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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 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

The W4 method: A new multi-dimensional root-finding scheme for nonlinear systems of equations

We propose a new class of method for solving nonlinear systems of equations, which, among other things, has four nice features: (i) it is inspired by the mathematical property of damped oscillators, (ii) it can be regarded as a simple extension to the Newton-Raphson (NR) method, (iii) it has the same local convergence as the NR method does, (iv) it has a significantly wider convergence region or the global convergence than that of the NR method. In this article, we present the evidence of these properties, applying our new method to some examples and comparing it with the NR method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Neural Ordinary Differential Equations for Nonlinear System Identification

Neural ordinary differential equations (NODE) have been recently proposed as a promising approach for nonlinear system identification tasks. In this work, we systematically compare their predictive performance with current state-of-the-art nonlinear and classical linear methods. In particular, we present a quantitative study comparing NODE's performance against neural state-space models and classical linear system identification methods and evaluate their inference speed and prediction performance on open-loop errors across eight different dynamical systems. The experiments show that NODEs can consistently improve the prediction accuracy by order of magnitude compared to benchmark methods. Besides improved accuracy, we also observed that NODEs are less sensitive to hyperparameters compared to neural state-space models by paying the cost of increased computation at the inference time.

machine leaning, system identification, physics in↗
DOE OSTI2020

Extended Lagrangian Born–Oppenheimer molecular dynamics using a Krylov subspace approximation

It is shown how the electronic equations of motion in extended Lagrangian Born–Oppenheimer molecular dynamics simulations can be integrated using low-rank approximations of the inverse Jacobian kernel. This kernel determines the metric tensor in the harmonic oscillator extension of the Lagrangian that drives the evolution of the electronic degrees of freedom. The proposed kernel approximation is derived from a pseudoinverse of a low-rank estimate of the Jacobian, which is expressed in terms of a generalized set of directional derivatives with directions that are given from a Krylov subspace approximation. The approach allows a tunable and adaptive approximation that can take advantage of efficient preconditioning techniques. The proposed kernel approximation for the integration of the electronic equations of motion makes it possible to apply extended Lagrangian first-principles molecular dynamics simulations to a broader range of problems, including reactive chemical systems with numerically sensitive and unsteady charge solutions. This can be achieved without requiring exact full calculations of the inverse Jacobian kernel in each time step or relying on iterative non-linear self-consistent field optimization of the electronic ground state prior to the force evaluations as in regular direct Born–Oppenheimer molecular dynamics. We note the low-rank approximation of the Jacobian is directly related to Broyden’s class of quasi-Newton algorithms and Jacobian-free Newton–Krylov methods and provides a complementary formulation for the solution of nonlinear systems of equations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Uncertainty estimation of bifurcated solutions in the Rayleigh–Bénard problem for advanced nuclear reactors applications

Multiphysics models of nuclear reactors frequently comprise nonlinear systems of equations. The nonlinear nature of these models could lead to solution bifurcations, where a small change in a certain parameter, e.g., the thermophysical properties of the coolant, can lead to a sudden change in the system’s behavior. At the point in parameter space where this happens, called a critical point, the Jacobian matrix of the model’s nonlinear operator becomes singular potentially permitting multiple solutions to coexist. In this paper, we perform uncertainty estimation (UE) in a parameter range that includes bifurcated solutions within the context of Rayleigh–Bénard problem. We perform this analysis assuming uncertain temperature difference, and tilt angle for the iterative solution algorithm with a unit Prandtl number (Pr = 1). Also, we perform this analysis under uncertain thermophysical properties for both FLiBe molten salt and liquid sodium as working fluid. We deploy two approaches to compute statistical moments for the resulting distributions of selected flow-field variables. The first approach is the blind computation of the mean and the standard deviation without any consideration of solution bifurcation, while the second approach utilizes k-means clustering to cluster each branch’s solutions together and compute separate statistical moments for each branch. The statistical distributions are obtained by perturbing the selected parameters about nominal values that correspond to a solution on one of the valid branches, and that solution is used as initial guess for the iterative solution algorithm. We found that perturbation of any parameter when its nominal value is close to its critical point always leads to branch jumping, i.e., the iterations converge to a solution on a branch different from the branch of the initial guess. This produces a statistical ensemble comprised of fundamentally different solutions leading to wrong mean values and uncertainty estimates, whereas clustering provides an efficient way to deal with this type of computation. This work is important for developing Gen IV nuclear systems because many of these systems rely on natural convection for cooling especially in accident conditions.

97 - MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Multilevel-in-width training for deep neural network regression

A common challenge in regression is that for many problems, the degrees of freedom required for a high-quality solution also allows for overfitting. Regularization is a class of strategies that seek to restrict the range of possible solutions so as to discourage overfitting while still enabling good solutions, and different regularization strategies impose different types of restrictions. In this paper, we present a multilevel regularization strategy that constructs and trains a hierarchy of neural networks, each of which has layers that are wider versions of the previous network's layers. Here, we draw intuition and techniques from the field of Algebraic Multigrid (AMG), traditionally used for solving linear and nonlinear systems of equations, and specifically adapt the Full Approximation Scheme (FAS) for nonlinear systems of equations to the problem of deep learning. Training through V-cycles then encourage the neural networks to build a hierarchical understanding of the problem. We refer to this approach as multilevel-in-width to distinguish from prior multilevel works which hierarchically alter the depth of neural networks. The resulting approach is a highly flexible framework that can be applied to a variety of layer types, which we demonstrate with both fully connected and convolutional layers. We experimentally show with PDE regression problems that our multilevel training approach is an effective regularizer, improving the generalize performance of the neural networks studied.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Hierarchical Network Partitioning for Solution of Potential-Driven, Steady-State Nonlinear Network Flow Equations

The solution of potential-driven steady-state flow in large networks is a task which manifests in various engineering applications, such as transport of natural gas or water through pipeline networks. The resultant system of nonlinear equations depends on the network topology, and in general, there is no numerical algorithm that offers guaranteed convergence to the solution (assuming a solution exists). Some methods offer guarantees in cases where the network topology satisfies certain assumptions, but these methods fail for larger networks. On the other hand, the Newton-Raphson algorithm offers a convergence guarantee if the starting point lies close to the (unknown) solution. It would be advantageous to compute the solution of the large nonlinear system through the solution of smaller nonlinear sub-systems wherein the solution algorithms (Newton-Raphson or otherwise) are more likely to succeed. Here, this letter proposes and describes such a procedure, a hierarchical network partitioning algorithm that enables the solution of large nonlinear systems corresponding to potential-driven steady-state network flow equations.

42 ENGINEERING↗
DOE OSTI2022

Asynchronous domain decomposition methods for nonlinear PDEs

One- and two-level parallel asynchronous methods for the numerical solution of nonlinear systems of equations, especially those arising from (nonlinear) partial differential equations, are studied. The proposed methods are based on domain decomposition techniques. Local convergence theorems are presented in several cases, with appropriate hypotheses. Computational results on a shared memory multiprocessor machine for various problems exhibiting nonlinearities are reported, illustrating the potential of these asynchronous methods, especially for heterogeneous clusters.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Development and Analysis of Optimal Multilevel Solvers on Advanced Computers. Final Report

Constrained optimization in the context of time dependent, partial differential equations (PDE) leads to a symmetric, block-tridiagonal system of nonlinear equations that must be solved repeatedly in an iterative solution strategy. The blocks represent spatial discretization, while the connection between the blocks represents a forward and backward integration in time. The focus of this project is to apply a parallel-in-time (PiT) solution technique to the large block-triangular system.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

An eigenvalue-based method for computing the relaxed pressure in compressible multiphase flow with N phases

The modeling of compressible multiphase flows is a decades-old area of study with many applications across various fields. Many of these application areas use stiff pressure relaxation. This process involves the solution of a nonlinear system with N + 1 equations and N + 1 unknowns, where N is the number of phases. The resolution of this system with general equations of state (EOSs) is difficult. Furthermore, nonlinear systems can admit multiple solutions, and current solution methods do not address this possibility. Very recently, a thermodynamic relaxation method was introduced, which effectively maps a relatively simple predictor equation of state onto a more complex target equation of state. In this context, the target EOSs are the chosen EOSs for the thermodynamic model. Furthermore, this thermodynamic relaxation has the benefit of simplifying the stiff pressure relaxation system of equations. In this article, we show this system reduces to a polynomial of degree N, which can be recast as an eigenvalue problem through the use of the associated companion matrix. We show that although this eigenvalue method is generally less efficient than Newton–Raphson iteration, it does not suffer from convergence issues and finds all N roots of the polynomial. Hence, the method provides a fail-safe for root-finding iterative methods and a way to address the issue of multiple solutions to the nonlinear system of equations in stiff pressure relaxation.

Eigenvalue algorithm↗
DOE OSTI2020

Recent advances in computational mathematics and applications

We are honored to bring you this special issue dedicated to recent advances in computational mathematics and applications in science and engineering. The papers in this special issue were presented in ”Conference on Computational Mathematics and Applications (CCMA)” held at the University of Nevada Las Vegas (UNLV) during October 25–27, 2019. The conference has attracted about 90 attendees from six countries. The 15 papers in the special issue comprise a diverse collection of theoretical numerical analysis as well as applications in various subjects. Topics cover optimal control of PDEs, discontinuous Galerkin methods, fluid flow in porous media, turbulence flow, static and moving interface problems, complex fluids composed by the mixture of Newtonian fluid and nematic (liquid crystal) flows, modeling of ocean–atmosphere system, image segmentation algorithm, uncertainty quantification problems for turbulence model and Maxwell’s equations, iterative solver for systems resulting from mixed methods, and novel methods for solving systems of nonlinear equations. As the Guest Editors we would like to thank Dr. Leland Jameson at NSF (National Science Foundation) for kindly supporting our conference proposal which made this conference possible. We would also like to thank Darren Sugrue from Elsevier who provided partial support for our conference, and Thennarasu Gunasekaran and his production team from Elsevier who produced this nice special issue. We are grateful to many people (Dr. Zhijian Wu, Lori Ornelas, Elsa Juarez, and many Ph.D. students) at the Department of Mathematical Sciences of UNLV who provided tremendous support for the conference. Finally, we very much appreciate all authors who contributed their precious results to this special issue.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Accurate and Efficient Parametric Model-Order Reduction for Turbulent Thermal Transport

This project produced new algorithms and software for low-cost exploration of turbulent thermal-fluids behavior under parametric variation by using reduced-order models (ROMs). The ROMs numerically solve the governing equations for fluid motion by using a small set of basis functions (typically, N=20-200 modes) to represent the solution. The base modes are computed as optimal combinations of solutions from expensive high-fidelity "anchor-point" solutions involving millions of unknowns, which are typically generated by solving the full Navier-Stokes equations on a supercomputer. The ROM solution is itself a combination of the base modes, where the basis coefficients are determined by evolving an NxN system of nonlinear equations. The overarching idea is to use the inexpensive ROM to predict solutions under conditions where the parameters differ from the anchor-point conditions. Several ingredients are required to make ROMs useful for thermal hydraulics analysis. These include: a stable and accurate ROM that is capable of reproducing the large-scale dynamics of turbulent flow, error indicators than can guide the choice of anchor points, and low-cost mechanisms for evaluating nonlinear terms in the reduced equations.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2020

DESC: A stellarator equilibrium solver

In this paper, the new code DESC is presented to solve for fixed-boundary ideal magnetohydrodynamic equilibria in stellarators. The approach directly solves the equilibrium force balance as a system of nonlinear equations in the form f(x) = 0. The independent variables x represent nested magnetic flux surfaces expressed in the inverse representation with toroidal flux coordinates, and the equations f(x) quantify equilibrium force balance errors at discrete points in real space. Discretizing with global Fourier–Zernike basis functions properly treats the magnetic axis and minimizes the number of coefficients needed to describe the flux surfaces. The pseudospectral method provides great flexibility in where the errors are evaluated, and the system of equations is efficiently solved with a Newton–Raphson iteration. Equilibria are computed and compared against VMEC for both axisymmetric and non-axisymmetric examples. The results show fast convergence rates and solutions with low errors throughout the plasma volume.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Tusas: A fully implicit parallel approach for coupled phase-field equations

In this study, we develop a fully-coupled, fully-implicit approach for phase-field modeling of solidification in metals and alloys. Predictive simulation of solidification in pure metals and metal alloys remains a significant challenge in the field of materials science, as microstructure formation during the solidification process plays a critical role in the properties and performance of the solid material. Our simulation approach consists of a finite element spatial discretization of the fully-coupled nonlinear system of partial differential equations at the microscale, which is treated implicitly in time with a preconditioned Jacobian-free Newton-Krylov method. The approach is algorithmically scalable as well as efficient due to an effective preconditioning strategy based on algebraic multigrid and block factorization. We implement this approach in the open-source Tusas framework, which is a general, flexible tool developed in C++ for solving coupled systems of nonlinear partial differential equations. The performance of our approach is analyzed in terms of algorithmic scalability and efficiency, while the computational performance of Tusas is presented in terms of parallel scalability and efficiency on emerging heterogeneous architectures. We demonstrate that modern algorithms, discretizations, and computational science, and heterogeneous hardware provide a robust route for predictive phase-field simulation of microstructure evolution during additive manufacturing.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Linear and Nonlinear Solvers for Simulating Multiphase Flow within Large-Scale Engineered Subsurface Systems

Simulation of multiphase flow in the subsurface is well-known to be computationally challenging. While there have been many studies that have explored approaches to overcoming these challenges, they often utilize relatively simple case studies. In this paper, we focus on the unique numerical challenges posed by modeling large-scale engineered subsurface systems, characterized by discrete features embedded in a heterogeneous natural subsurface setting. The man-made features such as shafts, tunnels, and barriers often cause multiple challenges in modeling the domain for multiphase porous media flow. This flow scenario can have a wide range of applications such as nuclear waste repositories, enhanced recovery of a petroleum reservoir, geothermal engineering, and carbon sequestration. An example of these severe numerical challenges is the case of performance assessment (PA) for Waste Isolation Pilot Plant (WIPP), the only operating deep geological repository in the US, which simulates extreme material properties of bedded salt rock formation and extreme contrast due to open excavation next to the formation. The models have extremes not only of permeability and porosity but also of the constitutive models needed for multiphase flow; additionally, they have process models like salt creep closure reducing porosity over time, fracturing in clay and anhydrite interbeds of the bedded salt, gas generation from the waste materials, and unintentional human borehole intrusions in some scenarios. Numerical simulations require the solution of coupled systems of nonlinear PDEs; in our work, we use the open-source simulator PFLOTRAN which is based on Finite Volume discretization. The solution of the nonlinear equations requires use of the Newton-Raphson iteration at each time step, which entails the solution of the linearized Jacobian system at each iteration. The effects of all the processes (i.e., large number of unknowns, highly nonlinear constitutive relations, large contrasts in material properties in short distances) lead to an ill-conditioned Jacobian matrix that severely challenges traditional linear solver, i.e., stabilized biconjugate gradient with block Jacobi incomplete LU preconditioner (BCGS-ILU) leading to non-convergence for traditional Newton-Raphson nonlinear solver causing unacceptably long computation time for each model. This paper presents linear solvers such as constrained pressure residual (CPR) two-stage preconditioner with alternate-block-factorization (ABF) and quasi- implicit pressure and explicit saturation (QIMPES) decouplers and flexible generalized residual solver (FGMRES). The new general-purpose nonlinear solver, Newton trust-region dogleg Cauchy (NTRDC), is also introduced to resolve extreme nonlinearities in the models. We demonstrate the effectiveness of each method relative to the default BCGS-Newton solver. The two best cases had nearly 50 times speed-up and achieved completion of a simulation in 14 hours that never completed due to non-convergence with the default solver. We also investigate the strong scalability of each method and discuss some of the deficiencies found for Block Jacobi preconditioner using parallel domain decomposition, and node packing effects of modern processor architecture.

Preconditioner, Nonlinear, Porous media, Multiphas↗
DOE OSTI2021

A moving discontinuous Galerkin finite element method with interface condition enforcement for compressible flows

A variation of moving discontinuous Galerkin finite element method with interface condition enforcement (MDG-ICE) is developed for solving the compressible Euler equations. The MDG-ICE method, originating from the work of Corrigan et al. [1], [2], [3], [4], is based on the space-time DG formulation, where both flow field and grid geometry are considered as independent variables and the conservation laws are enforced both on discrete elements and element interfaces. The element conservation laws are solved in the standard discontinuous solution space to determine conservative quantities, while the interface conservation is enforced using a variational formulation in a continuous space to determine discrete grid geometry. The resulting over-determined system of nonlinear equations arising from the MDG-ICE formulation can then be solved in a least-squares sense, leading to an unconstrained nonlinear least-squares problem that is regularized and solved by Levenberg-Marquardt method. A number of numerical experiments for both 1D unsteady and 2D steady state compressible flow problems are conducted to assess the accuracy and robustness of the MDG-ICE method. Numerical results obtained indicate that the MDG-ICE method is able to implicitly detect and track all types of discontinuities via interface conservation enforcement and satisfy the conservation law on both elements and interfaces via grid movement and grid management, demonstrating that an exponential rate of convergence for Sod and Lax-Harden shock tube problems can be achieved and highly accurate solutions without overheating to both double-rarefaction wave and Noh problems can be obtained.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Enhancing Photosynthesis Simulation Performance in ESMs with Machine Learning-Assisted Solvers

When simulating vegetation dynamics, photosynthesis accounts for a large fraction of the computational cost in most Earth System Models (ESMs). This is largely since photosynthesis is represented as a system of nonlinear equations, and the solution requires the use of an initial guess followed by many iterations of the numerical solver to obtain a solution. We use machine learning (ML) to replicate the response surface of the model’s numerical solver to improve the choice of initial guess, therefore requiring fewer iterations to obtain a final solution. We implemented this test on the leaf-level calculations as well as at the canopy scale, and for both we observed fewer iterations of the photosynthesis solver when a ML-based initial guess was implemented. The model tested here is the Energy Exascale Earth System Model - Land Model (ELM). The ML-based algorithms used here are trained on simulations from the model itself and used only to improve the initial guess for the solver; therefore, the model maintains its own set of physics to obtain the final solution. This work shows novel ways to utilize ML-based methods to improve the performance of numerical solvers in ESMs.

Massoud, Elias [ORNL] (ORCID:0000000217725361)↗
DOE OSTI2025

Complex Dependence of Calcite Crack Kinetics on Salinity: The Role of DLVO and Hydration Forces

Abstract Subcritical crack growth (SCG) plays an important role in many geological processes such as delayed earth rupture and rock weathering. The complex dependency of SCG on the in‐crack fluid chemistry, however, is still poorly understood. In this study, we utilize the newly developed surface force‐based fracture theory (SFFT) to elucidate the relative contributions of surface forces and solute transport to the crack growth kinetics of calcite in NaCl solutions. Expanding on Barenblatt's cohesive crack model, SFFT introduces an effective stress intensity at the crack tip that encompasses all the relevant intermolecular forces across the crack in addition to the external far‐field stresses. The nonlinear system of equations portraying the crack opening profile, the solute distribution in a propagating crack, and the crack growth velocity are numerically solved via an implicit scheme. After carefully calibrating the model for calcite‐water systems, the SFFT is used to predict the SCG response of calcite at different NaCl concentrations, based on various hypotheses. These predictions are then compared to existing SCG data from the literature. We demonstrate that the experimentally observed variation of SCG rate with NaCl concentration cannot be explained solely by DLVO forces (electrostatic and Van der Waals interactions). This can be remediated by introducing an exponentially decaying hydration force with a nonlinear, nonmonotonic dependence on NaCl concentration. Furthermore, we demonstrate that accounting for both diffusive and advective transport of ions is important in explaining the absence of a stage‐II SCG response for calcite in electrolyte solutions. Plain Language Summary Subcritical crack growth (SCG) refers to the slow propagation of cracks in materials under a stress below the threshold for catastrophic failure. SCG is a key process in many geological events, for example, delayed earth ruptures and rock weathering. New initiatives such as underground CO 2 and H 2 storage in carbonate reservoirs further call for better understanding of SCG in carbonate minerals subjected to varying fluid chemistry. This study examines the SCG of calcite, a key mineral found in carbonate rocks, intergranular cement in sandstones, and filling material in mineral veins and faults, determining their deformation and strength. A mathematical model is developed to describe how the crack opens and propagates, how solutes (like salts) distribute within the crack, and how the crack surfaces interact with each other. We used the model to predict calcite SCG in water at different salt concentrations and compared it with experimental data. Our results revealed that the hydration force is the dominating factor in determining the complex, non‐linear dependency of SCG on salinity. We also found that both the movement of ions by diffusion and by bulk water flow are crucial for explaining the SCG rates, especially when the cracks grow quickly. Key Points Surface Force‐Based Fracture Theory predicts the complex subcritical crack growth patterns of calcite crystals immersed in NaCl solutions Results highlight the dominant role of hydration forces in altering the fracture behavior of calcite compared to VdW and electric double‐layer forces Advective solute transport explains the absence of stages‐II and ‐III subcritical crack growth responses in solid‐liquid systems

DLVO↗
DOE OSTI2024

Classifying photonic topology using the spectral localizer and numerical K -theory

Recently, the spectral localizer framework has emerged as an efficient approach for classifying topology in photonic systems featuring local nonlinearities and radiative environments. In nonlinear systems, this framework provides rigorous definitions for concepts such as topological solitons and topological dynamics, where a system’s occupation induces a local change in its topology due to nonlinearity. For systems embedded in radiative environments that do not possess a shared bulk spectral gap, this framework enables the identification of local topology and shows that local topological protection is preserved despite the lack of a common gap. However, as the spectral localizer framework is rooted in the mathematics of C*-algebras, and not vector bundles, understanding and using this framework requires developing intuition for a somewhat different set of underlying concepts than those that appear in traditional approaches for classifying material topology. In this tutorial, we introduce the spectral localizer framework from a ground-up perspective and provide physically motivated arguments for understanding its local topological markers and associated local measure of topological protection. In doing so, we provide numerous examples of the framework’s application to a variety of topological classes, including crystalline and higher-order topology. We then show how Maxwell’s equations can be reformulated to be compatible with the spectral localizer framework, including the possibility of radiative boundary conditions. To aid in this introduction, we also provide a physics-oriented introduction to multi-operator pseudospectral methods and numerical K-theory, two mathematical concepts that form the foundation for the spectral localizer framework. Finally, we provide some mathematically oriented comments on the C*-algebraic origins of this framework, including a discussion of real C*-algebras and graded C*-algebras that are necessary for incorporating physical symmetries. Looking forward, we hope that this tutorial will serve as an approachable starting point for learning the foundations of the spectral localizer framework.

97 MATHEMATICS AND COMPUTING↗