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Matrix algebra translator and executor /MATE/ takes equations describing structural control system environmental interaction problem for flexible spacecraft components and loads them into self programming computer.
The supersonic flow field over a body of revolution incident to the free stream is simulated numerically on a large, array processor (the CDC Cyber 205). The configuration is composed of a cone-cylinder forebody followed by a conical afterbody from which emanates a centered, supersonic propulsive jet. The free-stream Mach number is 2, the jet-exit Mach number is 2.5, and the jet-to-free-stream static pressure ratio is 3. Both the external flow and the exhaust are ideal air at a common total temperature. The thin-layer approximation to the time-dependent, compressible, Reynolds-averaged Navier-Stokes equations are solved using an implicit finite-difference algorithm. The data base, of 5 million words, is structured in a 'pencil' format so that efficient use of the array processor can be realized. The computer code is completely vectorized to take advantage of the data structure. Turbulence closure is achieved using an empirical algebraic eddy-viscosity model. The configuration and flow conditions correspond to published experimental tests and the computed solutions are consistent with the experimental data.
The generalized eigenvalue problem provides a suitable framework for reliable solutions of many system theoretic, control, and estimation problems. A general algorithm for solving the matrix algebraic Riccati equation (ARE) which utilizes a pencil structure is described here. This algorithm avoids unnecessary inversion of cost or transition matrices, making it a numerically sound way to solve for the gains and/or ARE with singular quadratic costs, for cases satisfying detectability and stabilizability conditions. Examples are solution with discrete dead-beat control, noiseless measurements in Kalman filters and time-delays in discrete-time systems, which cause difficulties in the Hamiltonian standard eigenvalue problem formulation. The ARE algorithm implementatiton and numerical examples are shown.
Topics considered include advances in finite difference techniques for computational fluid dynamics, the spectral element methods for the incompressible Navier-Stokes equations, a review of recent developments in time integration, and advances and trends in element-by-element techniques. Also examined are the algebraic multigrid methods applied to problems in computational structural mechanics, grid generation for the solution of partial differential equations, advances in adaptive improvements, and new computing systems and their impact on computational mechanics.
We have proposed and implemented an extension of the structure-based model for weak deformations. It was shown that the extended model will correctly reduce to the form of standard k-e models for the case of equilibrium under weak mean strain. The realizability of the extended model is guaranteed by the method of its construction. The predictions of the proposed model were very good for rotating homogeneous shear flows and for irrotational axisymmetric contraction, but were seriously deficient in the case of plane strain and axisymmetric expansion. We have concluded that the problem behind these difficulties lies in the algebraic constitutive equation relating the Reynolds stresses to the structure parameters rather than in the slow model developed here. In its present form, this equation assumes that under irrotational strain the principal axes of the Reynolds stresses remain locked onto those of the eddy-axis tensor. This is correct in the RDT limit, but inappropriate under weaker mean strains, when the non-linear eddy-eddy interactions tend to misalign the two sets of principal axes and create some non-zero theta and gamma.
A method is developed for predicting the behavior of three-dimensional, turbulent boundary layers occurring in internal flows, including those on turbomachinery rotor blades. These boundary layers are complex, turbulent, and subject to Coriolis and centrifugal forces. The major thrust of this paper is the development and use of an algebraic Reynolds stress model (ARSM) that captures the changes in turbulent flow structure arising from curvature, rotation, and three dimensionality. The prediction of pressure-driven secondary flow agrees well with the measured data, and all three turbulence models (k-epsilon, algebraic eddy viscosity, and ARSM) show the same level of agreement. The prediction of boundary-layer development on rotor blades shows much better agreement with measurements with the ARSM. It is essential to employ higher-order turbulence models to capture the effects of rotation, curvature, and three dimensionality on boundary layers in turbomachinery.
A new method has been developed to calculate the steady flow and structural deformations for fluid/structure interaction problems. The discretized fluid dynamic and structural equations are regarded as a single set of coupled, nonlinear, algebraic equations. The equilibrium solution is directly obtained using Newton's method. The governing equations used for the fluid flow are the two-dimensional Navier-Stokes equations, and a finite-element model is used to represent the structure. This paper describes the analytical method and presents sample calculations demonstrating the technique. The results show rapid convergence and good agreement with experimental data.
The control of merging aircraft is formulated as a finite-time, quadratic optimal control problem of a linear system with state and control constraints. The purpose of this paper is to demonstrate that the Hybrid Air Traffic Controller (HAC), which has been previously developed as a solution to this problem, may be easily implemented. Use is made of both the properties of the algebraic solution to the matrix Riccati equation and the structure of the linear model. This approach results in a real-time synthesis procedure for the HAC which does not rely on iterative numerical integration techniques.
Structural dynamics is one of the critical disciplines for the successful design, development, and testing of space launch vehicles. It is applied from the smallest component (turbine blades), all the way to the entire vehicle, and has to be calculated for every phase of a mission, from ascent and orbit to landing. Successful application of structural dynamics requires extensive knowledge of Fourier techniques, linear algebra, random variables, finite element modeling, and essentials of SDOF and MDOF vibration theory. Working knowledge of fluid dynamics, statistics, and data analysis also extremely useful.
Multivariable systems such as a finite-element model of vibrating structures, control systems, and large-scale systems are often formulated in terms of differential equations which give rise to lambda matrices. The present investigation is concerned with the formulation of the algebraic theory of lambda matrices and the relationship of latent roots, latent vectors, and latent projectors to the eigenvalues, eigenvectors, and eigenprojectors of the companion form. The chain rule for latent projectors and eigenprojectors for the repeated latent root or eigenvalues is given.
Structural designers today face both opportunities and challenges in a vast array of available analysis and optimization programs. Some programs such as NASTRAN, are very general, permitting the designer to model any structure, to any degree of accuracy, but often at a higher computational cost. Additionally, such general procedures often do not allow easy implementation of all constraints of interest to the designer. Other programs, based on algebraic expressions used by designers one generation ago, have limited applicability for general structures with modem materials. However, when applicable, they provide easy understanding of design decisions trade-off. Finally, designers can also use specialized programs suitable for designing efficiently a subset of structural problems. For example, PASCO and PANDA2 are panel design codes, which calculate response and estimate failure much more efficiently than general-purpose codes, but are narrowly applicable in terms of geometry and loading. Therefore, the problem of optimizing structures based on simultaneous use of several models and computer programs is a subject of considerable interest. The problem of using several levels of models in optimization has been dubbed variable complexity modeling. Work under NASA grant NAG1-1808 has been concerned with the development of variable complexity modeling strategies with special emphasis on response surface techniques. In addition several modeling issues for the design of shells of revolution were studied.
Structural designers today face both opportunities and challenges in a vast array of available analysis and optimization programs. Some programs such as NASTRAN, are very general, permitting the designer to model any structure, to any degree of accuracy, but often at a higher computational cost. Additionally, such general procedures often do not allow easy implementation of all constraints of interest to the designer. Other programs, based on algebraic expressions used by designers one generation ago, have limited applicability for general structures with modem materials. However, when applicable, they provide easy understanding of design decisions trade-off. Finally, designers can also use specialized programs suitable for designing efficiently a subset of structural problems. For example, PASCO and PANDA2 are panel design codes, which calculate response and estimate failure much more efficiently than general-purpose codes, but are narrowly applicable in terms of geometry and loading. Therefore, the problem of optimizing structures based on simultaneous use of several models and computer programs is a subject of considerable interest. The problem of using several levels of models in optimization has been dubbed variable complexity modeling. Work under NASA grant NAG1-2110 has been concerned with the development of variable complexity modeling strategies with special emphasis on response surface techniques. In addition, several modeling issues for the design of shells of revolution were studied.
Approximation Concepts Code for Efficient Structural Synthesis (ACCESS3) combines methods to offer structural system designer powerful, efficient tool for synthesis of minimum-weight designs. Approximation concepts convert general structural synthesis problem into sequence of explicit problems of separable algebraic form; dual method exploits separable form to construct sequence of explicit dual functions.
A simultaneous analysis and design approach to the maximization of buckling or vibration eigenvalues is presented. Both unimodal and bimodal optimum solutions are considered. A discretization of the structure and response is used to obtain sets of nonlinear algebraic equations which are solved numerically. The formulation is applied to the solution of the optimum design of a beam-column supported by an elastic foundation for maximum buckling load. Two cases are considered: the optimum design of the column for a given foundation and the optimum design of the foundation for a given column. Results are compared to published solutions when possible.
Two self-dual (48,24;12) codes are constructed as 6 x 8 matrices whose columns add up to form an extended BCH-Hamming (8,4;4) code and whose rows sum to odd or even parity. The codes constructed have the identical weight structure of the extended quadratic residue code of length 48. Algebraic isomorphisms may exist between pairs of these three codes. However, because of their matrix form, the newly constructed codes are easily correctable for all five-error and many six-error patterns. The first code comes from restricting a binary cyclic (63,18;36) code to a 6 x 7 matrix and then adjoining six dimensions to the extended 6 x 8 matrix. These six dimensions are generated by linear combinations of row permutations of a 6 x 8 matrix of weight 12, whose sums of rows and columns add to one. The second code comes from a slight modification in the parity (eighth) dimension of the Reed-Solomon (8,4;5) code over GF(64). Error correction in both codes uses the row sum parity information to detect errors in the correction algorithm.
This viewgraph presentation discusses turbulence modeling requirements, development philosophy, and approach; two major areas of concentration (high speed and low speed turbulence modeling); high speed turbulence modeling; compressibility effects; turbulence models adapted to USA code; M = 9.2 flat plate flow; Mach 7.05 flow over axisymmetric flare; Mach 8.6 flow over cold wall edge; low speed turbulence modeling; turbulence models being assessed; turbulence model deck structure and integration with Navier-Stokes solver; nonlinear algebraic-stress model; rotation modified k-epsilon model; and Reynolds stress model.
Based on the algebraic Riccati equation approach, this paper presents a simple and flexible method for designing observer-based robust-H-infinity control laws for linear systems with structured parameter uncertainty. The observer-based robust-H-infinity output-feedback control law, obtained by solving three augmented algebraic Riccati equations, provides both robust stability and disturbance attenuation with H-infinity-norm bound for the closed-loop uncertain linear system. Several tuning parameters are embedded into the augmented algebraic Riccati equations so that flexibility in finding the symmetric positive-definite solutions (and hence, the robust-H-infinity control laws) is significantly increased. A benchmark problem associated with a mass-spring system, which approximates the dynamics of a flexible structure, is used to illustrate the design methodologies, and simulation results are presented.
A method is developed for predicting three-dimensional turbulent boundary layers occurring in internal flows, including rotor blades of turbomachinery. These boundary layers are complex, turbulent, and are subject to Coriolis and centrifugal forces. The algebraic Reynolds stress model (ARSM) developed in this paper satisfies the realizability conditions exactly and captures the changes in turbulenet structure arising from curvature and rotation. The prediction of pressure driven secondary flow agrees well with the data and all the three turbulent models (k-e, algebraic eddy viscosity, and ARSM) show the same level of agreement. The prediction of boundary layer on rotor blades shows much better agreement with the ARSM. It is essential to employ the higher order models to capture the effects of rotation and curvature and three-dimensional boundary layers in turbomachinery.