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NASA NTRS1989

A genuinely multi-dimensional upwind cell-vertex scheme for the Euler equations

The solution of the two-dimensional Euler equations is based on the two-dimensional linear convection equation and the Euler-equation decomposition developed by Hirsch et al. The scheme is genuinely two-dimensional. At each iteration, the data are locally decomposed into four variables, allowing convection in appropriate directions. This is done via a cell-vertex scheme with a downwind-weighted distribution step. The scheme is conservative, and third-order accurate in space. The derivation and stability analysis of the scheme for the convection equation, and the derivation of the extension to the Euler equations are given. Preconditioning techniques based on local values of the convection speeds are discussed. The scheme for the Euler equations is applied to two channel-flow problems. It is shown to converge rapidly to a solution that agrees well with that of a third-order upwind solver.

Powell, Kenneth G.↗

NASA NTRS2002

Multi-dimensional Upwind Fluctuation Splitting Scheme with Mesh Adaption for Hypersonic Viscous Flow

A multi-dimensional upwind fluctuation splitting scheme is developed and implemented for two-dimensional and axisymmetric formulations of the Navier-Stokes equations on unstructured meshes. Key features of the scheme are the compact stencil, full upwinding, and non-linear discretization which allow for second-order accuracy with enforced positivity. Throughout, the fluctuation splitting scheme is compared to a current state-of-the-art finite volume approach, a second-order, dual mesh upwind flux difference splitting scheme (DMFDSFV), and is shown to produce more accurate results using fewer computer resources for a wide range of test cases. A Blasius flat plate viscous validation case reveals a more accurate upsilon-velocity profile for fluctuation splitting, and the reduced artificial dissipation production is shown relative to DMFDSFV. Remarkably, the fluctuation splitting scheme shows grid converged skin friction coefficients with only five points in the boundary layer for this case. The second half of the report develops a local, compact, anisotropic unstructured mesh adaptation scheme in conjunction with the multi-dimensional upwind solver, exhibiting a characteristic alignment behavior for scalar problems. The adaptation strategy is extended to the two-dimensional and axisymmetric Navier-Stokes equations of motion through the concept of fluctuation minimization.

Wood, William A., III↗

DOE OSTI2024

Discrete-Element and Material-Point Method (DEM and MPM) Based Solvers for Sustainable Technologies

We present the use of discrete element method (DEM) and material point method (MPM) in three relevant green technology applications that include biomass feedstock handling, lithium-ion battery manufacturing, and high-pressure reverse osmosis. Our open-source DEM and MPM solvers are developed using performance portable grid and particle management library, AMReX, thus enabling superior performance on NVIDIA and AMD GPUs with > 100 million particles. Our DEM solver resolves the motion of individual particles in a granular system and includes a bonded sphere method for modeling non-spherical particles along with Hertzian and liquid bridge-based contact models. We simulate highly variable biomass feedstock flows in large-scale hoppers for biofuel production and electrode calendering in battery manufacturing using DEM. Our simulations predict flow blockage in large scale biomass hoppers and electrode microstructure variations, thus providing valuable information for biofuel and battery manufacturers, respectively. The second half of the talk will be on MPM and its application towards pore resolved simulations of reverse osmosis membranes under compressive loads. We present a validation study of our MPM simulations with membrane microscopy imaging thus providing useful insights on membrane stability under high pressure conditions. We also present a spectral stability analysis of using linear hat, quadratic and cubic spline basis in MPM indicating regions of numerical stability.

BIOMASS FUELS,MATHEMATICS AND COMPUTING↗

NASA NTRSDate not supplied

Surface-Normal Active Flow Control for Lift Enhancement and Separation Mitigation for High-Lift Common Research Model

This study explores the feasibility of using small surface-normal jets near the flap trailing edge as active aerodynamic load control on the high-lift Common Research Model wing/-body configuration. Chimera Grid Tools are utilized to generate structured curvilinear overset grids, and the Reynolds-averaged Navier-Stokes solver OVERFLOW is employed to solve for the flow-field around the geometry. The so-called microjet is initially employed across both the inboard and the outboard flaps on the pressure-side near the trailing edge. It is shown that implementing the microjet on the inboard flap is more effective compared to implementing it on the outboard flap. This is because, prior to microjet implementation, the flow on the outboard flap exhibits extensive separation, while the flow on the inboard flap exhibits moderate separation. For microjet implementation across the inboard flap only, the relationship between momentum coefficient of the microjet and lift-enhancement is found to be∆CL'1.66√Cμfor the rangeCμ= 0.00−0.012. We show that implementing a microjet with a jet velocity ratio of one, which corresponds toCμ= 0.003, can shift the linear region of the lift curve by ∆CL= 0.08. The linear shift in the lift curve is significant for enhancing airplane performance such as increasing its payload. Microjet implementation effects on the drag coefficient are also investigated through a drag decomposition analysis. Further, we employ an induced drag analysis based on the spanwise load distribution and show that the microjet-related increase in pressure drag coefficient is dominated by the increase in the induced drag while microjet implementation reduces the form drag. These preliminary results show that favorable changes in aerodynamic performance can be achieved by using the surface-normal jets presented in this study.

ARMD↗

DOE OSTI2024

ReEDS Performance Improvement

The Regional Energy Deployment System (ReEDS) is an open-source, spatially explicit, long-term capacity expansion model for the bulk electric power system of the contiguous United States, encompassing multiple scenarios with technological and political assumptions (see https://github.com/NREL/ReEDS-2.0). With the increased needs for capabilities, higher temporal and spatial resolutions to model the evolution of the power system with modern technologies and low-carbon pathways, ReEDS' model solution times have increased significantly from 4-6 hours in 2018 to 18-48+ hours in 2023 . Also, the model size for commonly-run ReEDS scenarios reached 22 and 28 million equations and variables, respectively. These runtimes can be especially challenging under certain scenario settings (e.g., very high temporal or spatial resolution) or with limited computational power. In this presentation, we will discuss several methods we used to improve model runtime, including data preparation, model modification, and solver tuning. The implementation of these methods shrank the model size to 7.2 and 7.3 million equations and variables, respectively. Furthermore, this led to a 77% reduction in the model's run time for commonly-run ReEDS scenarios. We will discuss the process of identifying areas for solve time improvements and how the specific enhancements for the ReEDS model might be applied to other similar large-scale models.

ENERGY PLANNING, POLICY, AND ECONOMY,MATHEMATICS A↗

DOE OSTI2023

Encoding trade-offs and design toolkits in quantum algorithms for discrete optimization: coloring, routing, scheduling, and other problems

Challenging combinatorial optimization problems are ubiquitous in science and engineering. Several quantum methods for optimization have recently been developed, in different settings including both exact and approximate solvers. Addressing this field of research, this manuscript has three distinct purposes. First, we present an intuitive method for synthesizing and analyzing discrete (i.e., integer-based) optimization problems, wherein the problem and corresponding algorithmic primitives are expressed using a discrete quantum intermediate representation (DQIR) that is encoding-independent. This compact representation often allows for more efficient problem compilation, automated analyses of different encoding choices, easier interpretability, more complex runtime procedures, and richer programmability, as compared to previous approaches, which we demonstrate with a number of examples. Second, we perform numerical studies comparing several qubit encodings; the results exhibit a number of preliminary trends that help guide the choice of encoding for a particular set of hardware and a particular problem and algorithm. Our study includes problems related to graph coloring, the traveling salesperson problem, factory/machine scheduling, financial portfolio rebalancing, and integer linear programming. Third, we design low-depth graph-derived partial mixers (GDPMs) up to 16-level quantum variables, demonstrating that compact (binary) encodings are more amenable to QAOA than previously understood. We expect this toolkit of programming abstractions and low-level building blocks to aid in designing quantum algorithms for discrete combinatorial problems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

Mapping Spiking Neural Networks to Heterogeneous Crossbar Architectures using Integer Linear Programming

Advances in novel hardware devices and architectures allow Spiking Neural Network (SNN) evaluation using ultra-low power, mixed-signal, memristor crossbar arrays. As individual network sizes quickly scale beyond the dimensional capabilities of single crossbars, networks must be mapped onto multiple crossbars. Crossbar sizes within modern Memristor Crossbar Architectures (MCAs) are determined predominately not by device technology but by network topology; more, smaller crossbars consume less area thanks to the high structural sparsity found in larger, brain-inspired SNNs. Motivated by continuing increases in SNN sparsity due to improvements in training methods, we propose utilizing heterogeneous crossbar sizes to further reduce area consumption. This approach was previously unachievable as prior compiler studies only explored solutions targeting homogeneous MCAs. Our work improves on the state-of-the-art by providing Integer Linear Programming (ILP) formulations supporting arbitrarily heterogeneous architectures. By modeling axonal interactions between neurons, our methods produce better mappings while removing inhibitive a priori knowledge requirements. We first show a 16.7-27.6% reduction in area consumption for square-crossbar homogeneous architectures. Then, we demonstrate 66.9-72.7% further reduction when using a reasonable configuration of heterogeneous crossbar dimensions. Next, we present a new optimization formulation capable of minimizing the number of inter-crossbar routes. When applied to solutions already near-optimal in area, an 11.9-26.4% routing reduction is observed without impacting area consumption. Finally, we present a profile-guided optimization capable of minimizing the number of runtime spikes between crossbars. Compared to the best-area-then-route optimized solutions, we observe a further 0.5-14.8% inter-crossbar spike reduction while requiring 1–3 orders of magnitude less solver time.

Pohl, Devin [ORNL] (ORCID:0009000040149027)↗

NASA NTRS2023

CFD Integrated Transition Modeling for High-Speed Flows via Coupled OVERFLOW-LASTRAC Analysis

This work details ongoing efforts at the NASA Langley Research Center to develop and validate a general-use CFD tool that includes built-in predictions of boundary-layer transition in high-speed flows. Existing tools for the coupling of the NASA OVERFLOW structured overset RANS solver and LASTRAC stability analysis code have been extended to capture boundary-layer transition in high-speed flows driven by either Mack’s first-mode or secondmode instabilities and/or crossflow instabilities. The efficacy of this coupled approach has been demonstrated by examining a variety of supersonic and hypersonic test cases for which experimental validation data is available. The intermittency prescription parameters inherited from prior low-speed applications are found to be suitable for some, but not all high-speed flow scenarios. The method is found to be robust for straight cone configurations and improvements necessary to accurately capture transition on geometries with spatially complex amplification factor envelopes with intermediate regions of slow N-factor variation are examined. The automated, coupled analysis is demonstrated for a 3d supersonic test case and found to perform well within the limits of the linear stability analysis on which it relies. Finally, a preliminary investigation of the method’s robustness to non-ideal CFD meshes is conducted.

High-speed flow↗

NASA NTRS2014

Adjoint Methods for Adjusting Three-Dimensional Atmosphere and Surface Properties to Fit Multi-Angle/Multi-Pixel Polarimetric Measurements

This paper derives an efficient procedure for using the three-dimensional (3D) vector radiative transfer equation (VRTE) to adjust atmosphere and surface properties and improve their fit with multi-angle/multi-pixel radiometric and polarimetric measurements of scattered sunlight. The proposed adjoint method uses the 3D VRTE to compute the measurement misfit function and the adjoint 3D VRTE to compute its gradient with respect to all unknown parameters. In the remote sensing problems of interest, the scalar-valued misfit function quantifies agreement with data as a function of atmosphere and surface properties, and its gradient guides the search through this parameter space. Remote sensing of the atmosphere and surface in a three-dimensional region may require thousands of unknown parameters and millions of data points. Many approaches would require calls to the 3D VRTE solver in proportion to the number of unknown parameters or measurements. To avoid this issue of scale, we focus on computing the gradient of the misfit function as an alternative to the Jacobian of the measurement operator. The resulting adjoint method provides a way to adjust 3D atmosphere and surface properties with only two calls to the 3D VRTE solver for each spectral channel, regardless of the number of retrieval parameters, measurement view angles or pixels. This gives a procedure for adjusting atmosphere and surface parameters that will scale to the large problems of 3D remote sensing. For certain types of multi-angle/multi-pixel polarimetric measurements, this encourages the development of a new class of three-dimensional retrieval algorithms with more flexible parametrizations of spatial heterogeneity, less reliance on data screening procedures, and improved coverage in terms of the resolved physical processes in the Earth's atmosphere.

linearization↗

NASA NTRS2014

Adjoint Methods for Adjusting Three-Dimensional Atmosphere and Surface Properties to Fit Multi-Angle Multi-Pixel Polarimetric Measurements

adjoint methods↗

DOE OSTI2020

Impact of Vibrational Nonequilibrium on the Supersonic Mode in Hypersonic Boundary Layers

The supersonic mode in hypersonic boundary layers has recently been shown to achieve a larger peak growth rate than the traditional second mode in particular circumstances relevant to free flight. In this environment, high levels of aerodynamic heating are encountered such that incorporation of chemical and vibrational nonequilibrium effects can become critical. The impact of thermal nonequilibrium (TNE) on the supersonic mode in particular has not been thoroughly and systematically investigated. This work uses thermochemical nonequilibrium (TCNE) direct numerical simulation (DNS) and TCNE linear stability theory (LST) to obtain a more complete investigation of the supersonic mode using both nonequilibrium and frozen gas models. The simulation is Mach 5 flow over a 1-mm-nose-radius axisymmetric cone 1 m in length, representative of experimental conditions in ground test facilities. Both LST and DNS results indicate that TNE effects are destabilizing compared with frozen flow and can lead to an N-factor difference of approximately +1 at the end of the cone. LST results also indicate that TNE effects are stabilizing to the second and supersonic modes in comparison with thermal equilibrium. The supersonic mode appeared to be slightly more affected by TNE than the second mode. In conclusion, collectively, the findings in this work suggest that even in ground test facilities where TCNE effects are relatively small compared with free flight, incorporating nonequilibrium models in both the mean flow and stability solvers can have a practical effect on hypersonic boundary-layer transition prediction analyses.

42 ENGINEERING↗

DOE OSTI2023

Probability of Initiation in Neutron Transport

We discuss the numerical solution of the nonlinear integro-differential equation for the probability of a divergent neutron chain in a stationary system (i.e., the probability of initiation (POI)). We follow the development described in Bell’s classic paper on the stochastic theory of neutron transport. As noted by Bell, the linearized form of this equation resembles the linear adjoint neutron transport equation. A matrix formalism for the discretized steady state (or forward) neutron equation in slab geometry is first developed and is then used to derive the discrete adjoint equation. A main advantage of this discrete development is that the resulting discrete adjoint equation does not depend upon how the multigroup cross sections for the forward problem are obtained. That is, we derive the discrete adjoint directly from the discrete forward equations rather than discretizing directly the adjoint equation. This also guarantees that the discrete adjoint operator is consistent with the inner product used to define the adjoint operator. We discuss three approaches for the numerical solution of the POI equations, and present numerical results on several test problems. The three solution methods are a simple fixed-point iteration, a second approach that is akin to a nonlinear Power iteration, and a third approach which uses a Newton-Krylov nonlinear solver. We also give sufficient conditions to guarantee the existence and uniqueness of nontrivial solutions to our discrete POI equations when the discrete system is supercritical, and that only the trivial solution exists when the discrete system is subcritical. Our approach is modeled after the analysis presented for the continuous POI equations by Mokhtar-Kharroubi and Jarmouni-Idrissi, and by Pazy and Rabinowitz.

42 ENGINEERING↗

DOE OSTI2025

Simulations of the churning mode: Toroidally symmetric plasma convection and turbulence around the X-points in a snowflake divertor

Using a reduced MHD model, extended to include field-aligned thermal conduction, we present numerical simulations of the churning mode (CM): a toroidally symmetric, non-linear plasma vortex in the vicinity of the null points in a snowflake (SF) divertor [D. D. Ryutov et al., Phys. Scr. 89, 088002 (2014)]. Simulations are carried out across a range of inter-null separations, d xx ⁠, and inter-null orientations, θ⁠, primarily in conditions relevant to the MAST-U tokamak. We find that, when d xx is small, the CM induces additional transport across the X-points when β pm ≳8%, where β pm is the ratio of the plasma pressure in the null region to poloidal magnetic pressure at the midplane. This transport also increases approximately linearly as d xx is reduced. A diffusive model of this transport is shown to predict the total transport across the null points, where diffusion coefficients of up to ∼ 10 2 m 2 s −1 centered on a small region around the X-points are used. However, the CM also results in significant changes to the flux surfaces in the null region, which is not captured by this diffusive model. The changes in magnetic geometry mean the fractional exhaust power delivered to each divertor leg is highly sensitive to β pm ⁠, d xx ⁠, and θ⁠. For small values of θ⁠, the CM can induce a change in topology, redirecting exhaust power from a secondary divertor leg on the high field side to one on the low field side. Similar behavior is found in the fraction of exhaust power going to the inner and outer divertor. Such changes in the flux surfaces may not be captured by Grad–Shafranov solvers and so may be a source of error in the magnetic reconstruction of SF experiments. We consistently find that the fractional exhaust power going to a secondary divertor leg on the high field side is small, consistent with SF experiments.

Computer simulation↗

DOE OSTI2025

Efficient shallow Ritz method for 1D diffusion problems

This paper studies the shallow Ritz method for solving the one-dimensional diffusion problem. It is shown that the shallow Ritz method improves the order of approximation dramatically for non-smooth problems. To realize this optimal or nearly optimal order of the shallow Ritz approximation, we develop a damped block Newton (dBN) method that alternates between updates of the linear and non-linear parameters. Per each iteration, the linear and the non-linear parameters are updated by exact inversion and one step of a modified, damped Newton method applied to a reduced non-linear system, respectively. The computational cost of each dBN iteration is $\mathcal{O}$(n). Starting with the non-linear parameters as a uniform partition of the interval, numerical experiments show that the dBN is capable of efficiently moving mesh points to nearly optimal locations. In conclusion, to improve the efficiency of the dBN further, we propose an adaptive damped block Newton (AdBN) method by combining the dBN with the adaptive neuron enhancement (ANE) method [28].

Diffusion problems↗

DOE OSTI2020

Mixed-precision iterative refinement using tensor cores on GPUs to accelerate solution of linear systems

Double-precision floating-point arithmetic (FP64) has been the de facto standard for engineering and scientific simulations for several decades. Problem complexity and the sheer volume of data coming from various instruments and sensors motivate researchers to mix and match various approaches to optimize compute resources, including different levels of floating-point precision. In recent years, machine learning has motivated hardware support for half-precision floating-point arithmetic. A primary challenge in high-performance computing is to leverage reduced-precision and mixed-precision hardware. We show how the FP16/FP32 Tensor Cores on NVIDIA GPUs can be exploited to accelerate the solution of linear systems of equations Ax = b without sacrificing numerical stability. The techniques we employ include multiprecision LU factorization, the preconditioned generalized minimal residual algorithm (GMRES), and scaling and auto-adaptive rounding to avoid overflow. We also show how to efficiently handle systems with multiple right-hand sides. On the NVIDIA Quadro GV100 (Volta) GPU, we achieve a 4×-5× performance increase and 5× better energy efficiency versus the standard FP64 implementation while maintaining an FP64 level of numerical stability.

GMRES↗

NASA NTRS2023

Evaluation of Transport-Equation-Based Transition Models for High-Speed Boundary Layers Using OVERFLOW

Accurate modeling of laminar-turbulent transition is crucial for the design of hypersonic flight systems. However, the current transition models used in production CFD codes are insufficient for high-speed flows. Many extensions to low-speed models have been suggested; however, a thorough verification and validation effort is needed before these models can be used in design settings. Challenges include potentially missing details of the model implementation requirements and/or a complete specification of the input parameters needed to replicate the test findings. A meaningful assessment of the generalization capability of these models is also hindered by a lack of information regarding the specific flow configurations and associated grids employed for model calibration. As a key first step toward model verification, we present an independent assessment of two recently proposed models for high-speed transition, namely, a model within the SST-$\gamma$ framework and a model based on the SST-$\gamma-\nu_L$ equations. These models are implemented in the NASA OVERFLOW 2.3e solver and their performance in predicting first mode, second mode, and crossflow transition has been evaluated for several test cases in the supersonic and hypersonic regimes. Besides the test cases employed by the model developers, which could have also been used for model calibration, the present assessment includes supplementary configurations that contribute to an unbiased assessment of the models. The outcomes presented in this study indicate the potential for the models to be applied to high-speed flight configurations. Key steps toward future improvements to these models are also outlined.

High-speed flow↗

DOE OSTI2026

A Scalable Reduced‐Order Model for the Steady Navier–Stokes Equations

Scaling up new scientific technologies from laboratory to industry often involves demonstrating performance on a larger scale. Computer simulations can accelerate design and predictions in the deployment process, though traditional numerical methods are computationally intractable even for intermediate pilot plant scales. Recently, the component reduced order modeling method has been developed to tackle this challenge by combining projection reduced order modeling and discontinuous Galerkin domain decomposition. However, while many scientific or engineering applications involve nonlinear physics, this method has only been demonstrated for various linear systems. In this work, the component reduced order modeling method is extended to steady Navier–Stokes flow, with application to general nonlinear physics in view. The large‐scale, global domain is decomposed into a combination of small‐scale unit component. Linear subspaces for flow velocity and pressure are identified via proper orthogonal decomposition over sample snapshots collected from each small‐scale unit component. Velocity bases are augmented with a pressure supremizer to satisfy the inf–sup condition for stable pressure prediction. Two different nonlinear reduced order modeling methods are employed and compared for efficient evaluation of nonlinear advection: A third‐order tensor projection operator and the empirical quadrature procedure. The proposed method is demonstrated on the flow over arrays of five different unit objects, achieving a 23‐fold speedup with less than 4% relative error in domains up to 256 times larger than the unit components. Furthermore, a numerical experiment with the pressure supremizer strongly indicates the need for a supremizer for stable pressure prediction. A comparison between the tensorial approach and the empirical quadrature procedure revealed a slight advantage of the empirical quadrature procedure. The framework is compared with an alternating Schwarz‐based reduced‐order approach, demonstrating improved efficiency and robustness for the DG‐based global solver while retaining flexibility for sub‐scale iterative solvers. The method is further extended to a coupled advection–diffusion and Navier–Stokes system, illustrating its applicability to multi‐physics problems and its potential for more general, inter‐coupled nonlinear systems.

42 ENGINEERING↗

DOE OSTI2024

Accuracy of kinetic equilibrium reconstruction of NSTX and NSTX-U plasmas and its impact on the transport and stability analysis

An accurate magnetohydrodynamic (MHD) equilibrium reconstruction is an essential starting point for stability and transport plasma analysis. Herein this work describes an approach for obtaining kinetic equilibrium reconstructions using the OMFIT framework, which has been applied for the first time to spherical tokamak data from NSTX and NSTX-U. The EFIT equilibrium solver is integrated with experimental data analysis procedures and subsequent TRANSP transport simulations to enhance the accuracy of the reconstruction, in particular, at the edge region, by adding constraints on the total pressure and current density profiles, based on the transport code solution. The accuracy of the equilibrium reconstruction depends on the uncertainty and number of constraints, as well as the choice of basis functions to represent the pressure and current density profiles. Improved fidelity of the equilibrium reconstruction is demonstrated by reducing the variability of the magnetic axis and boundary locations from several centimeters, for reconstructions based on magnetic and experimental pressure constraints, to only several millimeters, for kinetic reconstructions based on transport code constraints, when different representations of basis functions were tested. The variability of the safety factor on axis was reduced ten times in the same sensitivity study. The accuracy of the equilibrium reconstruction and subsequent mapping of the experimental kinetic profile data have a significant impact on the trapped gyro Landau fluid and linear CGYRO turbulence simulations, which predict different spectra of unstable modes and turbulent fluxes for cases with different numbers of constraints in the equilibrium reconstruction. Conversely, the stability analysis performed using the GATO code shows plasmas that are stable to n = 1 MHD modes in both equilibria using magnetic and experimental pressure constraints as well as the transport code constrained equilibrium. However, a scan of parameters away from these conditions shows considerable deviation in the threshold of unstable modes between these reconstructions. Therefore, for reliable plasma analysis and use in turbulence and stability calculations, a high-fidelity equilibrium reconstruction with accurate kinetic constraints based on transport code solutions is necessary.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗