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At least 19 records

All sources NASA NTRS DOE OSTI

DOE OSTI2022

U-splines: Splines over unstructured meshes

U-splines are a novel approach to the construction of a spline basis for representing smooth objects in Computer-Aided Design (CAD) and Computer-Aided Engineering (CAE). A spline is a piecewise-defined function that satisfies continuity constraints between adjacent cells in a mesh. U-splines differ from existing spline constructions, such as Non-Uniform Rational B-splines (NURBS), subdivision surfaces, T-splines, and hierarchical B-splines, in that they can accommodate local variation in cell size, polynomial degree, and smoothness simultaneously over more varied mesh configurations. Mixed cell types (e.g., triangle and quadrilateral cells in the same mesh) and T-junctions are also supported, although the continuity of interfaces with triangle and tetrahedral cells is limited in the present work. The U-spline algorithm introduces a new technique for using local null space solutions to construct basis functions for the global spline null space problem. The U-spline construction is presented for curves, surfaces, and volumes with higher dimensional generalizations possible. Lastly, a set of requirements are given to ensure that the U-spline basis is positive, forms a partition of unity, is complete, and is locally linearly independent.

42 ENGINEERING↗

NASA NTRS1985

Multivariate Spline Algorithms for CAGD

Two special polyhedra present themselves for the definition of B-splines: a simplex S and a box or parallelepiped B, where the edges of S project into an irregular grid, while the edges of B project into the edges of a regular grid. More general splines may be found by forming linear combinations of these B-splines, where the three-dimensional coefficients are called the spline control points. Univariate splines are simplex splines, where s = 1, whereas splines over a regular triangular grid are box splines, where s = 2. Two simple facts render the development of the construction of B-splines: (1) any face of a simplex or a box is again a simplex or box but of lower dimension; and (2) any simplex or box can be easily subdivided into smaller simplices or boxes. The first fact gives a geometric approach to Mansfield-like recursion formulas that express a B-spline in B-splines of lower order, where the coefficients depend on x. By repeated recursion, the B-spline will be expressed as B-splines of order 1; i.e., piecewise constants. In the case of a simplex spline, the second fact gives a so-called insertion algorithm that constructs the new control points if an additional knot is inserted.

Boehm, W.↗

NASA NTRS1993

Spline screw payload fastening system

A system for coupling an orbital replacement unit (ORU) to a space station structure via the actions of a robot and/or astronaut is described. This system provides mechanical and electrical connections both between the ORU and the space station structure and between the ORU and the ORU and the robot/astronaut hand tool. Alignment and timing features ensure safe, sure handling and precision coupling. This includes a first female type spline connector selectively located on the space station structure, a male type spline connector positioned on the orbital replacement unit so as to mate with and connect to the first female type spline connector, and a second female type spline connector located on the orbital replacement unit. A compliant drive rod interconnects the second female type spline connector and the male type spline connector. A robotic special end effector is used for mating with and driving the second female type spline connector. Also included are alignment tabs exteriorally located on the orbital replacement unit for berthing with the space station structure. The first and second female type spline connectors each include a threaded bolt member having a captured nut member located thereon which can translate up and down the bolt but are constrained from rotation thereabout, the nut member having a mounting surface with at least one first type electrical connector located on the mounting surface for translating with the nut member. At least one complementary second type electrical connector on the orbital replacement unit mates with at least one first type electrical connector on the mounting surface of the nut member. When the driver on the robotic end effector mates with the second female type spline connector and rotates, the male type spline connector and the first female type spline connector lock together, the driver and the second female type spline connector lock together, and the nut members translate up the threaded bolt members carrying the first type electrical connector up to the complementary second type connector for interconnection therewith.

Vranish, John M.↗

NASA NTRS1981

Theory, computation, and application of exponential splines

A generalization of the semiclassical cubic spline known in the literature as the exponential spline is discussed. In actuality, the exponential spline represents a continuum of interpolants ranging from the cubic spline to the linear spline. A particular member of this family is uniquely specified by the choice of certain tension parameters. The theoretical underpinnings of the exponential spline are outlined. This development roughly parallels the existing theory for cubic splines. The primary extension lies in the ability of the exponential spline to preserve convexity and monotonicity present in the data. Next, the numerical computation of the exponential spline is discussed. A variety of numerical devices are employed to produce a stable and robust algorithm. An algorithm for the selection of tension parameters that will produce a shape preserving approximant is developed. A sequence of selected curve-fitting examples are presented which clearly demonstrate the advantages of exponential splines over cubic splines.

Mccartin, B. J.↗

NASA NTRS1993

Split spline screw

A split spline screw type payload fastener assembly, including three identical male and female type split spline sections, is discussed. The male spline sections are formed on the head of a male type spline driver. Each of the split male type spline sections has an outwardly projecting load baring segment including a convex upper surface which is adapted to engage a complementary concave surface of a female spline receptor in the form of a hollow bolt head. Additionally, the male spline section also includes a horizontal spline releasing segment and a spline tightening segment below each load bearing segment. The spline tightening segment consists of a vertical web of constant thickness. The web has at least one flat vertical wall surface which is designed to contact a generally flat vertically extending wall surface tab of the bolt head. Mutual interlocking and unlocking of the male and female splines results upon clockwise and counter clockwise turning of the driver element.

John M Vranish↗

NASA NTRS1994

Splines and control theory

In this work, the relationship between splines and the control theory has been analyzed. We show that spline functions can be constructed naturally from the control theory. By establishing a framework based on control theory, we provide a simple and systematic way to construct splines. We have constructed the traditional spline functions including the polynomial splines and the classical exponential spline. We have also discovered some new spline functions such as trigonometric splines and the combination of polynomial, exponential and trigonometric splines. The method proposed in this paper is easy to implement. Some numerical experiments are performed to investigate properties of different spline approximations.

Zhang, Zhimin↗

NASA NTRS1984

Rational-spline approximation with automatic tension adjustment

An algorithm for weighted least-squares approximation with rational splines is presented. A rational spline is a cubic function containing a distinct tension parameter for each interval defined by two consecutive knots. For zero tension, the rational spline is identical to a cubic spline; for very large tension, the rational spline is a linear function. The approximation algorithm incorporates an algorithm which automatically adjusts the tension on each interval to fulfill a user-specified criterion. Finally, an example is presented comparing results of the rational spline with those of the cubic spline.

Schiess, J. R.↗

DOE OSTI2023

Extended isogeometric analysis of multi-material and multi-physics problems using hierarchical B-splines

Here, this paper presents an immersed, isogeometric finite element framework to predict the response of multi-material, multi-physics problems with complex geometries using locally refined discretizations. To circumvent the need to generate conformal meshes, this work uses an extended finite element method (XFEM) to discretize the governing equations on non-conforming, embedding meshes. A flexible approach to create truncated hierarchical B-splines discretizations is presented. This approach enables the refinement of each state variable field individually to meet field-specific accuracy requirements. To obtain an immersed geometry representation that is consistent across all hierarchically refined B-spline discretizations, the geometry is immersed into a single mesh, the XFEM background mesh, which is constructed from the union of all hierarchical B-spline meshes. An extraction operator is introduced to represent the truncated hierarchical B-spline bases in terms of Lagrange shape functions on the XFEM background mesh without loss of accuracy. The truncated hierarchical B-spline bases are enriched using a generalized Heaviside enrichment strategy to accommodate small geometric features and multi-material problems. The governing equations are augmented by a formulation of the face-oriented ghost stabilization enhanced for locally refined B-spline bases. We present examples for two- and three-dimensional linear elastic and thermo-elastic problems. The numerical results validate the accuracy of our framework. The results also demonstrate the applicability of the proposed framework to large, geometrically complex problems.

42 ENGINEERING↗

NASA NTRS1974

Preliminary results of flight tests of vortex attenuating splines

Flight tests have been conducted to evaluate the effectiveness of a wingtip vortex attenuating device, referred to as a spline. Vortex penetrations were made with a PA-28 behind a C-54 aircraft with and without wingtip 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 nautical miles (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 0.6 n. mi., or less. There was a slight increase in approach noise (approximately 4 db) with the splines installed due primarily to the higher engine power used during approach. Although splines significantly reduced the C-54 rate of climb, the rates available with four engines were acceptable for this test program. Splines did not introduce any noticeable change in the handling qualities of the C-54.

Hastings, E. C., Jr.↗

NASA NTRS1985

Examination of the Circle Spline Routine

The Circle Spline routine is currently being used for generating both two and three dimensional spline curves. It was developed for use in ESCHER, a mesh generating routine written to provide a computationally simple and efficient method for building meshes along curved surfaces. Circle Spline is a parametric linear blending spline. Because many computerized machining operations involve circular shapes, the Circle Spline is well suited for both the design and manufacturing processes and shows promise as an alternative to the spline methods currently supported by the Initial Graphics Specification (IGES).

Dolin, R. M.↗

NASA NTRS1988

Rational-Spline Subroutines

Smooth curves drawn among plotted data easily. Rational-Spline Approximation with Automatic Tension Adjustment algorithm leads to flexible, smooth representation of experimental data. "Tension" denotes mathematical analog of mechanical tension in spline or other mechanical curve-fitting tool, and "spline" as denotes mathematical generalization of tool. Program differs from usual spline under tension, allows user to specify different values of tension between adjacent pairs of knots rather than constant tension over entire range of data. Subroutines use automatic adjustment scheme that varies tension parameter for each interval until maximum deviation of spline from line joining knots less than or equal to amount specified by user. Procedure frees user from drudgery of adjusting individual tension parameters while still giving control over local behavior of spline.

Schiess, James R.↗

DOE OSTI2023

Fourier-informed knot placement schemes for B-spline approximation

Fitting B-splines to scientific data is especially challenging when the given data contain noise, jumps, or corners. Here, we describe how periodic data sets with these features can be efficiently approximated with B-splines by analyzing the Fourier spectrum of the data. Our method uses a collection of spectral filters to compute high-order derivatives, smoothed versions of noisy data, and the locations of jump discontinuities. Further, these quantities are then combined to choose knots that capture the qualitative features of the data, leading to accurate B-spline approximations with few knots. The method we introduce is direct and does not require any intermediate B-spline fitting before choosing the final knot distribution. Aside from fast Fourier transforms to transfer to and from Fourier space, the method runs in linear time with very little communication. We assess performance on several test cases in one and two dimensions, including data sets with jump discontinuities and noise. These tests show the method fits discontinuous data without spurious oscillations and remains effective in the presence of noise.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Soft x-ray spectrum unfold of K-edge filtered x-ray diode arrays using cubic splines

Cubic spline interpolation is able to recover temporally and spectrally resolved soft x-ray flux from an array of K-edge filtered x-ray diodes without the need for a priori assumptions about the spectrum or the geometry of the emitting volume. The mathematics of the cubic spline interpolation is discussed in detail. The analytic nature of the cubic spline solution allows for analytical error propagation, and the method of calculating the error for radiation temperature, spectral power, and confidence intervals of the unfolded spectrally resolved flux is explained. An unfold of a blackbody model demonstrates the accuracy of the cubic spline unfold. Tests of cubic spline performance using spectrally convolved detailed atomic model simulation results have been performed to measure the method's ability to conserve spectral power to within a factor of 2 or better in line-dominated regimes. The unfold is also demonstrated to work when information from the x-ray diode array is limited due to high signal-to-noise ratios or lack of signal due to over-attenuation or over-filtration of the x-ray diode signal. Here, the robustness of the unfold with respect to background subtraction and raw signal processing, signal alignment between diode traces, limited signal information, and initial conditions is discussed. Results from an example analysis of a halfraum drive are presented to demonstrate the capabilities of the unfold in comparison with previously established methods.

47 OTHER INSTRUMENTATION↗

NASA NTRS1983

Spline approximation of quantile functions

The study reported here explored the development and utility of a spline representation of the sample quantile function of a continuous probability distribution in providing a functional description of a random sample and a method of generating random variables. With a spline representation, the random samples are generated by transforming a sample of uniform random variables to the interval of interest. This is useful, for example, in simulation studies in which a random sample represents the only known information about the distribution. The spline formulation considered here consists of a linear combination of cubic basis splines (B-splines) fit in a least squares sense to the sample quantile function using equally spaced knots. The following discussion is presented in five parts. The first section highlights major results realized from the study. The second section further details the results obtained. The methodology used is described in the third section, followed by a brief discussion of previous research on quantile functions. Finally, the results of the study are evaluated.

Schiess, J. R.↗

NASA NTRS1987

An algorithm for surface smoothing with rational splines

Discussed is an algorithm for smoothing surfaces with spline functions containing tension parameters. The bivariate spline functions used are tensor products of univariate rational-spline functions. A distinct tension parameter corresponds to each rectangular strip defined by a pair of consecutive spline knots along either axis. Equations are derived for writing the bivariate rational spline in terms of functions and derivatives at the knots. Estimates of these values are obtained via weighted least squares subject to continuity constraints at the knots. The algorithm is illustrated on a set of terrain elevation data.

Schiess, James R.↗

NASA NTRS1992

Self-Aligning, Spline-Locking Fastener

Self-aligning, spline-locking fastener is two-part mechanism operated by robot, using one tool and simple movements. Spline nut on springloaded screw passes through mating spline fitting. Operator turns screw until vertical driving surfaces on spline nut rest against corresponding surfaces of spline fitting. Nut rides upward, drawing pieces together. Used to join two parts of structure, to couple vehicles, or to mount payload in vehicle.

Vranish, John M.↗

NASA NTRS1993

Radial spline assembly for antifriction bearings

An outer race carrier is constructed for receiving an outer race of an antifriction bearing assembly. The carrier in turn is slidably fitted in an opening of a support wall to accommodate slight axial movements of a shaft. A plurality of longitudinal splines on the carrier are disposed to be fitted into matching slots in the opening. A deadband gap is provided between sides of the splines and slots, with a radial gap at ends of the splines and slots and a gap between the splines and slots sized larger than the deadband gap. With this construction, operational distortions (slope) of the support wall are accommodated by the larger radial gaps while the deadband gaps maintain a relatively high springrate of the housing. Additionally, side loads applied to the shaft are distributed between sides of the splines and slots, distributing such loads over a larger surface area than a race carrier of the prior art.

Moore, Jerry H.↗

NASA NTRS2001

B-spline Method in Fluid Dynamics

B-spline functions are bases for piecewise polynomials that possess attractive properties for complex flow simulations : they have compact support, provide a straightforward handling of boundary conditions and grid nonuniformities, and yield numerical schemes with high resolving power, where the order of accuracy is a mere input parameter. This paper reviews the progress made on the development and application of B-spline numerical methods to computational fluid dynamics problems. Basic B-spline approximation properties is investigated, and their relationship with conventional numerical methods is reviewed. Some fundamental developments towards efficient complex geometry spline methods are covered, such as local interpolation methods, fast solution algorithms on cartesian grid, non-conformal block-structured discretization, formulation of spline bases of higher continuity over triangulation, and treatment of pressure oscillations in Navier-Stokes equations. Application of some of these techniques to the computation of viscous incompressible flows is presented.

Botella, Olivier↗