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NASA NTRSDate not supplied

Towards a Third-Order Accurate, Second-Derivative-Free, Shock-Capturing Finite-Volume Method for Hypersonic Flows on Tetrahedral Grids

In this paper, we report progress in the development of a third-order accurate, second-derivative-free, shock-capturing finite-volume solver for three-dimensional unstructured grids. The method is economical in the sense that the computation and storage of second derivatives are not required for third-order accuracy. It is based on point-valued numerical solutions stored at cells, gradients computed and stored at nodes, and an efficient projected-derivative formula that eliminates the need for second derivatives in a quadratic solution interpolation. The projected-derivative formula is also used to eliminate second derivatives from a high-order flux quadrature formula, so that it can be implemented conveniently in the form of a numerical flux at a face center plus a correction term. Similarly, a high-order source quadrature formula can also be implemented in the form of a cell-center point evaluation plus a similar correction term. These features make it relatively straightforward to extend an existing second-order finite-volume code to third-order. This paper reports progress of implementing the method in the NASA VULCAN-CFD code and discusses the implementation of a high-order accurate limiter for shock capturing.

Weighted Least-Squares↗
NASA NTRS1975

Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique

A general fourth order differencing scheme is developed and applied to three viscous test problems to verify the accuracy and applicability of the technique. The procedure is atypical since only three nodes are necessary to obtain the desired fourth order accuracy. This is accomplished by a differencing technique which considers the function and all necessary derivatives as unknowns. The relations for these derivatives yield simple tridiagonal equations which can be easily solved. Comparisons of the fourth order results with those computed using second order methods are presented for the test problems and clearly indicate that the accuracy achieved by these fourth order computations is always significantly better than current second order procedures.

Hirsh, R. S.↗
NASA NTRS1978

The stability of numerical methods for second order ordinary differential equations

An important characterization of a numerical method for first order ODE's is the region of absolute stability. If all eigenvalues of the linear problem dy/dt = Ay are inside this region, the numerical method is stable. If the second order system d/dt(dy/dt) = 2Ady/dt - By is solved as a first order system, the same result applies to the eigenvalues of the generalized eigenvalue problem (lambda-squared)I 2(lambda)A + B. No such region exists for general methods for second order equations, but in some cases a region of absolute stability can be defined for methods for the single second order equation d/dt(dy/dt) = 2ady/dt - by. The absence of a region of absolute stability can occur when different members of a system of first order equations are solved by different methods.

Gear, C. W.↗
NASA NTRS1984

Second-order-accurate spatial differencing for the transonic small-disturbance equation

Current methods for calculating transonic flows with the small-disturbance potential equation are typically only first-order accurate in the supersonic regions of the flow. However, calculations using the full-potential equation show significant improvements in accuracy when second-order methods are used instead of first-order methods. In this paper, algorithms for the small-disturbance equations are presented, for both steady and unsteady flows, with spatial differencing that is second-order accurate in both the subsonic and supersonic regions of the flow. These algorithms are stable, simple extensions of implicit monotone algorithms; the former are only first-order accurate spatially in the supersonic regions. Several calculations in one and two dimensions have been made, and the first- and second-order calculations are compared. In one dimension, the calculations include the four types of shock-wave motion. In two dimensions, the comparisons include steady flow over a Korn airfoil and unsteady flow over a pitching airfoil. All the comparisons in two dimensions show that the new methods are just as stable numerically as the old methods, but improve the accuracy of the solutions. The algorithm improvements can be implemented in present computer codes by making minor coding modifications.

Goorjian, P. M.↗
NASA NTRS1984

Substrate-induced orientational order in the isotropic phase of liquid crystals

Nematic order induced near a solid boundary in an otherwise isotropic liquid crystal is studied theoretically, at temperatures just above the bulk nematic-isotropic phase transition. Three distinct regimes are found, depending on the strength of orientational torques at the boundary: (1) strong orientational order, (2) strong orientational order followed by a first-order transition to a state of weak orientational order as temperature is raised, and (3) a state of weak orientational order.

Mauger, A.↗
NASA NTRS1987

Fixed order compensator design based on frequency shaped cost functionals

This paper treats the design of fixed order compensators using frequency shaped cost functionals. The formulation is done in an output feedback setting which exploits an observer canonical form to represent the compensator dynamics. The major advantages of the design approach are that the order of the compensator is fixed by the design process, and it is not necessary to realize the frequency shaping dynamics as an integral part of the compensator design. This avoids the two-step approach of full state feedback design followed by the design of an observer, which in the case of a full order observer requires a compensator of order n + nf where n is the order of the plant and nf is the order of the frequency shaping realization. The formulation also precludes the use of direct feedback of the plant output, thus reducing the effect of sensor noise and improving the robustness to high frequency unmodeled dynamics. A structural vibration model describing the fast dynamics of a lightweight flexible arm is used as an example to illustrate the design procedure.

Calise, Anthony J.↗
NASA NTRS1988

Finite-dimensional approximation for optimal fixed-order compensation of distributed parameter systems

In controlling distributed parameter systems it is often desirable to obtain low-order, finite-dimensional controllers in order to minimize real-time computational requirements. Standard approaches to this problem employ model/controller reduction techniques in conjunction with LQG theory. In this paper we consider the finite-dimensional approximation of the infinite-dimensional Bernstein/Hyland optimal projection theory. This approach yields fixed-finite-order controllers which are optimal with respect to high-order, approximating, finite-dimensional plant models. The technique is illustrated by computing a sequence of first-order controllers for one-dimensional, single-input/single-output, parabolic (heat/diffusion) and hereditary systems using spline-based, Ritz-Galerkin, finite element approximation. Numerical studies indicate convergence of the feedback gains with less than 2 percent performance degradation over full-order LQG controllers for the parabolic system and 10 percent degradation for the hereditary system.

Bernstein, Dennis S.↗
NASA NTRS1988

Third-order multi-dimensional Euler/Navier-Stokes solver

A nonoscillatory multidimensional advection code is presented which is applicable to the study of very high Reynolds number steady flow, and results are obtained for the oblique advection of a step profile. The scheme is based on an exponential upwinding or linear extrapolation refinement of a previous QUICK scheme, allowing sharp monotonic resolution of the step while retaining third-order accuracy. The resulting simple flux-limiting criteria have been applied to a higher order method and a formally third-order supercompressive algorithm. Results are presented for first-order, second-order, and third-order upwinding.

Leonard, B. P.↗
NASA NTRS1990

Robust eigensystem assignment for state estimators using second-order models

A novel design of a state estimator is presented using second-order dynamic equations of mechanical systems. The eigenvalues and eigenvectors of the state estimator are assigned by solving the second-order eigenvalue problem of the structural system. Three design method for the state estimator are given in this paper. The first design method uses collocated sensors to measure the desired signals and their derivatives. The second design method uses prefilters to shift signal phases to obtain estimates of the signal derivatives. These two methods are used to build a second-order state estimator model. The third design method is the conventional one which converts a typical second-order dynamic model to a first-order model, and then builds a state estimator based on the first-order model. It is shown that all the three designs for state estimation are similar. A numerical example representing a large space structure is given for illustration of the design methods presented in this paper.

Juang, Jer-Nan↗
NASA NTRS1990

A first-order Lyapunov robustness method for linear systems with uncertain parameters

A method for stability-robustness analysis based on a quadratic Liapunov function that varies linearly with uncertainty parameters is derived. Linear time-invariant systems with structured uncertainties are discussed. The Liapunov function is optimized numerically to maximize the robustness region in parameter space. Numerical results are given for four examples in which the first-order method is compared to previous Liapunov methods. While the zero-order method is slightly better than the first-order method for one example, the first-order method is clearly superior in the other three (more realistic) examples. The first-order method is especially superior for the active control of flexible structures, where robustness with respect to (1) unmodeled coupling between modeled modes and (2) unmodeled modes is important. For such applications, the first-order method is much better at detecting the increased robustness associated with increased separation between frequencies.

Leal, M. A.↗
NASA NTRS1994

High-order squeezing of the quantum electromagnetic field and the generalized uncertainty relations in two-mode squeezed states

It is found that two-mode output quantum electromagnetic field in two-mode squeezed states exhibits higher-order squeezing to all even orders. And the generalized uncertainty relations are also presented for the first time. The concept of higher-order squeezing of the single-mode quantum electromagnetic field was first introduced and applied to several processes by Hong and Mandel in 1985. Lately Li Xizeng and Shan Ying have calculated the higher-order squeezing in the process of degenerate four-wave mixing and presented the higher-order uncertainty relations of the fields in single-mode squeezed states. In this paper we generalize the above work to the higher-order squeezing in two-mode squeezed states. The generalized uncertainty relations are also presented for the first time.

Li, Xi-Zeng↗
NASA NTRS1993

Higher-order ionospheric effects on the GPS observables and means of modeling them

Based on relaistic modeling of the electron density of the ionosphere and using a dipole moment approximation for the earth magnetic field, we are able to estimate the effect of the ionosphere on the Global Positioning Systems (GPS) signal for a ground user. The lowest-order (1/f(exp 2)) effect, which is of the order of .1 - 30 meters of zenith group delay, is subtracted out by forming a linear combination of the dual frequencies of the GPS signal. One is left with second- (1/f(exp 3)) and third-order (1/f(exp 4)) effects which are estimated typically to be approximately 0 - 2 cm, and approximately 0 - 2 mm at zenith respectively, depending on the time of day, time of year, the solar cycle and the relative geometry of the magnetic field and the line of sight. Given the total electron content along a line of sight, we derive an approximation to the second-order term which is accurate to approximately 90% within the magnetic dipole moment model; this approximation can be used to reduce the second-order term to the millimeter level, thus potentially improving precise positioning in space and on the ground. The induced group delay, or phase advance, due to second- and third-order effects are examined for two ground receivers located at equatorial and mid-latitude regions tracking several GPS satellites.

Bassiri, Sassan↗
NASA NTRS1995

Testing higher-order Lagrangian perturbation theory against numerical simulations. 2: Hierarchical models

We present results showing an improvement of the accuracy of perturbation theory as applied to cosmological structure formation for a useful range of scales. The Lagrangian theory of gravitational instability of Friedmann-Lemaitre cosmogonies is compared with numerical simulations. We study the dynamics of hierarchical models as a second step. In the first step we analyzed the performance of the Lagrangian schemes for pancake models, the difference being that in the latter models the initial power spectrum is truncated. This work probed the quasi-linear and weakly non-linear regimes. We here explore whether the results found for pancake models carry over to hierarchical models which are evolved deeply into the non-linear regime. We smooth the initial data by using a variety of filter types and filter scales in order to determine the optimal performance of the analytical models, as has been done for the 'Zel'dovich-approximation' - hereafter TZA - in previous work. We find that for spectra with negative power-index the second-order scheme performs considerably better than TZA in terms of statistics which probe the dynamics, and slightly better in terms of low-order statistics like the power-spectrum. However, in contrast to the results found for pancake models, where the higher-order schemes get worse than TZA at late non-linear stages and on small scales, we here find that the second-order model is as robust as TZA, retaining the improvement at later stages and on smaller scales. In view of these results we expect that the second-order truncated Lagrangian model is especially useful for the modelling of standard dark matter models such as Hot-, Cold-, and Mixed-Dark-Matter.

Melott, A. L.↗
NASA NTRS1995

Spectral ordering techniques for incomplete LU preconditoners for CG methods

The effectiveness of an incomplete LU (ILU) factorization as a preconditioner for the conjugate gradient method can be highly dependent on the ordering of the matrix rows during its creation. Detailed justification for two heuristics commonly used in matrix ordering for anisotropic problems is given. The bandwidth reduction and weak connection following heuristics are implemented through an ordering method based on eigenvector computations. This spectral ordering is shown to be a good representation of the heuristics. Analysis and test cases in two and three dimensional diffusion problems demonstrate when ordering is important, and when an ILU decomposition will be ordering insensitive. The applicability of the heuristics is thus evaluated and placed on a more rigorous footing.

Clift, Simon S.↗
NASA NTRS2000

Extension of Efficient Low Dissipative High Order Schemes for 3-D Curvilinear Moving Grids

The efficient low dissipative high order schemes proposed by Yee et al. is formulated for 3-D curvilinear moving grids. These schemes consists of a high order base schemes combined with nonlinear characteristic filters. The amount of numerical dissipation is minimized by applying the schemes to the entropy splitting form of the inviscid flux derivatives. The analysis is given for a thermally perfect gas. The main difficulty in the extension of higher order schemes that were formulated in Cartesian coordinates to curvilinear moving grids is the higher order transformed metric evaluations. The higher order numerical evaluation of the transformed metric terms to insure freestream preservation is done in a coordinate invariant manner. The formulation is an improvement over existing formulation of high order scheme in curvilinear moving grids.

Vinokur, Marcel↗
NASA NTRS2001

A High-Order Finite Spectral Volume Method for Conservation Laws on Unstructured Grids

A time accurate, high-order, conservative, yet efficient method named Finite Spectral Volume (FSV) is developed for conservation laws on unstructured grids. The concept of a 'spectral volume' is introduced to achieve high-order accuracy in an efficient manner similar to spectral element and multi-domain spectral methods. In addition, each spectral volume is further sub-divided into control volumes (CVs), and cell-averaged data from these control volumes is used to reconstruct a high-order approximation in the spectral volume. Riemann solvers are used to compute the fluxes at spectral volume boundaries. Then cell-averaged state variables in the control volumes are updated independently. Furthermore, TVD (Total Variation Diminishing) and TVB (Total Variation Bounded) limiters are introduced in the FSV method to remove/reduce spurious oscillations near discontinuities. A very desirable feature of the FSV method is that the reconstruction is carried out only once, and analytically, and is the same for all cells of the same type, and that the reconstruction stencil is always non-singular, in contrast to the memory and CPU-intensive reconstruction in a high-order finite volume (FV) method. Discussions are made concerning why the FSV method is significantly more efficient than high-order finite volume and the Discontinuous Galerkin (DG) methods. Fundamental properties of the FSV method are studied and high-order accuracy is demonstrated for several model problems with and without discontinuities.

Wang, Z. J.↗
NASA NTRS2002

Efficient Reformulation of the Thermoelastic Higher-order Theory for Fgms

Functionally graded materials (FGMs) are characterized by spatially variable microstructures which are introduced to satisfy given performance requirements. The microstructural gradation gives rise to continuously or discretely changing material properties which complicate FGM analysis. Various techniques have been developed during the past several decades for analyzing traditional composites and many of these have been adapted for the analysis of FGMs. Most of the available techniques use the so-called uncoupled approach in order to analyze graded structures. These techniques ignore the effect of microstructural gradation by employing specific spatial material property variations that are either assumed or obtained by local homogenization. The higher-order theory for functionally graded materials (HOTFGM) is a coupled approach developed by Aboudi et al. (1999) which takes the effect of microstructural gradation into consideration and does not ignore the local-global interaction of the spatially variable inclusion phase(s). Despite its demonstrated utility, however, the original formulation of the higher-order theory is computationally intensive. Herein, an efficient reformulation of the original higher-order theory for two-dimensional elastic problems is developed and validated. The use of the local-global conductivity and local-global stiffness matrix approach is made in order to reduce the number of equations involved. In this approach, surface-averaged quantities are the primary variables which replace volume-averaged quantities employed in the original formulation. The reformulation decreases the size of the global conductivity and stiffness matrices by approximately sixty percent. Various thermal, mechanical, and combined thermomechanical problems are analyzed in order to validate the accuracy of the reformulated theory through comparison with analytical and finite-element solutions. The presented results illustrate the efficiency of the reformulation and its advantages in analyzing functionally graded materials.

Bansal, Yogesh↗
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↗