Numerical Jacobian Research Articles

An efficient solution of the multi-term multi-harmonic electron Boltzmann equation for use in global models

Solving the electron Boltzmann equation is an essential but costly step in simulating low-temperature plasma kinetics. This work addresses the problem by introducing a solution of the general multi-term multi-harmonic Boltzmann equation (MTMH-BE) optimized for electrons in time-dependent non-equilibrium gases and electric fields. This is accomplished by configuring the numerical Jacobian as the product of a time-independent sparse matrix and an array of time-dependent coefficients. As this approach requires a fixed energy grid, the MTMH-BE is discretized using a finite volume scheme with support for non-uniform cell size, enabling robust solutions across a wide range of reduced field strengths. The new MTMH-BE solver is tested against a range of benchmarks, demonstrating excellent agreement with existing methods. Performance with time-dependent electron energy is evaluated using LoKI-B's pulsed electric field model, with the new solver demonstrating improved wall time scaling and an improvement in grid convergence of over three orders of magnitude. Alternatively, performance with quasi-stationary electrons in an evolving non-equilibrium gas is tested using a nitrogen cross-section set with up to 58 vibrational states and 3,570 processes. Here, the MTMH-BE solver achieves a wall time reduction of up to 99.6% over MultiBolt for error-controlled simulations in the two-term limit, reducing the average solution time from 1-2 seconds to less than 10 ms. These results indicate that the new MTMH-BE solver can improve accuracy and performance in state-to-state global models without the use of grid refinement, cross-section pruning, and other time-saving procedures.

Inverse modeling of circular lattices via orbit response measurements in the presence of degeneracy

The number and relative placement of BPMs and steerers with respect to the quadrupoles in a circular lattice can lead to degeneracy in the context of inverse modeling of accelerator optics. Further, the measurement uncertainties introduced by beam position monitors can propagate by the inverse modeling process in ways that prohibit the successful estimation of model errors. In this contribution, the influence of BPM and steerer placement on the conditioning of the inverse problem is studied. An analytical version of the Jacobian, linking the quadrupole gradient errors along with BPM and steerer gain errors with the orbit response matrix, is derived. It is demonstrated that this analytical version of the Jacobian can be used in place of the numerically obtained Jacobian during the fitting procedure. The approach is first tested with simulations and the findings are verified by measurement data taken on SIS18 synchrotron at GSI. The results are crosschecked with the standard numerical Jacobian approach. The quadrupole errors causing tune discrepancies observed at SIS18 are identified.

PressPurt: network sensitivity to press perturbations under interaction uncertainty

While the use of networks to understand how complex systems respond to perturbations is pervasive across scientific disciplines, the uncertainty associated with estimates of pairwise interaction strengths (edge weights) remains rarely considered. Mischaracterizations of interaction strength can lead to qualitatively incorrect predictions regarding system responses as perturbations propagate through often counteracting direct and indirect effects. Here, we introduce PressPurt, a computational package for identifying the interactions whose strengths must be estimated most accurately in order to produce robust predictions of a network's response to press perturbations. The package provides methods for calculating and visualizing these edge-specific sensitivities (tolerances) when uncertainty is associated to one or more edges according to a variety of different error distributions. The software requires the network to be represented as a numerical (quantitative or qualitative) Jacobian matrix evaluated at stable equilibrium. PressPurt is open source under the MIT license and is available as both a Python package and an R package hosted at https://github.com/dkoslicki/PressPurt and on the CRAN repository https://CRAN.R-project.org/package=PressPurt.

Observability and sensitivity analysis of lightcurve measurement models for use in space situational awareness

ABSTRACTThis paper deals with the observability and sensitivity analysis of the lightcurve measurement for use in the estimation of instantaneous pose (orientation) and shape geometry. Since several reflectance models exist that are typically used in obtaining a synthetic lightcurve measurement, they are compared to assess the observability or/and ‘sensitivity’ to determine the ‘best model’ for use. These measurement models are nonlinear with implicit and non-trivial dependency on the states; a numerical Jacobian as well as an unscented Kalman filter derived observation matrix are synthesized for each choice of the measurement model and compared. As the linearization is about an attitude (orientation) trajectory, distinct cases are considered to elaborate the results. For the cases evaluated here, it is shown that the attitude and shape/size can indeed be estimated from the lightcurve, but this is also dependent upon the initial conditions (subsequent attitude trajectory). Linear observability analysis of a discretized system is also performed with respect to the attitude states and shape/size parameters using the singular value decomposition of the observability matrix synthesized for a batch of measurements. The results are summarized for various initial conditions of the resident space object's attitude and angular velocity states.

Implementation of Detailed Chemistry Mechanisms in Engine Simulations

The stiffness of large chemistry mechanisms has been proved to be a major hurdle toward predictive engine simulations. As a result, detailed chemistry mechanisms with a few thousand species need to be reduced based on target conditions so that they can be accommodated within the available computational resources. The computational cost of simulations typically increases super-linearly with the number of species and reactions. This work aims to bring detailed chemistry mechanisms within the realm of engine simulations by coupling the framework of unsteady flamelets and fast chemistry solvers. A previously developed tabulated flamelet model (TFM) framework for nonpremixed combustion was used in this study. The flamelet solver consists of the traditional operator-splitting scheme with variable coefficient ordinary differential equation (ODE) solver (VODE) and a numerical Jacobian for solving the chemistry. In order to use detailed mechanisms with thousands of species, a new framework with the Livermore solver for ODEs in sparse form (LSODES) chemistry solver and an analytical Jacobian was implemented in this work. Results from 1D simulations show that with the new framework, the computational cost is linearly proportional to the number of species in a given chemistry mechanism. As a result, the new framework is 2–3 orders of magnitude faster than the conventional variable coefficient ODE (VODE) solver for large chemistry mechanisms. This new framework was used to generate unsteady flamelet libraries for n-dodecane using a detailed chemistry mechanism with 2755 species and 11,173 reactions. The engine combustion network (ECN) spray A experiments, which consist of an igniting n-dodecane spray in turbulent, high-pressure engine conditions are simulated using large eddy simulations (LES) coupled with detailed mechanisms. A grid with 0.06 mm minimum cell size and 22 ×106 peak cell count was implemented. The framework is validated across a range of ambient temperatures against ignition delay and liftoff lengths (LOLs). Qualitative results from the simulations were compared against experimental OH and CH2O planar laser-induced fluorescence (PLIF) data. The models are able to capture the spatial and temporal trends in species compared to those observed in the experiments. Quantitative and qualitative comparisons between the predictions of the reduced and detailed mechanisms are presented in detail. The main goal of this study is to demonstrate that detailed reaction mechanisms (∼1000 species) can now be used in engine simulations with a linear increase in computation cost with number of species during the tabulation process and a small increase in the 3D simulation cost.

Effects of the Jacobian evaluation on Newton's solution of the Euler equations

Newton's method is developed for solving the 2-D Euler equations. The Euler equations are discretized using a finite-volume method with upwind flux splitting schemes. Both analytical and numerical methods are used for Jacobian calculations. Although the numerical method has the advantage of keeping the Jacobian consistent with the numerical residual vector and avoiding extremely complex analytical differentiations, it may have accuracy problems and need longer execution time. In order to improve the accuracy of numerical Jacobians, detailed error analyses are performed. A method is developed for calculating an optimal perturbation magnitude that can minimize the error in numerical Jacobians. The accuracy of the numerical Jacobians is improved significantly by using the optimal perturbation magnitude. The effects of the accuracy of numerical Jacobians on the convergence of the flow solver are also investigated. In order to reduce the execution time for numerical Jacobian evaluation, flux vectors with perturbed flow variables are calculated only for neighbouring cells. A sparse matrix solver that is based on LU factorization is used. Effects of different flux splitting methods and higher-order discretizations on the performance of the solver are analysed.

Comparison of simulation algorithms for accelerated determination of periodic steady state of switched networks

This paper presents a comparison of the following algorithms for accelerated determination of periodic steady state of switched networks: Newton's method with analytically determined Jacobian; Newton's method with numerically determined Jacobian; Newton's method with Broyden updates of an initial numerically determined Jacobian; Newton's method with a globally convergent strategy (and numerical Jacobian); Bukowski's method; and Skelboe's method. Each algorithm is incorporated into a very accurate power electronics' simulator (PECS) at source-code level and compared on a common basis. The results on several switching converters suggest the analytical Newton's method to be the most accurate and fastest. When analytical derivatives are not available, both Broyden's and Skelboe's methods are competitive.

Bifurcation and stability analyses for a two-phase Rayleigh–Benard problem in a cavity

Stability and bifurcation analyses of a partially melted or solidified material heated from below and cooled from above in a cavity, the so-called two-phase Rayleigh–Benard problem, are conducted by a finite-volume/Newton’s method. Bifurcation analysis techniques using a numerical Jacobian and an iterative matrix solver suitable to this large complicated system are adopted. The onset and evolution of melt flows coupling with the heat conduction in the solid and a deformable melt/solid interface are illustrated through detailed bifurcation diagrams, and the linear stability of each flow family is carefully examined. Some comparison with the one-phase system is performed. Results are presented for a variety of parameters of interest, including the Rayleigh number, aspect ratio, and tilt angle. Although most calculations are presented for the melt with a Prandtl number of one, the effects of Prandtl number on the onset of cellular convection and the sensitivity of symmetry breaking by tilting are examined. Furthermore, the dynamic responses of an unstable static state to stable solutions after small disturbances are illustrated, and the effect of heat of fusion is discussed.

Comparison of numerical and analytical Jacobians

The results are presented of comparison between numerical and analytical approaches for forming the Jacobian matrices that arise from implicit discretizations of the governing equations The derivation of the exact analytical Jacobians using the MACSYMA symbolic manipulation expert system is discussed in detail. The numerical differentiation procedure used to develop the numerical Jacobians is described. The Jacobian entries obtained from the preceding approaches were tested by comparing the density residuals obtained for selected test cases. The test cases include supersonic flat plate and compression corner geometries. The results indicate that the two approaches are virtually identical, although the numerical Jacobian approach may have some implementation advantages over the exact analytical approach.

Direct and iterative algorithms for the three-dimensional Euler equations

Direct and iterative algorithms have been developed for solving a finite volume discretization of the three-dimensional Euler equations in curvilinear coordinates. The Euler equations are discretized using numerical derivatives of the numerical flux vector for the Jacobians. Two direct solvers are formulated, one of which has a diagonal plane matrix structure with significantly lower memory requirements. The direct solvers are used as a benchmark in measuring the convergence rate and robustness of more computationally efficient solvers which include two factored approaches, a Newton-relaxation algorithm and a discretized Newton-relaxation algorithm, which uses numerical Jacobians. A diagonal plane formulation for the Newton-relaxation algorithms has also been developed that may have the potential for massive parallelization. It is demonstrated that the Newton-relaxation approach can give convergence rates and robustness equal to that of a direct solver for three-dimensional problems. As a demonstration of the robustness of both the Newton-relaxation algorithm and numerical Jacobians, quadratic convergence to machine zero is demonstrated

Simulation of Tokamak Divertor Plasmas including Cross‐Field Drifts

AbstractThe effects of plasma currents and cross‐field drifts on single null tokamak divertor operation are simulated using a fully implicit 2‐D fluid code. Equations solved are those for particle continuity, parallel momentum, electron energy, ion energy, electrostatic potential, and neutral gas diffusion. The core and scrape‐off layer regions are separated by a magnetic separatrix, both of which are included in the simulation. The core plasma is poloidally periodic, and the inner and outer private flux regions are properly connected. The code utilizes a fully implicit method‐of‐lines scheme to advance the variables in time with a Krylov technique or a direct Newton iteration with a numerical Jacobian. Results are presented on the effects of currents and cross‐field drifts for DIII‐D single‐null parameters.

Efficient Formation of Numerical Jacobian Used in Flame Codes

Abstract An efficient method for the formation of the numerical Jacobian in combustion codes has been applied to premixed laminar and counterflow diffusion flames. The improvements are portable to similar codes for treating ignition, two-dimensional flames and the general class of reactive-diffusive processes. Execution times were reduced by a factor of seven with no decrease in accuracy, for a mechanism containing 46 species and 213 reactions, when compared with the starting form of the premixed flame code developed at Sandia Laboratories (Smooke, 1982). The most important new feature is the efficient formation of the numerical Jacobian, reducing the CPU-time spent on computing chemical kinetics, thermodynamic and transport properties among the N unknowns (temperature, species mass fractions, and mass flow) in the system. Normally, the Jacobian is i formed in 3N + I full vector function evaluations requiring 90-95% of the computing time (Smooke, 1982). Every variable is perturbed simultaneously at every ...