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NASA NTRS1980

Buckling of periodic structures

Equations are developed for the buckling of a general lattice structure that has repetitive geometry. Equilibrium at a typical node is expressed using finite element techniques, and the only assumption is that the response is periodic. By basing the stiffness matrix on the exact solution of the beam column equation, accurate results are obtained for complex buckling behavior that would require a very large system of equations using conventional techniques. The present method requires the eigenvalues of only a 6x6 determinant. The results are used to study the buckling of isogrid cylinders, three-element truss columns and polygonal rings. Details of the analysis including expressions for all terms in the governing stability determinant are given.

Anderson, M. S.↗

NASA NTRS1993

Aspects of a high-resolution scheme for the Navier-Stokes equations

In this paper we emphasize the importance of the form of the numerical dissipation model in computing accurate viscous flow solutions. A high-resolution scheme for viscous flows based on three-point central differencing and a matrix dissipation is considered. The various components of this scheme, including 'entropy fix', limiter function, and boundary-point dissipation are discussed. By analyzing boundary-point dissipation stencils, we confirm that with the matrix dissipation model the normal numerical dissipation terms in the streamwise momentum equation are independent of the Reynolds number. Such independence is not achieved with a scalar dissipation form. The accuracy of the central-difference scheme, with and without matrix dissipation, and the flux-difference split scheme of Roe, which is classified as a high-resolution scheme, is compared. For this comparison, three high Reynolds number laminar flows are considered. Solutions of the Navier-Stokes equations are obtained for low-speed flow over a flat plate, transonic flow over an airfoil with transition near the leading edge, and hypersonic flow over a compression ramp. The emphasis of the comparison is primarily on the details of the viscous flows. The necessity of the high-resolution property is revealed.

Swanson, R. C.↗

NASA NTRS1992

Simplified micromechanical equations for thermal residual stress analysis of coated fiber composites

The fabrication of metal matrix composites poses unique problems to the materials engineer. The large thermal expansion coefficient mismatch between the fiber and matrix leads to high tensile residual stresses at the fiber/matrix interface which could lead to premature matrix cracking during cooldown. Fiber coating could be used to reduce thermal residual stresses. A simple closed-form analysis, based on a three-phase composite cylinder model, was developed to calculate thermal residual stresses in a fiber/interface/matrix system. Guidelines, in the form of simple equations, for the selection of appropriate material properties of the fiber coating, were also derived to minimize thermal residual stresses in the matrix during fabrication.

Naik, Rajiv A.↗

NASA NTRS1981

Reduced order feedback control equations for linear time and frequency domain analysis

An algorithm was developed which can be used to obtain the equations. In a more general context, the algorithm computes a real nonsingular similarity transformation matrix which reduces a real nonsymmetric matrix to block diagonal form, each block of which is a real quasi upper triangular matrix. The algorithm works with both defective and derogatory matrices and when and if it fails, the resultant output can be used as a guide for the reformulation of the mathematical equations that lead up to the ill conditioned matrix which could not be block diagonalized.

Frisch, H. P.↗

NASA NTRS1992

A finite element: Boundary integral method for electromagnetic scattering

A method that combines the finite element and boundary integral techniques for the numerical solution of electromagnetic scattering problems is presented. The finite element method is well known for requiring a low order storage and for its capability to model inhomogeneous structures. Of particular emphasis in this work is the reduction of the storage requirement by terminating the finite element mesh on a boundary in a fashion which renders the boundary integrals in convolutional form. The fast Fourier transform is then used to evaluate these integrals in a conjugate gradient solver, without a need to generate the actual matrix. This method has a marked advantage over traditional integral equation approaches with respect to the storage requirement of highly inhomogeneous structures. Rectangular, circular, and ogival mesh termination boundaries are examined for two-dimensional scattering. In the case of axially symmetric structures, the boundary integral matrix storage is reduced by exploiting matrix symmetries and solving the resulting system via the conjugate gradient method. In each case several results are presented for various scatterers aimed at validating the method and providing an assessment of its capabilities. Important in methods incorporating boundary integral equations is the issue of internal resonance. A method is implemented for their removal, and is shown to be effective in the two-dimensional and three-dimensional applications.

Collins, J. D.↗

NASA NTRS1982

Analysis of Feedback Control Systems

BLOCK IT implements algorithm that obtains reduced-order feedback control equations for use in both time and frequency domain analysis. BLOCK IT computes real nonsingular similarity-transformation matrix-reduces real nonsymmetric matrix to block-diagonal form; each block is real quasi-upper triangular matrix. Algorithm works with defective and derogatory matrices. If program fails, algorithm generates data output used as guide for reformulation of mathematical equations that lead to ill-conditioned matrix not block-diagonalized.

Frisch, H. P.↗

NASA NTRS1967

Matrix Analysis of Longitudinal and Torsional Vibrations in Nonuniform Multibranch Beams

Since longitudinal modes and frequencies provide basic data for dynamic analyses of arbitrary beam-like structures and since closed-form solutions for the modes are generally not feasible to obtain, an approximate method is developed for computing the natural frequencies and the corresponding mode shapes for a variable-section, unconstrained multibranch beam. A lumped mass analogy employing influence coefficients is used to represent the beam. The simultaneous equations of motion for the lumped mass system are derived in matrix form and algebraically manipulated to yield a classical eigenvalue equation solvable by standard procedures. The orthogonality relationship of the natural modes is derived and used to form the basis of an orthogonal sweeping process for determination of modes above the fundamental. Numerical examples including an application to a solid-fuel launch system are presented. Also, a detailed discussion is devoted to the theoretical verifications of the approximate modes and frequencies.

Wingate, Robert T.↗

NASA NTRS2001

Durability and Damage Development in Woven Ceramic Matrix Composites

Damage development in woven SiC/SiNC ceramic matrix composites (CMC's) under tensile and cyclic loading both at room and elevated temperatures have been investigated for the exhaust nozzle of high-efficient turbine engines. The ultimate strength, failure strain, proportional limit and modulus data at a temperature range of 23 to 1250 C are generated. The tensile strength of SiC/SiNC woven composites have been observed to increase with increased temperatures up to 1000 C. The stress/strain plot shows a pseudo-yield point at 25 percent of the failure strain (epsilon(sub r)) which indicates damage initiation in the form of matrix cracking. The evolution of damage beyond 0.25 epsilon(sub f), both at room and elevated temperature comprises multiple matrix cracking, interfacial debonding, and fiber pullout. Although the nature of the stress/strain plot shows damage-tolerant behavior under static loading both at room and elevated temperature, the life expectancy of SiC/SiNC composites degrades significantly under cyclic loading at elevated temperature. This is mostly due to the interactions of fatigue damage caused by the mechanically induced plastic strain and the damage developed by the creep strain. The in situ damage evolutions are monitored by acoustic event parameters, ultrasonic C-scan and stiffness degradation. Rate equations for modulus degradation and fatigue life prediction of ceramic matrix composites both at room and elevated temperatures are developed. These rate equations are observed to show reasonable agreement with experimental results.

Haque, A.↗

NASA NTRS1989

Parallel algorithms for computation of the manipulator inertia matrix

The development of an O(log2N) parallel algorithm for the manipulator inertia matrix is presented. It is based on the most efficient serial algorithm which uses the composite rigid body method. Recursive doubling is used to reformulate the linear recurrence equations which are required to compute the diagonal elements of the matrix. It results in O(log2N) levels of computation. Computation of the off-diagonal elements involves N linear recurrences of varying-size and a new method, which avoids redundant computation of position and orientation transforms for the manipulator, is developed. The O(log2N) algorithm is presented in both equation and graphic forms which clearly show the parallelism inherent in the algorithm.

Amin-Javaheri, Masoud↗

NASA NTRS1980

On time discretizations for spectral methods

New methods are introduced for the time integration of the Fourier and Chebyshev methods of solution for dynamic differential equations. These methods are unconditionally stable, even though no matrix inversions are required. Time steps are chosen by accuracy requirements alone. For the Fourier method both leapfrog and Runge-Kutta methods are considered. For the Chebyshev method only Runge-Kutta schemes are tested. Numerical calculations are presented to verify the analytic results. Applications to the shallow water equations are presented.

Gottlieb, D.↗

NASA NTRS1986

Simplified composite micromechanics for predicting microstresses

A unified set of composite micromechanics equations is summarized and described. This unified set is for predicting the ply microstresses when the ply stresses are known. The set consists of equations of simple form for predicting three-dimensional stresses (six each) in the matrix, fiber, and interface. Several numerical examples are included to illustrate use and computational effectiveness of the equations in this unified set. Numerical results from these examples are discussed with respect to their significance on microcrack formation and, therefore, damage initiation in fiber composites.

Chamis, C. C.↗

NASA NTRS1986

Simplified composite micromechanics for predicting microstresses

Chamis, Christos C.↗

NASA NTRS1987

Simplified composite micromechanics for predicting microstresses

Chamis, Christos C.↗

NASA NTRS1996

Multigrid Methods for Fully Implicit Oil Reservoir Simulation

In this paper we consider the simultaneous flow of oil and water in reservoir rock. This displacement process is modeled by two basic equations: the material balance or continuity equations and the equation of motion (Darcy's law). For the numerical solution of this system of nonlinear partial differential equations there are two approaches: the fully implicit or simultaneous solution method and the sequential solution method. In the sequential solution method the system of partial differential equations is manipulated to give an elliptic pressure equation and a hyperbolic (or parabolic) saturation equation. In the IMPES approach the pressure equation is first solved, using values for the saturation from the previous time level. Next the saturations are updated by some explicit time stepping method; this implies that the method is only conditionally stable. For the numerical solution of the linear, elliptic pressure equation multigrid methods have become an accepted technique. On the other hand, the fully implicit method is unconditionally stable, but it has the disadvantage that in every time step a large system of nonlinear algebraic equations has to be solved. The most time-consuming part of any fully implicit reservoir simulator is the solution of this large system of equations. Usually this is done by Newton's method. The resulting systems of linear equations are then either solved by a direct method or by some conjugate gradient type method. In this paper we consider the possibility of applying multigrid methods for the iterative solution of the systems of nonlinear equations. There are two ways of using multigrid for this job: either we use a nonlinear multigrid method or we use a linear multigrid method to deal with the linear systems that arise in Newton's method. So far only a few authors have reported on the use of multigrid methods for fully implicit simulations. Two-level FAS algorithm is presented for the black-oil equations, and linear multigrid for two-phase flow problems with strong heterogeneities and anisotropies is studied. Here we consider both possibilities. Moreover we present a novel way for constructing the coarse grid correction operator in linear multigrid algorithms. This approach has the advantage in that it preserves the sparsity pattern of the fine grid matrix and it can be extended to systems of equations in a straightforward manner. We compare the linear and nonlinear multigrid algorithms by means of a numerical experiment.

Molenaar, J.↗

NASA NTRS1990

Melt migration in a silicate liquid-olivine system - An experimental test of compaction theory

An experimentally derived melt migration profile is compared with those obtained from compaction theory. A couple is prepared in which a source-for-melt (a disk composed of a silicate glass and olivine) was placed in contact with a sink-for-melt (a disk of polycrystalline olivine) in order to induce melt migration under laboratory conditions. It is noted that melt infiltrates into the sink disk along triple junctions driven by capillary forces. In order to analyze the melt migration profile in terms of compaction theory, the equations developed by McKenzie (1984) to describe porous flow of a liquid in a deformable matrix were modified. The bouyancy force term is replaced with a capillary force term and governing equations are then solved numerically with the initial boundary conditions specified by the experimental design. Also, it is determined that compaction theory provides a good description of the experimental results provided that the permeability of these solid-liquid materials increases linearly with increasing liquid fraction.

Riley, G. N., Jr.↗

NASA NTRS2021

Multi-Frequency NMM3D Simulations of Wave Propagation in Vegetation for Remote Sensing of Soil Moisture

To investigate the feasibility of using multi-frequencies to fur- ther improve the soil moisture retrieval, the recently devel- oped hybrid method is used to perform full-wave scattering simulations of a wheat field at three different frequencies of L-, S- and C- bands. In the hybrid method, the multiple scattering within a single wheat plant is first captured using the T-matrix of vec- tor cylindrical waves calculated from the full-wave solutions of HFSS. In the second step, the scattering among different plants is considered using the Foldy-Lax equations of multiple scattering theory (MST) and the extracted T-matrix. The transmission of microwaves through wheat field at the L-, S- and C-bands are calculated using the hybrid method. Results show that: (1) the transmission obtained from the full-wave simulations is much larger than those computed from the radiative transfer equations (RTE) model, (2) the hybrid method transmission has a weaker frequency dependence than those of the RTE, and (3) the attenuation caused by the vegetation layer would saturate with an increase in frequency.

Yueh, Simon↗

NASA NTRS1998

Preconditioning the Helmholtz Equation for Rigid Ducts

An innovative hyperbolic preconditioning technique is developed for the numerical solution of the Helmholtz equation which governs acoustic propagation in ducts. Two pseudo-time parameters are used to produce an explicit iterative finite difference scheme. This scheme eliminates the large matrix storage requirements normally associated with numerical solutions to the Helmholtz equation. The solution procedure is very fast when compared to other transient and steady methods. Optimization and an error analysis of the preconditioning factors are present. For validation, the method is applied to sound propagation in a 2D semi-infinite hard wall duct.

Baumeister, Kenneth J.↗

NASA NTRS1989

An optimal momentum management controller for the Space Station

This paper presents a new sequential design procedure for determining an optimal CMG (control moment gyro) momentum management and attitude control system for the Space Station Freedom. First, the Space Station equations of motion are linearized and uncoupled, and the associated state space equations are defined. Next, a new sequential procedure is used for the development of a continuous quadratic regulator with eigenvalue placement in a specified region of the complex plane. The regional pole assignment method is utilized since it is best suited for tradeoffs between eigenvalue locations and robustness with respect to parameter variations, sensor failures, implementation accuracies and gain reductions. The matrix sign function is used for solving the Riccati equations which appear in the design procedure. Simulation results are given which show that the resultant design provides desired system performance.

Sunkel, J. W.↗