SEARCH · Search NASA

Results for “Nonlinear Solvers”

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.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 289 records · Page 16

All sources NASA NTRS DOE OSTI

DOE OSTI2026

Conditional diffusion machine-learning framework for mapping valence electron distribution from convergent beam electron diffraction

Quantitative convergent beam electron diffraction (CBED) enables determination of aspherical valence electron distributions through refinement of low-order structure factors, which are highly sensitive to chemical bonding and charge density variations. However, conventional quantitative CBED (QCBED) requires solving a highly nonlinear inverse problem with many coupled parameters, and computationally intensive dynamical diffraction calculations, making it time-consuming and difficult to apply to complex systems. More broadly, reconstructing charge density and orbital electron distribution from diffraction data has long been a central challenge in both x-ray and electron crystallography. Here, in this study, we introduce an artificial-intelligence (AI)-based framework that replaces traditional refinement with a data-driven inverse solver. Using a large synthetic CBED dataset generated by Bloch-wave simulations, we train a conditional diffusion model to directly infer crystal structural parameters and multipole density formalism parameters, and hence valence electron distributions, from CBED patterns alone. By learning from forward simulations across realistic parameter space, the model effectively solves the inverse problem. Compared with direct regression approaches, the diffusion-based framework provides posterior parameter distributions for rigorous uncertainty quantification while preserving quantitative fidelity and reducing analysis time by orders of magnitude. By eliminating the need for external single-crystal x-ray diffraction data and complex nonlinear refinement, this approach enables practical, high-throughput, and in situ quantitative CBED, enabling real-time mapping of valence electron distributions and their correlation with functional responses in quantum and energy materials.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

NASA NTRS1974

Fast direct numerical solution of the nonhomogeneous Cauchy-Riemann equations

A fast direct (noniterative) 'Cauchy-Riemann Solver' is developed for solving the finite-difference equations representing systems of first-order elliptic partial differential equations in the form of the nonhomogeneous Cauchy-Riemann equations. The method is second-order accurate and requires approximately the same computer time as a fast cyclic-reduction Poisson solver. The accuracy and efficiency of the direct solver are demonstrated in an application to solving an example problem in aerodynamics: subsonic inviscid flow over a biconvex airfoil. The analytical small-perturbation solution contains singularities, which are captured well by the computational technique. The algorithm is expected to be useful in nonlinear subsonic and transonic aerodynamics.

Lomax, H.↗

NASA NTRS1994

Prediction of Undsteady Flows in Turbomachinery Using the Linearized Euler Equations on Deforming Grids

A linearized Euler solver for calculating unsteady flows in turbomachinery blade rows due to both incident gusts and blade motion is presented. The model accounts for blade loading, blade geometry, shock motion, and wake motion. Assuming that the unsteadiness in the flow is small relative to the nonlinear mean solution, the unsteady Euler equations can be linearized about the mean flow. This yields a set of linear variable coefficient equations that describe the small amplitude harmonic motion of the fluid. These linear equations are then discretized on a computational grid and solved using standard numerical techniques. For transonic flows, however, one must use a linear discretization which is a conservative linearization of the non-linear discretized Euler equations to ensure that shock impulse loads are accurately captured. Other important features of this analysis include a continuously deforming grid which eliminates extrapolation errors and hence, increases accuracy, and a new numerically exact, nonreflecting far-field boundary condition treatment based on an eigenanalysis of the discretized equations. Computational results are presented which demonstrate the computational accuracy and efficiency of the method and demonstrate the effectiveness of the deforming grid, far-field nonreflecting boundary conditions, and shock capturing techniques. A comparison of the present unsteady flow predictions to other numerical, semi-analytical, and experimental methods shows excellent agreement. In addition, the linearized Euler method presented requires one or two orders-of-magnitude less computational time than traditional time marching techniques making the present method a viable design tool for aeroelastic analyses.

Clark, William S.↗

NASA NTRS1998

EUPDF: Eulerian Monte Carlo Probability Density Function Solver for Applications With Parallel Computing, Unstructured Grids, and Sprays

The success of any solution methodology used in the study of gas-turbine combustor flows depends a great deal on how well it can model the various complex and rate controlling processes associated with the spray's turbulent transport, mixing, chemical kinetics, evaporation, and spreading rates, as well as convective and radiative heat transfer and other phenomena. The phenomena to be modeled, which are controlled by these processes, often strongly interact with each other at different times and locations. In particular, turbulence plays an important role in determining the rates of mass and heat transfer, chemical reactions, and evaporation in many practical combustion devices. The influence of turbulence in a diffusion flame manifests itself in several forms, ranging from the so-called wrinkled, or stretched, flamelets regime to the distributed combustion regime, depending upon how turbulence interacts with various flame scales. Conventional turbulence models have difficulty treating highly nonlinear reaction rates. A solution procedure based on the composition joint probability density function (PDF) approach holds the promise of modeling various important combustion phenomena relevant to practical combustion devices (such as extinction, blowoff limits, and emissions predictions) because it can account for nonlinear chemical reaction rates without making approximations. In an attempt to advance the state-of-the-art in multidimensional numerical methods, we at the NASA Lewis Research Center extended our previous work on the PDF method to unstructured grids, parallel computing, and sprays. EUPDF, which was developed by M.S. Raju of Nyma, Inc., was designed to be massively parallel and could easily be coupled with any existing gas-phase and/or spray solvers. EUPDF can use an unstructured mesh with mixed triangular, quadrilateral, and/or tetrahedral elements. The application of the PDF method showed favorable results when applied to several supersonic-diffusion flames and spray flames. The EUPDF source code will be available with the National Combustion Code (NCC) as a complete package.

Raju, M. S.↗

NASA NTRS1987

A full potential flow analysis with realistic wake influence for helicopter rotor airload prediction

A 3-D, quasi-steady, full potential flow solver was adapted to include realistic wake influence for the aerodynamic analysis of helicopter rotors. The method is based on a finite difference solution of the full potential equation, using an inner and outer domain procedure for the blade flowfield to accommodate wake effects. The nonlinear flow is computed in the inner domain region using a finite difference solution method. The wake is modeled by a vortex lattice using prescribed geometry techniques to allow for the inclusion of realistic rotor wakes. The key feature of the analysis is that vortices contained within the finite difference mesh (inner domain) were treated with a vortex embedding technique while the influence of the remaining portion of the wake (in the outer domain) is impressed as a boundary condition on the outer surface of the finite difference mesh. The solution procedure couples the wake influence with the inner domain solution in a consistent and efficient solution process. The method has been applied to both hover and forward flight conditions. Correlation with subsonic and transonic hover airload data is shown which demonstrates the merits of the approach.

Egolf, T. Alan↗

NASA NTRS1985

A Numerical Method of Calculating Propeller Noise Including Acoustic Nonlinear Effects

Using the transonic flow fields(s) generated by the NASPROP-E computer code for an eight blade SR3-series propeller, a theoretical method is investigated to calculate the total noise values and frequency content in the acoustic near and far field without using the Ffowcs Williams - Hawkings equation. The flow field is numerically generated using an implicit three dimensional Euler equation solver in weak conservation law form. Numerical damping is required by the differencing method for stability in three dimensions, and the influence of the damping on the calculated acoustic values is investigated. The acoustic near field is solved by integrating with respect to time the pressure oscillations induced at a stationary observer location. The acoustic far field is calculated from the near field primitive variables as generated by NASPROP-E computer code using a method involving a perturbation velocity potential as suggested by Hawkings in the calculation of the acoustic pressure time-history at a specified far field observed location. the methodologies described are valid for calculating total noise levels and are applicable to any propeller geometry for which a flow field solution is available.

Korkan, K. D.↗

NASA NTRS1989

A natural flow wing design employing 3-D nonlinear analysis applied at supersonic speeds

A wing-design study has been conducted on a 65-deg-swept leading-edge delta wing in which a near-conical geometry was employed to take advantage of the naturally occurring conical flow which arises over such a wing in a supersonic flow field. Three-dimensional nonlinear analysis methods were used in the study. In preliminary design, wing planform, design conditions, and near-conical concept were derived and a baseline standard wing (conventional airfoil distribution) and a baseline near-conical wing were chosen. During the initial analysis, a full-potential solver was employed to determine the aerodynamic characteristics of the baseline standard delta wing and the near-conical delta wing. Modifications due to airfoil thickness, leading-edge radius, and camber were then applied to the baseline near-conical wing. The final design employed a Euler solver to analyze the best wing configurations found in the initial design, and to extend this study to develop a more refined wing. Benefits due to each modification are discussed, and a final natural flow wing geometry is chosen and its aerodynamic characteristics are compared with the baseline wings.

Bauer, Steven X. S.↗

NASA NTRS2020

Wind Tunnel Investigation of the Supersonic Stage Separation Aerodynamics of a Generic 0.0175-Scale Bimese Two-Stage-to-Orbit Reusable Launch Vehicle Configuration

A wind tunnel investigation was conducted of the supersonic stage separation aerodynamics of a generic two-stage-to-orbit bimese wingbody configuration in the NASA Langley Research Center Unitary Plan Wind Tunnel. Proximity and isolated model testing was conducted at Mach numbers of 2.3, 3.0, and 4.5 and a unit Reynolds number of 2.0 million per foot using 0.0175-scale models of the Langley Glide-Back Booster concept designated as the orbiter and booster in belly-to-belly and back-to-belly configurations. Longitudinal forces and moments were obtained on both models and surface static pressure measurements were obtained on the orbiter model at 328 relative proximity locations and at relative angles of attack of 0 degrees and 5 degrees. The test results supported a larger effort to develop and validate experimental and computational tools applicable to the design and simulation of stage separation and abort procedures for reusable launch vehicles composed of multiple bodies, including winged bodies. An initial proof-of-concept experiment featuring low-cost uninstrumented models was conducted to verify an emerging automated model control system and new support system hardware, and to identify potential model and support system blockage and unsteady aerodynamics/model dynamics prior to committing to higher-fidelity instrumented models. This investigation led to upgrades in the facility stage separation hardware, calibration and testing techniques and capabilities, and data analysis and documentation methodologies that have been extended to the more recent NASA Constellation and Space Launch System crew and cargo launch vehicle programs. A virtual diagnostics interface methodology was used to facilitate the design of the stage separation support hardware, to position the models in the test section, and to define the experimental test space. Advances in the facility automated model positioning system established a foundation for the development of a continuous-sweep data acquisition technique that is responsible for significant productivity improvements to the current NASA Space Launch System test program. The automated model positioning capability was leveraged to conduct a companion statistically-designed stage separation experiment requiring randomization of the relative proximity positions of the orbiter and booster models. The respective zones of influence and interference effects of the orbiter and booster were identified from three-dimensional scatter plots, contour and influence maps, and two-dimensional plotting methods. The highly-nonlinear, shock-dominated aerodynamic characteristics of the orbiter and booster in the Unitary Plan Wind Tunnel exhibited good agreement with independent test data obtained in a NASA Marshall Space Flight Center wind tunnel and with computational fluid dynamics predictions using a compressible, three-dimensional flow solver and an inviscid, unstructured Cartesian method.

Erickson, Gary E.↗

DOE OSTI2024

Direct prediction of saturated neoclassical tearing modes in slab using an equilibrium approach

We demonstrate for the first time that the nonlinear saturation of neoclassical tearing modes (NTMs) can be found directly using a variational principle based on Taylor relaxation, without needing to simulate the intermediate, resistivity-dependent dynamics. As in previous investigations of classical tearing mode saturation (Loizu et al 2020 Phys. Plasmas 27 070701; Loizu and Bonfiglio 2023 J. Plasma Phys. 89 905890507), we make use of Stepped Pressure Equilibrium Code (SPEC) (Hudson et al 2012 Phys. Plasmas 19 112502), an equilibrium solver based on the variational principle of the multi-region relaxed magnetohydrodynamics (MHDs), featuring stepped pressure profiles and arbitrary magnetic topology. We work in slab geometry and employ a simple bootstrap current model J bs = C$\boldsymbol{\nabla}$p to study the bootstrap-driven tearing modes, scanning over the asymptotic matching parameter Δ' and bootstrap current strength. Saturated island widths produced by SPEC agree well with the predictions of an initial value resistive MHDs code (Huang and Bhattacharjee 2016 Astrophys. J. 818 20) while being orders of magnitude faster to calculate. Additionally, we observe good agreement with a simple analytical modified Rutherford equation, without requiring any fitting coefficients. The match is obtained for both linearly unstable classical tearing modes in the presence of bootstrap current, and NTMs, which are linearly stable but nonlinear-unstable due to the effects of the bootstrap current.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2024

Assessing the impact of alpha particles on thermal confinement in JET D-T plasmas through global GENE-Tango simulations

The capability of the global, electromagnetic gyrokinetic GENE code interfaced with the transport Tango solver is exploited to address the impact of fusion alpha particles (in their dual role of fast particles and heating source) on plasma profiles and performance at JET in the discharges with the highest quasi-stationary peak fusion power during the DTE2 experimental campaigns. Employing radially global nonlinear electromagnetic GENE-Tango simulations, we compare results with/without alpha particles and alpha heating. Our findings reveal that alpha particles have a negligible impact on turbulent transport, with GENE-Tango converging to similar plasma profiles regardless of their inclusion as a kinetic species in GENE. On the other hand, alpha heating is found to contribute to the peaking of the electron temperature profiles, leading to a 1 keV drop on the on-axis electron temperature when alpha heating is neglected in Tango. The minimal impact of alpha particles on turbulent transport in this JET discharge–despite this being the shot with the highest fusion output–is attributed to the low content of fusion alpha in this discharge. To assess the potential impact of alpha particles on turbulent transport in regimes with higher alpha particle density, as expected in ITER and fusion reactors, we artificially increased the alpha particle concentration to levels expected for ITER. By performing global nonlinear GENE standalone simulations, we found that increasing the alpha particle density beyond five times the nominal value lead to significant overall turbulence destabilization. These results demonstrate that an increased alpha particle concentration can significantly impact transport properties under simulated JET experimental conditions. However, these findings cannot be directly extrapolated to ITER due to the substantial differences in parameters such as plasma size, magnetic field, plasma current, and thermal pressure.

energetic particles↗

NASA NTRS1987

A generalized procedure for constructing an upwind based TVD scheme

A generalized formulation for constructing second- and higher-order accurate TVD (total variation diminishing) schemes is presented. A given scheme is made TVD by limiting antidiffusive flux differences with some linear functions, so-called limiters. The general idea of the formulation and its mathematical proof of Harten's TVD conditions is shown by applying the Lax-Wendroff method to scalar nonlinear equations and a constant-coefficient system of conservation laws. For the system of equations, several definitions are derived for the argument used in the limiter function and present their performance in numerical experiments. The formulation is extended to the nonlinear system. It is demonstrated that the present procedure can easily convert existing central or upwind, and second- or higher-order differencing schemes to preserve monotonicity and yield physically admissible solutions. The formulation is simple mathematically as well as numerically; both matrix-vector multiplication and Riemann solver are avoided. Although the notion of TVD is based on the initial value problem, application to the steady Euler equations of the formulation is also made.

Liou, Meng-Sing↗

NASA NTRS1987

Studies of vortex dominated flows; Proceedings of the Symposium, Hampton, VA, July 9-11, 1985

Papers are presented on waves and bifurcations in vortex filaments, a ring-vortex representation of an axisymmetric vortex sheet, and comparison of experiment with the dynamics of the von Karman vortex trail. Also considered are force-free and loss-free transitions between vortex flow states, a vortex breakdown simulation based on a nonlinear inviscid method, and the prediction of highly vortical flows using an Euler equation model. Other topics include the theory of high-Reynolds-number flow past a blunt body, progress on the calculation of large-scale separation at high Reynolds numbers, and viscous-inviscid interaction solvers and computation of highly separated flows. Papers are also presented on simulation studies of vortex dynamics of a leading edge vortex flap, methods for numerical simulation of leading edge vortex flow, and comparison of measured and computed pitot pressures in a leading edge vortex from a delta wing.

Hussaini, M. Y.↗

NASA NTRS1992

A staggered mesh finite difference scheme for the computation of compressible flows

A simple high resolution finite difference technique is presented to approximate weak solutions to hyperbolic systems of conservation laws. The method does not rely on Riemann problem solvers and is therefore easy to extend to a wide variety of problems. The overall performance (resolution and CPU requirements) is competitive, with other state-of-the-art techniques offering sharp nonoscillatory shocks and contacts. Theoretical results confirm the reliability of the approach for linear systems and nonlinear scalar equations.

Sanders, Richard↗

NASA NTRS1996

Non-linear k-epsilon-v(sup 2)(bar) modeling with application to high-lift

The k-epsilon-v(sup 2)(bar) model has been investigated to quantify its predictive performance on two high-lift configurations: 2D flow over a single-element aerofoil, involving closed-type separation; 3D flow over a prolate spheroid, involving open-type separation. A 'code-friendly' modification has been proposed which enhances the numerical stability, in particular, for explicit and uncoupled flow solvers. As a result of introducing Reynolds-number dependence into a coefficient of the s-equation, the skin-friction distribution for the by-pass transitional flow over a flat plate is better predicted. In order to improve deficiencies arising from the Boussinesq approximation, a nonlinear stress-strain constitutive relation was adopted, in which the only one free constant is calibrated on the basis of DNS data, and the Reynolds-stress anisotropy near the wall is fairly well represented.

Lien, F. S.↗

DOE OSTI2026

Active operator learning with predictive uncertainty quantification for partial differential equations

With the increased prevalence of neural operators being used to provide rapid solutions to partial differential equations (PDEs), understanding the accuracy of model predictions and the associated error levels is necessary for deploying reliable surrogate models in scientific applications. Existing uncertainty quantification (UQ) frameworks employ ensembles or Bayesian methods, which can incur substantial computational costs during both training and inference. Here, we propose a lightweight predictive UQ method tailored for Deep operator networks (DeepONets) that also generalizes to other operator networks. Numerical experiments on linear and nonlinear PDEs demonstrate that the framework’s uncertainty estimates are unbiased and provide accurate out-of-distribution uncertainty predictions with a sufficiently large training dataset. Our framework provides fast inference and uncertainty estimates that can efficiently drive outer-loop analyses that would be prohibitively expensive with conventional solvers. We demonstrate how predictive uncertainties can be used in the context of Bayesian optimization and active learning problems to yield improvements in accuracy and data-efficiency for outer-loop optimization procedures. In the active learning setup, we extend the framework to Fourier Neural Operators (FNO) and describe a generalized method for other operator networks. To enable real-time deployment, we introduce an inference strategy based on precomputed trunk outputs and a sparse placement matrix, reducing evaluation time by more than a factor of five. Our method provides a practical route to uncertainty-aware operator learning in time-sensitive settings.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Domain-decomposition nonlinear manifold reduced order model

This software combines nonlinear-manifold reduced order models (NM-ROMs) with domain decomposition (DD) techniques. NM-ROMs, which utilize a shallow, sparse autoencoder trained with full order model (FOM) snapshot data, approximate the FOM state on a nonlinear manifold. These models offer advantages over linear-subspace ROMs (LS-ROMs) particularly in scenarios with slowly decaying Kolmogorov n-width. However, the training of NM-ROMs involves a number of parameters that scale with the size of the FOM, and storing high-dimensional FOM snapshots can significantly increase the cost of ROM training for extreme-scale problems. To mitigate these costs, the software employs DD to partition the FOM into smaller subdomains, computes NM-ROMs for each, and then integrates these to form a global NM-ROM. This strategy offers multiple benefits: it enables parallel training of subdomain NM-ROMs, reduces the number of parameters needed, decreases the dimensional requirements of subdomain FOM training data, and allows for customization to the unique characteristics of each FOM subdomain. The use of a shallow, sparse autoencoder architecture in each subdomain NM-ROM facilitates the application of hyper-reduction (HR), simplifying the nonlinear complexities and enhancing computational speed. This software marks the inaugural application of NM-ROM combined with HR to a DD problem. It features an algebraic DD reformulation of the FOM, training of NM-ROMs with HR for each subdomain, and employs a sequential quadratic programming (SQP) solver for the evaluation of the coupled global NMROM. The effectiveness of the DD NM-ROM with HR is numerically demonstrated on the 2D steady-state Burgers' equation, showing an order of magnitude improvement in accuracy over the DD LS-ROM with HR.

Diaz, AlejandroN↗

NASA NTRS2024

A Convexification-Based Outer-Approximation Method for Convex and Nonconvex MINLP

The advancement of domain reduction techniques has significantly enhanced the performance of solvers in mathematical programming. This paper delves into the impact of integrating convexification and domain reduction techniques within the Outer-Approximation method. We propose a refined convexification-based Outer-Approximation method alongside a Branch-and-Bound method for both convex and nonconvex Mixed-Integer Nonlinear Programming problems. These methods have been developed and incorporated into the open-source Mixed-Integer Nonlinear Decomposition Toolbox for Pyomo-MindtPy. Comprehensive benchmark tests were conducted, validating the effectiveness and reliability of our proposed algorithms. These tests highlight the improvements achieved by incorporating convexification and domain reduction techniques into the Outer-Approximation and Branch-and-Bound methods.

Optimization↗

NASA NTRS1992

Treating convection in sequential solvers

The treatment of the convection terms in the sequential solver, a standard procedure found in virtually all pressure based algorithms, to compute the flow problems with sharp gradients and source terms is investigated. Both scalar model problems and one-dimensional gas dynamics equations have been used to study the various issues involved. Different approaches including the use of nonlinear filtering techniques and adoption of TVD type schemes have been investigated. Special treatments of the source terms such as pressure gradients and heat release have also been devised, yielding insight and improved accuracy of the numerical procedure adopted.

Shyy, Wei↗