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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 181 records · Page 10
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Optimizing multigrid reduction-in-time and Parareal coarse-grid operators for linear advection

Parallel-in-time methods, such as multigrid reduction-in-time (MGRIT) and Parareal, provide an attractive option for increasing concurrency when simulating time-dependent partial differential equations (PDEs) in modern high-performance computing environments. While these techniques have been very successful for parabolic equations, it has often been observed that their performance suffers dramatically when applied to advection-dominated problems or purely hyperbolic PDEs using standard rediscretization approaches on coarse grids. In this paper, we apply MGRIT or Parareal to the constant-coefficient linear advection equation, appealing to existing convergence theory to provide insight into the typically nonscalable or even divergent behavior of these solvers for this problem. To overcome these failings, we replace rediscretization on coarse grids with improved coarse-grid operators that are computed by applying optimization techniques to approximately minimize error estimates from the convergence theory. Therefore, one of our main findings is that, in order to obtain fast convergence as for parabolic problems, coarse-grid operators should take into account the behavior of the hyperbolic problem by tracking the characteristic curves. Our approach is tested for schemes of various orders using explicit or implicit Runge–Kutta methods combined with upwind-finite-difference spatial discretizations. In all cases, we obtain scalable convergence in just a handful of iterations, with parallel tests also showing significant speed-ups over sequential time-stepping.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Optimal Polynomial Smoothers and One‐Sided V‐Cycles for Poisson Problems

The solution to the Poisson equation arising from the spectral element discretization of the incompressible Navier‐Stokes equations needs robust preconditioning strategies. One such strategy is multigrid. To realize the potential of multigrid methods, effective smoothing strategies are needed. Chebyshev polynomial smoothers, in conjunction with pointwise Jacobi or additive Schwarz methods (ASMs), prove to be an effective smoother. Other polynomial smoothers, however, may provide superior convergence to the multigrid preconditioner. The authors compare the standard Chebyshev polynomial smoothers to both the novel fourth‐kind Chebyshev polynomial smoothers proposed by Lottes as well as smoothers based on the polynomial of best uniform approximation to as proposed by Kraus, Vassilevski, and Zikatanov. At the cost of symmetry, further improvements may be made. For example, a order polynomial smoother on both sides of the V‐cycle may be substituted with an order polynomial smoother on one side at no additional cost. The choice of omitting the postsmoother in favor of higher‐order polynomial presmoothing is advantageous in cases where the multigrid approximation property constant is large. The authors consider a 2D model problem based on finite differences to motivate the choice of polynomial smoother, order, and whether to apply postsmoothing for the target application of high‐order ‐geometric multigrid methods for GPU architectures. Results from both domains demonstrate the substantial improvement of these approaches over the standard Chebyshev polynomial smoother with a symmetric V‐cycle.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Sparsity Applications for Gradient‐Based Optimization of Wind Farms

Optimizing wind farms is essential for designing efficient energy systems, especially as farms grow larger and span multiple sites. However, this optimization becomes increasingly challenging due to the rising computational cost associated with more turbines. Gradient‐based optimization methods scale better than gradient‐free approaches for large problems, but the most computationally expensive component remains the calculation of gradients for the objective function and constraint Jacobians. To address this, we propose leveraging sparsity to accelerate gradient evaluations and reduce the size of the constraint Jacobian. Wind farms naturally exhibit sparsity—many turbines do not influence each other under certain wind directions. However, unlike traditional sparse problems with fixed patterns, wind farm sparsity is dynamic, requiring new strategies to handle changing interactions efficiently. This paper presents a study of sparsity in wind farm optimization and introduces several methods to exploit it. These strategies are tested on multiple farms using the analytic Cumulative Curl model, with gradients computed via automatic differentiation (AD). The same sparsity‐aware techniques are also applicable to finite difference (FD) methods, where they can yield even greater speedups due to the high cost of directional evaluations. Results show that sparse methods achieve up to a 10x speedup with less than ± 5% variance in optimized wake losses compared to traditional methods. These findings suggest that sparsity‐aware optimization not only maintains solution quality but also scales efficiently with farm size, enabling more comprehensive design exploration at reduced computational cost.

17 WIND ENERGY↗
DOE OSTI2020

Quantum Spectral Methods for Differential Equations

Recently developed quantum algorithms address computational challenges in numerical analysis by performing linear algebra in Hilbert space. Such algorithms can produce a quantum state proportional to the solution of a d-dimensional system of linear equations or linear differential equations with complexity poly(logd). While several of these algorithms approximate the solution to within ϵ with complexity poly(log(1/ϵ)), no such algorithm was previously known for differential equations with time-dependent coefficients. In this work, we develop a quantum algorithm for linear ordinary differential equations based on so-called spectral methods, an alternative to finite difference methods that approximates the solution globally. Using this approach, we give a quantum algorithm for time-dependent initial and boundary value problems with complexity poly(logd, log(1/ϵ)).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Non-close-packed plasmonic Bravais lattices through a fluid interface-assisted colloidal assembly and transfer process

Abstract The assembly of colloids at fluid interfaces followed by their transfer to solid substrates represents a robust bottom-up strategy for creating colloidal monolayers over large, macroscopic areas. In this study, we showcase how subtle adjustments in the transfer process, such as varying the contact angle of the substrate and controlling deposition speed and direction, enable the realization of all five two-dimensional Bravais lattices. Leveraging plasmonic core–shell microgels as the building blocks, we successfully engineered non-close-packed plasmonic lattices exhibiting hexagonal, square, rectangular, centered rectangular, and oblique symmetries. Beyond characterizing the monolayer structures and their long-range order, we employed extinction spectroscopy alongside finite difference time domain simulations to comprehensively investigate and interpret the plasmonic response of these monolayers. Additionally, we probed the influence of the refractive index environment on the plasmonic properties by two methods: first, by plasma treatment to remove the microgel shells, and second, by overcoating the resulting gold nanoparticle lattices with a homogeneous refractive index polymer film. Graphical Abstract

Chemistry↗
DOE OSTI2022

Controls on the frequency content of near-source infrasound at open-vent volcanoes: a case study from Volcán Villarrica, Chile

Infrasound signals from open-vent volcanoes can contain information related to that volcano’s eruption dynamics and future activity. Studying a specific volcano’s acoustic signature during periods of relative quiescence is thus important for understanding potential warning signs of impending eruptions. Volcán Villarrica, located in southern Chile, has a long-lived active lava lake that produces continuous infrasound with spectral peaks centered around 1 Hz that vary by + / - ~ 0.2 Hz over day-to-week-long time scales. The infrasound frequency content has been shown to illuminate key volcanic properties such as eruption style and crater shape. Leading up to Villarrica’s most recent paroxysm in 2015, for instance, infrasound spectral changes coincided with a rise in Villarrica’s lava lake level. Quantifying and understanding the regular fluctuation in recorded infrasonic frequencies from Villarrica and other open-vent volcanoes is thus imperative during periods of relative calm. Our experiment entailed a week-long period of crater rim infrasound observations associated with stable, open-vent activity and revealed two independent source processes: spatter bursting events (relatively broadband infrasound) and lava lake-induced tremor (sustained signals peaked at 1 Hz). A comparison of these acoustic signals with results from 3-D finite-difference time-domain wave propagation models shows that while sound speed variability can influence Villarrica’s spectrum, it cannot explain the full extent of the observed frequency excursions. We conclude that source spectrum variability is primarily responsible for the frequency excursions. In conclusion, our work highlights the utility of data collected from a dense distribution of twenty infrasound sensors operating at the summit coupled with numerical modeling of sound radiation.

58 GEOSCIENCES↗
DOE OSTI2024

Numerical analysis of a time discretized method for nonlinear filtering problem with Lévy process observations

Abstract In this paper, we consider a nonlinear filtering model with observations driven by correlated Wiener processes and point processes. We first derive a Zakai equation whose solution is an unnormalized probability density function of the filter solution. Then, we apply a splitting-up technique to decompose the Zakai equation into three stochastic differential equations, based on which we construct a splitting-up approximate solution and prove its half-order convergence. Furthermore, we apply a finite difference method to construct a time semi-discrete approximate solution to the splitting-up system and prove its half-order convergence to the exact solution of the Zakai equation. Finally, we present some numerical experiments to demonstrate the theoretical analysis.

Mathematics↗
DOE OSTI2023

Efficient Multigrid Reduction-in-Time for Method-of-Lines Discretizations of Linear Advection

Parallel-in-time methods for partial differential equations (PDEs) have been the subject of intense development over recent decades, particularly for diffusion-dominated problems. It has been widely reported in the literature, however, that many of these methods perform quite poorly for advection-dominated problems. In this report we analyze the particular iterative parallel-in-time algorithm of multigrid reduction-in-time (MGRIT) for discretizations of constant-wave-speed linear advection problems. We focus on common method-of-lines discretizations that employ upwind finite differences in space and Runge-Kutta methods in time. Using a convergence framework we developed in previous work, we prove for a subclass of these discretizations that, if using the standard approach of rediscretizing the fine-grid problem on the coarse grid, robust MGRIT convergence with respect to CFL number and coarsening factor is not possible. This poor convergence and non-robustness is caused, at least in part, by an inadequate coarse-grid correction for smooth Fourier modes in space-time known as characteristic components. We propose an alternative coarse-grid operator that provides a better correction of these modes. This coarse-grid operator is related to previous work and uses a semi-Lagrangian discretization combined with an implicitly treated truncation error correction. Theory and numerical experiments show the proposed coarse-grid operator yields fast MGRIT convergence for many of the method-of-lines discretizations considered, including for both implicit and explicit discretizations of high order. Parallel results demonstrate speed-up over sequential time-stepping.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

A Particle-in-Cell Method for Plasmas with a Generalized Momentum Formulation, Part II: Enforcing the Lorenz Gauge Condition

In a previous paper Christlieb et al. (A particle-in-cell method for plasmas with a generalized momentum formulation, part I: Model formulation, 2024), we developed a new particle-in-cell (PIC) method for the relativistic Vlasov–Maxwell system in which the electromagnetic fields and the equations of motion for the particles were cast in terms of scalar and vector potentials through a Hamiltonian formulation. This new method evolved the potentials under the Lorenz gauge using integral equation methods. New methods to construct spatial derivatives of the potentials that converge at the same rates as the fields were also presented. The new particle method was compared against standard explicit discretizations, including the well-known FDTD-PIC method, for a range of applications involving sheaths and particle beams. Here, this paper extends this new class of methods by focusing on the enforcement the Lorenz gauge condition in both exact and approximate forms using co-located meshes. A time-consistency property of the proposed field solver for the vector potential form of Maxwell’s equations is established, which is shown to preserve the equivalence between the semi-discrete Lorenz gauge condition and the analogous semi-discrete continuity equation. Using this property, we present three methods to enforce a semi-discrete gauge condition. The first method introduces an update for the continuity equation that is consistent with the discretization of the Lorenz gauge condition. Both the finite difference and spectral implementations satisfy this discrete gauge condition to machine precision. The second approach we propose enforces a semi-discrete continuity equation using the boundary integral solution to the field equations. The potential benefit of this approach is that it eliminates spatial derivatives that appear on the particle data, namely the current density, which is often calculated by linear combinations of low-order spline basis functions. This method is ideally suited to boundary integral equation methods that invert multi-dimensional operators without dimensional splitting techniques and will be the subject of future work. The third approach introduces a gauge correcting method that makes direct use of the gauge condition to modify the scalar potential and uses local maps for both the charge and current densities. This results in a gauge error, as the maps do not enforce the continuity equation. The vector potential coming from the current density is taken to be exact, and using the Lorenz gauge, we compute a correction to the scalar potential that makes the two potentials satisfy the gauge condition. This method also enforces the gauge condition to machine precision. We demonstrate two of the proposed methods in the context of periodic domains. Problems defined on bounded domains, including those with complex geometric features remain an ongoing effort. However, this work shows that it is possible to design computationally efficient methods that can effectively enforce the Lorenz gauge condition in a non-staggered PIC formulation.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

A Particle-in-Cell Method for Plasmas with a Generalized Momentum Formulation, Part I: Model Formulation

Here, this paper formulates a new particle-in-cell method for the Vlasov–Maxwell system. Under the Lorenz gauge condition, Maxwell’s equations for the electromagnetic fields can be written as a collection of scalar and vector wave equations. The use of potentials for the fields motivates the adoption of a Hamiltonian formulation for particles that employs the generalized (conjugate) momentum. A notable advantage offered by the Hamiltonian formulation is the elimination of time derivatives in the Lorenz gauge formulation that are required by the standard Newton–Lorentz treatment of the particles. This allows the fields to retain the full time-accuracy guaranteed by the field solver. The resulting updates for particles require only knowledge of the fields and their spatial derivatives. An analytical method for constructing these spatial derivatives is presented that exploits the underlying integral solution used in the field solver for the wave equations. Moreover, these derivatives are demonstrated to converge at the same rate as the fields in both time and space. The Method of Lines Transpose field solver we consider in this work is globally first-order accurate in time and high-order accurate in space (e.g., fourth- and fifth-order) and belongs to a larger class of methods which are unconditionally stable, can address geometry, and leverage $\mathcal {O}(N)$ fast summation methods for efficiency. We demonstrate the method on several well-established benchmark problems on bounded domains, including a plasma sheath as well as a relativistic particle beam. The efficacy of the proposed formulation is established by comparing with a second-order accurate finite-difference time-domain method that employs a leapfrog time advance for particles and a charge conserving map suitable for bounded domains. The new method shows mesh-independent numerical heating properties even in cases where the plasma Debye length is smaller than the grid spacing. This is an important feature of the new method for problems defined on bounded domains, because it permits the use of coarser grids in space in the representation of the fields. Such a capability has significant implications for the simulation of plasmas in bounded domains with complex geometry, where the ratio between the largest and smallest cells can vary significantly. The use of high-order spatial approximations in the new method also means that fewer grid points are required in order to achieve a fixed accuracy. Our results also suggest that the new method can be used with fewer simulation particles per cell compared to the benchmark explicit method, which permits further computational savings.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Boundary Corrections for Kernel Approximation to Differential Operators

The kernel-based approach to operator approximation for partial differential equations has been shown to be unconditionally stable for linear PDEs and numerically exhibit unconditional stability for non-linear PDEs. These methods have the same computational cost as an explicit finite difference scheme but can exhibit order reduction at boundaries. In previous work on periodic domains, order reduction was addressed, yielding high-order accuracy. The issue addressed in this work is the elimination of order reduction of the kernel-based approach for a more general set of boundary conditions. Further, we consider the case of both first and second order operators. To demonstrate the theory, we provide not only the mathematical proofs but also experimental results by applying various boundary conditions to different types of equations. The results agree with the theory, demonstrating a systematic path to high order for kernel-based methods on bounded domains.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Adaptive sampling quasi-Newton methods for zeroth-order stochastic optimization

Here, we consider unconstrained stochastic optimization problems with no available gradient information. Such problems arise in settings from derivative-free simulation optimization to reinforcement learning. We propose an adaptive sampling quasi-Newton method where we estimate the gradients using finite differences of stochastic function evaluations within a common random number framework. We develop modified versions of a norm test and an inner product quasi-Newton test to control the sample sizes used in the stochastic approximations and provide global convergence results to the neighborhood of a locally optimal solution. We present numerical experiments on simulation optimization problems to illustrate the performance of the proposed algorithm. When compared with classical zeroth-order stochastic gradient methods, we observe that our strategies of adapting the sample sizes significantly improve performance in terms of the number of stochastic function evaluations required.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Vector-level feedforward control of LPBF melt pool area using a physics-based thermal model

Laser powder bed fusion (LPBF) is an additive manufacturing technique that has gained popularity thanks to its ability to produce geometrically complex, fully dense metal parts. However, these parts are prone to internal defects and geometric inaccuracies, stemming in part from variations in the melt pool. Here, this paper proposes a novel vector-level feedforward control framework for regulating melt pool area in LPBF. By decoupling part-scale thermal behavior from small-scale melt pool physics, the controller provides a scale-agnostic prediction of melt pool area and efficient optimization over it. This is done by operating on two coupled lightweight models: a finite-difference thermal model that efficiently captures vector-level temperature fields and a reduced-order, analytical melt pool model. Each model is calibrated separately with minimal single-track and 2D experiments, and the framework is validated on a complex 3D geometry in both Inconel 718 and 316L stainless steel. Results showed that feedforward vector-level laser power scheduling reduced geometric inaccuracy in key dimensions by 62%, overall porosity by 16.5%, and photodiode root-mean-squared deviation by 38.5% on average. Overall, this modular, data-efficient approach demonstrates that proactively compensating for known thermal effects can significantly improve part quality while remaining computationally efficient and readily extensible to other materials and machines.

Additive manufacturing↗
DOE OSTI2024

Neutron transport methods for multiphysics heterogeneous reactor core simulation in Griffin

Griffin is a reactor physics application based on the Multiphysics Object-Oriented Simulation Environment (MOOSE). This work discloses the methods, algorithms, and implementation for simulating heterogeneous reactor dynamics models. Griffin utilizes a discontinuous finite-element method with discrete ordinates (DFEM-S ) to discretize the field variable of the multigroup neutron transport equation. Multiphysics feedback is handled using two-step tabulated cross-section methodology. Feedback quantities are evaluated using the MOOSE-MultiApp system to couple various engineering phenomena, such as heat conduction and thermal fluids. The multiphysics DFEM-S system is solved using fixed-point iteration with a fully asynchronous parallel sweeper, unstructured coarse-mesh finite difference acceleration, and a multi-timescale improved quasi-static method scheme. The implementation is applied to a multiphysics microreactor model, with two transients: one initiated by a single heat-pipe failure and another by control drum rotation. Importantly, these examples demonstrate the ability of Griffin to tractably solve the neutron transport equation considering seven independent variables and feedback.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

The Transient Multi-Level method for Monte Carlo reactor statics calculations

The Transient Multi-Level (TML) method is applied to a time-dependent Monte Carlo transport solver to offload some of the computational burden of the expensive Monte Carlo solve to lower-order Coarse Mesh Finite Difference (CMFD) and Exact Point Kinetics Equations (EPKE) solvers via factorization of the neutron flux at the transport and CMFD levels using the Predictor Corrector Quasi-Static Method (PCQM). The Monte Carlo transient is solved by a modified fission source iteration scheme that introduces a single transient source bank. The method is implemented in the production-level Monte Carlo code, Shift, and verified with prescribed reactivity ramps from the two-dimensional version of the C5G7-TD reactor benchmark. The results show that, as compared to other quasi-static methods, the TML reduces the stochastic noise inherent to the transient Monte Carlo solver by factors of ~2 to 6 for various norm comparisons of the reactor power amplitude. Finally, the TML additionally reduces the number of Monte Carlo evaluations needed to simulate the transient, leading to roughly an order of magnitude improvement in CPU time relative to the standard PCQM for the problems tested.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2024

Computational modeling of microalgal biofilm growth in heterogeneous rotating algal biofilm reactors (RABRs) for wastewater treatment

Rotating algal biofilm reactors (RABRs) are innovative systems designed to cultivate microalgae biofilms efficiently. In this paper, we have developed a novel mathematical model to accurately capture the growth dynamics of algae biofilms within RABR. By considering the spatial heterogeneity of the RABR, we introduce a PDE-based model that addresses the spatial variations across the substratum, enabling a more accurate simulation of biofilm growth in RABRs. The photosynthesis process is modeled through reactive kinetics, driving the growth of the algae biofilm. To analyze the system's behavior, we employ finite difference numerical methods to solve the complex PDE model. We then conduct extensive numerical simulations to understand algae biofilm growth in the RABR environment under various operational factors and environmental conditions. One primary focus in these simulations is to investigate the impact of various harvesting strategies, harvesting frequencies, light intensity, and light exposure on the overall biomass productivity of the algae biofilm. The numerical results provide valuable insights into optimizing algae biofilm growth and designing harvesting techniques in RABR systems. Our proposed novel mathematical model provides an effective platform for the theoretical investigation and design of RABRs for wastewater treatment.

09 BIOMASS FUELS↗
DOE OSTI2025

Constrained or unconstrained? Neural-network-based equation discovery from data

Throughout many fields, practitioners often rely on differential equations to model systems. Yet, for many applications, the theoretical derivation of such equations and/or the accurate resolution of their solutions may be intractable. Instead, recently developed methods, including those based on parameter estimation, operator subset selection, and neural networks, allow for the data-driven discovery of both ordinary and partial differential equations (PDEs), on a spectrum of interpretability. The success of these strategies is often contingent upon the correct identification of representative equations from noisy observations of state variables and, as importantly and intertwined with that, the mathematical strategies utilized to enforce those equations. Specifically, the latter has been commonly addressed via unconstrained optimization strategies. Representing the PDE as a neural network, we propose to discover the PDE (or the associated operator) by solving a constrained optimization problem and using an intermediate state representation similar to a physics-informed neural network (PINN). The objective function of this constrained optimization problem promotes matching the data, while the constraints require that the discovered PDE is satisfied at a number of spatial collocation points. We present a penalty method and a widely used trust-region barrier method to solve this constrained optimization problem, and we compare these methods on numerical examples. Our results on several example problems demonstrate that the latter constrained method outperforms the penalty method, particularly for higher noise levels or fewer collocation points. This work motivates further exploration into using sophisticated constrained optimization methods in scientific machine learning, as opposed to their commonly used, penalty-method or unconstrained counterparts. For both of these methods, we solve these discovered neural network PDEs with classical methods, such as finite difference methods, as opposed to PINNs-type methods relying on automatic differentiation. Here, we briefly highlight how simultaneously fitting the data while discovering the PDE improves the robustness to noise and other small, yet crucial, implementation details.

Data-driven discovery↗
DOE OSTI2025

ExaCA v2.0: A versatile, scalable, and performance portable cellular automata application for additive manufacturing solidification

The previously established ExaCA software for performance portable alloy grain structure simulation has been updated to better represent the solidification behavior during complex alloy processing conditions, such as those encountered during metal additive manufacturing (AM), and for improved performance and scalability. Here, an extension to the time–temperature history input data format and the core ExaCA algorithm to include an arbitrary number of melting and solidification events yielded improved prediction of texture for various melt pool geometries, expanding the range of AM-relevant conditions that can be accurately simulated. Improved heat transport process simulation coupling, including the creation of large raster datasets from single track time–temperature history data and in-memory coupling with the new, performance portable finite difference code Finch, were also demonstrated in example studies on the effect of multilayer AM microstructure predictions on hatch spacing and cell size, respectively. Additional new features are detailed and demonstrated, including the ability to perform simulations using various interfacial response function forms, execute simulations on state-of-the-art hardware, improved usability through post-processing versatility, and improved strong and weak scaling performance. The performance, physics, and versatility improvements demonstrated here will further enable large-scale studies on AM process–microstructure relationships that were not previously possible. Furthermore, the usability improvements and ability to run coupled AM process–microstructure simulations using the Finch-ExaCA workflow will facilitate broader use of this open-source software by the computational materials community.

36 MATERIALS SCIENCE↗