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All sourcesNASA NTRSDOE OSTI
NASA NTRS1994

Preconditioned implicit solvers for the Navier-Stokes equations on distributed-memory machines

The GMRES method is parallelized, and combined with local preconditioning to construct an implicit parallel solver to obtain steady-state solutions for the Navier-Stokes equations of fluid flow on distributed-memory machines. The new implicit parallel solver is designed to preserve the convergence rate of the equivalent 'serial' solver. A static domain-decomposition is used to partition the computational domain amongst the available processing nodes of the parallel machine. The SPMD (Single-Program Multiple-Data) programming model is combined with message-passing tools to develop the parallel code on a 32-node Intel Hypercube and a 512-node Intel Delta machine. The implicit parallel solver is validated for internal and external flow problems, and is found to compare identically with flow solutions obtained on a Cray Y-MP/8. A peak computational speed of 2300 MFlops/sec has been achieved on 512 nodes of the Intel Delta machine,k for a problem size of 1024 K equations (256 K grid points).

Ajmani, Kumud↗
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 NTRS1993

Convergence acceleration of implicit schemes in the presence of high aspect ratio grid cells

The performance of Navier-Stokes codes are influenced by several phenomena. For example, the robustness of the code may be compromised by the lack of grid resolution, by a need for more precise initial conditions or because all or part of the flowfield lies outside the flow regime in which the algorithm converges efficiently. A primary example of the latter effect is the presence of extended low Mach number and/or low Reynolds number regions which cause convergence deterioration of time marching algorithms. Recent research into this problem by several workers including the present authors has largely negated this difficulty through the introduction of time-derivative preconditioning. In the present paper, we employ the preconditioned algorithm to address convergence difficulties arising from sensitivity to grid stretching and high aspect ratio grid cells. Strong grid stretching is particularly characteristic of turbulent flow calculations where the grid must be refined very tightly in the dimension normal to the wall, without a similar refinement in the tangential direction. High aspect ratio grid cells also arise in problems that involve high aspect ratio domains such as combustor coolant channels. In both situations, the high aspect ratio cells can lead to extreme deterioration in convergence. It is the purpose of the present paper to address the reasons for this adverse response to grid stretching and to suggest methods for enhancing convergence under such circumstances. Numerical algorithms typically possess a maximum allowable or optimum value for the time step size, expressed in non-dimensional terms as a CFL number or vonNeumann number (VNN). In the presence of high aspect ratio cells, the smallest dimension of the grid cell controls the time step size causing it to be extremely small, which in turn results in the deterioration of convergence behavior. For explicit schemes, this time step limitation cannot be exceeded without violating stability restrictions of the scheme. On the other hand, for implicit schemes, which are typically unconditionally stable, there appears to be room for improvement through careful tailoring of the time step definition based on results of linear stability analyses. In the present paper, we focus on the central-differenced alternating direction implicit (ADI) scheme. The understanding garnered from this analyses can then be applied to other implicit schemes. In order to systematically study the effects of aspect ratio and the methods of mitigating the associated problems, we use a two pronged approach. We use stability analyses as a tool for predicting numerical convergence behavior and numerical experiments on simple model problems to verify predicted trends. Based on these analyses, we determine that efficient convergence may be obtained at all aspect ratios by getting a combination of things right. Primary among these are the proper definition of the time step size, proper selection of viscous preconditioner and the precise treatment of boundary conditions. These algorithmic improvements are then applied to a variety of test cases to demonstrate uniform convergence at all aspect ratios.

Buelow, B. E. O.↗
NASA NTRS1995

An Adaptive Semi-Implicit Scheme for Simulations of Unsteady Viscous Compressible Flows

A numerical scheme for simulation of unsteady, viscous, compressible flows is considered. The scheme employs an explicit discretization of the inviscid terms of the Navier-Stokes equations and an implicit discretization of the viscous terms. The discretization is second order accurate in both space and time. Under appropriate assumptions, the implicit system of equations can be decoupled into two linear systems of reduced rank. These are solved efficiently using a Gauss-Seidel method with multigrid convergence acceleration. When coupled with a solution-adaptive mesh refinement technique, the hybrid explicit-implicit scheme provides an effective methodology for accurate simulations of unsteady viscous flows. The methodology is demonstrated for both body-fitted structured grids and for rectangular (Cartesian) grids.

Steinthorsson, Erlendur↗
NASA NTRS1996

Explicit and Implicit Resources: A Simplified Approach to User Requirements Modeling

The International Space Station (ISS) program is dependent on the concept of distributed planning, which places a large responsibility in the hands of the science users. The users assume the responsibility in submitting the operations scheduling requirements in the format required by the scheduling software. The concept of explicit and implicit resources, developed to facilitate the operations scheduling of ISS by non-scheduling experts, is presented. The proposed concepts are defined and examples of scheduling requirements, implemented using explicit and implicit resources, are described together with the requirements modeling and scheduler formatting functions. The software developed to support the implicit and explicit resources concept is discussed.

Hagopian, Jeff↗
NASA NTRS1997

Flowfield-Dependent Mixed Explicit-Implicit (FDMEL) Algorithm for Computational Fluid Dynamics

Despite significant achievements in computational fluid dynamics, there still remain many fluid flow phenomena not well understood. For example, the prediction of temperature distributions is inaccurate when temperature gradients are high, particularly in shock wave turbulent boundary layer interactions close to the wall. Complexities of fluid flow phenomena include transition to turbulence, relaminarization separated flows, transition between viscous and inviscid incompressible and compressible flows, among others, in all speed regimes. The purpose of this paper is to introduce a new approach, called the Flowfield-Dependent Mixed Explicit-Implicit (FDMEI) method, in an attempt to resolve these difficult issues in Computational Fluid Dynamics (CFD). In this process, a total of six implicitness parameters characteristic of the current flowfield are introduced. They are calculated from the current flowfield or changes of Mach numbers, Reynolds numbers, Peclet numbers, and Damkoehler numbers (if reacting) at each nodal point and time step. This implies that every nodal point or element is provided with different or unique numerical scheme according to their current flowfield situations, whether compressible, incompressible, viscous, inviscid, laminar, turbulent, reacting, or nonreacting. In this procedure, discontinuities or fluctuations of an variables between adjacent nodal points are determined accurately. If these implicitness parameters are fixed to certain numbers instead of being calculated from the flowfield information, then practically all currently available schemes of finite differences or finite elements arise as special cases. Some benchmark problems to be presented in this paper will show the validity, accuracy, and efficiency of the proposed methodology.

Garcia, S. M.↗
NASA NTRS1981

A Diagonal Form of an Implicit Approximate-Factorization Algorithm

A modification of an implicit approximate-factorization finite-difference algorithm applied to partial differential equations is presented. This algorithm is applied to the two- and three-dimensional Euler equations in general curvilinear coordinates. The modification transforms the coupled system of equations into an uncoupled diagonal form that requires less computational work. For steady-state applications, the resulting diagonal algorithm retains the stability and accuracy characteristics of the original algorithm. The diagonal algorithm reduces the storage requirement of the implicit solution process and therefore has an important effect on the application of implicit finite-difference schemes to vector processors. Results are presented for realistic two-dimensional transonic flow fields about airfoils. Computation costs are reduced 24-34%.

Pulliam, T. H.↗
NASA NTRS1999

Implicit Space-Time Conservation Element and Solution Element Schemes

Artificial numerical dissipation is in important issue in large Reynolds number computations. In such computations, the artificial dissipation inherent in traditional numerical schemes can overwhelm the physical dissipation and yield inaccurate results on meshes of practical size. In the present work, the space-time conservation element and solution element method is used to construct new and accurate implicit numerical schemes such that artificial numerical dissipation will not overwhelm physical dissipation. Specifically, these schemes have the property that numerical dissipation vanishes when the physical viscosity goes to zero. These new schemes therefore accurately model the physical dissipation even when it is extremely small. The new schemes presented are two highly accurate implicit solvers for a convection-diffusion equation. The two schemes become identical in the pure convection case, and in the pure diffusion case. The implicit schemes are applicable over the whole Reynolds number range, from purely diffusive equations to convection-dominated equations with very small viscosity. The stability and consistency of the schemes are analysed, and some numerical results are presented. It is shown that, in the inviscid case, the new schemes become explicit and their amplification factors are identical to those of the Leapfrog scheme. On the other hand, in the pure diffusion case, their principal amplification factor becomes the amplification factor of the Crank-Nicolson scheme.

Chang, Sin-Chung↗
NASA NTRS1998

Effects of sleep loss, time of day, and extended mental work on implicit and explicit learning of sequences

Tacit knowledge is part of many professional skills and can be studied experimentally with implicit-learning paradigms. The authors explored the effects of 2 different stressors, loss of sleep and mental fatigue, on implicit learning in a serial-response time (RT) task. In the 1st experiment, 1 night of sleep deprivation was shown to impair implicit but not explicit sequence learning. In the 2nd experiment, no impairment of both types of sequence learning was found after 1.5 hr of mental work. Serial-RT performance, in contrast, suffered from both stressors. These findings suggest that sleep deprivation induces specific risks for automatic, skill-based behavior that are not present in consciously controlled performance.

Learning↗
NASA NTRS2010

Development of Implicit Methods in CFD NASA Ames Research Center 1970's - 1980's

The focus here is on the early development (mid 1970's-1980's) at NASA Ames Research Center of implicit methods in Computational Fluid Dynamics (CFD). A class of implicit finite difference schemes of the Beam and Warming approximate factorization type will be addressed. The emphasis will be on the Euler equations. A review of material pertinent to the solution of the Euler equations within the framework of implicit methods will be presented. The eigensystem of the equations will be used extensively in developing a framework for various methods applied to the Euler equations. The development and analysis of various aspects of this class of schemes will be given along with the motivations behind many of the choices. Various acceleration and efficiency modifications such as matrix reduction, diagonalization and flux split schemes will be presented.

Pulliam, Thomas H.↗
NASA NTRS2010

Development and Verification of the Charring Ablating Thermal Protection Implicit System Solver

The development and verification of the Charring Ablating Thermal Protection Implicit System Solver is presented. This work concentrates on the derivation and verification of the stationary grid terms in the equations that govern three-dimensional heat and mass transfer for charring thermal protection systems including pyrolysis gas flow through the porous char layer. The governing equations are discretized according to the Galerkin finite element method with first and second order implicit time integrators. The governing equations are fully coupled and are solved in parallel via Newton's method, while the fully implicit linear system is solved with the Generalized Minimal Residual method. Verification results from exact solutions and the Method of Manufactured Solutions are presented to show spatial and temporal orders of accuracy as well as nonlinear convergence rates.

Amar, Adam J.↗
NASA NTRS2011

State-Based Implicit Coordination and Applications

In air traffic management, pairwise coordination is the ability to achieve separation requirements when conflicting aircraft simultaneously maneuver to solve a conflict. Resolution algorithms are implicitly coordinated if they provide coordinated resolution maneuvers to conflicting aircraft when only surveillance data, e.g., position and velocity vectors, is periodically broadcast by the aircraft. This paper proposes an abstract framework for reasoning about state-based implicit coordination. The framework consists of a formalized mathematical development that enables and simplifies the design and verification of implicitly coordinated state-based resolution algorithms. The use of the framework is illustrated with several examples of algorithms and formal proofs of their coordination properties. The work presented here supports the safety case for a distributed self-separation air traffic management concept where different aircraft may use different conflict resolution algorithms and be assured that separation will be maintained.

Narkawicz, Anthony J.↗
NASA NTRS2013

Making the Implicit Explicit: Towards an Assurance Case for DO-178C

For about two decades, compliance with Software Considerations in Airborne Systems and Equipment Certification (DO-178B) has been the primary means for receiving regulatory approval for using software on commercial airplanes. A new edition of the standard, DO-178C, was published in December 2011, and regulatory bodies have started the process towards recognizing this edition. The stated purpose of DO-178C remains unchanged from its predecessor: providing guidance “for the production of software for airborne systems and equipment that performs its intended function with a level of confidence in safety that complies with airworthiness requirements.” Within the text of the guidance, little or no rationale is given for how a particular objective or collection of objectives contributes to achieving this purpose. Thus the assurance case for the document is implicit. This paper discusses a current effort to make the implicit explicit. In particular, the paper describes the current status of the research seeking to identify the specific arguments contained in, or implied by, the DO-178C guidance that implicitly justify the assumption that the document meets its stated purpose.

Holloway, C. Michael↗
NASA NTRS2017

Implicitly Coordinated Detect and Avoid Capability for Safe Autonomous Operation of Small UAS

As the airspace becomes increasingly shared by autonomous small Unmanned Aerial Systems (UAS), there would be a pressing need for coordination strategies so that aircraft can safely and independently maneuver around obstacles, geofences, and traffic aircraft. Explicitly coordinating resolution strategies for small UAS would require additional components such as a reliable vehicle-to-vehicle communication infrastructure and standardized protocols for information exchange that could significantly increase the cost of deploying small UAS in a shared airspace. This paper explores a novel approach that enables multiple aircraft to implicitly coordinate their resolution maneuvers. By requiring all aircraft to execute the proposed approach deterministically, it is possible for all of them to implicitly agree on the region of airspace each will be occupying in a given time interval. The proposed approach lends itself to the construction of a suitable feedback mechanism that enables the real-time execution of an implicitly conflict-free path in a closed-loop manner dealing with uncertainties in aircraft speed. If a network infrastructure is available, the proposed approach can also exploit the benefits of explicit information.

Balachandran, Swee↗
NASA NTRS2020

Implicit Formulations of Bounded-Impulse Trajectory Models for Preliminary Interplanetary Low-Thrust Analysis

The bounded-impulse approach to low-thrust interplanetary trajectory optimization is widely used. In an effort to efficiently implement this approach using NASA’s OpenMDAO optimization software, the authors have implemented implicit formulations of the forward shooting/backwards-shooting methods commonly used in bounded-impulse models. These implicit approaches allow for vectorization of the underlying calculations which can significantly reduce runtime in interpreted languages. An implicit approach may be either converged by using an underlying nonlinear solver to converge the state propagation, or as a constraint in an optimizer-driven multiple-shooting approach. Significant computational efficiency gains are realized through the utilization of the modular approach to unified derivatives. Further computational efficiency is achieved by capitalizing on the sparsity of the constraint Jacobian matrix. This work demonstrates that a vectorized multiple-shooting approach for propagating a state-time history is superior in terms of computational efficiency as the number of segments in the state-propagation is increased.

Falck, Robert D↗
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↗
NASA NTRSDate not supplied

Towards Robust and Accurate Implicit Gradient Methods for Second- and Third-Order Nodal-Gradient Cell-Centered Finite-Volume Discretizations on Tetrahedral Grids

In this paper, we introduce implicit gradient methods as alternatives to conventional least-squares gradient methods for second- and third-order nodal-gradient cell-centered finite-volume discretizations, where solutions are stored at cells but gradients are stored at nodes. Because of the unique configuration of solutions and gradients, implicit gradient systems developed for the node-centered edge-based discretization method can be directly applied once the numerical solutions are interpolated from cells to nodes with sufficient accuracy. The resulting defect-correction solver can be loosely coupled with a flow-equation solver, and at convergence, solutions and gradients that satisfy the corresponding residual equations are obtained. Each iteration is relatively cheap compared with least-squares methods involving hundreds of neighbors. Numerical results are presented for accuracy verification studies and some simple but realistic flow problems.

Computational Fluid Dynamics↗
DOE OSTI2025

Robust Implicit Adaptive Low Rank Time-Stepping Methods for Matrix Differential Equations

In this work, we develop implicit rank-adaptive schemes for time-dependent matrix differential equations. The dynamic low rank approximation (DLRA) is a well-known technique to capture the dynamic low rank structure based on Dirac–Frenkel time-dependent variational principle. In recent years, it has attracted a lot of attention due to its wide applicability. Our schemes are inspired by the three-step procedure used in the rank adaptive version of the unconventional robust integrator (the so called BUG integrator) (Ceruti et al. in BIT Numer Math 62(4):1149–1174, 2022) for DLRA. First, a prediction (basis update) step is made computing the approximate column and row spaces at the next time level. Second, a Galerkin evolution step is invoked using an implicit solves for the small core matrix. Finally, a truncation is made according to a prescribed error threshold. Since the DLRA is evolving the differential equation projected on to the tangent space of the low rank manifold, the error estimate of the BUG integrator contains the tangent projection (modeling) error which cannot be easily controlled by mesh refinement. This can cause convergence issue for equations with cross terms. To address this issue, we propose a simple modification, consisting of merging the row and column spaces from the explicit step truncation method together with the BUG spaces in the prediction step. In addition, we propose an adaptive strategy where the BUG spaces are only computed if the residual for the solution obtained from the prediction space by explicit step truncation method, is too large. Here, we prove stability and estimate the local truncation error of the schemes under assumptions. We benchmark the schemes in several tests, such as anisotropic diffusion, solid body rotation and the combination of the two, to show robust convergence properties.

97 MATHEMATICS AND COMPUTING↗