreading. Shape sensitivities are derived using the material derivative approach. The resulting sensitivities depend on boundary data only, and can be computed very efficiently. Through numerical tests, we also prove that their accuracy is good. Next, the performance of two gradient-based optimisation algorithms and a so-called one-shot method is compared. Using the latter method, optimal solutions are obtained in a computational effort less than five times the time needed for a single analysis simulation.

Quantifying main chamber wall recycling, erosion and resulting material migration, at least on the basis of known or empirical far scrape-off layer (SOL) processes, is still highly uncertain, despite its relevance for ITER and fusion reactor design studies. This affects, for example, the design problem of first mirror performance of many optical diagnostics in the harsh ITER environment. Poor computational access is not least due to a fundamental technical limitation in apparently all current tokamak edge plasma fluid codes, which implicates a wide computationally unresolved gap between the outermost plasma layer treated in codes and the real vessel wall. We show how the current ITER version of the B2-EIRENE code (SOLPS-4.3) can be extended to cover also this far SOL, on the same footing as the rest of the plasma transport model. We discuss consequences of this new model for estimating plasma power and particle sink terms caused by a fairly realistic wall in ITER based on the conventional Bohm criterion along all plasma–wall interfaces. Corrections were made to this article on 14 July 2011. The authors have been assigned to the correct affiliations.

The plasma and neutral transport in the plasma edge of a nuclear fusion reactor is usually simulated using coupled finite volume (FV)/Monte Carlo (MC) codes. However, under conditions of future reactors like ITER and DEMO, convergence issues become apparent. This paper examines the convergence behaviour and the numerical error contributions with a simplified FV/MC model for three coupling techniques: Correlated Sampling, Random Noise and Robbins Monro. Also, practical procedures to estimate the errors in complex codes are proposed. Moreover, first results with more complex models show that an order of magnitude speedup can be achieved without any loss in accuracy by making use of averaging in the Random Noise coupling technique.

As magnetic confinement fusion progresses towards the development of first reactor-scale devices, computational tokamak divertor design is a topic of high priority. Presently, edge plasma codes are used in a forward approach, where magnetic field and divertor geometry are manually adjusted to meet design requirements. Due to the complex edge plasma flows and large number of design variables, this method is computationally very demanding. On the other hand, efficient optimization-based design strategies have been developed in computational aerodynamics and fluid mechanics. Such an optimization approach to divertor target shape design is elaborated in the present paper. A general formulation of the design problems is given, and conditions characterizing the optimal designs are formulated. Using a continuous adjoint framework, design sensitivities can be computed at a cost of only two edge plasma simulations, independent of the number of design variables. Furthermore, by using a one-shot method the entire optimization problem can be solved at an equivalent cost of only a few forward simulations. The methodology is applied to target shape design for uniform power load, in simplified edge plasma geometry.

The iterative Monte Carlo (MC) code EMC3-EIRENE is frequently used for plasma edge simulations in 3D applications. So far, a quantitative evaluation of the numerical quality of the code results remains an open issue. In this paper, we demonstrate a framework for the practical assessment of accuracy and convergence with EMC3-EIRENE. Moreover, we provide a first accuracy analysis with EMC3-EIRENE for a DIII-D divertor edge plasma case. First, we introduce post-processing averaging to efficiently reduce the variance of the statistical error. Then, we estimate the deterministic error contributions based on their theoretical reduction rates by comparing solutions with a different grid resolution, time step, or number of MC particles per iteration. Finally, using parameterized expressions for the error and the computational time, suitable numerical parameters are determined to achieve faster and/or more accurate results. We found that simulations can be more than twice as fast without losing accuracy by making use of post-processing averaging and choosing optimal parameters. In addition, we conclude that the discretization error is the dominant error contribution for the case selected in this paper, which demonstrates the importance of constructing an adequate mesh.

Automated methods based on optimization can greatly assist computational engineering design in many areas. In this paper an optimization approach to the magnetic design of a nuclear fusion reactor divertor is proposed and applied to a tokamak edge magnetic configuration in a first feasibility study. The approach is based on reduced models for magnetic field and plasma edge, which are integrated with a grid generator into one sensitivity code. The design objective chosen here for demonstrative purposes is to spread the divertor target heat load as much as possible over the entire target area. Constraints on the separatrix position are introduced to eliminate physically irrelevant magnetic field configurations during the optimization cycle. A gradient projection method is used to ensure stable cost function evaluations during optimization. The concept is applied to a configuration with typical Joint European Torus(JET) parameters and it automatically provides plausible configurations with reduced heat load. online at: http://stacks.iop.org/0029-5515/55/013001 ispartof: Nuclear Fusion vol:55 issue:1 status: published