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At least 379 records · Page 21

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DOE OSTI2021

A short note on the accuracy of the discontinuous Galerkin method with reentrant faces

In this work, we study the convergence of the discontinuous Galerkin (DG) method applied to the advection–reaction equation on meshes with reentrant faces. On such meshes, the upwind numerical flux is not smooth, and so the numerical integration of the resulting face terms can only be expected to be first-order accurate. Despite this inexact integration, we prove that the DG method converges with order $\mathscr{O}$(h p+1/2 ), which is the same rate as in the case of exact integration. Consequently, specialized quadrature rules that accurately integrate the non-smooth numerical fluxes are not required for high-order accuracy. These results are numerically corroborated on examples of linear advection and discrete ordinates transport equations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Leveraging Multitime Hamilton–Jacobi PDEs for Certain Scientific Machine Learning Problems

Hamilton-Jacobi partial differential equations (HJ PDEs) have deep connections with a wide range of fields, including optimal control, differential games, and imaging sciences. By considering the time variable to be a higher dimensional quantity, HJ PDEs can be extended to the multi-time case. In this paper, we establish a novel theoretical connection between specific optimization problems arising in machine learning and the multi-time Hopf formula, which corresponds to a representation of the solution to certain multi-time HJ PDEs. Through this connection, we increase the interpretability of the training process of certain machine learning applications by showing that when we solve these learning problems, we also solve a multi-time HJ PDE and, by extension, its corresponding optimal control problem. As a first exploration of this connection, we develop the relation between the regularized linear regression problem and the Linear Quadratic Regulator (LQR). We then leverage our theoretical connection to adapt standard LQR solvers (namely, those based on the Riccati ordinary differential equations) to design new training approaches for machine learning. Lastly, we provide some numerical examples that demonstrate the versatility and possible computational advantages of our Riccati-based approach in the context of continual learning, post-training calibration, transfer learning, and sparse dynamics identification.

97 MATHEMATICS AND COMPUTING↗

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 Machine Learning Initializer for Newton-Raphson AC Power Flow Convergence

Power flow computations are fundamental to many power system studies. Obtaining a converged power flow case is not a trivial task especially in large power grids due to the non-linear nature of the power flow equations. One key challenge is that the widely used Newton based power flow methods are sensitive to the initial voltage magnitude and angle estimates, and a bad initial estimate would lead to non-convergence. This paper addresses this challenge by developing a random-forest (RF) machine learning model to provide better initial voltage magnitude and angle estimates towards achieving power flow convergence. This method was implemented on a real ERCOT 6102 bus system under various operating conditions. By providing better Newton-Raphson initialization, the RF model precipitated the solution of 2,106 cases out of 3,899 non-converging dispatches. These cases could not be solved from flat start or by initialization with the voltage solution of a reference case. Finally, results obtained from the RF initializer performed better when compared with DC power flow initialization, Linear regression, and Decision Trees.

random forest↗

DOE OSTI2020

Development of the finite slab geometry in the NORA code

In this work we develop the complete linearization formalism for the radiation transfer equation in slab geometry. This development was motivated by an observed convergence instability arising from the NORA code’s inner boundary while modeling a suite of simple test problems in slab geometry presented at the Line Transfer Workshop held in LLNL in May of 2019. The convergence issue was seen to emerge early on in Newton-Raphson iterations. These test problems required parameters in the former NORA inner boundary to achieve the asymptotic limit of zero incoming intensity. In this work we derive the limiting case directly to gain insight into the former algorithm’s pathology.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Time-lapse approach to correct deficiencies of 2D soil zymography

Membrane zymography is commonly used in rhizosphere ecology to map enzyme activities in intact soil samples and plant roots. The method consists of incubating a membrane saturated with an enzyme-specific fluorogenic substrate on the soil/root surface followed by measurements of fluorescence intensity of the product in the membrane. The traditional zymography is based on assumptions that fluorescence on membrane images is linearly increasing with time during zymography and an increase in product content is numerically equal to enzyme activity in soil below the membrane. These assumptions are unlikely to hold in experimental settings. In this work we introduce a new zymography technique, time-lapse zymography (TLZ); the approach that eliminates the need for assumptions of the traditional zymography and provides more realistic estimates of enzymatic activities. We assessed the new technique in a series of laboratory and modeling experiments, including quantification of the fluorescent product diffusion (e.g. MUF: 4-methylumbelliferone) from plant roots, enzyme activity measurements on the roots, and HYDRUS-2D & HP2 software calibration with obtained data. The calibrated model was used to analyze the processes governing spatial and temporal dynamics of MUF contents in the membrane. The results indicated that the enzyme diffusion within the membrane-soil system was negligible, and measured zymograms were adequately reproduced solely by accounting for substrate and product diffusions and for catalytic enzyme reaction described by the Michaelis-Menten equation. TLZ enabled identifying and using linear parts on MUF time series and accounting for MUF losses in each zymogram pixel, considerably improving the accuracy as compared to the traditional method. Results demonstrated that enzymatic activity from only a thin soil layer (~0.2 mm) is adequately represented in zymograms.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2020

Low-dissipation centred schemes for hyperbolic equations in conservative and non-conservative form

We propose a one-parameter family of low-dissipation centred numerical schemes for solving hyperbolic equations in conservative or non-conservative form, using finite volume or discontinuous Galerkin finite element methods. The new schemes spring out from the multi-dimensional FORCE method and are determined by a single parameter α ≥ 1. Given an increasing sequence of real numbers 1 ≤ α 1 < α 2 < . . . < α K , there corresponds a sequence of numerical schemes with stability restriction associated to a decreasing sequence of Courant numbers 1 > c 1 > c 2 > . . . > c K > 0 and a decreasing sequence of corresponding numerical viscosity functions d 1 > d 2 > . . . > d K . For a given Courant number c k ≤ 1 there is a real number α k > 0 and a corresponding stable scheme with minimal numerical viscosity d k . The proposed schemes suit very well the family of high-order discontinuous Galerkin finite element methods and the recently proposed class of ADER-TR schemes, whose orders of accuracy define decreasing sequences of Courant numbers, as the order of accuracy increases. The centred methods of this paper are stable in 2D and 3D in the frame of simultaneous updating formulae, unlike other centred methods, such as 1D FORCE, which are not. Furthermore, the schemes are highly accurate for slowly-moving waves, which is precisely the kind of waves that traditional centred methods smear disastrously. Additional features of the proposed schemes include ease of implementation and applicability to any hyperbolic system either in conservative or non-conservative form. Here, the proposed schemes are analysed and computationally assessed through a suite of test problems for a linear model system, for the Euler equations in one and two space dimensions, and for the Baer-Nunziato equations for compressible two-phase flow.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Extending Analysis of Asay Windows to High Areal Mass Range [Slides]

The accretion layer addition to Asay window analysis results in new equations of motion that depend on the location of collisions within the layer. Asay window equations of motion can be represented in terms of experiment observables. Balance of linear momentum for variable mass systems provides a framework for developing equations of motion for Asay window diagnostics under a variety of operating assumptions. The growth in the accretion layer thickness during an experiment substantially influences mass and density measurements when high accumulation of areal mass is present.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Extending the Asay window measurement model to high areal mass ranges [Slides]

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Power Flow Geometry and Approximation

Here, the power flow equations are important in numerous power systems problems of practical interest which consider alternating current power flow (ACPF) physics. Perhaps the most well studied being the alternating current optimal power flow problem (ACOPF), seeking to optimize the operation of an electric power system. Due to their non-linearity, problems which include the power flow equations are typically challenging, particularly in optimization. Interestingly, the set of solutions to the power flow equations forms a smooth manifold. As a result, differential geometry can be used to describe and analyze this set of equations. This approach has proven effective in several engineering applications (e.g., solving ACOPF and analyzing the solution space boundary). Central to the success of this approach is an understanding of the power flow manifold's geometry. In this work, we develop the geometric and topological properties of this manifold using concepts from differential geometry. After demonstrating the convenience of this manifold's representation as a function's graph, computational methods are emphasized: we develop retractions, error bounds for linear approximation, and formulas for evaluating the Riemannian metric (including associated objects such as geodesics and the curvature tensor). Scalar curvature and the second fundamental form play a new role in quantifying the quality of linear approximations, like the popular direct current approximation. All functions are implemented in Julia and available in an online repository. Proofs are included for completeness.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2021

A multi-dimensional parametric study of variability in multi-phase flow dynamics during geologic CO 2 sequestration accelerated with machine learning

Successful geologic CO 2 storage projects depend on numerical simulations to predict reservoir performance during site selection, injection verification, and post-injection monitoring phases of the project. These numerical simulations solve non-linear sets of coupled partial differential equations, while accounting for multi-phase fluid dynamics on the basis of constitutive equations that are embedded into the solution scheme. As a consequence, individual simulations often require tens to hundreds of hours to complete on high-performance computing clusters. Moreover, laboratory experiments reveal that parametric functions for capillary pressure and relative permeability exhibit substantial variability, even within the same rock type. This combination of computational expense and wide-ranging parametric variability means that there remains substantial uncertainty in the behavior of multi-phase CO 2 -water systems, particularly in the context of feedbacks between relative permeability and capillary pressure. To bridge this knowledge gap, here we develop a novel workflow that utilizes physics-based numerical simulation to train an artificial neural network (ANN) emulator for interrogating the multivariate parameter space that governs both capillary pressure and relative permeability. With this approach, the ANN is trained to emulate both fluid pressure distribution and CO 2 saturation, which are then interrogated quantitatively to generate parametric response surface mappings with high-fidelity resolution. Results from this study initially show that capillary entry pressure is the dominant control on both CO 2 plume geometry and fluid pressure propagation when considering the combined effects of capillary pressure and relative permeability, particularly when phase interference is low and residual CO 2 saturation is high. Moreover, the ANN emulator provides tremendous computational speed-up by computing 2691 individual simulations in several minutes; whereas, the same simulation ensemble would have required ~3 years of simulation time using only physics-based simulation methods (25,000 times speed up).

58 GEOSCIENCES↗

DOE OSTI2024

Optimal control of the electron temperature profile in DIII-D using machine learning surrogate models

The viability of the tokamak as a potential fusion reactor depends on the ability to keep the plasma in a stable regime while achieving temperatures, densities, and confinement times that are as high as possible. Tokamak scenario development attempts to find plasma regimes that achieve all of these conditions and are accessible with a given set of hardware constraints. This requires the ability to control plasma properties such as the normalized beta, the internal inductance, safety factor, rotation, etc. One property that has received less attention than some of the others, but is no less critical to achieving high performance, is the electron temperature (T e ) profile. In this work, Linear Quadratic Integral (LQI) control is used to develop a controller for the electron temperature profile in DIII-D. The controller is based on a linearized model derived from the transport equation that describes the evolution of the electron temperature, and includes contributions from the neural network surrogate models NubeamNet and MMMnet. Furthermore, the controller is tested in simulation using COTSIM, and is proven capable of tracking a target T e profile.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2023

Novel method for accurately estimating membrane transport properties and mass transfer coefficients in reverse osmosis

Here, we present a simple and robust method to simultaneously characterize the water and salt permeability (A, B) of reverse osmosis (RO) membranes and mass transfer coefficient (k) in membrane modules. The proposed methodology comprises a set of RO experiments performed at different operating pressures or stages. The measured water and salt fluxes in each stage are simultaneously fitted to the RO transport equations by performing a non-linear regression, using A, B, and k as regression parameters. We first perform a systematic accuracy analysis of the proposed method across the full operational range of RO. The assessment shows that the method accuracy is substantially higher than current methods and increases with number of experimental stages and driving forces. This assessment is used to inform the design of an experimental protocol that minimizes errors in estimated A, B, and k. We then evaluate two commercial RO membranes following the new protocol. For both membranes, A and B parameters decrease by 17% and 15% from the dilute solution to seawater concentrations, whereas the k parameter remains constant. Our study demonstrates that the proposed method, informed by data-driven experimental designs, provides a new approach for accurately characterizing transport phenomenon in membrane processes with feeds of less than 100 g/L total dissolved solids.

36 MATERIALS SCIENCE↗

DOE OSTI2021

Resurgence of the QCD Adler function

Not Available

Non-linear ordinary differential equations↗

DOE OSTI2020

(Photo)Electrocatalytic CO 2 Reduction at the Defective Anatase TiO 2 (101) Surface

Excessive carbon dioxide (CO 2 ) emissions by combustion of fossil fuels are linked to global warming and rapid climate change. One promising route to lowering the concentration of CO 2 in the atmosphere is to reduce it to useful small molecules via photoelectrocatalytic hydrogenation, which would enable solar energy storage with a zero-carbon emission cycle and perform a more efficient separation of the photogenerated electron and hole pair than pure photocatalysis. Indeed, photoelectrocatalytic CO 2 reduction has been an intense focus of research. Using the density functional theory (DFT), we studied the CO 2 reduction reaction on the defective anatase TiO 2 (101) surface, at both the solvent/catalyst and the electrolyte/catalyst interfaces. The analysis of the electronic structure of the surface shows a contrast between the solvent/catalyst and the electrolyte/catalyst interfaces, which results in the two corresponding catalytic cycles being distinct. Furthermore, our study explains at the electronic and mechanistic levels why methanol is the main product in the presence of the electrolyte and why the overpotential is not only controlled by the reaction process but also by the diffusion process.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2020

High-speed x-ray phase contrast imaging and digital image correlation analysis of microscale shock response of an additively manufactured energetic material simulant

The performance of energetic materials subjected to dynamic loading significantly depends on their micro- and meso-scale structural morphology. The geometric versatility offered by additive manufacturing opens new pathways to tailor the performance of these materials. Additively manufactured energetic materials (AMEMs) have a wide range of structural characteristics with a hierarchy of length scales and process-inherent heterogeneities, which are hitherto difficult to precisely control. It is important to understand how these features affect AMEMs’ response under dynamic/shock loading. Therefore, temporally and spatially resolved measurements of both macroscopic behavior and micro- and meso-level processes influencing macroscopic behavior are required. In this paper, we analyze the shock compression response of an AMEM simulant loaded under several impact conditions and orientations. Furthermore, x-ray phase contrast imaging (PCI) is used to track features across the observed shock front and determine the linear shock velocity vs particle velocity equation of state, as well as to quantify the interior deformation fields via digital image correlation (DIC) analyses. Photon Doppler velocimetry is simultaneously used to measure the particle velocities of the specimens, which are consistent with those obtained from x-ray PCI. The DIC analyses provide an assessment of the average strain fields inside the material, showing that the average axial strain depends on the loading intensity and reaches as high as 0.23 for impact velocities up to 1.5 km/s. The overall results demonstrate the utility of x-ray PCI for probing “in-material” equation of state and interior strains associated with dynamic shock compression behavior of the AMEM simulant.

3D printing↗

DOE OSTI2025

Algorithmic Advancements for High-Order Self-Gravitating Hydrodynamics

Self-gravity plays a key role in the formation and evolution of many astronomical objects. Though gravity is often dominant at large scales, other forces (e.g., gas pressure gradients, radiation, and/or magnetic fields) often compete. It is therefore essential for numerical simulations to evaluate their interplay accurately and robustly. Hanawa & Mullen derived a 4th-order accurate finite volume scheme to solve the equations of self-gravitating hydrodynamics on a uniform Cartesian grid. In this work, we supply improvements to the algorithm that (1) mitigate spurious gravitational circulation and (2) greatly simplify the evaluation of the high order corrections. The proposed algorithm provides the gravitational acceleration (ρg) and the gravitational energy release (ρv · g) as source terms for the hydrodynamic equations, all while preserving conservation of linear momentum. Spurious heating and/or cooling associated with truncation error in the numerical evaluation of the gravitational energy release decreases in proportion to the fourth power of the cell width. We demonstrate fourth order convergence on smooth problems (e.g., 3D inclined sound wave propagation and 3D equilibria). An application test tracks the spherical collapse of a polytrope by an imposed, sudden decrease of the central gas pressure; a bounce and second collapse (associated with a spherical accretion shock) are robustly captured by the high order algorithm.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2022

Learning Optimal Multigrid Smoothers via Neural Networks

Multigrid methods are one of the most efficient techniques for solving large sparse linear systems arising from partial differential equations (PDEs) and graph Laplacians from machine learning applications. One of the key components of multigrid is smoothing, which aims at reducing high-frequency errors on each grid level. However, finding optimal smoothing algorithms is problem-dependent and can impose challenges for many problems. In this paper, we propose an efficient adaptive framework for learning optimized smoothers from operator stencils in the form of convolutional neural networks (CNNs). Here, the CNNs are trained on small-scale problems from a given type of PDEs based on a supervised loss function derived from multigrid convergence theories and can be applied to large-scale problems of the same class of PDEs. Numerical results on anisotropic rotated Laplacian problems and variable coefficient diffusion problems demonstrate improved convergence rates and solution time compared with classical hand-crafted relaxation methods.

97 MATHEMATICS AND COMPUTING↗