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Integrating matrices are derived for arbitrarily spaced grid points using either interpolating or least squares fit orthogonal polynomials. Several features of the equally spaced grid case are also discussed.
Techniques for defining a finite element model (FEM) for analysis of nonlinear vibrations in beam structures subjected to harmonic excitation are presented. The resulting model covers longitudinal deformation and inertial effects. The nonlinear oscillations of a beam element under forced excitation are modeled by a harmonic force matrix based on first order approximations of the Jacobian elliptic forcing function. Harmonic force and nonlinear stiffness matrices are derived and the nonlinear forced responses of beams are calculated under various boundary conditions. The results of FEM computations for simply-supported and clamped beams show that midplane stretching caused by large deflections increases the nonlinearity. Axially-restrained beams experience only hardening nonlinearity, while axially-free beams have reduced nonlinearity in deformation and inertia and an increase in linearity due to large deflection.
Finite element stiffness matrices are derived using an extended Hellinger-Reissner principle in which internal displacements are added to serve as Lagrange multipliers to introduce the equilibrium constraint in each element. In a consistent formulation the assumed stresses are initially unconstrained and complete polynomials and the total displacements are also complete such that the corresponding strains are complete in the same order as the stresses. Several examples indicate that resulting properties for elements constructed by this consistent formulation are ideal and are less sensitive to distortions of element geometries. The method has been used to find the optimal stress terms for plane elements, 3-D solids, axisymmetric solids, and plate bending elements.
A simple and effective linear C(0)-tiangular element for plates and shells is developed on the basis of the Hellinger-Reissner mixed variational principle with independently assumed stress and displacement fields. Two main features are emphasized in this development concerning the assumed transverse shear stress field. First, this assumption results in edge-type penalty constraints in the thin plate/shell regimes. Second, the element is used in the form of a four-triangle, cross-diagonal macroelement mesh. As a result, the element is shown to be free from shear locking in thin plate/shell applications, and it satisfies all the appropriate patch tests required for Kirchhoff plate models. In addition, the element exhibits good overall convergence properties in a variety of test problems for plates and shells. Finally, from the computational standpoint, the element is very efficient since all stiffness component matrices are derived in explicit forms.
This paper presents a finite element method for the solution of Navier-Stokes equations with the assumption of thermodynamic and chemical equilibrium. The method employs an upwind finite element technique with an implicit time-marching scheme for the solution, and uses an adaptively generated unstructured triangular mesh with several layers of quadrilateral elements near solid walls. The complexity associated with the assumption that the flow is in equilibrium is treated consistently, and the inviscid flux Jacobian matrices are derived. Several problems involving inviscid and viscous hypersonic flow past blunt are solved. Results are compared with other numerical results and experimental data, and generally good agreement is observed.
Load and stress recovery from transient dynamic studies are improved upon using an extended acceleration vector in the modal acceleration technique applied to structural analysis. Extension of the normal LTM (load transformation matrices) stress recovery to automatically compute margins of safety is presented with an application to the Hubble space telescope.
Fibers formed from blends of silsesquioxane polymers were characterized to study the pyrolytic conversion of these precursors to ceramics. The morphology of fibers pyrolyzed to 1400 C revealed primarily amorphous glasses whose conversion to beta-SiC is a function of both blend composition and pyrolysis conditions. Formation of beta-SiC crystallites within the glassy phase is favored by higher than stoichiometric C/Si ratios, while carbothermal reduction of Si-O bonds to form SiC with loss of SiO and CO occurs at higher methyl/phenylpropyl silsesquioxane (lower C/Si) ratios. As the carbothermal reduction is assumed to be diffusion controlled, the fibers can serve as model systems to gain understanding of the silsesquioxane pyrolysis behavior, and therefore are useful in the development of polysilsesquioxane-derived ceramic matrices and coatings as well.
The effect of additional unlabeled samples in improving the supervised learning process is studied in this paper. Three learning processes. supervised, unsupervised, and combined supervised-unsupervised, are compared by studying the asymptotic behavior of the estimates obtained under each process. Upper and lower bounds on the asymptotic covariance matrices are derived. It is shown that under a normal mixture density assumption for the probability density function of the feature space, the combined supervised-unsupervised learning is always superior to the supervised learning in achieving better estimates. Experimental results are provided to verify the theoretical concepts.
Observed correlations between atmospheric concentrations of CO2 and CO represent potentially powerful information for improving CO2 surface flux estimates through coupled CO2-CO inverse analyses. We explore the value of these correlations in improving estimates of regional CO2 fluxes in east Asia by using aircraft observations of CO2 and CO from the TRACE-P campaign over the NW Pacific in March 2001. Our inverse model uses regional CO2 and CO surface fluxes as the state vector, separating biospheric and combustion contributions to CO2. CO2-CO error correlation coefficients are included in the inversion as off-diagonal entries in the a priori and observation error covariance matrices. We derive error correlations in a priori combustion source estimates of CO2 and CO by propagating error estimates of fuel consumption rates and emission factors. However, we find that these correlations are weak because CO source uncertainties are mostly determined by emission factors. Observed correlations between atmospheric CO2 and CO concentrations imply corresponding error correlations in the chemical transport model used as the forward model for the inversion. These error correlations in excess of 0.7, as derived from the TRACE-P data, enable a coupled CO2-CO inversion to achieve significant improvement over a CO2-only inversion for quantifying regional fluxes of CO2.
This paper provides details of developments pertaining to vibration analysis of gyroscopic systems, that involves a finite element structural discretization followed by the solution of the resulting matrix eigenvalue problem by a progressive, accelerated simultaneous iteration technique. Thus Coriolis, centrifugal and geometrical stiffness matrices are derived for shell and line elements, followed by the eigensolution details as well as solution of representative problems that demonstrates the efficacy of the currently developed numerical procedures and tools.
Integrating and differentiating matrices allow the numerical integration and differential of functions whose values are known at points of a discrete grid. Previous derivations of these matrices were restricted to one dimensional grids or to rectangular grids with uniform spacing in at least one direction. Integrating and differentiating matrices were developed for grids with nonuniform spacing in both directions. The use of these matrices as operators to reformulate boundary value problems on rectangular domains as matrix problems for a finite dimensional solution vector is considered. The method requires nonuniform grids which include near boundary points. An eigenvalue problem for the transverse vibrations of a simply supported rectangular plate is solved to illustrate the method.
Integrating and differentiating matrices allow the numerical integration and differential of functions whose values are known at points of a discrete grid. Previous derivations of these matrices were restricted to one dimensional grids or to rectangular grids with uniform spacing in at least one direction. Integrating and differentiating matrices were developed for grids with nonuniform spacing in both directions. The use of these matrices as operators to reformulate boundary value problems on rectangular domains as matrix problems for a finite dimensional solution vector is considered. The method requires nonuniform grids which include near boundary points. An eigenvalue problem for the transverse vibrations of a simply supported rectangular plate is solved to illustrate the method.
The Asymptotic Waveform Evaluation (AWE) technique is applied to a generalized frequency domain electromagnetic problem. Most of the frequency domain techniques in computational electromagnetics result in a matrix equation, which is solved at a single frequency. In the AWE technique, the Taylor series expansion around that frequency is applied to the matrix equation. The coefficients of the Taylor's series are obtained in terms of the frequency derivatives of the matrices evaluated at the expansion frequency. The coefficients hence obtained will be used to predict the frequency response of the system over a frequency range. The detailed derivation of the coefficients (called 'moments') is given along with an illustration for electric field integral equation (or Method of Moments) technique. The radar cross section (RCS) frequency response of a square plate is presented using the AWE technique and is compared with the exact solution at various frequencies.
A prefiltering version of the Kalman filter is derived for both discrete and continuous measurements. The derivation consists of determining a single discrete measurement that is equivalent to either a time segment of continuous measurements or a set of discrete measurements. This prefiltering version of the Kalman filter easily handles numerical problems associated with rapid transients and ill-conditioned Riccati matrices. Therefore, the derived technique for extrapolating the Riccati matrix from one time to the next constitutes a new set of integration formulas which alleviate ill-conditioning problems associated with continuous Riccati equations. Furthermore, since a time segment of continuous measurements is converted into a single discrete measurement, Potter's square root formulas can be used to update the state estimate and its error covariance matrix. Therefore, if having the state estimate and its error covariance matrix at discrete times is acceptable, the prefilter extends square root filtering with all its advantages, to continuous measurement problems.
A continuum model for the SCOLE configuration has been derived using transfer matrices. Controller designs for distributed parameter systems have been analyzed. Pole-assignment controller design is considered easy to implement but stability is not guaranteed. An explicit transfer function of dynamic controllers has been obtained and no model reduction is required before the controller is realized. One specific LQG controller for continuum models had been derived, but other optimal controllers for more general performances need to be studied.
A spectrometric imager and a thermal imager, which achieve multiplexing by the use of binary optical encoding masks, were developed. The masks are based on orthogonal, pseudorandom digital codes derived from Hadamard matrices. Spatial and/or spectral data is obtained in the form of a Hadamard transform of the spatial and/or spectral scene; computer algorithms are then used to decode the data and reconstruct images of the original scene. The hardware, algorithms and processing/display facility are described. A number of spatial and spatial/spectral images are presented. The achievement of a signal-to-noise improvement due to the signal multiplexing was also demonstrated. An analysis of the results indicates both the situations for which the multiplex advantage may be gained, and the limitations of the technique. A number of potential applications of the spectrometric imager are discussed.
An imager and a spectrometric imager, which achieve multiplexing by the use of binary optical encoding masks, have been built and tested. The masks are based on orthogonal, pseudorandom digital codes derived from Hadamard matrices. The spatial (and/or spectral) data are therefore obtained in the form of a Hadamard transform of the spatial (and/or spectral) scene. Computer algorithms are used to decode the data and reconstruct images of the original scene. The hardware, algorithms processing and display facility are described. A number of spatial and spatial/spectral images, obtained in the laboratory, are presented.
Preconditioning matrices for Chebyshev derivative operators in several space dimensions; the Jacobi matrix technique in computational fluid dynamics; and Chebyshev techniques for periodic problems are discussed.