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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 217 records · Page 12
All sourcesNASA NTRSDOE OSTI
NASA NTRS1985

An implicit time-marching method for studying unsteady flow with massive separation

A fully implicit time-marching method is developed such that all spatial derivatives are approximated using central differences, but no use is made of any artificial dissipation. The numerical method solves the discretized equations using Alternating Direction Implicit-Block Gaussian Elimination technique. The method is implemented in the unsteady analysis, which solves the incompressible Navier-Stokes equations in terms of vorticity and stream function in generalized orthogonal coordinates. A clustered conformal C-grid is employed, and every effort is made to resolve the various length scales in the flow problem. The metric discontinuity at the branch-cut is treated appropriately using analytic continuation. Introduction of the BGE reordering permits implicit treatment of the branch cut in the numerical method. The vorticity singularity at the cusped trailing edge is also appropriately treated. This accurate and efficient implicit method is used to study flow at Re = 1000, past a 12-percent thick symmetric Joukowski airfoil at high angle of attack 30 and 53 deg.

Osswald, G. A.↗
NASA NTRS2000

Solving ODE Initial Value Problems With Implicit Taylor Series Methods

In this paper we introduce a new class of numerical methods for integrating ODE initial value problems. Specifically, we propose an extension of the Taylor series method which significantly improves its accuracy and stability while also increasing its range of applicability. To advance the solution from t (sub n) to t (sub n+1), we expand a series about the intermediate point t (sub n+mu):=t (sub n) + mu h, where h is the stepsize and mu is an arbitrary parameter called an expansion coefficient. We show that, in general, a Taylor series of degree k has exactly k expansion coefficients which raise its order of accuracy. The accuracy is raised by one order if k is odd, and by two orders if k is even. In addition, if k is three or greater, local extrapolation can be used to raise the accuracy two additional orders. We also examine stability for the problem y'= lambda y, Re (lambda) less than 0, and identify several A-stable schemes. Numerical results are presented for both fixed and variable stepsizes. It is shown that implicit Taylor series methods provide an effective integration tool for most problems, including stiff systems and ODE's with a singular point.

Scott, James R.↗
NASA NTRS1972

On stiffly stable implicit linear multistep methods.

The motivation to increase the step size with no degradation of numerical accuracy and stability has led to the discovery of particular members of the class of stiffly stable implicit linear multistep algorithms. Sufficient conditions for a consistent linear multistep method to be stiffly stable are given. These conditions involve properties of the stability mapping from the extended complex plane onto itself.

Cooke, C. H.↗
NASA NTRS1986

The effects of various implicit operators on a flux vector splitting method

Three different implicit operators in a numerical method for solving the two-dimensional steady Euler equations using flux vector splitting are investigated. These include the implementation of the true Jacobian matrices in the implicit part, the use of their splitting approximate form, and a modification of the implicit part of the scheme that uses the scalar form of the implicit part based on the spectral radii of the split Jacobian matrices. The three versions of the basic algorithm are compared with the results given by two common numerical methods using several two-dimensional test cases. Special attention is paid to the quality of results as well as computational efficiency and convergence properties.

Von Lavante, E.↗
NASA NTRS1985

A comparison between implicit and hybrid methods for the calculation of steady and unsteady inlet flows

Numerical simulation of steady and unsteady transonic diffuser flows using two different computer codes are discussed and compared with experimental data. The codes solve the Reynolds-averaged, compressible, Navier-Stokes equations using various turbulence models. One of the codes has been applied extensively to diffuser flows and uses the hybrid method of MacCormack. This code is relatively inefficient numerically. The second code, which was developed more recently, is fully implicit and is relatively efficient numerically. Simulations of steady flows using the implicit code are shown to be in good agreement with simulations using the hybrid code. Both simulations are in good agreement with experimental results. Simulations of unsteady flows using the two codes are in good qualitative agreement with each other, although the quantitative agreement is not as good as in the steady flow cases. The implicit code is shown to be eight times faster than the hybrid code for unsteady flow calculations and up to 32 times faster for steady flow calculations. Results of calculations using alternative turbulence models are also discussed.

Coakley, T. J.↗
NASA NTRS1999

Implicit and Multigrid Method for Ideal Multigrid Convergence: Direct Numerical Simulation of Separated Flow Around NACA 0012 Airfoil

Numerical investigation of flow separation over a NACA 0012 airfoil at large angles of attack has been carried out. The numerical calculation is performed by solving the full Navier-Stokes equations in generalized curvilinear coordinates. The second-order LU-SGS implicit scheme is applied for time integration. This scheme requires no tridiagonal inversion and is capable of being completely vectorized, provided the corresponding Jacobian matrices are properly selected. A fourth-order centered compact scheme is used for spatial derivatives. In order to reduce numerical oscillation, a sixth-order implicit filter is employed. Non-reflecting boundary conditions are imposed at the far-field and outlet boundaries to avoid possible non-physical wave reflection. Complex flow separation and vortex shedding phenomenon have been observed and discussed.

Liu, Chao-Qun↗
DOE OSTI2025

Snap-Through Buckling Pressure Prediction of Spherical Caps: A Comparison of Analytical, Implicit, and Explicit Methods

Snap-through buckling is a nonlinear and dynamic instability that occurs in curved shell structures such as domes, pressure vessels, and aerospace panels. Unlike classical linear buckling, it involves a sudden transition between equilibrium states caused by geometric nonlinearity and rapid strain-to-kinetic energy conversion (Timoshenko & Gere, 1961; Budiansky & Roth, 1962). This study investigates the snap-through behavior of thin spherical caps using both implicit and explicit solvers in ANSYS Workbench. While implicit analysis accurately captures quasi-static response, it struggles with convergence near instability. In contrast, explicit LS-DYNA simulation naturally handles the nonlinear dynamic event with minimal tuning and computational cost.

42 ENGINEERING↗
NASA NTRS2016

Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review

A review of diagonally implicit Runge-Kutta (DIRK) methods applied to rst-order ordinary di erential equations (ODEs) is undertaken. The goal of this review is to summarize the characteristics, assess the potential, and then design several nearly optimal, general purpose, DIRK-type methods. Over 20 important aspects of DIRKtype methods are reviewed. A design study is then conducted on DIRK-type methods having from two to seven implicit stages. From this, 15 schemes are selected for general purpose application. Testing of the 15 chosen methods is done on three singular perturbation problems. Based on the review of method characteristics, these methods focus on having a stage order of two, sti accuracy, L-stability, high quality embedded and dense-output methods, small magnitudes of the algebraic stability matrix eigenvalues, small values of aii, and small or vanishing values of the internal stability function for large eigenvalues of the Jacobian. Among the 15 new methods, ESDIRK4(3)6L[2]SA is recommended as a good default method for solving sti problems at moderate error tolerances.

Kennedy, Christopher A.↗
NASA NTRS1982

Transonic flow calculations using Euler's equations

An implicit finite difference method with implicit boundary conditions is employed to solve the steady Euler's equations for flows past arbitrary geometries. The resulting code is used to investigate in a systematic way various aspects of flow past airfoils at transonic speeds such as, method of solution, boundary conditions, grid stretching and generation, shock and sonic point operators, the Kutta condition, and smoothing. Results obtained are in good agreement with results of other codes. Moreover, it appears that the method of solution employed is such that Kutta's condition need not be invoked. This statement appears to be valid for other existing schemes employed in the solution of Euler's equations.

Atkins, H. L.↗
DOE OSTI2024

A B-spline based gradient-enhanced micropolar implicit material point method for large localized inelastic deformations

The quasi-brittle response of cohesive-frictional materials in numerical simulations is commonly represented by softening plasticity or continuum damage models, either individually or in combination. However, classical models, particularly when coupled with non-associated plasticity, often suffer from ill-posedness and a lack of objectivity in numerical simulations. Moreover, the performance of the finite element method significantly degrades in simulations involving finite strains when mesh distortion reaches excessive levels. This represents a challenge for modeling cohesive-frictional materials, given their tendency to experience strongly localized deformations, such as those occurring during shear band dominated failure. Hence, accurate modeling of the response of cohesive-frictional solids is a demanding task. To address these challenges, we present an extension of the material point method (MPM) for the unified gradient-enhanced micropolar continuum, aiming at the analysis of finite localized inelastic deformations in cohesive-frictional materials. The generalized gradient-enhanced micropolar continuum formulation is employed to tackle challenges related to localization and softening material behavior, while the MPM addresses issues arising from excessive deformations. The method utilizes a B-spline formulation for the rigid background mesh to mitigate the well-known cell crossing errors of the MPM. To demonstrate the performance of the method, 2D and 3D numerical studies on localized failure in sandstone in plane strain compression and triaxial extension tests are presented. A comparison with finite element results confirms the suitability of the formulation. Moreover, an efficient numerical implementation of the formulation is presented, and it is demonstrated that the additional MPM specific overhead is negligible.

B-spline↗
NASA NTRS1995

Nozzle Flow with Vibrational Nonequilibrium

Flow of nitrogen gas through a converging-diverging nozzle is simulated. The flow is modeled using the Navier-Stokes equations that have been modified for vibrational nonequilibrium. The energy equation is replaced by two equations. One equation accounts for energy effects due to the translational and rotational degrees of freedom, and the other accounts for the affects due to the vibrational degree of freedom. The energy equations are coupled by a relaxation time which measures the time required for the vibrational energy component to equilibrate with the translational and rotational energy components. An improved relaxation time is used in this thesis. The equations are solved numerically using the Steger-Warming flux vector splitting method and the Implicit MacCormack method. The results show that uniform flow is produced outside of the boundary layer. Nonequilibrium exists in both the converging and diverging nozzle sections. The boundary layer region is characterized by a marked increase in translational-rotational temperature. The vibrational temperature remains frozen downstream of the nozzle, except in the boundary layer.

Landry, John Gary↗
NASA NTRS1989

A class of high resolution explicit and implicit shock-capturing methods

An attempt is made to give a unified and generalized formulation of a class of high resolution, explicit and implicit shock capturing methods, and to illustrate their versatility in various steady and unsteady complex shock wave computations. Included is a systematic review of the basic design principle of the various related numerical methods. Special emphasis is on the construction of the basis nonlinear, spatially second and third order schemes for nonlinear scalar hyperbolic conservation laws and the methods of extending these nonlinear scalar schemes to nonlinear systems via the approximate Riemann solvers and the flux vector splitting approaches. Generalization of these methods to efficiently include equilibrium real gases and large systems of nonequilibrium flows are discussed. Some issues concerning the applicability of these methods that were designed for homogeneous hyperbolic conservation laws to problems containing stiff source terms and shock waves are also included. The performance of some of these schemes is illustrated by numerical examples for 1-, 2- and 3-dimensional gas dynamics problems.

Yee, H. C.↗
NASA NTRS1987

Three dimensional hypersonic flow simulations with the CSCM implicit upwind Navier-Stokes method

The Bardina and Lombard (1985) three-dimensional CSCM Navier-Stokes method is presently extended to the simulation of complex hypersonic reentry vehicle external flows at angle of attack. The robust stability of the method derives from the combination of conservative implicit upwind flux difference splitting with a three-dimensional diagonally-dominant approximate factorization and relaxation scheme and characteristic-based implicit boundary approximations. The method's efficiency derives from an implicit symmetric Gauss-Seidel 'method of planes' relaxation scheme with alternating directional space marching sweeps along the flow coordinate direction.

Bardina, Jorge↗
NASA NTRS1994

Global Asymptotic Behavior of Iterative Implicit Schemes

The global asymptotic nonlinear behavior of some standard iterative procedures in solving nonlinear systems of algebraic equations arising from four implicit linear multistep methods (LMMs) in discretizing three models of 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed using the theory of dynamical systems. The iterative procedures include simple iteration and full and modified Newton iterations. The results are compared with standard Runge-Kutta explicit methods, a noniterative implicit procedure, and the Newton method of solving the steady part of the ODEs. Studies showed that aside from exhibiting spurious asymptotes, all of the four implicit LMMs can change the type and stability of the steady states of the differential equations (DEs). They also exhibit a drastic distortion but less shrinkage of the basin of attraction of the true solution than standard nonLMM explicit methods. The simple iteration procedure exhibits behavior which is similar to standard nonLMM explicit methods except that spurious steady-state numerical solutions cannot occur. The numerical basins of attraction of the noniterative implicit procedure mimic more closely the basins of attraction of the DEs and are more efficient than the three iterative implicit procedures for the four implicit LMMs. Contrary to popular belief, the initial data using the Newton method of solving the steady part of the DEs may not have to be close to the exact steady state for convergence. These results can be used as an explanation for possible causes and cures of slow convergence and nonconvergence of steady-state numerical solutions when using an implicit LMM time-dependent approach in computational fluid dynamics.

Yee, H. C.↗
NASA NTRS2006

Application of p-Multigrid to Discontinuous Galerkin Formulations of the Poisson Equation

We investigate p-multigrid as a solution method for several different discontinuous Galerkin (DG) formulations of the Poisson equation. Different combinations of relaxation schemes and basis sets have been combined with the DG formulations to find the best performing combination. The damping factors of the schemes have been determined using Fourier analysis for both one and two-dimensional problems. One important finding is that when using DG formulations, the standard approach of forming the coarse p matrices separately for each level of multigrid is often unstable. To ensure stability the coarse p matrices must be constructed from the fine grid matrices using algebraic multigrid techniques. Of the relaxation schemes, we find that the combination of Jacobi relaxation with the spectral element basis is fairly effective. The results using this combination are p sensitive in both one and two dimensions, but reasonable convergence rates can still be achieved for moderate values of p and isotropic meshes. A competitive alternative is a block Gauss-Seidel relaxation. This actually out performs a more expensive line relaxation when the mesh is isotropic. When the mesh becomes highly anisotropic, the implicit line method and the Gauss-Seidel implicit line method are the only effective schemes. Adding the Gauss-Seidel terms to the implicit line method gives a significant improvement over the line relaxation method.

Helenbrook, B. T.↗
DOE OSTI2026

Explicit Monotone Stable Super-Time-stepping Methods for Finite Time Singularities

We explore a novel way to numerically resolve the scaling behavior of finite-time singularities in solutions of nonlinear parabolic PDEs. The Runge–Kutta–Legendre (RKL) and Runge–Kutta–Gegenbauer (RKG) super-time-stepping methods were originally developed for nonlinear complex physics problems with diffusion. These are multistage single step second-order, forward-in-time methods with no implicit solves. The advantage is that the time-step size for stability scales with stage number 𝑠 as $\mathcal{O}$⁡(𝑠 2 ). Many interesting nonlinear PDEs have finite-time singularities, and the presence of diffusion often limits one to using implicit or semi-implicit time-step methods for stability constraints. Finite-time singularities are particularly challenging due to the large range of scales that one desires to resolve, often with adaptive spatial grids and adaptive time steps. Here, in this study, we show two examples of nonlinear PDEs for which the self-similar singularity structure has time and space scales that are resolvable using the RKL and RKG methods, without forcing even smaller time steps. Compared to commonly used implicit numerical methods, we achieve a significantly smaller run time while maintaining comparable accuracy. We also prove numerical monotonicity for both the RKL and RKG methods under their linear stability conditions for the constant coefficient heat equation, in the case of infinite domain and periodic boundary condition, leading to a theoretical guarantee of the superiority of the RKL and RKG methods over traditional super-time-stepping methods, such as the Runge-Kutta-Chebyshev and the orthogonal Runge-Kutta-Chebyshev methods. Code can be found at https://github.com/ZT220501/SRK-Singularity.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1994

A Class of High-Resolution Explicit and Implicit Shock-Capturing Methods

The development of shock-capturing finite difference methods for hyperbolic conservation laws has been a rapidly growing area for the last decade. Many of the fundamental concepts, state-of-the-art developments and applications to fluid dynamics problems can only be found in meeting proceedings, scientific journals and internal reports. This paper attempts to give a unified and generalized formulation of a class of high-resolution, explicit and implicit shock capturing methods, and to illustrate their versatility in various steady and unsteady complex shock waves, perfect gases, equilibrium real gases and nonequilibrium flow computations. These numerical methods are formulated for the purpose of ease and efficient implementation into a practical computer code. The various constructions of high-resolution shock-capturing methods fall nicely into the present framework and a computer code can be implemented with the various methods as separate modules. Included is a systematic overview of the basic design principle of the various related numerical methods. Special emphasis will be on the construction of the basic nonlinear, spatially second and third-order schemes for nonlinear scalar hyperbolic conservation laws and the methods of extending these nonlinear scalar schemes to nonlinear systems via the approximate Riemann solvers and flux-vector splitting approaches. Generalization of these methods to efficiently include real gases and large systems of nonequilibrium flows will be discussed. Some perbolic conservation laws to problems containing stiff source terms and terms and shock waves are also included. The performance of some of these schemes is illustrated by numerical examples for one-, two- and three-dimensional gas-dynamics problems. The use of the Lax-Friedrichs numerical flux to obtain high-resolution shock-capturing schemes is generalized. This method can be extended to nonlinear systems of equations without the use of Riemann solvers or flux-vector splitting approaches and thus provides a large savings for multidimensional, equilibrium real gases and nonequilibrium flow computations.

Yee, H. C.↗
NASA NTRSDate not supplied

Updates to Implicit Edge-Based Gradient Methods

In this paper, we report updates to the implicit edge-based gradient methods originally introduced in [H. Nishikawa, AIAA Paper 2020-3048, 2020]. First, we clarify the relationship between gradient accuracy and truncation error and show that the quadratic method involves a free parameter. Then, we provide a complete description and a simplified matrix form of the implicit gradient systems including a consistent boundary treatment, and derive a set of parameters for achieving fourth-order gradient accuracy on regular tetrahedral grids. A stability analysis is performed for a relaxation scheme used to solve the implicit gradient systems, and the result serves as a guide for choosing parameters. Numerical results are shown for accuracy verification and also for realistic inviscid flow problems in three dimensions, including flows with shock waves.

Weighted Least-Squares↗