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DOE OSTI2026

Verification of an energy-conserving semi-implicit electrostatic particle-in-cell scheme for modeling high-density plasma at scale

A verification study of a semi-implicit energy-conserving electrostatic particle-in-cell algorithm is presented. The algorithm relaxes the time-step and mesh-size constraints that require resolution of the plasma period and Debye length associated with traditional explicit momentum-conserving particle-in-cell algorithms. Physical implications and applicability of using the semi-implicit scheme for modeling high-density plasmas are discussed. Where possible, numerical results are compared against analytical solutions. The simulation results indicate that the algorithm is stable at time steps larger than twice the inverse plasma frequency and cell sizes larger than the Debye length. It is found that the algorithm gives adequate results, provided that the distribution function and the spatiotemporal scales dictating the physics of the problem are resolved. As such, the algorithm may provide a robust method for kinetic modeling of high-density plasmas at scale.

Cyclotron resonance↗

DOE OSTI2019

A fourth-order adaptive mesh refinement algorithm for the multicomponent, reacting compressible Navier–Stokes equations

In this study, we present a fourth-order in space and time block-structured adaptive mesh refinement algorithm for the compressible multicomponent reacting Navier–Stokes equations. The algorithm uses a finite-volume approach that incorporates a fourth-order discretisation of the convective terms. The time-stepping algorithm is based on a multi-level spectral deferred corrections method that enables explicit treatment of advection and diffusion coupled with an implicit treatment of reactions. The temporal scheme is embedded in a block-structured adaptive mesh refinement algorithm that includes subcycling in time with spectral deferred correction sweeps applied on levels. Here we present the details of the multi-level scheme paying particular attention to the treatment of coarse–fine boundaries required to maintain fourth-order accuracy in time. We then demonstrate the convergence properties of the algorithm on several test cases including both non-reacting and reacting flows. Finally we present simulations of a vitiated dimethyl ether jet in 2D and a turbulent hydrogen jet in 3D, both with detailed kinetics and transport.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2013

PIROCK: A swiss-knife partitioned implicit–explicit orthogonal Runge–Kutta Chebyshev integrator for stiff diffusion–advection–reaction problems with or without noise

A partitioned implicit–explicit orthogonal Runge–Kutta method (PIROCK) is proposed for the time integration of diffusion–advection–reaction problems with possibly severely stiff reaction terms and stiff stochastic terms. The diffusion terms are solved by the explicit second order orthogonal Chebyshev method (ROCK2), while the stiff reaction terms (solved implicitly) and the advection and noise terms (solved explicitly) are integrated in the algorithm as finishing procedures. It is shown that the various coupling (between diffusion, reaction, advection and noise) can be stabilized in the PIROCK method. The method, implemented in a single black-box code that is fully adaptive, provides error estimators for the various terms present in the problem, and requires from the user solely the right-hand side of the differential equation. Numerical experiments and comparisons with existing Chebyshev methods, IMEX methods and partitioned methods show the efficiency and flexibility of our new algorithm.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2017

A tightly-coupled domain-decomposition approach for highly nonlinear stochastic multiphysics systems

Multiphysics simulations often involve nonlinear components that are driven by internally generated or externally imposed random fluctuations. When used with a domain-decomposition (DD) algorithm, such components have to be coupled in a way that both accurately propagates the noise between the subdomains and lends itself to a stable and cost-effective temporal integration. We develop a conservative DD approach in which tight coupling is obtained by using a Jacobian-free Newton–Krylov (JfNK) method with a generalized minimum residual iterative linear solver. This strategy is tested on a coupled nonlinear diffusion system forced by a truncated Gaussian noise at the boundary. Enforcement of path-wise continuity of the state variable and its flux, as opposed to continuity in the mean, at interfaces between subdomains enables the DD algorithm to correctly propagate boundary fluctuations throughout the computational domain. Reliance on a single Newton iteration (explicit coupling), rather than on the fully converged JfNK (implicit) coupling, may increase the solution error by an order of magnitude. Increase in communication frequency between the DD components reduces the explicit coupling's error, but makes it less efficient than the implicit coupling at comparable error levels for all noise strengths considered. Finally, the DD algorithm with the implicit JfNK coupling resolves temporally-correlated fluctuations of the boundary noise when the correlation time of the latter exceeds some multiple of an appropriately defined characteristic diffusion time.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2019

Reaction Dynamics of Cyanohydrins with Hydrosulfide in Water

We studied the reaction dynamics of a proposed prebiotic reaction theoretically. The chemical process involves acetone cyanohydrin or formalcyanohydrin reacting with hydrosulfide in an aqueous environment. Rate constants and populations of reactant and product bimolecular geometric orientations for the reactions were obtained by using density functional theory for the energies, transition-state theory for the rates, and matrix exponentiation as well as the hybrid tau-leaping algorithm for the population dynamics. The role of including the solvent explicitly versus implicitly was also investigated. We found that adding explicit water or hydrogen sulfide molecules lowers the activation energy barrier and leads to a more efficient reaction pathway. In particular, hydrogen sulfide was a better proton donor. Lastly, we studied the role of including more than one reactant and product bimolecular orientation geometry in the dynamics. Including all initial pathways, reactant to reactant, product to product, reactant to product, and product to reactant led to a larger reaction rate constant compared to the single minimum energy pathway approach. Overall, we found that most reactions which involve formalcyanohydrin occur more rapidly or at the same speed as reactions which involve acetone cyanohydrin at room temperature.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2021

Implicit highly-coupled single-ion Hall-MHD formulation for hybrid particle-in-cell codes

The rudiments of a particle-based single-fluid two-temperature magnetohydrodynamic (MHD) algorithm have been outlined in Thoma et al. (2013). The extension of this algorithm to include the effect of Hall physics is described in this paper. An implicit leapfrog version of the algorithm, which allows timesteps large compared to the resistive decay time and other relevant timescales, has recently been added to a hybrid particle-in-cell code. In standard MHD the Hall term in the generalized Ohm’s law can often be neglected when the Hall parameter is small. This term must, however, be retained in regimes where it is non-negligible. The retention of displacement current in Maxwell’s equations avoids the numerical difficulties associated with the whistler mode, which are encountered in standard explicit Hall-MHD codes, and allows the algorithm to be incorporated into hybrid particle-in-cell codes, for which particles may migrate from a kinetic to fluid to MHD description based upon local ambient plasma conditions. A highly-coupled implicit Hall-MHD formalism is presented, in which displacement current can either be retained or neglected. Even when displacement current is neglected, the highly-coupled implicit formalism avoids the restrictive timesteps for the whistler mode in explicit Hall-MHD codes. A comparison of numerical and analytic dispersion analysis demonstrates the feasibility of this approach and establishes relevant constraints to assure numerical stability. The implementation of the algorithm is described, and test simulation results in 1D and 2D in both linear and nonlinear regimes are presented.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Learning Constrained Parametric Differentiable Predictive Control Policies With Guarantees

We present differentiable predictive control (DPC), a method for offline learning of constrained neural control policies for nonlinear dynamical systems with performance guarantees. We show that the sensitivities of the parametric optimal control problem can be used to obtain direct policy gradients. Specifically, we employ automatic differentiation (AD) to efficiently compute the sensitivities of the model predictive control (MPC) objective function and constraints penalties. To guarantee safety upon deployment, we derive probabilistic guarantees on closed-loop stability and constraint satisfaction based on indicator functions and Hoeffding’s inequality. We empirically demonstrate that the proposed method can learn neural control policies for various parametric optimal control tasks. In particular, we show that the proposed DPC method can stabilize systems with unstable dynamics, track time-varying references, and satisfy nonlinear state and input constraints. Our DPC method has practical time savings compared to alternative approaches for fast and memory-efficient controller design. Specifically, DPC does not depend on a supervisory controller as opposed to approximate MPC based on imitation learning. We demonstrate that, without losing performance, DPC is scalable with greatly reduced demands on memory and computation compared to implicit and explicit MPC while being more sample efficient than model-free reinforcement learning (RL) algorithms.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Fully implicit crystal plasticity models representing orientations with modified Rodrigues parameters

Here, this work describes a crystal plasticity formulation combining several mathematical, numerical, and implementation choices to produce a highly efficient model. Specifically, the key choices in the implementation are (1) representing orientations with modified Rodrigues parameters, (2) implementing a fully coupled implicit time integration for the elastic stretch, the crystal orientations, and the model internal variables, (3) implementing the model in the NEML2 constitutive modeling framework, based on PyTorch, to vectorize the calculations and port the computation to GPUs and other hardware accelerators, and (4) an exact implementation of the consistent tangent matrix, even for arbitrary coupling to other field variables beyond the displacements, like temperature, neutron fluence, etc. The first two features of the model are, to our knowledge, novel. The paper considers each of these choices individually as well as the final model as a whole. This includes a full description of modified Rodrigues parameters, their advantages over other representations of orientations, the mathematical formulae and tools required to implement a model with modified Rodrigues parameters, and a detailed description of the geometry of the space of modified Rodrigues parameters (in an appendix). It also includes a description of a fully implicit time integration scheme for the orientations and the advantages in representing orientations with modified Rodrigues parameters in implementing such a model. The work then assess, via numerical examples, the advantages of fully coupled implicit time integration versus more common decoupled and explicit time integration schemes. These studies demonstrate the computational advantages of fully coupled integration versus other time integration algorithms, though the performance of the competing models depends on the complexity of the underlying single crystal model. The study concludes by demonstrating that the choice of time integration method affects the sharpness of the predicted texture, with explicit methods for integrating the orientations overestimating texture sharpness and implicit methods underestimating texture sharpness.

Crystal plasticity↗

DOE OSTI2020

Time-stepping DPG formulations for the heat equation

For a wide range of PDEs, the discontinuous Petrov–Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan provides discrete stability starting from a coarse mesh and minimization of the residual in a user-controlled norm, among other appealing features. Research on DPG for transient problems has mainly focused on spacetime discretizations, which has theoretical advantages, but practical costs for computations and software implementations. The sole examination of time-stepping DPG formulations was performed by Führer, Heuer, and Gupta, who applied Rothe’s method to an ultraweak formulation of the heat equation to develop an implicit time-stepping scheme; their work emphasized theoretical results, including error estimates in time and space. Here, we follow Führer, Heuer, and Gupta in examining the heat equation; our focus is on numerical experiments, examining the stability and accuracy of several formulations, including primal as well as ultraweak, and explicit as well as implicit and Crank–Nicolson time-stepping schemes. We are additionally interested in communication-avoiding algorithms, and we therefore include a highly experimental formulation that places all the trace terms on the right-hand side of the equation.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2017

A stable and accurate partitioned algorithm for conjugate heat transfer

We describe a new partitioned approach for solving conjugate heat transfer (CHT) problems where the governing temperature equations in different material domains are time-stepped in a implicit manner, but where the interface coupling is explicit. The new approach, called the CHAMP scheme (Conjugate Heat transfer Advanced Multi-domain Partitioned), is based on a discretization of the interface coupling conditions using a generalized Robin (mixed) condition. The weights in the Robin condition are determined from the optimization of a condition derived from a local stability analysis of the coupling scheme. The interface treatment combines ideas from optimized-Schwarz methods for domain-decomposition problems together with the interface jump conditions and additional compatibility jump conditions derived from the governing equations. For many problems (i.e. for a wide range of material properties, grid-spacings and time-steps) the CHAMP algorithm is stable and second-order accurate using no sub-time-step iterations (i.e. a single implicit solve of the temperature equation in each domain). In extreme cases (e.g. very fine grids with very large time-steps) it may be necessary to perform one or more sub-iterations. Each sub-iteration generally increases the range of stability substantially and thus one sub-iteration is likely sufficient for the vast majority of practical problems. The CHAMP algorithm is developed first for a model problem and analyzed using normal-mode the- ory. The theory provides a mechanism for choosing optimal parameters in the mixed interface condition. A comparison is made to the classical Dirichlet-Neumann (DN) method and, where applicable, to the optimized- Schwarz (OS) domain-decomposition method. For problems with different thermal conductivities and dif- fusivities, the CHAMP algorithm outperforms the DN scheme. For domain-decomposition problems with uniform conductivities and diffusivities, the CHAMP algorithm performs better than the typical OS scheme with one grid-cell overlap. Lastly, the CHAMP scheme is also developed for general curvilinear grids and CHT ex- amples are presented using composite overset grids that confirm the theory and demonstrate the effectiveness of the approach.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2021

Impact of Integration Scheme on Performance of Anisotropic Plasticity Models

Given the prevalent role of metals in a variety of industries, schemes to integrate corresponding constitutive models in finite element applications have long been studied. A number of formulations have been developed to accomplish this task; each with their own advantages and costs. Often the focus has been on ensuring the accuracy and numerical stability of these algorithms to enable robust integration. While important, emphasis on these performance metrics may often come at the cost of computational expense potentially neglecting the needs of individual problems. In the current work, the performance of two of the most common integration methods for anisotropic plasticity -- the convex cutting plane (CCP) and closest point projection (CPP) -- across a variety of metrics is assessed; including accuracy and cost. A variety of problems are considered ranging from single elements to large representative simulations including both implicit quasistatic and explicit transient dynamic type responses. The relative performance of each scheme in the different instances is presented with an eye towards guidance on when the different algorithms may be beneficial.

36 MATERIALS SCIENCE↗

DOE OSTI2021

Validation and Demonstration of Control System Functional Capabilities within the IES Plug-and-Play Simulation Environment

The concept of an integrated energy system (IES) is meant to combine different energy technologies in synergistic ways to achieve a more secure and economical energy supply. The RAVEN-based HYBRID framework is used to find the optimal installed capacity and the optimal economical dispatch of each component of the IES. The new RAVEN plugin for grid and capacity optimization (HERON) only addresses the limits that affect the production variables and the corresponding rates of variation (explicit constraints). However, other variables are subject to constraints, and the associated limits should be accounted for (implicit constraints). In particular, for the power dispatch problem, the optimization algorithm takes into account the limits on the electrical power output and the corresponding hourly power variations but does not consider other constraints on process variables whose response affects the service life of the IES. This report describes a scheme that allows accounting for implicit constraints without increasing the size of the optimization problem. To obtain a more accurate approximation of the nonlinear dynamic behavior, a parametric version of the dynamic mode decomposition with control (DMDc) algorithm was developed to derive the state-space representation matrices of the IES components at different scheduling parameter. Thanks to this approach, a more accurate approximation of the system response can be obtained, the limits imposed by thermal mechanical implicit constraints can be translated into power dispatch limits, and the feedbacks to HERON power dispatcher can be provided. To assess the developed methodology, a power dispatching test case composed of three power generating and storage units (Balance of Plant, Secondary Energy Source, Thermal Energy Storage) was developed. The power output of each one of the three units was optimized to meet the imposed time-dependent load demand trajectory and to maximize the IES profitability by meeting both the explicit and implicit constraints.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2013

Numerical Stability and Accuracy of Temporally Coupled Multi-Physics Modules in Wind-Turbine CAE Tools

In this paper we examine the stability and accuracy of numerical algorithms for coupling time-dependent multi-physics modules relevant to computer-aided engineering (CAE) of wind turbines. This work is motivated by an in-progress major revision of FAST, the National Renewable Energy Laboratory's (NREL's) premier aero-elastic CAE simulation tool. We employ two simple examples as test systems, while algorithm descriptions are kept general. Coupled-system governing equations are framed in monolithic and partitioned representations as differential-algebraic equations. Explicit and implicit loose partition coupling is examined. In explicit coupling, partitions are advanced in time from known information. In implicit coupling, there is dependence on other-partition data at the next time step; coupling is accomplished through a predictor-corrector (PC) approach. Numerical time integration of coupled ordinary-differential equations (ODEs) is accomplished with one of three, fourth-order fixed-time-increment methods: Runge-Kutta (RK), Adams-Bashforth (AB), and Adams-Bashforth-Moulton (ABM). Through numerical experiments it is shown that explicit coupling can be dramatically less stable and less accurate than simulations performed with the monolithic system. However, PC implicit coupling restored stability and fourth-order accuracy for ABM; only second-order accuracy was achieved with RK integration. For systems without constraints, explicit time integration with AB and explicit loose coupling exhibited desired accuracy and stability.

17 WIND ENERGY↗

DOE OSTI2015

Explicit integration with GPU acceleration for large kinetic networks

We demonstrate the first implementation of recently-developed fast explicit kinetic integration algorithms on modern graphics processing unit (GPU) accelerators. Taking as a generic test case a Type Ia supernova explosion with an extremely stiff thermonuclear network having 150 isotopic species and 1604 reactions coupled to hydrodynamics using operator splitting, we demonstrate the capability to solve of order 100 realistic kinetic networks in parallel in the same time that standard implicit methods can solve a single such network on a CPU. This orders-of-magnitude decrease in computation time for solving systems of realistic kinetic networks implies that important coupled, multiphysics problems in various scientific and technical fields that were intractable, or could be simulated only with highly schematic kinetic networks, are now computationally feasible.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2017

Extreme scale unstructured adaptive CFD for aerodynamic flow control

Understanding the flow of fluid, either liquid or gas, through and around solid bodies has challenged man since the dawn of scientific inquiry. Many of the great minds of science and mathematics have progressively built up a hierarchy of fluid models since fluid flow impacts our lives in so many fundamental ways - from the early days of flow of water in viaducts to cities, to today's flight of planes. This chapter is concerned with the computational modeling of turbulent flow around aerodynamic bodies such as planes and wind turbines. In this case, viscous effects near the solid bodies create very thin boundary layers that yield highly anisotropic (gradients normal to the surface may be 10 6 larger than gradients along the surface) solutions to the governing non-linear partial differential equations (PDE); the Navier-Stokes equations. Furthermore, turbulent flows develop extremely broad ranges of length and time scales motivating the use of discretization methods capable of employing adaptivity and implicit time integration. The combination of these features - non-linear, anisotropy, adaptivity, and implicit - dramatically raise the complexity of the discretization, posing large challenges to efficient scalable parallel implementation. However, through careful design, the more complex algorithms can provide great reductions in computational cost relative to simpler methods (e.g., Cartesian grids with explicit time integration) that are easier to mate efficiently to hardware. In this chapter, we not only describe our approach but we also address the fact that while complex algorithms may never be as efficient flop-for-flop as simple methods, in the important measure of science-per-core-hour, they can still win big by making complex features like adaptivity and implicit methods as efficient and scalable as possible.

42 ENGINEERING↗

DOE OSTI2017

Rythmos: Solution and Analysis Package for Differential-Algebraic and Ordinary-Differential Equations

Time integration is a central component for most transient simulations. It coordinates many of the major parts of a simulation together, e.g., a residual calculation with a transient solver, solution with the output, various operator-split physics, and forward and adjoint solutions for inversion. Even though there is this variety in these transient simulations, there is still a common set of algorithms and procedures to progress transient solutions for ordinary-differential equations (ODEs) and differential-alegbraic equations (DAEs). Rythmos is a collection of these algorithms that can be used for the solution of transient simulations. It provides common time-integration methods, such as Backward and Forward Euler, Explicit and Implicit Runge-Kutta, and Backward-Difference Formulas. It can also provide sensitivities, and adjoint components for transient simulations. Rythmos is a package within Trilinos, and requires some other packages (e.g., Teuchos and Thrya) to provide basic time-integration capabilities. It also can be coupled with several other Trilinos packages to provide additional capabilities (e.g., AztecOO and Belos for linear solutions, and NOX for non-linear solutions). The documentation is broken down into three parts: Theory Manual, User's Manual, and Developer's Guide. The Theory Manual contains the basic theory of the time integrators, the nomenclature and mathematical structure utilized within Rythmos, and verification results demonstrating that the designed order of accuracy is achieved. The User's Manual provides information on how to use the Rythmos, description of input parameters through Teuchos Parameter Lists, and description of convergence test examples. The Developer's Guide is a high-level discussion of the design and structure of Rythmos to provide information to developers for the continued development of capabilities. Details of individual components can be found in the Doxygen webpages.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Comparison of exponential integrators and traditional time integration schemes for the shallow water equations

We report the time integration scheme is probably one of the most fundamental choices in the development of an ocean model. In this paper, we investigate several time integration schemes when applied to the shallow water equations. This set of equations is accurate enough for the modeling of a shallow ocean and is also relevant to study as it is the one solved for the barotropic (i.e. vertically averaged) component of a three dimensional ocean model. We analyze different time stepping algorithms for the linearized shallow water equations. High order explicit schemes are accurate but the time step is constrained by the Courant-Friedrichs-Lewy stability condition. Implicit schemes can be unconditionally stable but, in practice lack accuracy when used with large time steps. In this paper we propose a detailed comparison of such classical schemes with exponential integrators. The accuracy and the computational costs are analyzed in different configurations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2026

Asymptotic-preserving semi-implicit finite volume scheme for extended magnetohydrodynamics

A Finite Volume (FV) scheme is developed for solving the extended magnetohydrodynamic (XMHD) equations, yielding accurate results in the ideal, resistive, and Hall MHD limits. This is accomplished by first re-writing the XMHD equations such that it allows the algorithm to retain the use of ideal MHD Riemann solvers and the constrained transport method to preserve divergence-free magnetic fields. Incorporation of electron inertia and displacement current introduces additional numerical stiffness which motivates a semi-implicit FV scheme that re-formulates the XMHD model as a relaxation system. The equations are then advanced in time using an explicit 2nd-order Runge–Kutta scheme with operator splitting applied to the implicit source term updates at each sub-stage. For additional numerical stability, a density-dependent slope limiter is implemented to increase flux diffusivity at low density regions where non-ideal effects become significant. The algorithm is subsequently implemented in a scalable adaptive mesh refinement (AMR) framework. As the new algorithm retains many aspects of the ideal MHD formulations, it asymptotes naturally to the ideal MHD limit. Moreover, it shows promising results at the resistive and Hall MHD limits. This is verified against reference test problems for ideal, resistive and Hall MHD.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗