SEARCH · Search NASA

Results for “Fast direct 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 55 records · Page 3

All sources NASA NTRS DOE OSTI

DOE OSTI2024

A Fast Algebraic Multigrid Solver and Accurate Discretization for Highly Anisotropic Heat Flux I: Open Field Lines

We present a novel solver technique for the anisotropic heat flux equation, aimed at the high level of anisotropy seen in magnetic confinement fusion plasmas. Such problems pose two major challenges: (i) discretization accuracy and (ii) efficient implicit linear solvers. We simultaneously address each of these challenges by constructing a new finite element discretization with excellent accuracy properties, tailored to a novel solver approach based on algebraic multigrid (AMG) methods designed for advective operators. We pose the problem in a mixed formulation, introducing the directional temperature gradient as an auxiliary variable. The temperature and auxiliary fields are discretized in a scalar discontinuous Galerkin space with upwinding principles used for discretizations of advection. We demonstrate the proposed discretization’s superior accuracy over other discretizations of anisotropic heat flux, achieving error 1000x smaller for anisotropy ratio of 10 9 , for closed field lines. The block matrix system is reordered and solved in an approach where the two advection operators are inverted using AMG solvers based on approximate ideal restriction, which is particularly efficient for upwind discontinuous Galerkin discretizations of advection. To ensure that the advection operators are nonsingular, in this paper we restrict ourselves to considering open (acyclic) magnetic field lines for the linear solvers. We demonstrate fast convergence of the proposed iterative solver in highly anisotropic regimes where other diffusion-based AMG methods fail.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1979

A fast Cauchy-Riemann solver

The inhomogeneous Cauchy-Riemann equations in a rectangle are discretized by a finite difference approximation. Several different boundary conditions are treated explicitly, leading to algorithms which have overall second-order accuracy. All boundary conditions with either u or v prescribed along a side of the rectangle can be treated by similar methods. The algorithms presented here have nearly minimal time and storage requirements and seem suitable for development into a general-purpose direct Cauchy-Riemann solver for arbitrary boundary conditions.

Ghil, M.↗

DOE OSTI2025

Conjugate Heat Transfer Modeling of Salt-Filled Fuel Pins for Stable Salt Reactor Safety Analysis

The Stable Salt Reactor (SSR) combines the proven structural design of light water reactor fuel assemblies with the inherent safety and fuel-cycle advantages of molten salt technology. In its fast reactor configuration, the SSR utilizes recycled nuclear waste as fuel, sealed within narrow salt-filled fuel pins and cooled by a surrounding liquid salt coolant. Reliable transfer of heat from the molten fuel salt through the cladding to the external coolant is essential for both reactor safety and performance. This work investigates conjugate heat transfer (CHT) in the SSR’s salt-filled fuel pins using NekRS, a high-fidelity spectral element computational fluid dynamics (CFD) solver. The analyses capture internal natural convection within the molten fuel salt and external forced convection in the coolant, under steady-state and transient operating conditions. Parametric studies evaluate how variations in reactor power and coolant flow rate influence heat transfer distributions and system response. The high-fidelity CFD results are time-averaged and post-processed for direct comparison with moderate-fidelity Reynolds-averaged Navier–Stokes (RANS) models, and for the development of reduced-order models within the SAM system code. These validated models support fast-running safety analyses of normal and off-normal transients, improving predictive capability for key safety margins. By integrating advanced CFD with system-level safety tools, this study strengthens the modeling framework for SSR design, reduces uncertainty in molten salt CHT simulations, and accelerates the engineering and licensing of next-generation nuclear reactors.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

NASA NTRS1975

A fast semidirect method for computing transonic aerodynamic flows

A fast, semidirect, iterative computational method, previously introduced for finite-difference solution of subsonic and slightly supercritical flow over airfoils, is extended both to apply to strongly supercritical conditions and to include full second-order accuracy in computing inviscid flows over airfoils. The nonlinear small-disturbance equations are solved iteratively by a direct, linear, elliptic solver. General, fully conservative, type-dependent difference equations are formulated, including parabolic- and shock-point transition operators that provide consistency with the integral conservation laws. These equations specialize to either first-order or to fully second-order-accurate equations. Various free parameters are evaluated for rapid convergence of the first-order scheme. Resulting pressure distributions and computing times are compared with the improved Murman-Cole line-relaxation method.

Martin, E. D.↗

DOE OSTI2024

Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes

In recent years, there has been widespread adoption of machine learning-based approaches to automate the solving of partial differential equations (PDEs). Among these approaches, Gaussian processes (GPs) and kernel methods have garnered considerable interest due to their flexibility, robust theoretical guarantees, and close ties to traditional methods. They can transform the solving of general nonlinear PDEs into solving quadratic optimization problems with nonlinear, PDE-induced constraints. However, the complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel, and its partial derivatives, a result of the PDE constraint and for which fast algorithms are scarce. The primary goal of this paper is to provide a near-linear complexity algorithm for working with such kernel matrices. We present a sparse Cholesky factorization algorithm for these matrices based on the near-sparsity of the Cholesky factor under a novel ordering of pointwise and derivative measurements. The near-sparsity is rigorously justified by directly connecting the factor to GP regression and exponential decay of basis functions in numerical homogenization. We then employ the Vecchia approximation of GPs, which is optimal in the Kullback-Leibler divergence, to compute the approximate factor. This enables us to compute ϵ-approximate inverse Cholesky factors of the kernel matrices with complexity O(N log d (N/ϵ)) in space and O(N log 2d (N/ϵ)) in time. We integrate sparse Cholesky factorizations into optimization algorithms to obtain fast solvers of the nonlinear PDE. We numerically illustrate our algorithm’s near-linear space/time complexity for a broad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, and Monge-Ampère equations. In summary, we provide a fast, scalable, and accurate method for solving general PDEs with GPs and kernel methods.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1991

Parallel variable-band Choleski solvers for computational structural analysis applications on vector multiprocessor supercomputers

A Choleski method used to solve linear systems of equations that arise in large scale structural analyses is described. The method uses a novel variable-band storage scheme and is structured to exploit fast local memory caches while minimizing data access delays between main memory and vector registers. Several parallel implementations of this method are described for the CRAY-2 and CRAY Y-MP computers demonstrating the use of microtasking and autotasking directives. A portable parallel language, FORCE, is also used for two different parallel implementations, demonstrating the use of CRAY macrotasking. Results are presented comparing the matrix factorization times for three representative structural analysis problems from runs made in both dedicated and multi-user modes on both the CRAY-2 and CRAY Y-MP computers. CPU and wall clock timings are given for the various parallel methods and are compared to single processor timings of the same algorithm. Computation rates over 1 GIGAFLOP (1 billion floating point operations per second) on a four processor CRAY-2 and over 2 GIGAFLOPS on an eight processor CRAY Y-MP are demonstrated as measured by wall clock time in a dedicated environment. Reduced wall clock times for the parallel methods relative to the single processor implementation of the same Choleski algorithm are also demonstrated for runs made in multi-user mode.

Poole, E. L.↗

NASA NTRS1989

Assessment of numerical methods for the solution of fluid dynamics equations for nonlinear resonance systems

The capability of accurate nonlinear flow analysis of resonance systems is essential in many problems, including combustion instability. Classical numerical schemes are either too diffusive or too dispersive especially for transient problems. In the last few years, significant progress has been made in the numerical methods for flows with shocks. The objective was to assess advanced shock capturing schemes on transient flows. Several numerical schemes were tested including TVD, MUSCL, ENO, FCT, and Riemann Solver Godunov type schemes. A systematic assessment was performed on scalar transport, Burgers' and gas dynamic problems. Several shock capturing schemes are compared on fast transient resonant pipe flow problems. A system of 1-D nonlinear hyperbolic gas dynamics equations is solved to predict propagation of finite amplitude waves, the wave steepening, formation, propagation, and reflection of shocks for several hundred wave cycles. It is shown that high accuracy schemes can be used for direct, exact nonlinear analysis of combustion instability problems, preserving high harmonic energy content for long periods of time.

Przekwas, A. J.↗

NASA NTRS1981

Shock-free configurations in two-and three-dimensional transonic flow

Efforts to replace Sobieczky's complicated analog computations of solutions to the hodograph equations by a fast elliptic solver in order to generate shock-free airfoil designs more effectively are described. The indirect design of airfoil and wing shapes that are free from shock waves even though the local flow velocity exceeds the speed of sound is described. The problem of finding an airfoil in two dimensional, irrotational flow that has a prescribed pressure distribution is as addressed. Sobieczky's suggestion to use a fictitious gas for finding shock-free airfoils directly in the physical plane was the basis for a more efficient procedure for achieving the same end.

Seebass, A. R.↗

NASA NTRS2001

Towards Optimal Multigrid Efficiency for the Navier-Stokes Equations

A fast multigrid solver for the steady incompressible Navier-Stokes equations is presented. Unlike time-marching schemes, this approach uses relaxation of the steady equations. Application of this method results in a discretization that correctly distinguishes between the advection and elliptic parts of the operator, allowing efficient smoothers to be constructed. Numerical solutions are shown for flow over a flat plate and a Karman-Trefftz airfoil. Using collective Gauss-Seidel line relaxation in both the vertical and horizontal directions, multigrid convergence behavior approaching that of O(N) methods is achieved. The computational efficiency of the numerical scheme is compared with that of a Runge-Kutta based multigrid method.

Swanson, R. C.↗

NASA NTRS2004

Textbook Multigrid Efficiency for the Steady Euler Equations

A fast multigrid solver for the steady incompressible Euler equations is presented. Unlike time-marching schemes, this approach uses relaxation of the steady equations. Application of this method results in a discretization that correctly distinguishes between the advection and elliptic parts of the operator, allowing efficient smoothers to be constructed. Solvers for both unstructured triangular grids and structured quadrilateral grids have been written. Computations for channel flow and flow over a nonlifting airfoil have computed. Using Gauss-Seidel relaxation ordered in the flow direction, textbook multigrid convergence rates of nearly one order-of-magnitude residual reduction per multigrid cycle are achieved, independent of the grid spacing. This approach also may be applied to the compressible Euler equations and the incompressible Navier-Stokes equations.

Roberts, Thomas W.↗

DOE OSTI2024

Rethinking materials simulations: Blending direct numerical simulations with neural operators

Abstract Materials simulations based on direct numerical solvers are accurate but computationally expensive for predicting materials evolution across length- and time-scales, due to the complexity of the underlying evolution equations, the nature of multiscale spatiotemporal interactions, and the need to reach long-time integration. We develop a method that blends direct numerical solvers with neural operators to accelerate such simulations. This methodology is based on the integration of a community numerical solver with a U-Net neural operator, enhanced by a temporal-conditioning mechanism to enable accurate extrapolation and efficient time-to-solution predictions of the dynamics. We demonstrate the effectiveness of this hybrid framework on simulations of microstructure evolution via the phase-field method. Such simulations exhibit high spatial gradients and the co-evolution of different material phases with simultaneous slow and fast materials dynamics. We establish accurate extrapolation of the coupled solver with large speed-up compared to DNS depending on the hybrid strategy utilized. This methodology is generalizable to a broad range of materials simulations, from solid mechanics to fluid dynamics, geophysics, climate, and more.

36 MATERIALS SCIENCE↗

NASA NTRS1994

Technical report series on global modeling and data assimilation. Volume 2: Direct solution of the implicit formulation of fourth order horizontal diffusion for gridpoint models on the sphere

High order horizontal diffusion of the form K Delta(exp 2m) is widely used in spectral models as a means of preventing energy accumulation at the shortest resolved scales. In the spectral context, an implicit formation of such diffusion is trivial to implement. The present note describes an efficient method of implementing implicit high order diffusion in global finite difference models. The method expresses the high order diffusion equation as a sequence of equations involving Delta(exp 2). The solution is obtained by combining fast Fourier transforms in longitude with a finite difference solver for the second order ordinary differential equation in latitude. The implicit diffusion routine is suitable for use in any finite difference global model that uses a regular latitude/longitude grid. The absence of a restriction on the timestep makes it particularly suitable for use in semi-Lagrangian models. The scale selectivity of the high order diffusion gives it an advantage over the uncentering method that has been used to control computational noise in two-time-level semi-Lagrangian models.

Max J. Suarez↗

NASA NTRS1991

Parallel-vector computation for linear structural analysis and non-linear unconstrained optimization problems

Several parallel-vector computational improvements to the unconstrained optimization procedure are described which speed up the structural analysis-synthesis process. A fast parallel-vector Choleski-based equation solver, pvsolve, is incorporated into the well-known SAP-4 general-purpose finite-element code. The new code, denoted PV-SAP, is tested for static structural analysis. Initial results on a four processor CRAY 2 show that using pvsolve reduces the equation solution time by a factor of 14-16 over the original SAP-4 code. In addition, parallel-vector procedures for the Golden Block Search technique and the BFGS method are developed and tested for nonlinear unconstrained optimization. A parallel version of an iterative solver and the pvsolve direct solver are incorporated into the BFGS method. Preliminary results on nonlinear unconstrained optimization test problems, using pvsolve in the analysis, show excellent parallel-vector performance indicating that these parallel-vector algorithms can be used in a new generation of finite-element based structural design/analysis-synthesis codes.

Nguyen, D. T.↗

NASA NTRS2004

Evaluation of a Multigrid Scheme for the Incompressible Navier-Stokes Equations

A fast multigrid solver for the steady, incompressible Navier-Stokes equations is presented. The multigrid solver is based upon a factorizable discrete scheme for the velocity-pressure form of the Navier-Stokes equations. This scheme correctly distinguishes between the advection-diffusion and elliptic parts of the operator, allowing efficient smoothers to be constructed. To evaluate the multigrid algorithm, solutions are computed for flow over a flat plate, parabola, and a Karman-Trefftz airfoil. Both nonlifting and lifting airfoil flows are considered, with a Reynolds number range of 200 to 800. Convergence and accuracy of the algorithm are discussed. Using Gauss-Seidel line relaxation in alternating directions, multigrid convergence behavior approaching that of O(N) methods is achieved. The computational efficiency of the numerical scheme is compared with that of Runge-Kutta and implicit upwind based multigrid methods.

Swanson, R. C.↗

NASA NTRSDate not supplied

Enabling Interoperability in Earth System Digital Twins (ESDT): Integrating Observations, Models, and AI for Actionable Insights Through NASA'S Intelligent Systems Technology Program

NASA’s Intelligent Systems Technology Program (IST) is driving a paradigm shift in Earth science through the development of Earth System Digital Twins (ESDT). These integrated information systems create a dynamic "digital replica" of the Earth by harmonizing continuous, multi-source observations with high-fidelity models and state-of-the-art artificial intelligence (AI) that enable “What now?”, “What next?”, and “What if?” scenario building. These scenarios are reflected in NASA IST’s series of ESDTs, from the Coastal Zone Digital Twin that integrates complex data on the current state of the Chesapeake Bay to the Terrestrial Environmental Rapid-Replication and Assimilation Hydrometeorological (TerraHydro) AI-based ESDT that forecasts water movement across Earth’s surface, to the Agriculture Land Information System (AgLIS) which can be used to assess optimal planting dates and crop yield estimates. By bridging the gap between vast data archives and actionable insights, these projects enable a system-of-systems approach to understanding complex, interacting Earth processes. This poster will highlight recent innovations and future directions from NASA’s ESDT initiatives: Continuous Data Assimilation & Multi-Source Fusion. A core requirement of the ESDT work is the transition from static models to dynamic "living" replicas. This involves creating frameworks for the continual assimilation of near-real-time data from uncoordinated, heterogeneous sources, including satellite observations and airborne assets, and ground-based Internet of Things (IoT) sensors. These systems link design, operational status, and environmental data, ensuring the digital twin accurately reflects the current state of the physical Earth system. High-Fidelity Hybrid Modeling & Computational Acceleration to enable interactive "what-if" explorations, programs are moving beyond traditional, slow physical solvers by developing fast surrogate machine learning models and Deep Generative Models (DGMs). These hybrid approaches use neural networks to emulate complex physics, such as cloud feedback or ocean dynamics, at a fraction of the original computing cost, often leveraging advanced hardware like Graphics Processing Units (GPUs) to achieve the necessary scale. Federated Ecosystems & Interoperable Frameworks rather than building isolated tools, NASA IST is moving toward federated ESDTs and reusable analytic collaborative frameworks. This theme focuses on interoperability standards and common ontologies that allow specialized digital twins to interact and share data. This system-of-systems architecture supports multi-discipline investigations, such as analyzing how upstream watershed changes impact downstream urban flooding or how wildfire emissions affect regional air quality. By leveraging these advancements, ESDTs empower researchers and decision-makers to conduct real-time analysis and run complex hypothetical scenarios, ultimately improving our understanding of Earth’s evolving systems and informing critical real-world applications.

Earth System↗

NASA NTRS1999

An O(Nm(sup 2)) Plane Solver for the Compressible Navier-Stokes Equations

A hierarchical multigrid algorithm for efficient steady solutions to the two-dimensional compressible Navier-Stokes equations is developed and demonstrated. The algorithm applies multigrid in two ways: a Full Approximation Scheme (FAS) for a nonlinear residual equation and a Correction Scheme (CS) for a linearized defect correction implicit equation. Multigrid analyses which include the effect of boundary conditions in one direction are used to estimate the convergence rate of the algorithm for a model convection equation. Three alternating-line- implicit algorithms are compared in terms of efficiency. The analyses indicate that full multigrid efficiency is not attained in the general case; the number of cycles to attain convergence is dependent on the mesh density for high-frequency cross-stream variations. However, the dependence is reasonably small and fast convergence is eventually attained for any given frequency with either the FAS or the CS scheme alone. The paper summarizes numerical computations for which convergence has been attained to within truncation error in a few multigrid cycles for both inviscid and viscous ow simulations on highly stretched meshes.

Thomas, J. L.↗

DOE OSTI2025

CUDO: closed-form universal dwell-time optimization for computer-controlled optical surfacing

Precision optical figuring demands fast and accurate dwell time optimization to reach nanometer- and sub-nanometer-level accuracy in next-generation optical systems. We introduce CUDO (closed-form universal dwell-time optimization), the first, to the best of our knowledge, unified closed-form analytical framework that supports both function-form and matrix-form dwell time models in computer-controlled optical surfacing (CCOS). In contrast to traditional methods, which rely on iterative optimization and hyperparameter tuning, our framework derives direct analytical solutions with no adjustable parameters. This approach unifies the solution principles of existing methods within a single mathematical model, delivering three key advantages: (1) accuracy on par with, or superior to, iterative solvers, (2) substantial reduction in computation time, and (3) numerical robustness. Comparative studies with prior art confirm that closed-form solutions achieve equivalent residual error while removing runtime bottlenecks. By simplifying the implementation and enabling real-time, scalable deployment, CUDO establishes a practical foundation for future deterministic fabrication of large-aperture and high-performance optics.

36 MATERIALS SCIENCE↗

NASA NTRS1980

An implicit algorithm for the transonic full-potential equation in conservative form

A fast, implicit approximate factorization algorithm for the solution of the conservative full-potential equation for transonic flow in two and three dimensions is presented. Stability in supersonic regions is maintained by the use of an upwind evaluation of the density coefficient along all coordinate directions, providing an effective upwind difference of the streamwise terms for any orientation of the velocity vector and thereby enhancing the reliability of the algorithm. The algorithm is shown to provide rapid convergence for the computation of certain difficult two-dimensional test cases, including cases with fishtail shock patterns, demonstrating the reliability and efficiency of the procedure. Surface pressure coefficient distributions obtained by the present method are also found to be in good agreement with those computed by successive-line overrelaxation and a hybrid direct-solver/successive-line overrelaxation scheme, with significant reductions in CPU time required. A three-dimensional solution for a swept wing mounted between parallel walls is also presented which demonstrates the high convergence rate of the algorithm in three dimensions as well as two.

Holst, T.↗