Equilibrium Solver Research Articles

Prizes on Crowdsourcing Platforms: An Equilibrium Analysis of Competing Contests

Crowdsourcing platforms typically take a passive approach and let solvers self-select which (if any) of the concurrently running contests they wish to participate into. Thus, firms which set prizes and organize contests on these platforms are competing among themselves (for solver participation and effort). We model this competition among contests, and solve for the equilibrium solver participation and characterize the equilibrium among the firms. Next we compare the outcomes to a setting where the platform can “nudge” the solvers toward a particular contest (i.e., platforms can recommend a contest for a given solver, but cannot enforce participation). Our main results reveal that solvers' self-sorting into contests hurts the solvers, the firms and the platform. Thus, platform policies to nudge solver entry improve overall welfare.

An inverse equilibrium tool to define axisymmetric plasma equilibria

In this paper, a numerical tool, called the inverse equilibrium tool (IET), for the solution of the inverse equilibrium engineering problem is presented. IET is developed in a MATLAB environment and allows for the computation of the coil currents needed to obtain a predetermined plasma shape with well defined plasma global parameters (i.e. total plasma current and total poloidal magnetic flux at the boundary) by solving a constrained minimization problem. IET can be used for the characterization of an existing plasma boundary or for the full design of a new one. Thus, it can be used to generate families of equilibrium configurations and to determine if the desired plasma shape is within the engineering capabilities of the device. The typical features of IET can be summarized as follows: (a) the plasma shape can be arbitrarily defined or characterized by means of a compact analytical functional form; (b) the fixed boundary plasma equilibrium solver allows the user to arbitrarily define the plasma current density profile for both tokamak and reversed field pinch magnetic configurations; (c) the minimization problem for the computation of the equilibrium coil currents is solved with both single and multi-objective optimization approaches. IET was validated considering both limiter and lower single null ITER-like plasma configurations and an upper single null experimental plasma in the RFX-mod tokamak. For all the cases under analysis, IET leads to accurate results for determining the currents that are able to reproduce the direct equilibrium reconstruction. The accuracy of the solution and the simplicity of both user interface and computational algorithm are the key features of the IET computational tool.

Equilibrium Chemistry in $${\text {BCl}}_3$$ BCl 3 – $${\text {H}}_2$$ H 2 –Ar Plasma

The approach, which was developed earlier for modeling chemical reactions in laser induced plasmas, is applied to radio-frequency discharge plasmas. The model is based on the assumption that all ionization processes and chemical reactions are at local thermodynamic equilibrium. A chemical composition of an argon-hydrogen plasma with an addition of boron trichloride is studied as a function of plasma temperature and mole ratio $${\text {H}}_2/{\text {BCl}}_3$$ . It is established that more than twenty simple and composite molecules and ions can be formed in the course of chemical reactions. The results are compared with those obtained earlier by means of another equilibrium model that uses ab-initio quantum chemical computations of thermochemical and kinetic data and a 0D thermochemical equilibrium solver.

Optimal MSE polarisation angle and q-profile estimation using Kalman filters and the plasma simulator RAPTOR

Accurate q-profile reconstruction is of importance for the development of advanced scenarios, but continues to be a challenge in tokamak research. To constrain the q-profile in the plasma centre the motional Stark effect diagnostic (MSE) is often used, however achieving routine measurements with the required accuracy proves to be difficult in many devices. We present a novel approach to obtain accurate estimates of the q-profile using an observer based approach. The observer combines MSE measurements with a model-based prediction of the system. For this the plasma transport simulator RAPTOR is coupled with a fixed-boundary equilibrium solver to create a model-based prediction of the MSE measurements. An extended Kalman filter is used to merge profile evolution predictions from the RAPTOR code with measurements. Using synthetic data we demonstrate accurate q-profile estimations in situations where the model is purposely disturbed and only erroneous MSE measurements are available. For shots at ASDEX Upgrade we show that by constraining RAPTOR with MSE measurements, the evolution of the model’s q-profile is in close proximity to reference profiles of reconstructed equilibria from an integrated diagnostic suite.

Plasma shape optimization for EAST tokamak using orthogonal method**Project supported by the National Natural Science Foundation of China (Grant No. 51677051) and the Institute of Plasma Physics, Chinese Academy of Sciences.

It is necessary to reduce the currents of poloidal field (PF) coils as small as possible, during the static equilibrium design procedure of Experimental Advanced Superconductive Tokamak (EAST). The quasi-snowflake (QSF) divertor configuration is studied in this paper. Starting from a standard QSF plasma equilibrium, a new QSF equilibrium with 300 kA total plasma current is designed. In order to reduce the currents of PF6 and PF14, the influence of plasma shape on PF coil current distribution is analyzed. A fixed boundary equilibrium solver based on a non-rigid plasma model is used to calculate the flux distribution and PF coil current distribution. Then the plasma shape parameters are studied by the orthogonal method. According to the result, the plasma shape is redefined, and the calculated equilibrium shows that the currents of PF6 and PF14 are reduced by 3.592 kA and 2.773 kA, respectively.

Linear analyses of peeling-ballooning modes in high beta pedestal plasmas

We present the linear simulations of edge plasma instabilities using the 3-field peeling-ballooning model and gyro-Landau-fluid model under the BOUT++ framework. A series of realistic equilibria of shifted circular geometry are generated by a global equilibrium solver CORSICA, where the Shafranov shift, elongation effects, and bootstrap current are included. The linear growth rate spectrum of the peeling-ballooning modes is shown in a wide range of pressure gradient and parallel current density in the pedestal region. The results show that the bootstrap current stabilizes high beta ballooning modes. The simulations with different fractions of bootstrap current indicate a trend for the existence of the high beta peeling-ballooning mode stability region. Taking the kinetic effects into account, the linear simulations of kinetic peeling-ballooning mode using the gyro-Landau-fluid model show that this region can be accessed.

A lattice Boltzmann–cellular automaton study on dendrite growth with melt convection in solidification of ternary alloys**Project supported by the National Natural Science Foundation of China (Grant Nos. 51728601 and 51771118).

A lattice Boltzmann (LB)–cellular automaton (CA) model is employed to study the dendrite growth of Al-4.0 wt%Cu–1.0 wt%Mg alloy. The effects of melt convection, solute diffusion, interface curvature, and preferred growth orientation are incorporated into the coupled model by coupling the LB–CA model and the CALPHAD-based phase equilibrium solver, PanEngine. The dendrite growth with single and multiple initial seeds was numerically studied under the conditions of pure diffusion and melt convection. Effects of initial seed number and melt convection strength were characterized by new-defined solidification and concentration entropies. The numerical result shows that the growth behavior of dendrites, the final microstructure, and the micro-segregation are significantly influenced by melt convection during solidification of the ternary alloys. The proposed solidification and concentration entropies are useful characteristics bridging the solidification behavior and the microstructure evolution of alloys.

Tokamak systems analyses using a fast 1D transport solver: Models and assessments

During the pre-conceptual design stage of nuclear fusion power plants, systems codes provide rapid evaluation of feasible design points and consequences of major design choices. They do this by optimizing simple models of all relevant power plant subsystems within physical laws and engineering limits. Typically these models use 0D scaling relations to describe the interactions between the physics and engineering design parameters. This can lead to oversimplifications due to the high non-linearity of the underlying physics models involved.In this work, we describe a novel approach that integrates a fast 1D transport and equilibrium solver in the PROCESS systems code. Validation of both the standalone version as well as the version integrated into PROCESS against simulations run with the ASTRA transport code show good agreement. The comparison between the previous 0D version of the PROCESS code and the new 1D version show logical differences due to the different models. In the near future, more detailed investigations of the effects of this model on the European demonstration power plant (DEMO) design are planned.

Quasi-axisymmetric magnetic fields: weakly non-axisymmetric case in a vacuum

An asymptotic expansion is performed to obtain quasi-axisymmetric magnetic configurations that are weakly non-axisymmetric. A large space of solutions is identified, which satisfy the condition of quasi-axisymmetry on a single magnetic flux surface, while (non-axisymmetric) globally quasi-axisymmetric solutions are shown to not exist, agreeing with the conclusions of previous theoretical work. The solutions found are shown to be geometrically constrained at low aspect ratio or high toroidal period number. Solutions satisfying the more general condition of omnigeneity (generalized quasi-axisymmetry) are also shown to exist, and it is found that quasi-axisymmetric deformations can be superposed with an omnigenous solution, while preserving the property of omnigeneity, effectively extending the space of ‘good’ configurations. A numerical solution of the first-order quasi-axisymmetry problem is demonstrated and compared with solutions found with a widely used magnetohydrodynamic equilibrium solver, independently verifying that quasi-axisymmetry is satisfied at the appropriate order. It is thereby demonstrated that approximately quasi-axisymmetric solutions can be directly constructed, i.e. without using numerical search algorithms.

3D equilibrium reconstruction with islands**Notice of Copyright. This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of

This paper presents the development of a 3D equilibrium reconstruction tool and the results of the first-ever reconstruction of an island equilibrium. The SIESTA non-nested equilibrium solver has been coupled to the V3FIT 3D equilibrium reconstruction code. Computed from a coupled VMEC and SIESTA model, synthetic signals are matched to measured signals by finding an optimal set of equilibrium parameters. By using the normalized pressure in place of normalized flux, non-equilibrium quantities needed by diagnostic signals can be efficiently mapped to the equilibrium. The effectiveness of this tool is demonstrated by reconstructing an island equilibrium of a DIII-D inner wall limited L-mode case with an n = 1 error field applied. Flat spots in Thomson and ECE temperature diagnostics show the reconstructed islands have the correct size and phase.

Scramjet performance for ideal combustion processes

Models used to analyse scramjet engine cycles are typically either very simple or focused on specific combustor geometries. This paper uses a quasi-one-dimensional, inviscid chemical equilibrium solver to examine scramjet engines defined by predetermined combustion processes. This solver is capable of rapidly analysing combustion processes without requiring predefined geometries and is validated against the Hyshot II flight experiment and a constant area combustor ground test. Combustion occurs using constant area, constant pressure, constant Mach number or constant temperature processes. Constant area combustion typically produces the highest specific impulse for given combustor entrance conditions. Maximum pressure and temperature limitations were introduced, and multiple engines with combinations of combustion process were examined. It was found that engines with a combination of all four combustion processes can have lower maximum values of pressure and temperature whilst maintaining high performance compared with constant area combustor engines.

Investigation of real gas effects on combustion and emissions in internal combustion engines and implications for development of chemical kinetics mechanisms

Real gas effects on combustion and emissions in internal combustion engines are investigated using three-dimensional computational fluid dynamics. The Peng–Robinson equation of state is implemented to describe pressure–volume–temperature relationships and to calculate thermodynamic properties and relevant partial derivatives. Four facilities are modeled, including non-reacting compression in a motoring engine, combustion in a conventional diesel combustion engine and in a reactivity controlled compression ignition engine, as well as for a non-reacting reflected wave in a shock tube. It is found that the real gas effects of gas mixtures in practical internal combustion engine operation are sensitive to the operating load and the amount of premixed fuel. Excellent agreement against experiments was found for engine simulations with the Peng–Robinson equation of state in terms of cylinder pressure and apparent heat release rate. However, discrepancies with predictions from the ideal gas law grow with increased load and larger amounts of premixed fuel. In particular, the predicted emissions of soot, NOx, CO and unburnt hydrocarbons show increasing sensitivity to real gas effects as a result of changes in combustion phasing. Fuel condensation is also modeled using a vapor–liquid phase equilibrium solver and significant dependency on the equation of state used is found. Therefore, it is recommended to include real gas effects in internal combustion engine modeling to capture combustion and emissions characteristics accurately. Additionally, the results emphasize the role of real gas effects on reaction rates. Shock tube simulations are used to demonstrate the importance of using the real gas equation of state in the interpretation of chemical kinetic measurements. Significantly different compressed gas temperatures behind the reflected shock are predicted when real gas effects are considered. This needs to be realized when developing chemical kinetic models and rate constants for engine applications from shock tube data.

Study of impurity effects on CFETR steady-state scenario by self-consistent integrated modeling

Impurity effects on fusion performance of China fusion engineering test reactor (CFETR) due to extrinsic seeding are investigated. An integrated 1.5D modeling workflow evolves plasma equilibrium and all transport channels to steady state. The one modeling framework for integrated tasks framework is used to couple the transport solver, MHD equilibrium solver, and source and sink calculations. A self-consistent impurity profile constructed using a steady-state background plasma, which satisfies quasi-neutrality and true steady state, is presented for the first time. Studies are performed based on an optimized fully non-inductive scenario with varying concentrations of Argon (Ar) seeding. It is found that fusion performance improves before dropping off with increasing , while the confinement remains at high level. Further analysis of transport for these plasmas shows that low-k ion temperature gradient modes dominate the turbulence. The decrease in linear growth rate and resultant fluxes of all channels with increasing can be traced to impurity profile change by transport. The improvement in confinement levels off at higher . Over the regime of study there is a competition between the suppressed transport and increasing radiation that leads to a peak in the fusion performance at (~2.78 for CFETR). Extrinsic impurity seeding to control divertor heat load will need to be optimized around this value for best fusion performance.

High efficiency dual-fuel combustion through thermochemical recovery and diesel reforming

A computational system optimization was conducted to explore the potential benefits of diesel reforming in dual-fuel combustion strategies. A comprehensive CFD model, validated against syngas (50/50 H2/CO by mole) metal engine experiments, was used to simulate the engine combustion process. The engine CFD solver was coupled with an equilibrium solver for the reformer process and three different reforming processes were investigated: Partial oxidation, steam reforming, and autothermal reforming. A system level approach was used to evaluate the impact of thermochemical recovery of exhaust energy and reformer losses. A design of experiments of simulations was conducted to explore the combustion system design space and a genetic algorithm was used to search the resulting response surface and find the optimal operating conditions. It was found that fuel reforming can increase engine net indicated efficiencies by as much as 9% due to a shorter combustion duration and reduction in heat transfer losses. The partial oxidation reforming system resulted in the lowest system efficiencies at 44% due to its exothermic nature, while steam reforming and autothermal reforming were able to achieve over 48% system efficiency, an improvement in global efficiency of 8% compared to a diesel baseline due to exhaust heat recovery.

A simplified analytic form for generation of axisymmetric plasma boundaries

An improved method has been formulated for generating analytic boundary shapes as input for axisymmetric MHD equilibria. This method uses the family of superellipses as the basis function, as previously introduced. The improvements are a simplified notation, reduction of the number of simultaneous nonlinear equations to be solved, and the realization that not all combinations of input parameters admit a solution to the nonlinear constraint equations. The method tests for the existence of a self-consistent solution and, when no solution exists, it uses a deterministic method to find a nearby solution. Examples of generation of boundaries, including tests with an equilibrium solver, are given.

Investigation of fast ion pressure effects in ASDEX Upgrade by spectral MSE measurements

High precision measurements of fast ion effects on the magnetic equilibrium in the ASDEX Upgrade tokamak have been conducted in a high-power (10 MW) neutral-beam injection discharge. An improved analysis of the spectral motional Stark effect data based on forward-modeling, including the Zeeman effect, fine-structure and non-statistical sub-level distribution, revealed changes in the order of 1% in . The results were found to be consistent with results from the equilibrium solver CLISTE. The measurements allowed us to derive the fast ion pressure fraction to be and variations of the fast ion pressure are consistent with calculations of the transport code TRANSP. The results advance the understanding of fast ion confinement and magneto-hydrodynamic stability in the presence of fast ions.

Pycalphad: CALPHAD-based Computational Thermodynamics in Python

The pycalphad software package is a free and open-source Python library for designing thermodynamic models, calculating phase diagrams and investigating phase equilibria using the CALPHAD method. It provides routines for reading thermodynamic databases and solving the multi-component, multi-phase Gibbs energy minimization problem. The pycalphad software project advances the state of thermodynamic modeling by providing a flexible yet powerful interface for manipulating CALPHAD data and models. The key feature of the software is that the thermodynamic models of individual phases and their associated databases can be programmatically manipulated and overridden at run-time without modifying any internal solver or calculation code. Because the models are internally decoupled from the equilibrium solver and the models themselves are represented symbolically, pycalphad is an ideal tool for CALPHAD database development and model prototyping.

Use of reconstructed 3D VMEC equilibria to match effects of toroidally rotating discharges in DIII-D

A technique for tokamak equilibrium reconstructions is used for multiple DIII-D discharges, including L-mode and H-mode cases when weakly 3D fields are applied. The technique couples diagnostics to the non-linear, ideal MHD equilibrium solver VMEC, using the V3FIT code, to find the most likely 3D equilibrium based on a suite of measurements. It is demonstrated that V3FIT can be used to find non-linear 3D equilibria that are consistent with experimental measurements of the plasma response to very weak 3D perturbations, as well as with 2D profile measurements. Observations at DIII-D show that plasma rotation larger than 20 krad s−1 changes the relative phase between the applied 3D fields and the measured plasma response. Discharges with low averaged rotation (10 krad s−1) and peaked rotation profiles (40 krad s−1) are reconstructed. Similarities and differences to forward modeled VMEC equilibria, which do not include rotational effects, are shown. Toroidal phase shifts of up to are found between the measured and forward modeled plasma responses at the highest values of rotation. The plasma response phases of reconstructed equilibra on the other hand match the measured ones. This is the first time V3FIT has been used to reconstruct weakly 3D tokamak equilibria.

Effect of resonant magnetic perturbations on microturbulence in DIII-D pedestal

Vacuum resonant magnetic perturbations (RMP) applied to otherwise axisymmetric tokamak plasmas produce in general a combination of non-resonant effects that preserve closed flux surfaces (kink response) and resonant effects that introduce magnetic islands and/or stochasticity (tearing response). The effect of the plasma kink response on the linear stability and nonlinear transport of edge turbulence is studied using the gyrokinetic toroidal code GTC for a DIII-D plasma with applied n = 2 vacuum RMP. GTC simulations use the 3D equilibrium of DIII-D discharge 158103 (Nazikian et al 2015 Phys. Rev. Lett. 114 105002), which is provided by nonlinear ideal MHD VMEC equilibrium solver in order to include the effect of the plasma kink response to the external field but to exclude island formation at rational surfaces. Analysis using the GTC simulation results reveal no increase of growth rates for the electrostatic drift wave instability and for the electromagnetic kinetic-ballooning mode in the presence of the plasma kink response to the RMP. Furthermore, nonlinear electrostatic simulations show that the effect of the 3D equilibrium on zonal flow damping is very weak and found to be insufficient to modify turbulent transport in the electrostatic turbulence.

Global kinetic ballooning mode simulations in BOUT++

We report on simulation results of a 3+1 gyro-Landau-fluid (GLF) model in BOUT++ framework, which contributes to increasing the physics understanding of the edge turbulence. We find that there is no second stability region of kinetic ballooning modes (KBM) in the concentric circular geometry. The first unstable β of KBM decreases below the ideal ballooning mode threshold with increasing . In order to study the KBM in the real tokamak equilibrium, we find that the approximation of shifted circular geometry () is not valid for a high β global equilibrium near the second stability region of KBM. Thus we generate a series of real equilibria from a global equilibrium solver CORSICA, including both Shafranov shift and elongation effects, but not including bootstrap current. In these real equilibria, the second stability region of KBM are observed in our global linear simulations. The most unstable mode for different β are the same while the mode number spectrum near the second stability region is wider than the case near the first stability region. The nonlinear simulations show that the energy loss of an ELM keeps increasing with β, because the linear drive of the turbulence remains strong for the case near the second stability region during profile evolution.