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NASA NTRS1995

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. The present method and the Arbitrary Lagrangian-Eulerian (ALE) method have a similarity in spirit-eliminating the cross-streamline numerical diffusion. For this purpose, we suggest a simple grid constraint condition and utilize an accurate discretization procedure. This grid constraint is only applied to the transverse cell face parallel to the local stream velocity, and hence our method for the steady state problems naturally reduces to the streamline-curvature method, without explicitly solving the steady stream-coordinate equations formulated a priori. Unlike the Lagrangian method proposed by Loh and Hui which is valid only for steady supersonic flows, the present method is general and capable of treating subsonic flows and supersonic flows as well as unsteady flows, simply by invoking in the same code an appropriate grid constraint suggested in this paper. The approach is found to be 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 NTRS1997

Runge-Kutta Methods for Linear Ordinary Differential Equations

Three new Runge-Kutta methods are presented for numerical integration of systems of linear inhomogeneous ordinary differential equations (ODES) with constant coefficients. Such ODEs arise in the numerical solution of the partial differential equations governing linear wave phenomena. The restriction to linear ODEs with constant coefficients reduces the number of conditions which the coefficients of the Runge-Kutta method must satisfy. This freedom is used to develop methods which are more efficient than conventional Runge-Kutta methods. A fourth-order method is presented which uses only two memory locations per dependent variable, while the classical fourth-order Runge-Kutta method uses three. This method is an excellent choice for simulations of linear wave phenomena if memory is a primary concern. In addition, fifth- and sixth-order methods are presented which require five and six stages, respectively, one fewer than their conventional counterparts, and are therefore more efficient. These methods are an excellent option for use with high-order spatial discretizations.

Zingg, David W.↗

NASA NTRS1975

Advances in Application of Fast Semidirect Computational Methods in Transonic Flow

This paper is intended as a review and summary of the advances made in a recently developed approach for rapid numerical solution of the equations of inviscid transonic aerodynamics. The investigation has been limited to two-dimensional, steady, inviscid flow over airfoils in a subsonic free stream, with emphasis on development of a rapid computational technique, rather than on generality of application. The approach uses finite-difference algorithms called "fast direct elliptic solvers" within an iteration scheme. "Direct" means that the entire computation field is solved at once, rather than in successive traverses over the field as in a point- or line-relaxation method. Such an iterative method is referred to as "semidirect." The iterative convergence can be faster than in other relaxation methods because changes are felt simultaneously at all points in each succeeding iteration. Direct elliptic solvers and semidirect methods have restrictions, but these are gradually being removed. Direct solvers were first developed for solving Poisson's equation on a rectangle without interior boundaries. A method to treat first-order systems, a direct Cauchy-Riemann solver has also been developed. Numerical treatment of part of a system of nonlinear equations by a Poisson solver has been reported. Also Poisson solvers in semidirect methods were used for nonseparable elliptic equations. The semidirect method was extended to the solution of a problem of mixed type, where the improved Murman-Cole transonic small-disturbance difference equations were solved. A slightly supercritical flow over a biconvex airfoil was treated successfully, but the iterations did not converge for more strongly supercritical conditions In another work the addition of terms ot both sides of the difference equations stabilized the iteration for supercritical conditions with large supersonic zones. For this, the Cauchy-Riemann solver was revised to incl,ude the needed terms. Most recently, the evaluation of parameters for rapid convergence and comparisons, with Murman's line-relaxation method was described. The method was extended to full second order accuracy in a fully conservative formulation in another work.

Martin, E. Dale↗

NASA NTRS2000

High Order Discontinuous Gelerkin Methods for Convection Dominated Problems with Application to Aeroacoustics

This project is about the investigation of the development of the discontinuous Galerkin finite element methods, for general geometry and triangulations, for solving convection dominated problems, with applications to aeroacoustics. On the analysis side, we have studied the efficient and stable discontinuous Galerkin framework for small second derivative terms, for example in Navier-Stokes equations, and also for related equations such as the Hamilton-Jacobi equations. This is a truly local discontinuous formulation where derivatives are considered as new variables. On the applied side, we have implemented and tested the efficiency of different approaches numerically. Related issues in high order ENO and WENO finite difference methods and spectral methods have also been investigated. Jointly with Hu, we have presented a discontinuous Galerkin finite element method for solving the nonlinear Hamilton-Jacobi equations. This method is based on the RungeKutta discontinuous Galerkin finite element method for solving conservation laws. The method has the flexibility of treating complicated geometry by using arbitrary triangulation, can achieve high order accuracy with a local, compact stencil, and are suited for efficient parallel implementation. One and two dimensional numerical examples are given to illustrate the capability of the method. Jointly with Hu, we have constructed third and fourth order WENO schemes on two dimensional unstructured meshes (triangles) in the finite volume formulation. The third order schemes are based on a combination of linear polynomials with nonlinear weights, and the fourth order schemes are based on combination of quadratic polynomials with nonlinear weights. We have addressed several difficult issues associated with high order WENO schemes on unstructured mesh, including the choice of linear and nonlinear weights, what to do with negative weights, etc. Numerical examples are shown to demonstrate the accuracies and robustness of the methods for shock calculations. Jointly with P. Montarnal, we have used a recently developed energy relaxation theory by Coquel and Perthame and high order weighted essentially non-oscillatory (WENO) schemes to simulate the Euler equations of real gas. The main idea is an energy decomposition under the form epsilon = epsilon(sub 1) + epsilon(sub 2), where epsilon(sub 1) is associated with a simpler pressure law (gamma)-law in this paper) and the nonlinear deviation epsilon(sub 2) is convected with the flow. A relaxation process is performed for each time step to ensure that the original pressure law is satisfied. The necessary characteristic decomposition for the high order WENO schemes is performed on the characteristic fields based on the epsilon(sub l) gamma-law. The algorithm only calls for the original pressure law once per grid point per time step, without the need to compute its derivatives or any Riemann solvers. Both one and two dimensional numerical examples are shown to illustrate the effectiveness of this approach.

Shu, Chi-Wang↗

NASA NTRS2001

Comparison of Two Acoustic Waveguide Methods for Determining Liner Impedance

Acoustic measurements taken in a flow impedance tube are used to assess the relative accuracy of two waveguide methods for impedance eduction in the presence of grazing flow. The aeroacoustic environment is assumed to contain forward and backward-traveling acoustic waves, consisting of multiple modes, and uniform mean flow. Both methods require a measurement of the complex acoustic pressure profile over the length of the test liner. The Single Mode Method assumes that the sound pressure level and phase decay-rates of a single progressive mode can be extracted from this measured complex acoustic pressure profile. No a priori assumptions are made in the Finite Element. Method regarding the modal or reflection content in the measured acoustic pressure profile. The integrity of each method is initially demonstrated by how well their no-flow impedances match those acquired in a normal incidence impedance tube. These tests were conducted using ceramic tubular and conventional perforate liners. Ceramic tubular liners were included because of their impedance insensitivity to mean flow effects. Conversely, the conventional perforate liner was included because its impedance is known to be sensitive to mean flow velocity effects. Excellent comparisons between impedance values educed with the two waveguide methods in the absence of mean flow and the corresponding values educed with the normal incident impedance tube were observed. The two methods are then compared for mean flow Mach numbers up to 0.5, and are shown to give consistent results for both types of test liners. The quality of the results indicates that the Single Mode Method should be used when the measured acoustic pressure profile is clearly dominated by a single progressive mode, and the Finite Element Method should be used for all other cases.

Jones, Michael G.↗

NASA NTRS2001

On Multifunctional Collaborative Methods in Engineering Science

Multifunctional methodologies and analysis procedures are formulated for interfacing diverse subdomain idealizations including multi-fidelity modeling methods and multi-discipline analysis methods. These methods, based on the method of weighted residuals, ensure accurate compatibility of primary and secondary variables across the subdomain interfaces. Methods are developed using diverse mathematical modeling (i.e., finite difference and finite element methods) and multi-fidelity modeling among the subdomains. Several benchmark scalar-field and vector-field problems in engineering science are presented with extensions to multidisciplinary problems. Results for all problems presented are in overall good agreement with the exact analytical solution or the reference numerical solution. Based on the results, the integrated modeling approach using the finite element method for multi-fidelity discretization among the subdomains is identified as most robust. The multiple method approach is advantageous when interfacing diverse disciplines in which each of the method's strengths are utilized.

Ransom, Jonathan B.↗

NASA NTRS1995

A General Method for Solving Systems of Non-Linear Equations

The method of steepest descent is modified so that accelerated convergence is achieved near a root. It is assumed that the function of interest can be approximated near a root by a quadratic form. An eigenvector of the quadratic form is found by evaluating the function and its gradient at an arbitrary point and another suitably selected point. The terminal point of the eigenvector is chosen to lie on the line segment joining the two points. The terminal point found lies on an axis of the quadratic form. The selection of a suitable step size at this point leads directly to the root in the direction of steepest descent in a single step. Newton's root finding method not infrequently diverges if the starting point is far from the root. However, the current method in these regions merely reverts to the method of steepest descent with an adaptive step size. The current method's performance should match that of the Levenberg-Marquardt root finding method since they both share the ability to converge from a starting point far from the root and both exhibit quadratic convergence near a root. The Levenberg-Marquardt method requires storage for coefficients of linear equations. The current method which does not require the solution of linear equations requires more time for additional function and gradient evaluations. The classic trade off of time for space separates the two methods.

Nachtsheim, Philip R.↗

NASA NTRS2002

Meshless Local Petrov-Galerkin Method for Bending Problems

Recent literature shows extensive research work on meshless or element-free methods as alternatives to the versatile Finite Element Method. One such meshless method is the Meshless Local Petrov-Galerkin (MLPG) method. In this report, the method is developed for bending of beams - C1 problems. A generalized moving least squares (GMLS) interpolation is used to construct the trial functions, and spline and power weight functions are used as the test functions. The method is applied to problems for which exact solutions are available to evaluate its effectiveness. The accuracy of the method is demonstrated for problems with load discontinuities and continuous beam problems. A Petrov-Galerkin implementation of the method is shown to greatly reduce computational time and effort and is thus preferable over the previously developed Galerkin approach. The MLPG method for beam problems yields very accurate deflections and slopes and continuous moment and shear forces without the need for elaborate post-processing techniques.

Phillips, Dawn R.↗

NASA NTRS2002

On the Formulation of Weakly Singular Displacement/Traction Integral Equations; and Their Solution by the MLPG Method

In this paper, a very simple method is used to derive the weakly singular traction boundary integral equation based on the integral relationships for displacement gradients. The concept of the MLPG method is employed to solve the integral equations, especially those arising in solid mechanics. A moving Least Squares (MLS) interpolation is selected to approximate the trial functions in this paper. Five boundary integral Solution methods are introduced: direct solution method; displacement boundary-value problem; traction boundary-value problem; mixed boundary-value problem; and boundary variational principle. Based on the local weak form of the BIE, four different nodal-based local test functions are selected, leading to four different MLPG methods for each BIE solution method. These methods combine the advantages of the MLPG method and the boundary element method.

Atluri, Satya N.↗

NASA NTRS2002

Three-Dimensional High-Order Spectral Finite Volume Method for Unstructured Grids

Many areas require a very high-order accurate numerical solution of conservation laws for complex shapes. This paper deals with the extension to three dimensions of the Spectral Finite Volume (SV) method for unstructured grids, which was developed to solve such problems. We first summarize the limitations of traditional methods such as finite-difference, and finite-volume for both structured and unstructured grids. We then describe the basic formulation of the spectral finite volume method. What distinguishes the SV method from conventional high-order finite-volume methods for unstructured triangular or tetrahedral grids is the data reconstruction. Instead of using a large stencil of neighboring cells to perform a high-order reconstruction, the stencil is constructed by partitioning each grid cell, called a spectral volume (SV), into 'structured' sub-cells, called control volumes (CVs). One can show that if all the SV cells are partitioned into polygonal or polyhedral CV sub-cells in a geometrically similar manner, the reconstructions for all the SVs become universal, irrespective of their shapes, sizes, orientations, or locations. It follows that the reconstruction is reduced to a weighted sum of unknowns involving just a few simple adds and multiplies, and those weights are universal and can be pre-determined once for all. The method is thus very efficient, accurate, and yet geometrically flexible. The most critical part of the SV method is the partitioning of the SV into CVs. In this paper we present the partitioning of a tetrahedral SV into polyhedral CVs with one free parameter for polynomial reconstructions up to degree of precision five. (Note that the order of accuracy of the method is one order higher than the reconstruction degree of precision.) The free parameter will be determined by minimizing the Lebesgue constant of the reconstruction matrix or similar criteria to obtain optimized partitions. The details of an efficient, parallelizable code to solve three-dimensional problems for any order of accuracy are then presented. Important aspects of the data structure are discussed. Comparisons with the Discontinuous Galerkin (DG) method are made. Numerical examples for wave propagation problems are presented.

Liu, Yen↗

NASA NTRS2003

Deployment Simulation Methods for Ultra-Lightweight Inflatable Structures

Two dynamic inflation simulation methods are employed for modeling the deployment of folded thin-membrane tubes. The simulations are necessary because ground tests include gravity effects and may poorly represent deployment in space. The two simulation methods are referred to as the Control Volume (CV) method and the Arbitrary Lagrangian Eulerian (ALE) method. They are available in the LS-DYNA nonlinear dynamic finite element code. Both methods are suitable for modeling the interactions between the inflation gas and the thin-membrane tube structures. The CV method only considers the pressure induced by the inflation gas in the simulation, while the ALE method models the actual flow of the inflation gas. Thus, the transient fluid properties at any location within the tube can be predicted by the ALE method. Deployment simulations of three packaged tube models; namely coiled, Z-folded, and telescopically-folded configurations, are performed. Results predicted by both methods for the telescopically-folded configuration are correlated and computational efficiency issues are discussed.

Wang, John T.↗

NASA NTRS1998

Single-Transducer, Ultrasonic Imaging Method for High-Temperature Structural Materials Eliminates the Effect of Thickness Variation in the Image

NASA Lewis Research Center's Life Prediction Branch, in partnership with Sonix, Inc., and Cleveland State University, recently advanced the development of, refined, and commercialized an advanced nondestructive evaluation (NDE) inspection method entitled the Single Transducer Thickness-Independent Ultrasonic Imaging Method. Selected by R&D Magazine as one of the 100 most technologically significant new products of 1996, the method uses a single transducer to eliminate the superimposing effects of thickness variation in the ultrasonic images of materials. As a result, any variation seen in the image is due solely to microstructural variation. This nondestructive method precisely and accurately characterizes material gradients (pore fraction, density, or chemical) that affect the uniformity of a material's physical performance (mechanical, thermal, or electrical). Advantages of the method over conventional ultrasonic imaging include (1) elimination of machining costs (for precision thickness control) during the quality control stages of material processing and development and (2) elimination of labor costs and subjectivity involved in further image processing and image interpretation. At NASA Lewis, the method has been used primarily for accurate inspections of high temperature structural materials including monolithic ceramics, metal matrix composites, and polymer matrix composites. Data were published this year for platelike samples, and current research is focusing on applying the method to tubular components. The initial publicity regarding the development of the method generated 150 requests for further information from a wide variety of institutions and individuals including the Federal Bureau of Investigation (FBI), Lockheed Martin Corporation, Rockwell International, Hewlett Packard Company, and Procter & Gamble Company. In addition, NASA has been solicited by the 3M Company and Allison Abrasives to use this method to inspect composite materials that are manufactured by these companies.

Roth, Don J.↗

NASA NTRS2001

Development of a Hybrid RANS/LES Method for Turbulent Mixing Layers

Significant research has been underway for several years in NASA Glenn Research Center's nozzle branch to develop advanced computational methods for simulating turbulent flows in exhaust nozzles. The primary efforts of this research have concentrated on improving our ability to calculate the turbulent mixing layers that dominate flows both in the exhaust systems of modern-day aircraft and in those of hypersonic vehicles under development. As part of these efforts, a hybrid numerical method was recently developed to simulate such turbulent mixing layers. The method developed here is intended for configurations in which a dominant structural feature provides an unsteady mechanism to drive the turbulent development in the mixing layer. Interest in Large Eddy Simulation (LES) methods have increased in recent years, but applying an LES method to calculate the wide range of turbulent scales from small eddies in the wall-bounded regions to large eddies in the mixing region is not yet possible with current computers. As a result, the hybrid method developed here uses a Reynolds-averaged Navier-Stokes (RANS) procedure to calculate wall-bounded regions entering a mixing section and uses a LES procedure to calculate the mixing-dominated regions. A numerical technique was developed to enable the use of the hybrid RANS-LES method on stretched, non-Cartesian grids. With this technique, closure for the RANS equations is obtained by using the Cebeci-Smith algebraic turbulence model in conjunction with the wall-function approach of Ota and Goldberg. The LES equations are closed using the Smagorinsky subgrid scale model. Although the function of the Cebeci-Smith model to replace all of the turbulent stresses is quite different from that of the Smagorinsky subgrid model, which only replaces the small subgrid turbulent stresses, both are eddy viscosity models and both are derived at least in part from mixing-length theory. The similar formulation of these two models enables the RANS and LES equations to be solved with a single solution scheme and computational grid. The hybrid RANS-LES method has been applied to a benchmark compressible mixing layer experiment in which two isolated supersonic streams, separated by a splitter plate, provide the flows to a constant-area mixing section. Although the configuration is largely two dimensional in nature, three-dimensional calculations were found to be necessary to enable disturbances to develop in three spatial directions and to transition to turbulence. The flow in the initial part of the mixing section consists of a periodic vortex shedding downstream of the splitter plate trailing edge. This organized vortex shedding then rapidly transitions to a turbulent structure, which is very similar to the flow development observed in the experiments. Although the qualitative nature of the large-scale turbulent development in the entire mixing section is captured well by the LES part of the current hybrid method, further efforts are planned to directly calculate a greater portion of the turbulence spectrum and to limit the subgrid scale modeling to only the very small scales. This will be accomplished by the use of higher accuracy solution schemes and more powerful computers, measured both in speed and memory capabilities.

Georgiadis, Nicholas J.↗

NASA NTRS2008

Object Recognition using Feature- and Color-Based Methods

An improved adaptive method of processing image data in an artificial neural network has been developed to enable automated, real-time recognition of possibly moving objects under changing (including suddenly changing) conditions of illumination and perspective. The method involves a combination of two prior object-recognition methods one based on adaptive detection of shape features and one based on adaptive color segmentation to enable recognition in situations in which either prior method by itself may be inadequate. The chosen prior feature-based method is known as adaptive principal-component analysis (APCA); the chosen prior color-based method is known as adaptive color segmentation (ACOSE). These methods are made to interact with each other in a closed-loop system to obtain an optimal solution of the object-recognition problem in a dynamic environment. One of the results of the interaction is to increase, beyond what would otherwise be possible, the accuracy of the determination of a region of interest (containing an object that one seeks to recognize) within an image. Another result is to provide a minimized adaptive step that can be used to update the results obtained by the two component methods when changes of color and apparent shape occur. The net effect is to enable the neural network to update its recognition output and improve its recognition capability via an adaptive learning sequence. In principle, the improved method could readily be implemented in integrated circuitry to make a compact, low-power, real-time object-recognition system. It has been proposed to demonstrate the feasibility of such a system by integrating a 256-by-256 active-pixel sensor with APCA, ACOSE, and neural processing circuitry on a single chip. It has been estimated that such a system on a chip would have a volume no larger than a few cubic centimeters, could operate at a rate as high as 1,000 frames per second, and would consume in the order of milliwatts of power.

Duong, Tuan↗

NASA NTRS2010

Comparison of Iterative and Non-Iterative Strain-Gage Balance Load Calculation Methods

The accuracy of iterative and non-iterative strain-gage balance load calculation methods was compared using data from the calibration of a force balance. Two iterative and one non-iterative method were investigated. In addition, transformations were applied to balance loads in order to process the calibration data in both direct read and force balance format. NASA's regression model optimization tool BALFIT was used to generate optimized regression models of the calibration data for each of the three load calculation methods. This approach made sure that the selected regression models met strict statistical quality requirements. The comparison of the standard deviation of the load residuals showed that the first iterative method may be applied to data in both the direct read and force balance format. The second iterative method, on the other hand, implicitly assumes that the primary gage sensitivities of all balance gages exist. Therefore, the second iterative method only works if the given balance data is processed in force balance format. The calibration data set was also processed using the non-iterative method. Standard deviations of the load residuals for the three load calculation methods were compared. Overall, the standard deviations show very good agreement. The load prediction accuracies of the three methods appear to be compatible as long as regression models used to analyze the calibration data meet strict statistical quality requirements. Recent improvements of the regression model optimization tool BALFIT are also discussed in the paper.

Ulbrich, N.↗

NASA NTRS2010

Comparison of Traditional Design Nonlinear Programming Optimization and Stochastic Methods for Structural Design

Structural design generated by traditional method, optimization method and the stochastic design concept are compared. In the traditional method, the constraints are manipulated to obtain the design and weight is back calculated. In design optimization, the weight of a structure becomes the merit function with constraints imposed on failure modes and an optimization algorithm is used to generate the solution. Stochastic design concept accounts for uncertainties in loads, material properties, and other parameters and solution is obtained by solving a design optimization problem for a specified reliability. Acceptable solutions were produced by all the three methods. The variation in the weight calculated by the methods was modest. Some variation was noticed in designs calculated by the methods. The variation may be attributed to structural indeterminacy. It is prudent to develop design by all three methods prior to its fabrication. The traditional design method can be improved when the simplified sensitivities of the behavior constraint is used. Such sensitivity can reduce design calculations and may have a potential to unify the traditional and optimization methods. Weight versus reliabilitytraced out an inverted-S-shaped graph. The center of the graph corresponded to mean valued design. A heavy design with weight approaching infinity could be produced for a near-zero rate of failure. Weight can be reduced to a small value for a most failure-prone design. Probabilistic modeling of load and material properties remained a challenge.

Patnaik, Surya N.↗

NASA NTRS2011

Collocation and Galerkin Time-Stepping Methods

We study the numerical solutions of ordinary differential equations by one-step methods where the solution at tn is known and that at t(sub n+1) is to be calculated. The approaches employed are collocation, continuous Galerkin (CG) and discontinuous Galerkin (DG). Relations among these three approaches are established. A quadrature formula using s evaluation points is employed for the Galerkin formulations. We show that with such a quadrature, the CG method is identical to the collocation method using quadrature points as collocation points. Furthermore, if the quadrature formula is the right Radau one (including t(sub n+1)), then the DG and CG methods also become identical, and they reduce to the Radau IIA collocation method. In addition, we present a generalization of DG that yields a method identical to CG and collocation with arbitrary collocation points. Thus, the collocation, CG, and generalized DG methods are equivalent, and the latter two methods can be formulated using the differential instead of integral equation. Finally, all schemes discussed can be cast as s-stage implicit Runge-Kutta methods.

Huynh, H. T.↗

NASA NTRS2011

Flight Test Results of a GPS-Based Pitot-Static Calibration Method Using Output-Error Optimization for a Light Twin-Engine Airplane

As part of the NASA Aviation Safety Program (AvSP), a novel pitot-static calibration method was developed to allow rapid in-flight calibration for subscale aircraft while flying within confined test areas. This approach uses Global Positioning System (GPS) technology coupled with modern system identification methods that rapidly computes optimal pressure error models over a range of airspeed with defined confidence bounds. This method has been demonstrated in subscale flight tests and has shown small 2- error bounds with significant reduction in test time compared to other methods. The current research was motivated by the desire to further evaluate and develop this method for full-scale aircraft. A goal of this research was to develop an accurate calibration method that enables reductions in test equipment and flight time, thus reducing costs. The approach involved analysis of data acquisition requirements, development of efficient flight patterns, and analysis of pressure error models based on system identification methods. Flight tests were conducted at The University of Tennessee Space Institute (UTSI) utilizing an instrumented Piper Navajo research aircraft. In addition, the UTSI engineering flight simulator was used to investigate test maneuver requirements and handling qualities issues associated with this technique. This paper provides a summary of piloted simulation and flight test results that illustrates the performance and capabilities of the NASA calibration method. Discussion of maneuver requirements and data analysis methods is included as well as recommendations for piloting technique.

Martos, Borja↗