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This paper presents a framework for the study of the convergence properties of optimal control algorithms and illustrates its use by means of two examples. The framework consists of an algorithm prototype with a convergence theorem, together with some results in relaxed controls theory.
The minimum commitment method has been applied to the analysis of creep-rupture data. The method is based on a parameter representing the focal point of convergence of all isothermals when extended to the long or short times necessary for such convergence to occur. The technique may be applied by manual-graphic analysis on computer code. It is illustrated for the nickel-base alloy Astroloy.
A study of accuracy and convergence of linear functional finite element solution to linear parabolic and hyperbolic partial differential equations is presented. A variable-implicit integration procedure is employed for the resultant system of ordinary differential equations. Accuracy and convergence is compared for the consistent and two lumped assembly procedures for the identified initial-value matrix structure. Truncation error estimation is accomplished using Richardson extrapolation.
An experiment was conducted at static conditions to determine the internal performance effects of nozzle throat contouring, the result of increasing the circular-arc throat radius. Five nonaxisymmetric converging-diverging nozzles were tested at nozzle pressure ratios up to 9.0. Data are presented as internal thrust ratios, discharge coefficients, and static-pressure distributions. Comparisons of internal performance data for the five nozzles show that throat contouring increases the value of discharge coefficient but has no significant effect on internal thrust ratio except in cases of internal flow separation. To illustrate the use of the two dimensional converging-diverging (2-D C-D) nozzle data base, a two dimensional inviscid theory was applied to the five configurations. The generally good agreement of data with theoretical results indicates that two-dimensional inviscid theory can be successfully applied to the prediction of 2-D C-D nozzle internal flow.
Descriptions of the intertropical convergence zones (ITCZ) in the tropical Pacific have been obtained from shipboard measurements of Rn-222, CO, and CH4 in combination with conventional meteorological data and satellite images. The intertropical convergence zone is marked by light shifting waves near an area of heavy cloud cover and precipitation, and appears to be located north and south of the south equatorial current. A 'second' ITCZ with the same atmospheric features was encountered just north of the south equatorial current in the Southern Hemisphere. Atmospheric Rn-222 increases north of the ITCZ and serves as a sensitive indicator for this atmospheric boundary.
The finite element method is applied for the solution of transonic potential flows through a cascade of airfoils. Convergence characteristics of the solution scheme are discussed. Accuracy of the numerical solutions is investigated for various flow regions in the transonic flow configuration. The design of an efficient finite element computational grid is discussed for improving accuracy and convergence.
A computer code based on the wave-envelope technique is used to perform detailed numerical calculations for the intensification and refraction of sound in converging hard walled and lined circular ducts carrying high mean Mach number flows. The results show that converging ducts produce substantial refractions toward the duct center for waves propagating against near choked flows. As expected, the magnitude of the refraction decreases as the real part of the admittance increases. The pressure wave pattern is that of interference among the different modes, and hence the variation of the magnitude of pressure refraction with frequency is not monotonic.
The Galerkin-weighted residuals formulation is employed to derive an implicit finite element solution algorithm for a generally non-linear initial-boundary value problem. Solution accuracy and convergence with discretization refinement are quantized in several error norms, for the non-linear parabolic partial differential equation system governing laminar boundary layer flow, using linear, quadratic and cubic functions. Richardson extrapolation is used to isolate integration truncation error in all norms, and Newton iteration is employed for all equation solutions performed in double-precision. The mathematical theory supporting accuracy and convergence concepts for linear elliptic equations appears extensible to the non-linear equations characteristic of laminar boundary layer flow.
Variations in generalized forces calculated by different computer programs are traced to improper mathematical modeling techniques. Comparisons of generalized forces calculated by three theoretical methods are presented to illustrate difficulties involved in obtaining prediction convergence for increasing wave number. Use of a sufficiently dense chordwise paneling arrangement, in finite panel methods, results in predictions that are essentially identical to predictions of converged solutions. Procedural modifications are suggested for application in finite panel methods to increase prediction accuracy and reduce computer usage costs.
The Galerkin-Weighted Residuals formulation is employed to derive an implicit finite element solution algorithm for the nonlinear parabolic partial differential equation system governing turbulent boundary layer flow. Solution accuracy and convergence with discretization refinement are quantized in several error norms using linear and quadratic basis functions. Richardson extrapolation is used to isolate integration truncation error in all norms, and Newton iteration is employed for all equation solutions performed in double-precision. The mathematical theory supporting accuracy and convergence concepts for linear elliptic equations appears extensible to the nonlinear equations characteristic of turbulent boundary layer flow.
Effect of nonlinearity on convergence of the local linear and global linear adaptive controllers is evaluated. A nonlinear helicopter vibration model is selected for the evaluation which has sufficient nonlinearity, including multiple minimum, to assess the vibration reduction capability of the adaptive controllers. The adaptive control algorithms are based upon a linear transfer matrix assumption and the presence of nonlinearity has a significant effect on algorithm behavior. Simulation results are presented which demonstrate the importance of the caution property in the global linear controller. Caution is represented by a time varying rate weighting term in the local linear controller and this improves the algorithm convergence. Nonlinearity in some cases causes Kalman filter divergence. Two forms of the Kalman filter covariance equation are investigated.
A multiple-grid algorithm for use in efficiently obtaining steady solution to the Euler and Navier-Stokes equations is presented. The convergence of a simple, explicit fine-grid solution procedure is accelerated on a sequence of successively coarser grids by a coarse-grid information propagation method which rapidly eliminates transients from the computational domain. This use of multiple-gridding to increase the convergence rate results in substantially reduced work requirements for the numerical solution of a wide range of flow problems. Computational results are presented for subsonic and transonic inviscid flows and for laminar and turbulent, attached and separated, subsonic viscous flows. Work reduction factors as large as eight, in comparison to the basic fine-grid algorithm, were obtained. Possibilities for further performance improvement are discussed.
The effects of geometric design parameters on two dimensional convergent-divergent nozzles were investigated at nozzle pressure ratios up to 12 in the static test facility. Forward flight (dry and afterburning power settings), vectored-thrust (afterburning power setting), and reverse-thrust (dry power setting) nozzles were investigated. The nozzles had thrust vector angles from 0 deg to 20.26 deg, throat aspect ratios of 3.696 to 7.612, throat radii from sharp to 2.738 cm, expansion ratios from 1.089 to 1.797, and various sidewall lengths. The results indicate that unvectored two dimensional convergent-divergent nozzles have static internal performance comparable to axisymmetric nozzles with similar expansion ratios.
Semi-discrete generalizations of the second order extension of Godunov's scheme, known as the MUSCL scheme, are constructed, starting with any three point ""E" scheme. They are used to approximate scalar conservation laws in one space dimension. For convex conservation laws, each member of a wide class is proven to be a convergent approximation to the correct physical solution. Comparison with another class of high resolution convergent schemes is made.
Semi-discrete generalizations of the second order extension of Godunov's scheme, known as the MUSCL scheme, are constructed, starting with any three point E scheme. They are used to approximate scalar conservation laws in one space dimension. For convex conservation laws, each member of a wide class is proven to be a convergent approximation to the correct physical solution. Comparison with another class of high resolution convergent schemes is made.
The stochastic, infinite time, discrete output feedback problem for time invariant linear systems is examined. Two sets of sufficient conditions for the existence of a stable, globally optimal solution are presented. An expression for the total change in the cost function due to a change in the feedback gain is obtained. This expression is used to show that a sequence of gains can be obtained by an algorithm, so that the corresponding cost sequence is monotonically decreasing and the corresponding sequence of the cost gradient converges to zero. The algorithm is guaranteed to obtain a critical point of the cost function. The computational steps necessary to implement the algorithm on a computer are presented. The results are applied to a digital outer loop flight control problem. The numerical results for this 13th order problem indicate a rate of convergence considerably faster than two other algorithms used for comparison.
A multiple-grid algorithm for use in efficiently obtaining steady solution to the Euler and Navier-Stokes equations is presented. The convergence of a simple, explicit fine-grid solution procedure is accelerated on a sequence of successively coarser grids by a coarse-grid information propagation method which rapidly eliminates transients from the computational domain. This use of multiple-gridding to increase the convergence rate results is substantially reduced work requirements for the numerical solution of a wide range of flow problems. Computational results are presented for subsonic and transonic inviscid flows and for laminar and turbulent, attached and separated, subsonic viscous flows. Work reduction factors as large as eight, in comparison to the basic fine-grid algorithm, were obtained. Possibilities for further performance improvement are discussed. Previously announced in STAR as N83-21847
Analyses of the imaging properties of several designs for varied-line space gratings in converging beams of light in grazing-incidence spectrometers are presented. An explicit model is defined for the case of a plane-reflection grating intercepting light that converges and is reflected to a stigmatic point associated with the zero-order image of the grating. Smooth spatial variation of the grating constant then permits aberration correction. The aberrations are expressed as polynomials in the grating lens coordinates using power series expansions. Application of the model is illustrated in terms of aberrations experienced with the short wavelength spectrometer on the EUVE satellite. Attention is given to straight and parallel in-plane grooves, curved groove in-plane designs and off-plane grooves. Aberrations due to dispersions and misalignment are also considered.