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NASA NTRS1988

An improved V-Lambda solution of the matrix Riccati equation

The authors present an improved algorithm for computing the V-Lambda solution of the matrix Riccati equation. The improvement is in the reduction of the computational load, results from the orthogonality of the eigenvector matrix that has to be solved for. The orthogonality constraint reduces the number of independent parameters which define the matrix from n-squared to n (n - 1)/2. The authors show how to specify the parameters, how to solve for them and how to form from them the needed eigenvector matrix. In the search for suitable parameters, the analogy between the present problem and the problem of attitude determination is exploited, resulting in the choice of Rodrigues parameters.

Bar-Itzhack, Itzhack Y.↗

NASA NTRS1989

Stability robustness improvement of direct eigenspace assignment based feedback systems using singular value sensitivities

A methodology to improve the stability robustness of feedback control systems designed using direct eigenspace assignment techniques is presented. The method consists of considering the sensitivity of the minimum singular value of the return difference transfer matrix at the plant input to small changes in the desired closed-loop eigenvalues and the specified elements of the desired closed-loop eigenvectors. Closed-form expressions for the gradient of the minimum return difference singular value with respect to desired closed-loop eigenvalue and eigenvector parameters are derived. Closed-form expressions for the gradients of the control feedback gains with respect to the specified eigenspace parameters are obtained as an intermediate step. The use of the gradient information to improve the guaranteed gain and phase margins in eigenspace assignment based designs is demonstrated by application to an advanced fighter aircraft.

Garg, Sanjay↗

NASA NTRS1990

More About Multiple-Boundary-Condition Testing

Measured shapes of vibrational modes used to update mathematical models. Report extends discussion of multiple-boundary-condition vibrational testing techniques. Emphasis on further refinement of mathematical model by use of differences between measured eigenvectors and eigenvectors predicted by model to be refined.

Kuo, Chin-Po↗

NASA NTRS1990

Computational aspects of sensitivity calculations in linear transient structural analysis

A study was performed focusing on the calculation of sensitivities of displacements, velocities, accelerations, and stresses in linear, structural, transient response problems. One significant goal of the study was to develop and evaluate sensitivity calculation techniques suitable for large-order finite element analyses. Accordingly, approximation vectors such as vibration mode shapes are used to reduce the dimensionality of the finite element model. Much of the research focused on the accuracy of both response quantities and sensitivities as a function of number of vectors used. Two types of sensitivity calculation techniques were developed and evaluated. The first type of technique is an overall finite difference method where the analysis is repeated for perturbed designs. The second type of technique is termed semi-analytical because it involves direct, analytical differentiation of the equations of motion with finite difference approximation of the coefficient matrices. To be computationally practical in large-order problems, the overall finite difference methods must use the approximation vectors from the original design in the analyses of the perturbed models. In several cases this fixed mode approach resulted in very poor approximations of the stress sensitivities. Almost all of the original modes were required for an accurate sensitivity and for small numbers of modes, the accuracy was extremely poor. To overcome this poor accuracy, two semi-analytical techniques were developed. The first technique accounts for the change in eigenvectors through approximate eigenvector derivatives. The second technique applies the mode acceleration method of transient analysis to the sensitivity calculations. Both result in accurate values of the stress sensitivities with a small number of modes and much lower computational costs than if the vibration modes were recalculated and then used in an overall finite difference method.

Greene, William H.↗

NASA NTRS1990

Dynamic substructuring by the boundary flexibility vector method of component mode synthesis

Component mode synthesis (CMS) is a method of dynamic analysis, for structures having a large number of degrees of freedom (DOF). These structures often required lengthy computer CPU time and large computer memory resources, if solved directly by the finite-element method (FEM). In CMS, the structure is divided into independent components in which the DOF are defined by a set of generalized coordinates defined by displacement shapes. The number of the generalized coordinates are much less than the original number of physical DOF, in the component. The displacement shapes are used to transform the component property matrices and any applied external loads, to a reduced system of coordinates. Reduced system property matrices are assembled, and any type of dynamic analysis is carried out in the reduced coordinate system. Any obtained results are back transformed to the original component coordinate systems. In all conventional methods of CMS, the mode shapes used for components are dynamic mode shapes, supplemented by static deflected shapes. Historically, all the dynamic mode shapes used in conventional CMS are the natural modes (eigenvectors) of components. A new method of CMS, namely the boundary flexibility vector method of CMS, is presented. The method provides for the incorporation of a set of static Ritz vectors, referred to as boundary flexibility vectors, as a replacement and/or supplement to conventional eigenvectors, as displacement shapes for components. The generation of these vectors does not require the solution of a costly eigenvalue problem, as in the case of natural modes in conventional CMS, and hence a substantial saving in CPU time can be achieved. The boundary flexibility vectors are generated from flexibility (or stiffness) properties of components. The formulation presented is for both free and fixed-interface components, and for both the free and forced vibration problems. Free and forced vibration numerical examples are presented to verify the accuracy of the method and the saving in CPU time. Compared to conventional methods of CMS, the results indicate that by using the new method, more accurate results can be obtained with a substantial saving in CPU time.

Abdallah, Ayman Ahmed↗

NASA NTRS1990

Parameter identification of aeroelastic modes of rotary wings from transient time histories

A novel moving-block technique is presented for parameter-identification tasks concerning the aeroelastic modes of rotary wings, using a least-squares solver in the time domain to estimate dampings and eigenvector components. The least-squares moving-block (LSMB) technique is shown to yield superior estimates to those of the previous moving-block technique for four different analytical test cases. An additional advantage of the LSMB approach is its ability to estimate eigenvector components with good accuracy; this feature is useful in assessing the coupling of the different modes. The method can also be used to perform periodic modal identification, and is applicable to experimental data for the extraction of modal information.

Amrani, Ahmed Omar↗

NASA NTRS1992

Robust Assignment Of Eigensystems For Flexible Structures

Improved method for placement of eigenvalues and eigenvectors of closed-loop control system by use of either state or output feedback. Applied to reduced-order finite-element mathematical model of NASA's MAST truss beam structure. Model represents deployer/retractor assembly, inertial properties of Space Shuttle, and rigid platforms for allocation of sensors and actuators. Algorithm formulated in real arithmetic for efficient implementation. Choice of open-loop eigenvector matrix and its closest unitary matrix believed suitable for generating well-conditioned eigensystem with small control gains. Implication of this approach is that element of iterative search for "optimal" unitary matrix appears unnecessary in practice for many test problems.

Juang, Jer-Nan↗

NASA NTRS1992

Conservative-variable average states for equilibrium gas multi-dimensional fluxes

Modern split component evaluations of the flux vector Jacobians are thoroughly analyzed for equilibrium-gas average-state determinations. It is shown that all such derivations satisfy a fundamental eigenvalue consistency theorem. A conservative-variable average state is then developed for arbitrary equilibrium-gas equations of state and curvilinear-coordinate fluxes. Original expressions for eigenvalues, sound speed, Mach number, and eigenvectors are then determined for a general average Jacobian, and it is shown that the average eigenvalues, Mach number, and eigenvectors may not coincide with their classical pointwise counterparts. A general equilibrium-gas equation of state is then discussed for conservative-variable computational fluid dynamics (CFD) Euler formulations. The associated derivations lead to unique compatibility relations that constrain the pressure Jacobian derivatives. Thereafter, alternative forms for the pressure variation and average sound speed are developed in terms of two average pressure Jacobian derivatives. Significantly, no additional degree of freedom exists in the determination of these two average partial derivatives of pressure. Therefore, they are simultaneously computed exactly without any auxiliary relation, hence without any geometric solution projection or arbitrary scale factors. Several alternative formulations are then compared and key differences highlighted with emphasis on the determination of the pressure variation and average sound speed. The relevant underlying assumptions are identified, including some subtle approximations that are inherently employed in published average-state procedures. Finally, a representative test case is discussed for which an intrinsically exact average state is determined. This exact state is then compared with the predictions of recent methods, and their inherent approximations are appropriately quantified.

Iannelli, G. S.↗

NASA NTRS1990

Intercomparison between microwave and infrared observations of precipitable water

The possible utility of observations by the TIROS-N Vertical Sounder system as a data base for a global climatology of atmospheric precipitable water is assessed. A comparison is described between the atmospheric precipitable water content (PWC) derived from IR observations by the High-Resolution Infrared Radiation Sounder (HIRS/2) of the TOVS system using the eigenvector regression technique and derived from microwave observations by the Special Sensor Microwave/Imager (SSM/I). It is suggested that the difference between the two data sets can largely be attributed to the eigenvector method used to retrieve PWC from the HIRS/2 observations. It is shown that in the regions of subsidence associated with the air motion around the subtropical zone of high pressure, the HIRS/2 tends to overestimate the PWC.

Tjemkes, Stephen A.↗

NASA NTRS1992

Structural damage detection using a subspace rotation algorithm

A computationally inexpensive algorithm is developed to provide an insight to the location of structural damage, using the original finite element model and a subset of measured eigenvalues and eigenvectors. The computational requirements of the algorithm may make the technique suitable for real-time implementation. With damage location determined, a second algorithm is developed to determine the extent of damage. The algorithms are demonstrated using two classes of structural models. The effects of eigenvector measurement error is demonstrated and techniques to overcome the effects of noise are discussed.

Zimmerman, David C.↗

NASA NTRS1992

Stability investigations of airfoil flow by global analysis

As the result of global, non-parallel flow stability analysis the single value of the disturbance growth-rate and respective frequency is obtained. This complex value characterizes the stability of the whole flow configuration and is not referred to any particular flow pattern. The global analysis assures that all the flow elements (wake, boundary and shear layer) are taken into account. The physical phenomena connected with the wake instability are properly reproduced by the global analysis. This enhances the investigations of instability of any 2-D flows, including ones in which the boundary layer instability effects are known to be of dominating importance. Assuming fully 2-D disturbance form, the global linear stability problem is formulated. The system of partial differential equations is solved for the eigenvalues and eigenvectors. The equations, written in the pure stream function formulation, are discretized via FDM using a curvilinear coordinate system. The complex eigenvalues and corresponding eigenvectors are evaluated by an iterative method. The investigations performed for various Reynolds numbers emphasize that the wake instability develops into the Karman vortex street. This phenomenon is shown to be connected with the first mode obtained from the non-parallel flow stability analysis. The higher modes are reflecting different physical phenomena as for example Tollmien-Schlichting waves, originating in the boundary layer and having the tendency to emerge as instabilities for the growing Reynolds number. The investigations are carried out for a circular cylinder, oblong ellipsis and airfoil. It is shown that the onset of the wake instability, the waves in the boundary layer, the shear layer instability are different solutions of the same eigenvalue problem, formulated using the non-parallel theory. The analysis offers large potential possibilities as the generalization of methods used till now for the stability analysis.

Morzynski, Marek↗

NASA NTRS1992

Improved procedures for eigenvalue approximation and sensitivities for active structures

The equations of motion of actively controlled structures are usually reduced to first order form, a procedure which obscures symmetries that are present in the second order form of these equations. The loss of symmetry requires the calculation of both left and right eigenvectors for obtaining derivatives of the stability eigenvalues of the system. The paper shows that, for some control laws, only right eigenvectors are required if derivatives are obtained from the second order form of the equations of motion. The paper also examines reduced basis approximations for the eigenvalues and their derivatives. It is shown that including Ritz vectors representing the effect of local actuation forces in the reduced basis improves the accuracy of the eigenvalues and eigenvalue derivatives. Two active truss examples are used for demonstrating the improved accuracy.

Ponslet, Eric↗

NASA NTRS1993

Controls-structures integrated design optimization with shape variations

The shape design variables have been introduced into the set of design variables of the Controls-Structure Integrated (CSI) Design of space-structures. The importance of the shape variations in improving the design (obtained with only control and sizing variables) has been aptly illustrated. Two different types of design variables that describe the shape variations of the structure have been introduced. In the first case, the nodal coordinates have been considered as design variables. This has the inherent difficulty of having too many design variables. This not only is time consuming but also memory intensive and may not yield a manufacturable shape to the structure. The second approach has been introduced to overcome this difficulty. The structure is allowed to vary in a particular pre defined pattern. The coefficients of these patterns are considered as the shape design variables. The eigenvalue and eigenvector sensitivity equations with respect to these coefficient design variables have been developed and are used to approximate the eigenvalues and eigenvectors in a perturbed design.

Koganti, Gopichand↗

NASA NTRS1988

A proposed method for enhanced eigen-pair extraction using finite element methods: Theory and application

The paper covers two distinct parts: theory and application. The goal of this work was the reduction of model size with an increase in eigenvalue/vector accuracy. This method is ideal for the condensation of large truss- or beam-type structures. The theoretical approach involves the conversion of a continuum transfer matrix beam element into an 'Exact' dynamic stiffness element. This formulation is implemented in a finite element environment. This results in the need to solve a transcendental eigenvalue problem. Once the eigenvalue is determined the eigenvectors can be reconstructed with any desired spatial precision. No discretization limitations are imposed on the reconstruction. The results of such a combined finite element and transfer matrix formulation is a much smaller FEM eigenvalue problem. This formulation has the ability to extract higher eigenvalues as easily and as accurately as lower eigenvalues. Moreover, one can extract many more eigenvalues/vectors from the model than the number of degrees of freedom in the FEM formulation. Typically, the number of eigenvalues accurately extractable via the 'Exact' element method are at least 8 times the number of degrees of freedom. In contrast, the FEM usually extracts one accurate (within 5 percent) eigenvalue for each 3-4 degrees of freedom. The 'Exact' element results in a 20-30 improvement in the number of accurately extractable eigenvalues and eigenvectors.

Jara-Almonte, J.↗

NASA NTRS1993

Application of the target decomposition theorem to a polarimetric random media model

With advances in polarimetric radar measurements of land surfaces, the need for understanding the underlying scattering mechanisms and dominant target features has become the focus of many studies. In particular, the maximum use of the polarimetric information to identify and/or separate parameters related to the surface features such as vegetation thickness, structure, water content, and soil surface characteristics will enhance the possibility of using polarimetric radars for monitoring the earth's surface from space. In this paper, Cloude's decomposition theorem is applied to a polarimetric random media model to simulate the radar measurements of vegetated canopies. The vegetated canopies are modeled as a three layer discrete random medium with leaves and branches in the first layer, tree trunks in the second layer, and a half space of homogeneous ground with rough interface as the bottom layer. The distorted born approximation (DBA) has been used to compute full Mueller matrix of the canopy, using canonical dielectric objects such as thin discs and cylinders as leaves, branches, and trunks, respectively. The Mueller matrix and the derived covariance matrix contain information on the second order statistics of radar signals at various polarizations from the canopy. To decompose the covariance matrix to its constituent targets, the eigenvalues and eigenvectors of the covariance matrix are computed in terms of the physical parameters of the canopy. In addition, each eigenvector explicitly shows the scattering mechanisms such as odd and even reflections in the canopy. Cloude's decomposition theorem is applied using the Pauli spin matrices as a basis and an expression for the degree of disorder or the entropy for the vegetated surface is found. Then, the physical parameters estimated from in situ measurements are used in the random media to obtain realistic covariance matrices. As a result, the sensitivity of the eigenvalue spectrums and the coefficients resulting from the target decomposition theorem to the physical parameters of the canopy are examined and the possible use of Cloude's theorem to estimate vegetation and soil parameters is discussed.

Saatchi, Sasan S.↗

NASA NTRS1994

The algebraic multigrid projection for eigenvalue problems; backrotations and multigrid fixed points

The periods of the theorem for the algebraic multigrid projection (MGP) for eigenvalue problems, and of the multigrid fixed point theorem for multigrid cycles combining MGP with backrotations, are presented. The MGP and the backrotations are central eigenvector separation techniques for multigrid eigenvalue algorithms. They allow computation on coarse levels of eigenvalues of a given eigenvalue problem, and are efficient tools in the computation of eigenvectors.

Costiner, Sorin↗

NASA NTRS1995

Modeling of nonequilibrium space plasma flows

Godunov-type numerical solution of the 20 moment plasma transport equations. One of the centerpieces of our proposal was the development of a higher order Godunov-type numerical scheme to solve the gyration dominated 20 moment transport equations. In the first step we explored some fundamental analytic properties of the 20 moment transport equations for a low b plasma, including the eigenvectors and eigenvalues of propagating disturbances. The eigenvalues correspond to wave speeds, while the eigenvectors characterize the transported physical quantities. In this paper we also explored the physically meaningful parameter range of the normalized heat flow components. In the second step a new Godunov scheme type numerical method was developed to solve the coupled set of 20 moment transport equations for a quasineutral single-ion plasma. The numerical method and the first results were presented at several national and international meetings and a paper describing the method has been published in the Journal of Computational Physics. To our knowledge this is the first numerical method which is capable of producing stable time-dependent solutions to the full 20 (or 16) moment set of transport equations, including the full heat flow equation. Previous attempts resulted in unstable (oscillating) solutions of the heat flow equations. Our group invested over two man-years into the development and implementation of the new method. The present model solves the 20 moment transport equations for an ion species and thermal electrons in 8 domain extending from a collision dominated to a collisionless region (200 km to 12,000 km). This model has been applied to study O+ acceleration due to Joule heating in the lower ionosphere.

Gombosi, Tamas↗

NASA NTRS1993

A spectral algorithm for envelope reduction of sparse matrices

A new algorithm for reducing the envelope of a sparse matrix is presented. This algorithm is based on the computation of eigenvectors of the Laplacian matrix associated with the graph of the sparse matrix. A reordering of the sparse matrix is determined based on the numerical values of the entries of an eigenvector of the Laplacian matrix. Numerical results show that the new reordering algorithm can in some cases reduce the envelope by more than a factor of two over the current standard algorithms such as Gibbs-Poole-Stockmeyer (GPS) or SPARSPAK's reverse Cuthill-McKee (RCM).

Barnard, Stephen T.↗