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NASA NTRS1991

The L sub 1 finite element method for pure convection problems

The least squares (L sub 2) finite element method is introduced for 2-D steady state pure convection problems with smooth solutions. It is proven that the L sub 2 method has the same stability estimate as the original equation, i.e., the L sub 2 method has better control of the streamline derivative. Numerical convergence rates are given to show that the L sub 2 method is almost optimal. This L sub 2 method was then used as a framework to develop an iteratively reweighted L sub 2 finite element method to obtain a least absolute residual (L sub 1) solution for problems with discontinuous solutions. This L sub 1 finite element method produces a nonoscillatory, nondiffusive and highly accurate numerical solution that has a sharp discontinuity in one element on both coarse and fine meshes. A robust reweighting strategy was also devised to obtain the L sub 1 solution in a few iterations. A number of examples solved by using triangle and bilinear elements are presented.

Jiang, Bo-Nan↗

NASA NTRS1990

Accuracy analysis of the semi-analytical method for shape sensitivity calculation

The semianalytical method of design sensitivity analysis that is widely used for calculating derivatives of static response with respect to design variables for structures modeled by finite elements is studied. It is shown that the method can have serious accuracy problems for shape design variables in structures modeled by beam, plate, truss, frame, and solid elements. Errors are shown to be associated with an incompatibility of the sensitivity field with the structure. An error index is developed to test the accuracy of the semianalytical method. It characterizes the difference in errors between a general finite-difference method and the semianalytical method. A method for improving the accuracy of the semianalytical method (when possible) is provided. Examples are presented to demonstrate the use of the error index.

Barthelemy, Bruno↗

NASA NTRS1992

Divergence preserving discrete surface integral methods for Maxwell's curl equations using non-orthogonal unstructured grids

Several new discrete surface integral (DSI) methods for solving Maxwell's equations in the time-domain are presented. These methods, which allow the use of general nonorthogonal mixed-polyhedral unstructured grids, are direct generalizations of the canonical staggered-grid finite difference method. These methods are conservative in that they locally preserve divergence or charge. Employing mixed polyhedral cells, (hexahedral, tetrahedral, etc.) these methods allow more accurate modeling of non-rectangular structures and objects because the traditional stair-stepped boundary approximations associated with the orthogonal grid based finite difference methods can be avoided. Numerical results demonstrating the accuracy of these new methods are presented.

Madsen, Niel K.↗

NASA NTRS1992

Comparison of epifluorescent viable bacterial count methods

Two methods, the 2-(4-Iodophenyl) 3-(4-nitrophenyl) 5-phenyltetrazolium chloride (INT) method and the direct viable count (DVC), were tested and compared for their efficiency for the determination of the viability of bacterial populations. Use of the INT method results in the formation of a dark spot within each respiring cell. The DVC method results in elongation or swelling of growing cells that are rendered incapable of cell division. Although both methods are subjective and can result in false positive results, the DVC method is best suited to analysis of waters in which the number of different types of organisms present in the same sample is assumed to be small, such as processed waters. The advantages and disadvantages of each method are discussed.

Rodgers, E. B.↗

NASA NTRS1992

Applications of a direct/iterative design method to complex transonic configurations

The current study explores the use of an automated direct/iterative design method for the reduction of drag in transport configurations, including configurations with engine nacelles. The method requires the user to choose a proper target-pressure distribution and then develops a corresponding airfoil section. The method can be applied to two-dimensional airfoil sections or to three-dimensional wings. The three cases that are presented show successful application of the method for reducing drag from various sources. The first two cases demonstrate the use of the method to reduce induced drag by designing to an elliptic span-load distribution and to reduce wave drag by decreasing the shock strength for a given lift. In the second case, a body-mounted nacelle is added and the method is successfully used to eliminate increases in wing drag associated with the nacelle addition by designing to an arbitrary pressure distribution as a result of the redesigning of a wing in combination with a given underwing nacelle to clean-wing, target-pressure distributions. These cases illustrate several possible uses of the method for reducing different types of drag. The magnitude of the obtainable drag reduction varies with the constraints of the problem and the configuration to be modified.

Smith, Leigh Ann↗

NASA NTRS1992

The solution of coupled Schroedinger equations using an extrapolation method

In this paper, extrapolation to the limit in a finite-difference method is applied to solve a system of coupled Schroedinger equations. This combination results in a method that only requires knowledge of the potential energy functions for the system. This numerical procedure has several distinct advantages over the more conventional methods. Namely, initial guesses for the term values are not needed; assumptions need be made about the behavior of the wavefunctions, such as the slope or magnitude in the nonclassical region; and the algorithm is easy to implement, has a firm mathematical foundation, and provides error estimates. Moreover, the method is less sensitive to round-off error than other methods since a small number of mesh points is used and it can be implemented on small computers. A comparison of the method with another numerical method shows results agreeing within 1 part in 10 exp 4.

Goorvitch, D.↗

NASA NTRS1992

The structures, binding energies and vibrational frequencies of Ca3 and Ca4: An application of the CCSD(T) method

The Ca3 and Ca4 metallic clusters have been investigated using state-of-the-art ab initio quantum mechanical methods. Large atomic natural orbital basis sets have been used in conjunction with the singles and doubles coupled-cluster (CCSD) method, a coupled-cluster method that includes a perturbational estimate of connected triple excitations, denoted CCSD(T), and the multireference configuration interaction (MRCI) method. The equilibrium geometries, binding energies and harmonic vibrational frequencies have been determined with each of the methods so that the accuracy of the coupled-cluster methods may be assessed. Since the CCSD(T) method reproduces the MRCI results very well, cubic and quartic force fields of Ca3 and Ca4 have been determined using this approach and used to evaluate the fundamental vibrational frequencies. The infrared intensities of both the e' mode of Ca3 and the t2 mode of Ca4 are found to be small. The results obtained in this study are compared and contrasted with those from our earlier studies on small Be and Mg clusters.

Lee, Timothy J.↗

NASA NTRS1992

An alternate method for achieving temperature control in the -130 C to 75 C range

Thermal vacuum testing often requires temperature control of chamber shrouds and heat exchangers within the -130 C to 75 C range. There are two conventional methods which are normally employed to achieve control through this intermediate temperature range: (1) single-pass flow where control is achieved by alternately pulsing hot gaseous nitrogen (GN2) and cold LN2 into the feed line to yield the setpoint temperature; and (2) closed-loop circulation where control is achieved by either electrically heating or LN2 cooling the circulating GN2 to yield the setpoint temperature. A third method, using a mass flow ratio controller along with modulating control valves on GN2 and LN2 lines, provides excellent control but equipment for this method is expensive and cost-prohibitive for all but long-term continuous processes. The single-pass method provides marginal control and can result in unexpected overcooling of the test article from even a short pulse of LN2. The closed-loop circulation method provides excellent control but requires an expensive blower capable of operating at elevated pressures and cryogenic temperatures. Where precise control is needed (plus or minus 2 C), single-pass flow systems typically have not provided the precision required, primarily because of overcooling temperature excursions. Where several individual circuits are to be controlled at different temperatures, the use of expensive cryogenic blowers for each circuit is also cost-prohibitive, especially for short duration of one-of-a-kind tests. At JPL, a variant of the single-pass method was developed that was shown to provide precise temperature control in the -130 C to 75 C range while exhibiting minimal setpoint overshoot during temperature transitions. This alternate method uses a commercially available temperature controller along with a GN2/LN2 mixer to dampen the amplitude of cold temperature spikes caused by LN2 pulsing. The design of the GN2/LN2 mixer, the overall control system configuration, the operational procedure, and the prototype system test results are described.

Johnson, Kenneth R.↗

NASA NTRS1992

Runge-Kutta methods combined with compact difference schemes for the unsteady Euler equations

Recent development using compact difference schemes to solve the Navier-Stokes equations show spectral-like accuracy. A study was made of the numerical characteristics of various combinations of the Runge-Kutta (RK) methods and compact difference schemes to calculate the unsteady Euler equations. The accuracy of finite difference schemes is assessed based on the evaluations of dissipative error. The objectives are reducing the numerical damping and, at the same time, preserving numerical stability. While this approach has tremendous success solving steady flows, numerical characteristics of unsteady calculations remain largely unclear. For unsteady flows, in addition to the dissipative errors, phase velocity and harmonic content of the numerical results are of concern. As a result of the discretization procedure, the simulated unsteady flow motions actually propagate in a dispersive numerical medium. Consequently, the dispersion characteristics of the numerical schemes which relate the phase velocity and wave number may greatly impact the numerical accuracy. The aim is to assess the numerical accuracy of the simulated results. To this end, the Fourier analysis is to provide the dispersive correlations of various numerical schemes. First, a detailed investigation of the existing RK methods is carried out. A generalized form of an N-step RK method is derived. With this generalized form, the criteria are derived for the three and four-step RK methods to be third and fourth-order time accurate for the non-linear equations, e.g., flow equations. These criteria are then applied to commonly used RK methods such as Jameson's 3-step and 4-step schemes and Wray's algorithm to identify the accuracy of the methods. For the spatial discretization, compact difference schemes are presented. The schemes are formulated in the operator-type to render themselves suitable for the Fourier analyses. The performance of the numerical methods is shown by numerical examples. These examples are detailed. described. The third case is a two-dimensional simulation of a Lamb vortex in an uniform flow. This calculation provides a realistic assessment of various finite difference schemes in terms of the conservation of the vortex strength and the harmonic content after travelling a substantial distance. The numerical implementation of Giles' non-refelctive equations coupled with the characteristic equations as the boundary condition is discussed in detail. Finally, the single vortex calculation is extended to simulate vortex pairing. For the distance between two vortices less than a threshold value, numerical results show crisp resolution of the vortex merging.

Yu, Sheng-Tao↗

NASA NTRS1992

An extended Lagrangian method

A unique formulation of describing fluid motion is presented. The method, referred to as 'extended Lagrangian method', is interesting from both theoretical and numerical points of view. The formulation offers accuracy in numerical solution by avoiding numerical diffusion resulting from mixing of fluxes in the Eulerian description. Meanwhile, it also avoids the inaccuracy incurred due to geometry and variable interpolations used by the previous Lagrangian methods. Unlike the Lagrangian method previously imposed which is valid only for supersonic flows, the present method is general and capable of treating subsonic flows as well as supersonic flows. The method proposed in this paper is robust and stable. It automatically adapts to flow features without resorting to clustering, thereby maintaining rather uniform grid spacing throughout and large time step. Moreover, the method is shown to resolve multi-dimensional discontinuities with a high level of accuracy, similar to that found in one-dimensional problems.

Liou, Meng-Sing↗

NASA NTRS1993

A new flux conserving Newton's method scheme for the two-dimensional, steady Navier-Stokes equations

A new numerical method is developed for the solution of the two-dimensional, steady Navier-Stokes equations. The method that is presented differs in significant ways from the established numerical methods for solving the Navier-Stokes equations. The major differences are described. First, the focus of the present method is on satisfying flux conservation in an integral formulation, rather than on simulating conservation laws in their differential form. Second, the present approach provides a unified treatment of the dependent variables and their unknown derivatives. All are treated as unknowns together to be solved for through simulating local and global flux conservation. Third, fluxes are balanced at cell interfaces without the use of interpolation or flux limiters. Fourth, flux conservation is achieved through the use of discrete regions known as conservation elements and solution elements. These elements are not the same as the standard control volumes used in the finite volume method. Fifth, the discrete approximation obtained on each solution element is a functional solution of both the integral and differential form of the Navier-Stokes equations. Finally, the method that is presented is a highly localized approach in which the coupling to nearby cells is only in one direction for each spatial coordinate, and involves only the immediately adjacent cells. A general third-order formulation for the steady, compressible Navier-Stokes equations is presented, and then a Newton's method scheme is developed for the solution of incompressible, low Reynolds number channel flow. It is shown that the Jacobian matrix is nearly block diagonal if the nonlinear system of discrete equations is arranged approximately and a proper pivoting strategy is used. Numerical results are presented for Reynolds numbers of 100, 1000, and 2000. Finally, it is shown that the present scheme can resolve the developing channel flow boundary layer using as few as six to ten cells per channel width, depending on the Reynolds number.

Scott, James R.↗

NASA NTRS1991

Multiple methods integration for structural mechanics analysis and design

A new research area of multiple methods integration is proposed for joining diverse methods of structural mechanics analysis which interact with one another. Three categories of multiple methods are defined: those in which a physical interface are well defined; those in which a physical interface is not well-defined, but selected; and those in which the interface is a mathematical transformation. Two fundamental integration procedures are presented that can be extended to integrate various methods (e.g., finite elements, Rayleigh Ritz, Galerkin, and integral methods) with one another. Since the finite element method will likely be the major method to be integrated, its enhanced robustness under element distortion is also examined and a new robust shell element is demonstrated.

Housner, J. M.↗

NASA NTRS1993

A Discrete Probability Function Method for the Equation of Radiative Transfer

A discrete probability function (DPF) method for the equation of radiative transfer is derived. The DPF is defined as the integral of the probability density function (PDF) over a discrete interval. The derivation allows the evaluation of the PDF of intensities leaving desired radiation paths including turbulence-radiation interactions without the use of computer intensive stochastic methods. The DPF method has a distinct advantage over conventional PDF methods since the creation of a partial differential equation from the equation of transfer is avoided. Further, convergence of all moments of intensity is guaranteed at the basic level of simulation unlike the stochastic method where the number of realizations for convergence of higher order moments increases rapidly. The DPF method is described for a representative path with approximately integral-length scale-sized spatial discretization. The results show good agreement with measurements in a propylene/air flame except for the effects of intermittency resulting from highly correlated realizations. The method can be extended to the treatment of spatial correlations as described in the Appendix. However, information regarding spatial correlations in turbulent flames is needed prior to the execution of this extension.

Sivathanu, Y. R.↗

NASA NTRS1992

Consensus theoretic classification methods

Consensus theory is adopted as a means of classifying geographic data from multiple sources. The foundations and usefulness of different consensus theoretic methods are discussed in conjunction with pattern recognition. Weight selections for different data sources are considered and modeling of non-Gaussian data is investigated. The application of consensus theory in pattern recognition is tested on two data sets: 1) multisource remote sensing and geographic data and 2) very-high-dimensional remote sensing data. The results obtained using consensus theoretic methods are found to compare favorably with those obtained using well-known pattern recognition methods. The consensus theoretic methods can be applied in cases where the Gaussian maximum likelihood method cannot. Also, the consensus theoretic methods are computationally less demanding than the Gaussian maximum likelihood method and provide a means for weighting data sources differently.

Benediktsson, Jon A.↗

NASA NTRS1991

The L sub 1 finite element method for pure convection problems

Jiang, Bo-Nan↗

NASA NTRS1992

Accelerated complete-linearization method for calculating NLTE model stellar atmospheres

Two approaches to accelerating the method of complete linearization for calculating NLTE model stellar atmospheres are suggested. The first one, the so-called Kantorovich variant of the Newton-Raphson method, consists of keeping the Jacobi matrix of the system fixed, which allows us to calculate the costly matrix inversions only a few times and then keep them fixed during the subsequent computations. The second method is an application of the Ng acceleration. Both methods are extremely easy to implement with any model atmosphere code based on complete linearization. It is demonstrated that both methods, and especially their combination, yield a rapidly and globally convergent algorithm, which takes 2 to 5 times less computer time, depending on the model at hand and the required accuracy, than the ordinary complete linearization. Generally, the time gain is more significant for more complicated models. The methods were tested for a broad range of atmospheric parameters, and in all cases they exhibited similar behavior. Ng acceleration applied on the Kantorovich variant thus offers a significant improvement of the standard complete-linearization method, and may now be used for calculating relatively involved NLTE model stellar atmospheres.

Hubeny, I.↗

NASA NTRS1993

Parallel methods for dynamic simulation of multiple manipulator systems

In this paper, efficient dynamic simulation algorithms for a system of m manipulators, cooperating to manipulate a large load, are developed; their performance, using two possible forms of parallelism on a general-purpose parallel computer, is investigated. One form, temporal parallelism, is obtained with the use of parallel numerical integration methods. A speedup of 3.78 on four processors of CRAY Y-MP8 was achieved with a parallel four-point block predictor-corrector method for the simulation of a four manipulator system. These multi-point methods suffer from reduced accuracy, and when comparing these runs with a serial integration method, the speedup can be as low as 1.83 for simulations with the same accuracy. To regain the performance lost due to accuracy problems, a second form of parallelism is employed. Spatial parallelism allows most of the dynamics of each manipulator chain to be computed simultaneously. Used exclusively in the four processor case, this form of parallelism in conjunction with a serial integration method results in a speedup of 3.1 on four processors over the best serial method. In cases where there are either more processors available or fewer chains in the system, the multi-point parallel integration methods are still advantageous despite the reduced accuracy because both forms of parallelism can then combine to generate more parallel tasks and achieve greater effective speedups. This paper also includes results for these cases.

Mcmillan, Scott↗

NASA NTRS1993

Compressible turbulent flow simulation with a multigrid multiblock method

We describe a multigrid multiblock method for compressible turbulent flow simulations and present results obtained from calculations on a two-element airfoil. A vertex-based spatial discretization method and explicit multistage Runge-Kutta time-stepping are used. The slow convergence of a single grid method makes the multigrid method, which yields a speed factor of about 20, indispensable. The numerical predictions are in good agreement with experimental results. It is shown that the convergence of the multigrid process depends considerably on the ordering of the various loops. If the block loop is put inside the stage loop, the process converges more rapidly than if the block loop is situated outside the stage loop when a three-stage Runge-Kutta method is used. If a five-stage scheme is used, the process does not converge in the latter block ordering. Finally, the process based on the five-stage method is about 60 percent more efficient than with the three-stage method, if the block loop is inside the stage loop.

Kuerten, Hans↗