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NASA NTRS2023

On Hermite Interpolation using Bernstein Polynomials for Trajectory Generation

This work presents a solution to the two-point Hermite interpolation problem using Bernstein polynomials. The Hermite interpolation problem is of particular interest in aerospace applications where boundary conditions for trajectories often specify derivative constraints. In the examples shown, a trajectory will be generated between an initial condition and a final condition. For example, a trajectory is generated that connects an aircraft’s current position and velocity with a point on the runway at a desired landing velocity. The numerical stability of the proposed algorithms is analyzed empirically.

Bezier curves↗
NASA NTRS1979

Time optimal control of a jet engine using a quasi-Hermite interpolation model

This work made preliminary efforts to generate nonlinear numerical models of a two-spooled turbofan jet engine, and subject these models to a known method of generating global, nonlinear, time optimal control laws. The models were derived numerically, directly from empirical data, as a first step in developing an automatic modelling procedure.

Comiskey, J. G.↗
NASA NTRS1990

GRID2D/3D: A computer program for generating grid systems in complex-shaped two- and three-dimensional spatial domains. Part 2: User's manual and program listing

An efficient computer program, called GRID2D/3D, was developed to generate single and composite grid systems within geometrically complex two- and three-dimensional (2- and 3-D) spatial domains that can deform with time. GRID2D/3D generates single grid systems by using algebraic grid generation methods based on transfinite interpolation in which the distribution of grid points within the spatial domain is controlled by stretching functions. All single grid systems generated by GRID2D/3D can have grid lines that are continuous and differentiable everywhere up to the second-order. Also, grid lines can intersect boundaries of the spatial domain orthogonally. GRID2D/3D generates composite grid systems by patching together two or more single grid systems. The patching can be discontinuous or continuous. For continuous composite grid systems, the grid lines are continuous and differentiable everywhere up to the second-order except at interfaces where different single grid systems meet. At interfaces where different single grid systems meet, the grid lines are only differentiable up to the first-order. For 2-D spatial domains, the boundary curves are described by using either cubic or tension spline interpolation. For 3-D spatial domains, the boundary surfaces are described by using either linear Coon's interpolation, bi-hyperbolic spline interpolation, or a new technique referred to as 3-D bi-directional Hermite interpolation. Since grid systems generated by algebraic methods can have grid lines that overlap one another, GRID2D/3D contains a graphics package for evaluating the grid systems generated. With the graphics package, the user can generate grid systems in an interactive manner with the grid generation part of GRID2D/3D. GRID2D/3D is written in FORTRAN 77 and can be run on any IBM PC, XT, or AT compatible computer. In order to use GRID2D/3D on workstations or mainframe computers, some minor modifications must be made in the graphics part of the program; no modifications are needed in the grid generation part of the program. The theory and method used in GRID2D/3D is described.

Bailey, R. T.↗
NASA NTRS1990

GRID2D/3D: A computer program for generating grid systems in complex-shaped two- and three-dimensional spatial domains. Part 1: Theory and method

An efficient computer program, called GRID2D/3D was developed to generate single and composite grid systems within geometrically complex two- and three-dimensional (2- and 3-D) spatial domains that can deform with time. GRID2D/3D generates single grid systems by using algebraic grid generation methods based on transfinite interpolation in which the distribution of grid points within the spatial domain is controlled by stretching functions. All single grid systems generated by GRID2D/3D can have grid lines that are continuous and differentiable everywhere up to the second-order. Also, grid lines can intersect boundaries of the spatial domain orthogonally. GRID2D/3D generates composite grid systems by patching together two or more single grid systems. The patching can be discontinuous or continuous. For continuous composite grid systems, the grid lines are continuous and differentiable everywhere up to the second-order except at interfaces where different single grid systems meet. At interfaces where different single grid systems meet, the grid lines are only differentiable up to the first-order. For 2-D spatial domains, the boundary curves are described by using either cubic or tension spline interpolation. For 3-D spatial domains, the boundary surfaces are described by using either linear Coon's interpolation, bi-hyperbolic spline interpolation, or a new technique referred to as 3-D bi-directional Hermite interpolation. Since grid systems generated by algebraic methods can have grid lines that overlap one another, GRID2D/3D contains a graphics package for evaluating the grid systems generated. With the graphics package, the user can generate grid systems in an interactive manner with the grid generation part of GRID2D/3D. GRID2D/3D is written in FORTRAN 77 and can be run on any IBM PC, XT, or AT compatible computer. In order to use GRID2D/3D on workstations or mainframe computers, some minor modifications must be made in the graphics part of the program; no modifications are needed in the grid generation part of the program. This technical memorandum describes the theory and method used in GRID2D/3D.

Shih, T. I.-P.↗
NASA NTRS1991

Algebraic grid generation for complex geometries

An efficient computer program called GRID2D/3D has been developed to generate single and composite grid systems within geometrically complex two- and three-dimensional (2D and 3D) spatial domains that can deform with time. GRID2D/3D generates single grid systems by using algebraic grid generation methods based on transfinite interpolation. The distribution of grid points within the spatial domain is controlled by stretching functions and grid lines can intersect boundaries of the spatial domain orthogonally. GRID2D/3D generates composite grid systems by patching together two or more single grid systems. The patching can be discontinuous or continuous. For 2D spatial domains the boundary curves are constructed by using either cubic or tension spline interpolation. For 3D spatial domains the boundary surfaces are constructed by using a new technique, developed in this study, referred to as 3D bidirectional Hermite interpolation.

Shih, T. I.-P.↗
NASA NTRS2002

Computing Aerodynamic Performance of a 2D Iced Airfoil: Blocking Topology and Grid Generation

The ice accrued on airfoils can have enormously complicated shapes with multiple protruded horns and feathers. In this paper, several blocking topologies are proposed and evaluated on their ability to produce high-quality structured multi-block grid systems. A transition layer grid is introduced to ensure that jaggedness on the ice-surface geometry do not to propagate into the domain. This is important for grid-generation methods based on hyperbolic PDEs (Partial Differential Equations) and algebraic transfinite interpolation. A 'thick' wrap-around grid is introduced to ensure that grid lines clustered next to solid walls do not propagate as streaks of tightly packed grid lines into the interior of the domain along block boundaries. For ice shapes that are not too complicated, a method is presented for generating high-quality single-block grids. To demonstrate the usefulness of the methods developed, grids and CFD solutions were generated for two iced airfoils: the NLF0414 airfoil with and without the 623-ice shape and the B575/767 airfoil with and without the 145m-ice shape. To validate the computations, the computed lift coefficients as a function of angle of attack were compared with available experimental data. The ice shapes and the blocking topologies were prepared by NASA Glenn's SmaggIce software. The grid systems were generated by using a four-boundary method based on Hermite interpolation with controls on clustering, orthogonality next to walls, and C continuity across block boundaries. The flow was modeled by the ensemble-averaged compressible Navier-Stokes equations, closed by the shear-stress transport turbulence model in which the integration is to the wall. All solutions were generated by using the NPARC WIND code.

Chi, X.↗
NASA NTRS1995

Elliptic surface grid generation on minimal and parmetrized surfaces

An elliptic grid generation method is presented which generates excellent boundary conforming grids in domains in 2D physical space. The method is based on the composition of an algebraic and elliptic transformation. The composite mapping obeys the familiar Poisson grid generation system with control functions specified by the algebraic transformation. New expressions are given for the control functions. Grid orthogonality at the boundary is achieved by modification of the algebraic transformation. It is shown that grid generation on a minimal surface in 3D physical space is in fact equivalent to grid generation in a domain in 2D physical space. A second elliptic grid generation method is presented which generates excellent boundary conforming grids on smooth surfaces. It is assumed that the surfaces are parametrized and that the grid only depends on the shape of the surface and is independent of the parametrization. Concerning surface modeling, it is shown that bicubic Hermite interpolation is an excellent method to generate a smooth surface which is passing through a given discrete set of control points. In contrast to bicubic spline interpolation, there is extra freedom to model the tangent and twist vectors such that spurious oscillations are prevented.

Spekreijse, S. P.↗
NASA NTRS1995

Elliptic surface grid generation on minimal and parametrized surfaces

An elliptic grid generation method, which generates boundary conforming grids in a two dimensional physical space, is presented. The method is based on the composition of an algebraic and elliptic transformation. The composite mapping obeys the Poisson grid generation system with control functions specified by the algebraic transformation. It is shown that the grid generation on a minimal surface in a three dimensional space is equivalent to the grid generation in a two dimensional domain in physical space. A second elliptic grid generation method, which generates boundary conforming grids on smooth surfaces, is presented. Concerning surface modeling, it is shown that bicubic Hermit interpolation is an excellent method to generate a smooth surface crossing a discrete set of control points.

Spekreijse, S. P.↗
NASA NTRS1995

Multiresolution Representation Using Biorthogonal Multiwavelets

We generalize Harten's multiresolution representation to biorthogonal multiwavelets. Several variants are considered. For example, a given array of discrete point values is transformed to point values and derivatives or point 'values and cell averages'. Compact Hermite interpolation is used in the decomposition and reconstruction algorithm. The resulting basis functions that are symmetric or skewsymmetric, compact, and smooth with optimal order accuracy. Harten's approach has several advantages: the multiresolution scheme is inherently discrete, non-periodic boundary conditions are easy to implement, and the representation can be extended to unstructured grids in bounded domains. We demonstrate the compression features of the new mutliwavelets by application to variable scale piecewise smooth functions with jump discontinuities typical of numerical solutions of nonlinear hyperbolic conservation laws.

Warming, Robert F.↗
NASA NTRS2004

High Order And High Resolution Methods For a Model CAA Problem

The initial value problem for the first order linear wave equation in one space dimension is treated for two cases with specified initial data and grid, and data from solutions at t = 400 and t = 800 are presented, as prescribed for Problem 1 in Category 1. Results are shown from computations with a sequence of recently developed high order and high resolution methods which combine Hermite interpolation, Cauchy-Kowaleskya recursion for time derivatives, and Taylor series time advancement. These methods have the same order of accuracy in time as in space. Results are shown from methods that range from third to nineteenth order. The stated problems with the prescribed coarse grid can be simulated with errors that are at the level of machine accuracy if the method is sufficiently high order. In addition, the growth of the maximum absolute error out to t = 100,000 is given for simulations with the stated problem data.

Goodrich, John W.↗
NASA NTRS1986

Interactive algebraic grid-generation technique

An algebraic grid generation technique and use of an associated interactive computer program are described. The technique, called the two boundary technique, is based on Hermite cubic interpolation between two fixed, nonintersecting boundaries. The boundaries are referred to as the bottom and top, and they are defined by two ordered sets of points. Left and right side boundaries which intersect the bottom and top boundaries may also be specified by two ordered sets of points. when side boundaries are specified, linear blending functions are used to conform interior interpolation to the side boundaries. Spacing between physical grid coordinates is determined as a function of boundary data and uniformly space computational coordinates. Control functions relating computational coordinates to parametric intermediate variables that affect the distance between grid points are embedded in the interpolation formulas. A versatile control function technique with smooth-cubic-spline functions is presented. The technique works best in an interactive graphics environment where computational displays and user responses are quickly exchanged. An interactive computer program based on the technique and called TBGG (two boundary grid generation) is also described.

Smith, R. E.↗
NASA NTRS1990

Algebraic Generation Of Two-Dimensional Grids

Two-Boundary Grid Generation (TBGG) computer program applies interactive algebraic grid-generation technique in two dimensions. Incorporates algebraic equations that relate uniform rectangular computational domain to arbitrary physical domain. Hermite cubic interpolation performed between fixed, nonintersecting boundaries. Applied to variety of problems in which discrete techniques used to solve such partial differential equations as governing equations of fluid flow. Written in FORTRAN 77.

Smith, Robert E.↗
NASA NTRS1993

Grid and design variables sensitivity analyses for NACA four-digit wing-sections

Two distinct parameterization procedures are developed for investigating the grid sensitivity with respect to design parameters of a wing-section example. The first procedure is based on traditional (physical) relations defining NACA four-digit wing-sections. The second is advocating a novel (geometrical) parameterization using spline functions such as NURBS (Non-Uniform Rational B-Splines) for defining the wing-section geometry. An interactive algebraic grid generation technique, known as Hermite Cubic Interpolation, is employed to generate C-type grids around wing-sections. The grid sensitivity of the domain with respect to design and grid parameters has been obtained by direct differentiation of the grid equations. A hybrid approach is proposed for more geometrically complex configurations. A comparison of the sensitivity coefficients with those obtained using a finite-difference approach has been made to verify the feasibility of the approach. The aerodynamic sensitivity coefficients are obtained using the compressible two-dimensional thin-layer Navier-Stokes equations.

Sadrehaghighi, Ideen↗
NASA NTRS1977

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.↗
NASA NTRS2000

An Automated Approach to Very High Order Aeroacoustic Computations in Complex Geometries

Computational aeroacoustics requires efficient, high-resolution simulation tools. And for smooth problems, this is best accomplished with very high order in space and time methods on small stencils. But the complexity of highly accurate numerical methods can inhibit their practical application, especially in irregular geometries. This complexity is reduced by using a special form of Hermite divided-difference spatial interpolation on Cartesian grids, and a Cauchy-Kowalewslci recursion procedure for time advancement. In addition, a stencil constraint tree reduces the complexity of interpolating grid points that are located near wall boundaries. These procedures are used to automatically develop and implement very high order methods (>15) for solving the linearized Euler equations that can achieve less than one grid point per wavelength resolution away from boundaries by including spatial derivatives of the primitive variables at each grid point. The accuracy of stable surface treatments is currently limited to 11th order for grid aligned boundaries and to 2nd order for irregular boundaries.

Dyson, Rodger W.↗
NASA NTRS2001

Automated Approach to Very High-Order Aeroacoustic Computations

Computational aeroacoustics requires efficient, high-resolution simulation tools. For smooth problems, this is best accomplished with very high-order in space and time methods on small stencils. However, the complexity of highly accurate numerical methods can inhibit their practical application, especially in irregular geometries. This complexity is reduced by using a special form of Hermite divided-difference spatial interpolation on Cartesian grids, and a Cauchy-Kowalewski recursion procedure for time advancement. In addition, a stencil constraint tree reduces the complexity of interpolating grid points that am located near wall boundaries. These procedures are used to develop automatically and to implement very high-order methods (> 15) for solving the linearized Euler equations that can achieve less than one grid point per wavelength resolution away from boundaries by including spatial derivatives of the primitive variables at each grid point. The accuracy of stable surface treatments is currently limited to 11th order for grid aligned boundaries and to 2nd order for irregular boundaries.

Dyson, Rodger W.↗
NASA NTRS1986

Spectral analysis of linear, shift-invariant interpolants

The use of a spectral analysis technique to evaluate the reconstruction/interpolation performance of linear, shift-invariant interpolants is examined. The technique was utilized to measure the performance of cubic hermite, quintic hermite, exponential, cubic, spline, Nu, PCC, Keys cubic, and BAWA cubic interpolants. The performance criterion is based upon the mean square error of the difference between the sampled and reconstructed functions. The reconstruction properties, interpolation functions, and reconstruction filters for the interpolants are studied and compared. It is noted that the spectral analysis technique is applicable to reconstruction algorithms used in signal and image processes, and interpolants used in numerical analysis, computer-aided design, and computer graphics.

Lansing, Donald L.↗
NASA NTRS1987

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.↗