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DOE OSTI2023

Demonstrating Computational Equivalence Between Continuous and Discrete Adjoint Methods by Calculating Time-Dependent Adjoint Solutions with Neutron Diffusion Models

The continuous adjoint method and the discrete adjoint method are two alternative approaches used to calculate adjoint solutions for adjoint systems. The continuous adjoint method derives adjoint equations analytically from continuous forward equations and then solves the adjoint equations either analytically or numerically in a discretized form whereas the discrete adjoint method calculates the adjoint solutions directly from the discretized forward equations. With regard to the methodology development and calculation procedure, distinct differences are well recognized between the two methods. For certain reasons, both methods are exclusively preferred and commonly used by different computational communities, but limited studies clarify the connections between the two adjoint methods from either of the communities. Herein, this paper demonstrates the computational equivalence between the continuous and discrete adjoint methods by investigating time-dependent adjoint solutions to the two-group neutron diffusion model in nuclear reactor analysis problems using both methods. Adjoint solutions can be used to estimate system parameters for reactor safety analysis. Appropriate final state conditions for the adjoint systems are specified in both of the methods, and the conditions are clarified with proper physical explanations. With the help of an event-based case study on neutron diffusion models, the accuracy of the time-dependent adjoint fluxes obtained from both methods is verified, and the pros and cons of both adjoint methods are examined. More importantly, the computational equivalence of both methods is demonstrated when they are applied to multigroup neutron diffusion systems. The advantage of calculating time-dependent adjoint fluxes by directly solving time-dependent adjoint systems rather than taking steady-state approximations as in common practice is also demonstrated.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2021

Rare Event Sampling Improves Mercury Instability Statistics

Due to the chaotic nature of planetary dynamics, there is a non-zero probability that Mercury's orbit will become unstable in the future. Previous efforts have estimated the probability of this happening between 3 and 5 billion years in the future using a large number of direct numerical simulations with an N-body code, but were not able to obtain accurate estimates before 3 billion years in the future because Mercury instability events are too rare. In this paper we use a new rare-event sampling technique, Quantile Diffusion Monte Carlo (QDMC), to estimate that the probability of a Mercury instability event in the next 2 billion years is approximately 10 –4 in the REBOUND N-body code. We show that QDMC provides unbiased probability estimates at a computational cost of up to 100 times less than direct numerical simulation. QDMC is easy to implement and could be applied to many problems in planetary dynamics in which it is necessary to estimate the probability of a rare event.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2024

Finite domain solution of a KGD hydraulic fracture in the viscosity-dominated regime

This paper describes a numerical algorithm for solving the classic problem of a plane strain (KGD) fracture propagating in an impermeable elastic medium with zero toughness. The method, which takes advantage of the self-similar nature of the solution, combines a domain-based scheme to solve the elasticity equations and a finite volume method to solve the nonlinear lubrication equation. This work represents a first step towards developing a model able to account for pore pressure diffusion in the medium and corresponding poroelastic effects, noting that these processes are more efficiently solved using a domain-based rather than a boundary integral method. To enhance the efficiency and accuracy of the numerical scheme, the far-field crack asymptotics is embedded in the discretized elastic relationship between the fluid pressure and the crack opening, while the coupled fluid-solid tip asymptote is enforced in a weak form when solving the nonlinear lubrication equation. The proposed technique yields results that closely match the analytical solution, even with a coarse mesh. This approach offers potential for addressing more complex hydraulic fracturing problems in the future.

Domain-based method↗

DOE OSTI2023

Theoretical estimates of flammability bounds for thin condensed fuel diffusion flames in microgravity using detailed models of chemistry and radiation

Recently, U-shape flammability maps have been constructed showing minimal oxygen vs. flame strain for opposed flame spread in micro-gravity by Olson and Ferkul. The U-shape defines the limiting flammability bounds from radiative extinction and flame blow-off. Here, the minimum of the U corresponds to the minimum possible oxidizer concentration where burning can occur, and is an important quantity of interest for fire safety. While high strain extinction bounds have been well analyzed, low strain radiative extinction has not. To estimate low strain extinction, in this study an analytical theory is developed based on thin flame theory coupled with a heat and mass transfer model for solid fuels. A reaction progress variable based on the Damköhler number is adapted in the theory to account for incomplete combustion at high strain rates and enable the capturing of the full flammability map. The analytical model is compared to a one dimensional numerical model w/ detailed chemical kinetics and coupled radiation heat transfer in planar and spherical geometries. The flammability maps are then qualitatively compared to experimental extinguishment data compiled by Olson and Ferkul for cylindrical rods of PMMA showing similar trends. The results show the newly developed analytics capture the radiative extinction bound compared to the numerical model and qualitatively agrees with microgravity data.

38 RADIATION CHEMISTRY, RADIOCHEMISTRY, AND NUCLEA↗

DOE OSTI2022

Less can be more: Insights on the role of electrode microstructure in redox flow batteries from two-dimensional direct numerical simulations

Understanding how to structure a porous electrode to facilitate fluid, mass, and charge transport is key to enhancing the performance of electrochemical devices, such as fuel cells, electrolyzers, and redox flow batteries (RFBs). Here, using a parallel computational framework, direct numerical simulations are carried out on idealized porous electrode microstructures for RFBs. Strategies to improve an electrode design starting from a regular lattice are explored. By introducing vacancies in the ordered arrangement, it is possible to achieve higher voltage efficiency at a given current density, thanks to improved mixing of reactive species, despite reducing the total reactive surface. Careful engineering of the location of vacancies, resulting in a density gradient, outperforms disordered configurations. Our simulation framework is a new tool to explore transport phenomena in RFBs, and our findings suggest new ways to design performant electrodes.

25 ENERGY STORAGE↗

DOE OSTI2025

Continuum shock mixture models for Ni+Al multilayers: Individual layers and bulk equations of state

Continuum shock mixture models are reviewed and applied to determine the equations of state for five different compositions of Ni x Al y ⁠, as well as bulk Ni+Al reactive multilayers, by combining the fundamental property data for elemental nickel and aluminum. From the literature, we down-select and evaluate two analytical models for the mixture Hugoniot, i.e., the well-known method of kinetic energy averaging (KEA) and a recent model proposed by Jordan and Baer [J. Appl. Phys. 111, 083516 (2012)]. Fundamentally, the former method assumes pressure equilibrium, whereas the latter assumes a common particle velocity and mixture sound speed from compressible two-phase cavitating flows. Additionally, we construct thermodynamically complete equations of state by fitting Einstein oscillator series models for the specific heat at constant volume. Finally, the solid solution approximation is invoked for intermetallic compositions, which are not strictly physical mixtures. Overall, the KEA model provides a better fit to the available Ni x Al y and Ni+Al multilayer shock compression data; however, there are combinations of material properties where the performance of these two models is thought to be reversed. Moreover, the results of this work include the first analytical solution of Jordan–Baer that does not require numerical root finding, as well as proposed modifications to the Einstein oscillator series to incorporate some effects of local pressure–temperature equilibrium and reaction–diffusion. Future work is planned that will use these equations of state in mesoscale simulations to study shock-induced reaction in Ni+Al multilayers, and the intended application is illustrated with a brief 2D hydrocode example.

36 MATERIALS SCIENCE↗

DOE OSTI2023

Exponential Runge-Kutta Parareal for non-diffusive equations

Parareal is a well-known parallel-in-time algorithm that combines a coarse and fine propagator within a parallel iteration. It allows for large-scale parallelism that leads to significantly reduced computational time compared to serial time-stepping methods. However, like many parallel-in-time methods it can fail to converge when applied to non-diffusive equations such as hyperbolic systems or dispersive nonlinear wave equations. Here, this paper explores the use of exponential integrators within the Parareal iteration. Exponential integrators are particularly interesting candidates for Parareal because of their ability to resolve fast-moving waves, even at the large stepsizes used by coarse propagators. This work begins with an introduction to exponential Parareal integrators followed by several motivating numerical experiments involving the nonlinear Schrödinger equation. These experiments are then analyzed using linear analysis that approximates the stability and convergence properties of the exponential Parareal iteration on nonlinear problems. The paper concludes with two additional numerical experiments involving the dispersive Kadomtsev-Petviashvili equation and the hyperbolic Vlasov-Poisson equation. These experiments demonstrate that exponential Parareal methods offer improved time-to-solution compared to serial exponential integrators when solving certain non-diffusive equations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

A meshless stochastic method for Poisson–Nernst–Planck equations

A plethora of biological, physical, and chemical phenomena involve transport of charged particles (ions). Its continuum-scale description relies on the Poisson–Nernst–Planck (PNP) system, which encapsulates the conservation of mass and charge. The numerical solution of these coupled partial differential equations is challenging and suffers from both the curse of dimensionality and difficulty in efficiently parallelizing. We present a novel particle-based framework to solve the full PNP system by simulating a drift–diffusion process with time- and space-varying drift. We leverage Green’s functions, kernel-independent fast multipole methods, and kernel density estimation to solve the PNP system in a meshless manner, capable of handling discontinuous initial states. The method is embarrassingly parallel, and the computational cost scales linearly with the number of particles and dimension. We use a series of numerical experiments to demonstrate both the method’s convergence with respect to the number of particles and computational cost vis-à-vis a traditional partial differential equation solver.

Chemistry↗

DOE OSTI2020

Where did the tumor start? An inverse solver with sparse localization for tumor growth models

In this work, we present a numerical scheme for solving an inverse problem for parameter estimation in tumor growth models for glioblastomas, a form of aggressive primary brain tumor. The growth model is a reaction–diffusion partial differential equation (PDE) for the tumor concentration. We use a PDE-constrained optimization formulation for the inverse problem. The unknown parameters are the reaction coefficient (proliferation), the diffusion coefficient (infiltration), and the initial condition field for the tumor PDE. Segmentation of magnetic resonance imaging (MRI) scans drive the inverse problem where segmented tumor regions serve as partial observations of the tumor concentration. Like most cases in clinical practice, we use data from a single time snapshot. Moreover, the precise time relative to the initiation of the tumor is unknown, which poses an additional difficulty for inversion. We perform a frozen-coefficient spectral analysis and show that the inverse problem is severely ill-posed. We introduce a biophysically motivated regularization on the structure and magnitude of the tumor initial condition. In particular, we assume that the tumor starts at a few locations (enforced with a sparsity constraint on the initial condition of the tumor) and that the initial condition magnitude in the maximum norm is equal to one. We solve the resulting optimization problem using an inexact quasi-Newton method combined with a compressive sampling algorithm for the sparsity constraint. Our implementation uses PETSc and AccFFT libraries. We conduct numerical experiments on synthetic and clinical images to highlight the improved performance of our solver over a previously existing solver that uses standard two-norm regularization for the calibration parameters. The existing solver is unable to localize the initial condition. Our new solver can localize the initial condition and recover infiltration and proliferation. In clinical datasets (for which the ground truth is unknown), our solver results in qualitatively different solutions compared to the two-norm regularized solver.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

An Efficient Numerical Algorithm for Solving Data Driven Feedback Control Problems

The goal of this paper is to solve a class of stochastic optimal control problems numerically, in which the state process is governed by an Itô type stochastic differential equation with control process entering both in the drift and the diffusion, and is observed partially. The optimal control of feedback form is determined based on the available observational data. In this work, we call this type of control problems the data driven feedback control. The computational framework that we introduce to solve such type of problems aims to find the best estimate for the optimal control as a conditional expectation given the observational information. To make our method feasible in providing timely feedback to the controlled system from data, we develop an efficient stochastic optimization algorithm to implement our computational framework.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms

Here in this work we present a non-intrusive model reduction method to learn low-dimensional models of dynamical systems with non-polynomial nonlinear terms that are spatially local and that are given in analytic form. In contrast to state-of-the-art model reduction methods that are intrusive and thus require full knowledge of the governing equations and the operators of a full model of the discretized dynamical system, the proposed approach requires only the non-polynomial terms in analytic form and learns the rest of the dynamics from snapshots computed with a potentially black-box full-model solver. The proposed method learns operators for the linear and polynomially nonlinear dynamics via a least-squares problem, where the given non-polynomial terms are incorporated on the right-hand side. The least-squares problem is linear and thus can be solved efficiently in practice. The proposed method is demonstrated on three problems governed by partial differential equations, namely the diffusion–reaction Chafee–Infante model, a tubular reactor model for reactive flows, and a batch-chromatography model that describes a chemical separation process. The numerical results provide evidence that the proposed approach learns reduced models that achieve comparable accuracy as models constructed with state-of-the-art intrusive model reduction methods that require full knowledge of the governing equations.

42 ENGINEERING↗

DOE OSTI2024

PINN surrogate of Li-ion battery models for parameter inference, Part II: Regularization and application of the pseudo-2D model

Bayesian parameter inference is useful to improve Li-ion battery diagnostics and can help formulate battery aging models. However, it is computationally intensive and cannot be easily repeated for multiple cycles, multiple operating conditions, or multiple replicate cells. To reduce the computational cost of Bayesian calibration, numerical solvers for physics-based models can be replaced with faster surrogates. A physics-informed neural network (PINN) is developed as a surrogate for the pseudo-2D (P2D) battery model calibration. For the P2D surrogate, additional training regularization was needed as compared to the PINN single-particle model (SPM) developed in Part I. Both the PINN SPM and P2D surrogate models are exercised for parameter inference and compared to data obtained from a direct numerical solution of the governing equations. A parameter inference study highlights the ability to use these PINNs to calibrate scaling parameters for the cathode Li diffusion and the anode exchange current density. By realizing computational speed-ups of ~2250x for the P2D model, as compared to using standard integrating methods, the PINN surrogates enable rapid state-of-health diagnostics. Finally, in the low-data availability scenario, the testing error was estimated to ~2 mV for the SPM surrogate and ~10 mV for the P2D surrogate which could be mitigated with additional data.

25 ENERGY STORAGE↗

DOE OSTI2024

Large eddy simulation of a supersonic lifted hydrogen flame: Impacts of Lewis, turbulent Schmidt and Prandtl numbers

Parametric large eddy simulations (LES) of a supersonic lifted hydrogen flame are reported. The emphases are on two aspects: impacts of (1) Lewis number (Lei of the ith species) and (2) turbulent Schmidt and Prandtl numbers (⁠Sc t and Pr t ⁠) on supersonic turbulent flame and flow structures. Five cases are considered: species-specific Le i , Sc t = Sr t = 1.0 (C 0 ); unity Le i , Sc t = Pr t = 1.0 (C 1 ); species-specific Sc t = 0.5, Pr t = 1.0 (C 2 ); species-specific Le i , Sc t = 1.0, Pr t = 0.5 (C 3 ); and species-specific Le i , Sc t = Pr t = 0.5 (C 4 ). Numerical results of instantaneous and/or time-averaged species mole fractions, mixture fraction, heat release rate, flame base location, and mixed modes of premixed and diffusion combustion are compared between cases C 0 and C 1 . Differences in auto-ignition locations and strengths and flame structures and stabilization specify the impacts of Lewis number. They are triggered by different predictions of species mass and thermal diffusions at fuel-coflow and/or coflow-ambient air mixing layers. These differences are rationalized by a scale analysis of mass/thermal diffusion and convection for case C 0 , which suggests the relatively low but non-negligible former against the latter. Cases C 0 and C 2 –C 4 barely see differences in terms of instantaneous and/or time-averaged temperature, velocity, and mixed combustion modes except for further downstream areas where combustion occurs. Both Sc t and Pr t impose less significant influences than Lewis number, as sub-grid scale mass/thermal diffusion is subordinate to its resolved counterpart according to their scale analysis for case C 4 .

42 ENGINEERING↗

DOE OSTI2020

Grassy ELM regime at low pedestal collisionality in high-power tokamak plasma

Spontaneous mitigation of giant ELMs and appearance of grassy ELMs have been observed repeatedly at low pedestal collisionality (${\nu }_{\mathrm{e},\mathrm{p}\mathrm{e}\mathrm{d}}^{{\ast}}\sim 0.15$) in the high-power (P inj > 13 MW) hybrid scenario in DIII-D. Higher β p and higher q 95 appear to be beneficial to achieving the grassy ELM regime. The grassy ELM H-mode plasma shows high energy confinement performance (H 98y2 up to over 1.5) under the conditions of high neutral beam torque and high core rotation. The pedestal width appears to exceed the EPED1.0 model prediction by more than 50%. Pedestal stability analysis performed with the ELITE code indicates that the stability against low-n kink/peeling modes is improved with increased auxiliary heating power and the operational point in the grassy ELM regime is located near the ballooning boundary. The pedestal stability characteristics during the grassy ELM crashes have been investigated in comparison with the giant ELM crashes based on plasma profiles experimentally measured with high resolution and accuracy. It has been found that the underlying mechanism for the observed small-amplitude ELM crashes is mainly the expansion of the ballooning stability boundary induced by an initial radially localized collapse in the pedestal, which helps to stop the growth of instabilities and further collapse of the pedestal. The effect of electron density pedestal on mitigating edge localized instabilities has been analyzed by numerical simulation, suggesting that the electron density pedestal characterized by high n e,sep /n e,ped and low density gradient helps to stabilize peeling-ballooning modes because of a low pressure pedestal gradient and to lower the ballooning boundary mainly because of a low ion diamagnetic frequency in the pedestal region, thus triggering ballooning instabilities and producing the intrinsic grassy ELMs. Numerical simulation of the Chinese fusion engineering test reactor (CFETR) with the SOLPS code indicates that the separatrix density might be insensitive to the electron diffusivity in the pedestal region and increase with the power flowing from the core region to the edge region. Furthermore, pedestal stability analysis suggests that the flat density pedestal with high separatrix density obtained in the high-power plasma in CFETR would make the operational point close to the ballooning boundary, which is considered to help destabilize ballooning instabilities and facilitate the access to the grassy ELM regime.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2021

Randomized Sketching Algorithms for Low-Memory Dynamic Optimization

This paper develops a novel limited-memory method to solve dynamic optimization problems. The memory requirements for such problems often present a major obstacle, particularly for problems with PDE constraints such as optimal flow control, full waveform inversion, and optical tomography. In these problems, PDE constraints uniquely determine the state of a physical system for a given control; the goal is to find the value of the control that minimizes an objective. While the control is often low dimensional, the state is typically more expensive to store. This paper suggests using randomized matrix approximation to compress the state as it is generated and shows how to use the compressed state to reliably solve the original dynamic optimization problem. Concretely, the compressed state is used to compute approximate gradients and to apply the Hessian to vectors. The approximation error in these quantities is controlled by the target rank of the sketch. This approximate first- and second-order information can readily be used in any optimization algorithm. As an example, we develop a sketched trust-region method that adaptively chooses the target rank using a posteriori error information and provably converges to a stationary point of the original problem. Numerical experiments with the sketched trust-region method show promising performance on challenging problems such as the optimal control of an advection-reaction-diffusion equation and the optimal control of fluid flow past a cylinder.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Semi-Analytic Functions to Calculate the Deposition Coefficients for Ice Crystal Vapor Growth in Bin and Bulk Microphysical Models

Numerical cloud models require estimates of the vapor growth rate for ice crystals. Current bulk and bin microphysical parameterizations generally assume that vapor growth is diffusion limited, though some parameterizations include the influence of surface attachment kinetics through a constant deposition coefficient. A parameterization for variable deposition coefficients is provided herein. The parameterization is an explicit function of the ambient ice supersaturation and temperature, and an implicit function of crystal dimensions and pressure. The parameterization is valid for variable surface types including growth by dislocations and growth by step nucleation. Deposition coefficients are predicted for the two primary growth directions of crystals, allowing for the evolution of the primary habits. Comparisons with benchmark calculations of instantaneous mass growth indicate that the parameterization is accurate to within a relative error of 1%. Parcel model simulations using Lagrangian microphysics as a benchmark indicate that the bulk parameterization captures the evolution of mass mixing ratio and fall speed with typical relative errors of less than 10%, whereas the average axis lengths can have errors of up to 20%. The bin model produces greater accuracy with relative errors often less that 10%. The deposition coefficient parameterization can be used in any bulk and bin scheme, with low error, if an equivalent volume spherical radius is provided.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

Improved Prediction of Cold-Air Pools in the Weather Research and Forecasting Model Using a Truly Horizontal Diffusion Scheme for Potential Temperature

The terrain-following vertical coordinate system used by many atmospheric models, including the Weather Research and Forecasting (WRF) Model, is prone to errors in regions of complex terrain. These errors stem, in part, from the calculation of horizontal gradients within the diffusion term of the momentum or scalar evolution equations. In WRF, such gradients can be calculated along coordinate surfaces, or using metric terms that help account for grid skewness. However, neither of these options ensures a truly horizontal gradient calculation, especially if a grid cell is skewed enough that the heights of the neighboring grid points used in the calculation fall outside the vertical range of the cell. In this work, an improved scheme that uses Taylor series approximations to vertically interpolate variables to the level necessary for a truly horizontal gradient calculation is implemented in WRF for the diffusion of potential temperature. The scheme is validated using an atmosphere-at-rest configuration, in which spurious flows develop only as a result of numerical errors and can thus be used as a proxy for model performance. Following validation, the method is applied to the simulation of cold-air pools (CAPs), which occur in regions of complex terrain and are characterized by strong near-surface temperature gradients. Using the truly horizontal scheme, idealized simulations demonstrate reduced numerical mixing in a quiescent CAP, and a realistic case study in the Columbia River basin shows a reduction in positive wind speed bias by up to roughly 20% compared to observations from the Second Wind Forecast Improvement Project.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2024

Impact of a Bluff Body on a Pressurized Oxy-fuel Flame

Designing an effective burner is vital for the development of pressurized oxy-fuel combustion technologies. In the present work, turbulent jet diffusion burners are adopted for a pressurized oxy-combustor, with a bluff-body employed to create a recirculation zone, thereby stabilizing the flame in such a combustor. The objective of this numerical study is to perform a systematic analysis of the characteristics of such a pressurized non-premixed flame. Specifically, a 15-bar pressurized oxy-fuel combustor of power 100 kWth is modeled by means of the Ansys FLUENT commercial platform, using the Reynolds-averaged Navier-Stokes (RANS) approach. The present work focuses on identifying the aerothermodynamic features of the pressurized oxy-fuel burner with a disk-shape bluff-body. It is shown that the fuel-to-oxidizer stream momentum ratio has a great impact on the temperature profile of the down-fired, co-axial, pressurized oxy-coal, diffusion flame. A parametric study of the blockage ratio of the burner identifies an optimal range for this pressurized burner.

Li, Lei↗