Numerical Heating Research Articles

Stochastic weighted particle control for electrostatic particle-in-cell Monte Carlo collision simulations in an axisymmetric coordinate system

The non-uniform grids in the axisymmetric coordinate system pose a significant challenge for electrostatic particle-in-cell/Monte Carlo collision (PIC/MCC) simulations because they require numerous macroparticles to manage numerical heating around the mid-axis. To address this, we have developed a stochastic weighted particle control method that selectively samples small-weight particles, effectively controlling the particle number without inducing numerical heating. This method is based on a rejection-acceptance probability merging scheme, which is easy to implement and has a low time complexity. We have also made essential modifications, including a corrected density deposition scheme, an energy conservation scheme, and the introduction of target weights. By applying this particle control method, the number of macroparticles in the simulation can be reduced by more than one order of magnitude, significantly reducing the required computing time and storage. Furthermore, appropriately setting target weights also enables enhanced resolution of dilute regions with an acceptable increase in computational cost.

Numerical challenges in modeling gravothermal collapse in Self-Interacting Dark Matter halos

When dark matter has a large cross section for self scattering, halos can undergo a process known as gravothermal core collapse, where the inner core rapidly increases in density and temperature.To date, several methods have been used to implement Self-Interacting Dark Matter (SIDM) in N-body codes, but there has been no systematic study of these different methods or their accuracy in the core-collapse phase. In this paper, we compare three different numerical implementations of SIDM, including the standard methods from the GIZMO and Arepo codes, by simulating idealized dwarf halos undergoing significant dark matter self interactions (σ/m = 50 cm2/g).When simulating these halos, we also vary the massresolution, time-stepping criteria, and gravitational force-softening scheme. The various SIDM methods lead to distinct differences in a halo's evolution during the core-collapse phase, as each results in spurious scattering rate differences and energy gains/losses.The use of adaptive force softening for gravity can lead to numerical heating that artificially accelerates core collapse, while an insufficiently small simulation time step can cause core evolution to stall or completely reverse. Additionally, particle numbers must be large enough to ensure that the simulated halos are not sensitive to noise in the initial conditions. Even for the highest-resolution simulations tested in this study (106 particles per halo), we find that variations of order 10% in collapse time are still present.The results of this work underscore the sensitivity of SIDM modeling on the choice of numerical implementation and motivate a careful study of how these results generalize to halos in a cosmological context.

Kinematic morphology of low-mass galaxies in IllustrisTNG

ABSTRACT The origin of diverse kinematic morphologies observed in low-mass galaxies is unclear. In this study, we investigate the kinematic morphologies of central galaxies with stellar mass $10^{8.5-9.0}{\rm M}_{\odot }$ at $z=0$ in the TNG50-1 cosmological simulation. The majority of the low-mass galaxies in TNG50-1 are dispersion-dominated, consistent with observations. By tracing the evolutionary histories of simulated low-mass galaxies, we find that while most stars form in rotating cold gas discs, the orientation of the star-forming discs relative to the galaxies may evolve with cosmic time. If the cold gas disc remains aligning with the galaxy during its evolution, stars formed at different times share the same rotational direction, leading to a rotation-dominated system. On the contrary, frequent misalignment of cold gas disc would result in a dispersion-dominated system. In addition, we also find that the two-body scattering can have a non-negligible numerical heating effect on the simulated galaxy morphology, especially at central regions of galaxies and for relatively low-mass galaxies. By comparing results of simulations with different resolutions, our results suggest that the simulated morphology of galaxies is roughly reliable when their number of stellar particles exceeds about $10^{4}$, and bulge morphology of galaxies can not be resolved robustly at the resolution level of TNG50-1.

Decentralized PI-control and anti-windup in resource sharing networks

We consider control of multiple stable first-order agents which have a control coupling described by an M-matrix. These agents are subject to incremental sector-bounded input nonlinearities. We show that such plants can be globally asymptotically stabilized to a unique equilibrium using fully decentralized proportional–integral controllers equipped with anti-windup and subject to local tuning rules. In addition, we show that when the nonlinearities correspond to the saturation function, the closed loop asymptotically minimizes a weighted 1-norm of the agents state mismatch. The control strategy is finally compared to other state-of-the-art controllers on a numerical district heating example.

Scaling relations in laser-induced vaporization of thin free-flying liquid metal sheets

We experimentally study the vaporization of free-flying liquid tin sheets when exposed to a 100-ns laser pulse with an intensity between 0.2×107 and 4×107 W/cm2, a case inspired by current developments around “advanced target shaping” in industrial laser-produced plasma sources for extreme ultraviolet (EUV) nanolithography machines. Our findings reveal a gradual vaporization and a linear relationship between the average vaporization rate and laser pulse intensity (with a prefactor of 1.0(3)×10−7ms−1/Wcm−2), for various targets ranging from 20nmto200nm in thickness. We introduce a numerical one-dimensional heating and vaporization model based on Hertz-Knudsen evaporation and find excellent agreement between simulations and experimental data. We furthermore demonstrate that the amount of vaporization of liquid tin targets in the investigated laser intensity range is governed solely by the deposited fluence, and collapse all data onto a single nondimensional curve, enabling the accurate prediction of vaporization dynamics in applications in future development of EUV sources. Published by the American Physical Society 2024

Research on the optimization of the temperature field distribution of a multi-source microwave agent system based on event-triggered

Aiming at the problem of optimizing temperature distribution and heating efficiency in a multi-source microwave agent system, a collaborative tracking strategy using an event-triggering approach is explored. Firstly, setting the control object as a switch to optimize the system control process, a new cooperative switching strategy of event-triggered is designed to intensify the process of energy distribution by changing the time–frequency distribution of power. Secondly, exploring multi-agent architecture with an event-triggered multi-microwave source system to minimize communication frequency and energy consumption for faster agreement. Then, The material grouping strategy divides all materials into four categories based on their characteristics. Two materials, potato (group (a)) and SiC (group (c)) are selected for experimentation. Finally, a simple and easy-to-implement multi-source microwave numerical heating model was constructed using finite element analysis. The heating law described in this article was tested through experiments. The results showed that using this method increased the utilization rate of microwave energy in SiC by a maximum of 9.67% compared to traditional heating methods. Furthermore, it improved the temperature uniformity while maintaining excellent quality. Additionally, the energy utilization efficiency of potatoes was maximized by 8.76%, and the temperature uniformity was improved mostly by 12.04%.

Direct implicit and explicit energy-conserving particle-in-cell methods for modeling of capacitively coupled plasma devices

Achieving large-scale kinetic modeling is a crucial task for the development and optimization of modern plasma devices. With the trend of decreasing pressure in applications, such as plasma etching, kinetic simulations are necessary to self-consistently capture the particle dynamics. The standard, explicit, electrostatic, momentum-conserving particle-in-cell method suffers from restrictive stability constraints on spatial cell size and temporal time step, requiring resolution of the electron Debye length and electron plasma period, respectively. This results in a very high computational cost, making the technique prohibitive for large volume device modeling. We investigate the direct implicit algorithm and the explicit energy conserving algorithm as alternatives to the standard approach, both of which can reduce computational cost with a minimal (or controllable) impact on results. These algorithms are implemented into the well-tested EDIPIC-2D and LTP-PIC codes, and their performance is evaluated via 2D capacitively coupled plasma discharge simulations. The investigation reveals that both approaches enable the utilization of cell sizes larger than the Debye length, resulting in a reduced runtime, while incurring only minor inaccuracies in plasma parameters. The direct implicit method also allows for time steps larger than the electron plasma period; however, care must be taken to avoid numerical heating or cooling. It is demonstrated that by appropriately adjusting the ratio of cell size to time step, it is possible to mitigate this effect to an acceptable level.

A high-efficiency gas–liquid coupled heat-driven thermoacoustic heat pump

The search for a diversified and sustainable technology for energy utilisation is crucial for reducing the consumption of fossil energy. This research proposed a novel heat-driven thermoacoustic heat pump with a gas–liquid coupled resonator that exhibits high efficiency, compact structure and easy fabrication. A series of theoretical and experimental studies has been performed in this work, including numerical and experimental domestic heating performance, onset characteristics as well as the effects of different liquid masses and high and ambient temperatures on the performance of a heat pump. Results show that the theoretical and experimental heat pump performance exhibit good agreement, with a standard error of about 5%. Moreover, the gas–liquid coupled heat-driven thermoacoustic heat pump has a low onset temperature (70 °C at 3 MPa), offering a wide range of applications for low-grade thermal energy utilisation. With high, ambient and domestic heating temperatures of 300 °C, 10 °C and 50 °C, respectively, a maximum domestic heating power of 18.4 kW is obtained, corresponding to the maximum coefficient of performance of domestic heating based on heating power (COPh) of 1.48. Furthermore, a maximum relative Carnot efficiency of 50.9% and a COPh of 1.31 with a hot-end temperature of 300 °C are achieved under ambient and domestic heating temperatures of −20 °C and 50 °C, respectively. These results present a new domestic heating approach for the utilisation of medium- and low-grade thermal energy.

The fragility of thin discs in galaxies – I. Building tailored N-body galaxy models

ABSTRACT Thin stellar discs on both galactic and nuclear, sub-kpc scales are believed to be fragile structures that would be easily destroyed in major mergers. In turn, this makes the age dating of their stellar populations a useful diagnostics for the assembly history of galaxies. We aim at carefully exploring the fragility of such stellar discs in intermediate- and low-mass encounters, using high-resolution N-body simulations of galaxy models with structural and kinematic properties tailored to actually observed galaxies. As a first but challenging step, we create a dynamical model of FCC 170, a nearly edge-on galaxy in the Fornax cluster with multiple galactic components and including both galactic-scale and nuclear stellar discs (NSDs), using detailed kinematic data from the Multi Unit Spectroscopic Explorer and a novel method for constructing distribution function-based self-consistent galaxy models. We then create N-body realizations of this model and demonstrate that it remains in equilibrium and preserves its properties over many Gyr, when evolved with a sufficiently high particle number. However, the NSD is more prone to numerical heating, which gradually increases its thickness by up to 22 per cent in 10 Gyr even in our highest resolution runs. Nevertheless, these N-body models can serve as realistic representations of actual galaxies in merger simulations.

Heated wind-tunnel experiments and numerical investigations onhypersonic blunt cone aerodynamic heating

Aero-thermal analysis of hypersonic vehicles is of fundamental interest for structural design and material selection. Therefore, we investigated the aerodynamic heating characteristics of a blunt cone made from stainless steel in a combustion-heated hypersonic wind tunnel with a free stream of Mach 6. During ground testing, the temperature rise of the model from room temperature to thermal equilibrium under the influence of hypersonic flow was measured with thermocouples, and the equilibrium temperature of the blunt section was 1130–1150 K. In addition, the transient thermal response of the model to the aerothermodynamic load was simulated by fluid-thermal coupled analysis and compared with the measurement data. The obtained results correspond well to typical thermocouple data, and the temperature predicts the measurement with an accuracy on the order of 8.8%. On this foundation, some factors affecting the radiation equilibrium temperature of the blunt cone are also discussed. The analysis indicates that emissivity has an important influence, while turbulence intensity at the nozzle inlet has minimal impact.

On numerical energy conservation for an implicit particle-in-cell method coupled with a binary Monte-Carlo algorithm for Coulomb collisions

Conventional particle-in-cell (PIC) methods suffer from enhanced numerical heating (explicit PIC) or cooling (semi-implicit PIC) when coupled with a binary Monte-Carlo algorithm for Coulomb collisions. In this work, a fully-implicit θ-PIC scheme (with adjustable time-biasing parameter 1/2≤θ≤1) is considered. The discrete change in energy of a closed system after a time step for this scheme scales with (1/2−θ)Cθ, where Cθ is a positive definite quantity that depends on the frequency spectrum of the energy in the fields. Collisions lead to additional energy in the field fluctuations associated with high-frequency light waves produced by a numerical Bremsstrahlung process, which can result in a large increase in the numerical cooling rate for θ>1/2. However, for θ=1/2, energy is exactly conserved. The energy in the field fluctuations on long time scales agrees with that calculated using the equipartition theorem for a classical system in thermodynamic equilibrium.

An Energy Conserving Vlasov Solver That Tolerates Coarse Velocity Space Resolutions: Simulation of MMS Reconnection Events

AbstractVlasov solvers that operate on a phase‐space grid are highly accurate but also numerically demanding. Coarse velocity space resolutions, which are largely unproblematic in particle‐in‐cell (PIC) simulations, can lead to numerical heating or oscillations in continuum Vlasov methods. To address this issue, we present a new dual Vlasov solver which is based on an established positivity preserving advection scheme for the update of the distribution function and an energy conserving partial differential equation solver for the kinetic update of mean velocity and temperature. The solvers work together via moment fitting during which the maximum entropy part of the distribution function is replaced by the solution from the partial differential equation solver. This numerical scheme makes continuum Vlasov methods competitive with PIC methods concerning computational cost and enables us to model large scale reconnection in Earth's magnetosphere with a fully kinetic continuum method. The simulation results agree well with measurements by the MMS spacecraft.

An energy- and charge-conserving electrostatic implicit particle-in-cell algorithm for simulations of collisional bounded plasmas

The present work investigates the applicability of the implicit energy- and charge-conserving particle-in-cell method to collisional bounded plasmas. The possible modifications might require modeling of the charge dynamics associated with a particle impinging on a bounding surface of different materials and coupling to an external RLC circuit. The former requires no change if the lowest order shape function is used for the current density assignment, whereas the method needs to be altered when the shape function is linear and particle recombination at the surface of a perfect conductor is assumed. The proposed fix consists of adding a virtual oppositely charged particle symmetrically with respect to the conductor boundary. A newly derived numerical form of the power balance equation contains only physical terms and demonstrates that the numerical heating is controlled solely by the accuracy of the energy-conserving iteration procedure. The external circuit's contribution is incorporated into the field iterations through Kirchhoff's laws for a system consisting of the plasma reactor and the external circuit. Due to the implicit nature of the proposed algorithm modification, it does not lead to numerical instabilites, and thus is more robust than explicit analogs. The energy-conserving PIC method can be employed for numerical modeling of a wide range of technological bounded plasmas, including those with high densities, which is demonstrated with 1D simulations of a capacitively coupled plasma discharge in Cartesian geometry. It is found that the algorithm remains stable even if the iterative procedure needed for the consistent update of the particle orbits and the electric field in the framework of an implicit method is stopped after the first iteration and the lowest order shape function is used for the electric field interpolation along with the conduction current assignment. The resulting solutions prove reasonably close to the more precise counterparts obtained with the energy conservation ensured to much better accuracy and with higher order shape functions. Since this makes the computational complexity of the new algorithm comparable to that of the explicit PIC code whereas utilizing much smaller grids is possible, the new algorithm should be preferred over the latter when simulating high-density plasmas.

Numerical simulation of the cathode erosion by ion sputtering in hollow cathode arc discharges

Cathode erosion by ion sputtering is one of the primary erosion mechanisms in hollow cathode discharges. In this paper, the moving mesh approach coupling with a two-dimensional plasma fluid model was employed to simulate the cathode erosion process for 100 h in a hollow cathode arc discharge. In the numerical model, plasma transport, arc flow and heating, and cathode thermal balance were considered. In addition, the electron emission, ion bombardment, and surface radiation were taken into account on the cathode wall. The normal erosion depth on the cathode surface was defined to characterize the degree of cathode erosion. The results showed that the ion sputtering erosion was sensitive to the ion flux toward the cathode wall and the distribution of the plasma potential. The external magnetic field could mitigate the erosion process. The changes of erosion depth and position under different discharge currents, gas flow rates, and intensities of the applied magnetic field were also investigated in this paper.

Application of sparse grid combination techniques to low temperature plasmas Particle-In-Cell simulations. II. Electron drift instability in a Hall thruster

Three-dimensional simulations of partially magnetized plasma are real challenges that actually limit the understanding of the discharge operations such as the role of kinetic instabilities using explicit Particle-In-Cell (PIC) schemes. The transition to high performance computing cannot overcome all the limits inherent to very high plasma densities and thin mesh sizes employed to avoid numerical heating. We have applied a recent method proposed in the literature [L. F. Ricketson and A. J. Cerfon, Plasma Phys. Controlled Fusion 59, 024002 (2017)] to model low temperature plasmas. This new approach, namely, the sparse grid combination technique, offers a gain in computational time by solving the problem on a reduced number of grid cells, hence allowing also the reduction of the total number of macroparticles in the system. We have modeled the example of the two-dimensional electron drift instability, which was extensively studied in the literature to explain the anomalous electron transport in a Hall thruster. Comparisons between standard and sparse grid PIC methods show an encouraging gain in the computational time with an acceptable level of error. This method offers a unique opportunity for future three-dimensional simulations of instabilities in partially magnetized low temperature plasmas.

Implementation of higher-order velocity mapping between marker particles and grid in the particle-in-cell code XGC

The global total- $f$ gyrokinetic particle-in-cell code XGC, used to study transport in magnetic fusion plasmas or to couple with a core gyrokinetic code while functioning as an edge gyrokinetic code, implements a five-dimensional continuum grid to perform the dissipative operations, such as plasma collisions, or to exchange the particle distribution function information with a core code. To transfer the distribution function between marker particles and a rectangular two-dimensional velocity-space grid, XGC employs a bilinear mapping. The conservation of particle density and momentum is accurate enough in this bilinear operation, but the error in the particle energy conservation can become undesirably large and cause non-negligible numerical heating in a steep edge pedestal. In the present work we update XGC to use a novel mapping technique, based on the calculation of a pseudo-inverse, to exactly preserve moments up to the order of the discretization space. We describe the details of the implementation and we demonstrate the reduced interpolation error for a tokamak test plasma using first- and second-order elements with the pseudo-inverse method and comparing with the bilinear mapping.

Higher-order particle representation for particle-in-cell simulations

In this paper we present an alternative approach to the representation of simulation particles for unstructured electrostatic and electromagnetic PIC simulations. In our modified PIC algorithm we represent particles as having a smooth shape function limited by some specified finite radius, r0. A unique feature of our approach is the representation of this shape by surrounding simulation particles with a set of virtual particles with delta shape, with fixed offsets and weights derived from Gaussian quadrature rules and the value of r0. As the virtual particles are purely computational, they provide the additional benefit of increasing the arithmetic intensity of traditionally memory bound particle kernels. The modified algorithm is implemented within Sandia National Laboratories' unstructured EMPIRE-PIC code, for electrostatic and electromagnetic simulations, using periodic boundary conditions. We show results for a representative set of benchmark problems, including electron orbit, a transverse electromagnetic wave propagating through a plasma, numerical heating, and a plasma slab expansion. Good error reduction across all of the chosen problems is achieved as the particles are made progressively smoother, with the optimal particle radius appearing to be problem-dependent.

Analysis of hyperbolic Pennes bioheat equation in perfused homogeneous biological tissue subject to the instantaneous moving heat source

The one-dimensional hyperbolic Pennes bioheat equation under instantaneous moving heat source is solved analytically based on the Eigenvalue method. Comparison with results of in vivo experiments performed earlier by other authors shows the excellent prediction of the presented closed-form solution. We present three examples for calculating the Arrhenius equation to predict the tissue thermal damage analysis with our solution, i.e., characteristics of skin, liver, and kidney are modeled by using their thermophysical properties. Furthermore, the effects of moving velocity and perfusion rate on temperature profiles and thermal tissue damage are investigated. Results illustrate that the perfusion rate plays the cooling role in the heating source moving path. Also, increasing the moving velocity leads to a decrease in absorbed heat and temperature profiles. The closed-form analytical solution could be applied to verify the numerical heating model and optimize surgery planning parameters.

Numerical heating in particle-in-cell simulations with Monte Carlo binary collisions.

The binary Monte Carlo (MC) collision algorithm is a standard and robust method to include binary Coulomb collision effects in particle-in-cell (PIC) simulations of plasmas. Here we show that the coupling between PIC and MC algorithms can give rise to (nonphysical) numerical heating of the system that significantly exceeds that observed when these algorithms operate independently. We argue that this deleterious effect results from an inconsistency between the particle motion associated with MC collisions and the work performed by the collective electromagnetic field on the PIC grid. This inconsistency manifests as the (artificial) stochastic production of electromagnetic energy, which ultimately heats the plasma particles. The MC-induced numerical heating can significantly impact the evolution of the simulated system for long simulation times (≳10^{3} collision periods, for typical numerical parameters). We describe the source of the MC-induced numerical heating analytically and discuss strategies to minimize it.

Simulating pitch angle scattering using an explicitly solvable energy-conserving algorithm.

Particle distribution functions evolving under the Lorentz operator can be simulated with the Langevin equation for pitch-angle scattering. This approach is frequently used in particle-based Monte-Carlo simulations of plasma collisions, among others. However, most numerical treatments do not guarantee energy conservation, which may lead to unphysical artifacts such as numerical heating and spectra distortions. We present a structure-preserving numerical algorithm for the Langevin equation for pitch-angle scattering. Similar to the well-known Boris algorithm, the proposed numerical scheme takes advantage of the structure-preserving properties of the Cayley transform when calculating the velocity-space rotations. The resulting algorithm is explicitly solvable, while preserving the norm of velocities down to machine precision. We demonstrate that the method has the same order of numerical convergence as the traditional stochastic Euler-Maruyama method. The numerical scheme is benchmarked by simulating the pitch-angle scattering of a particle beam and comparing with the analytical solution. Benchmark results show excellent agreement with theoretical predictions, showcasing the remarkable long-time accuracy of the proposed algorithm.