Three-dimensional fluid flow calculations using a flux-spline method
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Karki, K. C.↗
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Domain identification in impedance computed tomography by spline collocation method
A method for estimating an unknown domain in elliptic boundary value problems is considered. The problem is formulated as an inverse problem of integral equations of the second kind. A computational method is developed using a splice collocation scheme. The results can be applied to the inverse problem of impedance computed tomography (ICT) for image reconstruction.
Kojima, Fumio↗
Modeling of topographic effects on Antarctic sea ice using multivariate adaptive regression splines
The role of seafloor topography in the spatial variations of the southern ocean sea ice cover as observed (every other day) by the Nimbus 7 scanning multichannel microwave radiometer satellite in the years 1980, 1983, and 1984 is studied. Bottom bathymetry can affect sea ice surface characteristics because of the basically barotropic circulation of the ocean south of the Antarctic Circumpolar current. The main statistical tool used to quantify this effect is a local nonparametric regression model of sea ice concentration as a function of the depth and its first two derivatives in both meridional and zonal directions. First, we model the relationship of bathymetry to sea ice concentration in two sudy areas, one over the Maud Rise and the other over the Ross Sea shelf region. The multiple correlation coefficient is found to average 44% in the Maud Rise study area and 62% in the Ross Sea study area over the years 1980, 1983, and 1984. Second, a strategy of dividing the entire Antarctic region into an overlapping mosaic of small areas, or windows is considered. Keeping the windows small reduces the correlation of bathymetry with other factors such as wind, sea temperature, and distance to the continent. We find that although the form of the model varies from window to window due to the changing role of other relevant environmental variables, we are left with a spatially consistent ordering of the relative importance of the topographic predictors. For a set of three representative days in the Austral winter of 1980, the analysis shows that an average of 54% of the spatial variation in sea ice concentration over the entire ice cover can be attributed to topographic variables. The results thus support the hypothesis that there is a sea ice to bottom bathymetry link. However this should not undermine the considerable influence of wind, current, and temperature which affect the ice distribution directly and are partly responsible for the observed bathymetric effects.
De Veaux, Richard D.↗
Spline-Based Smoothing of Airfoil Curvatures
Constrained fitting for airfoil curvature smoothing (CFACS) is a splinebased method of interpolating airfoil surface coordinates (and, concomitantly, airfoil thicknesses) between specified discrete design points so as to obtain smoothing of surface-curvature profiles in addition to basic smoothing of surfaces. CFACS was developed in recognition of the fact that the performance of a transonic airfoil is directly related to both the curvature profile and the smoothness of the airfoil surface. Older methods of interpolation of airfoil surfaces involve various compromises between smoothing of surfaces and exact fitting of surfaces to specified discrete design points. While some of the older methods take curvature profiles into account, they nevertheless sometimes yield unfavorable results, including curvature oscillations near end points and substantial deviations from desired leading-edge shapes. In CFACS as in most of the older methods, one seeks a compromise between smoothing and exact fitting. Unlike in the older methods, the airfoil surface is modified as little as possible from its original specified form and, instead, is smoothed in such a way that the curvature profile becomes a smooth fit of the curvature profile of the original airfoil specification. CFACS involves a combination of rigorous mathematical modeling and knowledge-based heuristics. Rigorous mathematical formulation provides assurance of removal of undesirable curvature oscillations with minimum modification of the airfoil geometry. Knowledge-based heuristics bridge the gap between theory and designers best practices. In CFACS, one of the measures of the deviation of an airfoil surface from smoothness is the sum of squares of the jumps in the third derivatives of a cubicspline interpolation of the airfoil data. This measure is incorporated into a formulation for minimizing an overall deviation- from-smoothness measure of the airfoil data within a specified fitting error tolerance. CFACS has been extensively tested on a number of supercritical airfoil data sets generated by inverse design and optimization computer programs. All of the smoothing results show that CFACS is able to generate unbiased smooth fits of curvature profiles, trading small modifications of geometry for increasing curvature smoothness by eliminating curvature oscillations and bumps (see figure).
Li, W.↗
Quick acting gimbal joint
An improved linkage assembly provides for locking the pivotable position of one member relative to another member. The linkage assembly permits the unlocking of these members, their subsequent repositioning, and the reliable relocking in a desired position without utilizing tooling. The first member includes a pair of spaced apart support flanges each having a through aperture therein defining a flange female spline. The second member has an end positioned between the pair of support flanges and also includes a through aperture defining a pair of second member female splines. A pair of control members are each slidably moveable along the common axis and extend through the respective flanges. Each control member has a male splined portion which normally engages both the respective flange female spline and the second member female spline to lock the rotatable position of the first and second members. When each of the control members are pressed inward simultaneously, the male splined portions disengage the flange female splines to permit rotation of the first member about the common axis with respect to the second member.
William B Wood↗
Hierarchical Control and Trajectory Planning
Most of the time on this project was spent on the trajectory planning problem. The construction is equivalent to the classical spline construction in the case that the system matrix is nilpotent. If the dimension of the system is n then the spline of degree 2n-1 is constructed. This gives a new approach to the construction of splines that is more efficient than the usual construction and at the same time allows the construction of a much larger class of splines. All known classes of splines are reconstructed using the approach of linear control theory. As a numerical analysis tool control theory gives a very good tool for constructing splines. However, for the purposes of trajectory planning it is quite another story. Enclosed in this document are four reports done under this grant.
Martin, Clyde F.↗
Polynominal Interpolation Methods for Viscous Flow Calculations
Higher-order collocation procedures resulting in tridiagonal matrix systems are derived from polynomial spline interpolation and by Hermitian (Taylor series) finite-difference discretization. The similarities and special features of these different developments are discussed. The governing systems apply for both uniform and variable meshes. Hybrid schemes resulting from two different polynomial approximations for the first and second derivatives lead to a nonuniform mesh extension of the so-called compact or Pad? difference technique (Hermite 4). A variety of fourth-order methods are described and the Hermitian approach is extended to sixth-order (Hermite 6). The appropriate spline boundary conditions are derived for all procedures. For central finite differences, this leads to a two-point, second-order accurate generalization of the commonly used three-point end-difference formula. Solutions with several spline and Hermite procedures are presented for the boundary layer equations, with and without mass transfer, and for the incompressible viscous flow in a driven cavity. Divergence and nondivergence equations are considered for the cavity. Among the fourth-order techniques, it is shown that spline 4 has the smallest truncation error. The spline 4 procedure generally requires one-quarter the number of mesh points in a given coordinate direction as a central finite-difference calculation of equal accuracy. The Hermite 6 procedure leads to remarkably accurate boundary layer solutions.
Rubin, S. G.↗
The Construction of Curves and Surfaces Using Numerical Optimization Techniques
Numerical optimization techniques are playing an increasing role in curve and surface construction. Often difficult problems in curve and surface construction, especially when some aspect of shape control is involved, can be phrased as a constrained optimization problem. Four such classes of problems are explored: parametric curve fitting with non-linear shape constraints; explicit surface fitting with linear shape constraints; surface fitting to scattered data giving rise to ill-posed problems; finally, variable knot problems. In each of these problems there is a nonlinear aspect: either the shape of the curve or surface is important for manufacturing or engineering reasons or the shape affects the convergence of numerical algorithms which use the curve or surface or the placement of knots affects the accuracy of the fits. In all cases the class of functions used is that of parametric spline curves and tensor or direct product spline surfaces. The reason for choosing this class is that splines provide flexible models that are easily evaluated and stored. Furthermore, the B-spline representation of splines leads to convenient expressions for shape control over regions.
Ferguson, D. R.↗
Quantitative analysis of the reconstruction performance of interpolants
The analysis presented provides a quantitative measure of the reconstruction or interpolation performance of linear, shift-invariant interpolants. The performance criterion is the mean square error of the difference between the sampled and reconstructed functions. The analysis is applicable to reconstruction algorithms used in image processing and to many types of splines used in numerical analysis and computer graphics. When formulated in the frequency domain, the mean square error clearly separates the contribution of the interpolation method from the contribution of the sampled data. The equations provide a rational basis for selecting an optimal interpolant; that is, one which minimizes the mean square error. The analysis has been applied to a selection of frequently used data splines and reconstruction algorithms: parametric cubic and quintic Hermite splines, exponential and nu splines (including the special case of the cubic spline), parametric cubic convolution, Keys' fourth-order cubic, and a cubic with a discontinuous first derivative. The emphasis in this paper is on the image-dependent case in which no a priori knowledge of the frequency spectrum of the sampled function is assumed.
Lansing, Donald L.↗
Direct numerical simulation of incompressible axisymmetric flows
In the present work, we propose to conduct direct numerical simulations (DNS) of incompressible turbulent axisymmetric jets and wakes. The objectives of the study are to understand the fundamental behavior of axisymmetric jets and wakes, which are perhaps the most technologically relevant free shear flows (e.g. combuster injectors, propulsion jet). Among the data to be generated are various statistical quantities of importance in turbulence modeling, like the mean velocity, turbulent stresses, and all the terms in the Reynolds-stress balance equations. In addition, we will be interested in the evolution of large-scale structures that are common in free shear flow. The axisymmetric jet or wake is also a good problem in which to try the newly developed b-spline numerical method. Using b-splines as interpolating functions in the non-periodic direction offers many advantages. B-splines have local support, which leads to sparse matrices that can be efficiently stored and solved. Also, they offer spectral-like accuracy that are C(exp O-1) continuous, where O is the order of the spline used; this means that derivatives of the velocity such as the vorticity are smoothly and accurately represented. For purposes of validation against existing results, the present code will also be able to simulate internal flows (ones that require a no-slip boundary condition). Implementation of no-slip boundary condition is trivial in the context of the b-splines.
Loulou, Patrick↗
Student Support for Research in Hierarchical Control and Trajectory Planning
Generally, classical polynomial splines tend to exhibit unwanted undulations. In this work, we discuss a technique, based on control principles, for eliminating these undulations and increasing the smoothness properties of the spline interpolants. We give a generalization of the classical polynomial splines and show that this generalization is, in fact, a family of splines that covers the broad spectrum of polynomial, trigonometric and exponential splines. A particular element in this family is determined by the appropriate control data. It is shown that this technique is easy to implement. Several numerical and curve-fitting examples are given to illustrate the advantages of this technique over the classical approach. Finally, we discuss the convergence properties of the interpolant.
Martin, Clyde F.↗
Results of full-scale vortex attenuation flight experiments
Flight tests have been conducted to evaluate the effectiveness of a wing-tip vortex attenuating device, referred to as a spline. Vortex penetrations were made with a PA-28 behind a C-54 aircraft with and without wing-tip splines attached, and the resultant rolling acceleration was measured and related to the roll acceleration capability of the PA-28. Tests were conducted over a range of separation distances from about 5 n mi to less than 1 n mi. Preliminary results indicate that, with the splines installed, there was a significant reduction in the vortex-induced roll acceleration experienced by the PA-28 probe aircraft, and the distance at which the PA-28 roll control became ineffective was reduced from 2.5 n mi to less than 1 n mi. Although splines significantly reduced the C-54 rate of climb, the rates available with four engines were acceptable for this test program.
Hastings, E. C., Jr.↗
The separated turbulent boundary layer over a wavy wall
A study and application of the fourth order spline collocation procedure, numerical solution of boundary layer like differential equations, is presented. A simple inversion algorithm for the simultaneous solution of the resulting difference equations is given. Particular attention is focused on the boundary condition representation for the spline second derivative approximations. Solutions using the spline procedure, as well as the three point finite difference method, are presented for several model problems in order to assess and improve the spline numerical scheme. Application of the resulting algorithm to the incompressible laminar self similar boundary layer equations is presented.
Polak, A.↗
Time Varying Compensator Design for Reconfigurable Structures Using Non-Collocated Feedback
Analysis and synthesis tools are developed to improved the dynamic performance of reconfigurable nonminimum phase, nonstrictly positive real-time variant systems. A novel Spline Varying Optimal (SVO) controller is developed for the kinematic nonlinear system. There are several advantages to using the SVO controller, in which the spline function approximates the system model, observer, and controller gain. They are: The spline function approximation is simply connected, thus the SVO controller is more continuous than traditional gain scheduled controllers when implemented on a time varying plant; ft is easier for real-time implementations in storage and computational effort; where system identification is required, the spline function requires fewer experiments, namely four experiments; and initial startup estimator transients are eliminated. The SVO compensator was evaluated on a high fidelity simulation of the Shuttle Remote Manipulator System. The SVO controller demonstrated significant improvement over the present arm performance: (1) Damping level was improved by a factor of 3; and (2) Peak joint torque was reduced by a factor of 2 following Shuttle thruster firings.
Scott, Michael A.↗
The inversion of aureole measurements to derive aerosol size distributions
An iterative method to invert size distributions from simulated scattered radiance measurements at small angles from the sun has been investigated. The inferred size distributions were represented by piecewise linear and cubic spline functions. Various relevant characteristics were investigated and it was found that: (1) the inverted size distribution was insensitive to the number of knots in the piecewise linear spline; (2) within the range of sensitivity, the choice of initial guess had little effect on the inverted size distribution; (3) five per cent random noise in the simulated radiances appreciably deteriorated the result but variations are still tolerable when compared with other methods for determining size distributions; (4) the inverted distribution was insensitive to the index of refraction used in the kernel for particle radii greater than 1 micron; (5) the choice of wavelength between 0.40 and 0.70 microns has a negligible effect on the inverted distribution; (6) a range of tropospheric aerosol size distributions gives acceptable inverted results; and (7) the cubic spline representation can give reasonable inverted distributions, but may become unstable.
Twitty, J. T.↗
Polynomial interpolation methods for viscous flow calculations
Higher-order collocation procedures which result in block-tridiagonal matrix systems are derived from (1) Taylor series expansions and from (2) polynomial interpolation, and the relationships between the two formulations, called respectively Hermite and spline collocation, are investigated. A Hermite block-tridiagonal system for a nonuniform mesh is derived, and the Hermite approach is extended in order to develop a variable-mesh sixth-order block-tridiagonal procedure. It is shown that all results obtained by Hermite development can be recovered by appropriate spline polynomial interpolation. The additional boundary conditions required for these higher-order procedures are also given. Comparative solutions using second-order accurate finite difference and spline and Hermite formulations are presented for the boundary layer on a flat plate, boundary layers with uniform and variable mass transfer, and the viscous incompressible Navier-Stokes equations describing flow in a driven cavity.
Rubin, S. G.↗
Inversion of the strain-life and strain-stress relationships for use in metal fatigue analysis
The paper presents closed-form solutions (collocation method and spline-function method) for the constants of the cyclic fatigue life equation so that they can be easily incorporated into cumulative damage analysis. The collocation method involves conformity with the experimental curve at specific life values. The spline-function method is such that the basic life relation is expressed as a two-part function, one applicable at strains above the transition strain (strain at intersection of elastic and plastic lines), the other below. An illustrative example is treated by both methods. It is shown that while the collocation representation has the advantage of simplicity of form, the spline-function representation can be made more accurate over a wider life range, and is simpler to use.
Manson, S. S.↗
Numerical computations on one-dimensional inverse scattering problems
An approximate method to determine the index of refraction of a dielectric obstacle is presented. For simplicity one dimensional models of electromagnetic scattering are treated. The governing equations yield a second order boundary value problem, in which the index of refraction appears as a functional parameter. The availability of reflection coefficients yield two additional boundary conditions. The index of refraction by a k-th order spline which can be written as a linear combination of B-splines is approximated. For N distinct reflection coefficients, the resulting N boundary value problems yield a system of N nonlinear equations in N unknowns which are the coefficients of the B-splines.
Dunn, M. H.↗