Showing papers on "Piecewise published in 1991"

Stabilized Finite Element Formulations for Incompressible Flow Computations

Abstract: Publisher Summary This chapter discusses stabilized finite element formulations for incompressible flow computations. Finite element computation of incompressible flows involve two main sources of potential numerical instabilities associated with the Galerkin formulation of a problem. The stabilization techniques that are reviewed more extensively than others are the Galerkin/ least-squares (GLS), streamline-upwind/ Petrov–Galerkin (SUPG), and pressure-stabilizing/Petrov–Galerkin (PSPG) formulations. The SUPG stabilization for incompressible flows is achieved by adding to the Galerkin formulation a series of terms, each in the form of an integral over a different element. These integrals involve the product of the residual of the momentum equation and the advective operator acting on the test function. The natural boundary conditions are the conditions on the stress components, and these are the conditions assumed to be imposed at the remaining part of the boundary. The interpolation functions used for velocity and pressure are piecewise bilinear in space and piecewise linear in time. These computations involve no global coefficient matrices, and therefore need substantially less computer memory and time compared to noniterative solution of the fully discrete equations. It is suggested that for two-liquid flows, the solution and variational function spaces for pressure should include the functions that are discontinuous across the interface.

Piecewise surface flattening for non-distorted texture mapping

Abstract: This paper introduces new techniques for interactive piecewise flattening of parametric 3-D surfaces, leading to a non-distorted, hence realistic, texture mapping. Cuts are allowed on the mapped texture and we make a compromise between discontinuities and distortions. These techniques are based on results from differential geometry, more precisely on the notion of "geodesic curvature": isoparametric curves of the surface are mapped, in a constructive way, onto curves in the texture plane with preservation of geodesic curvature at each point. As an application, we give a concrete example which is a first step towards an efficient and robust CAD tool for shoe modeling.

Stability of higher-order Hood-Taylor methods

Abstract: The stability of a higher-order Hood–Taylor method for the approximation of the stationary Stokes equations using continuous piecewise polynomials of degree 3 to approximate velocities and continuo...

Nonconforming finite element methods for the equations of linear elasticity

Abstract: In the adaptation of nonconforming finite element methods to the equations of elasticity with traction boundary conditions, the main difficulty in the analysis is to prove that an appropriate discrete version of Korn's second inequality is valid. Such a result is shown to hold for nonconforming piecewise quadratic and cubic finite elements and to be false for nonconforming piecewise linears. Optimal-order error estimates, uniform for Poisson ratio v E [0, 1/2), are then derived for the corresponding P2 and P3 methods. This contrasts with the use of C finite elements, where there is a deterioration in the convergence rate as v -1/2 for piecewise polynomials of degree < 3. Modifications of the continuous methods and the nonconforming linear method which also give uniform optimal-order error estimates are discussed.

Sensitivity Analysis for Mean-Variance Portfolio Problems

Abstract: This paper shows how to perform sensitivity analysis for Mean-Variance MV portfolio problems using a general form of parametric quadratic programming. The analysis allows an investor to examine how parametric changes in either the means or the right-hand side of the constraints affect the composition, mean, and variance of the optimal portfolio. The optimal portfolio and associated multipliers are piecewise linear functions of the changes in either the means or the right-hand side of the constraints. The parametric parts of the solution show the rates of substitution of securities in the optimal portfolio, while the parametric parts of the multipliers show the rates at which constraints are either tightening or loosening. Furthermore, the parametric parts of the solution and multipliers change in different intervals when constraints become active or inactive. The optimal MV paths for sensitivity analyses are piecewise parabolic, as in traditional MV analysis. However, the optimal paths may contain negatively sloping segments and are characterized by types of kinks, i.e., points of nondifferentiability, not found in MV analysis.

A Computational Framework for Determining Stereo Correspondence from a

Abstract: We present a computational framework for stereopsis based on the outputs of linear spatial filters tuned to a range of orientations and scales. This approach goes beyond edge-based and area-based approaches by using a richer image description and incorporating several stereo cues that previously have been neglected in the computer vision literature. A technique based on using the pseudo-inverse is presented for characterizing the information present in a vector of filter responses. We show how in our framework viewing geometry can be recovered to determine the locations of epipolar lines. An assumption that visible surfaces in the scene are piecewise smooth leads to differential treatment of image regions corresponding to binocularly visible surfaces, surface boundaries, and occluded regions that are only monocularly visible. The constraints imposed by viewing geometry and piecewise smoothness are incorporated into an iterative algorithm that gives good results on random-dot stereograms, artificially generated scenes, and natural grey-level images.

Optimal finite-thrust spacecraft trajectories using collocation and nonlinear programming

Abstract: A new method is described for the determination of optimal spacecraft trajectories in an inverse-square field using finite, fixed thrust. The method employs a recently developed direct optimization technique that uses a piecewise polynomial representation for the state and controls and collocation, thus converting the optimal control problem into a nonlinear programming problem, which is solved numerically. This technique has been modified to provide efficient handling of those portions of the trajectory that can be determined analytically, i.e., the coast arcs. Among the problems that have been solved using this method are optimal rendezvous and transfer (including multirevolution cases) and optimal multiburn orbit insertion from hyperbolic approach.

Smooth interpolation of a mesh of curves

Abstract: The interpolation of a mesh of curves by a smooth regularly parametrized surface with one polynomial piece per facet is studied. Not every mesh with a well-defined tangent plane at the mesh points has such an interpolant: the curvature of mesh curves emanating from mesh points with an even number of neighbors must satisfy an additional “vertex enclosure constraint.” The constraint is weaker than previous analyses in the literature suggest and thus leads to more efficient constructions. This is illustrated by an implemented algorithm for the local interpolation of a cubic curve mesh by a piecewise [bi]quarticC 1 surface. The scheme is based on an alternative sufficient constraint that forces the mesh curves to interpolate second-order data at the mesh points. Rational patches, singular parametrizations, and the splitting of patches are interpreted as techniques to enforce the vertex enclosure constraint.

Piecewise surface flattening for non-distorted texture mapping

Abstract: This paper introduces new techniques for interactive piecewise flattening of parametric 3-D surfaces, leading to a non-distorted, hence realistic, texture mapping. Cuts are allowed on the mapped te...

A data model for scientific visualization with provisions for regular and irregular grids

Abstract: This paper presents a mathematical data model for scientific visualization based on the mathematics of fiber bundles. The findings of previous works are extended to the case of piecewise field representations (associated with grid-based data representations), and a general mathematical model for piecewise representations of fields on irregular grids is presented. A discussion of the various types of regularity that can be found in computational grids and techniques for compact field representation based on each form of regularity are presented. These techniques can be combined to obtain efficient methods for representing fields on grids with various regular or partially regular structures.

A method of univariate interpolation that has the accuracy of a third-degree polynomial

Abstract: A method of interpolation that accurately interpolates data values that satisfy a function is said to have the accuracy of that function. The desired or required properties for a univariate interpolation method are reviewed, and the accuracy of a third–degree polynomial is found to be one of the desirable properties. A method of univariate interpolation having the accuracy of a third–degree polynomial while retaining the desirable properties of the method developed earlier by Akima( J. ACM 17, PP. 589–602, 1970) has been developed. The newly developed method is based on a piecewise function composed of a set of polynomials, each of degree three, at most, and applicable to successive intervals of the given data points. The method estimates the first derivative of the interpolating function (or the slope of the curve) at each given data point from the coordinates of seven data points. The resultant curve looks natural in many cases when the method is applied to curve fitting. The method is presented with examples. Possible use of a higher–degree polynomial in each interval is also examined.

Error bounded variable distance offset operator for free form curves and surfaces

Abstract: Most offset approximation algorithms for freeform curves and surfaces may be classified into two main groups. The first approximates the curve using simple primitives such as piecewise arcs and lines and then calculates the (exact) offset operator to this approximation. The second offsets the control polygon/mesh and then attempts to estimate the error of the approximated offset over a region. Most of the current offset algorithms estimate the error using a finite set of samples taken from the region and therefore can not guarantee the offset approximation is within a given tolerance over the whole curve or surface. This paper presents new methods to globally bound the error of the approximated offset of freeform curves and surfaces and then automatically derive new approximations with improved accuracy. These tools can also be used to develop a global error bound for a variable distance offset operation and to detect and trim out loops in the offset.

Exponential stabilization of mobile robots with nonholonomic constraints

Abstract: The authors present an exponentially stable controller for a two degree of freedom robot with nonholonomic constraints. They propose a piecewise smooth controller to make the origin exponentially stable for any initial condition in the state space. The main difference with respect to other approaches can be summarized as follow. The proposed scheme does not seek to render the discontinuous surface invariant, as opposed to the principles of sliding control, but rather to make this surface non-attractive. Infinite switching in the control law and the undesirable chattering phenomenon, can thus be avoided. Furthermore, this control law yields exponential stability so that the convergence can be chosen arbitrarily fast. >

An inner product space approach to image coding by contractive transformations

Abstract: Consideration is given to modeling of arbitrary digital gray tone images as the fixed points of contractive transformations on the image space. The transformation is designed to exploit a form of image redundancy called piecewise self-similarity. Results from inner product space theory are used to optimize the transformation within a class of affine transformations. The resulting transformation is described by a small amount of information compared to the original image, and its piecewise self-similar fixed point, found by successively iterating the transformation from any initial image, is close to the original image. Results are improved compared to earlier results with this technique. >

Global properties of continuous piecewise linear vector fields. Part I: Simplest case in ℝ2

Abstract: Among non-linear vector fields, the simplest that can be studied are those which are continuous and piecewise linear. Associated with these types of vector fields are partitions of the state space into a finite number of regions. In each region the vector field is linear. On the boundary between regions it is required that the vector field be continuous from both regions in which it is linear. This presentation is devoted to the analysis in two dimensions of the simplest possible types of continuous piecewise linear vector fields, namely linear vector fields possessing only one boundary condition. As a practical concern the analysis will attempt to ask and answer questions raised about the existence of steady state solutions. Since the local theory of fixed points in a linear vector field is sufficient to determine the stability of fixed points in a piecewise linear vector field, most of the steady state behaviour to be studied will be towards limited cycles. the results will present sufficient conditions for the existence, or non-existence as the case may be, of limit cycles. Particular attention will be paid to the domain of attraction whenever possible. With these results qualitative statements may be made for piecewise linear models of many physical systems.

Sensitivity computation in piecewise approximate circuit simulation

Abstract: Both direct and adjoint methods are applied to the computation of time-domain transient sensitivities in the already efficient SPECS piecewise approximate circuit simulation environment. By exploiting the event-driven nature of SPECS, the computation, storage, and interpolation of the Jacobians that specify the appropriate linearized circuit during the forward simulation is not required to obtain sensitivities. Both direct and adjoint methods require only a marginal computational overhead and provide equivalent results. The choice of the method to use depends on the ratio of the number of outputs to the number of parameters. More parameters favor the adjoint method and more outputs favor the direct. The overall efficiency of both SPECS and its time-domain sensitivity extension allows it to be applied to realistic circuits that, due to their large size, had previously made such analysis impractical. Simulation results on industrial circuits of up to 1500 MOS transistors show that transient sensitivities can be efficiently computed for large circuits. >

Super spline spaces of smoothnessr and degreed≥3r+2

Abstract: The problem of computing the dimension of spaces of splines whose elements are piecewise polynomials of degreed withr continuous derivatives globally has attracted a great deal of attention recently. We contribute to this theory by obtaining dimension formulae for certain spaces of super splines, including the case where varying amounts of additional smoothness is enforced at each vertex. We also explicitly construct minimally supported bases for the spaces. The main tool is the Bernstein-Bezier method.

Shocks and other nonlinear filtering applied to image processing

Abstract: Two new filters for image enhancement are developed, extending the early work of the authors. One filter uses a new nonlinear time dependent partial differential equation and its discretization, the second uses a discretization which constrains the backwards heat equation and keeps it variation bounded. The evolution of the initial image as t increases through U(x,y,t) is the filtering process. The processed image is piecewise smooth, nonoscillatory and apparently an accurate reconstruction. The algorithms are fast and easy to program.

On oscillatory motion of a spring-mass system subjected to piecewise constant forces

Abstract: Abstract Oscillatory motion of a spring-mass system subjected to a piecewise constant force of the form f ( x ([ t ])) or f ([ t ]) is studied. Solutions of a damped spring-mass system subjected to linear, sinusoidal or non-linear piecewise constant forces are derived. Results are graphically presented and compared with those of the corresponding systems subjected to continuous forces. Oscillatory properties of an undamped spring-mass system subjected to a piecewise constant force are studied in detail. When in a non-linear system the non-linear function is treated as a piecewise constant force in a small enough time interval, the resulting response agrees with the solution of the original non-linear equation. To this effect, examples of linear, non-linear and chaotic systems have been treated in the text.

Generalized Riemann zeta -function regularization and Casimir energy for a piecewise uniform string.

Abstract: The generalized $\ensuremath{\zeta}$-function techniques will be utilized to investigate the Casimir energy for the transverse oscillations of a piecewise uniform closed string. We find that the $\ensuremath{\zeta}$-function regularization method can lead straightforwardly to a correct result.

CAM motion synthesis using cubic splines

Abstract: The application of minimum norm principle, similar to the principle of minimum potential energy, is presented for the general synthesis of cam motion. The approach involves the use of piecewise cubic spline functions for representing the follower displacement. The cubic splines are more convenient and simpler to use compared to general spline functions and also result in smaller peak acceleration and jerk due to the application of the minimum norm principle. A general procedure is presented for application to any cam-follower system. The effectiveness of the approach is illustrated by comparing the results given by the present method with those given by other approaches for a disk cam-translating follower.

A finite element scheme based on the velocity correction method for the solution of the

Abstract: SUMMARY In this paper a finite element solution for two-dimensional incompressible Viscous flow is considered. The velocity correction method (explicit forward Euler) is applied for time integration. Discretization in space is carried out by the Galerkin weighted residual method. The solution is in terms of primitive variables, which are approximated by piecewise bilinear basis functions defined on isoparametric rectangular elements. The second step of the obtained algorithm is the solution of the Poisson equation derived for pressure. Emphasis is placed on the prescription of the proper boundary conditions for pressure in order to achieve the correct solution. The scheme is completed by the introduction of the balancing tensor viscosity; this makes this method stable (for the advection-dominated case) and permits us to employ a larger time increment. Two types of example are presented in order to demonstrate the performance of the developed scheme. In the first case all normal velocity components on the boundary are specified (eg. lid-driven cavity flow). In the second type of example the normal derivative of velocity is applied over a portion of the boundary (e.g. flow through sudden expansion). The application of the described method to non-isothermal flows (forced convection) is also included.