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NASA NTRS2003

Implicit Spacecraft Gyro Calibration

This paper presents an implicit algorithm for spacecraft onboard instrument calibration, particularly to onboard gyro calibration. This work is an extension of previous work that was done where an explicit gyro calibration algorithm was applied to the AQUA spacecraft gyros. The algorithm presented in this paper was tested using simulated data and real data that were downloaded from the Microwave Anisotropy Probe (MAP) spacecraft. The calibration tests gave very good results. A comparison between the use of the implicit calibration algorithm used here with the explicit algorithm used for AQUA spacecraft indicates that both provide an excellent estimation of the gyro calibration parameters with similar accuracies.

Harman, Richard↗

NASA NTRS1979

Digital adaptive controllers for VTOL vehicles. Volume 2: Software documentation

The VTOL approach and landing test (VALT) adaptive software is documented. Two self-adaptive algorithms, one based on an implicit model reference design and the other on an explicit parameter estimation technique were evaluated. The organization of the software, user options, and a nominal set of input data are presented along with a flow chart and program listing of each algorithm.

Hartmann, G. L.↗

NASA NTRS1983

Time-dependent finite-difference simulation of unsteady interactive flows

The solution of the time-dependent, Reynolds-averaged, Navier-Stokes equations for unsteady, interacting flows by finite-difference algorithms is discussed. Specific examples include (1) unsteady transonic flow over a thick biconvex airfoil, (2) determination of buffet boundaries for a transonic lifting airfoil, (3) the simulation of aileron buzz and (4) dynamic stall. Algorithms considered include explicit methods, mixed (or hybrid) methods, and fully implicit methods. Consideration of time scales for computational stability, computational accuracy, and physical accuracy and the use of time-dependent adaptive meshing to realize computational efficiency are also discussed.

Deiwert, G. S.↗

NASA NTRS1984

Multiple steady states for characteristic initial value problems

The time dependent, isentropic, quasi-one-dimensional equations of gas dynamics and other model equations are considered under the constraint of characteristic boundary conditions. Analysis of the time evolution shows how different initial data may lead to different steady states and how seemingly anamolous behavior of the solution may be resolved. Numerical experimentation using time consistent explicit algorithms verifies the conclusions of the analysis. The use of implicit schemes with very large time steps leads to erroneous results.

Salas, M. D.↗

NASA NTRS1985

Implicit/explicit analysis of interactive phenomena in: supersonic, chemically-reacting, mixing and boundary layer problems

An efficient implicit/explicit procedure for solving the parabolized Navier-Stokes (PNS) equations to analyze supersonic, viscous, chemically-reacting flows is presented. The approach combines a fully-implicit algorithm for solving the parabolic mixing/boundary layer equations with an explicit algorithm for determining the hyperbolic pressure field. The unified model, SPLITP, contains provisions for treating bounding/embedded subsonic regions using a pressure-splitting procedure which suppresses elliptic (upstream influence) effects without recourse to sublayer approximations. Several two-equation turbulence models with compressibility and curvature-correction terms are incorporated into SPLITP. Applications to a variety of flow problems are presented which include: ducted supersonic mixing and combustion flowfields, free jets and plumes, and, curved boundary layers and wall jets.

Dash, S. M.↗

NASA NTRS1984

Exponential-fitted methods for integrating stiff systems of ordinary differential equations: Applications to homogeneous gas-phase chemical kinetics

Conventional algorithms for the numerical integration of ordinary differential equations (ODEs) are based on the use of polynomial functions as interpolants. However, the exact solutions of stiff ODEs behave like decaying exponential functions, which are poorly approximated by polynomials. An obvious choice of interpolant are the exponential functions themselves, or their low-order diagonal Pade (rational function) approximants. A number of explicit, A-stable, integration algorithms were derived from the use of a three-parameter exponential function as interpolant, and their relationship to low-order, polynomial-based and rational-function-based implicit and explicit methods were shown by examining their low-order diagonal Pade approximants. A robust implicit formula was derived by exponential fitting the trapezoidal rule. Application of these algorithms to integration of the ODEs governing homogenous, gas-phase chemical kinetics was demonstrated in a developmental code CREK1D, which compares favorably with the Gear-Hindmarsh code LSODE in spite of the use of a primitive stepsize control strategy.

Pratt, D. T.↗

NASA NTRS1995

Analysis of composite ablators using massively parallel computation

In this work, the feasibility of using massively parallel computation to study the response of ablative materials is investigated. Explicit and implicit finite difference methods are used on a massively parallel computer, the Thinking Machines CM-5. The governing equations are a set of nonlinear partial differential equations. The governing equations are developed for three sample problems: (1) transpiration cooling, (2) ablative composite plate, and (3) restrained thermal growth testing. The transpiration cooling problem is solved using a solution scheme based solely on the explicit finite difference method. The results are compared with available analytical steady-state through-thickness temperature and pressure distributions and good agreement between the numerical and analytical solutions is found. It is also found that a solution scheme based on the explicit finite difference method has the following advantages: incorporates complex physics easily, results in a simple algorithm, and is easily parallelizable. However, a solution scheme of this kind needs very small time steps to maintain stability. A solution scheme based on the implicit finite difference method has the advantage that it does not require very small times steps to maintain stability. However, this kind of solution scheme has the disadvantages that complex physics cannot be easily incorporated into the algorithm and that the solution scheme is difficult to parallelize. A hybrid solution scheme is then developed to combine the strengths of the explicit and implicit finite difference methods and minimize their weaknesses. This is achieved by identifying the critical time scale associated with the governing equations and applying the appropriate finite difference method according to this critical time scale. The hybrid solution scheme is then applied to the ablative composite plate and restrained thermal growth problems. The gas storage term is included in the explicit pressure calculation of both problems. Results from ablative composite plate problems are compared with previous numerical results which did not include the gas storage term. It is found that the through-thickness temperature distribution is not affected much by the gas storage term. However, the through-thickness pressure and stress distributions, and the extent of chemical reactions are different from the previous numerical results. Two types of chemical reaction models are used in the restrained thermal growth testing problem: (1) pressure-independent Arrhenius type rate equations and (2) pressure-dependent Arrhenius type rate equations. The numerical results are compared to experimental results and the pressure-dependent model is able to capture the trend better than the pressure-independent one. Finally, a performance study is done on the hybrid algorithm using the ablative composite plate problem. It is found that there is a good speedup of performance on the CM-5. For 32 CPU's, the speedup of performance is 20. The efficiency of the algorithm is found to be a function of the size and execution time of a given problem and the effective parallelization of the algorithm. It also seems that there is an optimum number of CPU's to use for a given problem.

Shia, David↗

NASA NTRS1985

Solution of partial differential equations on vector and parallel computers

The present status of numerical methods for partial differential equations on vector and parallel computers was reviewed. The relevant aspects of these computers are discussed and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. Application areas utilizing these computers are briefly discussed.

Ortega, J. M.↗

NASA NTRS1985

Solution of partial differential equations on vector and parallel computers

Ortega, J. M.↗

NASA NTRS1991

Computational methods for structural load and resistance modeling

An automated capability for computing structural reliability considering uncertainties in both load and resistance variables is presented. The computations are carried out using an automated Advanced Mean Value iteration algorithm (AMV +) with performance functions involving load and resistance variables obtained by both explicit and implicit methods. A complete description of the procedures used is given as well as several illustrative examples, verified by Monte Carlo Analysis. In particular, the computational methods described in the paper are shown to be quite accurate and efficient for a material nonlinear structure considering material damage as a function of several primitive random variables. The results show clearly the effectiveness of the algorithms for computing the reliability of large-scale structural systems with a maximum number of resolutions.

Thacker, B. H.↗

NASA NTRS1995

Flow simulations about steady-complex and unsteady moving configurations using structured-overlapped and unstructured grids

The limiting factor in simulating flows past realistic configurations of interest has been the discretization of the physical domain on which the governing equations of fluid flow may be solved. In an attempt to circumvent this problem, many Computational Fluid Dynamic (CFD) methodologies that are based on different grid generation and domain decomposition techniques have been developed. However, due to the costs involved and expertise required, very few comparative studies between these methods have been performed. In the present work, the two CFD methodologies which show the most promise for treating complex three-dimensional configurations as well as unsteady moving boundary problems are evaluated. These are namely the structured-overlapped and the unstructured grid schemes. Both methods use a cell centered, finite volume, upwind approach. The structured-overlapped algorithm uses an approximately factored, alternating direction implicit scheme to perform the time integration, whereas, the unstructured algorithm uses an explicit Runge-Kutta method. To examine the accuracy, efficiency, and limitations of each scheme, they are applied to the same steady complex multicomponent configurations and unsteady moving boundary problems. The steady complex cases consist of computing the subsonic flow about a two-dimensional high-lift multielement airfoil and the transonic flow about a three-dimensional wing/pylon/finned store assembly. The unsteady moving boundary problems are a forced pitching oscillation of an airfoil in a transonic freestream and a two-dimensional, subsonic airfoil/store separation sequence. Accuracy was accessed through the comparison of computed and experimentally measured pressure coefficient data on several of the wing/pylon/finned store assembly's components and at numerous angles-of-attack for the pitching airfoil. From this study, it was found that both the structured-overlapped and the unstructured grid schemes yielded flow solutions of comparable accuracy for these simulations. This study also indicated that, overall, the structured-overlapped scheme was slightly more CPU efficient than the unstructured approach.

Newman, James C., III↗

NASA NTRS2000

A Finite-Element Approach for Modeling Inviscid and Viscous Compressible Flows using Prismatic Grids

The Galerkin finite-element method is used to solve the Euler and Navier-Stokes equations on prismatic meshes. It is shown that the prismatic grid is advantageous for correctly and efficiently capturing the boundary layers in high Reynolds number flows. It can be captured accurately because of the ability to cluster grid points normal to the body. The efficiency derives from the implicit treatment of the normal direction. To treat the normal direction implicitly, a semi-implicit Runge-Kutta time stepping scheme is developed. The semi-implicit algorithm is validated on simple geometries for inviscid and viscous flows and its convergence history is compared to that of the explicit Runge-Kutta scheme. The semi-implicit scheme is shown to be a factor of 3 to 4 faster in terms of CPU time to convergence.

Pandya, S. A.↗

NASA NTRS1988

High-order low-pass implicit tangent filters for use in finite area calculations

High-order implicit tangent filters are developed. The implicit tangent filters possess a highly selective amplitude response function and they can be applied relatively close to a boundary. Comparisons are made between the implicit and the traditional or explicit filters. Numerical simulations are performed to test the response in a limited-area model. An algorithm to maximize computing efficiency is presented. All tests indicate the desirability and utility of the implicit tangent filter.

Raymond, William H.↗

NASA NTRS1983

A finite element method for diffusion dominated unsteady viscous flows

A general conforming finite element scheme for computing viscous flows is presented which is of second-order accuracy in space and time. Viscous terms are treated implicitly and advection terms are treated explicitly in the time marching segment of the algorithm. A method for solving the algebraic equations at each time step is given. The method is demonstrated on two test problems, one of them being a plane vortex flow for which asymptotic methods are used to obtain suitable numerical boundary conditions at each time step.

Gunzburger, M. D.↗

NASA NTRS1987

A barely implicit correction for flux-corrected transport

In virtue of adding one elliptic equation that must be implicitly solved, the 'barely implicit correction' method presented removes the stringent sidestep limit imposed by sound speed in the explicit methods. The barely implicit method is presently combined with a flux-corrected transport algorithm, which has been rendered fourth-order by the appropriate subtraction of corrected fluxes, in order to accurately represent the sharp gradients in subsonic flows. The multigrid MGRID technique (DeVore, 1984) is used to efficiently solve the elliptic pressure equation.

Patnaik, G.↗

NASA NTRS1988

A computational procedure for large rotational motions in multibody dynamics

A computational procedure suitable for the solution of equations of motion for multibody systems is presented. The present procedure adopts a differential partitioning of the translational motions and the rotational motions. The translational equations of motion are then treated by either a conventional explicit or implicit direct integration method. A principal feature of this procedure is a nonlinearly implicit algorithm for updating rotations via the Euler four-parameter representation. This procedure is applied to the rolling of a sphere through a specific trajectory, which shows that it yields robust solutions.

Park, K. C.↗

NASA NTRS2008

A Formal Framework for the Analysis of Algorithms That Recover From Loss of Separation

We present a mathematical framework for the specification and verification of state-based conflict resolution algorithms that recover from loss of separation. In particular, we propose rigorous definitions of horizontal and vertical maneuver correctness that yield horizontal and vertical separation, respectively, in a bounded amount of time. We also provide sufficient conditions for independent correctness, i.e., separation under the assumption that only one aircraft maneuvers, and for implicitly coordinated correctness, i.e., separation under the assumption that both aircraft maneuver. An important benefit of this approach is that different aircraft can execute different algorithms and implicit coordination will still be achieved, as long as they all meet the explicit criteria of the framework. Towards this end we have sought to make the criteria as general as possible. The framework presented in this paper has been formalized and mechanically verified in the Prototype Verification System (PVS).

Butler, RIcky W.↗

NASA NTRS2009

Formally Verified Practical Algorithms for Recovery from Loss of Separation

In this paper, we develop and formally verify practical algorithms for recovery from loss of separation. The formal verification is performed in the context of a criteria-based framework. This framework provides rigorous definitions of horizontal and vertical maneuver correctness that guarantee divergence and achieve horizontal and vertical separation. The algorithms are shown to be independently correct, that is, separation is achieved when only one aircraft maneuvers, and implicitly coordinated, that is, separation is also achieved when both aircraft maneuver. In this paper we improve the horizontal criteria over our previous work. An important benefit of the criteria approach is that different aircraft can execute different algorithms and implicit coordination will still be achieved, as long as they all meet the explicit criteria of the framework. Towards this end we have sought to make the criteria as general as possible. The framework presented in this paper has been formalized and mechanically verified in the Prototype Verification System (PVS).

Butler, Ricky W.↗