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At least 1,693 records · Page 94

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NASA NTRS1988

Investigating a hybrid perturbation-Galerkin technique using computer algebra

A two-step hybrid perturbation-Galerkin method is presented for the solution of a variety of differential equations type problems which involve a scalar parameter. The resulting (approximate) solution has the form of a sum where each term consists of the product of two functions. The first function is a function of the independent field variable(s) x, and the second is a function of the parameter lambda. In step one the functions of x are determined by forming a perturbation expansion in lambda. In step two the functions of lambda are determined through the use of the classical Bubnov-Gelerkin method. The resulting hybrid method has the potential of overcoming some of the drawbacks of the perturbation and Bubnov-Galerkin methods applied separately, while combining some of the good features of each. In particular, the results can be useful well beyond the radius of convergence associated with the perturbation expansion. The hybrid method is applied with the aid of computer algebra to a simple two-point boundary value problem where the radius of convergence is finite and to a quantum eigenvalue problem where the radius of convergence is zero. For both problems the hybrid method apparently converges for an infinite range of the parameter lambda. The results obtained from the hybrid method are compared with approximate solutions obtained by other methods, and the applicability of the hybrid method to broader problem areas is discussed.

Andersen, Carl M.↗

NASA NTRS2003

Improved Collision-Detection Method for Robotic Manipulator

An improved method has been devised for the computational prediction of a collision between (1) a robotic manipulator and (2) another part of the robot or an external object in the vicinity of the robot. The method is intended to be used to test commanded manipulator trajectories in advance so that execution of the commands can be stopped before damage is done. The method involves utilization of both (1) mathematical models of the robot and its environment constructed manually prior to operation and (2) similar models constructed automatically from sensory data acquired during operation. The representation of objects in this method is simpler and more efficient (with respect to both computation time and computer memory), relative to the representations used in most prior methods. The present method was developed especially for use on a robotic land vehicle (rover) equipped with a manipulator arm and a vision system that includes stereoscopic electronic cameras. In this method, objects are represented and collisions detected by use of a previously developed technique known in the art as the method of oriented bounding boxes (OBBs). As the name of this technique indicates, an object is represented approximately, for computational purposes, by a box that encloses its outer boundary. Because many parts of a robotic manipulator are cylindrical, the OBB method has been extended in this method to enable the approximate representation of cylindrical parts by use of octagonal or other multiple-OBB assemblies denoted oriented bounding prisms (OBPs), as in the example of Figure 1. Unlike prior methods, the OBB/OBP method does not require any divisions or transcendental functions; this feature leads to greater robustness and numerical accuracy. The OBB/OBP method was selected for incorporation into the present method because it offers the best compromise between accuracy on the one hand and computational efficiency (and thus computational speed) on the other hand.

Leger, Chris↗

NASA NTRS1976

Solution of linear systems by a singular perturbation technique

An approximate solution is obtained for a singularly perturbed system of initial valued, time invariant, linear differential equations with multiple boundary layers. Conditions are stated under which the approximate solution converges uniformly to the exact solution as the perturbation parameter tends to zero. The solution is obtained by the method of matched asymptotic expansions. Use of the results for obtaining approximate solutions of general linear systems is discussed. An example is considered to illustrate the method and it is shown that the formulas derived give a readily computed uniform approximation.

Ardema, M. D.↗

NASA NTRS1987

Approximate polynomial preconditioning applied to biharmonic equations on vector supercomputers

Applying a finite difference approximation to a biharmonic equation results in a very ill-conditioned system of equations. This paper examines the conjugate gradient method used in conjunction with the generalized and approximate polynomial preconditionings for solving such linear systems. An approximate polynomial preconditioning is introduced, and is shown to be more efficient than the generalized polynomial preconditionings. This new technique provides a simple but effective preconditioning polynomial, which is based on another coefficient matrix rather than the original matrix operator as commonly used.

Wong, Yau Shu↗

NASA NTRS1992

An engineering aerodynamic heating method for hypersonic flow

A capability to calculate surface heating rates has been incorporated in an approximate three-dimensional inviscid technique. Surface streamlines are calculated from the inviscid solution, and the axisymmetric analog is then used along with a set of approximate convective-heating equations to compute the surface heat transfer. The method is applied to blunted axisymmetric and three-dimensional ellipsoidal cones at angle of attack for the laminar flow of a perfect gas. The method is also applicable to turbulent and equilibrium-air conditions. The present technique predicts surface heating rates that compare favorably with experimental (ground-test and flight) data and numerical solutions of the Navier-Stokes (NS) and viscous shock-layer (VSL) equations. The new technique represents a significant improvement over current engineering aerothermal methods with only a modest increase in computational effort.

Riley, Christopher J.↗

NASA NTRS1992

An Engineering Aerodynamic Heating Method for Hypersonic Flow

Riley, Christopher J.↗

NASA NTRS1988

A direct interaction approximation treatment of high Rayleigh number convective turbulence and comparison with experiment

A method is proposed for treating steady-state, thermally driven convection using the full direct interaction approximation to treat the nonlinear energy transfer and a prescribed instability function to model the eenrgy input. The instability function used is the growth rate of the mechanism generating the turbulence. This methodology can be easily generalized and applied to other driving mechanisms. The 1/3 power law form of the N vs. R relation for water is duplicated here and the coefficient is computed, using a two-point closure, to be less than about 0.08.

Hartke, Gregory J.↗

NASA NTRS1978

On a method computing transient wave propagation in ionospheric regions

A consequence of an exoatmospheric nuclear burst is an electromagnetic pulse (EMP) radiated from it. In a region far enough away from the burst, where nonlinear effects can be ignored, the EMP can be represented by a large-amplitude narrow-time-width plane-wave pulse. If the ionosphere intervenes the origin and destination of the EMP, frequency dispersion can cause significant changes in the original pulse upon reception. A method of computing these dispersive effects of transient wave propagation is summarized. The method described is different from the standard transform techniques and provides physical insight into the transient wave process. The method, although exact, can be used in approximating the early-time transient response of an ionospheric region by a simple integration with only explicit knowledge of the electron density, electron collision frequency, and electron gyrofrequency required. As an illustration of the method, it is applied to a simple example and contrasted with the corresponding transform solution.

Gray, K. G.↗

NASA NTRS2009

Local-in-Time Adjoint-Based Method for Optimal Control/Design Optimization of Unsteady Compressible Flows

.We study local-in-time adjoint-based methods for minimization of ow matching functionals subject to the 2-D unsteady compressible Euler equations. The key idea of the local-in-time method is to construct a very accurate approximation of the global-in-time adjoint equations and the corresponding sensitivity derivative by using only local information available on each time subinterval. In contrast to conventional time-dependent adjoint-based optimization methods which require backward-in-time integration of the adjoint equations over the entire time interval, the local-in-time method solves local adjoint equations sequentially over each time subinterval. Since each subinterval contains relatively few time steps, the storage cost of the local-in-time method is much lower than that of the global adjoint formulation, thus making the time-dependent optimization feasible for practical applications. The paper presents a detailed comparison of the local- and global-in-time adjoint-based methods for minimization of a tracking functional governed by the Euler equations describing the ow around a circular bump. Our numerical results show that the local-in-time method converges to the same optimal solution obtained with the global counterpart, while drastically reducing the memory cost as compared to the global-in-time adjoint formulation.

Yamaleev, N. K.↗

NASA NTRS2014

A Novel Method for Estimating Shortwave Direct Radiative Effect of Above-Cloud Aerosols Using CALIOP and MODIS Data

This paper describes an efficient and unique method for computing the shortwave direct radiative effect (DRE) of aerosol residing above low-level liquid-phase clouds using CALIOP and MODIS data. It addresses the overlap of aerosol and cloud rigorously by utilizing the joint histogram of cloud optical depth and cloud top pressure while also accounting for subgrid-scale variations of aerosols. The method is computationally efficient because of its use of grid-level cloud and aerosol statistics, instead of pixel-level products, and a pre-computed look-up table based on radiative transfer calculations. We verify that for smoke over the southeast Atlantic Ocean the method yields a seasonal mean instantaneous (approximately 1:30PM local time) shortwave DRE of above cloud aerosol (ACA) that generally agrees with more rigorous pixel-level computation within 4 percent. We also estimate the impact of potential CALIOP aerosol optical depth (AOD) retrieval bias of ACA on DRE. We find that the regional and seasonal mean instantaneous DRE of ACA over southeast Atlantic Ocean would increase, from the original value of 6.4 W m(-2) based on operational CALIOP AOD to 9.6 W m(-2) if CALIOP AOD retrieval are biased low by a factor of 1.5 (Meyer et al., 2013) and further to 30.9 W m(-2) if CALIOP AOD retrieval are biased low by a factor of 5 as suggested in (Jethva et al., 2014). In contrast, the instantaneous ACA radiative forcing efficiency (RFE) remains relatively invariant in all cases at about 53 W m(-2) AOD(-1), suggesting a near linear relation between the instantaneous RFE and AOD. We also compute the annual mean instantaneous shortwave DRE of light-absorbing aerosols (i.e., smoke and polluted dust) over global oceans based on 4 years of CALIOP and MODIS data. We find that the variability of the annual mean shortwave DRE of above-cloud light-absorbing aerosol is mainly driven by the optical depth of the underlying clouds. While we demonstrate our method using CALIOP and MODIS data, it can also be extended to other satellite data sets, as well as climate model outputs.

CALIOP and MODIS↗

NASA NTRS1993

A variable-order laminated plate theory based on the variational-asymptotical method

The variational-asymptotical method is a mathematical technique by which the three-dimensional analysis of laminated plate deformation can be split into a linear, one-dimensional, through-the-thickness analysis and a nonlinear, two-dimensional, plate analysis. The elastic constants used in the plate analysis are obtained from the through-the-thickness analysis, along with approximate, closed-form three-dimensional distributions of displacement, strain, and stress. In this paper, a theory based on this technique is developed which is capable of approximating three-dimensional elasticity to any accuracy desired. The asymptotical method allows for the approximation of the through-the-thickness behavior in terms of the eigenfunctions of a certain Sturm-Liouville problem associated with the thickness coordinate. These eigenfunctions contain all the necessary information about the nonhomogeneities along the thickness coordinate of the plate and thus possess the appropriate discontinuities in the derivatives of displacement. The theory is presented in this paper along with numerical results for the eigenfunctions of various laminated plates.

Lee, Bok W.↗

NASA NTRS1981

Forebody and base region real gas flow in severe planetary entry by a factored implicit numerical method. II - Equilibrium reactive gas

The factored-implicit, gas-dynamic algorithm has been adapted to the numerical simulation of equilibrium reactive flows. Changes required in the perfect gas version of the algorithm are developed, and the method of coupling gas-dynamic and chemistry variables is discussed. A flow-field solution that approximates a Jovian entry case was obtained by this method and compared with the same solution obtained by HYVIS, a computer program much used for the study of planetary entry. Comparison of surface pressure distribution and stagnation line shock-layer profiles indicates that the two solutions agree well.

Davy, W. C.↗

NASA NTRS1986

Efficient solution methods for the Navier-Stokes equations

Implicit finite difference schemes for solving two-dimensional and three-dimensional Euler and thin layer Navier-Stokes equations are addressed. The methods are demonstrated in fully vectorized codes for a Cray type architecture. The Beam and Warming implicit approximate factorization algorithm in generalized coordinates is used. The methods are either time accurate or accelerated non-time accurate steady state schemes. Acceleration and efficiency modifications such as matrix reduction, diagonalization, and flux split schemes are presented. Two dimensional inviscid and viscous calculations (e.g., airfoils with a deflected spoiler, circulation control airfoils, and unsteady buffeting) and of three dimensional viscous elliptical bodies, exhausting boattails, and generic oblique wing computations are discussed.

Pulliam, T. H.↗

NASA NTRS1977

An implicit-iterative solution of the heat conduction equation with a radiation boundary condition

For the problem of predicting one-dimensional heat transfer between conducting and radiating mediums by an implicit finite difference method, four different formulations were used to approximate the surface radiation boundary condition while retaining an implicit formulation for the interior temperature nodes. These formulations are an explicit boundary condition, a linearized boundary condition, an iterative boundary condition, and a semi-iterative boundary method. The results of these methods in predicting surface temperature on the space shuttle orbiter thermal protection system model under a variety of heating rates were compared. The iterative technique caused the surface temperature to be bounded at each step. While the linearized and explicit methods were generally more efficient, the iterative and semi-iterative techniques provided a realistic surface temperature response without requiring step size control techniques.

Williams, S. D.↗

NASA NTRS1971

A thin-shock-layer solution for nonequilibrium, inviscid hypersonic flows in earth, Martian, and Venusian atmospheres

An approximate inverse solution is presented for the nonequilibrium flow in the inviscid shock layer about a vehicle in hypersonic flight. The method is based upon a thin-shock-layer approximation and has the advantage of being applicable to both subsonic and supersonic regions of the shock layer. The relative simplicity of the method makes it ideally suited for programming on a digital computer with a significant reduction in storage capacity and computing time required by other more exact methods. Comparison of nonequilibrium solutions for an air mixture obtained by the present method is made with solutions obtained by two other methods. Additional cases are presented for entry of spherical nose cones into representative Venusian and Martian atmospheres. A digital computer program written in FORTRAN language is presented that permits an arbitrary gas mixture to be employed in the solution. The effects of vibration, dissociation, recombination, electronic excitation, and ionization are included in the program.

Grose, W. L.↗

NASA NTRS1974

The modified equation approach to the stability and accuracy analysis of finite-difference methods

The stability and accuracy of finite-difference approximations to simple linear partial differential equations are analyzed by studying the modified partial differential equation. Aside from round-off error, the modified equation represents the actual partial differential equation solved when a numerical solution is computed using a finite-difference equation. The modified equation is derived by first expanding each term of a difference scheme in a Taylor series and then eliminating time derivatives higher than first order by certain algebraic manipulations. The connection between 'heuristic' stability theory based on the modified equation approach and the von Neumann (Fourier) method is established. In addition to the determination of necessary and sufficient conditions for computational stability, a truncated version of the modified equation can be used to gain insight into the nature of both dissipative and dispersive errors.

Warming, R. F.↗

NASA NTRS1992

Eigenvalue error analysis of viscously damped structures using a Ritz reduction method

The efficient solution of the eigenvalue problem that results from inserting passive dampers with variable stiffness and damping coefficients into a structure is addressed. Eigenanalysis of reduced models obtained by retaining a number of normal modes augmented with Ritz vectors corresponding to the static solutions resulting from the load patterns introduced by the dampers has been empirically shown to yield excellent approximations to the full eigenvalue problem. An analysis of this technique in the case of a single damper is presented. A priori and a posteriori error estimates are generated and tested on numerical examples. Comparison theorems with modally truncated models and a Markov parameter matching reduced-order model are derived. These theorems corroborate the heuristic that residual flexibility methods improve low-frequency approximation of the system. The analysis leads to other techniques for eigenvalue approximation. Approximate closed-form solutions are derived that include a refinement to eigenvalue derivative methods for approximation. An efficient Newton scheme is also developed. A numerical example is presented demonstrating the effectiveness of each of these methods.

Chu, Cheng-Chih↗

NASA NTRS1992

Calculation of fully three-dimensional separated flow with an unsteady viscous-inviscid interaction method

Three previous papers have shown that the viscous inviscid numerical methods were capable of calculating separated turbulent flows. The 'Semi-inverse method' and the models of the first author allow the computation of massive separations, stalled flows, and shock wave-boundary layer interactions, in two-dimensional or quasi-three-dimensional conditions, from low speeds to supersonic speeds. The 'Semi-implicit' method for time-consistent coupling allows for the computation of time-accurate transonic separated flow and buffer computations in two-dimensions. The present paper shows that the viscous-inviscid interaction approach is also able to compute the fully three-dimensional flow separation phenomena. The method is based on a thin-layer approximation of the theory of 'Defect-Formulation' that provides the viscous-inviscid splitting of the Navier-Stokes equations. A parametric analytical modelling of the 3D-turbulent velocity profiles is involved. Numerically, the 3D-velocity profiles are discretized in the normal z-direction and driven by parametric integral 3D-equations in direct or inverse modes in the x-direction. The viscous-inviscid coupling is fully 3D and solved the time-consistency problem with an extension of the 'Semi-implicit' method previously suggested in two-dimensions. A 3D inviscid subroutine with TSP approximation is used. Results are obtained for transonic steady flows over wings with shock-induced transonic separation. The method provides results for highly three-dimensional flow separations, such as induced by a three-dimensional through at the wall of a flat plate. The 3D viscous-inviscid coupling and the 3D model of the velocity field provide three-dimensional instantaneous skin-friction lines whose patterns exhibit the same complex topology as Navier-Stokes solvers, with foci, nodes, and saddle-points.

Leballeur, J. C.↗