Implicit Interpolation Research Articles

Three-Dimensional Implicit Curve Interpolation

An approach to 3D implicit curve interpolation in computer numerical systems (CNC) is presented. A 3D implicit curve is the intersection of two implicit surfaces. Geometric entities of 3D implicit curves are related to motion entities along the curve. The results are then used to develop real-time 3D implicit curve interpolation. An improved interpolation scheme has been proposed to augment the initial interpolation scheme to eliminate the cumulative point deviations. Simulations of implicit curve interpolation have been carried out to verify the effectiveness of the proposed algorithm. They also demonstrate that under typical machining conditions, the interpolation error for the improved scheme is well within the accuracy requirements of typical machines. The example has an application potential for tool-path interpolation when tool paths are obtained by intersecting an implicit design surface by drive cylinders. These show that the proposed algorithm is potentially useful for the machining of implicit surfaces.

An Algorithm for Assembling Overlapping Grid Systems

A general purpose algorithm for assembling overlapping grid systems for solving partial differential equations (PDEs) is described. The method combines and improves on the techniques implemented in the codes Beggar, CMPGRD, DCF3D, and PEGSUS. The present algorithm, which has been implemented for the two-dimensional case in the code Xcog, starts by calculating a global definition of the boundary of the computational domain based on the location of the physical boundaries in all component grids. The global boundary description isused to identify all grid points outside of the computational domain. The remaining points are classified according to the priority of the component grids in the overlapping grid. If the overlap between the components is sufficiently large, all remaining points can either interpolate from an overlapping grid or be used to discretize the PDE. The algorithm can therefore construct an overlapping grid for implicit (coupled) interpolation in one pass and needs only to iterate to ensure explicit (decoupled) interpolation. If the algorithm fails due to insufficient overlap between the components, the main parts of the grid will still be valid. This enables the code to report inconsistent grid points to the user, to facilitate an improvement of the input. Discrepancies between the boundary representations where the grids overlap are handled by a mismatch tolerance, which is estimated automatically. The interpolation data are corrected for the boundary mismatch, and it is shown how the interpolation error in a thin boundary layer is substantially reduced by this correction.

Implicit interpolation in reverse‐time migration

Reverse‐time migration applies finite‐difference wave equation solutions by using unaliased time‐reversed recorded traces as seismic sources. Recorded data can be sparsely or irregularly sampled relative to a finely spaced finite‐difference mesh because of the nature of seismic acquisition. Fortunately, reliable interpolation of missing traces is implicitly included in the reverse‐time wave equation computations. This implicit interpolation is essentially based on the ability of the wavefield to “heal itself” during propagation. Both synthetic and real data examples demonstrate that reverse‐time migration can often be performed effectively without the need for explicit interpolation of missing traces.

The effect of grid irregularity on truncation error for discretizations of Laplaces's equation

AbstractThe discretization of differential equations to obtain numerical solutions introduces truncation error (TE). The form of the TE depends on the differential equation, the discretization scheme, and the discretization grid in a fairly complicated fashion. The character and magnitude of the TE determines in part the discrepancy between the approximate and exact solutions to the PDE. In this study, a symbolic manipulator was used to investigate the geometrical dependence of the TE in discrete approximations to Laplace's equation. The effect of various grid irregularities on the leading TE terms was investigated for two discretization schemes. A nine point centered stencil was varied in aspect ratio, skewness, and non‐uniformity (irregular spacing). The first scheme was the common centered finite difference (FD) method, applied by transforming to a uniform, orthogonal computational space. The second scheme was the implicit interpolation method (II), developed for discretization on irregular grids. Both discretization methods had leading TE that was second order (inversely proportional to the number of grid points squared) and had a sharp rise in TE for skewness angles beyond sixty degrees. The FD method skewness errors were greater than those for the II method, but high aspect ratio errors were less. For non‐linear transformations the FD method TE contained second derivatives, the same order as the governing differential equation, while the II method's lowest order TE derivatives were third order. Thus while both methods have the same order of accuracy, the nature of the numerical error in the discrete solution would be different.

New ALE applications in non‐linear fast‐transient solid dynamics

The arbitrary Lagrangian—Eulerian (ALE) formulation, which is already well established in the hydrodynamics and fluid‐structure interaction fields, is extended to materials with memory, namely, non‐ linear path‐dependent materials. Previous attempts to treat non‐ linear solid mechanics with the ALE description have, in common, the implicit interpolation technique employed. Obviously, this implies a numerical burden which may be uneconomical and may induce to give up this formulation, particularly in fast‐transient dynamics where explicit algorithms are usually employed. Here, several applications are presented to show that if adequate stress updating techniques are implemented, the ALE formulation could be much more competitive than classical Lagrangian computations when large deformations are present. Moreover, if the ALE technique is interpreted as a simple interpolation enrichment, adequate—in opposition to distorted or locally coarse—meshes are employed. Notice also that impossible computations (or at least very involved numerically) with a Lagrangian code are easily implementable in an ALE analysis. Finally, it is important to observe that the numerical examples shown range from a purely academic test to real engineering simulations. They show the effective applicability of this formulation to non‐linear solid mechanics and, in particular, to impact, coining or forming analysis.

A fast spotlight-mode synthetic aperture radar imaging system

The author reformulates the standard spotlight mode synthetic aperture radar (SAR) problem as a (/spl tau/,p) Radon transform. This results in a new algorithm for SAR that uses a linear chirp FM source signal that is a function of the geometry between the source and the object. While the close relation between computer-aided-tomography (CAT) and SAR has been understood for over a decade, no accompanying computational advantages could be realized. Recently, Kelley and Madisetti (1991), have proposed a new approach to image reconstruction from projections using the so-called fast Radon transform (FRT). He derives, in the (/spl tau/,p) domain, a FRT-based solution to the SAR problem. The FRT offers implicit interpolation and parallel processing advantages not found in conventional back projection operations. He proposes its efficient application in the computationally intensive problems in telecommunications and remote sensing, especially in military applications.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Global solution of the generalized Abel integral equation by implicit interpolation

The construction of a (global) approximate solution for a given generalized Abel integral equation may be viewed as a problem of (implicit) interpolation in a prescribed linear space. In this paper, piecewise polynomials (extended spline functions) of a given degree and of class C are used to generate such an approximating function. Results on convergence and error bounds are given, and the practical application of this method is illustrated by a numerical example.

Global Solution of the Generalized Abel Integral Equation by Implicit Interpolation

Global Solution of the Generalized Abel Integral Equation by Implicit Interpolation