First-order Interpolation Research Articles

Characterization and construction of helical polynomial space curves

Helical space curves are characterized by the property that their unit tangents maintain a constant inclination with respect to a fixed line, the axis of the helix. Equivalently, a helix exhibits a circular tangent indicatrix, and constant curvature/torsion ratio. If a polynomial space curve is helical, it must be a Pythagorean-hodograph (PH) curve. The quaternion representation of spatial PH curves is used to characterize and construct helical curves. Whereas all spatial PH cubics are helical, the helical PH quintics form a proper subset of all PH quintics. Two types of PH quintic helix are identified: (i) the “monotone-helical” PH quintics, in which a scalar quadratic factors out of the hodograph, and the tangent exhibits a consistent sense of rotation about the axis; and (ii) general helical PH quintics, which possess irreducible hodographs, and may suffer reversals in the sense of tangent rotation. First-order Hermite interpolation is considered for both helical PH quintic types. The helicity property offers a means of fixing the residual degrees of freedom in the general PH quintic Hermite interpolation problem, and yields interpolants with desirable shape features.

The three-dimensional numerical aerodynamics of a movable block burner

Computational fluid-dynamics techniques were employed to study the aerodynamics of a movable block swirl burner, developed by the International Flame Research Foundation, IFRF, which is characterized by the ability to adjust continuously and dynamically the intensity of the swirl by means of the simultaneous rotation of eight movable blocks, inserted between eight fixed blocks. Five three-dimensional grids were constructed for the burner, corresponding to five positions of the movable blocks. Both the k-e and RNG k-e isotropic turbulence models were applied. Only the latter described the existence of a central reverse flow along the annular duct. The employment of first-order and second-order interpolation schemes provided distinct results. The later provided results closer to the experimental tests. The swirl number decayed in the annular duct. The predicted swirl numbers for this movable block swirl burner were lower than the corresponding IFRF's experimental data, as was also observed by other researchers. This gave rise to the suspicion of some possible measurement error in the IFRF's experiments. On the other hand, the lack of agreement between the experimental data and the predictions regarding swirling flows could be attributed to the possible inadequate performance of the k-e model, as a consequence of its isotropic approximation. Still another possible explanation could be a phenomenon called bifurcation, in which one given swirl number can be associated with two distinct conditions of steady state flow. In addition, this complex flows requires a scrupulous development of the grids for the boundary condition and the employment of adequate interpolation schemes.

A zonal grid algorithm for DNS of turbulent boundary layers

A zonal grid algorithm for direct numerical simulation (DNS) of incompressible turbulent flows within a Finite-Volume framework is presented. The algorithm uses fully coupled embedded grids and a conservative treatment of the grid-interface variables. A family of conservative prolongation operators is tested in a 2D vortex dipole and a 3D turbulent boundary layer flow. These tests show that both, first- and second-order interpolation conserves the overall second-order spatial accuracy of the scheme. The first-order conservative interpolation has a smaller damping effect on the solution but the second-order conservative interpolation has better spectral properties. The application of this algorithm in boundary layer flow separating and reattaching due to the presence of a streamwise pressure gradient reveals the power and usefulness of the presented algorithm. This simulation has been made possible by the zonal grid algorithm by reducing the required number of grid points from about 500 × 10 6 to 130 × 10 6 grid cells.

On a bitangential interpolation problem for contractive-valued functions on the unit ball

We solve the bitangential interpolation problem with a finite number of interpolation nodes for a multivariable analogue of the Schur class consisting of matrix-valued analytic functions on the ball. The interpolation conditions are formulated via a generalized functional calculus with operator argument, thereby generalizing in a compact way the simple, first-order interpolation conditions considered for this class of functions in earlier work. The description of all solutions is given via a Redheffer transform whose entries are given explicitly in terms of the interpolation data.

A chromatographic method for detection of end points in chemical reactions

Gas chromatography is recognized as an analytical technique possessing desirable advantages for a wide range of applications. The purpose of this work is to demonstrate the usefulness of GC instrumentation for determining values of end-points for a variety of reactions. Gas chromatographic titrimetry may be characterized as that in which the change of peak-height ratio of the reaction during a titration is recorded as a function of titrant added. The experimental data are fitted to simple first-order interpolation equations, so that the titration curve, when extrapolated, intersects the volume axis, the point of intersection being the equivalence-point volume. The end-point values are in good agreement with those reported in the literature and with control measurements. From the results it is concluded that the method can be used with high confidence to determine end-point values in a fast and easy way.

Feasibility studies of virtual laryngoscopy by CT and MRI-from data acquisition, image segmentation, to interactive visualization

Virtual endoscopy concept has been applied to study the larynx, as well as other hollow organs in recent years, assuming a clean lumen. In this work, we investigated the feasibility of virtual laryngoscopy by (1) studying currently available imaging protocols, (2) developing a suitable image segmentation method, and (3) constructing art efficient visualization system. By utilizing helical computed tomography (CT), the images for laryngeal volume can be obtained during a breath hold with 0.3 mm resolution. A fast pulse sequence using 1.5 T magnetic resonance (MR) imager can achieve 1 mm resolution within few minutes. The gain in tissue contrast on MR images is at the cost of resolution, and motion artifacts must be considered during image segmentation. A first-order Lagrange interpolation was applied to mitigate the reduced resolution, as well as partial volume effect and noise on the MR images. An automatic segmentation algorithm was adapted to extract the wall volume of the larynx. The algorithm considers local voxel property and classifies voxels based on the local property in the KL (Karhunen-Loeve) space. A visualization system was constructed for examining the mucosa and wall geometry with anatomical references in three dimensions. It navigates inside the lumen, as well as outside the larynx interactively with capability of inspecting and zooming into the regions of interest. It can also cut the larynx in any orientation to open the whole volume for viewing the entire inner surface. The procedure was tested on 2 volunteers and 2 patients. The segmentation performed consistently for all the studies and showed to be relatively insensitive to mild respiratory motion artifacts in the MR images. Image processing was accomplished within a few minutes on PC and low-end SGI platforms. These studies demonstrated the feasibility of virtual laryngoscopy for diagnosis of laryngeal abnormalities.

Interactive 3-D registration of ultrasound and magnetic resonance images based on a magnetic position sensor.

The use of stereotactic systems has been one of the main approaches for image-based guidance of the surgical tool within the brain. The main limitation of stereotactic systems is that they are based on preoperative images that might become outdated and invalid during the course of surgery. Ultrasound (US) is considered the most practical and cost-effective intraoperative imaging modality, but US images inherently have a low signal-to-noise ratio. Integrating intraoperative US with stereotactic systems has recently been attempted. In this paper, we present a new system for interactively registering two-dimensional US and three-dimensional magnetic resonance (MR) images. This registration is based on tracking the US probe with a dc magnetic position sensor. We have performed an extensive analysis of the errors of our system by using a custom-built phantom. The registration error between the MR and the position sensor space was found to have a mean value of 1.78 mm and a standard deviation of 0.18 mm. The registration error between US and MR space was dependent on the distance of the target point from the US probe face. For a 3.5-MHz phased one-dimensional array transducer and a depth of 6 cm, the mean value of the registration error was 2.00 mm and the standard deviation was 0.75 mm. The registered MR images were reconstructed using either zeroth-order or first-order interpolation. The ease of use and the interactive nature of our system (approximately 6.5 frames/s for 344 x 310 images and first-order interpolation on a Pentium II 450 MHz) demonstrates its potential to be used in the operating room.

A bounded convection scheme for the overlapping control volume approach

This paper introduces a flux-limited scheme FLOCV for the overlapping control volume (OCV) approach to 2D steady and unsteady convection–diffusion problems on structured non-orthogonal grids. FLOCV switches from second- to first-order interpolation in the presence of extrema. Smooth switching between the two is ensured by weighted average second- and first-order upwind differencing, with the weights being dynamically determined. Five convective test problems are solved using this scheme and results are compared with known analytical solutions. It is found that FLOCV approximately retains second-order accuracy of the base discretization scheme on uniform grids and smooth non-uniform orthogonal grids. It is also found effective in removing oscillations for problems with discontinuities on both orthogonal and non-orthogonal grids, with little degradation of accuracy. © 1997 John Wiley & Sons, Ltd.

A Lagrangian dynamic subgrid-scale model of turbulence

The dynamic model for large-eddy simulation of turbulence samples information from the resolved velocity field in order to optimize subgrid-scale model coefficients. When the method is used in conjunction with the Smagorinsky eddy-viscosity model, and the sampling process is formulated in a spatially local fashion, the resulting coefficient field is highly variable and contains a significant fraction of negative values. Negative eddy viscosity leads to computational instability and as a result the model is always augmented with a stabilization mechanism. In most applications the model is stabilized by averaging the relevant equations over directions of statistical homogeneity. While this approach is effective, and is consistent with the statistical basis underlying the eddy-viscosity model, it is not applicable to complex-geometry inhomogeneous flows. Existing local formulations, intended for inhomogeneous flows, are most commonly stabilized by artificially constraining the coefficient to be positive. In this paper we introduce a new dynamic model formulation, that combines advantages of the statistical and local approaches. We propose to accumulate the required averages over flow pathlines rather than over directions of statistical homogeneity. This procedure allows the application of the dynamic model with averaging to in-homogeneous flows in complex geometries. We analyse direct numerical simulation data to document the effects of such averaging on the Smagorinsky coefficient. The characteristic Lagrangian time scale over which the averaging is performed is chosen based on measurements of the relevant Lagrangian autocorrelation functions, and on the requirement that the model be purely dissipative, guaranteeing numerical stability when coupled with the Smagorinsky model. The formulation is tested in forced and decaying isotropic turbulence and in fully developed and transitional channel flow. In homogeneous flows, the results are similar to those of the volume-averaged dynamic model, while in channel flow, the predictions are slightly superior to those of the spatially (planar) averaged dynamic model. The relationship between the model and vortical structures in isotropic turbulence, as well as ejection events in channel flow, is investigated. Computational overhead is kept small (about 10% above the CPU requirements of the spatially averaged dynamic model) by using an approximate scheme to advance the Lagrangian tracking through first-order Euler time integration and linear interpolation in space.

Interpolation with meromorphic matrix functions

A complete solution is given to a first-order pole-zero meromorphic matrix function interpolation problem on a closed Riemann surface. The solution to the interpolation problem is constructed from the solution to a natural linear homogeneous system.

Pythagorean-hodograph space curves

We investigate the properties of polynomial space curvesr(t)={x(t), y(t), z(t)} whose hodographs (derivatives) satisfy the Pythagorean conditionx′2(t)+y′2(t)+z′2(t)≡σ2(t) for some real polynomial σ(t). The algebraic structure of thecomplete set of regular Pythagorean-hodograph curves in ℝ3 is inherently more complicated than that of the corresponding set in ℝ2. We derive a characterization for allcubic Pythagoreanhodograph space curves, in terms of constraints on the Bezier control polygon, and show that such curves correspond geometrically to a family of non-circular helices. Pythagorean-hodograph space curves of higher degree exhibit greater shape flexibility (the quintics, for example, satisfy the general first-order Hermite interpolation problem in ℝ3), but they have no “simple” all-encompassing characterization. We focus on asubset of these higher-order curves that admits a straightforward constructive representation. As distinct from polynomial space curves in general, Pythagorean-hodograph space curves have the following attractive attributes: (i) the arc length of any segment can be determined exactly without numerical quadrature; and (ii) thecanal surfaces based on such curves as spines have precise rational parameterizations.

A direct approach for the development of plane elements

The derivation of an element is based on the differential equations of the problem. The second-order derivatives of equations are approximated by first-order interpolation function derivatives. The equations derived in this way, multiplied by the determinant of the Jacobian transformation matrix, represent a basis for the derivation of the element matrix. The determinant of the Jacobian is attached to the geometrical derivatives at nodes, and as a result of this, the determinant does not appear in the denominator. Consequently, explicit integration is possible. By defining the derivatives at the nodes, and at certain points inside the element, the element matrix is derived directly. Several quadrilateral plane stress elements developed by the proposed approach are explicitly defined; they give converging results of improved accuracy. This method of development of finite elements can be applied to the development of finite elements for the solution of any problem governed by similar or higher-order differential equations.

On the zero- and first-order interpolation in synthesized sine waves for testing purposes

The authors deal with the quality requirements of synthesized sine waves reconstructed through a zero- or first-order hold for testing purposes, particularly when a switching demodulator is used. In this case, the interest is to have great spectral purity near the carrier rather than in all of the band. It is shown that the best method to obtain great spectral purity in the frequency band close to the synthesized frequency is by using zero-order hold instead of a first-order hold. Results show that even though a first-order hold implies a decrease of total harmonic amplitude distortion (THD), the measured spurious harmonics are kept lower or equal when using a zero-order hold in the 15 closest components. It is concluded that in testing applications a zero-order hold (in spite of its greater THD) yields better results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Curved composite beams—optimum lay-up for buckling by ranking

Instability of laminated curved composite beams made of repeated sublaminate construction is studied using finite element method. In repeated sublaminate construction, a full laminate is obtained by repeating a basic sublaminate which has a smaller number of plies. This paper deals with the determination of optimum lay-up for buckling by ranking of such composite curved beams (which may be solid or sandwich). For this purpose, use is made of a two-noded, 16 degress of freedom curved composite beam finite element. The displacements u, v, w of the element reference axis are expressed in terms of one-dimensional first-order Hermite interpolation polynomials, and line member assumptions are invoked in formulation of the elastic stiffness matrix and geometric stiffness matrix. The nonlinear expressions for the strains, occurring in beams subjected to axial, flexural and torsional loads, are incorporated in a general instability analysis. The computer program developed has been used, after extensive checking for correctness, to obtain optimum orientation scheme of the plies in the sublaminate so as to achieve maximum buckling load for typical curved solid/sandwich composite beams.

A comparison of line integral algorithms

Integrals along lines passing through arrays of integer data are used in many applications. Existing algorithms employ parametric methods and floating point calculations to determine the data values that contribute to the line integrals, along with some form of interpolation to weight these data values. A new, nonparametric integer-arithmetic noninterpolating algorithm (NI0) and an extension using first-order interpolation (NI1) are presented here. These algorithms are compared for accuracy and speed with both Siddon’s [Med. Phys. 12, 252 (1985)] parametric floating point algorithm using no interpolation (PF0) and our extension using first-order interpolation (PF1). NI1 gives line integral values significantly closer to those of PF1 than does PF0 and runs nine times faster on a VAXstation 2000. NI0 gives less accurate line integral values than does PF0 but runs 19 times faster.

Thermal deformation and stress analysis of composite laminated plates by finite element method

Thermal deformations and stresses, induced by a non-uniform temperature field in composite laminated plates, are studied by the finite element method. The stiffness matrix and load vector are derived based on the minimum potential energy. All three displacements of the middle surface are expressed as products of one-dimensional first-order Hermite interpolation polynomials. Numerical results are presented as non-dimensional curves. The effects of boundary conditions, temperature distributions, aspect ratios, and other parameters on the thermal deformations and stresses are investigated.

A compact representation of proofs

A structure which generalizes formulas by including substitution terms is used to represent proofs in classical logic. These structures, called expansion trees, can be most easily understood as describing a tautologous substitution instance of a theorem. They also provide a computationally useful representation of classical proofs as first-class values. As values they are compact and can easily be manipulated and transformed. For example, we present an explicit transformations between expansion tree proofs and cut-free sequential proofs. A theorem prover which represents proofs using expansion trees can use this transformation to present its proofs in more human-readable form. Also a very simple computation on expansion trees can transform them into Craig-style linear reasoning and into interpolants when they exist. We have chosen a sublogic of the Simple Theory of Types for our classical logic because it elegantly represents substitutions at all finite types through the use of the typed λ-calculus. Since all the proof-theoretic results we shall study depend heavily on properties of substitutions, using this logic has allowed us to strengthen and extend prior results: we are able to prove a strengthen form of the firstorder interpolation theorem as well as provide a correct description of Skolem functions and the Herbrand Universe. The latter are not straightforward generalization of their first-order definitions.

Solution of some plate bending problems using the boundary element method

Some fundamental aspects of the boundary element method of the Kirchhoff theory of thin plate flexure are given. The direct boundary integral equation method with higher conforming properties (using first-order Hermitian interpolation for plate displacement ω, and zero-order Hermitian interpolation for angle of rotation θ, the moment M andthe equivalent shear V) are used for several computational examples. They are: square plate with simply-supported or clamped edges, the same square plate with square central opening and the cantilevered triangular plates. The results of computation as compared with some known experimental and theoritical results showed that the numerical schemes seemed to be satisfactory for the practical applications.

Finite element analysis of curved anisotropic laminated hollow bars

Curved hollow bars of laminated anisotropic construction are used as structural members in many industries. They are used in order to save weight without loss of stiffness in comparison with solid sections. In this paper are presented the details of the development of the stiffness matrices of laminated anisotropic curved hollow bars under line member assumptions for two typical sections, circular and square. They are 16dof elements which make use of one-dimensional first-order Hermite interpolation polynomials for the description of assumed displacement state. Problems for which analytical or other solutions are available are first solved using these elements. Good agreement was found between the results. In order to show the capability of the element, application is made to carbon fibre reinforced plastic layered anisotropic curved hollow bars.

Approximate FM Demodulation Using Zero Crossings

This paper deals with approximate methods for demodulating FM signals using their zero crossings. A first-order (linear) interpolation method is devised and analyzed. Computer programs were then prepared to compare this first-order interpolator to the more usual zero-order interpolator. Higher order interpolation is briefly discussed, and the results of using a second-order interpolator are compared to those using the lower order methods.