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NASA NTRS1987

The method of projected characteristics for the evolution of magnetic arches

A numerical method of solving fully nonlinear MHD equation is described. In particular, the formulation based on the newly developed method of projected characteristics (Nakagawa, 1981) suitable to study the evolution of magnetic arches due to motions of their foot-points is presented. The final formulation is given in the form of difference equations; therefore, the analysis of numerical stability is also presented. Further, the most important derivation of physically self-consistent, time-dependent boundary conditions (i.e. the evolving boundary equations) is given in detail, and some results obtained with such boundary equations are reported.

Nakagawa, Y.↗

NASA NTRS1990

A comparison between progressive extension method (PEM) and iterative method (IM) for magnetic field extrapolations in the solar atmosphere

This paper presents a comparison between two numerical methods for the extrapolation of nonlinear force-free magnetic fields, viz the Iterative Method (IM) and the Progressive Extension Method (PEM). The advantages and disadvantages of these two methods are summarized, and the accuracy and numerical instability are discussed. On the basis of this investigation, it is claimed that the two methods do resemble each other qualitatively.

Wu, S. T.↗

DOE OSTI2024

Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints

The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop a scalable, mesh-independent algorithm for optimal design with pointwise bound constraints. This paper also introduces the latent variable proximal point (LVPP) algorithm, from which the proximal Galerkin method derives. When analyzing the classical obstacle problem, we discover that the underlying variational inequality can be replaced by a sequence of second-order partial differential equations (PDEs) that are readily discretized and solved with, e.g., the proximal Galerkin method. Throughout this work, we arrive at several contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field, density-based topology optimization. The complete proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. Open-source implementations of our methods accompany this work to facilitate reproduction and broader adoption.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1974

The solution of three-variable duct-flow equations

This paper establishes a numerical method for the solution of three-variable problems and is applied here to rotational flows through ducts of various cross sections. An iterative scheme is developed, the main feature of which is the addition of a duplicate variable to the forward component of velocity. Two forward components of velocity result from integrating two sets of first order ordinary differential equations for the streamline curvatures, in intersecting directions across the duct. Two pseudo-continuity equations are introduced with source/sink terms, whose strengths are dependent on the difference between the forward components of velocity. When convergence is obtained, the two forward components of velocity are identical, the source/sink terms are zero, and the original equations are satisfied. A computer program solves the exact equations and boundary conditions numerically. The method is economical and compares successfully with experiments on bent ducts of circular and rectangular cross section where secondary flows are caused by gradients of total pressure upstream.

Stuart, A. R.↗

NASA NTRS2001

Adaptive Numerical Dissipation Controls for High Order Methods

A numerical scheme for direct numerical simulation of shock-turbulence interactions of high speed compressible flows would ideally not be significantly more expensive than the standard fourth or sixth-order compact or non-compact central differencing scheme. It should be possible to resolve all scales down to scales of order of the Kolmogorov scales of turbulence accurately and efficiently, while at the same time being able to capture steep gradients occurring at much smaller scales efficiently. The goal of this lecture is to review the progress and new development of the low dissipative high order shock-capturing schemes proposed by Yee et al. Comparison on the efficiency and accuracy of this class of schemes with spectral and the fifth-order WENO (weighted essentially nonoscillatory) scheme will be presented. A new approach to dynamically sense the appropriate amount of numerical dissipation to be added at each grid point using non-orthogonal wavelets will be discussed.

Yee, Helen C.↗

NASA NTRS1973

Time-split finite-volume method for three-dimensional blunt-body flow

An efficient numerical method for calculating plane, axisymmetric, and fully three-dimensional blunt-body flow is presented. It is a second-order-accurate, time-dependent finite-volume procedure that solves the Euler equations in integral conservation-law form. These equations are written with respect to a Cartesian coordinate system in which an embedded mesh adjusts in time to the motion of the bow shock that is automatically captured as part of the weak solution. With such an adjusting mesh, oscillations in flow properties near the shock are shown to be virtually eliminated. The scheme uses a time-splitting concept that accelerates the convergence appreciably. Comparisons are made between computed and experimental results.

Rizzi, A. W.↗

NASA NTRS1983

Second-order explicit finite-difference methods for transient-flow analysis

Three second-order accurate numerical methods - MacCormack's method, Lambda scheme and Gabutti scheme - are introduced to solve the quasi-linear, hyperbolic partial differential equations describing transient flows in closed conduits. The details of these methods and the treatment of boundary conditions are presented and the results computed by using these methods for a typical piping system are compared. It is shown that for the same accuracy, second-order methods require considerably lesser number of computational nodes and computer time as compared to those required by the first-order methods.

Chaudhry, M. H.↗

DOE OSTI2025

Chapter 4 - Recent Advances in Identification of Differential Equations from Noisy Data: IDENT Review

Differential equations and numerical methods are extensively used to model various real-world phenomena in science and engineering. With modern developments, we aim to find the underlying differential equation from a single observation of time-dependent data. If we assume that the differential equation is a linear combination of various linear and nonlinear differential terms, then the identification problem can be formulated as solving a linear system. The goal then reduces to finding the optimal coefficient vector that best represents the time derivative of the given data. We review some recent works on the identification of differential equations. We find some common themes for the improved accuracy: (i) The formulation of linear system with proper denoising is important, (ii) how to utilize sparsity and model selection to find the correct coefficient support needs careful attention, and (iii) there are ways to improve the coefficient recovery. We present an overview and analysis of recent developments on the topic.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1988

A mixed finite difference/Galerkin method for three-dimensional Rayleigh-Benard convection

A fast and accurate numerical method, for nonlinear conservation equation systems whose solutions are periodic in two of the three spatial dimensions, is presently implemented for the case of Rayleigh-Benard convection between two rigid parallel plates in the parameter region where steady, three-dimensional convection is known to be stable. High-order streamfunctions secure the reduction of the system of five partial differential equations to a system of only three. Numerical experiments are presented which verify both the expected convergence rates and the absolute accuracy of the method.

Buell, Jeffrey C.↗

NASA NTRS1979

Numerical solution of compressible viscous flows

The presented review is concerned with the problem of calculating compressible viscous flows. Basic numerical considerations and problems associated with calculating viscous flows are examined and current numerical approaches toward the solution of the Navier-Stokes equations are discussed. It is pointed out that the numerical solution of the full time-dependent equations for turbulent flow is not practical with present computers. Therefore, turbulence effects must be accounted for by modeling. Developments related to turbulence modeling are described. In connection with a discussion of numerical methods for solving viscous flow equations, attention is given to numerical domains of dependence of typical explicit and implicit methods, the diffusion problem, the convection-diffusion problem, and the split-hybrid method.

Maccormack, R. W.↗

NASA NTRS1981

Finite-analytic numerical solution of heat transfer in two-dimensional cavity flow

Heat transfer in cavity flow is numerically analyzed by a new numerical method called the finite-analytic method. The basic idea of the finite-analytic method is the incorporation of local analytic solutions in the numerical solutions of linear or nonlinear partial differential equations. In the present investigation, the local analytic solutions for temperature, stream function, and vorticity distributions are derived. When the local analytic solution is evaluated at a given nodal point, it gives an algebraic relationship between a nodal value in a subregion and its neighboring nodal points. A system of algebraic equations is solved to provide the numerical solution of the problem. The finite-analytic method is used to solve heat transfer in the cavity flow at high Reynolds number (1000) for Prandtl numbers of 0.1, 1, and 10.

Chen, C.-J.↗

DOE OSTI2024

Open‐Source Anaerobic Digestion Modeling Platform, Anaerobic Digestion Model No. 1 Fast (ADM1F)

An open‐source modeling platform, called Anaerobic Digestion Model No. 1 Fast (ADM1F), is introduced to achieve fast and numerically stable simulations of anaerobic digestion processes. ADM1F is compatible with an iPython interface to facilitate model configuration, simulation, data analysis, and visualization. Faster simulations and more stable results are accomplished by implementing an advanced open‐source library of numerical methods called Portable Extensive Toolkit for Scientific Computation (PETSc) to solve the ADM1 system of equations. Leveraging PETSc, ADM1F can consistently complete a steady‐state simulation under 0.2 s, over 99% faster than a benchmark ADM1 model implemented with MATLAB while achieving agreement of model outputs within 1% of those obtained with the benchmark model. For dynamic simulations, however, ADM1F has a computational speed advantage only when the influent characteristics update more frequently than every 4 h. The ability of ADM1F to be useful as a tool to study anaerobic digestion systems is demonstrated through two example implementations of ADM1F: (1) a two‐phase co‐digestion scenario evaluating the impact of the organic loading rate and the substrate composition on reactor performance and stability, and (2) a conventional digester scenario assessing the effectiveness of recovery strategies after disruptions that led to instability. These examples demonstrate how the high simulation speed and the convenience of the iPython interface allow ADM1F to complete complex analyses within minutes, much faster than computational strategies currently reported in the literature.

anaerobic co-digestion↗

NASA NTRS1972

Force-free magnetic-field structures and their role in solar activity.

Magnetic-field structures in solar active regions are expected to be substantially force-free. A method is proposed for calculating such structures by numerical methods. The method is applied to the study of the magnetic-field pattern associated with a sunspot of one polarity surrounded by a magnetic region of opposite polarity when the sunspot rotates with respect to the surrounding region. Rotation introduces a toroidal component of magnetic field, and the associated pressure leads to inflation of the magnetic field pattern. If the differential rotation exceeds about 180 deg, the force-free magnetic field has energy greater than that of an open magnetic-field configuration with the same photospheric boundary conditions. It is concluded that, beyond this point, the force-free field structure is metastable and can be converted into an open field structure by an explosive MHD instability.

Barnes, C. W.↗

NASA NTRS1974

Multilateration - A nondegenerate method of obtaining station coordinates and satellite ephemerides

A technique for the determination of three-dimensional station coordinates and satellite ephemerides is developed which is based on the principle of multilateration. The method makes use of a system of six ground stations which simultaneously measure the slant range between each station and one or two satellites. It is demonstrated that a minimum of six stations is required in order to yield a system which will be free of mathematical degeneracies. It will be seen that the method of multilateration is not dependent upon the position of the satellite or any other dynamical considerations in the equations used to determine the relative station coordinates. In fact, the satellite coordinates are obtained as a direct by-product of the method. Numerical results are presented which indicate that the method of multilateration can determine the relative three-dimensional station coordinates with an accuracy that is limited only by the hardware measurement system. If a highly accurate laser ranging system is used, then accuracies in the 1-cm range can be expected.

Ong, K. M.↗

NASA NTRS1989

CSM Testbed Development and Large-Scale Structural Applications

A research activity called Computational Structural Mechanics (CSM) conducted at the NASA Langley Research Center is described. This activity is developing advanced structural analysis and computational methods that exploit high-performance computers. Methods are developed in the framework of the CSM Testbed software system and applied to representative complex structural analysis problems from the aerospace industry. An overview of the CSM Testbed methods development environment is presented and some new numerical methods developed on a CRAY-2 are described. Selected application studies performed on the NAS CRAY-2 are also summarized.

Knight, Norman F., Jr.↗

NASA NTRS1988

Parameter estimation technique for boundary value problems by spline collocation method

A parameter-estimation technique for boundary-integral equations of the second kind is developed. The output least-squares identification technique using the spline collocation method is considered. The convergence analysis for the numerical method is discussed. The results are applied to boundary parameter estimations for two-dimensional Laplace and Helmholtz equations.

Kojima, Fumio↗

NASA NTRS1999

Supersonic Aftbody Closure Wind-Tunnel Testing, Data Analysis, and Computational Results

This paper reports on the model, test, and results from the Langley Supersonic Aftbody Closure wind tunnel test. This project is an experimental evaluation of the 1.5% Technology Concept Aircraft (TCA) aftbody closure model (Model 23) in the Langley Unitary Plan Wind Tunnel. The baseline TCA design is the result of a multidisciplinary, multipoint optimization process and was developed using linear design and analysis methods, supplemented with Euler and Navier-Stokes numerical methods. After a thorough design review, it was decided to use an upswept blade attached to the forebody as the mounting system. Structural concerns dictated that a wingtip support system would not be feasible. Only the aftbody part of the model is metric. The metric break was chosen to be at the fuselage station where prior aft-sting supported models had been truncated. Model 23 is thus a modified version of Model 20. The wing strongback, flap parts, and nacelles from Model 20 were used, whereas new aftbodies, a common forebody, and some new tails were fabricated. In summary, significant differences in longitudinal and direction stability and control characteristics between the ABF and ABB aftbody geometries were measured. Correcting the experimental data obtained for the TCA configuration with the flared aftbody to the representative of the baseline TCA closed aftbody will result in a significant reduction in longitudinal stability, a moderate reduction in stabilizer effectiveness and directional stability, and a moderate to significant reduction in rudder effectiveness. These reductions in the stability and control effectiveness levels of the baseline TCA closed aftbody are attributed to the reduction in carry-over area.

Allen, Jerry↗

DOE OSTI2024

Review of recent activities with MOOSE, an open-source finite element & finite volume multi-fidelity simulation framework

Modeling and simulation are an increasing part of engineering. This is undoubtedly driven by the high costs of constructing experimental facilities, but also enabled by the exponential increase in computing powers over the last decades, which allows computational models to be closer than ever to reality. One of the main drivers for the development of MOOSE is supporting advanced nuclear reactor simulations. A challenging aspect of modeling advanced nuclear reactors is the plurality of physics involved, including neutronics, thermal hydraulics and fuel performance. These physics are all coupled to some extent and are generally solved in a sequential but iterative fashion. The United States (U.S.) national laboratories have been developing MOOSE, an open source multiphysics framework since its inception at the Idaho National Laboratory (INL) in 2008. This framework enables seamless coupling of multiphysics simulations and facilitates the implementation of new physics and material governing laws. It is continuously expanded with novel numerical methods and new pre-implemented physics module. Numerous applications, developed within the Department of Energy (DOE) laboratories, academia, and industry, including outside of nuclear engineering, have been developed to study specialized physics problems. International collaborations are welcome on this open-source modeling and simulation project.

22 - GENERAL STUDIES OF NUCLEAR REACTORS↗