A 5D gyrokinetic full-f global semi-lagrangian code for flux-driven ion turbulence simulations

International audienceThis paper addresses non-linear gyrokinetic simulations of ion temperature gradient (ITG) turbulence in tokamak plasmas. The electrostatic Gysela code is one of the few international 5D gyrokinetic codes able to perform global, full-f and flux-driven simulations. Its has also the numerical originality of being based on a semi-Lagrangian (SL) method. This reference paper for the Gysela code presents a complete description of its multi-ion species version including: (i) numerical scheme, (ii) high level of parallelism up to 500k cores and (iii) conservation law properties

Vérification de Codes et Réduction de Modèles : Application au Transport dans les Plasmas Turbulents

Numerical analysis is now a key component of research, especially for the understanding and the control of complex systems. Simulations of magnetic confinement plasmas fall within this approach. One of the difficulties of this field is the wide range of scales. Spatial scales stretch from the millimeter to the meter, time scales stretch from the microsecond to the second.The chaotic nature of plasmas and the strong anisotropies require advanced numerical methods. Each of the two parts of my thesis take place in the frame of numerical simulation and fusion plasmas. I have set up the PoPe method, which is a general procedure for code verification. I have applied this method to two simulation tools: a fluid turbulence code and a kinetic turbulence code. Using these tools, I have studied the turbulent transport which determines performances of fusion plasmas.The principle of the PoPe method is to determine the equations which have generated a set of data. If the data was produced by a simulation tool, finding these equations and comparing them to the ones theoretically implemented is equivalent to verifying this simulation tool. The accuracy of this procedure allows to characterize the numerical error and to recover the order of each numerical scheme used. The first originality of this work is its area of applicability: no restrictive frame is imposed, simulations can be performed in any number of dimensions, in nonlinear regimes, and even in chaotic regimes. The second strength of this method is its low overall cost: the analytical work is elementary, the computation time overhead is marginal and the programming is minimal. Moreover, it does not interfere with the evolution of the simulation. This allows to use PoPe on all production simulations. Therefore, this is a powerful tool for code development, especially in a research environment where codes evolve continuously to explore new behaviors.The second part of my thesis deals with the study of turbulent transport in a fluid model and in a kinetic model restricted to the low frequency instabilities. In the first model, the transport is essentially in the form of chaotic avalanches: this is a quasi-ballistic transport where structures descend a gradient. The strong sensitivity of those avalanches to the mechanisms triggering and maintaining them make avalanches chaotic phenomena. A study of the link between turbulent transport intensity and the degree of chaos is carried out by changing the viscosity of the system. Turbulent transport appears only at a sufficiently high degree of chaos and its efficiency decreases when the degree of chaos increases above a second threshold. Thus, an optimal degree of chaos maximizing transport exists. The study of the avalanche-like turbulent transport is undertaken using the PoPe method, in order to find a reduced set of equations able to simulate this behavior. The common paradigm of diffusion - convection is analyzed and the conclusions are in agreement with past studies. The PoPe method introduces a systematic framework for these results.The second model of turbulent transport is characterized by two regimes of transport. The first highly efficient regime is based on electric potential structures elongated in the radial direction, while the second regime is based on structures elongated in the direction perpendicular to the radius, which fully insulate the system.Avalanches had been observed in previous work simulating the same model. This is shown to be the result of numerical errors rather than physical processes. Several modifications of the model are undertaken in order to recover avalanches.L'étude numérique est un outil de recherche qui est devenu incontournable, en particulier pour la compréhension et le contrôle des systèmes complexes. La simulation des plasmas de fusion par confinement magnétique s'inscrit parfaitement dans cette démarche. Une des difficultés de cette tâche est le rapport d'échelle, que ce soit les échelles d'espace du millimètre au mètre, ou les échelles de temps de la micro seconde à la seconde. La nature chaotique des plasmas et les très fortes anisotropies imposent l'utilisation de méthodes numériques avancées. C'est dans ce cadre que les deux volets de ma thèse s'inscrivent. J'ai mis en place la méthode PoPe, une procédure générale de vérification de codes que j'ai appliquée à deux outils de simulations : un code de turbulence fluide et un code de turbulence en régime cinétique. Avec ces outils j'ai réalisé l'étude du transport turbulent qui détermine la performance des plasmas de fusion.Le principe de la méthode PoPe est de déterminer les équations qui ont permis de générer un ensemble de données : si les données sont issues d'un code de simulation, retrouver ces équations et les comparer au modèle théoriquement implémenté est équivalent à vérifier le code. La précision de la procédure permet de caractériser l'erreur commise jusqu'à retrouver l'ordre des schémas numériques employés. La première originalité de ce travail est le domaine d'applicabilité de cette procédure : aucun cadre restrictif n'est imposé, les simulations peuvent être en dimension quelconque, en régime pleinement non linéaire, voire chaotique. Le second intérêt est le faible coût de cette méthode : le travail analytique est élémentaire, le surcoût en temps de calcul est marginal, les développements informatiques sont minimaux et ne perturbent pas l'évolution des simulations ce qui permet d'appliquer PoPe à toutes les simulations de production. C'est donc un outil puissant pour le développement de codes de recherche car ils sont en perpétuelle évolution et utilisés pour sonder des comportements nouveaux.Le second volet de ma thèse s'articule autour de l'étude du transport turbulent dans un modèle de bord fluide et un modèle cinétique restreint aux instabilités basse fréquence. Le transport du premier modèle est essentiellement sous forme d'avalanches chaotiques : c'est un transport quasiment balistique où des structures descendent des gradients. La forte sensibilité de ces avalanches par rapport aux conditions qui les déclenchent et qui les maintiennent rend le processus chaotique. Une étude du lien entre l'intensité du transport turbulent et le degré de chaos est menée en modifiant la viscosité du milieu. Le transport turbulent apparait lorsque le degré de chaos dépasse un seuil, puis toute augmentation du degré de chaos après avoir dépassé un second seuil diminue l'efficacité du transport : il existe un degré de chaos maximisant l'intensité du transport. La compréhension du transport turbulent par avalanches est abordée sous l'angle de la réduction de modèle en cherchant, à l'aide de la méthode PoPe, un modèle capable de décrire ce comportement.Le paradigme usuel de la diffusion - convection est ainsi analysé et les conclusions obtenues sont en accord avec des études déjà réalisées, la méthode PoPe y apporte un cadre systématique.Le second modèle de transport turbulent se caractérise par une description du transport comme résultant de deux régimes exclusifs : un régime de fort transport dû à des structures de potentiel électrique allongées dans la direction radiale d'une part, et d'autre part un régime de fort confinement dû à des structures allongées dans la direction perpendiculaire au rayon. Il est montré qu'un comportement de type avalanche précédemment vu dans ce modèle n'est pas d'origine physique mais est déclenché par des imprécisions numériques. Diverses modifications du modèle sont explorées pour retrouver le phénomène d'avalanche

Verification of codes and reduction of models : application to the transport in turbulent plasmas

L'étude numérique est un outil de recherche qui est devenu incontournable, en particulier pour la compréhension et le contrôle des systèmes complexes. La simulation des plasmas de fusion par confinement magnétique s'inscrit parfaitement dans cette démarche. Les larges rapports d'échelle en temps et espace, la nature chaotique des plasmas et les très fortes anisotropies imposent l'utilisation de méthodes numériques avancées. C'est dans ce cadre que les deux volets de ma thèse s'inscrivent.Le premier volet est l’originalité de ma thèse, la mise en place la méthode PoPe, une procédure générale de vérification de codes et réduction de modèles. Le principe de cette méthode est de déterminer les équations qui ont permis de générer un ensemble de données : si les données sont issues d'un code de simulation, retrouver ces équations et les comparer au modèle théoriquement implémenté est équivalent à vérifier le code. La précision de la procédure permet de caractériser l'erreur commise jusqu'à retrouver l'ordre des schémas numériques employés, même en régime chaotique.Le second volet de ma thèse se consacre à l’étude du transport turbulent qui détermine la performance des plasmas de fusion. L’étude du transport sous forme d’avalanches dans un modèle de bord fluide est entreprise en quantifiant l’impact du chaos sur l’auto-organisation. Pour un modèle cinétique restreint aux instabilités basse fréquence, la capacité de se bloquer dans deux régimes exclusifs, l’un isolant, l’autre conducteur, est étudiée. Ce modèle est amélioré pour permettre des relaxations entre ces deux états. Pour ces modèles fluide et cinétique, des modèles réduits obtenus avec la méthode PoPe sont proposés.Numerical analysis is now a key component of research, especially for the understanding and the control of complex systems. Simulations of magnetic confinement plasmas fall within this approach. One of the difficulties of this field is the wide range of spatial scales, time scales, the chaotic nature of plasmas and the strong anisotropies require advanced numerical methods. Each of the two parts of my thesis takes place in this frame of numerical simulation and fusion plasmas.The first part of my thesis is dedicated to the method PoPe, a general method for code verification and model reduction. The principle of this method is to determine the equations which have generated a set of data. If the data was produced by a simulation tool, finding these equations and comparing them to the ones theoretically implemented is equivalent to verifying this simulation tool. The accuracy of this procedure allows to characterize the numerical error and to recover the order of each numerical scheme used.The second part of my thesis deals with the study of turbulent transport which determines the efficiency of fusion plasma. The chaotic avalanches of a fluid model are studied considering the impact of the chaos on the self-organization. For a kinetic model restricted to the low frequency instabilities, the ability to block itself in two regimes, one insulating and the other conducting, is studied. Upgrades of this model are undertaken in order to introduce the possibility of relaxations between the two previous states. For both the fluid and the kinetic model, reduce models are proposed thank to the PoPe method

Verification and accuracy check of simulations with PoPe and iPoPe

International audienceThe theoretical background of the PoPe and iPoPe verification scheme is presented. Verification is performed using the simulation output of production runs. The computing overhead is estimated to be at most 10%. PoPe or iPoPe calculations can be done offline provided the necessary data is stored, for example additional time slices, or online where iPoPe is more effective. The computing overhead is mostly that of storing the necessary data. The numerical error is determined and split into a part proportional to the operators, which are combined to form the equations to be solved, thus modifying their control parameters, completed by a residual error orthogonal to these operators. The accuracy of the numerical solution is determined by this modification of the control parameters. The PoPe and iPoPe methods are illustrated in this paper with simulations of a simple mechanical system with chaotic trajectories evolving into a strange attractor with sensitivity to initial conditions. We show that the accuracy depends on the particular simulation both because the properties of the numerical solution depend on the values of the control parameter, and because the target accuracy will depend on the problem that is addressed. One shows that for a case close to bifurcations between different states, the accuracy is determined by the level of detail of the bifurcation phenomena one aims at describing. A unique verification index, the PoPe index, is proposed to characterise the accuracy, and consequently the verification, of each production run. The PoPe output allows one to step beyond verification and analyse for example the numerical scheme efficiency. For the chosen example at fixed PoPe index, therefore at fixed numerical error, one finds that the higher order integration scheme, comparing order 4 to order 2 Runge-Kutta time stepping, reduces the computation cost by a factor 4

Simulation Verification with PoPe

We present the theoretical background of the PoPe and iPoPe verification scheme. The verification that is performed uses the output of actual simulations of production runs. With a small computing overhead it is possible to check that the problem that is solved numerically is consistent with the equations that are to be addressed. In fact, one shows that the numerical error determined by both procedures can be split into a part proportional to the existing operators of the equations, thus modifying their control parameters, completed by a residual error orthogonal to these operators. The accuracy of the numerical solution can be tested on the error as well as on the modification of the control parameters. To illustrate the method, the evolution equation of a simple mechanical system with two conjugate degrees of freedom is used as simulation test bed. Importantly, although dissipative, the trajectory equations evolve towards a chaotic attractor, a strange attractor, characterised by a positive Lyapunov exponent and therefore sensitivity to initial conditions. It is shown that the chaotic state cannot be verified with the standard Method of Manufactured Solution. We present different facets of the PoPe verification method applied to this test case. We show that the evaluation of the accuracy is case dependent for two reasons. First, the error that is generated depends on the values of the control parameter and not only on the numerical scheme. Second, the target accuracy will depend on the problem one wants to address. In a case characterised by bifurcations between different states, the accuracy is determined by the level of detail of the bifurcation phenomena one wants to achieve. A unique verification index is proposed to characterise the accuracy, and consequently the verification, of any given simulation in the production runs. This PoPe index then gives a level of confidence of each simulation. A PoPe index of zero characterises a situation with 100% error level. One finds that although the accuracy is poor the robust features of the solution can still be recovered. The maximum PoPe index is determined by machine precision, typically in the range of 12 to 14. As an illustration this PoPe index is used to choose between a high order integration scheme and a reduced order integration scheme that is less precise but requires less operations. For the chosen example the PoPe index indicates that the high order scheme leads to a reduction of computer resources up to a factor 4 at given accuracy

Simulation Verification with PoPe

We present the theoretical background of the PoPe and iPoPe verification scheme. The verification that is performed uses the output of actual simulations of production runs. With a small computing overhead it is possible to check that the problem that is solved numerically is consistent with the equations that are to be addressed. In fact, one shows that the numerical error determined by both procedures can be split into a part proportional to the existing operators of the equations, thus modifying their control parameters, completed by a residual error orthogonal to these operators. The accuracy of the numerical solution can be tested on the error as well as on the modification of the control parameters. To illustrate the method, the evolution equation of a simple mechanical system with two conjugate degrees of freedom is used as simulation test bed. Importantly, although dissipative, the trajectory equations evolve towards a chaotic attractor, a strange attractor, characterised by a positive Lyapunov exponent and therefore sensitivity to initial conditions. It is shown that the chaotic state cannot be verified with the standard Method of Manufactured Solution. We present different facets of the PoPe verification method applied to this test case. We show that the evaluation of the accuracy is case dependent for two reasons. First, the error that is generated depends on the values of the control parameter and not only on the numerical scheme. Second, the target accuracy will depend on the problem one wants to address. In a case characterised by bifurcations between different states, the accuracy is determined by the level of detail of the bifurcation phenomena one wants to achieve. A unique verification index is proposed to characterise the accuracy, and consequently the verification, of any given simulation in the production runs. This PoPe index then gives a level of confidence of each simulation. A PoPe index of zero characterises a situation with 100% error level. One finds that although the accuracy is poor the robust features of the solution can still be recovered. The maximum PoPe index is determined by machine precision, typically in the range of 12 to 14. As an illustration this PoPe index is used to choose between a high order integration scheme and a reduced order integration scheme that is less precise but requires less operations. For the chosen example the PoPe index indicates that the high order scheme leads to a reduction of computer resources up to a factor 4 at given accuracy

Data-Driven Risk Matrices for CERN’s Accelerators

A risk matrix is a common tool used in risk assessment, defining risk levels with respect to the severity and probability of the occurrence of an undesired event. Risk levels can then be used for different purposes, e.g. defining subsystem reliability or personnel safety requirements. Over the history of the Large Hadron Collider (LHC), several risk matrices have been defined to guide system design. Initially, these were focused on machine protection systems, more recently these have also been used to prioritize consolidation activities. A new data-driven development of risk matrices for CERN’s accelerators is presented in this paper, based on data collected in the CERN Accelerator Fault Tracker (AFT). The data-driven approach improves the granularity of the assessment, and limits uncertainty in the risk estimation, as it is based on operational experience. In this paper the authors introduce the mathematical framework, based on operational failure data, and present the resulting risk matrix for LHC

Machine learning for early fault detection in accelerator systems

With the development of systems based on a combination of mechanics, electronics and – more and more - software components, increasing system complexity is a de facto trend in the engineering world. Particle accelerators make no exception to this paradigm. The continuous push for higher energies driven by particle physics implies that next generation machines will be at least one order of magnitude larger and more complex than present ones, posing unprecedented challenges in terms of beam performance and availability. The two most promising approaches CERN discusses as next generation projects are the Future Circular Collider (FCC) and the Compact Linear Collider (CLIC), with a size of 100 km and 48 km, respectively (see Fig.1 and Fig. 2)

Evaluating Kernels on Xeon Phi to accelerate Gysela application

This work describes the challenges presented by porting parts of the Gysel